Mathematics X

PATRONS
KENDRIYA VIDYALAYA
SANGATHAN JAIPUR REGION
Sh. B. L. MORODIA
Deputy
Commissioner
KVS JAIPUR
REGION
Sh. D. R. MEENA Sh. G.S. MEHTA Sh. MADHO

SINGH
Assistant Assistant Assistant
Commissioner Commissioner Commissioner
KVS JAIPUR KVS JAIPUR KVS JAIPUR
REGION REGION REGION
Mr. Atul Vyas
Principal
Kendriya Vidyalaya
Bharatpur (Rajasthan)
Coordinator
Mr. Navratan Mittal

Principal
Kendriya Vidyalaya
CISF Deoli,Tonk (Rajasthan)
Coordinator
Mrs. Durga Chauhan

Venue Principal
Kendriya Vidyalaya
No.2 AFS,Jodhpur (Rajasthan)
editorial
board
1. Mr. Narendra Singh Poonia

PGT (Maths) KV No.2 AFS Jodhpur
2. Mr. Vinod kumar

TGT(Maths) KV NO.2 AFS Jodhpur
3. Mrs. Arti sharma

TGT(Maths) KV NO.2 AFS Jodhpur
CHAPTER 1
REAL NUMBERS
The Fundamental Theorem of Arithmetic
Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order
in which the prime factors occur.
The prime factorisation of a natural number is unique, except for the order of its factors.
Property of HCF and LCM of two positive integers ‘a’ and ‘b’:
𝐇𝐂𝐅 (𝐚, 𝐛) × 𝐋𝐂𝐌 (𝐚, 𝐛) = 𝐚 × 𝐛
PRIME FACTORISATION METHOD TO FIND HCF AND LCM
HCF (a, b) = Product of the smallest power of each common prime factor in the numbers.
LCM (a, b) = Product of the greatest power of each prime factor, involved in the numbers.
PRACTICE QUESTIONS:
1. Complete the missing entries in the following factor tree:
(a) 42 and 21
(b) 24 and 12
(c) 7 and 3
(d) 84 and 42
2 The H.C.F. and the L.C.M. of 12, 21, 15 respectively are:
(a) 3, 140 (b) 12, 420 (c) 3, 420 (d) 420, 3
3 The H.C.F. of smallest prime number and the smallest composite number is .......... .
(a) 1 (b) 2 (c) 4 (d) none of these
4 225 can be expressed as

(a) 52× 32 (b) 52 × 5 × 32 (c) 52× 3 (d)52× 325
5 The largest number which divides 70 and 125, leaving remainders 5 and 8 respectively, is
(a) 13 (b) 65 (c) 875 (d) 1750
6 Prove that 7 + √2 is an irrational number.
7 Prove that √7 is an irrational number.
8 Check whether 5 × 3 × 11 + 11 𝑎𝑛𝑑 5 × 7 + 7 × 3 are composite number and justify.

9 Check whether 8n can end with the digit 0, where n is any natural number
10 Given that LCM (26,169) =338, find HCF (26,169).
11 Find the HCF and LCM of 6,72 and 120 using the prime factorization method
12 A class of 20 boys and 15 girls is divided into n groups so that each group has x boys and y girls. Find x, y
and n?
13 Assertion: √2 is an irrational number
Reason: 2 is the smallest prime number
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion
(A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true
14 Assertion: HCF of two number is 4 and their product is 192 then LCM is 48.
Reason: 𝐹𝑜𝑟 𝑎𝑛𝑦 𝑡𝑤𝑜 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑎 𝑎𝑛𝑑 𝑏 ,HCF (a, b)×LCM (a, b) =a× 𝑏
(A).
15 Assertion : √3 +√5 is an irrational number
Reason: sum of any two irrational number is always irrational
(A).
(d) Assertion (A) is false but reason (R) is true.
16 What is the ratio of LCM and HCF of the least composite and least prime number?
17 The LCM of 12 and 42 is 10 m + 4 then find the value of m.
18 In a school for primary classes duration of period was 40 minutes and for secondary classes duration was
1 hour. If the 1st bell rang at 8 AM for both the sections, then at what time will they ring together?
19 SCOUT RAJYA PURASKAR TESTING CAMP-2023

A Rajya Puraskar Testing Camp was held at KV-1, AFS, Jodhpur. In this camp 60 escorts, 84 Guides and
108 scouts participated. The quarter master has made the arrangement such that in each room same
number of escorts, scouts and guides are to be accommodated. Separate rooms are allotted for escorts,
scouts and guides.
(i) How many rooms are required for Scouts?

(ii) How many rooms are required for Guides?
(iii) How many rooms were required for all?
OR
If two officials visited and stayed at the vidyalaya and allotted a separate room then how many rooms will
be required now?
20 Rakesh works as a librarian in KVS.
He ordered for books on English, Hindi and Maths. He received 96 English books 240 Hindi books and
336 Maths books. He wishes to arrange these books on stacks such that each stack consist of the books on
only one subject and the number of books in each stack is the same.
He also wishes to keep the number of stacks minimum.
(i) Find the number of books in each stack.

(ii) Find the total number of stacks formed.
(iii) How many stacks of Maths books will be formed?
OR
If each Hindi book weighs 1.5 kg, then find the weight of books in a stack of Hindi books.
21. If the smallest prime factor of ‘a’ is 3 and the smallest prime factor of ‘b’ is 7. Then find the smallest prime
factor of (a+b).
22. Find the LCM of 𝑥 2 − 4 𝑎𝑛𝑑 𝑥 4 − 16.
23. Find the HCF of 𝑥 3 − 3𝑥 + 2 𝑎𝑛𝑑 𝑥 2 − 4𝑥 + 3.
24. If p is factor of q then find the HCF and LCM of p and q.
25. Is it possible to have two numbers such that their HCF is 15 and LCM is 365?
26. Prove that √7 +√5 is an irrational number.
27. Prove that 2 − √3 is an irrational number.
28. Prove that 3√2 is an irrational number
Answer Key
Q. No. Answers Q. No. Answers
1 a 14 (a) Both assertion (A) and reason (R) are true and reason
(R) is the correct explanation of assertion (A).
2 c 15 (c) Assertion (A) is true but reason (R) is false.
3 b 16 2:1
4 a 17 m=8
5 a 18 At 10 AM
8 composite number 19. (i) 9 rooms (ii) 7 rooms
(iii) 21 rooms OR 22 rooms
9 8n cannot be end with digit zero 20. (i) 48 (ii) 14
(iii) 7 OR 72 kg
10 HCF (26,169) = 13 21. a is odd, b is odd, a+b is even. Smallest prime factor of
a+b =2
11 HCF=6 LCM=360 22. 𝑥 2 − 4 = ( 𝑥 + 2)(𝑥 − 2) 𝑎𝑛𝑑
𝑥 4 − 16 = (𝑥 2 + 4)( 𝑥 + 2)(𝑥 − 2)
LCM = (𝑥 2 + 4)( 𝑥 + 2)(𝑥 − 2) = 𝑥 4 − 16
12 x=4,y=3 and n=5 23. 𝑥 3 − 3𝑥 + 2 = (𝑥 − 1) ( 𝑥 − 1)(𝑥 + 2)𝑎𝑛𝑑
𝑥 2 − 4𝑥 + 3 = (𝑥 − 1)(𝑥 − 3).
HCF = (x – 1)
13 (b) Both assertion (A) and reason 24. HCF = p and LCM = q
(R) are true but reason (R) is not
the correct explanation of assertion
(A).
25. No
CHAPTER - 2
POLYNOMIALS
An expression of the form p(x) = a 0 + a1x + a 2 x 2 + ••• ........... +anxn where a n ≠ 0 is called a polynomial in one variable x of degree
n, where; a 0 , .................................................................................................. an, are constants and they are called the coefficients of x 0 , x, x 2
.............................................................................................................................. x n . Each power of x is a non-negative integer.
Eg: - 5x 2 - 5x + 1 is a polynomial of degree 2.
Note: √𝒙 + 3 is not a polynomial
• A polynomial p(x) = ax + b of degree 1 is called a linear polynomial Eg: 5x - 3,2x etc.
• A polynomial p(x) = ax 2 + bx + c of degree 2 is called a quadratic polynomial
Eg: 5x 2 + x - 1
• A polynomial p(x) = ax 3 + bx 2 + cx + d of degree 3 is called a cubic polynomial.
Eg: √3x3 - x + 4, x 3 - 1 etc.
Zeroes of a polynomial: A real number k is called a zero of polynomial p(x) if p(k)=0. If the graph of y= p(x)
intersects the X-axis n times, the number of zeroes of y= p(x) is n. If the graph of x= p(y) intersects the Y-axis n
times, the number of zeroes of x= p(y) is n.
• A linear polynomial has only one zero.
• A quadratic polynomial has at most two zeroes.
• A cubic polynomial has at most three zeroes.
• A polynomial of degree n has at most n zeroes.
Relationship between zeroes and coefficients of a quadratic polynomial:
𝑓𝑜𝑟 𝑞𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐 𝑝𝑜𝑙𝑦𝑛𝑜𝑚𝑖𝑎𝑙 ∶ 𝑖𝑓 𝛼 & 𝛽 𝑎𝑟𝑒 𝑧𝑒𝑟𝑜𝑒𝑠 𝑜𝑓𝑎 𝑞𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐 𝑝𝑜𝑙𝑦𝑛𝑜𝑚𝑖𝑎𝑙 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 then
𝑏 − (𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑥 )
1. 𝑆𝑢𝑚 𝑜𝑓 𝑧𝑒𝑟𝑜𝑒𝑠 = 𝛼 + 𝛽 = − =
𝑎 (𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑥 2 )
𝑐 (𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑡𝑒𝑟𝑚)
2. 𝑃𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑧𝑒𝑟𝑜𝑒𝑠 = 𝛼𝛽 = =
𝑎 (𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑥 2 )
A quadratic polynomial whose zeroes are 𝛼 and 𝛽 is given by: 𝑥 2 − (𝛼 + 𝛽 )𝑥 + 𝛼𝛽= x2 – Sx +P.
Q.No. Multiple choice questions

1. Find the zeroes of the polynomial f(x) = x 2-x–6
(a) - 3, 2 (b) - 3, - 2 (c) 3, 2 (d) 3, - 2

2. For what value of k is - 4 a zero of the polynomial f(x) = x 2 - x - (2k + 2)?
(a) 6 (b) – 6 (c) 9 (d) – 9
3. If a and b are the zeroes of a polynomial such that a + b = - 6 and ab = - 4, then write the polynomial.
(a) x2 - 6x - 4 = 0 (b) x2 + 6x - 4 = 0 (c) x2 + 6x + 4 = 0 (d) x2 - 6x + 4 = 0
4. The zeroes of the polynomial x 2 - 3x - m(m + 3) are:
(a) m, m + 3 (b) - m, m + 3 (c) m, - (m + 3) (d) - m, - (m + 3)
5. The zeroes of the quadratic polynomial x + 99x + 127 are:
2
(a) both positive (b) both negative

(c) one positive and one negative (d) both equal
6. If one of the zeroes of the quadratic polynomial (k–1)x2 + kx + 1 is –3, then the value of k
4 4 2 2
(a) 3 (b) - 3 (c) 3 (d) -3
7. If α, β be the zero of the polynomial 2x2+ 5x + k such that 𝛼 2 + 𝛽 2 + 𝛼𝛽 = 21 , find k
4
(a) 3 (b) – 3 (c) – 2 (d) 2
8. What is the difference between the values of the polynomial 7x -3x2 + 7 at x= 1 and x=2
(a) -2 (b) 2 (c) 3 (d) none of these
9. If one of the zeroes of a quadratic polynomial of the form x +ax + b is the negative of the other, then it
2
(A)has no linear term and the constant term is negative.

(B)has no linear term and the constant term is positive.
(C)can have a linear term but the constant term is negative.
(D)can have a linear term but the constant term is positive.
10. The graph of the expression ax2+bx+c is an upward parabola, if
(a) a > 0 (b) a < 0 (c) a = 0 (d) none of these
ASSERTION REASON QUESTIONS:
11. ASSERTION: The graph of quadratic polynomial P(x) intersects x-axis at two points.
REASON: Degree of quadratic polynomial is 2.
a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
c) Assertion (A) is true but reason (R) is false. d)Assertion (A) is false but reason (R) is true.
12. ASSERTION: 2 + √3 is one zero of a quadratic polynomial then other zero will be 2 - √3.
REASON: Irrational zeroes always occurs in pairs.
c) Assertion (A) is true but reason (R) is false.
d) Assertion (A) is false but reason (R) is true
13. ASSERTION: The graph of a linear polynomial intersects the x-axis at most 1 point.
REASON: For polynomial P(x) of degree ‘n’ the graph of y = P(x) intersect x-axis at most n points.
d) Assertion (A) is false but reason (R) is true.
14. ASSERTION: 5√t - 7 is a linear polynomial.
REASON: A polynomial of degree 1 is called a linear polynomial.
15. ASSERTION: A quadratic polynomial can have at most two zeroes.
REASON: x2 +7x +12 has no real zeroes.
Short Answer Questions
16. If one zero of the polynomials (𝑎2 + 4)𝑥 2 + 9𝑥 + 4𝑎 is the reciprocal of the other, find the value of a.
17. If 𝛼 𝑎𝑛𝑑 𝛽 are the zeros of the quadratic polynomial f(x)= a𝑥 2 + 𝑏𝑥 + 𝑐 ,then evaluate:
1 1
+ -2𝛼𝛽.
𝛼 𝛽
18. If the square of difference of the zeroes of the quadratic polynomial f(x)= x 2 + px +45 is equal to 144, find the
value of p.
19. If 𝛼 and 𝛽 are the zeroes of the polynomial x2 + 6x + 2, find the value of 𝛼 −1 + 𝛽 −1 .
20. If zeros of x2-kx+6 are in the ratio 3:2, find k.
Long Answer Type Questions
21. If the sum of the zeroes of the quadratic polynomial f(x)= kx2 +2x +3k is equal to their product, find the value
of k.
22. If p and q are the zeroes of polynomial f(x)=2x2 -7x +3. Find the value of p2 + q2.
23. If 𝛼 and 𝛽 are the zeroes of the polynomial x2 -4√3x +3, find the value of 𝛼 + 𝛽 − 𝛼 𝛽.
24. Find the zeroes of the quadratic polynomial 4u2 + 8u. verify the relationships between the zeroes and the
coefficients.
25. If a and p are zeros of y2+5y+m, find the value of m such that (a+p)2 - ap = 24
26. If a and p are zeros of x2-x-2, find a polynomial whose zeros are (2a+1) and (2p+1)
Case study question
27. Online shopping is trending now a days and many customers prefer online shopping as it is very convenient,
less time consuming and any time accessible. A survey was done on a small scale among a group of people
and represented in a form of quadratic polynomial x2-60x +800, if 𝛼 are the number of customers who prefer
online shopping and 𝛽 be the number of customers who prefer market shopping, then answer the following
questions:-
1. Find the number of customers who prefer online shopping.
2. Find the number of customers who prefer market shopping.
3. Write a quadratic polynomial whose zeroes are (𝛼 -3) and (𝛽 + 2).
Or
Determine whether the given value of x is a zero of the polynomials or not,
2
P(x)=(2x+3) (3x-2); x=3.
28. Thermal Degradation:
Plastic materials subjected to prolonged exposure to high temperature will lose their strength and
toughness, become more prone to cracking, chipping and melting and I experienced such a degradation once
and that plastic piece got melted and transformed in a mathematical curve shown in the graph below:
1. From the above graph, how many numbers of zeroes are there?
2. Find the zeroes of the polynomial.
3. Find the sum and product of the zeroes of the quadratic polynomial: px 2+qx+pq.
OR
1
Write a polynomial whose sum of zeroes and product of zeroes are √2 and 3.
29. The Gateway of India is an arch-monument built in the early 20th century in the city of Mumbai (Bombay),
India. It was erected to commemorate the landing of King-Emperor George V, the first British monarch to
visit India, in December 1911 at Strand Road near Wellington Fountain.
1. If the slope of the gateway is represented by quadratic polynomial x2 -2x -8, then find its zeroes.
2. Find the quadratic equation for parabolic curve whose sum of zeroes is 6 and product of zeroes is 0
respectively.
3. Find the number of real zeroes of the polynomial f(x)= (x-2)2 + 4.
OR
1 1
Write a quadratic polynomial whose zeroes are 2 and 2.
ANSWER KEY
Q1. D Q2. C Q3. B Q4. B Q5. B Q6. A Q7. D Q8. B
Q9. A Q10. A Q11. A Q12. A Q13. A Q14. D Q15 C
1
Q16. 𝑍𝑒𝑟𝑜𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑜𝑙𝑦𝑛𝑜𝑚𝑖𝑎𝑙 𝑎𝑟𝑒 𝛼 𝑎𝑛𝑑 𝛼 , 𝛼 × 𝛼 =
1
Q17. 𝑠𝑢𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑜𝑙𝑦𝑛𝑜𝑚𝑖𝑎𝑙𝑠 =
𝑏 𝑐 1 1
4𝑎
, 𝑎2 + 4 = 4𝑎 , 𝑎 = 2 − 𝑎 , 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑜𝑙𝑦𝑛𝑜𝑚𝑖𝑎𝑙𝑠 = 𝑎 , 𝛼 + 𝛽 − 2𝛼𝛽 =
𝑎 2 +4
{(𝛼+𝛽)−2(𝛼𝛽)2 } 𝑏 2𝑐
= − (𝑐 + )
𝛼𝛽 𝑎
Q18. 𝛼 + 𝛽 = −𝑝 , 𝛼𝛽 = 45 −𝑏±√𝑏2 −4𝑎𝑐
Q19. 𝑞𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 =
⟹ 𝑔𝑖𝑣𝑒𝑛, (𝛼 − 𝛽 )2 = 144 2𝑎
(𝛼 + 𝛽 )2 − 4𝛼𝛽 = 144 𝛼 = −3 + √7 𝑎𝑛𝑑 𝛽 = −3 − √7
⟹ (−𝑝)2 − 4 × 45 = 144 1 1 1 1
∴ 𝛼 −1 + 𝛽 −1 = + ⟹ + ⟹ −3
⟹ 𝑝 = ±18 𝛼 𝛽 (−3 + √7) −3 − √7
𝛼 3
Q20. = ⟹ 𝛼 = 𝛽
3
Q21. 𝛼 + 𝛽 = 𝛼𝛽
𝛽 2 2 𝑏 𝑐
𝑐 3 ⟹ − =
⟹ 𝛼𝛽 = = 6 ⟹ 𝛽 2 = 6 ⟹ 𝛽 = ±2 𝑎𝑛𝑑 𝛼 = ±3 𝑎 𝑎
𝑎 2 2 3𝑘
𝑏 −𝑘 ⟹ − =
∴ 𝛼+𝛽 = − = − = 𝑘 ⟹ ±2 ± 3 = 𝑘 ⟹ 𝑘 𝑘 𝑘
𝑎 1 2
= ±5 ⟹ 𝑘= −
3
7 3 7 2 𝑏
Q23. 𝛼 + 𝛽 = − 𝑎 = 4√3 𝑎𝑛𝑑 𝛼𝛽 = 3 ⟹ 𝛼 + 𝛽 − 𝛼𝛽 =
Q22. 𝑝 + 𝑞 = 2 ⟹ 𝑝𝑞 = 2 ⟹ (𝑝 + 𝑞 )2 = (2)
37 4√3 − 3 ⟹ √3(4 − √3)
⟹ 𝑝2 + 𝑞 2 =
4
Q24. 4𝑢2 + 8 Q25. 𝑦 2 + 5𝑦 + 𝑚
⟹ 𝑏𝑦 𝑞𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 − 1 ± 1 ⟹ 𝑎 + 𝑝 = −5 ⟹ 𝑎𝑝 = 𝑚
⟹ 𝑒𝑖𝑡ℎ𝑒𝑟 − 1 + 1 = 0 𝑜𝑟 − 1 − 1 = −2 ⟹ (𝑎 + 𝑝)2 − 𝑎𝑝 = 24
𝑏 8 ⟹ (−5)2 − 𝑚 = 24
⟹ 𝑎𝑠 𝑤𝑒 𝑘𝑛𝑜𝑤 𝑡ℎ𝑎𝑡 𝛼 + 𝛽 = − ⟹ 0 + (−2) = −
𝑎 4 ⟹ 25 − 𝑚 = 24
𝑐 ⟹𝑚=1
⟹ −2 = −2 𝑎𝑛𝑑 𝛼𝛽 = ⟹ 0 × (−2)
𝑎
0
= ⟹ 0 = 0 𝐻. 𝑃.
4
Q26. 𝑥 2 − 𝑥 − 2 Q27. (1) 40
⟹ 𝑎 + 𝑝 = 1 𝑎𝑛𝑑 𝑎𝑝 = 2 (2) 20
⟹ 𝑜𝑡ℎ𝑒𝑟 𝑧𝑒𝑟𝑜𝑒𝑠 𝑎𝑟𝑒 2𝑎 + 1 𝑎𝑛𝑑 2𝑝 + 1 (3) 𝑥 2 − 59𝑥 + 814
⟹ 𝛼 + 𝛽 = 2𝑎 + 1 + 2𝑝 + 1 ⟹ 2(𝑎 + 𝑝 + 1) OR
⟹ 2(1 + 1) ⟹ 4 (3) Yes, 2/3 is a zero of the given polynomial.
𝛼𝛽 = (2𝑎 + 1)(2𝑝 + 1)
⟹ 2(𝑎 + 𝑝) + 4𝑎𝑝 + 1 ⟹ 2 × 1 + 4 × (−2) + 1 = −5
∴ 𝑁𝑒𝑤 𝑝𝑜𝑙𝑦𝑛𝑜𝑚𝑖𝑎𝑙 = 𝑥 2 − (𝛼 + 𝛽 )𝑥 + 𝛼𝛽
⟹ 𝑥 2 − 4𝑥 − 5
Q28. (1) 3 Q29. (1) By the middle term splitting or quadratic formula
(2) -2,0,2 𝑥 = 4, −2
𝑞
(3) Sum of zeroes = − 𝑝 and product of zeroes = q (2) 𝑥 2 − (𝛼 + 𝛽 )𝑥 + 𝛼𝛽
= 𝑥 2 − 6𝑥 + 0
OR
1 = 𝑥 2 − 6𝑥
(3) = 𝑥 2 − √2𝑥 + 3 (3) Number of real zeroes = 0, because D< 0
= 3𝑥 2 − 3√2𝑥 + 1 OR
(3) 𝑥 2 − (𝛼 + 𝛽 )𝑥 + 𝛼𝛽
1
= 𝑥2 − 𝑥 + 4
= 4𝑥 2 − 4𝑥 + 1
CHAPTER 3
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES
ALGEBRAIC INTERPRETATION OF PAIR OF LINEARE QUATIONS IN TWO VARIABLES
The pair of linear equations represented by these lines a1x+b1y+c1=0 and a2x +b2y +c2 = 0
S. Pair of lines Compare the Graphical Algebraic

No. ratios representation Interpretation
a1x + b1y + c1 = 0 𝑎1
≠
𝑏1 Intersecting lines Unique Solution
𝑎2 𝑏2
1 (Exactly one
a2x + b2y + c2 = 0 (consistent)
solution)
a1x + b1y + c1 = 0 𝑎1
=
𝑏1
= 𝑐1
𝑐 Coincident lines Infinitely many
𝑎2 𝑏2
2 a2x + b2y + c2 = 0
2 solutions
(consistent)
a1x + b1y + c1 = 0 𝑎1
=
𝑏1
≠
𝑐1 Parallel lines No solution
𝑎2 𝑏2 𝑐2
3 a2x + b2y + c2 = 0 (inconsistent)

MCQ
If a pair of linear equations is consistent, then the lines will be:

1
(a) parallel (b) always coincident (c) intersecting or coincident (d) always intersecting
2
The pair of equation x + 2y + 5 = 0 and - 3x - 6y + 1 = 0 have:
(a) a unique solution (b) exactly two solutions (c) infinitely many solutions (d) no
solution
The pair of linear equations 2x + 3y = 4 and 3x + 4y = 9 has:
3 (a) infinitely many solutions (b) no solution (c) a unique solution (d) two solutions
The value of k for which the system of linear equations x + 2y = 3, 5x + ky + 7 = 0 is inconsistent is equal to:
4 (a) 3 (b) – 4 (c) 4 (d) 10
The values of x and y in 2x + 3y = 2 and x - 2y = 8 are:
5
(a) - 4, 2 (b) - 4, - 2 (c) 4, - 2 (d) 4, 2
Sum of two numbers is 35 and their difference is 13, then the numbers are:
6 (a) 24 and 12 (b) 24 and 11 (c) 12 and 11 (d) none of these
x and y are two different digits. If the sum of the original number and the number formed by reversing the digits
7 is a perfect square, then value of x + y is
(a) 10 (b) 11 (c) 12 (d) 13
One equation of a pair of dependent linear equations −5x + 7y = 2. The second equation can be
8 (a) 10x + 14y + 4 = 0 (b) -10x -14y - 4 = 0
(c) -10x + 14y + 4 = 0 (d) 10x – 14y = - 4
9 If x = a and y = b is the solution of the equations x – y = 2 and x + y = 4, then the values of a2 and b2 are,
respectively
(a) 9 and 4 (b) 1and 4 (c) 9 and 1 (d) 4 and 25
10 The pair of equations x = b and y = a graphically represents lines which are
(a) parallel (b) intersecting at (b, a)
(c) coincident (d) intersecting at (a, b)
ASSERTION REASONING QUESTIONS
DIRECTION : In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice
as:
(a)Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b)Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c)Assertion (A) is true but reason (R) is false.
(d)Assertion (A) is false but reason (R) is true.
Q.1 Assertion : The linear equations x – 2y – 3 = 0 and 3x + 4y – 20 = 0 have exactly one solution.
Reason : If the pair of lines are parallel, then the pair has no solution and is called inconsistent pair of equations.
Q.2 Assertion : The graph of the linear equations 3x + 2y = 12 and 5x – 2y = 4 gives a pair of intersecting lines.
a b1
Reason : The graph of linear equations a1x+b1y+c1=0 and a2x +b2y +c2 = 0 gives a pair of intersecting lines if a1 ≠
2 b2
Q.3 Assertion : A pair of linear equations has no solution (s) if it is represented by intersecting lines graphically.
Reason: If the pair of lines are intersecting, then the pair has unique solution and is called consistent pair of equations.
CASE BASED STUDY QUESTIONS
Q1 MASK: Masks are an additional step to help prevent people from getting and spreading COVID-19. They
provide a barrier that keeps respiratory droplets from spreading. Wear a mask and take every day preventive
actions in public settings.
Due to ongoing Corona virus outbreak, Wellness Medical store has started selling masks of decent quality. The
store is selling two types of masks currently type A and type B. The cost of type A mask is Rs.15 and of type B
mask is Rs.20. In the month of April, 2020, the store sold 100 masks for total sales of Rs.1650
(i) How many masks of type A were sold in the month of April?
(ii) How many masks of type B were sold in the month of April?
(iii) If the store had sold 50 masks of each type, what would be its sale in the month of April?
OR
If the cost of type A mask would have been Rs.20 and of type B mask be Rs.25, then find the total sale for the
month of April.
Q2 At some point, it’s time to gently ease, kids off the parental gravy train. The circle graph shows the percentage
of parents who think significant financial support should end at various milestones
The difference in the percentage who would end this support after completing college and after completing
high school is 6%.
(i) What is the percentage of parents who would end financial support after a child completes college.
(ii) What is the percentage of parents who would end financial support after a child completes high school.
(iii) What is the total combined percentage of parents who would end financial support after a child completes high
school and after getting a full time job?
OR
What is the total combined percentage of parents who would end financial support after a child completes
college and after getting married?
IMPORTANT QUESTIONS FROM NCERT BOOK

𝑎1 𝑏1 𝑐1
1. On comparing the ratios , and find out whether the lines representing the following pairs of linear equations intersect
𝑎2 𝑏2 𝑐2
at a point, are parallel or coincident:
(i) 5x – 4y + 8 = 0, 7x + 6y – 9 = 0
(ii) 9x + 3y + 12 = 0, 18x + 6y + 24 = 0
(iii) 6x – 3y + 10 = 0, 2x – y + 9 = 0
𝑎 𝑏 𝑐1
2. On comparing the ratios 𝑎1 , 𝑏1 and find out whether the following pair of linear equations are consistent, or inconsistent.
2 2 𝑐2
(i) 5x – 3y = 11; – 10x + 6y = –22

(ii) 2x – 3y = 8; 4x – 6y = 9
3. Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the
garden.
4. The difference between two numbers is 26 and one number is three times the other. Find them.
5. The larger of two supplementary angles exceeds the smaller by 18 degrees. Find them.
6. The coach of a cricket team buys 7 bats and 6 balls for Rs.3800. Later, she buys 3 bats and 5 balls for
₹1750. Find the cost of each bat and each ball.
7.The taxi charges in a city consist of a fixed charge together with the charge for the distance covered. For adistance of 10 km, the
charge paid is ₹105 and for a journey of 15 km, the charge paid is ₹ 155. What are the fixed charges and the charge per km?
How much does a person have to pay for travelling a distanceof 25 km?
9
8.A fraction becomes , if 2 is added to both the numerator and the denominator. If, 3 is added to both the numerator and the
11
5
denominator it becomes 6 . Find the fraction.
9. Five years hence, the age of Jacob will be three times that of his son. Five years ago, Jacob’s age was seven times that of his
son. What are their present ages?
10. The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by
reversing the order of the digits. Find the number.
11. Meena went to a bank to withdraw ₹ 2000. She asked the cashier to give her ₹50 and ₹100 notes only. Meena got
25 notes in all. Find how many notes of ₹ 50 and ₹ 100 she received.
12. A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Sarita paid ₹
27 for a book kept for seven days, while Susila paid ₹ 21 for the book she kept for five days. Find the fixed charge and the
charge for each extra day.
ADDITIONAL QUESTIONS
1. Solve graphically the system of linear equations 4x – 5y + 16 = 0 and 2x + y – 6 = 0. Determine the vertices of the triangle
formed by these lines and the x-axis and find the area of the triangle so formed.
2. Solve the following system of linear equations graphically: 4x – 5y – 20 = 0 and 3x + 5y – 15 = 0.Determine the vertices of the
triangle formed by the lines representing the above equations and the y-axis.
3. Solve for x and y: 0.4x – 1.5y = 6.5, 0.3x – 0.2y = 0.9.
4. Find the values of k for which the system of equations x – 2y = 3, 3x + ky =1 has a unique solution.
5. Find the value of k for which the following pair of linear equations has infinitely many solutions:
2x – 3y = 7, (k + 1)x + (1 – 2k)y = (5k – 4) .
6. Find the values of k for which the pair of linear equations kx + 3y = k – 2 and 12x + ky = k has no solution.
7. Find the values of k for which the system of equations kx – y = 2, 6x – 2y = 3 has (i) a unique solution, (ii) no solution. (iii) Is there a
value of k for which the given system has infinitely many solutions?
8. The sum of the digits of a two-digit number is 12. The number obtained by interchanging its digits exceeds the given number by 18.
Find the number.
HOTS QUESTIONS
1. If a bag containing red and white balls, half the number of white balls is equal to one-third the numbers of red balls. Thrice the
total number of balls exceeds seven times the number of white balls by 6. How many balls of each colour does the bag contain?
2. A and B are two points 150 km apart on a highway. Two cars start A and B at the same time. If they move in the same direction they
meet in 15 hours. But if they move in the opposite direction, they meet in 1hour. Find their speeds.
3. Find the value of x and y: 99x + 101y =499
101x + 99y =501
4. In the figure below ABCDE is a pentagon with BE || CD and BC || DE. BC is perpendicular to DC. If the perimeter of ABCDE is 21
cm, find the values of x and y.
5. It can take 12 hours to fill a swimming pool using two pipes. If the pipe of larger diameter is used for 4 hours and the pipe of
smaller diameter for 9 hours only half the pool can be filled. How long would it take for each pipe to fill the pool separately?
6. Find the value of x and y of the following pair of linear equation and justify your answer.
2ax + by =a, 4ax + 2by -2a =0; a, b≠ 0
ANSWERS
MCQ
1. C 2. D 3.C 4.D 5.C 6.B

7. (B) 11 (The numbers that can be formed are (10x + y) and (10y + x). Hence, (10x + y) + (10y + x) = 11(x + y). If this is a perfect
square than x + y = 11.)
8. D 9. C 10. B

1. B 2. A 3. D
CASE STUDY QUESTIONS
1. (i) (a) 70, (ii) (c) 30 (iii) (b) 1750 OR (d) 2150
2. (i) (c) 28% (ii) (a) 22% (iii) (c) 52% OR (d) 34%
IMPORTANT QUESTIONS FROM NCERT BOOK

1. (i) Intersect at a point (ii) Coincident (iii) Parallel
2. (i) Consistent (ii) inconsistent
3. Length = 20 m and breadth = 16 m.
4. 39, 13.
5. 99, 81.
6. Cost of one bat = ₹500, cost of one ball = ₹50.
7. Fixed charge =₹5, charge per km = ₹10; travelling charge = ₹255.
9
8. Fraction = 7
9. Jacob’s age = 40 years, son’s age = 10 years.
10. 18
11. Number of Rs.50 = 10, Number of Rs.100 notes = 15.
12. Fixed charge per day = ₹ 15and the additional charge per day = ₹3
ADDITIONAL QUESTION
1. Vertices are (1, 4), (–4, 0) and (3, 0), 14 sq. unit
2. (0, –4), (0, 3) and (5, 0).
3. x = 5 and y = –3.
4. All real values of k, other than –6.
5. k = 5
6. k = 6 or k = – 6.
7. (i) k ≠ 3, (ii) k =3, (iii) no real value of k
8. 57
HOTS QUESTIONS
1. Let the number of red balls be x and white balls be yAccording to the question,
2x - 3y = 0 (1)
And 3(x + y) – 7y = 6
3x - 4y = 6 (2)
Solving (1) from (2) we have
y = 12 and x = 18
Hence, number of red balls = 18 and number of white balls = 12
2. Let the speed of car I from A be x and speed of car II from B be y.

Same Direction: Distance covered by car I = 150 + Distance covered by car II
15x = 150 + 15y
x - y = 10 (1)
Opposite Direction: Distance covered by car I + Distance covered by car II =150
x + y = 150 (2)
x = 80 and y = 70
Speed of car I from A = 80 km/h and speed of car II from B = 70km/h.
3.x= 3 , y = 4
4. Since BE || CD and BC || DE with BC is perpendicular to DC, bcde is arectangle.

BE = CD,
x+y=5 (1)
and DE = BE = x – y
Since perimeter of ABCDE is 21,AB + BC + CD + DE + EA = 21
3 + x – y + x + y + x – y + 3 = 216 + 3x – y = 21
3x – y = 15 (2)
x = 5 and y=0
5. The pipe of larger diameter alone can fill the pool in 20 hours and the pipe of smaller diameter alone can fill the pool in 30 hours.
6. Many solutions
Chapter 4
QUADRATIC EQUATIONS
An equation of the form p(x) =0, where p(x) is a quadratic polynomial (polynomial of degree 2) is a called quadratic equation.
In general form, ax2 + bx + c = 0, a ≠ 0 is a quadratic equation in variable x.
SOLUTION OF A QUADRATIC EQUATION
Roots of the quadratic equation ax2 + bx + c = 0 are called the solutions of the quadratic equation. Solutions of a Quadratic Equation
can be found by using following methods:
(i) By Factorisation Method: To find the solution of a quadratic equation by factorisation method, first represent the given
equation as a product of two linear factors by splitting the middle term or by using identities and then equate each of the factor equal
to zero to get the desired roots.
(ii) By Quadratic Formula: For a quadratic equation ax2 + bx + c = 0, we have
−𝒃±√𝐃
x= , where D (∆) = b2 - 4ac and D is called discriminant of the quadratic equation.
𝟐𝒂
The following cases arise:
i. If D = b2 - 4ac>0 then the roots of the equation are real and distinct.
−𝒃
ii. If D = b2 - 4ac=0 then roots of the equation are equal and real. α= β= 𝟐𝒂
iii. If D = b2 - 4ac<0 then there does not exist any real root.
iv. If D = b2 - 4ac>0 and perfect square, then the roots are real (rational) and unequal.
v. If D = b2- 4ac > 0 and not a perfect square, then the roots are real(irrational) and unequal
QUADRATIC EQUATION WHEN THE ROOTS ARE GIVEN
The quadratic equation whose roots are a and b is given as x2 - (a +b) x +ab = 0
(Question numbers from 1 to 10 carry 1 mark each)

Q1 The quadratic equation kx2 – 4kx + 2k + 1 = 0 has repeated roots, if k equal to
(a) 0 (b) 1/2 (c) 2 (d) 4
Q2 If r=3 is a root of the quadratic equation kr2- kr –3=0 then value of k is
(a) 6 (b) 2 (c) 1/2 (d) -1/2
Q3 The two roots of a quadratic equation are 2 and - 1. The equation is
(a) x2 + 2x - 2 = 0 (b) x2 + x + 2 = 0 (c) x2 - 2x + 2 = 0 (d) x2 - x - 2 = 0
Q4 ax2 + bx + c = 0, a > 0, b = 0, c > 0 has
(a) Two equal roots (b) one real roots (c) two distinct real roots (d) no real roots
Q5 Find the value of k for the which equation x +kx+64=0 and x -8x+k = 0 will have real and equal roots
2 2
(a) 8 (b)-8 (c) 16 (d) -16

Q6.one root of: x – (√3+1) x +√3=0 is
2
(a) 2 (b) -1 (c) √2 (d) √3

Assertion and reasoning based question (from Q7 to Q10)
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement
Q7. Assertion: 2x2 - 4x + 3 = 0 is a quadratic equation.
Reason: All polynomials of degree n, when n is a whole number can be treated as quadratic equation.
Q8. Assertion: 3y2 + 17y - 30 = 0 have distinct roots.
Reason: The quadratic equation ax 2 + bx + c = 0 have distinct roots (real) if D > 0.
Q9. Assertion: Both the roots of the equation x2 - x + 1 = 0 are real.
Reason: The roots of the equation ax2 + bx + c = 0 are real and distinct if and only if b2 - 4ac > 0
Q10. Assertion: The discriminant of the quadratic equation 2x2 -4x+3 =0, is -8 and hence the nature of its root is no real roots.
Reason: If b2 -4ac <0 then nature of roots is not real.
(Question numbers from 11 to 14 carry 2 marks each)
Q11. Find value of p for which the product of roots of the quadratic equation px 2 + 6x + 4p = 0 is equal to the sum of the roots.
Q12. A fast train takes 3 hours less than a slow train for a journey of 600 km. If the speed of the slow train is 10 km/h less than the
fast train, find the speeds of the two trains.
Q13. If the roots of the quadratic equation (b - c) x2 + (c - a) x + (a - b) = 0 are equal, prove that 2b = a + c.
Q14. A two digit number is such that the product of its digits is 18. When 63 is subtracted from the number, the digit interchange their
places. Find the number.
(Question numbers from 15 to 17 carry 3 marks each )
Q15. Solve for x: √3x2-2√2x-2√3 =0
1
Q16. Find the discriminant of the equation: 3x2 -2x + 3 =0 and hence find the nature of the roots Find the roots if they are real.
3
Q17. Two water taps together can fill a tank in 9 8 hours. The tap of larger diameter takes 10 hours less than the smaller one to fill
separately. Find the time in which each tap can separately fill the tank.
(Question numbers from 18 to 20 carry 5 marks each)
Q18. Sum of the areas of the two squares is 468 cm 2. If the difference of their perimeters is 24 m find the sides of the two squares
Q19 The age of the man is twice the square of the age of his son .Eight years hence the age of the man will be 4 more than three times
the age of his son,. Find their present ages.
1 1 1 1
Q 20.Solve for x: 𝑎+𝑏+𝑥 =𝑎 + 𝑏 + 𝑥 ; a≠0, b≠0, x ≠0
(Case based study: Question numbers from 21 to 23 carry 4 marks each)

Q21. Ram and Shyam are very close friends. Ram owns a Honda city and Shyam owns Toyota corolla. They go together for a picnic by
their cars. Ram’s car travels at x km/h while Shyam’s car travels at 5km/h more than Ram’s car. Shyam’s car takes 1 hour less than
ram’s car in covering 360km. Answer the following
(a) What will be distance covered by Ram’s car in 5 hours?
(b) Write the quadratic equation that describe the condition
(c) How much time did ram take to cover 360km?
Or
How much time did Shyam take to cover 360km?
Q22. Kavita and her mother went for small picnic. After having morning breakfast, Kavita insisted to travel in a motor boat .The speed
of the motor boat was 18km/h in still water. Kavita being a Mathematics student wanted to know the speed of current .So, she noted
the time for upstream and downstream, She found that for covering the distance 24 km, the boat took 1 hour more for upstream than
downstream.
Answer the following questions:
(a) Let the speed of the stream be x km/h, then find the speed of motorboat upstream.
(b) Write the correct quadratic equation for given condition
(c) What is the speed of stream
Or
How much time did the motorboat take going downward?
Q23.Ravi interested to install rectangular swimming pool in their backyard.
There is concrete sidewalk around the pool of width x m and the outer edges of the sidewalk width is 7m and length is 12 m and area
of the pool is 36 m2.
Answer the following questions:
(a) Based on the above information form quadratic equations.
(b) Find the width of the pool
Or
Find the length of the pool
(c) What will be area of pool if width of concrete walk side is 2.5m if dimensions remain same?
Answer
MCQ / ASSERTION
Q1. (b) Q2. (c)Q3. (d)Q4. (d)Q5. (c)Q6. (d)Q7. (c)Q8. (a)Q9. (d)Q10. (a)
2MARKS QUESTIONS
Q11. Sum of roots = product of roots
-6/p = 4p/p
p=-3/2
Q12. Let speed of fast train is x km/h,

600/x-10 – 600/x = 3
x2 -10x-2000 = 0
x=50, -40(-40 not possible because speed cannot be negative)
Speed of fast train is 50km/h and speed of slow train is 40km/h
Q13. For equal and real root is quadratic eqn. D=0

(c-a)2- 4 (b-c)(a-b)=0
b2+c2-2bc-4ac+4a2 +4bc -4ab=0
4a2+b2+c2-4ab+2bc -4ac =0
(2a-b-c)2 =0
2a = b+c
Q14.let ten placed of digits is x and unit place is x/18
ATQ
18 180
10x + 𝑥 -63 = 𝑥 + x
x2-7x-18=0
x= 9, -2 (negative value ignored)
Ten placed is 9 and unit placed is 2 number is 92
3 MARKS QUESTIONS
−√2
Q15. x= √6, √3
Q16. 9x2-6x+1 = 0
D=0 so, real roots exist
X=1/3, 1/3
Q17. Let the smaller tap fill the tank in x hr and the tap with larger diameter fill the tank in x-10 hr
ATQ
1 1 8
+ =
𝑥 𝑥−10 75
4x2 -115x+375=0
X = 15/4, 25 (15/4 not possible)
Smaller diameter tap fill in 25 hours and larger diameter tap fill in 15 hrs.
5 MARKS QUESTIONS
Q18. Side of 1st square is x cm and side of 2nd square is x+6 cm
ATQ
x +( x+6)2 = 468
2
x2 +6x-432=0
x= -18, 12 (negative value ignored)
Side of 1st square 12cm and side 2nd square is 18cm
Q19. Let present age of son is x years and father age is 2x2
ATQ
2x2 -3x-20=0
X=4, -5/2 (age cannot be negative)
Present age of son is 4 year and father age is 32years
1 1 1 1
Q20.𝑎+𝑏+𝑥 - = 𝑎 +𝑏
𝑥
x2+(a+b)x + ab =0
x = -a, -b
CASE BASED STUDY
Q21. (a) 225km (b) x2+5x-1800 (c) 9 hours or 8 hours
Q22. (a) 12km/h (b) x2+48x-324 (c) 6km/h or 1 hour
Q23. (a) 2x2-19x+24 (b) width = 4m or length 9 m (c) 14m2
CHAPTER 5 ARITHMETIC PROGRESSION
Things to remember:
 Standard form of an AP is given as: a, a+d, a+2d, a+3d...., where a is the first term and d is the common difference.
 nth term 𝒂𝒏 of the AP with first term a and common difference d is given by 𝒂𝒏 = a+ (n-1)d
 nth term from the end of an AP is given by: an=l - (n-1)d , where l is the last term of A.P., d is the common difference and an is
the nth term
𝒂+𝒄
 Arithmetic mean : b = , where a, b, c are in A.P. and b is arithmetic mean
𝟐
 Sum of first n positive integers: Let Sn = 1 + 2 + 3 +……..n, here a = 1, last term, l = n is

𝒏( 𝒏+𝟏)
Sn = 𝟐
𝒏
 Sum of n terms of an AP: Sn = { 2a+(n-1)d}
𝟐
𝒏
 Sum of n terms of an AP : Sn = 𝟐{ a+ l }, where l is last term of the series
 Also, an =Sn — Sn-1
 3 numbers in A.P. are taken as : a-d , a , a+d
 4 numbers in A.P. are taken as: a – 3d , a - d , a + d , a +3d
 5 numbers in A.P. are taken as: a – 2d, a – d, a , a + d, a + 2d
Very short answer type questions
1. Which term of the A.P. 27, 24, 21……. is 0?
(a) 9th (b) 10th (c) 11th (d) 12th
2. How many two digit numbers are divisible by 3?
(a) 28 (b) 29 (c) 30 (d) 31
3. First term of A.P. is p and common difference is q, then its 10 term is:
th
(a) q + 9p (b) p - 9q (c) p + 9q (d) 2p + 9q

4. What is the common difference of an A.P. in which a18 – a14 = 32
(a) 8 (b) -8 (c) -4 (d) 4
5. Find whether 100 is a term of A.P. 25, 28, 31……..or not?
(a) yes (b) no (c) Cannot be determined (d) none of these
6. Assertion: If the n term of an A.P. is 7- 4n, then its common difference is -4.
th
Reason: common difference of an A.P. is given by d = an+1 - an.

(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
7. Assertion: 10th term of the A.P.: -5, -1, 3, 7 ….. is 31
Reason: nth term of the A.P: a, a+d, a+ 2d……is given by 𝒂𝒏 = a+ (n-1)d.
(c) An assertion (A) is true but reason (R) is false.
8. If the sum of the first 4 terms of an A.P. is 40 and that of first 14 terms is 280. Find the sum of its first n terms.
(a) 6n2+6 (b) 6n+ n2 (c) 7n2+7 (d) 6n2+6n
9. 30th term of the A.P.; 10, 7, 4 ……is:
(a) 97 (b) 77 (c) -77 (d) -88
10. Which term of the A.P. 21, 42, 63, 84…… is 210?
(a) 9th (b) 10th (c) 11th (d) 12th
Short answer type questions
11. How many terms of the A.P.: 65, 60, 55, ……… be taken so that their sum is 0?
12. Find the common difference (d) of an A.P. whose first term is 10 and the sum of the first 14 terms is 1505
13. The 19th term of an A.P. is equal to 3 times its 6th term. If the 9th term is 19, find the A.P.
14. Find the sum of all 11 terms of an A.P. whose middle term is 30.
15. The sum of the first 7 terms of an A.P. is 63 and that of its next 7 terms is 161. Find the A.P.
16. Find the next term of the A.P. √𝟔, √𝟐𝟒, √𝟓𝟒,….
𝟏
17. The 26th, 11th and the last term of an AP are 0, 3 and - , respectively. Find the common difference and the number of terms.(Hint:
𝟓
a26= a + 25d=0 ; a11 = a +10d = 3))
18. If Sn denotes the sum of first n terms of an A.P., prove that S30 = 3[S20 - S10]
𝟐𝟎 𝟏𝟎
(Hint: S30= 3 [ (a + 19 d) - (a + 9d)]
𝟐 𝟐
19. What is the second negative term of the A.P.: 51, 48, 45, ………?
20. Find the sum of last ten terms of the A.P.: 8, 10, 12 …..126
Long answer type questions
21. The sum of 4 consecutive numbers in A.P. is 32 and the ratio of the product of the first and the last term to the product of two
middle terms is 7:15. Find the numbers.
(Hint: 4 consecutive no.s: (a-3d), (a-d), (a+d), (a + 3d))
22. 200 logs are stacked in the following manner: 20 logs in the bottom row, 19 in the next row, 18 in the row next to it and so on. In
how many rows are the 200 logs placed and how many logs are in the top row?
23. If m times the mth term of A.P. is equal to n times of nth term and m≠n, show that (m+n)th term of the A.P. is 0.
[Hint: step 1: m (am) = n (an); step 2: m{ a + (m-1)d} = n{ a + (n-1)d}]
24. A man repays a loan of ₹ 3250 by paying ₹ 20 in the first month and then increases the payment by ₹ 15 every month. How long
will it take him to clear the loan? (Hint: 𝒏𝟐{2x20+(n-1)15} = 3250)
25. A thief runs with a uniform speed of 100 m/minute. After one minute a policeman runs after the thief to catch him. He goes with a
speed of 100m/minute in the first minute and increased his speed by 10 m/minute every succeeding minute. After how many
minutes the policeman will catch the thief. Hint: [distance = 100(n +1) = 𝒏𝟐 {2 x 100+(n-1)10}]
26. A sum of ₹ 1000 is to be used to give five cash prizes to students of a school for their overall academic performance. If each prize
is ₹ 50 less than its preceding prize, find the value of each of the prizes
27. A contract on construction job specifies a penalty for delay of completion beyond a certain date as follows: ₹250 for the first day,
₹ 300 for the second day, ₹ 350 for the third day, etc., the penalty for each succeeding day being ₹ 50 more than for the preceding
day. How much money the contractor has to pay as penalty, if he has delayed the work by 30 days?
Case Study Based Questions
28. Seating Capacity:
The Fox Theatre creates a “theatre in the round” when it shows any of Rabindranath Tagore’s plays. The first row has 80 seats,
he second row has 88, the third row has 96, and so on.
(i) How many seats are in the 10th row?
(ii) How many seats are in the 25th row?
(iii) If there is a room for 25 rows, how many chairs will be needed to set up the theatre?
Or
How many people can sit in the second last row?
29. Contest Prizes:
A contest offers 15 prizes. The 1st prize is ₹ 5000, and each successive prize is ₹ 250 less than the preceding prize.
(i) What is the value of the 15th prize?
(ii) What is the total amount of money distributed in prizes?
(iii) What is the sum of first and last prize?
OR
By how much does the amount for 1st prize exceed the amount for the 6th prize?
30. Increasing Salary:
A teacher has a salary of ₹ 8,00,000 during the first year and gets an increment of ₹ 5000 each year.
(i) What will be his salary in the seventh year?
(ii) What is the total salary for 7 years of work for the teacher?
(iii) What is the difference of the salary of 7th year and 2nd year?
OR
What will be his salary after 10 years of service? ( Hint: find 11th term)
31. Let us practice Piano :
Suppose you practice the piano 45 min on the first day of the semester and increase your practice time by 5 min each day.
(i) How much total time will you devote to practicing during the first 15 days of the semester? (ii) How much time will you
devote to practicing during the 35th day of the semester?
(iii) How can you relate music with Mathematics?
Or
Write one benefit of learning music.
32. TOWER OF PISA :

To prove that objects of different weights fall at the same rate, Galileo dropped two objects with different weights from the
Leaning Tower of Pisa in Italy. The objects hit the ground at the same time. When an object is dropped from a tall building, it
falls about 16 feet in the first second, 48 feet in the second, and 80 feet in the third second, regardless of its weight.
(i) How many feet would an object fall in the sixth second?
(ii) How many feet would an object fall in the six second?
(Hint: Find the sum)
(iii) How many feet would an object fall in the 8th second?
OR
How many feet would the object fall from 6th second to 8th second?
Answers:
1. b 2. c 3. c 4. a 5. a
6. a 7. a 8. b 9. c 10. b
11. 27 12. 15 13. 3, 5, 7, 9 14. 330 15. 3,5,7,9
16. √𝟗𝟔 17. d = -1/5, n= 27 18. Refer hint 19. -6 20. 1170
21. 2, 6, 10,14 or 14, 22. 16 rows, 5 logs 23. Refer hint 24. 20 months 25. 5 min
10,6,2
26. 300, 250, 200, 150, 100 27. ₹ 29250 28. i. 152 29.i. ₹1500 30. i. ₹8,30, 000
ii. 272 ii. ₹48750 ii. ₹ 57, 05, 000
iii. 4400
iii. ₹3500 iii. ₹ 25000
264 people
₹1250 ₹ 850000
31.i. 20 hours 32. i. 176
ii. 215 min ii. 576
iii. open ended answer iii. 240 feet
64 feet
CHAPTER 6 (Triangles)
Similar Triangles: Two triangles are said to be similar if their corresponding angles are equal and their corresponding
sides are proportional.
Criteria for Similarity: In ΔABC and ΔDEF
(i) AAA Similarity: ΔABC ~ ΔDEF when
∠A=∠D, ∠B=∠E and ∠C=∠F
(ii) SAS Similarity:
ΔABC~ΔDEF when
∠A=∠D and AB/DE=AC/DF
(iii) SSS Similarity: ΔABC ~ ΔDEF if: AB/DE = AC/DF = BC/EF
Basic Proportionality Theorem: If a line is drawn parallel to one side of a triangle to intersect the other two sides
in distinct points, the other two sides are divided in the same ratio.
S.NO. QUESTIONS
1 If in two triangles Δ DEF and Δ PQR, ∠D = ∠Q and ∠R = ∠E, then which of the following is not true?
(a) EF/PR = DF/PQ (b) DE/PQ = EF/RP (c) DE/QR = DF/PQ (d) EF/RP = DE/QR
2 If ΔABC ~ ΔEDF and ΔABC is not similar to ΔDEF, then which of the following is not true?
(a) BC . EF = AC . FD (b) AB . EF = AC . DE (c) BC . DE = AB . EF (d) BC . DE = AB . FD
3 D and E are respectively the points on the sides AB and AC of a triangle ABC such that AD = 2 cm, BD = 3 cm, BC = 7.5
cm and DE || BC. Then, length of DE (in cms) is:
(a) 2.5 (b) 3 (c) 5 (d) 6
4 A vertical pole of length 6 m casts a shadow 4 m long on the ground and at the same time a tower casts a shadow 28
m long. Find the height of the tower (in m).
(a) 42 (b) 32 (c) 5 (d)16
5 In this figure if DE || BC. Then find the value of x :
(a) 10 (b) 11 (c) 12 (d)13
6 In Δ ABC, D and E are mid-points of AC and BC respectively such that DE || AB. If AD = 2x, BE = 2x – 1,
CD = x + 1 and CE = x – 1, then find the value of x:
7 Students of a school decided to participate in ‘Save girl child’ campaign. They decided to decorate a
triangular path as shown. If AB = AC and BC2 = AC × CD, then prove that BD = BC.
8 The perimeters of two similar triangles are 25 cm and 15 cm respectively. If one side of the first triangle is 9 cm, find
the corresponding side of the second triangle.
9 In ΔABC, D and E are points on the sides of AB and AC such that DE || BC. If AD = 2.5 cm, BD =3 cm, AE = 3.75 cm, find the
length of AC.
10 Prove that a line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side.
11 ABCD is a trapezium in which AB || DC and its diagonals intersect each other at the point O. Show that 𝐴𝑂/𝐵𝑂 = 𝐶𝑂/𝐷𝑂
12 Diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at the point O.
Using a similarity criterion for two triangles, show that OA/OC=OB/OD∙
13 In the given figure, altitudes AD and CE of ∆ABC intersect each other at the point P.
Show that:
(i) ∆AEP ~ ∆CDP
(ii) ∆ABD ~ ∆CBE
(iii) ∆AEP ~ ∆ADB
(iv) ∆PDC ~ ∆BEC
14 In the given figure, ABC and AMP are two right triangles, right angled at B and M respectively. Prove that:
(i) ∆ABC~ ∆AMP
(ii)CA/PA = BC/MP
15 CD and GH are respectively the bisectors of ∠ACB and ∠EGF such that D and H lie on sides AB and FE of ∆ABC and
∆EFG respectively. If ∆ABC ~ ∆FEG, show that
(i) ∆DCB~ ∆HGE (ii) C D / G H = A C / F G
16 Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM
of ∆PQR. Show that ∆ABC ~ ∆PQR.
17 State and prove Basic proportionality theorem.
HOTS QUESTIONS
1. In Fig , OB is the perpendicular bisector of the line segment DE, FA ⊥ OB and FE intersects OB at the point C.
Prove that: (1/ OA) + (1/ OB) = 2/ OC
2. In the figure, ABCD is a parallelogram and E divides BC in the ratio 1: 3. DB and AE intersect at F.
Show that DF = 4 FB and AF = 4 FE.
3. In Fig., ΔFEC ≅ΔGDB and ∠1 = ∠2. Prove that ΔADE ~ ∆ABC.
4. Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and median PM of another triangle
PQR. Show that ∆ ABC ~ ∆ PQR
CASE STUDY BASED QUESTIONS:
1) CASE STUDY
In the hot Indian summers, we are all glad to have the trees and their shade. But do you know how shadows are formed? Shadows are
formed when light, for example, sunlight, falls on opaque objects. Consider the shadows of two trees A and B. The shadow of two trees
A and B formed at 6 pm on a particular day is given in the diagram.
The height of tree A is 5m and the height of tree B is 7m. The length of the shadow of tree B is 21m.
1) What is the length of the shadow of tree A?
2) What concept is used for finding the height of the tree?
3) What is the value of x.
2) CASE STUDY 2- SCALE FACTOR

A scale drawing of an object is the same shape of the object but of different size. The scale of a drawing is a comparison of the length
used on a drawing to the length it represents. The scale is written as a ratio. The ratio of two corresponding sides in similar figures is
called the scale factor.
Scale factor= length in image / corresponding length in object
If one shape can become another using revising, then the shapes are similar.
Hence, two shapes are similar when one can become the other after a resize, flip, slide or turn.
In the photograph below showing the side view of a train engine. Scale factor is 1:200.
This means that a length of 1 cm on the photograph above corresponds to a length of 200cm or 2 m, the actual engine.
The scale can also be written as the ratio of two lengths.
Q1. If the length of the model is 11cm, then what is the overall length of the engine in the photograph above,
including the couplings (mechanism used to connect)?
Q2. What is the actual width of the door if the width of the door in photograph is 0.35cm?
Q3. The length of AB in the given figure:
3. CASE STUDY
Vijay is trying to find the average height of a tower near his house.
He is using the properties of similar triangles.
The height of Vijay’s house if 20m when Vijay’s house casts a shadow 10m long on the ground.
At the same time, the tower casts a shadow 50m long on the ground and the house of Ajay casts 20m .
shadow on the ground.
1. What is the height of the tower?
2. What will be the length of the shadow of the tower when Vijay’s house casts a shadow of 12m?
3. What is the height of Ajay’s house?
.
4. CASE STUDY
Mountaineering is the perfect activity for adventure lovers.
Every year, several mountaineers attempt to climb the Mount Everest.
The path of two mountaineers from the base camps B and C are shown above.
D and E are two mid camping areas in between their paths.
The line joining D and E is parallel to the line joining B and C.
1) Find the distance between E and C.
2) What is the ratio of the distance between DE and BC?
3) If AD/DB= 5/9 and EC = 180 m then find AE.
DIRECTION :
In the following questions a statement of Assertion (A) is followed by a statement of Reason (R).
Choose the correct option as:
1) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
2) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
3) Assertion (A) is true but reason (R) is false.
4) Assertion (A) is false but reason (R) is true.
1. Assertion: D and E are points on the sides AB and AC respectively of a ΔABC such that DE║BC then the value of
x is 4, when AD = x cm, DB = (x – 2) cm, AE = (x + 2) cm and EC = (x – 1) cm.
Reason: If a line is parallel to one side of a triangle, then its divides the other two sides in the same ratio.
2. Assertion (A): E and F are points on the sides PQ and PR respectively of a triangle PQR, if PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and
PF = 0.36 cm, then, EF || QR by converse of BPT.
Reason (R): Converse of basic proportionality theorem states that if a line divides any two sides of a triangle in the
same ratio, then the line is parallel to the third side.
3. Assertion (A): If ΔABC & ΔPQR are congruent triangles, then they are also similar triangles.
Reason (R): All congruent triangles are similar but the similar triangles need not be congruent.
4. Assertion (A): In the given fig. PA || QB || RC || SB
Reason (R ) : If three or more line segments are perpendicular to one line , then they are parallel to each other.
ANSWERS:
1) B 2) C 3) B 4) A 5) B
6) x=1/3 7) correct proof 8)DE=5.4cm 9) AC=8.25cm
Hots Questions
1.(Hint) ΔOFA ~ ΔODB
∠A = ∠B=90
2. (Hint) ΔADF ~ ΔEBF by AA
3. (Hint) DE || BC by converse of BPT
∠1= ∠3, ∠2 = ∠4 corresponding angles
Case study
1. (i) 15m (ii) similarity of triangle (iii) x = 2, -5 (neglect negative value)
2. (i) 22m (ii) 0.7 m (iii) 4cm
3. (i) 100m (ii)24m (iii)40m
4. (i)330km (ii) 5 : 11 (iii) 100m
ASSERTION REASONING QUESTIONS-

1. A 2) A 3) A 4) A
CHAPTER 7
COORDINATE GEOMETRY
The system of geometry where the position of points on the plane is described using an ordered pair of numbers.
DISTANCE FORMULA:
Distance between two given points A(x1,y1) and B(x2,y2)
AB = √(𝒙𝟐 − 𝒙𝟏 )𝟐 + (𝒚𝟐 − 𝒚𝟏 )𝟐
Note: Distance of a point P(x, y) from origin is √𝑥 2 + 𝑦 2
SECTION FORMULA:
 The coordinates of the point P(x, y) which divides the line segment joining the points A(x1,y1) And B(x2,y2) internally in the
ratio m1 : m2 are
𝑚 𝑥 +𝑚 𝑥 𝑚 𝑦 +𝑚 𝑦
x= 1𝑚2 +𝑚2 1 , y= 1𝑚2 +𝑚2 1
1 2 1 2
 If the ratio in which P divides AB is k:1, Then the coordinates of the point P will be:
𝑘𝑥2 +𝑥1 𝑘𝑦2 +𝑦1
x= 𝑘+1 , y= 𝑘+1
 The coordinates of point M(x,y) which is the midpoint of point A(x1,y1) and B(x2,y2) are
𝑥 +𝑥 𝑦 +𝑦
x = 22 1 , y = 22 1
CENTROID OF TRIANGLE: The coordinates of the vertices of a triangle are A(x1, y1) , B(x2,y2) and C(x3,y3) then centroid of given
triangle ABC can be find out using:
𝑥 +𝑥 +𝑥 𝑦 +𝑦 +𝑦
x= 1 32 3 , y= 1 32 3
Problems based on distance formula : To show that a given figure is a
 Parallelogram – prove that the opposite sides are equal.
 Rectangle – prove that the opposite sides are equal and the diagonals are equal.
 Parallelogram but not rectangle – prove that the opposite sides are equal and the diagonals are not equal.
 Rhombus – prove that all the four sides are equal
 Square – prove that all the four sides are equal and the diagonals are equal.
 Rhombus but not square – prove that all the four sides are equal and the diagonals are not equal.
 Isosceles triangle – prove any two sides are equal.
 Equilateral triangle – prove that all three sides are equal.
 Right triangle – prove that sides of triangle satisfy Pythagoras theorem.
MULTIPLE CHOICE QUESTIONS
SECTION - A
1. The distance between the point P (1, 4) and Q (4, 0) is:
(a) 4 (b) 5 (c) 6 (d) 3√3
2. AOBC is a rectangle whose three vertices are A (0, 3), O (0, 0) and B (5, 0). The length of its diagonal is:
(a) 5 (b)√34 (c) 34 (d) 4
3. Find the coordinates of the mid-point of the line segment joining A(3, 0) and B(−5, 4) is:
(a) (1, 2) (b) (1, -2) (c) (-1, 2) (d) (-1, -2)
4. The point which divides the line segment joining the points (7, –6) and (3, 4) in ratio 1:2 internally lies in the:
(a) I quadrant (b) II quadrant (c) III quadrant (d) IV quadrant
5. The distance of the point P (–6, 8) from the origin is:
(a) 2 (b) -10 (c)10 (d) -2
6. The distance of the point (5,3) from the y-axis is:
(a) 2 (b) 3 (c) 1 (d) 5
7. The point (-1, 6) divides the line segment joining the points (-3, 10) and (6, - 8) in the ratio:
(a) 2 : 3 (b) 2 : 5 (c) 2 : 7 (d) 2 : 9
8. The point which lies on the perpendicular bisector of the line segment joining the points A (–2, – 5) and B (2, 5) is:
(a) (0, 0) (b) (0, 2) (c) (2, 0) (d) (–2, 0)
9. If the distance between the points (2, –2) and (–1, x) is 5, one of the values of x is:
(a) 8 (b) 2 (c) 10 (d) 6
10. The distance of the point P (2, 3) from the x-axis is:
(a)2 (b) 3 (c) 1 (d) 5
11. The center of circle with end-points of the diameters A (2, 6) and B (2, -6)
(a) (2,6) (b) (2,0) (c) (0,6) (d) (0,0)
SECTION - B
1. The values of y, for which the distance between the points P(2,-3) and Q(10,y) is 10 units, are:
(a) 9, 6 (b) 3, -9 (c) -3, 9 (d) 9, -6
2. The perpendicular bisector of the line segment joining the points A (1, 5) and B (4, 6) cuts the y-axis at:
(a) (0, 13) (b) (0, –13) (c) (0, 12) (d) (13, 0)
3. A line intersects the Y-axis and X-axis at the points P and Q, respectively. If (2, - 5) is the mid-point of PQ, then the
coordinates of P and Q are respectively:
(a) (0, -5) and (2, 0) (b) (0, 10) and (- 4, 0) (c) (0, 4) and (- 10, 0) (d) (0, – 10) and (4, 0)
4. The point on X- axis which is equidistant from (2, -5) and (-2, 9) is:
(a) (–2,7) (b) (-7,0) (c) (–1,0) (d) (7,0)
5. In what ratio the line x – y – 2 = 0 divides the line segment joining (3, –1) and (8, 9)?
(a)1: 2 (b) 2: 1 (c) 2: 3 (d) 1: 3
6. The fourth vertex D of a parallelogram ABCD whose three vertices are A (–2, 3), B (6, 7) and C (8, 3) is:
(a) (0, 1) (b) (0, –1) (c) (–1, 0) (d) (1, 0)
7. The distance between A (a + b, a – b) and B (a – b, -a – b) is:
(a) 2a+2b (b) 2√(a2 + b2) (c) 2a (d)a2 - b2
8. The line segment joining the points (3, -1) and (-6, 5) is trisected. The coordinates of point of trisection are
(a) (0,1)(3, 3) (b) (0, 1)(- 3, 3) (c) (1, 0)(3, – 3) (d) (1, 0)(-3, -3)
9. The line 3x + y – 9 = 0 divides the line joining the points (1, 3) and (2, 7) internally in the ratio:
(a) 3: 4 (b) 3: 2 (c) 2: 3 (d) 4: 3
10. The coordinates of the centroid of a triangle whose vertices are (0, 6), (8,12) and (8, 0) is:
(a) (4, 6) (b) (16, 6) (c) (8, 6) (d) (16/3, 6)
11. The distance between (0, sin𝜃) and (-cos𝜃, 0) is:
(a) 0 (b)-1 (c) sin𝜃. 𝑐𝑜𝑠𝜃 (d) 1
12. If (a/3, 4) is the mid-point of the segment joining the points P (-6, 5) and R(-2, 3), then the value of ‘a’ is:
(a)12 (b) -6 (c) -12 (d) -4
13. The points (1,1), (-2, 7) and (3, -3) are:
(a) vertices of an equilateral Δ (b) collinear (c) vertices of an isosceles Δ (d) None of these
14.The perimeter of a triangle with vertices (0, 4), (0, 0) and (3, 0) is:
(a) 5 (b) 12 (c) 11 (d) 7
14. The coordinates of the point P dividing the line segment joining the points A(1,3), and B(4,6) in the ratio 2:1 is:
(a) (2,4) (b) (3,5) (c) (4,2) (d) (5,3)
Assertion- Reason Questions:
Direction for questions 1 & 5: In question numbers 1 and 5, a statement of Assertion (A) is followed by a statement of Reason (R).
Choose the correct option.
A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
B. Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
C. Assertion (A) is true but Reason (R) is false.
D. Assertion(A) is false but Reason (R) is correct.
1. Assertion: if the coordinate of the mid-points of sides AB and AC of triangle ABC are D(3,5) and E(-3,-3) respectively then
BC= 20 units
Reason: The line segment joining the mid points of two sides of a triangle is parallel
2. Assertion: The perimeter of ΔAOB where O is origin, A(3,0), B(0, 4) is 7 units.
Reason: Perimeter of a triangle is the sum of all three sides of the triangle.
3. Assertion: Three points A, B, C are such that AB + BC > AC, then they are collinear.
Reason: Three points are collinear if they lie on a straight line.
4. Assertion :The distance point P(2,3) from the x-axis is 3.
Reason: The distance from x-axis is equal to its ordinate.
HOTS
1. The mid-points D , E , F of sides AB , BC , and CA respectively of the sides of a triangle ABC are D(3,4) , E (8,9) and F (6,7).
Find the coordinates of the vertices of the triangle.
2. If the mid-points of the line segment joining the points A(3,4) and B(k,6) is P(x, y) and x+y-10=0, find the value of k.
3. Name the type of triangle PQR formed by the points P (√𝟐, √𝟐) , Q(−√𝟐, −√𝟐) and R (−√𝟔, √𝟔).
4. If P(𝟗𝒂 − 𝟐 , −𝒃) divides line segment joining A(𝟑𝒂 + 𝟏, −𝟑) and B (𝟖𝒂, 𝟓) in the ratio 3:1 , find the values of a and b.
5. Find the ratio in which the point (x, 1) divides the line segment joining the points (-3,5) and (2,-5). Also find the value of x.
CASE STUDY BASED QUESTIONS

CASE STUDY 1
Aditya Starts walking from his house to office. Instead of going to the office directly, he goes to a bank first, from there to his
daughter's school and then reaches the office. (Assume that all distances covered are in straight lines). If the house is situated at (2,
4), bank at (5, 8), school at (13, 14) and office at (13, 26) and coordinates are in km.
1. What is the distance between house and bank?
2. What is the approximate distance between house and office?
3. What is the total distance travelled by Aditya to reach the office?

OR
What is the extra distance travelled by Aditya?
CASE STUDY 2 : Social Distancing
In an examination hall, students are seated at a distance of 2m from each other to maintain social distance due to Covid -19 Pandemic.
Let three students are sitting in three points A (4, - 3), B (7,3) and C (8,5). Based on this information, answer the following questions
below.
1. If an invigilator at a point P, lying on the straight line joining B and C, such that it divides the distance between them in the
ratio of 1:2. Find the coordinates of P.
2. Find the coordinates of the mid-point of the line segment joining A and C?
3. Find the ratio in which B divides the line segment joining A and C.
OR
Whether the points A, B and C are collinear or not?
CASE STUDY 3: Circular Park
In a city, a circular park is situated with centre O(3 , 3). There are two exit gates P and Q which are opposite to each other. The
location of exit gate ‘P’ is (5 , 3)
1. What will be the location of exit gate ‘Q’?

2. If a pole R (x, 5) is fixed on a boundary of circular park which is equidistant from P and Q then, find the value of the value of
‘x’?
3. In what ratio does the centre O (3 , 3) divides the line segment joining the points P and Q?
OR
Find the distance between the points P and Q?
CASE STUDY 4 : Morning assembly : Morning Assembly is an integral part of the school’s schedule. A good school is always particular
about their morning assembly schedule. Morning assembly is important for a child’s development. It is essential to understand that
morning assembly is not just about standing in long queues and singing prayers or national anthem, but it’s something beyond just
prayers. All the activities carried out in morning assembly by the school staff and students have a great influence in every point of life.
The positive effects of attending school assemblies can be felt throughout life Suppose a school have 100 students and they all
assemble in prayer in 10 rows as given above, Here A,B,C and D are the positions of four friends Amar, Bharath, Colin and Dravid
respectively.
1. What is the distance between Amar and Bharath?

2. Which type of triangle is formed if we joint the positions of A, B and C?
3. Which type of quadrilateral is obtained by joining the points ABCD?
OR
If Imran is standing exactly in the middle of Bharath and Dravid, find the position of Imran.
ANSWERS
SECTION A SECTION B
Question Answer Question Answer
1 (b) 5 1 (b) 3,-9
2 (b) √34 2 (a) (0,13)
3 (c) (-1, 2) 3 (d)(0,-10) &(4,0)
4 (d) IV quadrant 4 (b) (-7,0)
5 (c)10 5 (c)2:3
6 (d) 5 6 (b) (0, -1)
7 (c) 2:7 7 (b)
8 (d) (0, 0) 8 (0,1) (-3,3)
9 (b) 2 9 (a) 3:4
10 (b) 3 10 (d) (16/3,6)
11 (b) (2, 0) 11 (d) 1
12 (c)-12
13 (b) collinear
14 (b) 12
15 (b) (3,5)
Assertion- Reason Questions: 1. (b) 2. (d) 3. (d) 4. (a)
Hot’s answers:
1. By mid-point theorem DFEB is a parallelogram. Let coordinates of B is (x,y)
X=3+8-5=5 and y=4+9-7=6. So B is (5,6). Similarly A(1,2) and C (11,12)
𝟑+𝒌 𝟑+𝒌
2. mid point is ( , 𝟓) = (𝒙, 𝒚) then x= and y=5 using x+y-10=0 get k=7.
𝟐 𝟐
3. PQ=PR=RQ=4 units. Triangle PQR is equilateral triangle.
4. By using section formula a=1 and b= -13/4.
5. k=3/2,or x=0
Answers of Case Based questions

Case study 1:
Q1 : 5 Q2 : 24.6 Q3 : 27 OR 2.4
Case study 2:: Social Distancing
22 11
Q1 : ( 3 , 3 ) Q2 :(6,1) Q3 : 3:1 OR collinear
Case study 3: circular park
Q1 : (1,3) Q2 : 3 Q3 : 1:1 OR 4
Case study 4: Morning Assembly
Q1 : 3√2 Q2 :an Isosceles triangle Q3 :Square OR (6,4)
Chapter 8
Introduction to Trigonometry
Important points:
Sometimes we observe imaginary triangle in nature,
e.g. if we look at the top of a tower , a right angle can be imagined. As shown in figure
We need to find height BC or distance AB or AC.
These all can be found by using mathematical techniques which comes under a branch ofmathematics called Trigonometry.
Consider a right angled triangle ABC right angled at B.
Fig 1
Observing the above two triangles we see that one side i.e. hypotenuse (longest side of right triangle) is fixed it is opposite to right
angle, but other sides varies in respect of angle under consideration. Here it is to note that we write:
Side opposite to given angle as PERPENDICULAR (P), Side adjacent to given angle as BASE (B)
And the longest side HYPOTENUSE (H).
TRIGONOMETRIC RATIOS
𝒔𝒊𝒅𝒆 𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆 𝒕𝒐 ∠𝑨 BC
𝒔𝒊𝒏 𝒐𝒇 ∠𝑨 = 𝐡𝐲𝐩𝐨𝐭𝐞𝐧𝐮𝐬𝐞
= AC
𝒔𝒊𝒅𝒆 𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕 𝒕𝒐∠𝑨 AB

𝒄𝒐𝒔𝒊𝒏𝒆 𝒐𝒇 ∠𝑨 = =
𝐡𝐲𝐩𝐨𝐭𝐞𝐧𝐮𝐬𝐞 AC
𝒔𝒊𝒅𝒆 𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆 𝒕𝒐 ∠𝑨 BC
𝒕𝒂𝒏𝒈𝒆𝒏𝒕 𝒐𝒇 ∠𝑨 = =
𝒔𝒊𝒅𝒆 𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕 𝒕𝒐 ∠𝑨 AB
𝟏 AC
𝒄𝒐𝒔𝒆𝒄𝒂𝒏𝒕 𝒐𝒇 ∠𝑨 = =
𝐬𝐢𝐧𝐞 𝐨𝐟 ∠𝑨 BC
𝟏 AC
𝒔𝒆𝒄𝒂𝒏𝒕 𝒐𝒇 ∠𝑨 = =
𝐜𝐨𝐬𝐢𝐧𝐞 𝐨𝐟 ∠𝑨 AB
𝟏 AB
𝒄𝒐𝒕𝒂𝒏𝒈𝒆𝒏𝒕 𝒐𝒇 ∠𝑨 = =
𝐭𝐚𝐧𝐠𝐞𝐧𝐭 𝐨𝐟 ∠𝑨 BC
TRIGNOMETRIC TABLE
0° 30° 45° 60° 90°
sin 0 1/2 1/√𝟐 √𝟑/𝟐 1
cos 1 √𝟑/𝟐 1/√𝟐 1/2 0
tan 0 1/√𝟑 1 √𝟑 Not Defined
cot Not Defined √𝟑 1 1/√𝟑 0
sec 1 2/√𝟑 √𝟐 2 Not Defined
cosec Not Defined 2 √𝟐 2/√𝟑 1
QUESTIONS
s.no
1 If x = 2 sin2𝜃 and y=2cos2 𝜃 +1 then x + y is equal to
(a) 3 (b) 2 (c) 1 (d) 0
2. Given in ∆ABC right angled at B, If tan A = 4/3, then the value of cos C is
(a) 3/4 (b)4/5 (c) 1 (d) none of these
3. In ∆OPQ, right-angled at P, OP = 7 cm and OQ - PQ = 1 cm, then the values of sin Q.
(a)7/25 (b) 24/25 (c) 1 (d) none of these
4. Given in ∆ABC right angled at B, 15 cot A = 8, then sin C =
(a) 0 (b) 8/17 (c) 1 (d) none of these
5. In ∆PQR, right-angled at Q, PR + QR = 25 cm and PQ = 5 cm, then the value of sin P is
5 12 13
(a) 13 (b) 13 (c) 12 (d) 0
6. 1
If 𝐶𝑜𝑠𝜃=2 then𝑪𝒐𝒔𝜽 − 𝑺𝒆𝒄𝜽 is equal to:
1 1 1 2
(a) −1 2 (b) − 2 (c) 1 2 (d) 3
7. If sin𝜽 = x and sec𝜽 =y, then tan𝜽 is equal to:

𝑥 𝑦 𝑥
(a) 𝑦(b) 𝑥 (c) 𝑥𝑦 (d) 𝑥+𝑦
8. 3
If cos A = 5 ,find the value of 9 +9 tan2 A :
9 25 1
(a) 25 (b) 25 (c) (d)
9 25
9. 3
If 3𝑥 = 𝑠𝑒𝑐𝜃 and𝑥 = 𝑡𝑎𝑛𝜃 , then 9(𝑥 2 − 𝑥 2 ) is :
1
1
(a) 9 (b) 3 (c) (d) 1
9
10. If sin(𝐴 − 𝐵) =
1 1
𝑎𝑛𝑑 cos(𝐴 + 𝐵) = 2 , 𝑡ℎ𝑒𝑛 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝐵 𝑖𝑠:
2
(a) 45° (b) 60° (c) 15° (d) 0°
11. Assertion: The value of sin600 cos300 + sin300 cos600 is 1.
Reason: sin900=1 and cos900=0
(c) Assertion (A) is true but reason(R) is false.
(d) Assertion (A) is false but reason(R) is true.
12. Assertion: In a right ∆ABC, right angled at B, if tanA=1, then 2sinA.cosA=1
Reason: cosec A is the abbreviation used for cosecant of angle A.
Assertion: sin(A+B)=sin A + sin B
13.
Reason: For any value of 𝜃, 1+tan2 𝜃 = sec2 𝜃
(c) Assertion (A) is true but reason(R) is false.
14. Assertion:- sin252° + 𝑐𝑜𝑠 2 52° = 1
Reason:- For any value of 𝜽, sin2𝜃 + 𝑐𝑜𝑠 2 𝜃 = 1
15. 𝟓
Assertion:-The value of𝒄𝒐𝒔𝜽 = 𝟑 , is not possible.
Reason: - In a right-angled triangle, hypotenuse is the largest side.
16. If 3cot A = 4, find the value of (cosec2 A + 1)/( cosec2 A - 1).
17. If tan (3x - 15°) = 1 then find the value of x.
18. 2𝑡𝑎𝑛𝐴
In a right ∆𝐴𝐵𝐶, right angled at B, the ratio of AB to AC is 1:√2. Find the value of 1+𝑡𝑎𝑛2𝐴.
19. Evaluate sin 60° cos 30° + sin 30° cos 60°.
20. Evaluate 2tan2 45° + cos2 30° - sin2 60°.
21. If secθ + tan θ = p, then find the value of cosecθ.
22. 𝑡𝑎𝑛𝜃−𝑐𝑜𝑡𝜃
Prove that 𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃 = 𝑡𝑎𝑛2 𝜃 − 𝑐𝑜𝑡 2 𝜃
23. Prove that: sin𝜃 (1 + tan𝜃) + cos𝜃 (1 + cot𝜃) = sec𝜃+ cosec𝜃.

24. If sin (A+B) =1 and tan (A-B) =
1
. Find the value of :
√3
(i) tan A + cot B
(ii) sec A + cosec B
HOTS
25. If 𝑠𝑒𝑐𝜃 = 𝑥 +
1
, prove that sec𝜃 + tan𝜃= 2x or 1/2x.
4𝑥
26. 𝑠𝑒𝑐𝜃+𝑡𝑎𝑛𝜃−1 𝑐𝑜𝑠𝜃

Prove that = .
𝑡𝑎𝑛𝜃−𝑠𝑒𝑐𝜃+1 1−𝑠𝑖𝑛𝜃
27. Prove that: sin6𝜃+ cos6𝜃= 1 - 3 sin2𝜃cos2𝜃.

28. m2 −1
If cosec A + cot A = m, Show that = cos A.
m2 +1
29. Case Study Based Question:-
Mohan, a class X student is a big foodie. Once his mother has made a sandwich for him. A thought has come into his mind
by seeing a piece of sandwich. He thought if he increases the base length and height, he can eat a bigger piece of
sandwich.
Answer the following questions accordingly:

(i) If the length of the base is 12 cm and the height is 5 cm then find the length of the hypotenuse of that
sandwich?
(ii) If he increase the base length to 15 cm and the hypotenuse to 17 cm, then find the height of the sandwich?
OR
If he decreases 7cm from the given base length and height is remain same as 5 cm. What will be the angle
between height and hypotenuse?
(iii) What will be the value of cosine of the angle between hypotenuse and the height of sandwich?
30. Case Study Based Question:-
In an educational institution a programme on ‘Sarva Shiksha Abhiyan’ was organized, for which members of institution
were given triangular shaped cardboards for writing slogans. Here PQ= 29 cm, PR=21 cm, QR= 20 cm.
Answer the following questions accordingly:
(i) If ∠PQR = θ, then find the value of sin2 θ − cos2 θ
(ii) If sin(𝑃 + 𝑄) = 1 and cos(𝑃 − 𝑄) = 1, then find P and Q
OR
√3
If sin(𝑃 + 𝑄 ) = cos(𝑃 − 𝑄 ) = , then
2
find P and Q
(iii) ∠𝑃𝑄𝑅 = 𝜃, then find the value of (𝑠𝑖𝑛𝜃 + 𝑐𝑜𝑠𝜃)2 .
Answers:-
(1) A (2) B (3) A (4) B (5) B (6) A
(7) C (8) A (9) D (10) C (11) B (12) B
(13) D (14) A (15) A (16) 544/81 (17) 20° (18) 1
𝑃 2 +1
(19) 1 (20) 2 (21)
𝑃 2 −1
Q.22-23 correct proof

Q.24. (i) 2√3 (ii) 4 ; A= 60°, B=30°
Q.25-28 correct proof
Q.29. (i) 13 cm (ii) 8 cm or 45° (iii) 5/13
Q.30. (i) 41/841 (ii) P= Q=45° or P=45°, Q =15° (iii) 1681/841
Chapter 9
Applications of Trigonometry
Important Points: -
Trigonometry can be used to measure the height of a building or trees, mountains etc. It is the study of relationship between the
ratios of the right-angled triangle's sides and its angles. Trigonometry is being used for finding the heights and distances of various
objects without measuring them. In solving problems of heights and distances two types of angles are involved:
1. The angle of Elevation
2. The angle of Depression
Before knowing these angles, it is necessary to know about the following terms.
> Horizontal Plane: A plane parallel to the earth is called the Horizontal Plane
> Horizontal Line: A line drawn parallel to horizontal plane is called a horizontal line.
Example of:
Angle of Elevation
Angle of Depression
Use of Right-angled Triangle in Trigonometry: -

Angle AMP is right angled at M, PM is perpendicular, AM is
base and AP is hypotenuse and angle PAM = 𝜽
Practice Questions
MCQ
1. The length of the shadow of a tower on the plane ground is √3 times the height of the tower. The angle of elevation of sun is:
(a) 45° (b) 30° (c) 60° (d) 90°
2. The tops of the poles of height 16 m and 10 m are connected by a wire of length l meters. If the wire makes an angle of 30° with
the horizontal, then l is equal to:
(a) 26 m (b) 16 m (c) 12 m (d) 10 m
3. A pole of height 6 m casts a shadow 2 √3 m long on the ground. the angle of elevation of the sun is:
(a) 30° (b) 60° (c) 45° (d) 90°
4. A ladder leaning against a wall makes an angle of 60° with the horizontal. If the foot of the ladder is 2.5 m away from the wall,
then the length of the ladder is:
(a) 3 m (b) 4 m (c) 5 m (d) 6 m
5. If a tower is 30 m height, costs a shadow 10√3 m long on the ground, then the angle of elevation of the sun is:
(a) 30° (b) 45° (c) 60° (d) 90°
Very Short Answers

6. A tower is 50 m high. When the sun's altitude is 45° then what will be the length of its shadow?
7. The length of shadow of a pole 50 m high is 50 m. Find the sun's altitude.
8. Find the angle of elevation of a point which is at a distance of 30 m from the base of a tower 10 √3 m high.
9. A kite is flying at a height of 50√3 m from the horizontal. It is attached with a string and makes an angle 60° with the
horizontal. Find the length of the string.
10. The upper part of a tree broken over by the wind makes an angle of 30° with the ground and the distance of the root from the
point where the top touches the ground is 25 m. What was the total height of the tree?
Long Answers
11. A man standing on the deck of a ship, 10 m above the water level observes the angle of elevation of the top of a hill as 60° and
angle of depression the bottom of a hill as 30°. Find the distance of the hill from the ship and height of the hill. (√3 = 1.732)
12. A bird is sitting on the top of a tree, which is 80 m high. The angle of elevation of the bird, from a point on the ground is 45°.
The bird flies away from the point of observation horizontally and remains at a constant height. After 2 seconds, the angle of
elevation of the bird from the point of observation becomes 30°. Find the speed of flying of the bird. (√3 = 1.732)
13. The shadow of a tower standing on a level ground is found to be 40 m longer when the Sun's altitude is 30° than when it is 60°.
Find the height of the tower
14. The angles of depression of the top and the bottom of an 8 m tall building from the top of a multi storeyed building are 30° and
45°, respectively. Find the height of the multi storeyed building and the distance between the two buildings.
15. From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are 30° and 45°,
respectively. If the bridge is at a height of 3 m from the banks, find the width of the river.
16. From the top of a 7 m high building, the angle of elevation of the top of a cable tower is 60° and the angle of depression of its
foot is 45°. Determine the height of the tower.
17. A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression
of 30°, which is approaching the foot of the tower with a uniform speed. Six seconds later, the angle of depression of the car is
found to be 60°. Find the time taken by the car to reach the foot of the tower from this point.
Case Study Questions
18. A hot air balloon is a type of aircraft. It is lifted by heating the air inside the balloon, usually with fire. Hot air weighs less than
the same volume of cold air (it is less dense), which means that hot air will rise up or float when there is cold air around it, just
like a bubble of air in a pot of water. The greater the difference between the hot and the cold, the greater the difference in
density, and the stronger the balloon will pull up.
Lakshman is riding on a hot air balloon. After reaching at height x at point P, he spots a lorry parked at B on the ground at an
angle of depression of 30°. The balloon rises further by 50 metres at point Q and now he spots the same lorry at an angle of
depression of 45° and a car parked at C at an angle of depression of 30°.
(i) What is the relation between the height x of the balloon at point P and distance d between point A and B?
(ii) When balloon rises further 50 meters, then what is the relation between new height y and d?
(iii) What is the new height of the balloon at point Q?
OR
What is the distance AB on the ground?
19. Observe the picture.
From a point A, h m above from water level, the angle of elevation of top of Chhatri (point B) is 45° and angle of depression of
its reflection in water (point C) is 60°. If the height of Chhatri above water level is (approximately) 10 m, then
(i) Draw a well-labelled figure based on the above information;
(ii) Find ∠ BAC
(iii) Find the height (h) of the point A above water level.
OR
Find distance AB.
20. To explain how trigonometry can be used to measure the height of an inaccessible object, a teacher gave the following example
to students: A TV tower stands vertically on the bank of a canal. From a point on the other bank directly opposite the tower,
the angle of the elevation of the top of the tower is 60° . From another point 20 m away from this point on the line joining this
point to the foot of the tower, the angle of elevation of the top of the tower is 30° (as shown in Figure).
Based on the above, answer the following questions:
(i) The width of the canal is
(ii) The angle formed by the line of sight with the horizontal when it is above the horizontal line is known as?
(iii) Find the height of the tower.
OR
Find the distance of the foot of the tower from the point D
21. A group of students of class X visited India Gate on an educational trip. The teacher and
students had interest in history as well. The teacher narrated that India Gate, official name Delhi Memorial, originally called
All-India War Memorial, monumental sandstone arch in New Delhi, dedicated to the troops of British India who died in wars
fought between 1914 and 1919. The teacher also said that India Gate, which is located at the eastern end of the Kartvaya-Path
(formerly called the Rajpath), is about 138 feet (42 metres) in height.
(i)What is the angle of elevation if they are standing at a distance of 42m away from
the monument?
(ii) They want to see the tower at an angle of 60°. So, they want to know the distance where they should stand and hence find
the distance.
(iii) If the altitude of sun is 60° then find the height of vertical tower that will cast a shadow of length 20 m.
OR
If the altitude of sun is 45° then find the height of vertical tower that will cast a shadow of length 20 m
Assertion & Reason

22. Assertion: The line of sight is the line drawn from the eye of an observer to the point of the object viewed by the observer.
Reason: Trigonometric ratios are used to find height or length of an object or distance between 2 distant objects.
A. Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
B. Both assertion (A) and reason (R) are true but reason (R) is NOT the correct explanation of assertion (A).
C. Assertion (A) is true but reasons (R) is false.
D. Assertion (A) is false but reasons (R) is true.
23 Assertion: If the length of shadow of a vertical pole is equal to its height, then the angle of
elevation of the sun is 45°.
Reason: According to Pythagoras theorem 𝒉𝟐 = 𝒍𝟐 + 𝒃𝟐 , where h = hypotenuse, l = length and b = base.
A. Both assertion (A)and reason (R) are true and reason (R) is the correct explanation of assertion (A).
B. Both assertion (A)and reason (R) are true but reason (R) is NOT the correct explanation of assertion (A).
24 Assertion: In the fig. if BC =20m, then height AB is 11.56m.
Reason: tan 𝜃 = AB/BC = PERPENDICULAR/BASE, where 𝜃 is the ∠ ACB.
A. Both assertion (A)and reason (R) are true and reason (R) is the correct explanation of assertion (A).
B. Both assertion (A)and reason (R) are true but reason (R) is NOT the correct explanation of assertion (A).
25 Assertion: If the angle of elevation of Sun, above a perpendicular line (tower) decreases, then the shadow of tower
increases.
Reason: It is due to decrease in slope of line of sight.
A. Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
B. Both assertion (A) and reason (R) are true but reason (R) is NOT the correct explanation of assertion (A).
ANSWER
1. B 2. C 3. B 4. C 5. C 6. 50m
7. 45O 8. 30O 9. 100m 10. 25√𝟑 m 11. 17.32m, 12. 29.28 m/sec
40m
13. h=20√𝟑 m 14. 4(3+√𝟑 ) m, 15. 3(√𝟑 +1)m 16. 7(√𝟑 +1) m 17. 3sec
12(√𝟑 + 1) m
18. (i) d = x √𝟑 (ii) d = y (iii) AQ=25(√3 +3)m OR 25(√3 +3)m
19. (i) draw yourself (ii) 105° (iii) 20 - 10√3 m OR Hint: use sin𝜃
20. (i) 10 m , (ii) angle of elevation (iii) 10√𝟑 m OR 30 m
21. (i) 45°, (ii) 14√𝟑 𝒎, (iii) 20√𝟑 m OR 20 m
22. B 23. B 24. A 25. A
Chapter 10 (CIRCLES)
BASIC TERMINOLOGY:
RADIUS: Distance from the centre to any point on the surface of a circle is called “Radius”.
SECANT: A secant to a circle is a line that cuts the circle at two distinct points.
CHORD: A chord is a line segment whose end points lie on the circle itself. Diameter is the longest
chord in a circle.
TANGENT: A tangent to a circle is a line that touches the circle at exactly one point. For every
point on the circle, there is a unique tangent passing through it. The point where the tangent
touches the circle is called “Point of contact”.
Key points:
1. No tangent can be drawn to a circle which passes through a point that lies inside it.
2. When a point of tangency lies on the circle, there is exactly one tangent to a circle that passes through it.
3. When the point lies outside of the circle, there are accurately two tangents to a circle through it.
THEOREMS:
1. The perpendicular from the centre of the circle to a chord bisects the chord.
2. The angle subtended by an arc at the centre of the circle is double the angle subtended by it at the remaining part of the circle.
3. Angles in the same segment of the circle are always equal.

4. Angle formed in a semicircle is always 90.
5. The sum of all the angles of a quadrilateral is 360.
IMPORTANT THEOREMS (WITH PROOF):

Theorem 1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Given: A circle with centre O and a tangent XY at point “P”.
To prove: OP is perpendicular to tangent XY.
Construction: A point Q on the tangent line XY, other than P. Join the points OQ.
Proof:
Point Q should lie outside the circle. Because if point Q lies inside the circle, XY will not be a tangent to the circle and XY would
become a secant of a circle. Now, OQ should be greater than the radius of the circle OP as it lies outside the circle.
Thus, OQ > OP.
As, this condition is obeyed for all points on line XY except P, OP should be the shortest of all distances from the centre of the circle
“O” to the points on line XY. Therefore, OP is perpendicular to XY.
Hence Proved.
Theorem 2: The lengths of tangents drawn from an external point to a circle are equal.
Given: A circle with centre O and two tangents say PQ and PR from an external point P.
To prove: PQ = PR
Construction: Join OQ, OR and OP.
Proof: In ∆OQP and ∆ORP,
OQ = OR = r (radii of the same circle)
OP = OP (common)
∠OQP = ∠ORP = 90
(Radius is always perpendicular to the point of contact)
∆OQP ≅ ∆ORP (By RHS congruence)
Thus, PQ = PR (By CPCT)
Hence proved.
Q1. Two balls of equal size are touching each other externally at point C and AB is common tangent to the balls. Then ∠ ACB=
(a) 600 (b) 450 (c) 300 (d) 900
Q2. Radha and Shyama were arguing that how many parallel tangents can a circle have? Can you help them?
(a) 1 (b) 2 (c) infinite (d) none of these
Q3. Three friends Ram, Shyam and Rahim are playing in a triangular park in which there is a circular rose garden as shown in
the fig. Three friends are standing at points A,B and C respectively. By the information given in the figure, can you calculate
perimeter of the park?
(a) 30 cm (b) 60cm (c) 45cm (d) 15cm
Q4. (i)If four sides of the quadrilateral ABCD are tangents to a circle , then:
(a)AC+AD=BD+CD (b) AB+CD=BC+AD (c) AB+CD=AC+BC (d)AC+AD=BC+DB
(ii) At one end A of a diameter AB of a circle of radius 5 cm, tangent XAY is drawn to the circle.
The length of the chord CD parallel to XY and at a distance 8 cm from A is:
(a) 4 cm (b) 5 cm (c) 6 cm (d) 8 cm
Q5. AP and AQ are tangents drawn from a point A to a circle with centre O and radius 9 cm. If OA=15 cm, then AP+AQ=
(a) 12cm (b) 18cm (c) 24cm (d) 36cm
Q6. If common tangents AB and CD of two wheels with centre O and O’ intersect at E, then find OEO’=?
(a) a triangle (b) a line (c) an arc (d) none of these
Q7. PQ and PR are two tangents from P to a circle with centre A. If

∠QAR=1300, find ∠QPR=?
(a) 40° (b) 50° (c)60° (d)20°
Q8. In two concentric circles, if length of one chord AB touching inner circle is 12cm then find the length of chord CD?
(a) 10cm (b) 15cm (c) 12cm (d) 6cm
Q9. The length of the tangent from a point which is at a distance of 10cm from the centre of the circle having radius 6cm is?
(a) 8cm (b) 10cm (c) 4cm (d)16cm
Q10. i) If AB= 14cm and PE=5cm, then AE=?
(a) 7 cm (b) 8 cm (c) 19 cm (d) 9cm
ii) The distance between two parallel tangents to a circle of radius 5cm is:
(a) 10 cm (b) 11 cm (c) 12 cm (d) 14 cm
Q11. Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.
Q12. Prove that the lengths of tangents drawn from an external point to a circle are equal.
Q13. Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that ∠ PTQ = 2 ∠ OPQ.
Q14. PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T (see Fig). Find the
length TP
Q15. Prove that the tangents drawn at the ends of a diameter of a circle are parallel.
Q16. A quadrilateral ABCD is drawn to circumscribe a circle (see Fig).
Prove that AB + CD = AD + BC
Q17. Prove that the parallelogram circumscribing a circle is a rhombus.

Q18. i) In figure XY and X′Y′ are two parallel tangents to a circle with centre O and another tangent AB with point of contact C
intersecting XY at A and X′Y′ at B. Prove that ∠ AOB = 90°.
ii) Prove that the intercept of a tangent between two parallel tangents to a circle subtends a right angle at the centre.
Q19. A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided
by the point of contact D are of lengths 8 cm and 6 cm respectively (see Fig.). Find the sides AB and AC.
Q20. Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.
Q21. A Ferris wheel (or a big wheel in the United Kingdom) is an amusement ride consisting of a rotating upright wheel with
multiple passengers carrying components (commonly referred to as passenger cars, cabins, tubs, capsules, gondolas, or
pods) attached to the rim in such a way that as the wheel turns, they are kept upright, usually by gravity. After taking a ride
in Ferris wheel, Aarti came out from the crowd and was observing her friends who were enjoying the ride. She was curious
about the different angles and measures that the wheel will form.
She forms the figure as given below.
i) Find ∠RSQ.
ii) Find ∠ORP.
iii) If PQ=40 m and OQ=30 m then find PO.
OR
Find ∠RQP.
Q22. A student draws two circles that touch each other externally at point K with centres A and B and radii 6 cm and 4 cm
respectively as shown in the figure:
i) If two circles touch externally, then the number of common tangents that can be drawn is____
ii) Find the length of PA.
iii) Find the length of PQ.
OR
Find the length of QY
Q23. Arun recently bought a gold coin from a jewellery shop. To protect it, he placed the gold coin in a triangular box, The edge
of the triangle touches the gold coin. In mathematical form, the given statement is defined with the adjoining figure such
that BP = 7 cm, CP =4 cm, AQ=5 cm and ∠ OBP =300.
i) Find the radius of the gold coin.

ii) Find the area of ∆OPB.
iii) If AB, BC and CA are the lengths of the tangents of ∆ ABC,
then check whether AB > BC and BC > AC are correct.
OR
Find the Perimeter of ∆ ABC.
Q24. In a Bike, both tyres touch the ground and when those points where they touch the ground are joined, we get a straight line.
The Straight Line formed can be considered a common tangent for both the circles.
Mathematically, In the fig, AB and CD are two common tangents of two equal circles which touch each other at D and AB =
8cm and CD = 4 cm.
i) Find length BC.
ii) Find ∠ADB.
iii) Find the area of Quadrilateral O1ACD.
OR
iv) Find the number of common tangents in the given
figure.
ASSERTION REASON QUESTIONS

In the following questions, a statement of assertion (A) is followed by a statement of
Reason (R). Choose the correct answer out of the following choices.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Q25. Assertion: The length of tangent drawn from a point at a distance of 13 cm from the centre of a circle of radius 5 cm is 10
cm.
Reason: A tangent to a circle is perpendicular to the radius through the point of contact.
Q26. Assertion: If PA and PB are tangents drawn from external point P to a circle with centre O, then the quadrilateral AOBP is
cyclic.
Reason: The angle between two tangents drawn from an external point to a circle is supplementary to the angle subtended
by the line segments joining the point of contact at the centre.
Q27. Assertion: A tangent to a circle is perpendicular to the radius through the point of contact.
Reason: The lengths of tangents drawn from an external point to a circle are equal.
Q28. Assertion:
In Fig. PQ and PR are tangents drawn from an external point P to a circle with centre O. If QPR = 800, then QOR = 100O
Reason: The angle between two tangents drawn from an external point to a circle is supplementary to the angle subtended
by the line segments joining the point of contact to the centre.
Q29. If AB, AC, PQ are tangents in given Fig. and AB = 5 cm, find the perimeter of  APQ.
Q30. ∆ABC is right-angled at B such that BC = 6 cm and AB = 8 cm. Find the radius of its incircle.
ANSWERS
Q1) d Q2) b Q 3) a Q4) (i) b (ii) d
Q 5) c Q6) b Q7) b Q8) c
Q9) a Q10) (i) d (ii) a Q(11 to 13) correct proof
Q14) 20/3 cm Q(15 to 18 ) correct proof Q19) AB=15 cm ,AC=13 cm Q20) correct proof
Q21) i) 750 ii) 900 iii) 50 cm OR 750 Q22) (i) 3 (ii) PA=10 cm (iii) PQ=25 cm OR 1cm
7 √3 √3
Q23) (i) 𝑐𝑚 (ii) 49 6 𝑠𝑞. 𝑐𝑚 (iii) AB= 12cm ,BC=11cm ,AC=9 cm ,statement is correct OR 32 cm
3
Q24) (i) BC=4cm (ii) 900 (iii) 16 sq.cm OR 2 common tangents
Q25) d Q26) a Q27) b Q28) a
Q29) Perimeter = AQ+QP+AP=10 cm

Q30) Area of  ABC = 24 cm2
Area of  ABC = Area of  OAB + Area of  OBC + Area of  OAC
r = 2cm
Chapter 11 (Areas Related to Circles)

IMPORTANT FORMULAS & CONCEPTS
1. Circumference of a circle = 2 π r.
2. Area of a circle = π r2
θ
3. Length of an arc of a sector of a circle with radius r and angle with degree measure θ is ×2πr
360
θ
4. Area of a sector of a circle with radius r and angle with degree measure θ is 360× πr2
5. Area of segment of a circle = Area of the corresponding sector – Area of the corresponding triangle
θ 1
=360× πr2- 2 r2sin 𝜃
MCQs
22
1. If the difference between the circumference and the radius of a circle is 37 cm, then using π = 7 the radius of the circle (in cm) is:
(a) 154 (b) 44 (c) 14 (d) 7
2. The ratio of the outer and inner perimeters of a circular path is 23:22. If the path is 5 meters wide, the diameter of the inner circle is
(a) 55 m (b) 110 m (c) 220 m (d) 230 m
3. The area of the sector of a circle of radius 6 cm whose central angle is 30°.
(a) 94.2 cm2 (b) 7.42 cm2 (c) 8.42 cm2 (d) 9.42 cm2
4. If π is taken as 22/7 the distance (in meters) covered by a wheel of diameter 35 cm, in one revolution is:
(a) 2.2 m (b) 1.1 m (c) 9.625 m (d) 96.25 m
5. The circumference of a circular field is 528 cm. Then the radius will be:
(a) 84 cm (b) 64 cm (c) 55 cm (d) 45 cm
6. If the circumference and the area of a circle are numerically equal, then diameter of the circle is:
(a) 5 (b) 8 (c) 2 (d) 4
7. The diameter of the driving wheel of a bus is 140 cm. How many revolutions per minute must the wheel make in order to keep a
speed of 66 kmph?
(a) 200 (b) 240 (c) 250 (d) 260
8. The perimeter of a quadrant of diameter 14 m.
(a) 11 m (b) 25 m (c) 12 m (d) 10 m
9. The inner circumference of a circular race track, 14 m wide, is 440 m. Find the radius of the outer circle:
(a) 85 m (b) 82 m (c) 80 m (d) 84 m
10. If the radius of a circle is doubled, its area is increased by:
(a) 100% (b) 200% (c) 300% (d) 400%
11. The length of the minute hand of a clock is 14 cm. The area swept by the minute hand in 5 minutes is:
(a) 153.9 cm² (b) 102.6 cm² (c) 51.3 cm² (d) 205.2 cm²
12. If the ratio of the circumferences of two circles is 3:1, then find the ratio of their areas.
(a) 8 : 1 (b) 9 : 16 (c) 9 : 1 (d) 16 : 9
13. A horse is tied to a peg at one corner of a square shaped grass field of side 15 m by means of a 7 m long rope. The area of that part
of the field in which the horse can graze, is:
(a) 77 cm² (b) 77/2 cm² (c) 154 cm² (d) 77/4 cm²
14. The area of the circle that can be inscribed in a square of side 6 cm, is:
(a) 18π cm² (b) 12π cm² (c) 9π cm² (d) 14π cm²
15. If the circumference of a circle and the perimeter of a square areequal, then:
(a) Area of the circle = Area of the square
(b) Area of the circle > Area of the square
(c) Area of the circle < Area of the square
(d) Nothing definite can be said about the relation between the areasof the circle and square.
SHORT AND LONG ANSWER QUESTIONS
1. Area of a sector of a circle of radius 36 cm is 54π cm 2. Find the length of the corresponding arc of the sector.
2. The length of the minute hand of a clock is 5 cm. Find the area swept by the minute hand during the time period 6:05 to 6:40 am.
3. The area of a sector is one-twelfth that of the complete circle. Find the angle of the sector.
4. Find the area of the largest triangle that can be inscribed in a semicircle of radius r unit.
5. The cost of fencing a circular field at the rate of Rs 24 per metre is Rs 5280. The field is to be ploughed at the rate of Rs 0.50 per m2.
Find the cost of ploughing the field
6. A car has two wipers which do not overlap. Each wiper has a blade of length 25 cm sweeping through an angle of 115°. Find the
total area cleaned at each sweep of the blades.
7. A horse is tied to a peg at one corner of a square shaped grass field of side 15 m by means of a 5 m long rope.
Find (i) The area of that part of the field in which the horse can graze.
(ii) The increase in the grazing area if the rope were 10 m long instead of 5m. (Use π = 3.14)
8. An umbrella has 8 ribs which are equally spaced. Assuming umbrella to be a flat circle of radius 45 cm, find the area between the
two consecutive ribs of the umbrella.
9. Find the radius of a circle whose circumference is equal to the sum of the circumferences of two circles of radii 15cm and 18cm.
10. A square tank has its side equal to 40 m. There are four semi-circular grassy plots all around it. Find the cost of turfing the plot at
Rs 1.25 per square meter.
HIGH ORDER THINKING QUESTIONS
1. Find the area of the shaded region given in Fig ( Use π = 3.14)
2. In Fig , a square is inscribed in a circle of diameter d and another square is circumscribing the circle. Is the area of the outer
square four times the area of the inner square? Give reasons for your answer.
3. Find the area of the flower bed (with semi-circular ends) shown in Fig.

1. Mr Sunil purchased a plot QRUT to build his house. He leave space of two congruent semicircles for gardening and a rectangular area
of breadth 3 cm for car parking.
Based on the above information, answer the following questions.
(i) Find the Area of square PQRS.
(ii) Find the Area of rectangle left for car parking.
(iii) Find the Radius of semi-circle
OR
Find the Area of a semi-circle.
2. To find the polluted region in different areas of Varanasi a survey was conducted by the students of class X. It was found that the
shaded region is the polluted region, where O is the centre of the circle.
(i) Find the radius of the circle
(ii) Find the area of the circle.
(iii) If D lies at the middle of arc BC, then find the area of region COD.
OR
Find the Area of the ΔBAC.
3. A brooch is a small piece of jewellery which has a pin at the back so it can be fastened on a dress, blouse or coat. Designs of some
brooch are shown below. Observe them carefully.
Design A: Brooch A is made with silver wire in the form of a circle with a diameter of 28mm. The wire is used for making 4
diameters which divides the circle into 8 equal parts.
Design B: Brooch b is made of two colours – Gold and silver. The outer part is made of Gold. The circumference of the silver
part is 44mm and the gold part is 3mm wide everywhere.
Refer to Design A
i) Find the total length of silver wire required.
ii) Find the area of each sector of the brooch.
Refer to Design B
iii) Find the circumference of outer part (golden).
OR
Find the difference of areas of golden and silver parts.
4. Director of a company select a round glass trophy for awarding their employees on annual function. Design of each trophy is made
as shown in the figure, where its base ABCD is golden plated from the front side at the rate of Rs. 6per cm 2.
i) Find the area of the sector ODCO.

ii) Find the area of ∆AOB.
iii) Find the total cost of golden plating.
OR
Find the area of the major sector formed in the given region.

Directions: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice
as:
Q.1. Assertion (A): In a circle of radius 6 cm, the angle of a sector is 60°. Then the area of the sector is 132/7 cm 2.
Reason (R): Area of the circle with radius r is πr2.
Q.2. Assertion (A): If the circumference of a circle is 176 cm, then its radius is 28 cm.
Reason (R): Circumference = 2π × radius.
Q.3. Assertion (A): If the outer and inner diameter of a circular path is 10 m and 6 m respectively, then area of the path is 16π m2.
Reason (R): If R and r be the radius of outer and inner circular path respectively, then area of circular path = π(R 2 – r2).
Q.4. Assertion (A): The length of the minute hand of a clock is 7 cm. Then the area swept by the minute hand in 5 minute is 77/6 cm 2.
Reason (R): The length of an arc of a sector of angle 𝜃 and radius r is given by l=360
θ
×2πr
ANSWERS
MCQ
1) D 2) C 3) D 4) B 5) A 6) D 7) C
8) B 9) D 10) C 11) C 12) C 13) B 14) C 15) B
SHORT AND LONG ANSWER QUESTIONS

1) 3π 2) 45.8cm2 3)300 4) r 2unit 5) Rs 1925 6)1254.96cm2
7) Area of quadrant=19.625 m2, increase in area=58.875m2 8) 795.5cm2 9) 33CM 10) Rs. 3140
HIGH ORDER THINKING QUESTIONS

1. 154.88 cm2
2. We can conclude that Area of outer square is not equal to 4 times the inner square.it is not true to say that area of a segment of a
circle is less than the area of its corresponding sector.
3. (380 + 25π) cm2

1. i) 729 cm2 ii) 81 cm2 iii) 6.75cm iv) 71.59cm2
2. i) 12.5 cm ii) 491.07cm2 iii) 122.76cm2 iv) 84cm2
3. i) 200mm ii) 77m2 iii) 62.86mm iv) 51π
4. i) 154 cm2 ii) 200cm2 iii) Rs.276 iv) 462 cm2

1. B 2. A 3. A 4. B
CHAPTER-12 (SURFACE AREA AND VOLUME)
MCQ (1MARK)
1. A piece of paper is in the shape of a semi-circular region of radius 10 cm. It is rolled to form a right circular cone. The slant
height is:
(a) 5 cm (b) 10 cm (c) 15 cm (d) 20 cm
2. The total surface area of a solid hemisphere of radius 7 cm is:
(a) 447 cm2 (b) 239 cm2 (c) 174 cm2 (d) 462cm2
3. A cylinder and a cone are of same base radius and of same height. The ratio of the volume of the cylinder to that of the cone is:
(a) 2 : 1 (b) 3 : 1 (c) 2 : 3 (d) 3 : 2
4. A solid formed on revolving a right angled triangle about its height is:
(a) cylinder (b) sphere (c) right circular cone (d) two cones
5. Volumes of two spheres are in the ratio 64:27. The ratio of their surface areas is:
(a) 3 : 4 (b) 4 : 3 (c) 9 : 16 (d) 16 : 9
6. The ratio of the total surface area to the lateral surface area of a cylinder with base radius 80 cm and height 20 cm is:
(a) 1 : 2 (b) 2 : 1 (c) 3 : 1 (d) 5 : 1
7. The radii of the base of a cylinder and a cone are in the ratio 3 : 4 and their heights are in the ratio 2 : 3, then ratio of their
volumes is:
(a) 9 : 8 (b) 9 : 4 (c) 3 : 1 (d) 27 : 64
8. If two cubes, each of edge 4 cm are joined end to end, then the surface area of the resulting cuboid is:
(a) 100 cm2 (b) 160 cm2 (c) 200 cm2 (d) 80 cm2
9. The curved surface area of a cylinder is 264 m and its volume is 924 m . The ratio of its diameter to its height is:
2 3
(a) 3 : 7 (b) 7 : 3 (c) 6 : 7 (d) 7 : 6

10. The curved surface area of a cone is 308 cm2 and its slant height is 14 cm. The radius of its base is:
(a) 8 cm (b) 7 cm (c) 9 cm (d) 12 cm
11. Base radius of two cylinder are in the ratio 2 : 3 and their heights are in the ratio 5 : 3. The ratio of their volumes is:
(a) 27 : 20 (b) 25 : 24 (c) 20 : 27 (d) 15 : 20
12. If base radius and height of a cylinder are increased by 100% then its volume increased by:
(a) 30% (b) 40% (c) 42% (d) 33.1%
13. The total surface area of a cylinder is 40π cm2. If height is 5.5 cm then its base radius is:
(a) 5cm (b) 2.5cm (c) 1.5cm (d) 10cm
14. A conical pit of top diameter 3.5 m is 12 m deep. What is its capacity in kilolitres?
(a) 38.5 kl (b) 48.5 kl (c) 39.5 kl (d) 47.5 kl
15. It costs Rs 2200 to paint the inner curved surface of a cylindrical vessel 10 m deep. If the cost of painting is at the rate of
Rs 20 per m2, find radius of the base.
(a) 1.75 m (b) 1.85 m (c) 1.95 m (d) 1.65 m
ASSERTION AND REASONING
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement
1. Assertion: Total surface area of the cylinder having radius of the base 14 cm and height 30 cm is 3872 cm 2.
Reason: If r be the radius and h be the height of the cylinder, then total surface area = (2πrh + 2πr 2).
2. Assertion: From a solid cylinder, whose height is 12 cm and diameter 10 cm a conical cavity of same height and same
diameter is hollowed out. Then, volume of the cone is 2200/7 cm3.
3. Reason: If a conical cavity of same height and same diameter is hollowed out from a cylinder of height h and base radius r,
then volume of the cone will be half of the volume of the cylinder.
4. Assertion: If the height of a cone is 24 cm and diameter of the base is 14 cm, then the slant height of the cone is 15 cm.
Reason: If r be the radius and h the slant height of the cone, then slant height = √(h 2+r2)
5. Assertion: If the radius of a cone is halved and volume is not changed, then height remains same.
Reason: If the radius of a cone is halved and volume is not changed then height must become four times of the original height.
VERY SHORT ANSWER TYPE QUESTIONS – 2 MARKS

1. Two cubes each of volume 343 cm3 are joined end to end to form a solid. Find the surface area of the solid.
2. Volume and surface area of a solid hemisphere are numerically equal. What is the diameter of hemisphere?
3. If the total surface area of a solid hemisphere is 462 cm 2, find its radius in m.
4. The base radii of two right circular cones of the same height are in the ratio 3:5. Find the ratio of their volumes.
5. Find the volume of the largest right circular cone that can be cut out of a cube whose edge is 9cm.
2
6. The TSA of a solid cylinder is 231cm2. If its CSA is of its TSA. Find its radius and height.
3
7. A cone and a cylinder of same radius 3.5cm have same CSA. If height of the cylinder is 14cm then find the slant height of the
cone.
8. A bird-bath in a garden is in the shape of a cylinder with a hemi-spherical depression at one end. The height of the hollow
cylinder is 1.45m and its radius is 30cm. find the TSA of the bird-bath.
9. A vessel in the shape of a hollow hemi-sphere mounted by a hollow cylinder. The diameter of the hemi-sphere is 0.14m and
the total height of the vessel is 13cm. find the inner surface area of the vessel.
SHORT ANSWER TYPE QUESTIONS – 3 MARKS
Q1. A toy is in the form of a cone mounted on a hemisphere of same radius 0.07m. If the total height of the toy is 31 cm, find its
total surface area.
Q2. Due to heavy floods in a state, thousands were rendered homeless. 50 schools collectively decided to provide place and the
canvas for 1500 tents and share the whole expenditure equally. The lower part of each tent is cylindrical with base radius
2.8 m and height 3.5 m and the upper part is conical with the same base radius, but of height 2.1 m. If the canvas used to
make the tents costs ₹120 per m2 , find the amount shared by each school to set up the tents.
Q3. An ice - cream cone consists of a cone surmounted by a hemisphere. The radius of the hemisphere is 3.5 cm and height of
the ice - cream cone is 12.5 cm. Calculate the volume of the ice – cream in the cone.
Q4. The sum of the radius of base and height of a solid right circular cylinder is 37 cm. If the total surface area of the solid
cylinder is 1628 sq. cm, find the volume of the cylinder
Q5. The radius and height of a solid right circular cone are in the ratio of 5 : 12. If its volume is 314 cm 3, find its total surface
area. (Use 𝜋 = 3.14 )
Q6. A petrol tank is a cylinder of base diameter 21 cm and length 18 cm fitted with a conical end of length 9 cm. Determine the
capacity of the tank in liters.
Q7. A rocket is in the form of a cylinder, closed at the lower end, has a cone attached to its top. If each one has a radius 20 cm
and height 0.21 m, find the surface area of the rocket.
LONGANSWER QUESTIONS ( 5 MARKS)
1. Two solid cones A and B are placed in a cylindrical tube as shown in the Fig. The ratio of their capacities is 2:1. Find the heights
and capacities of cones. Also, find the volume of the remaining portion of the cylinder.
2. Water flows through a cylindrical pipe, whose inner radius is 1 cm, at the rate of 80 m/sec in an empty cylindrical tank, the
radius of whose base is 40 cm. What is the rise of water level in tank in half an hour?
3. There are two identical solid cubical boxes of side 7cm. From the top face of the first cube a hemisphere of diameter equal to
the side of the cube is scooped out. This hemisphere is inverted and placed on the top of the second cube’s surface to form a
dome. Find the ratio of the total surface area of the two new solids formed.
4. Water is flowing at the rate of 15 km/h through a pipe of diameter 14 cm into a cuboidal pond which is 50 m long and 44 m
wide. In what time will the level of water in pond rise by 21 cm?
1. The Great Stupa at Sanchi is one of the oldest stone structures in India, and an important monument of Indian Architecture. It
was originally commissioned by the emperor Ashoka in the 3rd century BCE. Its nucleus was a simple hemispherical brick
structure built over the relics of the Buddha. .It is a perfect example of combination of solid figures. A big hemispherical dome
with a cuboidal structure mounted on it. (Take π= 22/7 )
i) Find the volume of the hemispherical dome if the height of the dome is 21 m.
ii) Find the cloth require covering the hemispherical dome if the radius of its base is 14m.
iii) The total surface area of the combined figure i.e. hemispherical dome with radius 14m and cubical shaped top
with dimensions 8m ×6m ×4m is
OR
Find the volume of the cuboidal shaped top with dimensions 8m ×6m ×4m.
2. Isha’s father brought an ice-cream brick, empty cones and scoop to pour the ice-cream into cones for all the family members.
Dimensions of the ice-cream brick are (30cm×25cm×10cm) and radius of hemi-spherical scoop is 3.5 cm. Also the radius and
height of cone are 3.5 cm and 15cm respectively. Based on above information, answer the following questions
i) What is the volume of hemispherical scoop?
ii) Find volume of the ice-cream cone.
iii) Find the minimum number of scoops required to fill one cone up to brim.
OR
Find the number of cones that can be filled up to brim using the whole brick.
3. Alok and his family went for a vacation to Manali. There they had a stay in tent for a night. Alok found that the tent in which
they stayed is in the form of a cone surmounted on a cylinder. The total height of the tent is 42m.diameter of the base is 42m
and height of the cylinder is 22m. . Based on above information, answer the following questions
i) How much canvas is needed to make the tent?

ii) If each person needs 126 m2 of floor, then how many persons can be
accommodated in the tent? 1
iii) Find the cost of the canvas used to make the tent, if the cost of 100m of
2
canvas is rupees 425. 2

OR
Find the volume of the tent. And find the number of persons that can be accommodate in tent, if each person needs
1892 m3 of space.
4. In the month of December 2020, it rained heavily throughout the day over the city of Hyderabad. Anil observed the raindrops
as they reached him. Each raindrop was in the shape of a hemisphere surmounted by a cone of the same radius of 1 mm.
Volume of one of such drops is 3.14 mm³. Anil collected the rain water in a pot having a capacity of 1099 cm³. [Use √2 = 1.4]
(a) Find the total height of the drop.

(b) Find the curved surface area of the drop.
(c) As the drop fell into the pot, it changed into a sphere. What was the radius of this sphere?
OR
How many drops will fill the pot completely?
ANSWER KEY
1. (b) 10 cm 2. (d) 462cm2 3. (b) 3 : 1 4. (c) Right circular 5. (d) 16 : 9
6. (d) 5 : 1 7. (a) 9 : 8 8. (b) 160 cm2 9. (a) 3 : 7 10. (b) 7 cm
11. (c) 20 : 27 12 .(d) 33.1% 13. (b) 2.5cm 14. (a) 38.5 kl 15. (a) 1.75 m
Assertion and Reason
1. (a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
2. (c)Assertion is correct statement but Reason is wrong statement.
3. (d) Assertion is wrong statement but Reason is correct statement
4. (d) Assertion is wrong statement but Reason is correct statement
VERY SHORT ANSWER TYPE (2MARKS)
1. 290 cm2 [volume of cube=a3 ,area of resulting cuboid=2(lb+bh+hl)]

2
2. 3 𝜋𝑅2 = 3𝜋𝑅2 , R=9/2 ,D=9
3. 3πr2=462 ⇒r=7
1
4. 9:25(volume of cone =3 𝜋𝑟 2 𝐻
5. h=9cm&r=diameter2=9/2=4.5cm, Volume=1/3×π×(4.5)29=190.85cm3
6. volume=269.5 cm3 (TSA of Cylinder=2Πr(r+h), CSA of Cylinder=2πrh , volume of Cylinder= πr2h)
7. 28cm (CSA of cone= CSA of cylinder)
8. 3.3 m2 (Total surface are of the bird-bath = curved surface area of the cylinder + curved surface area of the hemisphere)
9. 572 cm 2 . (inner surface area of vessel=CSA of cylindrical part +surface area of hemispherical part)
SHORT ANSWER TYPE (3 MARKS)
1. 858 sq. m (total surface area of toy= 2πr2+πrl = πr(2r+l)

2. Rs 332640, (Area of canvas used to make tent
= CSA of cylinder + CSA of cone) and Cost of 1500 tents at Rs 120 per sq .m= 1500×120×92.4= Rs 16,632,000
3. V. of ice cream in the cone =205.33 cm2. (V. of ice cream in the cone = volume of conical part +V. of hemi sphere)
4. Circumference = 2πr=44cm,Volume of the cylinder= 4620cm3 (hint- Given r+h=37 cm
Total surface area of the solid cylinder = 2πr(r+h)=1628cm2 )
5. Radius (r)=5x=5×1=5 m, now slant height=√ r2+h2 = 13 m
6. capacity of the tank=8316 cm3
7. A=412.334m2, (A=π×r2+2πrh+πrl)
LONG ANSWER QUESTIONS (5 MARKS)
1. 396 cm3 (hint- Required volume of the remaining portion = (Vol. of the cylinder) – (Vol. of cone A + Vol. of cone B) =594 cm3 –
(132 +66 ) cm3
2. H=89.999cm or 90cm (Hint- Volume of cylinder pipe πr2h= π(1)2(80)= 251.4285 cm3/sec, In 30 minutes volume of water is =
251.4285×30×60= 452571.5 cm3, Volume of cylinder tank =π(40)2h=5028.5714×h cm3 )
3. 1:1(total Surface rea of solid 1=TSA of cube +CSA of HS-base area of HS & total Surface rea of solid 2=TSA of cube +CSA of
HS-base area of HS)
4. the cylindrical tank will be filled in 120 min (hint- So, the water flow through the pipe in t hours will equal to the volume of the
tank∴πr2×v×t=πR2H)
CASE STUDY
1. SANCHI STUP
i) 19404 CU. M
ii) 1232 sq. m
iii) 1392 sq. m or 192 cu. M
2. Ice cream
i) 539/6 cu. Cm
ii) 192.5 cu. Cm
iii) No. of scoops to fill a cone= 2 or approx. 39
3. Tent
i) 4818 sq. m
ii) 11
iii) Rs. 20476.50 Or 39732 cu. M and no. of person=21
4. Water drop
i) 2 mm
ii) 10.68 sq. mm
1
iii) 𝑟=
3 3
(4) or no. of drops = 35000
Chapter 13 (Statistics)
Mean:
The arithmetic mean of a given data is the sum of the values of all the observations divided by the total number of observations.
There are two different formulas for calculating the mean for ungrouped data and the mean for grouped data.
Type 1 (Mean of raw data)
Suppose we have n values in a set of data namely as x1, x2, x3, ………….xn, then the mean of the data is given by using the formula
𝑠𝑢𝑚 𝑜𝑓 𝑎𝑙𝑙 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠 𝑥1 +𝑥2 +⋯……..+𝑥𝑛

𝑴𝒆𝒂𝒏 = ; 𝑥̅ =
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠 𝑛
Type -2 Mean of grouped data (If Frequency and variable are given)
The mean of grouped data deals with the frequencies of different observations or variables that are grouped together. If the
values of the observations are x1, x2, x3 ,………….xn and their corresponding frequencies are f1, f2, f3 ,………….fn, the sum of the
values of all the observations = f1x1 + f2x2 + . . . + fnxn, and the number of observations = f1 + f2 + . . . + fn . So, mean
∑ 𝑓𝑖 𝑥𝑖
𝑥̅ =
∑ 𝑓𝑖
Type- 3 Mean for grouped data. (If Frequency and Class Intervals are given)
To calculate the mean of grouped data, we have three different methods
1. Direct method
∑ 𝑓𝑖 𝑥𝑖 𝑙𝑜𝑤𝑒𝑟 𝑙𝑖𝑚𝑖𝑡+𝑢𝑝𝑝𝑒𝑟 𝑙𝑖𝑚𝑖𝑡

𝑥̅ = ; where, class marks (𝑥𝑖 ) =
∑ 𝑓𝑖 2
2. Assumed mean method
∑ 𝑓𝑑
𝑥̅ = 𝑎 + ∑ 𝑓𝑖 𝑖 Where, di = xi – a and a = assumed mean
𝑖
∑ 𝑓𝑖 𝑢𝑖 𝑥𝑖 −𝑎
3. Step Deviation Method 𝑥̅ = 𝑎 + ( ∑ 𝑓𝑖
) × h where 𝑢𝑖 = ℎ
Example The table below gives information about the percentage distribution of female employees in a company of various branches
and a number of departments. Find the mean percentage of female employee by the assumed mean method.
Female Employee 5-15 15-25 25-35 35-45 45-55 55-65 65-75

No of Departments 1 2 4 4 7 11 6
Solution:
(No of di = xi - a fidi
Female Employee(C I) Departments fi ) Class Marks xi
5-15 1 10 -30 -30
15-25 2 20 -20 -40
25-35 4 30 -10 -40
35-45 4 40 0 0
45-55 7 50 10 70
55-65 11 60 20 220
65-75 6 70 30 180
Total ∑fi =35 ∑fidi = 360
Assumed mean = a = 40
Mean = a+ (∑fidi /∑fi) = 40+ (360/35) = 40+(72/7) = 40 + 10.28 = 50.28
Hence, the mean percentage of female employees is 50.28.
Median
Algorithm
 Obtain the frequency distribution
 Prepare the cumulative frequency column
n
 Obtain n = (∑ fi ) and
2
n
 See the cumulative frequency just greater than (nearer to) 2 and determine the corresponding class. This class is known
as median class
 Obtain the values of the following from the frequency distribution table
l = lower limit of the median class
f = frequency of median class
h = width(size) of the median class
cf = cumulative frequency of the class preceding the median class
Substitute the values in the following formula
n/2 −c f
Median = 𝑙 + ( )h
f
Example: Find the median of the following data is
Marks obtained 0-10 10-20 20-30 30-40 40-50
No of Students 5 7 4 8 6
Solution:
Marks Cumulative
obtained No of students frequency
0-10 5 5
10-20 7 5+7=12
20-30 4 12+4=16
30-40 8 16+8=24
40-50 6 24+6=30
Total N=30
Now N/2 = 30/2 = 15

15 lies in the in cumulative frequency having class intervals 20-30 20-30 is the median class Now
using formula:
𝑛
− 𝑐𝑓
𝑀𝑒𝑑𝑖𝑎𝑛 = 𝑙 + ( 2
)h
𝑓
We have
Lower limit of median class l = 20
Cumulative frequency preceding the median class cf =12 Frequency of the median class f=4
Class size= difference of limits h = 10
15−12
Median = 20 +( )×10 =20 +30/4 = 20 + 7.5 = 27.5
4
Mode
COMPUTATION OF MODE FOR A CONTINOUS FREQUENCY DISTRIBUTION
 Obtain the continuous frequency distribution
 Determine the class of maximum frequency either by inspection or by grouping method
 This class is called the modal class
 Obtain the values of the following from the frequency distribution table
𝑙 = 𝑙𝑜𝑤𝑒𝑟 𝑙𝑖𝑚𝑖𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑚𝑜𝑑𝑎𝑙 𝑐𝑙𝑎𝑠𝑠
𝑓1 = 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝑚𝑜𝑑𝑎𝑙 𝑐𝑙𝑎𝑠𝑠
ℎ = 𝑤𝑖𝑑𝑡ℎ(𝑠𝑖𝑧𝑒) of the modal class
𝑓0 = 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑙𝑎𝑠𝑠 𝑝𝑟𝑒𝑐𝑒𝑑𝑖𝑛𝑔 𝑡ℎ𝑒 𝑚𝑜𝑑𝑎𝑙 𝑐𝑙𝑎𝑠𝑠
𝑓2 = 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑙𝑎𝑠𝑠 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 𝑡ℎ𝑒 𝑚𝑜𝑑𝑎𝑙 class
𝑓 −𝑓
Mode = 𝑙 + (2𝑓 −𝑓
1 0
)ℎ
1 −𝑓0 2
THE EMPIRICAL RELATIONSHIP BETWEEN THE THREE MEASURES OF CENTRAL TENDENCY
3 median = mode + 2 mean
EXAMPLE
The following data gives the information on the observed life times (in hours) of 150 electrical components. Find the mode
Life time( In Hours) 0-20 20-40 40-60 60-80 80-100

Frequency 15 10 35 50 40
SOLUTION:
The class 60-80 have the maximum frequency as 50
Modal class is 60-80
So, l = 60, f0 = 35, f1 = 50, f2 = 40, h = 20
𝑓 −𝑓 50−35
Mode = 𝑙 + (2𝑓 −𝑓
1 0
)ℎ = 60 + ( ) × 20 = 72
1 −𝑓0 2 2×50−35−40
QUESTIONS
Type 1 MCQ
1. Find the class marks of classes 10-20 and 35-55:
(a) 10, 35 (b) 20, 55 (c) 15, 45 (d) 17.5, 45
2. If di = xi - 13, ∑fidi = 30 and ∑fi=120 , then mean is equal to:
(a) 13 (b) 12.75 (c) 13.25 (d) 14.25
3. The mean of first ten odd natural numbers is:
(a) 5 (b) 10 (c) 20 (d) 19
4. If the mean of x, x + 3, x + 6, x + 9 and x + 12 is 10, then x equals:
(a) 1 (b) 2 (c) 4 (d) 6
5. For a frequency distribution, mean, median and mode are connected by the relation:
(a) Mode = 3 Mean - 2 Median (b) Mode = 2 Median - 3 Mean
(c) Mode = 3 Median - 2 Mean (d) Mode = 3 Median + 2 Mean.
5𝑛
6. The mean of first n natural numbers is 9 then n is:
(a) 5 (b) 9 (c) 4 (d) 10
7. If Median of data 16,18, 20, 24-x, 20 + 2x, 28, 30, 32 is 24 then x is:
(a) 4 (b) 18 (c) 16 (d) 20
8. The Mean of five numbers is 15. If we include one more number, the mean of 6 numbers become 17. The included number is:
(a) 24 (b) 26 (c) 25 (d) 27
Type 2 ASSERTION- REASAON QUESTIONS

NOTE - In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct
choice as:
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A). (b) Both assertion (A)
and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
9. Assertion: Consider the following frequency distribution: The modal class is 10-15.
Class interval 10-15 15-20 20-25 25-30 30-35
frequency 5 9 12 6 8
Reason: The class having maximum frequency is called the modal class.
10. Assertion (A): If the number of runs scored by 11 players of a cricket team of India are 5, 19, 42, 11, 50, 30, 21, 0, 52, 36, 27
then median is 30.
𝑛+1 𝑡ℎ
Reason (R): Median = ( 2 ) observation, if n is odd.
11. Assertion (A): If the value of mode and mean is 60 and 66 respectively, then the value of median is 64.
Reason (R): Median = (mode + 2 mean).
12. Assertion: the mode of the call received on 7 consecutive days 11,13,13,17,19,23,25 is 13.
Reason: Mode is the value that appears most frequent.
13. Find the median of the following data:

CI 0-10 10-20 20-30 30-40 40-50 TOTAL
Frequency 8 16 36 34 6 100
14. Find the modal literacy rate of 40 cities:
Literacy rate 30-40 40-50 50-60 60-70 70-80 80-90
No of cities 6 7 10 6 8 3
15. Find the missing frequencies f1, f2 and f3 in the following frequency distribution, when it is given that f2 : f3 = 4 : 3, and mean =
50:
CI 0-20 20-40 40-60 60-80 80-100 TOTAL
Frequency 17 f1 f2 f3 19 120
16. The following distribution shows the daily pocket allowance of children of a locality. The mean pocket allowance is Rs. 18.
Find the missing frequency f:
Daily pocket allowance 11-13 13-15 15-17 17-19 19-21 21-23 23-25
No of children 7 6 9 13 f 5 4
17. A class teacher has the following absentee record of 40 students of a class for the whole term. Find the mean number of days a
student was absent:
No of days 0-6 6-10 10-14 14-20 20-28 28-38 38-40
No of students 11 10 7 4 4 3 1
18. If the median of the distribution given below is 28.5, find the values of x and y:
CI 0-10 10-20 20-30 30-40 40-50 50-60 TOTAL
FREQUENCY 5 X 20 15 Y 5 60
19. A life insurance agent found the following data for distribution of ages of 100 policy holders. Calculate the median age, if
policies are given only to persons having age 18 years onwards but less than 60 year:
Age (in years) Number of policy holders
Below 20 2
Below 25 6
Below 30 24
Below 35 45
Below 40 78
Below 45 89
Below 50 92
Below 55 98
Below 60 100
20. An agency has decided to install customized playground equipment’s at various colony parks. For that they decided to study
the age group of children playing in a park of the particular colony The classification of children according to their ages,
playing in a park is shown in the following table:
Age group of children (in 6-8 8-10 10-12 12-14 14-16

years)
Number of children 43 58 70 42 27
Based on the above information answer the following:
(i) In which age group, will the maximum number of children belong?
(ii) Find the class marks of the modal class
(iii) Find the mode of the ages of children playing in the park.
Or
Find the mean of the ages of children playing in the park.
21. Electricity energy consumption is the form of energy consumption that uses electric energy. Global electricity consumption
continues to increase faster than world population, leading to an increase in the average amount of electricity consumed per
person (per capita electricity consumption).
A survey is conducted for 56 families of a Colony A. The following table gives the weekly consumption of electricity of these families.
Weekly consumption (in units) 0-10 10-20 20-30 30-40 40-50 50-60
No. of families 16 11 19 6 4 0
i. Find the difference between upper limit of the modal class and lower limit of median class.
ii. Calculate mean of the data.
iii. Calculate the mode of this data.
Or
Calculate the cumulative frequency and also find the median group.
22. On a particular day, National Highway Authority of India (NHAI) checked the toll tax collection of a particular toll plaza in
Rajasthan. The following table shows the toll tax paid by drivers and the number of vehicles on that particular day.
Weekly consumption (in units) 30-40 40-50 50-60 60-70 70-80

No. of families 80 110 120 70 40

(i) What is the class mark of 50-60
(ii) Write the model class of the given distribution.
(iii) Find the mean of toll tax received by NHAI by direct method.
Or
Find the mean of toll tax received by NHAI by assumed mean method.
23. Distance Analysis of Public Transport Buses Transport department of a city wants to buy some Electric buses
for the city. For which they want to analyse the distance travelled by existing public transport buses in a day.
Daily distance travelled (in km) 200-209 210-219 220-229 230-239 240-249
Number of buses 4 14 26 10 6
(i) Find the difference between upper limit of a class and lower limit of its succeeding class.
(ii) Find the median class.
(iii) The cumulative frequency of the class preceding the median class is :
Or
Find the median of distance travelled.
24. If mode of the following distribution is 55, then find the value of x.
Class 0-15 15-30 30-45 45-60 60-75 75-90
Frequency 10 7 x 15 10 12
25. Find mean and mode of the given data. Also find median using Empirical Formula.
Class 20-30 30-40 40-50 50-60 60-70
Frequency 25 40 42 33 10
ANSWER KEY:
1. C 2. C 3. B 4. C
2. 5. C 6. B 7. A 8. D
9. D 10. D 11. C 12. A
13. 27.22 14. 58.57 15. F1 = 28, F2 = 32, F3 = 24 16. 20
17. 12.48 19. 35.76
20. (i) 10-12 (ii) 10.6 (iii) 11
21. (i) 10 (ii) 19.8 (iii) 23.8
22. (i) 55 (ii) 50-60 (iii) 52.14
23. (i) 1 (ii) 219.5-229.5 (iii) 18 OR 224.6
24. X = 5
25. MODE = 42.22, MEAN = 31.3,
Chapter 14 (Probability)
1. The theoretical (classical) probability of an event E, written as P(E), is defined as
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑓𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑡𝑜 𝐸
P(E) =
𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑡𝑜 𝐸
Where, we assume that the outcomes of the experiment are equally likely.
2. The probability of a sure event (or certain event) is 1.
3. The probability of an impossible event is 0.
4. The probability of an event E is P(E) such that 0 ≤ P(E) ≤1
5. An event having only one outcome is called an elementary event. The sum of the probabilities of all the elementary events of an
experiment is 1.
̅)=1
6. For any event E, P( E ) + P ( E
Where E stands for ‘not E'. E and ̅
̅ E are called complementary events.
7. Playing cards:
It consists of 52 cards which are divided into 4 suits of 13 cards each:
(i) Spade ♠ (Black colour) (iii) Club ♣ (Black colour)
(ii) Diamond ♦ (Red colour) (iv) Heart ♥ (Red colour)
Face cards: King, Queen, Jack
Questions:
1. Which of the following cannot be the probability of an event?
2
(A) 0.7 (B) 3 (C) - 1.5 (D) 15%
2. Out of one digit prime numbers, one number is selected at random. The probability of selecting an even number is:
1 1 4 2
(A) 2 (B) 4 (C) 9 (D) 5
3. When a die is thrown, the probability of getting an odd number less than3 is:
1 1 1
(A) 6 (B) 3 (C) 2 (D) 0
4. A card is drawn from a well shuffled pack of 52 playing cards. The event E is that the card drawn is not a face card. The number
of outcomes favourable to the event E is
(A) 51 (B) 40 (C) 36 (D) 12
5. In a family of 3 children, the probability of having at least one boy is:
7 1 5 3
(A) 8 (B) 8 (C) 8 (D) 4
6. The probability of a number selected at random from the numbers 1, 2, 3, .... 15 is a multiple of 4 is:
4 2 1 1
(A) 15 (B) 15 (C) 5 (D) 3
7. A bag contains 6 red and 5 blue balls. One ball is drawn at random. The probability that the ball is blue is:
2 5 5 6
(A) 11 (B) 6 (C) 11 (D) 11
8. Two dice, one blue and one grey, are thrown at the same time. Write down all the possible outcomes. What is the probability
that the sum of the two numbers appearing on the top of the dice is
(i) 8? (ii) 13? (iii) Less than or equal to 12?
9. 12 defective pens are accidentally mixed with 132 good ones. It is not possible to just look at a pen and tell whether or not it is
defective. One pen is taken out at random from this lot. Determine the probability that the pen taken out is a good one.
10. A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random from the box, find the probability
that it bears (i) a two-digit number (ii) a perfect square number (iii) a number divisible by 5
11. A lot consists of 144 ball pens of which 20 are defective and the others are good. Nuri will buy a pen if it is good, but will not
buy if it is defective. The shopkeeper draws one pen at random and gives it to her. What is the probability that
(i) She will buy it? (ii) She will not buy it?
12. A letter of English alphabet is chosen at random. Determine the probability that the chosen letter is a consonant
13. A card is drawn from a well-shuffled deck of 52 playing cards. Then what is the probability that the card will not be a
diamond?
14. The probability of selecting a rotten apple randomly from a heap of 900 apples is 0.18. What is the number of rotten apples in
the heap?
15. Cards bearing numbers 2 to 21 are placed in a bag and mixed thoroughly. A card is taken out of the bag at random. What is the
probability that the number on the card taken out is an even number?
16. A card is drawn out from a well-shuffled deck of 52 cards. What is the probability of getting a red queen?
17. Two different dice are tossed together. Find the probability that (i) the number on each dice is odd, and (ii) the sum on the
numbers, appearing on the two dice, is 5.
18. Rahim tosses two different coins simultaneously. Find the probability of getting at least one tail.
19. A card is drawn at random from a well-shuffled pack of 52 cards. Find the probability that the card is neither a red card nor a
jack.
20. A box contains 100 red cards, 200 yellow cards and 50 blue cards. If a card is drawn at random from the box, then find the
probability that it will be :
(i) a blue card,
(ii) not a yellow card, and
(iii) neither yellow nor a blue card.
Answers:
1: C 2: C 3: A 4: B 5: A 6: C
7: C 8: (i) 5/36 (ii) 0 (iii) 1 9: 11/12
10 (i) 9/10 (ii) 1/10 (iii) 1/5 11: (i) 31/36 (ii) 5/36 12: 21/26 13: 3/4
14: 162 15: 1/2 16: 1/26 17: (i) 1/4 (ii) 1/9 18: 3/4
19: 6/13 20: (i) 1/7 (ii) 3/7 (iii) 2/7
Assertion and Reasoning Questions:
In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
1. Assertion (A): If a box contains 5 white, 2 red and 4 black marbles, then the probability of not drawing a white marble from the
5
box is 11.
Reason (R): P(E̅) = 1 – P(E), where E is any event.
1
2. Assertion (A): When two coins are tossed simultaneously then the probability of getting no tail is 4.
1
Reason (R): The probability of getting a head (i.e., no tail) in one toss of a coin is .
2
3. Assertion (A): If a die is thrown, the probability of getting a number less than 3 and greater than 2 is zero.
Reason (R): Probability of an impossible event is zero.
1
4. Assertion (A): In a simultaneously throw of a pair of dice, the probability of getting a doublet is 6.
Reason (R): Probability of an event may be negative.
5. Assertion (A): The probability of winning a game is 0.4, then the probability of losing it, is 0.6.
Reason (R): P( E ) + P(E̅) = 1.
Answers:
Q no. 1 2 3 4 5
Answer D B A C A
Case Study Based Questions :

Rohit wants to distribute chocolates in his class on his birthday. The chocolates are of three types: Milk chocolate, white
chocolate and dark chocolate. If the total number of students in the class is 54 and everyone gets a chocolate then answer the
following questions:
1
(i) If the probability of distributing milk chocolate is 3, then find the number of milk chocolate Rohit has. 1 mark
4
(ii) If the probability of distributing dark chocolate is 9 then find the number of dark chocolate Rohit has. 1 mark
(iii) Find the probability of distributing both milk and white chocolate. 2 marks
OR
Find the probability of distributing both white and dark chocolate.
1. Four friends are playing with cards. One of them hides all that 2’s, 5’s and Jacks from the deck of 52 cards and then shuffles
the remaining cards. Now, he tells to one of his friend to pick a card at random from the remaining cards.
On the basis of above information answer the following questions:

(i) Find the probability of getting ‘6 of spade’. 1 marks
(ii) Find the probability of getting a black diamond. 1 marks
(iii) Find the probability of getting a face card. 2 marks
OR
Find the probability of getting not a face card.
2. Three persons toss 3 coins simultaneously and note the outcomes. Then, they ask few questions to one another. Help them in
finding the answers of the following questions :
(i) Find the probability of getting exactly 3 tails. 1 marks

(ii) Find the probability of getting exactly 1 head. 1 marks
(iii) Find the probability of getting at most 3 heads. 2 marks
OR
Find the probability of getting at least two heads
3. Two friends A and B are playing a game. They roll a pair of fair dice one by one. The game starts with A.
(i) Now B rolls the dice, find the chances that he will get consecutive numbers. 1 marks
(ii) Now A rolls the dice again, find the chance that he gets more than 4 on each die. 1 marks
(iii) Now B rolls the dice again, find the chance that he gets even numbers on both dice. 2 marks
OR
Now B rolls the dice again, find the chance that he gets odd numbers on both dice.
Answers:
1. (i) 18 (ii) 24 (iii) 30 OR 36
2. (i) 1/10 (ii) 0 (iii) 1/5 OR 4/5
3. (i) 1/8 (ii) 3/8 (iii) 1 OR 1/2
4. (i) 5/18 (ii)1/9 (iii) 1/4 OR 1/4
Questions based on higher order thinking skills:

1. If 65% of the population have Black eyes, 25% have brown eyes and the remaining has blue eyes. the probability that a person
selected at random has :
(i) Blue Eyes (iii) Brown or black eyes
(ii) Blue or black eyes (iv) Neither blue nor brown eyes
2. A bag has 8 red balls and x blue balls, the odd against drawing a blue ball are 2 : 5. What is the value of x?
3. Find the probability of having 53 Sundays in
(i) a leap year (ii) a non-leap year
4. A number x is chosen at random from the numbers -3,-2,-1, 0, 1, 2, 3. What is the probability that |𝑥 | < 2?
Answers:
1 9 3 13
1. (i) (ii) (iii) (iv)
10 10 4 20
2. 20
2 1
3. (i) (ii)
7 7
3
4.
7
………………THANK YOU……………

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PATRONS

Sh. D. R. MEENA Sh. G.S. MEHTA Sh. MADHO

Mr. Navratan Mittal

Mrs. Durga Chauhan

1. Mr. Narendra Singh Poonia

2. Mr. Vinod kumar

3. Mrs. Arti sharma

𝐇𝐂𝐅 (𝐚, 𝐛) × 𝐋𝐂𝐌 (𝐚, 𝐛) = 𝐚 × 𝐛

PRIME FACTORISATION METHOD TO FIND HCF AND LCM

4 225 can be expressed as

8 Check whether 5 × 3 × 11 + 11 𝑎𝑛𝑑 5 × 7 + 7 × 3 are composite number and justify.

19 SCOUT RAJYA PURASKAR TESTING CAMP-2023

(i) How many rooms are required for Scouts?

(i) Find the number of books in each stack.

Q.No. Multiple choice questions

(a) - 3, 2 (b) - 3, - 2 (c) 3, 2 (d) 3, - 2

(a) both positive (b) both negative

(A)has no linear term and the constant term is negative.

S. Pair of lines Compare the Graphical Algebraic

3 a2x + b2y + c2 = 0 (inconsistent)

If a pair of linear equations is consistent, then the lines will be:

ASSERTION REASONING QUESTIONS

CASE BASED STUDY QUESTIONS

IMPORTANT QUESTIONS FROM NCERT BOOK

at a point, are parallel or coincident:

(i) 5x – 3y = 11; – 10x + 6y = –22

1. C 2. D 3.C 4.D 5.C 6.B

ASSERTION REASONING QUESTIONS

CASE STUDY QUESTIONS

IMPORTANT QUESTIONS FROM NCERT BOOK

2. Let the speed of car I from A be x and speed of car II from B be y.

4. Since BE || CD and BC || DE with BC is perpendicular to DC, bcde is arectangle.

(Question numbers from 1 to 10 carry 1 mark each)

(a) 8 (b)-8 (c) 16 (d) -16

(a) 2 (b) -1 (c) √2 (d) √3

(Case based study: Question numbers from 21 to 23 carry 4 marks each)

Q12. Let speed of fast train is x km/h,

Q13. For equal and real root is quadratic eqn. D=0

CHAPTER 5 ARITHMETIC PROGRESSION

 Sum of first n positive integers: Let Sn = 1 + 2 + 3 +……..n, here a = 1, last term, l = n is

(a) q + 9p (b) p - 9q (c) p + 9q (d) 2p + 9q

Reason: common difference of an A.P. is given by d = an+1 - an.

32. TOWER OF PISA :

3. In Fig., ΔFEC ≅ΔGDB and ∠1 = ∠2. Prove that ΔADE ~ ∆ABC.

2) CASE STUDY 2- SCALE FACTOR

2. (i) 22m (ii) 0.7 m (iii) 4cm

3. (i) 100m (ii)24m (iii)40m

4. (i)330km (ii) 5 : 11 (iii) 100m

ASSERTION REASONING QUESTIONS-

CASE STUDY BASED QUESTIONS

1. What is the distance between house and bank?

2. What is the approximate distance between house and office?

3. What is the total distance travelled by Aditya to reach the office?

1. What will be the location of exit gate ‘Q’?

1. What is the distance between Amar and Bharath?

Assertion- Reason Questions: 1. (b) 2. (d) 3. (d) 4. (a)

Answers of Case Based questions

𝒔𝒊𝒅𝒆 𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕 𝒕𝒐∠𝑨 AB

7. If sin𝜽 = x and sec𝜽 =y, then tan𝜽 is equal to: