Maths PDF
Maths PDF
Maths PDF
Session : 2019-20
CLASS X
MATHEMATICS
Class –X
Mathematics
Session: 2019-20
1
A WORD TO MY DEAR STUDENTS
It gives me great pleasure in presenting the Students ‘Support Material to all KV students of
class X. The material has been prepared keeping in mind your needs when you are preparing
for final exams and wish to revise and practice questions or when you want to test your ability
to complete the question paper in the time allotted or when you come across a question while
studying that needs an immediate answer but going through the textbook will take time or
when you want to revise the complete concept or idea in just a minute or try your hand at a
question from a previous CBSE Board exam paper or the Competitive exam to check your
understanding of the chapter or unit you have just finished. This material will support you in
any way you want to use it.
A team of dedicated and experienced teachers with expertise in their subjects has prepared
this material after a lot of exercise. Care has been taken to include only those items that are
relevant and are in addition to or in support of the textbook. This material should not be taken
as a substitute to the NCERT textbook but it is designed to supplement it.
The Students ‘Support Material has all the important aspects required by you; a design of the
question paper, syllabus, all the units/chapters or concepts in points, mind maps and
information in tables for easy reference, sample test items from every chapter and question
papers for practice along with previous years Board exam question papers.
I am sure that the Support Material will be used by both students and teachers and I am
confident that the material will help you perform well in your exams.
Happy learning!
2
FOREWORD
The Students' Support Material is a product of an in-house academic exercise undertaken by our
subject teachers under the supervision of subject expert at different levels to provide the students
a comprehensive, yet concise, learning support tool for consolidation of your studies. It
consists of lessons in capsule form, mind maps, concepts with flowcharts, pictorial representation
of chapters wherever possible, crossword puzzles, question bank of short and long answer type
questions with previous years' CBSE question papers.
The material has been developed keeping in mind latest CBSE curriculum and question
paper design. This material provides the students a valuable window on precise information and it
covers all essential components that are required for effective revision of the subject.
In order to ensure uniformity in terms of content, design, standard and presentation of the
material, it has been fine-tuned at KVS HQRS level.
I hope this material will prove to be a good tool for quick revision and will serve the purpose
of enhancing students ‘confidence level to help them perform better. Planned study blended with
hard work, good time management and sincerity will help the students reach the pinnacle of
success.
Best of Luck.
3
Mathematics
CONTENT TEAM
REVIEW TEAM
4
INDEX
S.NO TOPIC
PART -1
1 Real Numbers
2 Polynomials
3 A pair of linear equations in two variables
4 Quadratic Equations
5 Arithmetic Progressions
6 Triangles
7 Coordinate Geometry
8 Introduction to Trigonometry
9 Some Applications of Trigonometry
10 Circles
11 Constructions
12 Area Related to Circles
13 Surface Areas and Volumes
14 Statistics
15 Probability
PART – 2
16 Activities
PART- 3
17 ModelQuestionPaper1 (Standard)with blue print and marking scheme
18 ModelQuestionPaper2 (Standard)with blue print and marking scheme
19 Model Question Paper 3 (Basic)
20 Model Question Paper 4 (Basic)
5
COURSE STRUCTURE CLASS -X
As per the CBSE Circular F.1002/CBSE/Dir. (Acad.)/Mathematics dated 10/01/2019 the two levels of
Examination will be held in the subject of Mathematics in the Board Examination for class X in the year 2020
6
MATHEMATICS Standard Code (041)
QUESTION PAPER DESIGN CLASS – X (2019-20)
Time: 3 Hours Max. Marks: 80
Very Short
Answer- Short Short Long
Objective %
S. Typology of Questions Answer-I Answer- Answer Total Weightage
No. type (SA)(2 II (SA) (3 (LA)(4 Marks
(VSA) (1 (approx.)
Marks) Marks) Marks)
Mark)
7
MATHEMATICS-Basic Code (241)
QUESTION PAPER DESIGN
CLASS – X (2019-20)
Time: 3 Hours Max. Marks: 80
Very
Short
Short Short Long
Answer- %
S. Answer-I Answer- Answer Total
Typology of Questions Objective Weightage
No. (SA) II (SA) (LA) Marks
type (approx.)
(VSA) (2Marks) (3Marks) (4Marks)
(1Mark)
Understanding: Demonstrate
understanding of facts and ideas
by organizing, comparing, 7 1 1 4 28 35
2
translating, interpreting, giving
descriptions, and stating main
ideas
Applying: Solve problems to
new situations by applying 5 2 1 - 12 15
3 acquired knowledge, facts,
Techniques and rules in a
different way.
Analyzing :
Examine and break information
into parts by identifying motives
or causes. Make inferences and
find evidence to support
generalizations
Evaluating:
Present and defend opinions by
3 1 1 - 8 10
4 making judgments about
information, validity of ideas, or
quality of work based on a set of
criteria.
Creating:
Compile information together in
a different way by combining
elements in a new pattern or
proposing alternative solutions
20x1 =20 6x2 =12 8x3=24 6x4=24 80 100
Total
8
TOPIC WISE ANALYSIS OF EXAMPLES AND QUESTIONS
1 Real Number 11 18 29
2 Polynomials 09 08 17
Pair of linear equations in two
3 variables
19 21 40
4 Quadratic Equation 18 24 42
5 Arithmetic Progression 16 44 60
6 Triangle 14 55 69
7 Coordinate Geometry 15 25 40
8 Introduction to trigonometry 15 27 42
10 Circle 3 17 20
11 Construction 2 14 16
14 Statistics 09 25 34
15 Probability 13 25 38
9
DETAILS OF THE CONCEPTS TO BE MASTERED BY EVERY CHILD OF CLASS X
WITH EXERCISE AND EXAMPLES OF NCERT TEXT BOOKS.
SYMBOLS USED
TG/LG is idea identified error analysis of answers of Q.P of last three year.
* - Important Question a - Low T.G-Teaching Gap
** -Very Important Question b - Average L.G-Learning Gap
*** -Very Very Important Question c - Higher
DIFFICULTY LEVEL
IMPORTAN
DEGREEOF
REFERENCES(NCER
Sl. No
DEGREE
TOPIC CONCEPTS
TG/LG
T BOOK)
12
Real Numbers
(Key Points)
Real
Numbers
RATIONAL IRRATIONAL
NUMBERS (Q) NUMBERS (I)
NEGATIVE
INTEGERS (-1,-2,-3,)
ZERO (0) POSITIVE INTEGERS
(1, 2, 3…)
1. Euclid’s Division lemma:-Given Positive integers a and b there exist unique integer’s q and r
satisfying a=bq+r, where 0≤r<b, where a,b,q and r are respectively called as dividend, divisor,
quotient and remainder.
2. Euclid’s division Algorithm:-To obtain the HCF of two positive integers say c and d, with c>d, follow
the steps below:
Step I: Apply Euclid’s division lemma, to c and d, so we find whole numbers, q and n r such that
c=dq +r, 0
Step II: If r =0, d is the HCF of c and d. If r ≠ 0, apply the division lemma to d and r.
Step III: Continue the process still the remainder is zero. The divisor at this stage will be the required HCF
Note: - Let a and b be positive integers .If a=b q +r, 0≤r<b, then HCF (a, b) = HCF (b, r)
p
Theorem: x = Let x be a rational number such that the prime factorization of q is not of the form of
q
2 nx 5m, where n,m are non-negative integers. Then x has a decimal expansion which is non-
terminating repeating (recurring).
Ex:-
LEVEL-I
6. Write one rational and one irrational number lying between 0.25 and 0.32.
7. Express107 in the form of 4q+3 for some positive integer q.
8. Write whether the rational number will have at erminating decimal expansion or a non
14
LEVEL-II
1. Use Euclid’s division algorithm to find the HCF of 1288 and 575.
2. Check whether 5x3x11+11 and 5x7+7X3 are composite number and justify.
3. Check whether 6n can end with the digit 0, where n is any natural number.
4. Given that LCM (26,169) =338, write HCF (26,169).
5. Find the HCF and LCM of 6,72 and 120 using the prime factorization method.
6 Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or
3m+1for some integer m.
7. Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m+1 or
9m+8 for some integer m.
LEVEL-III
1. Show that √3 is an irrational number.
4. Find the LCM & HCF of 26 and 91 and verify that HCF x LCM=product of the two numbers
6. Show that one and only one out of n, n+2, n+4 is divisible by 3, where n is any positive integer.
7. Find the HCF of 65&117 and express it in the form of 65m+117n.
6. Show that there is no positive integer n for which n1 n1 is rational.
7. Using prime factorization method, find the HCF and LCM of 72,126 and 168. Also show that
HCF X LCM ≠ product of three numbers.
8. Three sets of English, Mathematics and Science books containing 336, 240 and 96 books
respectively have to be stacked in such away that all the books are stored subject wise and the height
of each stack is the same. How many stacks will be there?
15
9. A person wanted to distribute 96 apples and 112 oranges among poor children in an orphanage. He packed all
the fruits in boxes in such away that each box contains fruits of the same variety, and also every box
contains an equal number of fruits.
• Find the maximum number of boxes in which all the fruits can be packed.
• Which concept have you used to find it?
10. Teacher draws the factor tree, given in figure and ask the students to find the value
of x without finding the value of y and z. Shaurya gives the answer x=136
x
a) Is his answer correct?
2 z
2 17
Answer Level-I
1. 1.42 (flexible)
2. After 4 places of decimal.
3. 19000
10.14/55
Level-II
1. 23
2. Composite number and justification show that numbers are having more than two factors
3. No,6n cannot end with the digit 0.
4. 13
5. HCF=6, LCM = 360
6. ncert book.ex1.1/q.n.4
16
LEVEL-III
a. See
textbook. ex
9 p.no 13
2. See textbook ex 1.3
q.no2
3. See textbook
4. LCM = 182 HCF=13
Product =2366
5 See textbook
7. HCF =13
2. 2658= 2x3x443
5. Number =13
7. HCF = 6 LCM = 504 NOT EQUAL
17
18
19
20
21
8. Check whether x2 +3x +1 is a factor of 3x4+5x3 -7x2+2x+2?
9. Can x-7 be the remainder on division of a polynomial P(x) by 7x+2? Justify
your answer.
10. What must be subtracted from the polynomial f(x) =x4+2x3-13x2-12x+21 s0 that the
resulting polynomial is exactly divisible by x2 -4x +3.
11. Write the degree of the zero polynomial.
12. Find the zeros of the quadratic polynomial 6x2-7x-3 and verify the relationship
between the zeros and the coefficients.
13. Find the quadratic polynomial sum of whose zeros is 2√3 and their product is 2.
Level II
9. If the sum of squares of zeros of the polynomial 6x2+x+k is 25/36 find the value of k.
10. If one zero of the quadratic polynomial f(x) =4x2-8kx-9 is negative of the other, find the
value of k.
11. Find the value of k for which the quadratic polynomial 9x2-3kx+k has equal zeros
12. On dividing 3x3 -2x2+5x-5 by the polynomial p(x), the quotient and remainder are
x2 -x+2 and -7 respectively. Find p(x)
13. Find all the zeros of polynomial x4+x3-9x2-3x+18 if the two of its zeros are √3 and -√3.
14. If α and β are the zeros of quadratic polynomial p(x) =x2-(k-6) x+ (2k+1)
Find the value of k if α+ =αβ.
15. If the zeros of polynomial x2-5x+k are the reciprocal of the zeros then find the value of k.
16. If α and β are the zeros of the quadratic polynomial x2 -6x +a find the value of ‘a’
If 3α +2β=20
17. On dividing 3x3+4x2+5x-13by a polynomial g(x) ,the quotient and remainder
are 3x+10and 16x-43 respectively. Find the polynomial g(x).
18. If α and β are the zeroes of a polynomial x2-x-30, then form a quadratic
Polynomial whose zeroes are 2-α and 2- β
22
26. If the number of apples and mangoes are the zeroes of polynomial 3x2-8x-2k+1.
and the number of apples are 7 times the number of mangoes,
then find the number of zeroes and value of k.
23
ANSWERS
LEVEL I
1. A) 1 polynomial x-1
B) -2, 0 polynomial x2+2x
2. K(2x2 +x-15)
6. 0.
8. x2-2√3x+2
LEVEL II
9. k= -2
10. k= 0
11. k= 0 or k=4
12. 3x-1
13. Other Zeroes 2 and -3
14. K= -7
15. K=1
16. a= -16
LEVEL III
17. g(x) =x 2-2x+3
18. x2-3x-28
19. 5
20. (i) 4
(ii) 1
(iii). -6
24
22. -2
23. Other zero 1 and P= -6
24. a=3
25. 20x2-9x+1
26. 2 zeroes, K= -2/3
The general form for a pair of linear equations in two variables x and y is
a1x+b1y+c1=0
a2x + b 2y + c2 = 0
Where a1,b 1,c1,a2,b 2,c 2 are all real numbers and a1≠0,b 1≠0,a2≠0,b2≠0. Example
x + 3y – 6 = 0
2x – 3y – 12=0
Graphical representation of a pair of linear equations in two variables:
a1x+b1y+c1=0
a2x + b 2y + c2 = 0
I.e.unique solution. And these types of equations are called consistent pair of linear equations.
Ex: x – 2y = 0
3x + 4y – 20 = 0
Ex:2x+3y–9=0
4x+6y–18=0
25
The graph is Coincident lines,
1. Find the value of ‘a’ so that the point (2,9) lies on the line represented by ax-3y=5
2. Find the value of k so that the lines 2x–3y = 9 and kx - 9y = 18 will be parallel.
3. Find the value of k for which x + 2y = 5,3x + ky+15 = 0 is inconsistent
5. Determine the value of ‘a’ if the system of linear equations 3x+2y-4=0and ax–y–3=0 will
represent intersecting lines.
6. Write any one equation of the line which is parallel to2x–3y=5
7. Find the point of intersection of line -3x+7y=3 with x-axis
8. For what value of k the following pair has infinite number of solutions.
(k-3)x+3y=k
K (x + y) =12
9. Write the condition so that a 1x + b 1y = c1 and a 2x + b2y = c2 have unique solution.
26
Level -II
1. 5 pencils and 7 pens together cost Rs.50 whereas 7 pencils and 5 pens together cost Rs.46.Find the cost of
one pencil and that of one pen.
3x – 5y = -1
x– y =-1
6. Solve the equations:
- = -1
+ =8; Where, x ≠ 0 , y ≠ 0
x–y=7
5x + 12y =7
Level -III
1. Draw the graph of the equations
4x – y= 4
4x + y = 12
Determine the vertices of the triangle formed by the lines representing these equations and the x-axis.
Shade the triangular region so formed
2. Solve Graphically
x – y = –1
3x + 2y = 12.
Calculate the area bounded by these lines and the x- axis.
4u–v=14uv
3u + 2v = 16uv where u≠0, v≠ 0
27
4. Ritu can row downstream 20 km in 2 hours, and upstream 4km in 2hours. Find her speed of rowing in still
water and the speed of the current. (HOTS)
5. In a , =3 B=2( A+ B) find the these angle. (HOTS)
6. 8 men and 12 boys can finish a piece of work in 10 days while 6 men and 8 boys can finish it in14 Days.
Find the time taken by 1 man alone and that by one boy alone to finish the work. (HOTS)
7. Find the value of K for which the system of linear equations 2x + 5y = 3, (k+1)x + 2(k+2)y = 2 K will
have infinite number of solutions. (HOTS)
SELF EVALUTION
1. Solve for x and y:
x + y = a+b
ax – by = −
2. For what value of k will the equation x +5y – 7 = 0 and 4x + 20y + k = 0 represent coincident lines?
3. Solve graphically:
3x + y + 1 = 0
2x - 3y +8 = 0
4. The sum of digits of a two digit number is 9.If 27 is subtracted from the number, the digits are
reversed. Find the number.
5. Draw the graph of x+2y–7=0and 2x–y – 4 =0. Shade the area bounded by these lines and Y-axis.
6. Students of a class are made to stand in rows. If one student is extra in a row, there would be 2rows less.
If one student is less in a row there would be 3 rows more. Find the number of the students in the class.
7. Aman travels 370 km partly by train and remaining by car. If he covers 250 km by train and the rest by the
car it takes him 4hours, but if he travels 130 km by train and th rest by car, he takes18minutes longer.
Find the speed of the train and that of the car.
8. Given linear equation 2x+3y – 8 = 0, write an other linear equation such that the geometrical
representation of the pairs of lines are (i) intersecting lines, (ii) Parallel Lines.
28
9. Solve for x and y.
(a-b)x +(a +b)y = a2 - 2ab – b2
(a+b)(x+y) = a 2 + b 2 (CBSE 2004, ’07C,’08)
10. The sum of two numbers is 8 and the sum of the irreciprocalis 8/15.Find the numbers.
(CBSE 2009)
11. The owner of a taxi cab company decides to run all the cars he has on CNG fuel instead
Of petrol/diesel. The car hire charges in city comprises of fixed charges together with the charge for the
Distance covered. For a journey of 12km, the charge paid Rs.89 and for a journey of 20km, the charge
paid is Rs.145.
i. What will a person have to pay for travelling a distance of 30km?
ii. Which concept has been used to find it?
iii. Which values of the owner have been depicted here?
12. Riya decides to use public transport to cover a distance of 300km.She travels this distance partly
by train and remaining by bus. She takes 4 hours if she travels 60 km by bus and the remaining by train. If
she travels100kmby bus and the remaining by train, she takes 10minutes more.
i. Finds peed of train and bus separately.
ii. Which concept has been used to solve the above problem?
iii. Which values of Riya have been depicted here?
29
ANSWER
LEVEL-I
Q1 a=16
Q2. k=6
Q3 .k=6
Q4. Consistent
Q5. a ≠ 3/2
Q6. (May be another solution also)
Q7. (-1,0)
Q8. k= 6
Q9. ≠
LEVEL-II
Q6.
Q7.
LEVEL-III
Q1. (2, 4)(1,0 )(3,0 )
Q2. x = 2, y = 3 and area = 7.5 unit 2
Q3. u = ½ , v = ¼
Q4.Speed of the rowing in still water = 6km/hr
Speed of the current=4km/hr.
30
Q7.Speed of the train=100km/h, speed of the car=80km/h
Q8.(i)4x-3y-8=0(may be another equation also)
(ii) 4x+6y+16 =0 (may be another equation also)
Q9. X= a+b, y= - 2ab/(a+b)
Q10. 3,5
31
QUADRATIC EQUATIONS
KEY POINTS
1. The general form of a quadratic equation is ax2+bx+c=0, a≠o.a,b and c are real numbers.
2. A real number is said to be a root of quadratic equation ax2+bx+c=0 where a≠0 if a2+b+c=0.The zeroes of the
quadratic polynomial ax 2+bx+c and the roots of the corresponding quadratic equation a x2 + b x + c = 0 are the same.
3. Discriminant:-The expression b2 - 4ac is called Discriminant of the equation ax2+bx+c = 0 and is usually denoted
by D.Thus Discriminant D = b2 - 4ac.
4. Every quadratic equation has at most two roots which may be real, coincident or no real roots.
Product of roots
9. Irrational roots always occur in conjugate pairs. If 2+√3 is one of the roots of the quadratic equation then other
root is2-√3.
10. If a.b>0 the na>0and b>0 or a<0 and b<0 If a.b<0 the na>0 and b<0 or a<0 and b>0.
LEVEL-I
1. If is a root of the equation x2 + kx - = 0, then find the value of K.
32
6. Find the values of K for which the .equation 9x2 + 2kx +1 = 0 have real roots.
7. Find the Value of K if the equation x2- 2(k+1)x + k2 = 0 has equal roots.
8. For what value of k, x = a is a solution of equation x2 -(a+b)x+k=0?
LEVEL -II
1. If the x = 2 and x = 3 are roots of the equation 3 +2 −2 = 0, find the value of k and m.
6. Find the roots of Quadratic equation 16x2–24x –1= 0 by using the quadratic formula.
7. Find the Discriminant of the Quadratic equation 2x2-4x+3=0and hence find the nature of its roots.
LEVEL – III
1. In a class test ,the sum of Shefali‘s marks in math’s and English is 30.Had she got2 marks more in math’s
and 3 marks less in English, the product of their marks would have been 210. Find her marks in the two
subjects.
2. A two digit number is such that the product of its digit is 35.When18 is added to the number, the
digits interchange the places. Find the number.
3. Solve 3x2-23x-110 = 0
4. Solve the following equation for ‘x’, 9x2-9(a+b)x+2a 2+5ab+2b2 = 0
5. If the roots of the equation (a-b)x 2+ (b-c)x + (c-a) = 0 are equal, prove that 2a = b+c.
Self-Evaluation
1. Find the value of ‘p’ so that the equation 3x 2 – 5x – 2p = 0has equal roots. Also find the roots.
2. The sum of two numbers is15.If the sum of their reciprocals is . Find the two numbers.
8. If the price of petrol is increased by Rs.2 per litre,a person had to buy1 litre less petrol for Rs.1740. Find
the Original price of the petrol at that time.
a) Why do you think the price of petrol is increasing day by day?
b) What should we do to save petrol?
9. Ramesh wants to design a rectangular park of perimeter 80m and area 400m 2for jogging and morning walk
for the people of his colony. Is it possible to design the park? If so find the length and breadth of the park.
Which value of Ramesh his depicted here?
Answer
LEVEL-I
1. 2
±
2. .
3. 5
4. -4
5. - 2
6. -3≤ ≤
7. ±
8. K=ab
9. x2+4x-60 =0
10. -2 and 5
11. 21,23
LEVEL-II
1. m = 9 and k=-15/2
2. 3/2 or -5
3. 3/2 or 1
4. q2/p2 or -1
5. 50
±√
6.
7. -8, and roots are not real.
LEVEL-III
34
Self-Evaluation
1. 25/24
2. (10,5)or(5,10)
3. x2-4x +1=0
4. 20 days.
5. (13,16) or (16,13)
6. (a+b)/3 and (a- b)/3
7. Rs 58 per litre
8. Yes,l=20mandb=20m.
35
ARITHMETIC PROGRESSION
KEY CONCEPT
An AP is a list of number in which difference of a term to its preceding term is always constant. The
constant is called common difference (d) of AP. d = an+1-an
If ’a’ is the first term and ‘d’ is the common difference of an AP, then the AP is a,a+d,a+2d,a+3d…..
The nth term of an AP is denoted by an
an = a + (n-1) d where a = first term and d = common difference n= number of term
nth term from the end = l - (n-1)d Where l = last term
Various terms in an AP can be chosen in following manner.
LEVEL- II
11. Find the middle term of A.P 6,13,20,………..216
12. The 6t hterm of an A.P is -10 and its 10thterm is -26.Determine the15thterm of an A.P
13. The 8thterm of an A.P is 0 prove that its38t h term is triple its18th term.
14. The sum of three numbers in A.P is 21and their product is 231 find the numbers.
15. Find the sum of 25th term of an AP which nth term is given by tn=(7-3n)
16. Find the sum of all two digit odd positive numbers
17. Find the sum of three digits numbers which are divisible by 11
18. The sum of first 6 term of A.P is 42.The ratio its 10th term to 30thterm is1:3.
Calculate the first and 13th term of the A.P
19. How many term of the A.P: 17,15,13,…must be added to get the sum 72?
Explain the double answer.
20. The sums of n,2n,3n term of an A.P are S1,S2,and S3 respectively.Prove that S3=3(S 2-S1) 36
37
ANSWER
LEVEL-I
1. 14
2. 17
3. 209
4. no
5. 5
6. 948
7. 23rdterm
8. 3
9. 5√2
10. 7/5
LEVEL- II
11. 111
12. -46
14. (-3,7,11)
13. -800
14. 2475
15. 44550
16. 2, 26
17. 6,12
LEVEL- III
21.
22. n=0
23. ..
24. 5 years
38
TRIANGLES
Key Points
Similar Figures: Two figures having similar shapes (size may or may not same), called Similar
figures.
Pairs of all regular polygons, containing equal number of sides are examples of Similar
Figures.
SimilarTriangles:Two Triangles are said to be similar if
(a) Their corresponding angles are equal (also called Equiangular Triangles)
(b) Ratio of their corresponding sides are equal/proportional
All congruent figures are similar but similar figures may/may not congruent
Conditions for similarity of two Triangles
(a) AAA criterion/A-A corollary
(b) SAS similarity criterion
(c) SSS similarity criterion(where ‘S’ stands for ratio of corresponding sides of two
Triangles)
39
LEVEL I
(1) In the figure XY ∥QR,PQ/XQ=7/3 and PR=6.3cm then find YR
(2) If ∆ABC ~ ∆DEF and their areas be 64cm 2 & 121cm2 respectively ,then find BC if EF=15.4cm
(3) ABC is an isosceles ∆,right angled at C then prove that AB 2=2AC 2
(4) If ∆ABC~∆DEF,∠ A=46 0,∠ E = 620 then the measure of ∠ C = 720 .Is it true? Give reason.
(5) Theratioofthecorrespondingsidesoftwosimilartrianglesis16:25thenfindtheratioof their
perimeters.
(6) A man goes 24km in due east and then He goes 10km in due north. How far is he from the
starting point?
(7) The length of the diagonal so far hombus is 16cm & 12cm respectively then find the perimeter of the
rhombus.
(8) In the figure LM||CB and LN||CD then prove that AM/AB=AN/AD
(9) Whichoneisthesidesofarightangledtrianglesamongthefollowing(a)6cm,8cm&11cm
(b) 3cm,4cm & 6cm (c) 5cm , 12cm & 13cm
Level II
(1) InthefigureABDisatrianglerightangledatAandACisperpendiculartoBDthenshow that AC2=BCxDC
(2) Twopolesofheight10m&15mstandverticallyonaplaneground.Ifthedistancebetween
theirfeetis5√3mthenfindthedistancebetweentheirtops. 40
(3) D &E are the points on the sides AB & AC of ∆ABC ,as shown in the figure .If∠B=∠AED then show
that ∆ABC~∆AED
7) that∠A=∠C
8) Using converse of B.P.T. Prove that the line joining the mid points of any two sides of a triangle is
parallel to the third side of the triangle.
41
9) In the given figure ∆ABC & ∆DBC are on the same base BC .If AD intersect BC at O then prove
that ar(∆ABC)/ar(∆DBC)=AO/DO
Level III
(1) A point O is in the interior of a rectangle ABCD, is joined with each of the vertices A,B,C&D.
Prove that OA2 +OC 2 = OB2+OD2
(2) In an equilateral triangle ABCD is a point on the base BC such that BD=(1/3)BC ,then show that
9AD2=7AB2
(3) Prove that in a rhombus, sum of squares of the sides is equal to the sum of the squares of its
diagonals
(4) In the adjoining figure ABCD is a parallelogram. Through the mid -point M of the side CD ,a line is
drawn which cuts diagonal AC at Land AD produced at E. Prove that EL=2BL
(5) ABC & DBC are two triangle so the same base BC and on the same side of BC with
∠A=∠D=900.If CA &BD meet each other at E then show that AE x EC = BE x ED
(6) ABC is a triangle, right angle at C and p is the length of the perpendicular drawn from C to AB. By
expressing the area of the triangle in two ways show that (i) pc = ab (ii) 1 / p2 = 1 / a2 + 1 / b 2
42
(7) Prove that the ratio of the area so two similar triangles is equal to the ratio of their
corresponding sides.
(1) Find the value of x for which DE||BC in the adjoining figure
(2) In an equilateral triangle prove that three times the square of one side is equal to four times the
square of one of its altitude.
(3) The perpendicular from A on the side BC of a triangle ABC intersect BC at D such that DB=3CD.
Prove that 2AB2= 2AC2+ BC2
(4) In the adjoining figure P is the mid point of BC and Q is the mid point of AP .If BQ when
produced meets AC at R ,then prove that RA=(1/3)CA
(5) BL and CM are medians of triangle ABC, right angled at A then prove that 4(BL2 + CM2) = 5BC2
(6) In ∆ABC if AB = 6√3cm , AC = 12cm and BC = 6cm then show that ∠B=900
43
(7) In the adjoining figure QRP=90 0, PMR=900,QR = 26cm ,PM=8cm and MR=6cm then find the area of
∆PQR
(8) If the ratio of the corresponding sides of two similar triangles is 2:3 then find the ratio of
their corresponding altitudes.
(9) In the adjoining figure ABC is a ∆ right angle d at C. P & Q are the points on the sides CA &CB
respectively which divides these sides in the ratio2:1,then prove that 9(AQ 2+BP2)=13 AB2
(10) The adjoining figure AB || PQ || CD,AB =x unit, CD =y unit & PQ = z unit then prove that 1/x + 1/y = 1/z
(11) State and prove Pythagoras theorem .Using this theorem find the distance between the tops of two vertical
poles of height 12m &18m respectively fixed at a distance of 8m apart from each other.
(12) In the adjoining figure DEFG is a square &∠BAC=900 then prove that
(a) ∆AGF ~ ∆DBG (b) ∆ AGF ~ ∆EFC ( c) ∆ DBG ~∆EFC (d) DE 2 = BD XEC
44
45
25
45
46
48
49
50
51
52
LEVEL - I
6. Find the ratio in which the y-axis divides the segment joining (-3,6) and(12,-3).
7. Find the coordinates of a point A, where AB is diameter of a circle whose Centre is(2,-3)and B is(1,4)
8. Find the centroid of triangle whose vertices are (3,-7), (-8,6) and (5,10).
LEVEL - II
1. If A (-2,4),B(0,0),C(4,2) are the vertices of a ∆ABC, then find the length of median through the vertex A.
2. Find the value of x for which the distance between the points P(4,-5) and Q (12,x) is10 units.
3. If the points A (4,3) and B(x,5) are on the circle with Centre O(2,3) then find the value of x.
SELF EVALUATION
1. Find the center of a circle passing through the points (6,-6), (3,-7)and(3,3).
2. If the distance between the points (3,0) and (0,y)is 5 units and y is positive ,what is the value of y?
3. If the points(x ,y),(-5,-2)and(3,-5)are collinear r, then prove that 3x+8y+31=0.
4. Find the ratio in which the Y-axis divides the line segment joining the points (5,-6) and (-1,-4). Also find
the coordinates of the point of division.
5. By distance formula, show that the points (1,-1), (5,2) and (9,5) are collinear.
6. Show that the three points (a,a) ,(-a,-a) &(-a√3 , a√3) are the vertices of an equilateral triangle.
Board Questions
Q: 1) Find the value of k, if the point P (2, 4) is equidistant from the points (5, k) and (k, 7).
(CBSE: 2012)
Q:2)If the point A(0,2)is equidistant from the points B (3,p) and C(p,5),find p .Also find the length of AB.
(CBSE: 2014)
Q:3)Find the ratio in which the point P (x,2) divides the line-segments joining the points A(12,5)and B(4,-
3).Also ,find the value of x.
(CBSE: 2014)
Q:4)If the points A (-2, 1),B (a, b) and C(4,-1) are collinear and a-b=1.Find the value of a and b.
(CBSE: 2014)
Q: 5) In what ratio does the point (-4, 6) divides the line segment joining the points A (-6, 10) &B (3,-8)
(CBSE: 2012)
54
ANSWERKEY
LEVEL-I
1. 5
2. -12
3. (0,-10) and(4,0)
4. ±4
5. 2a=b
6. ¼
7. (3,-10)
8. (0,3)
LEVEL-II
1. 5 units
2. 1, -11
3. 2
4. √2 c
5. -1
6. Proof
7. K=-8
8. 4x - 5y - 3=0
LEVEL-III
1. 2:3
2. Proof
3. 1:4
4. a=2/7
5. B(-5,-2)
6. Proof
7. p=1
8. 5
SELFEVALUATION
1. (3,-2)
2. 4
3. Proof
4. 5:1, (0,-13/3)
5. Proof
6. Proof
BOARDQUESTIONS
1.K=3
2. P=1, AB=√10
3. 3:5, x=9
4. a=1, b=0
5.2/7
55
56
57
4. Trigonometric Identity----An equation involving trigonometric ratios of an angle is called a
trigonometric identity if it is true for all values of the angle.
Important trigonometric identities:
(i) sin2+ cos2= 1
(ii) 1+ tan2= sec 2
(iii) 1 + cot2= cosec2
5. Trigonometric Ratios of some specific angles
1. If θ and 3θ-30° are acute angles such that sin θ = cos (3θ-30°) , then find the value of tan θ. (1/ 3)
° °
2. Find the value of ( 3 /2)
° °
8. In a triangle ABC, it is given that ∠C=90˚and tan A=1/√3,find the value of (sinAcosB+cosAsinB)
(1) (CBSE2008)
2
9. Find the value of cosec 67˚- tan2 23˚. (1)
10. If cos x = cos60°cos30°+sin60°sin30°,then find the value of x (300)
59
(Ans. Of Question no. 1 :12/25)
4. Prove that .+ =2 cosec A
5. Prove that (sinθ + cosecθ)²+(cosθ+secθ)²=7+tan²θ+cot²θ. (CBSE 2008,2009C)
6. Evaluate: -- _
Ans: (1)
10. If cos θ+ sin θ = 1 and sinθ – cos θ=1, prove that x²/a²+y²/b²=2.
---
11. Prove that =tan θ.
Level - III
60
10. Prove that (1+cotθ-cosecθ)(1+tanθ+secθ)=2 (CBSE2005,07,08)
11. Evaluate: -
Ans: -25/6
2
12. If sinθ+cosθ = m and secθ+cosecθ=n, then prove that n (m –1) =2m.
Self-Evaluation
4. If tan A=n tan B, sin A=m sin B, prove that cos²A= (m²-1)/(n²-1).
seccosec(90°-)-tancot(90°-)
5. Prove + sin²55°+ sin²35° =2
that tan10°tan20°tan45°tan70°tan80°
7. Prove that - = - .
61
SOMEAPPLICATIONS OF TRIGONOMETRY
KEY POINTS
Line of sight
Line segment joining the object to the eye
of the observer is called the line of sight.
Angle of elevation
When an observer sees an object situated in
upward direction, the angle formed by line of
sight with horizontal line is called angle of
elevation.
Angle of depression
When an observer sees an object situated in
downward direction the angle formed by line of
sight with horizontal line is called angle of
depression.
62
Level – I (These questions can be taken as MCQ/FILL UP/TRUE AND FALSEi.e.1 mark each)
1. A pole 3m high casts a shadow m long on the ground, then find the sun’s elevation?
2. If length of the shadow and height of a tower are in the ratio1:1.Then find the angle of elevation.
3. An observer 1.5m tall is 20. 5 metres away from a tower 22m high. Determine the angle of
elevation of the top of the tower from the eye of the observer.
4. A ladder 15m long just reaches the top of vertical wall. If the ladder makes an angle 60 0 with the wall,
find the height of the wall.
5. In a rectangle ABCD, AB=20cmBAC= 60 0 then find the length of the side AD.
6. Find the angle of elevation of the sun’s altitude when the height of the shadow of a vertical pole is
equal to its height:
7. From a point 20 m away from the foot of a tower, the angle of elevation of top of the tower is 30°, find
the height of the tower.
8. In the adjacent figure, what are the angles of elevation and depression of the top and bottom of
a pole from the top of a tower h (m) high: Q O
0
Ans 45 , 60 0
0
30
450
A M
B L
LEVEL –II
9. The length of the shadow of a pillar is times its height. Find the angle of elevation of the
source of light.
10. A vertical pole10m long casts a shadow m long. At the same time tower casts a shadow 90m
long. Determine the height of the tower.
11. A ladder 50m long just reaches the top of a vertical wall. If the ladder makes an angle of 60 0 with the
wall, find the height of the wall.
12. Two poles of height 6m and 11m stands vertically on the ground. If the distance between their feet
is12m. Find the distance between their tops.
13. The shadow of tower, when the angle of elevation of the sun is 45 o is found to be 10m longer than
when it is 60 o.Find the height of the tower.
63
LEVEL –III
14. The angle of depression of the top and bottom of a tower as seen from the top of a100 m high cliff are
300 and 600 respectively. Find the height of the tower.
15. From a window (9m above ground) of a house in a street, the angles of elevation and depression of the
top and foot of another house on the opposite side of the street are 30 0and600 respectively.
Find the height of the opposite house and width of the street.
16. From the top of a hill, the angle of depression of two consecutive kilometer stones due east are found to
be 300 and45 0.Find the height of the hill.
17. Two poles of equal heights are standing opposite each other on either side of the road, which is 80m
wide. From a point between the road the angles of elevation of the top of the poles are 600and 300.
Find the heights of pole and the distance of the point from the poles.
18. The angle of elevation of a jetfighter from a point A on the ground is 60 0.After a flight of 15
seconds, the angle of elevation changes to 300. If the jet is flying at a speed of 720km/hr, find the
constant height at which the jet is flying.
19. A window in a building is at a height of 10 m above the ground. The angle of depression of a point P on
the ground from the window is 300.The angle of elevation of the top of the building from the point P is
600. Find the height of the building.
20. A boy, whose eye level is 1.3 m from the ground, spots a balloon moving with the wind in a
horizontal line at same height from the ground. The angle of elevation of the balloon from the eyes
of the boy at any instant is 600.After 2 seconds, the angle of elevation reduces to 300 if the speed of
the wind at that moment is m/ then find the height of the balloon from the ground.
21. A man on the deck on a ship14m above water level observes that the angle of elevation of the top of
a cliff is 600 and the angle of depression of the base of the cliff is 300.Calculate the distance of the
cliff from the ship and the height of the cliff.
22. A tower is 50m high. It’s shadow is x m shorter when the sun’s altitude is 45o than when it is 30o. Find
x correct to the nearest 10.
SELF EVALUATION / HOTS
23. An airplane when flying at a height of 3125m from the ground passes vertically below another Plane at
an instant when the angle of elevation of the two planes from the same point on the ground are 30°
and 60° respectively. Find the distance between the two planes at that instant.
24. From the top of a building 60m high, the angels of depression of the top and bottom of a vertical lamp
post are observed to be 30° and 60°respectively. Find
[i] horizontal distance between the building and the lamp post
[ii] height of the lamp post.
25. A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height
‘h’ m. At a point on the plane, the angles of elevation of the bottom and the top of the flag staff are
64
α and β respectively. Prove that the height of the tower is (h tan α) / (tanβ -tanα).
26. The angle of elevation of a cloud from a point 60m above a lake is 30° and the angle of depression of the
reflection of the cloud in the lake is 60°. find the height of the cloud from the surface of the lake.
27. A round balloon of radius ‘r’ subtends an angle α at the eye of the observer whose angle of elevation of
centre is β. Prove that the height of the centre of the balloon is (rsinβcosecα/2).
28. A person standing on the bank of a river observes that the angle of elevation of top of building of an
organization working for conservation of wildlife. Standing on the opposite bank is 60°.When he moves
40 m away from the bank he finds the angle of elevation to be 30°.Find the height of the building and
width of the river.
ANSWERS
Level 1:-
1)600 (2) 450 (3) 450 (4) 7.5 3 m (5) 20 3 m (6) 450 (7) 20 / √3 m (8) 45 0,60 0
Level 2:-
(9)30 0 (11) 25√3 m (12) 13 m (13) 5(√3 + 1) m
(10)30√3 m
Level 3:-
(14) 66.67m (15) 3m ,3 3 m (16)1.37km (17) 20 3 m, 20m and 16m (18) 500 3 m (19) 30m
(20) 87m (21)14 3 m and 56m (22) 36.6m
Self Evaluation
(23) 6250m (24) 20 3, 40m (26) 120m (28) 20 3 , 20m
65
CIRCLES
1. Circle: A circle is a collection of all points in a plane which are at a constant distance
(radius) from a fixed point (centre).
2. Secant & Tangent to a Circle: In fig.1 the line PQ and the circle have no common
point. Line PQ is called non-intersecting.Infig.2 line PQ a secant to a circle.Infig.3, there is
only 1 point A, which is common to the line PQ and the circle .The line is called a
tangent to the circle.
1
2 3
3. Tangent to a Circle:
It is a line that intersects the circle at only one point. There is only one tangent at a point
of the circle. The tangent to a circle is a special case of the secant, when the two end
points of its corresponding chord coincide.
4. Theorems:
1. The tangent at any point of a circle is perpendicular to the radius through the point
of contact.
2. The lengths of tangents drawn from an external point to a circle are equal.
66
5. Number of tangents from a point on a circle-
(i) There is no tangent to a circle passing through a point lying inside the circle.
(ii) There is one and only one tangent to a circle passing through a point lying on the circle.
(iii)There are exactly two tangent to a circle through a point lying outside the circle.
LEVEL I
(These questions can be taken in the form of MCQ/FILL UP/TRUE AND FALSE i.e.1 mark
each)
1. In the given fig. O is the centre of the circle and PQ is tangent then POQ + QPO is equal to
3. In the given fig. PQ and PR are tangents to the circle, QOP= 70°, then QPR is equal to
67
4. In the given fig. QS is a tangent to the circle, OS = 8 cm, OQ = 6 cm then the length of QS is
5. In the given fig PQ is tangent to outer circle and PR is tangent to inner circle. If PQ = 4 cm, OQ = 3 cm
and OR = 2 cm then the length of PR is
6. In the given fig. P, Q and R are the points of contact. If AB = 4 cm, BP = 2 cm then the perimeter of
ABC is
7. The distance between two tangent parallel to each other to a circle is 12cm. The radius of circle is
8. The chord of a circle of radius 10cm subtends a right angle at its centre. Find the length of the chord.
9. How many tangents can a circle have?
68
10. How many tangents can be drawn from a given exterior point to a circle?
LEVEL - II
11. Two concentric circles of radii a & b (a>b) are given. Find the length of the chord of the larger circle
which touches the smaller circle
12. From a point P outside the circle with centre O, tangents PA and PB are drawn to the circle. Prove that
OP is the right bisector of the line segment AB.
13. A circles is inscribed in a triangle ABC, touching BC, CA and AB at P, Q and R respectively. If AB= 10
cm AQ =7cm CQ =5 cm. Find BC.
14. A Quadrilateral ABCD is drawn to circumscribe a circle, as shown in the figure. Prove that AB+CD
= AD + BC
15. Two concentric circles are of radii 7 cm and r cm respectively, where r>7. A chord of the larger circle
of length 46 cm, touches the smaller circle. Find the value of ‘r’.
16. Prove that the tangent at any point of a circle is perpendicular to the radius through the point of
contact.
69
LEVEL - III
17. Prove that the length of tangents drawn from an external point to a circle are equal.
18. Prove that the tangents at the extremities of any chord of a circle, make equal angle with the chord.
19. PA and PB are tangents to the circle with the centre O from an external point P, touching the circle at A
and B respectively. Show that the quadrilateral AOBP is cyclic.
20. Prove that the parallelogram circumscribing a circle is a rhombus.
21. In the given figure, XY and X’Y’ are two parallel tangents to a circle with centre O and
another tangent AB with point of contact C intersects XY at A and X’Y’ at B. Prove that
AOB=900.
Q.22Two roads starting from P are touching a circular path at A and B. Sarita runs
from P to A, 20km and A to O, 15km and Reeta runs from P to O directly.
(a) Find the distance covered by Reeta.
(b) Who will win the race?
SELF EVALUATION
1. Draw a circle and two lines parallel to a given line such that one is a tangent and the
other, a secant to the circle.
2. Prove that perpendicular at the point of contact to the tangent to a circle passes
through the centre.
70
3. Prove that the angle between the two tangents drawn from an external point to a
circle is supplementary to the angle subtended by the lines segment joining the
points of contact at the centre.
4. The length of a tangent from a point A at a distance 5cm from the centre of the circle
is 4cm.Find the radius of the circle. Ans:12cm
5. Two concentric circles are of radii 6.5cm and 2.5cm.Find the length of the chord of
larger circle which touches the smaller circle. Ans:3cm
6. From a point P, 10cm away from the centre of the circle, a tangent PT of length 8cm
is drawn. Find the radius of the circle. Ans:6cm
71
72
Level III
Quad. OAPB,
∠AOB+ ∠OAP+∠ APB+∠ OBP=3600Or,
∠AOB+900+∠APB+900=360 0
Or, ∠AOB+ ∠APB+1800=3600Or,
∠AOB+ ∠APB=1800
Hence, quad. OAPB is cyclic.
20.
73
AP=AS …………….(i) [Tangents from A]
BP=BQ…………..(ii)[Tangents from B]
CR=CQ……………(iii)[Tangents from c]
DR=DS……………(iv)[Tangents from D]
Now, AB+CB=AP+BP+CR+DR
= AS + BQ + CQ + DS [From (i), (ii), (iii), (iv)]
= (AS + DS ) + (BQ + CQ)
=AD+BC
Or, AB+CD=AD+BC Or,
2AB =2AD
Or, AB = AD
Hence, AB = BC = CD = AD
Hence, ABCD is a rhombus.
21. In quad. APQB
∠APO+ ∠BQO+ ∠QBC+ ∠PAC=3600Or,90
0
+900+∠QBC+ ∠PAC=3600
Or, ∠QBC+ ∠PAC=180 0……………………………(i) We
have,∠CAO=½∠ PAC
And ∠ CBO = ½ ∠ QBC
Now, ∠ CAO + ∠ CBO = ½ ( ∠ PAC + ∠ QBC )
= ½ X 1800 (from eq. i)
=900……………………………… (ii)
In triangle AOB,
∠ CAO + ∠AOB + ∠ CBO = 180 0
Or, ∠ AOB + 90 0 = 1800 (from eq. ii)
74
Or, ∠AOB=9
0022.(i)
In triangle OAP,
OP2=OA2+A
P2(By
Pythagoras
Theorem)
Or,OP2=(15
)2+(20)2
Or, OP 2=
625 Or,
OP=25km
(ii) Dis
tance
covered by
Reeta = 25
km Distance
covered by
Sarita=20km
+15km
= 35 km
So, Rita will
win the
race.
75
CONSTRUCTIONS
1. Division of a line segment in the given ratio.
2. Construction of triangles:-
76
LEVEL – I
1. Draw a line segment AB=8cm and divide it in the ratio 4:3.
3. Draw a circle of radius 4cm.Take a point P on it. Draw tangent to the given circle at P.
LEVEL –II
1. Construct a triangle of sides 4cm, 5cm and 6cm and then construct a triangle similar to it whose sides are 2/3 of the
corresponding sides of the first triangle. (CBSE2013)
2. Construct a triangle similar to a given ∆ABC such that each of its sides is 2/3rd of the corresponding sides of
∆ABC. It is given that AB=5cm BC=6cm and AC=7cm. Also write the steps of construction.
3. Draw a pair of tangents to a circle of radius 4cm, which are inclined to each other at an angle of 600.
(CBSE2013)
4. Draw a circle of radius 5cm. From a point 8cm away from its centre construct the pair of tangents to the circle and
measure their lengths.
6. Draw a line segment AB=7.5cm and locate a point P on AB such that AP=3/7AB.Give justification of the
construction.
77
LEVEL-III
1. Draw a circle with centre O and radius 3.5cm. Take a horizontal diameter. Extend it to both sides to point P and Q
such that OP=OQ=7cm.Draw tangents PA and QB, one above the diameter and the other below the diameter. Is
PA||BQ.
2. Construct a ∆ABC in which AB=6cm, A=30°and B=60°.Construct another ∆AB’C ’similar to ∆ABC with base
AB’=8cm. (CBSE2015)
3. Draw a right triangle ABC in whichB=900,AB=5cm,BC=4cm,then construct another triangle A’BC’ whose sides are 5/3
times the corresponding sides of ∆ABC.Is the new triangle also a right triangle?
4. Draw a line segment AB of length 8cm.Taking A as centre, draw a circle of radius 4cm and taking B as centre,
draw another circle of radius 3cm.Construct tangents to each circle from the centre of the other circle.
5. Draw a line segment AB of length 7cm.Using ruler and compasses, find a point P on AB such that AP/AB= 3/5.
(CBSE2011)
6. Construct an isosceles triangle whose base is 8cm.and altitude 4cm. and then construct another triangle
Whose sides are ¾ times the corresponding sides of the isosceles triangle. (CBSE 2011)
7. ABC is a right triangle in which AB=5.4cm, BC=7cm and ∠B=900.Draw BD perpendicular on AC and a circle
through B, C ,D. Construct a pair of tangents from A to this circle.
8. Construct a triangle ABC in which AB=5cm, ∠B=600 and altitude CD=3cm.Construct a triangle PQR similar to
∆ABC such that each side of ∆AQR is 1.5 times that of the corresponding sides of ∆ABC.
9. Construct a tangent to a circle of radius 3.5 from a point on the concentric circle of radius6.5cmand measure
its length. Also, verify the measurement by actual calculation.
78
Self-Evaluation
1. Draw a line segment of length 7cm. Find a point P on it which divides it in the ratio 3:5.
2. Draw an isosceles triangle ABC in which AB=AC=6cm and BC=5cm.Construct a triangle PQR similar to ∆ABC in
which PQ=8cm. Also justify the construction.
3. Two line segments AB and AC include an angle of 60 0 where AB=5cmand AC=7CM.Locate points P and Q on AB and
AC respectively such that AP=3/4AB and AQ=1/4AC.Join P and Q and measure the length PQ.
4. Draw a triangle ABC in which AB=4cm, BC=6cm and AC=9cm.Construct a triangle similar to ∆ABC with scale factor
3/2.Justify your construction.
5. Draw a pair of tangents to a circle of radius 4.5cm, which are inclined to each other at an angle of 450.
6. Draw a line segment AB of length 7cm.Taking A as centre, draw a circle of radius 3cm and taking B as centre
another circle of radius 2.5cm.Construct tangents to each circle from the centre of the other circle.
7. Two trees are to be planted at two positions A and B in the middle of a park and the third tree is to be planted at
a position C in such a way that AC: BC=3:4.How it can be done?
8. Draw a circle of radius 5cm.Draw tangents from the end points of its diameter. What do you observe?
79
AREAS RELATED TO CIRCLES
KEY POINTS
1. Circle: The set of points which are at a constant distance from a fixed point in a plane is called a circle.
R
r
o
6.Semi-Circle:-A diameter divides a circle in to two equal arcs. Each of these two arcs is called a semi- circle.
7.Segment :- A segment of a circle is the region bounded by an arc and a chord, of a circle.
8.Sector of a circle: The region enclosed by an arc of a circle and its two bounding radii is called a sector of the
circle.
9. Quadrant :- One fourth of a circle/circular disc is called a quadrant. The central angle of a quadrant is 90 0.
2 r or
1/2 r²
+2
Semi- circle
2.
L+2r= r+2r r²
4. Sector of a circle
5.
Area of Segment r+2r sin r²- r²
of a circle
sin
80
a. Length of an arc AB= 2
LEVEL-I
1. If the perimeter of a circle is equal to that of square, then the ratio of their areas is
i. 22/7
ii. 14/11
iii. 7/22
iv. 11/14
2. The area of the square that can be inscribed in a circle of radius 8cm is
i. 256cm2
ii. 128cm2
iii. 64√2cm2
iv. 64cm2
3. Area of sector of a circle of radius 36cm is 54 cm2.Find the length of the corresponding arc of the
circle is
i. 6 cm
ii. 3 cm
iii. 5 cm
iv. 8 cm
81
4. A wheel has diameter 84cm.The number of complete revolution it will take to cover 792m is.
i. 100
ii. 150
iii. 200
iv. 300
5. The length of an arc of a circle with radius 12cm is10 cm. The central angle of this arc is.
i. 1200
ii. 60
iii. 750
iv. 1500
7. In figure ‘o’ is the centre of a circle. The area of sector OAB is 5/18 of the area of the circle find x.
LEVEL – II
1. Find the area of the shaded region in the figure if AC=24cm,BC=10cm and 'o' is the center of the circle
(use = 3.14)
B C
2. The inner circumference of a circular track is 440m.The track is 14m wide. Find the diameter of the outer circle of
the track. [Take =22/7]
82
3. Find the area of the shaded region.
4. A copper wire when bent in the form of a square encloses an area of 121cm2.If the same wire is bent in to the form
of a circle, find the area of the circle(use =22/7)
5. A wire is looped in the form of a circle of radius 28cm.It is rebent in to a square form. Determine the side of the
square(use =22/7)
LEVEL-III
1. Three horses are tethered with 7m long ropes at the three corners of a triangular field having sides 20m, 34m
42m.Find the area of the plot.
i. Grazed by horses
ii. Remains un grazed by horses
2. Calculate the area of shaded region in given figure where ABCD is square of side16cm.
3. ABC is a quadrant of circle of radius14cm and a semi-circle is drawn with BC as diameter. Find the area of
Shaded region.
83
4. The length of a minor arc is 2/9 of the circumference of the circle. Write the measure of the angle
subtended by the arc at the centre of the circle.
5. The area of an equilateral triangle is 49√3cm2. Taking each angular point as centre, circle is drawn with radius
equal to half the length of the side of the triangle. Find the area of triangle not included in the circles.
[Take √3=1.73]
SELF EVALUATION
1. Two circles touch externally the sum of the areas is 130 cm2 and distance between there centre is14cm. Find
the radius of circle.
2. Two circle touch internally. The sum of their areas is116π cm²distance between their centres is 6cm.
Find the radii of circles.
3. A pendulum swings through an angle of 300 and describes an arc 8.8cm in length. Find length of
pendulum.
4. The side of a square is10cm find the area of circumscribed and inscribed the circle.
5. An Umbrella has 8 ribs which are equally spaced. Assume Umbrella to be flat circle of radius 45cm find the
area between two consecutive ribs of umbrella.(use π =3.14)
6. A child prepare a poster on “save energy” on a square sheet whose each side measure 60cm. at each corner
of the sheet, she draw a quadrant of radius17.5cm in which she shows the ways to save energy at the centre.
She draws a circle of diameter 21cm and writes a slogan in it. Find the area of remaining sheet.(use
=3.14)
7. A birthday cake is circular in shape. This cake is equally divided among six friends where radius of the cake is
60cm.(use =3.14)
FInd the area of each piece of cake.
. ANSWER
LEVEL-I
1. (ii). 14/11
2. (ii). 128 cm2
3. (ii) 3π cm
4. (iv) 300
5. (iv) 1500
6. (ii)π/4
7. 1000
8. 36 cm
9. 3000
10. 154/3 cm2
84
LEVEL- II
2
1. 145.33 cm
2. D= 160 m
3. 4.71 cm2
4. 154 cm2
5. 44 cm
LEVEL- III
2
1. (i) 77 m
(ii) 259 m2
2. 109.7 cm2
3. 98 cm2
4. 800
5. 7.77 cm2
SELFEVALUATION
1. 11 cm and 3
cm
2. 4 cm and 10
cm
3. 16.8 cm
4. 50 π cm2, 25
π cm2
5. 794.81 cm2
85
SURFACE AREA AND VOLUMES
KEY CONCEPTS
1. CUBOID:
(I) TOTAL SURFACE AREA OF A CUBOID:2(LB+ BH +HL)
(II) Volume of a cuboid=L x B x H sq .units
(III) Diagonal of cuboid= l2 b2h2 units
CUBE :
(IV) Total Surface Area of a Cube = 6a2 sq. units
(V) Volume of the Cube=a3 cubic units
(VI) Diagonal of cube is= 3 a units
2. Right Circular Cylinder:
(I) Curved Surface Area=2π rh
(II) Total Surface Area=2 r ( h +r )
(III) Volume = r2h
4. Sphere:
(I) Surface Area =4 r2
4
(II) Volume= r³
3
5. Hemisphere:
(I) Curved Surface Area=2 r2
(II) Total Surface Area=3 r2
2
(III) Volume= r3
3
(III) Total Surface Area of the Frustum of the cone= [R 2+r2+l (R +r )] sq. units
LEVEL-I
1. The Surface Area of a Sphere is 616cm2. Find its radius.
2. The slant height of the frustum of a cone is 5cm. if the difference between the radii of its two circular ends is 4cm,
write height of the frustum.
3. A cylinder and a cone are of the same base radius and of the same height. Find the ratio of the curved surface area
of the cylinder to that of cone.
4. Two cones have their heights in the ratio1:3 and radii 3:1. What is the ratio of their volumes?
5. The radii of two cones are in the ratio 2:1 and their volumes are equal. What is the ratio their heights?
6. The diameter of a sphere is 6cm. it is melted and drawn into a wire of diameter 2mm. Find the length of the wire.
7. Find the curved surface area of a right circular cone of height 15 cm and base diameter is16cm.
8. Find the maximum volume of a cone that can be out of a solid hemisphere of radius r.
9. The diameters of the ends of a frustum of a cone are32cm and 20cm. If its slant height is 10cm. Find the
lateral surface area.
LEVEL-II
1. Metallic sphere of radii 6cm, 8cm and10cm respectively, are melted to form a single solid sphere. Find the radius of
the resulting sphere.
2. A 20m deep well with diameter7misdugandtheearth from digging evenly spread out to form a platform 22m
by14m. Find the height of the platform.
3. Two cubes of volume 64cm3 are joined end to end. Find the volume of the sphere.
4. The largest sphere is curved out of a cube of a side 7cm. Find the volume of the sphere.
5. A circus tent is cylindrical up to a height of 3m and conical above it. If the diameter of the base is 105 m and the
slant height of the conical part is 53m. Find the total canvas used in making the tent.
6. A vessel is in the form of a hemispherical bowl mounted by a hollow cylinder. The diameter of the sphere is 14cm
and the total height of the vessel is13cm.Find its capacity?
7. A solid toy is in the form of a right circular cylinder with a hemispherical shape at one end and a cone at the
other end. Their common diameter is 4.2cm and the height of the cylindrical and conical position is12cm and
7cm respectively. Find the volume of the solid toy.
87
8. A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to 1cm and the
height of the cone is equal to its radius. Find the volume of the solid in terms of
Level-III
1. A hemispherical depression is cut from one face of the cubical wooden block such that the diameter l of the
hemisphere is equal to the edge of the cube. Determine the surface area of the remaining solid.
2. A juice seller was serving his customers using glasses. The inner diameter of the cylindrical glass was 5cm, but
the bottom of the glass had a hemispherical raised portion which reduced the capacity of the glass. If the
height of glass was 10cm, find what the apparent capacity of the glass was and what the actual capacity
was.(use = 3.14)
3. The height of a cone is 30cm. A small cone is cut off at the top by a plane parallel to the base of its
volume be 1/27 of the volume of the given cone, at what height above the base is the section made?
4. An oil funnel of tin sheet consists of a cylindrical portion 10cm long attached to a frustum of a cone. If the
total height be 22cm, diameter of the cylindrical portion be 8cm and the diameter of the top of the funnel
be18cm. Find the area of the tin required to make the funnel.
5. A solid wooden toy is in the shape of a right circular cone mounted on a hemisphere. If the radius of the
hemisphere is 4.2cm and the total height of the toy is 10.2cm. Find the volume of the wooden toy.
SELF-EVALUATION
1. A tent is of the shape of a right circular cylinder up to a height of 3m and then becomes a right circular cone with a
maximum height of 13.5m, above the ground. Calculate the cost of painting the inner side of the tent at the
rate of Rs.2 per sq. metre, if the radius of the edge is14 metres.
2. A bucket is in the form of frustum of a cone and holds 28.490 litres of water. The radii of the top and bottom are
28cm and 21cm respectively. Find the height of the bucket.
3. The perimeters of the ends of a frustum is48cm and 36cm.If the height of the frustum be 11cm, find its
volume.
4. If the radii of the circular ends of a conical bucket which is 45cm high are 28cm and 7cm. Find the capacity of the
bucket.
5. A pen stand made of wood is in the shape of a cuboid with four conical depression’s to hold pens. The
dimensions of the cuboid are 15cm by 10cm by 3.5cm. The diameter of each of the depression is 1cm and the
depth is 1.4cm. Find the volume of the wood in the entire stand.
6. Three cubes each of side 5cm are joined end to end. Find the surface area of the resulting cuboid.
7. The diameter of a metallic sphere is 6cm. The sphere is melted and drawn into a wire of uniform cross-
section. If the length of the wire is 36m. Find its radius.
8. If the diameter of cross-section of a wire is decreased by 5%. How much percent will the length be increased so
that the volume remains the same?
88
ANSWER
LEVEL-I
1. 7 cm
2. 3cm
3. √2:1
4. 3:1
5. 1:4
6. 3600cm or 36m
7. 427.04 cm²
8. 1/3 r³
9. 816.4cm²
LEVEL- II
1. 12cm
2. 2.5 m
3. 128cm³
4. 179.67cm³
5. 9735m²
6. 1642.67cm³
7. 218.064cm³
8.
LEVEL- III
1. ² (24+ )
2. 163.54cm³
3. 20 cm
4. 249
5. 266.12cm³
SELF EVALUATION
1. Total area 1034m²,
Cost of painting Rs.2068
2. Height 15cm
3. 1554cm³
4. 48510cm3
5. 523.53 cm3
6. 350 cm2
7. 1mm
8. 10.8%
89
STATISTICS
Mode=l+ ×ℎ
Where
l=lower limit of modal class h=size of
class interval
f1 = Frequency of the modal class
fo =frequency of class preceding the modal class
f2=frequency of class succeeding the modal class
Ogive
Ogive is the graphical representation of the cumulative frequency distribution. It is of two types:
(i) Less than type ogive.
(ii) More than type ogive
Median by graphical method
The x-coordinated of the point of intersection of ‘less than ogive’ and‘ more than ogive’ gives the
median.
90
STATISTICS
Mean= =a+
Where a = assumed mean h = class size
And ui= (Xi – a)/h
Mode=l+
Where
l=lower limit of modal class
h=size of class interval
f1 = Frequency of the modal class
fo=frequency of class preceding the modal class
f2=frequency of class succeeding the modal class
Ogive
Ogive is the graphical representation of the cumulative frequency distribution. It is of two types:
(i) Less than type ogive.
(ii) More than type ogive
Median by graphical method
The x-coordinated of the point of intersection of ‘less than ogive’ and more than ogive’ gives the
median.
91
LEVEL – I
S. No Questions
1 What is the mean of first ten prime numbers?
2 What measure of central tendency is represented by the abscissa of the point where less than ogive
and more than ogive intersect?
3 If the mode of a data is 45 and mean is 27, then median is .
4 Find the mode of the following
Xi 35 38 40 42 44
fi 5 9 10 7 2
5 Write the median class of the following distribution.
Class 0-10 10-20 20-30 30-40 40-50 50-60 60-70
Frequency 4 4 8 10 12 8 4
6 The wickets taken by a bowler in10 cricket matches are as follows:2,6,4,5,0,2,1,3,2,3 Find the
mode of the data
7. How one can find median of a frequency distribution graphically
8. What important information one can get by the abscissa of the point of intersection of the less than type and the
more than type cumulative frequency curve of a group data
LEVEL – II
2. Given below is the distribution of IQ of the 100 students .Find the median of IQ 106.1
IQ 75-84 85-94 95-104 105-114 115-124 125-134 135-144
Frequency 8 11 26 31 18 4 2
4 A class teacher has the following absentee records of 40 students of a class for the whole term
No of 0-6 6-10 10-14 14-20 20-28 28-38 38-40
days
No of 11 10 7 4 4 3 1
students
Write the above distribution as less than type cumulative frequency distribution
92
5 Using the assumed mean method find the mean of the following data 27.2
Class interval 0-10 10-20 20-30 30-40 40-50
frequency 7 8 12 13 10
LEVEL – III
SN Question Ans
1 If the mean distribution is 25 P=16
Frequency 5 18 15 P 6
Then find P
2 Find the mean of the following frequency distribution using step deviation method 25
7 12 13 10 8
Frequency
3 Find the value of p if the median of the following frequency distribution is 50 P=10
Marks Less Less than Less than Less Less than Less than Less than Less than
than 10 30 50 than70 90 110 130 150
Frequency 0 10 25 43 65 87 96 100
5 Compare the modal ages of two groups of students appearing for entrance examination.
Age in yrs 16-18 18-20 20-22 22-24 24-
A=18.93
B=18.83
26
Group A 50 78 46 28 23
93
SELF – EVALUATION
1. What is the value of the median of the data using the graph in figure of less than ogive and more than
ogive?
4. Find the mean, mode and median for the following data.
Classes 0-10 10-20 20-30 30-40 40-50 50-60 60-70
frequency 5 8 15 20 14 8 5
94
5. The median of the following data is 52.5. Find the value of x and y, if the total frequency is100.
Class Interval 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100
frequency 2 5 X 12 17 20 Y 9 7 4
6. Draw‘ less than ogive’ and‘ more than ogive’ for the following distribution and hence find its median.
Marks Below Below Below Below Below Below Below Below Below Below
10 20 30 40 50 60 70 80 90 100
No. of 5 9 17 29 45 60 70 78 83 85
students
8. The following table shows age distribution of persons in a particular region. Calculate the median age.
9. If the median of the following data is 32.5. Find the value of x and y.
10. The following are ages of 300 patients getting medical treatment in a hospital on a particular day.
Age( in years) 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60 60 – 70
Number of 60 42 55 70 53 20
patients
95
Draw:
Number of 4 5 13 20 14 8 4
consumers
Mr.Sharma always saves electricity by switching off all the electrical equipments just immediately after their uses. So,
his family belongs to the group 65-85.
(i) Find the median of the above data
(ii) How many families consumed 125 or more units of electricity during a month?
(iii) What moral values of Mr.Sharma have been depicted in this situation?
Q2.The mileage (km per litre) of 50 cars of the same models is tested by manufacturers and details are tabulated as
given below:-
12
No. of cars 7 18 13
iii. The manufacturer claims that the mileage of the model is16km/litre. Do you agree with this claim?
iv. Which values do you think them manufacturer should imbibe in his life?
96
ANSWER
1. 12.9
2. MEDIAN
3. 33
4. MODE =40
5. MEDIAN=30-40
6. 2
7. OGIVE
8. Median
Level II
Q1 167
Q2 106.1
Q.3 28.5l
Q4
No. of 11 21 28 32 36 39 40
students
Q5 27.2
97
PROBABILITY
KEY POINTS
1. Probability:-The theoretical probability of an event E, written as P(E) is defined as.
P(E)=Number of out comes Favourable to E
Number of all possible out comes of the experiment
When we assume that the out comes of the experiment are equally likely.
LEVEL-I
1. The probability of getting bad egg in a lot of 400 is 0.035. Then find the number of bad eggs in the lot.
2. Write the probability of a sure event.
3. What is the probability of an impossible event?
4. When a dice is thrown, and then find the probability of getting an odd number less than 3.
5. A girl calculates that the probability of her winning the third prize in a lottery is 0.08. If 6000 tickets are sold, how
many ticket has she bought.
6. What is probability that a non-leap year selected at random will contain 53 Sundays.
7. A jar contains 54 marbles each of which is blue ,green or white . The probability of selecting a blue marble at
random from the jar is 1/3, and the probability of selecting a green marble at random is 4/9 . How many white
8. two coins are tossed simultaneously. Find the probability of getting exactly one head.
9. A card is drawn from a well shuffled deck of 52cards. Find the probability of getting an ace.
10. In a lottery, there are10 prizes and 25 blanks. Find the probability of getting a prize.
98
LEVEL-II
1. Find the probability of prime numbers selected at random from the numbers 3,4,5,6...25 .
2. A bag contains 5 red, 4 blue and 3 green balls. A ball is taken out from the bag at random. Find the probability that
the selected ball is (a) of red colour (b) not of green colour.
3. A card is drawn at random from a well-shuffled deck of playing cards. Find the probability of drawing
(a) A face card(b)card which is neither a king nor a red card
4. A dice is thrown once. What is the probability of getting a number greater than4?
5. Two dice are thrown at the same time. Find the probability that the sum of two numbers appearing on the top of the
dice is more than 9.
6. Two dice are thrown at the same time. Find the probability of getting different numbers on both dice.
7. A coin is tossed two times. Find the probability of getting almost one head.
8. Cards with numbers 2 to 101 are placed in a box. A card selected at random from the box. Find the probability that the
card which is selected has a number which is a perfect square.
9. Find the probability of getting the letter M in the word “MATHEMATICS”.
LEVEL-III
1. Cards bearing numbers 3,5,7…35 are kept in a bag. A card is drawn at random from the bag. Find the probability of
getting a card bearing (a) a prime number less than 15(b) a number divisible by 3 and 5.
2. Two dice are thrown at the same time. Find the probability of getting (a) same no. on the both side(b) different no. on
both dices.
3. A child game has 8 triangles of which three are blue and rest are red and ten squares of which six are blue and rest
are red. One piece is drawn at random. Find the probability of that is (a) A square (b) A triangle of red colour.
4. Two dice are thrown simultaneously. What is the probability that:
(a) 5 will not come up either of them? (b) 5 will come up on at least one? (c) 5 will come at both dice?
5. The king, queen and jack of clubs are removed from a deck of 52 playing cards and remaining cards are shuffled. A
card is drawn from the remaining cards. Find the probability of getting a card of (a) heart(b) queen(c) clubs
6. A game consists of tossing a one-rupee coin 3 times and noting its outcome each time. Hanif wins if all the tosses ,give
the same result,i.e.,3 heads or three tails and loses otherwise. Calculate the probability that Hanif will lose the game.
7. Cards bearing numbers 1,3,5…37 are kept in a bag. A card is drawn at random from the bag. Find the
Probability of getting a card bearing
99
(a) A prime number less than15
Self-Evaluation/HOTS
1. Two dice are thrown simultaneously. Find the probability of getting an even number as the sum.
2. Cards marked with the number 2 to 101 are placed in a box and mixed thoroughly. One card is drawn from the
box. Find the probability that the number on the card is:
(i) An even number
(ii) A number less than14
(iii) A number is perfect square
(iv) A prime number less than 20
3. Find the probability that a leap year selected at random will contain 53 Sundays.
Value based Question
Q1.In a survey, it was found that 40% people use petrol, 35% uses diesel and remaining uses CNG for their vehicles.
Find the probability that a person uses CNG at random.
(a) Which fuel out of above 3 is appropriate for the welfare of the society?
Level -I
1. A die is thrown once. What is probability of getting a number greater than4?
2. A bag contains 4 red and 6 blackballs. A ball is taken out of the bag at random. Find the probability of getting a
blackball?
3. A die is thrown once. Find the probability of getting.
a) Prime number
b) A number divisible by 2.
Level -II
1. A bag contains card which are numbered from 2 to 90. A card is drawn at random from the bag. Find
the probability that it bears.
100
a.) A Two digit number
b.) A number which is perfect square.
2. Two dice are rolled once. Find the probability of getting such numbers on the two dice whose product is12.
Level – III
1. Red queens and black jacks are removed from a pack of 52 playing cards. A card is drawn at random from the
remaining card, after reshuffling them. Find the probability that the drawn card is:
(i) King ii) of red colour iii) a face card iv) queen
2. All the red face cards are removed from a pack of 52 playing cards. A card is drawn at random from the
remaining cards after reshuffling them. Find the probability that the card drawn is
(i) Of red colour ii) a queen iii) an ace iv) a face card.
3. In a family of 3 children, find the probability of having at least 1 boy.
4. Three unbiased coins are thrown simultaneously. Find the probability of getting.
i. Exactly two heads.
ii. At least two heads.
iii. At most two heads.
101
ANSWER
LEVEL-I
1. 14
2. 1
3. 0
4. 1/6
5. 480
6. 1/7
7. 12
8. ½
9. 1/13
10. 2/7
LEVEL - II
1. 8/23
2. A. 5/12 B. ¾
3. A. 3/13 B. 6/13
4. 1/3
5. 1/6
6. 5/6
7. ¾
8. 9/100
9. 2/11
LEVEL– III
1. A.5/17 B.1/17
2. A. 1/6 B. 5/6
3. A. 5/9 B. 5/18
4. A. 25/36 B. 11/36 C. 1/36
5. A. 13/49 B. 3/49, C 10/49
6. ¾
7. A.5/19 B.1/19
8. A. 6 scores B. 1/3
102
SELF EVALUATION
1. ½
2. A. ½ B 3/25 C 9/100, D. 2/25
3. 2 / 7
BOARD QUESTION
LEVEL-I
1.1/3
2.3/5
3. ½, ½
LEVEL- II
1. 81/89, 8/89
2. 1/9
LEVEL- III
4. 3/8, ½, 7/8
*****************
103
ACTIVITIES
Activity1: To find the HCF of two Numbers Experimentally Based on Euclid Division Lemma
Activity2: To Draw the Graph of a Quadratic Polynomial and observe:
i. The shape of the curve when the coefficient of x2 is positive
ii. The shape of the curve when the coefficient of x2 is negative
iii. Its number of zero
Activity8: To Draw a System of Similar Squares, Using two intersecting Strips with nails
Activity9: To Draw a System of similar Triangles Using Y shaped Strips with nails
Activity10: To verify Basic proportionality theorem using parallel line board
Activity11: To verify the theorem Ratio of the Areas of Two Similar Triangles is Equal to the
Ratio of the Squares of their corresponding sides through paper cutting.
Activity12: To verify Pythagoras Theorem by paper cutting, paper folding and adjusting (Arranging)
Activity13: Verify that two figures (objects) having the same shape (and not necessarily the same
size) are similar figures. Extend the similarity criterion to Triangles.
Activity14: To find the distance between two objects by physically demonstrating the position
of the two objects say two Boys in a Hall, taking a set of reference axes with the
corner of the hall as origin.
Activity15: Division of line segment by taking suitable points that intersects the axes at some Points
and then verifying section formula.
Activity16: To verify the formula for the area of a triangle by graphical method.
Activity17: (a ) To make mathematical instrument clinometers (or sextant )for
measuring the angle of elevation/depression of an object
Activity 18: To calculate the height of an object making use of clinometers (or sextant)
104
Activity20: To find the number of tangent from a point to the circle
Activity21: To verify that lengths of tangents drawn from an external Point to a circle are
equal by using method of paper cutting, paper folding and pasting.
Activity22: To Draw a quadrilateral similar to a given quadrilateral as per given scale factor (Less than1)
Activity29: To get familiar with the idea of probability of an event through a double color card experiment.
Activity30: To verify experimentally that the probability of getting two tails when two coins are tossed
simultaneously is ¼ = (0.25) By eighty tosses of two coins.
105
ORAL TEST (REAL NUMBERS)
QUIZ
(POLYNOMIALS)
Answer the following questions:
1. What is a quadratic polynomial?
2. What is the degree of a quadratic polynomial?
3. What are the zeroes of a polynomial?
4. What is the shape of curve of a quadratic polynomial graph?
5. State remainder theorem.
ORAL TEST
1. If P(x) is a polynomial in x, the highest power of x in P(x) is called the _of the
polynomial P(x).
2. A polynomial of degree 2 is called a .
3. The linear polynomial ax + b, a≠0, has exactly one zero, namely, the x-coordinate of the
point where the graph of y= ax+ b intersects the _.
4. A polynomial P(x) of degree n has atmost _zeroes.
5. The sum and the product of the zeroes of a quadratic polynomial x 2+7x+10 is _
and _.
QUIZ
(Pair of linear equations in two variables)
Answer the following questions:
1. What is a pair of line of equations in two variables?
2. Give the general form of a pair of linear equation?
3. What are the methods of solving a pair of linear equation in two variables?
4. What is the condition for inconsistent solution?
5. What is the shape of curve in graph of a linear equation?
106
Oral Test
QUIZ
(Triangles)
ORAL TEST
107
5. State whether the following quadrilateral are similar or not.
QUIZ
(Introduction to Trigonometry)
1. What is trigonometry?
2. What are trigonometric ratios of an acute angle in a right triangle?
3. From the figure find the value of cos A.
ORAL TEST
108
QUIZ
(STATISTICS)
1. Name the measures of central tendency.
2. What is cumulative frequency?
3. How will you represent the cumulative frequency distribution graphically?
4. How will you find the median of a grouped data graphically with the help of one
ogive?
5. How will you find the median of a grouped data graphically with the help of both
ogives (i.e of the less than type and of more than type)?
ORAL TEST
1. _ _is the sum of the values of all the observations divided by the total
number of observations.
2. Class mark = _/2.
3. The formula for finding the mean using the step deviation method is .
4. The formula for finding the mode in a grouped frequency distribution is .
5. The formula for finding the median of grouped data is _.
109
KENDRIYA VIDYALAYA SANGATHAN
MODEL QUESTION PAPER 1 (STANDARD)
2019-20
Blue Print
Class- X Time-3 Hours
Subject-Mathematics Marks: 80
NUMBER SYSTEM 6
ALGEBRA 20
GEOMETRY 15
TRIGNOMETRY 12
11 STATISTICS 1(1)
4(1)
12 PROBABILITY 1(1) 2(1) 3(1) 11(5)
MENSURATION 10
Typology of
S. VSA SA1 SA2 LA TOTAL WEIGHTAGE
Question
no
1 REMEMBERING 6 2 2 1 20 25%
2 UNDERSTANDING 6 1 1 3 23 29%
3 APPLYING 5 2 2 1 19 24%
ANALYZING,
4 EVALUATING & 3 1 3 1 18 22%
CREATING
TOTAL 20x1=20 6x2=12 8x3=24 6x4=24 80 100%
111
KENDRIYA VIDYALAYA
SANGATHAN MODEL
QUESTION PAPER - 1
2019-20
Mathematics- STANDARD (041)
Class:-X Time: 3 Hrs
Marks: 80
General Instructions:
(i) All questions are compulsory.
(ii) The question paper comprises of five sections, A, B, C, D and E. You are to attempt all the
sections.
(iii) Section A comprises of 10 multiple choice questions (MCQs) one mark each. Section B
comprises of 10 questions of one mark each. Section C comprises of 6 questions of 2 marks
each. Section D comprises of 8 questions of 3 marks each. Section E comprises of 6 questions
of 4 marks each.
(iv) There is no overall choice. However, an internal choice has been provided in Section
C, D and E.
(v) Use of calculator and other electronic device is not permitted.
SECTION – A
Question Numbers 1 to 10 carry one mark each.
1. The product of three consecutive integers is divisible by
(a) 2 (b) 3 (c) 5 (d) 6
2. The midpoint of the line segment joining the points (-2, 8)and (-6,-4)
(a) (-4, -6) (b) (2, 6) (c) (-4, 2) (d) (4, 2)
3. If the lines given by 3x+2ky=2 and 2x+5y+1=0 are parallel, then value of k is
(a) (b) (c) (d)
4. Nature of roots of the equation + 2 + 4 = 0 is
(a) Real and equal (b) real and unequal (c) not real (d) none of these
5. If nth term of an AP is an + b then its common difference is
(a) b (b) a (c) n (d) – a
6. Complement angle of Cosine is
(a)Secant (b) Sine (c) Tangent (d) Cosecant
7. At point A on a diameter AB of a circle of radius 10cm, tangent XAY is drawn in the circle. The
length of the chord CD parallel to XY at a distance 16cm from A is
(a) 8 cm (b) 10cm (c) 16cm (d) 18cm
8. If two towers of height h1 and h2 subtend angles 600 and 300 respectively at the mid -point of
line joining their bases then h1:h2 is
(a) 3 :1 (b) 1 :2 (c)√3 : 1 (d) 1 : √3
9. A wire can be bent in the form of a circle of radius 7cm. if it bent in the form of a square, then its
area will be
(a) 351cm2 (b) 516 cm2 (c) 218 cm2 (d) 121 cm2
112
10. Class mark of the Class 10 -25 is
(a) 10 (b) 25 (c) 15 (d) 17.5
SECTION – B
Question Numbers 11 to 20 carry one mark each.
The following questions (11 to 15) consist of two statements – Assertion (A) and Reason(R).
Answer these questions selecting the appropriate option given below:
(a)Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c)A is true but R is False.
(d) A is False but R is true.
11. Assertion(A): The line segment joining the mid points of the sides of a triangle form four triangles
similar to original triangle.
Reason (R): Two polygons are similar if their corresponding sides are proportional.
12. Assertion(A): If 2 Sin2 = √3 then = 30 .
√
Reason (R): The value of sin60 = .
13. Assertion(A): If the surface area of a sphere is 616 cm2.Then its radius is 6cm.
Reason (R): Surface area of sphere = 4 .
14. Assertion(A): A circle can have more than two parallel tangents, parallel to the given chord.
Reason (R): The point at which a line touches the circle, is called the point of contact.
15. Assertion(A): If a number x is chosen at random from the numbers -3,-2,-1,0,1,2,3 then the
probability of | | < 2 .
Reason (R):| |<2 ⟹ −2 < < 2 ⟹ = {−1,0,1}, ℎ .
Fill in the Blanks. (Q. No. 16 to 20)
16. Acute angle satisfying √3 = is ………………..
17. Degree of polynomial + 5 − 12 − 2 is ……………..
18. Volume of a frustum of a cone having base radii r1 and r2 and height h is …………..
19. The ratio of the areas of two similar triangles is equal to the square of the ratio of their
……………….
20. The value of − is …………….
SECTION – C
(Question numbers 21 to 26 carry 2 marks each)
21. Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m+1 for some integer m.
22. For what value of ,will the system of linear equations + 2 = 5 and 3 + − 15 = 0 has unique solution?
23. Find the value of k for which the lines (k+1) x+3ky+15=0 and 5x+ ky + 5 =0 are coincident.
24. Find how many integers between 200 and 500 are divisible by 8.
OR
The general term of a sequence is given by an= 4n + 15. Is the sequence forming an AP? If so, find its 15th term.
25. Find the ratio in which the line segment joining the points (-2, 3) and (3, - 2) is divided by y-axis.
OR
Find the value of y for which the distance between the points P (2, -3) and Q (10, y) is 10 units.
26. A coin is tossed twice. What is the probability of getting at most one tail? 113
SECTION – D
(Question numbers 27 to 34 carry 3 marks each)
27 . Prove that 2 − √3 is an irrational number.
28. Solve the pair of equations: + =2 ; − =1
OR
If x +1 is a factor of 2 + + 2 + 1, then find the values of a and b given that 2a – 3b = 4
29. If A, B and C are the interior angles of ∆ABC, then prove that
+
tan = cot
2 2
OR
Prove that
1+
+ =2
1+
30. How many terms of the A.P. 24, 21, 18,…must be taken so that their sum is 78?.
31. If the points (-2,-1), (1, 0), (x, 3) and (3, y) form a parallelogram, find the values of x and y.
OR
The co-ordinates of the points A, B and C are (6, 3), (-3, 5) and (4,-2) respectively. P(x, y) is any
(∆ )
point in the plane. Show that (∆ )
= .
32. Prove that the tangents drawn from an external point to the circle are equal in length.
33. Find the area of the shaded region in the given figure. If O is the centre of the two concentric circles
and radii of the circles are 7cm and 14 cm respectively along with angle AOC =400.
34. Cards numbered from 11 to 60 are kept in a box. If a card is drawn at random from the box, find the probability that the
number on the drawn card is:
(i) an odd number (ii) a perfect square number (iii) a prime number less than 20.
114
SECTION – E
(Question numbers 35 to 40 carry 4 marks each)
35. Two water taps together can fill a tank in 9 hours. The tap of larger diameter takes 10 hours less than
the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.
OR
Solve for x: + = ; ≠ 1, .
36.In an equilateral triangle ABC, D is a point on side BC such that = . Prove that 9 =7 .
37. Construct a right triangle in which sides (other than the hypotenuse) are 8 cm and 6 cm. Then construct another triangle
whose sides are times the corresponding sides of the right triangle.
38. If the angle of a cloud from a point h metres above a lake is and the angle of depression of its reflection in the lake is ,
( )
prove that the height of the cloud is .
OR
The angle of elevation of an aeroplane from a point A on the ground is 600 .After a flight of 30 seconds, the angle of elevation
changes to 30 0. If the aeroplane is flying at a constant height of 3600√3 metres then find the speed of the aeroplane.
39. A container shaped like a right circular cylinder having diameter 12 cm and height 15 cm is full of ice cream. The ice cream is
to be filled into cones of height 12 cm and diameter 6 cm, having a hemispherical shape on the top. Find the number of such
cones which can be filled with ice cream.
OR
A metallic right circular cone 20 cm high whose vertical angle is 600 which is cut into two parts at the middle of its height by a
plane parallel to its base. If the frustum so obtained be drawn into a wire of diameter cm, find the length of the wire.
Draw the less than type and a more than type ogive from the given data. Hence find the median value.
115
KENDRIYA VIDYALAYA SANGATHAN
STANDARD (041)
2019-20
Answer Key
Class- X Time-3Hours
Subject- Mathematics
Marks: 80
Question Answer Question No. Answer Question Answer Question Answer
No. No. No.
1. d 11. c 21. Correct 31. X=6, y= 2
Proof Or
Correct
Proof
2. c 12. a 22. k≠6 32. Correct
Proof
3. c 13. d 23. K=14 33. 51.33 sq.cm
4. c 14. d 24. 38 34. 1/2,
Or 2/25,
Yes, 75 2/25
116
KENDRIYA VIDYALAYA SANGATHAN
2019-2020
CLASS:-X
Time Allowed: 03Hours Maximum Marks: 80
TOTAL
S.NO UNITS VSA SA-I SA-II LA
MARKS
Number
1 1(1) 2(1) 3 (1) - 6(3)
Systems
117
KENDRIYA VIDYALAYA SANGATHAN
MODEL QUESTION PAPER - 2
2019-20
Mathematics- STANDARD (041)
Class:-X Time: 3 Hrs
Marks: 80
General Instructions:
(i) All questions are compulsory.
(ii) The question paper comprises of five sections, A, B, C, D and E. You are to attempt all the sections.
(iii) Section A comprises of 10 multiple choice questions (MCQs) one mark each. Section B comprises of
10 questions of one mark each. Section C comprises of 6 questions of 2 marks each. Section D
comprises of 8 questions of 3 marks each. Section E comprises of 6 questions of 4 marks each.
(iv) There is no overall choice. However, an internal choice has been provided in Section C,D and E.
(v) Use of calculator and other electronic device is not permitted.
SECTION – A
Question Numbers 1 to 10 carry one mark each.
1. How many rational number are there between two rational number
a) one b) two c) three d) infinite
2. The sum and product of the zeroes of a quadratics polynomial are 2 and -15 respectively the quadratic
polynomial is
a) x2-2x+ 5 b) x2 -2x-15 c) x2 +2x-15 d) x2 -2x+15
3. For what value of k , the pair of linear equations 2x-y-3=0 and 2kx +y -2= 0 has 103olution = 1 and y= -1
a) 3 b) 3/2 c) 4 d) -4
4. If the point ( a ,0) , ( 0 , b) and ( 1 , 1) are collinear then 1/a +1/b is
a) 1 b) 2 c) 0 d) 3
5. If sin A= 1/2 then angle A is
a) 300 b) 600 c) 450 d) 900
6. The sum of first five natural number is
a) 10 b) 12 c) 15 d) 20
7. The ratio of radii of two sphere is 4:3. The ratio of their volume is
a) 64:27 b) 27:64 c) 16:9 d) 9:16
8. The probability of getting a even number when die is thrown is
a) 1/4 b) 3/4 c) 1/2 d)3/5
9. The mode of the observation 2, 3, 6, 7, 2, 8 ,2 is
a) 3 b) 2 c) 4 d) 6
10. The distance of the point ( 3, 4 ) from origin is
a) 5 b) 6 c) 10 d) 12
118
SECTION – B
Question Numbers 11 to 20 carry one mark each.
The following questions (11 to 15) consist of two statements- Assertion ( A ) and Reason( R) .
Answer the questions selecting the appropriate option given below:
a) Both A and R are true and R is the correct explanation for A
b) Both A and R are true and R is not correct explanation for A
c) A is true but R is false
d) A is false but R is true
11. Assertion (A) : The probability of getting a red card from a pack , when a card is drawn at random, is 1/4
Reason (R): A pack of card contain 26 red and 26 black cards.
12. Assertion (A) : Area of right isosceles triangle with one of the equal side of length 12cm is 72cm2
Reason (R): Area of triangle = 1/2 x base x height
13. Assertion (A) : The fourth angle of a quadrilateral with two right angles , and the measure of angles are 800
and 1000
Reason (R): The sum of angles of a quadrilateral is 3600
14. Assertion (A) : The points ( a/3 , a) lies on x= 3y
Reason (R): Every point on the graph of a linear equation in two variables is a solution of linear equation.
15. Assertion (A): 3x3 -2 is a binomial.
Reason (R): The degree of a cubic polynomial is 3.
18. A geometrical construction is the process of drawing a geometrical figure with the un graduated ruler and
a......................
20. In a frequency distribution, the mid value of a class is 120 and width is 20. The upper limit of the class
is.....................
SECTION – C
(Question numbers 21 to 26 carry 2 marks each)
Q.22: Write the condition to be satisfied by q so that a rational number p/q has a terminating
119
Q.23: If the mid-point of a segment joining A ( , ) and B ( X+1 ,Y-3 ) IS C ( 5 , - 2 ) find x and y .
OR
Q.25: Do the equation 5x+7y=8 and 10x+14y = 4 represent a pair of coincident lines? Justify your answer.
Q.26: If the system of equation 4x+y = 3 and (2k-1) x + (k-1) y = 2k+1 is inconsistent, then find k.
SECTION – D
(Question numbers 27 to 34 carry 3 marks each)
Q.27: Prove that √7 is irrational number.
Q28: Draw the graph of the linear equation x-y = -1 and 3x+2y -1= 0.
Q29: In an AP the first term is 2, the last term is 29 and the sum of the terms is 155, find the common difference of
AP.
Q.30: IF B (x, y) is a point on the line segment joining the point A (a, b) and C (b, a), then prove that x + y = a + b
Q.32: In a fig, AB and CD are two diameters of a circle (with centre O ) perpendicular to each other and OD is
the diameter of smaller circle. If OA =7cm. find the area of shaded region.
Q.33: Which term of the AP 121, 117, 113 ............... is its first negative term?
2x+3y -13 = 0
5x -4y + 2 = 0
120
SECTION – E
(Question numbers 35 to 40 carry 4 marks each)
Q.35: Reshma wishes to fit three rods in the shape of a right triangle. The hypotenuse is to be 2cm longer
than The base and 4cm longer than the altitude. What should be the length of rods?
OR
Solve for x: + = , x ≠ 0 , -1 , 2
( )
Q.36: Draw a right triangle ABC in which AC= AB= 4cm and ∠A = 600. Draw a triangle similar to ∆ABC with its sides
equal to of the corresponding sides of ∆ABC.
Q.37: Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their
Corresponding sides.
Q.38: Find the missing frequencies in the following frequency distribution if it is known that the mean of the
Frequency 5 F1 10 F2 7 8
Q.39: From a solid circular cylinder with height 10 cm and radius of the base 6 cm , a right circular cone of the
same height and same base is removed .find the volume of the remaining solid. Also calculate the whole
Surface area.
Q.40: The angle of elevation of the top ‘A ‘of a vertical tower AB from a point ‘P’ on the ground is 600.
At a point ‘Q’ 40 m vertically above ‘P’, the angle of elevation is 450. Find the height of the tower AB
OR
An aeroplane when flying at a height of 4000 m from the ground passes vertically above another
aeroplane at an instant when the angles of the elevation of the two planes from the same point on
The ground are 600 and 450, respectively. Find the vertical distance between the aero planes at that
Instant.
121
KENDRIYA VIDYALAYA SANGATHAN
STANDARD (041)
MODEL QUESTION PAPER - 2
2019-20
Answer Key
Class- X Time-3Hours
Subject- Mathematics
Marks: 80
Question Answer Question No. Answer Question Answer Question Answer
No. No. No.
1. d 11. d 21. x=3,2 31. Correct Proof
2. b 12. a 22. Factor 2 and 32. 66.5 cm2
5
3. b 13. a 23. x=6 & y= -1 33. 32nd term
or
√5 Unit
4. a 14. d 24. 1/9 34. x= 2 & y= 3
5. a 15. d 25. No 35. 8, 6 & 10 cm
Or
4, -23/11
6. c 16. k=-2 26. 3/2 36. Right
construction
7. a 17. Bisect 27. Correct 37. Correct Proof
Proof
8. c 18. Compass 28. Correct 38. F1=8, F2 =12
Graph
9. b 19. 36 29. 3 39. Volume= 240
cm3
Surface area=
(156 + 12√34
10. a 20. 100 30. Correct 40. Height of tower=
Proof 54.64 & PA =
109.3 mt.
Or
√3 − 1
4000( )
√3
122
KENDRIYA VIDYALAYA SANGATHAN
MATHEMATICS-Basic
Code (241)
CLASS - X
Blue Print of Question Paper- 3
Note: Number of questions are given within brackets and marks outside the brackets
123
KENDRIYA VIDYALAYA SANGATHAN
MATHEMATICS-Basic
Code (241)
Class-X
Model Question Paper- 3
Time allowed: 3 hours Maximum Marks: 80
General Instructions:
1. All questions are compulsory.
2. The question paper comprises of five sections, A, B, C, D and E. You are to attempt all the sections.
3. Section A comprises of 10 multiple choice questions (MCQs) one mark each. Section B comprises of
10 questions of one mark each. Section C comprises of 6 questions of 2 marks each. Section D
comprises of 8 questions of 3 marks each. Section E comprises of 6 questions of 4 marks each.
4. There is no overall choice. However, an internal choice has been provided in Section C, D and E.
5. Use of calculator and other electronic device is not permitted.
SECTION – A
Question Numbers 1 to 10 carry one mark each.
1. The value of k, so that quadratic equation 2x2 + kx + 3 = 0 has two equal roots.
(a) 2√6 (b) ± 2√6 (c) ± 2√3 (d) none of these
2. If P (E) = 0.05, what is the probability of ‘not E’?
(a) 0.95 (b) 0.05 (c) 1.05 (d) 1
3. In the figure, if angle ATO = 40°, find angle AOB.
(a) 120° (b) 90° (c) 150° (d) 100°
124
8. The area of the triangle formed by the line + = 1with the co-ordinate axes is
(a) ab (b) 2ab (c) ab (d) ab
9. The sum areas of a minor sector and the corresponding major sector of a circle is equal to
(a) Area of the circle (b) Area of the circle (c) Area of the circle (d) Area of the circle
10. The pair of equations x = 0 , y =0 represents
(a) Parallel lines (b) coincident lines (c) perpendicular lines (d) non-intersecting lines
SECTION – B
Question Numbers 11 to 20 carry one mark each.
State whether the following statements are true or false. (Q No. 11 to 15)
11. A polynomial cannot have more than one zero.
12. If a line divides any two sides of a triangle in same ratio the it is parallel to the third side.
13. In ∆ABC sin ( ) = cos
14. If the graph of a polynomial intersects x-axis at one point then it is a quadratic polynomial.
15. Mode + 2 mean = 3 median
Fill in the Blanks. (Q No. 16 to 20)
16. A tangent to a circle intersects it in.................... point(s).
17. The volume of frustum of a cone whose height is h and radius of bases be r1 and r2 is...........
18. The total surface area of a solid hemisphere having radius R is...............
19. cosec (90 – ) =............
20. The sum of first n natural number is......................
SECTION – C
(Question numbers 21 to 26 carry 2 marks each)
21. Given that HCF (306, 657) = 9, Find LCM (306, 657).
°
22. Evaluate ° °
23. Find a quadratic polynomial whose sum and product of zeroes are and -1 respectively.
24. Which term of the AP: 3, 8, 13, 18......... is 78?
OR
Find the 20 term from the last term of the AP: 3, 8, 13…………….253.
th
25. If the points (1, 2), (4, y), (x, 6) and3, 5) are the vertices of a parallelogram taken in order, find x and y.
26. Savita and Hamida are friends. What is the probability that both will have (a) different birthdays?
(b) the same birthday ? (ignoring a leap year)
125
SECTION – D
(Question numbers 27 to 34 carry 3 marks each)
27. Use Euclid’s division algorithm to find the HCF of 4052 and 12576.
28. Solve 6x +3y= 6xy , 2x + 4y =5xy
OR
For what values of k will the following pair of linear equations have infinitely many solutions?
Kx + 3y – (K-3) =0 12x + ky –k = 0
35. Two water tapes together can fill a tank in 9 hours. The tap of larger diameter takes 10 hours less than the
smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.
OR
In a class test, the sum of Shefali’s, marks in Mathematics and English is 30. Had she got 2 marks more in
Mathematics and 3 marks less in English, the product of their marks would have been 210. Find the
marks in the two subjects.
126
36. Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their
corresponding sides. Let ∆ABC ~∆DEF and their areas be 64 cm2 and 121 cm2 respectively. If EF = 15.4
cm, Find BC.
37. Draw a triangle ABC with side BC = 7 cm, ∠B =45°, ∠A = 105° . Then construct a triangle whose sides are
times the corresponding sides of ∆ABC. (Also write steps of construction)
38. The angle of elevation of an aeroplane from a point on the ground is 60° . After a flight of 30 second the angle of
elevation becomes 30°. If the aeroplane is flying at a constant height of 3000√3 m, find the speed of
theaeroplane.
OR
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.
39. A sphere, of diameter 12 cm, is dropped in a right circular cylindrical vessel, partly filled with water. If the sphere
is completely submerged in water, the water level in the cylindrical vessel rises by 3 cm. Find the diameter of
the cylindrical vessel.
OR
From a solid cylinder whose height is 2.4 cm and diameter 1.4 cm, a conical cavity of the same height and same
diameter is hollowed out. Find the total surface area of the remaining solid to the nearest cm2.
40. Find the values of the frequencies x and y in the following frequency distribution table if median is 32.
127
MATHEMATICS-Basic
Code (241)
Class-X
Model Question Paper- 3
Question Answer Question Answer Question Answer Question Answer
No. No. No. No.
1 b 11. False 21 22338 31 proof
2 a 12 True 22 √
-
√ 32 1
(iii)P(not a
green ball) =
5 b 15 True 25 X = 6, y= 3 35 25 h & 15 h
OR
Math=12,
English =18
or
math’s=13,
English =17
6 d 16 one 26 (i) 36 11.2 cm
(ii)
7 d 17. h(r12 +r22 27 4 37 construction
+r1r2)
8 c 18 3 28 X= 1 ,y =2 38 720 km/h
Or OR
K= 6 7(√3 + 1) m
9 a 19 sec 29 960 39 18 cm
OR
18 cm2
10 c 20 ( + 1) 30 1:1 & (-3/2,0) 40 X= 9, y=16
2 OR
28 q. unit
128
KENDRIYA VIDYALAYA SANGATHAN
MATHEMATICS-Basic
Code (241)
Class-X
Model Question Paper- 4
General Instructions:
1. All questions are compulsory.
2. The question paper comprises of five sections, A, B, C, D and E. You are to attempt all the sections.
3. Section A comprises of 10 multiple choice questions (MCQs) one mark each. Section B comprises of
10 questions of one mark each. Section C comprises of 6 questions of 2 marks each. Section D
comprises of 8 questions of 3 marks each. Section E comprises of 6 questions of 4 marks each.
4. There is no overall choice. However, an internal choice has been provided in Section C, D and E.
5. Use of calculator and other electronic device is not permitted.
SECTION – A
Question Numbers 1 to 10 carry one mark each.
129
9. The quadratic equation 2x2 - √5 x + 1 = 0 has
A) Two distinct roots B) Two equal roots C) No real roots D) More than 2 two real roots
SECTION – C
(Question numbers 21 to 26 carry 2 marks each)
21. Find the area of the shaded region in the given figure, if ABCD is a square of side 14 cm and APD
and BPC are semicircles.
23. For what value of pare 2p+1, 13, 5p-3, three consecutive terms of and A.P.?
24. What is the distance between the points A (5, 2) and B (2,-2).
25. A card is drawn at random from a well shuffled deck of 52 cards. Find the probability of getting:
(i) a king (ii) a king of red suit
26. Use Euclid`s division algorithm to find the HCF of 867 and 255.
130
SECTION – D
(Question numbers 27 to 34 carry 3 marks each)
29. The sum of 5th and 9th terms of an A.P. is 72 and the sum of 7th and 12th terms is 97. Find the A.P.
30. In Fig., XY and X′Y′ are two parallel tangents to a circle with centre O andanother tangent AB with
point of
31. In fig. 10, PQ=24cm, PR=7cm and O is the centre of the circle. Find the area of the shaded region.
(Take π=3.14).
131
32. The king, queen and jack of clubs are removed from a deck of 52 playing cards and remaining cards
are shuffled. A card is drawn from the remaining cards. Find the probability of getting a card of
(i) heart (ii) queen (iii) clubs.
33. Prove that.
(sin A + cosec A)2 + (cos A + sec A)2 = 7 + tan2 A + cot2 A
34. Metallic spheres of radii 6 cm, 8 cm and 10 cm, respectively, are melted to form a single
solid sphere. Find the radius of the resulting sphere.
SECTION – E
(Question numbers 35 to 40 carry 4 marks each)
35. Construct a triangle ABC in which BC=6.5cm, AB=4.5cm and ∠ABC=600. Construct atriangle
similar to this triangle whose sides are of the corresponding sides of the triangle ABC.
36. A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have
taken 1 hour less for the same journey. Find the speed of the train.
OR
The sum of the reciprocals of Rehman’s ages, (in years) 3 years ago and 5 years from now is .Find his
present age.
37. Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their
corresponding sides.
OR
In an equilateral triangle ABC, D is a point on side BC such that BD = BC. Prove that9 AD2 =
7AB2.
38. From a point on the ground, the angles of elevation of the bottom and top of a transmission tower
fixed at the top of a 20 m high building are 45° and 60° respectively. Find height of the tower.
OR
Two poles of equal heights are standing opposite each other on either side of the road, which is
80 m wide. From a point between them on the road, the angles of elevation of the top of the
poles are 60° and 30°, respectively. Find the height of the poles and the distances of the point
from the Poles.
132
39. A drinking glass is in the shape of a frustum of acone of height 14 cm. The diameters of its two
circular ends are 4 cm and 2 cm. Find the capacity of the glass.
40. The median of the following data is 525. Find the values of x and y, if the total frequency is 100.
133
MATHEMATICS-Basic
Code (241)
Class-X
Model Question Paper- 4
Question Answer Question Answer Question Answer Question Answer
No. No. No. No.
1. D 11. T 21. Area = 42 cm2 31. Area = 4523/28
cm2 = 161.53
cm2
2. D 12. T 22. Correct proof 32. 13/49, 3/49,
10/49
3. A 13. F 23. P=4 33. Correct steps of
proof
4. A 14. T 24. 5 unit 34. Radius = 12cm
5. C 15. F 25. 1/13,1/26 35. Correct
construction
6. B 16. 2/3 26. HCF =51 36. Speed = 40 km/h
or present age =
7years
7. A 17. SECANT 27. Correct proof 37. Statement, Fig.,
proof or correct
steps of proof
8. C 18. -35 28. Fraction = 5/12 38. Height of tower
OR Fixed charge = 20 (√3-1) m or
= Rs. 400 and Height = 20√3
per day charge m, Distance 20 m
Rs. 30 and 60 m
9. C 19. 2 29. a = 6, d = 5, A. 39. Capacity = 102
P. : 6, 11, 16, ….
cm3
10. B 20. IRRATIONAL 30. Correct proof 40. x = 9, y = 15
***
134
SET-1
Series JMS/2 H$moS> Z§.
Code No. 30/2/1
amob Z§. narjmWu H$moS >H$mo CÎma-nwpñVH$m Ho$ _wI-n¥ð
Roll No. >na Adí` {bIo§ &
Candidates must write the Code on the
title page of the answer-book.
J{UV
MATHEMATICS
30/2/1 1 P.T.O.
gm_mÝ` {ZX}e :
(i) g^r àíZ A{Zdm`© h¢ &
(ii) Bg àíZ-nÌ _| 30 àíZ h¢ Omo Mma IÊS>m| A, ~, g Am¡a X _| {d^m{OV h¢ &
(iii) IÊS> A _| EH$-EH$ A§H$ dmbo 6 àíZ h¢ & IÊS> ~ _| 6 àíZ h¢ {OZ_| go àË`oH$ 2 A§H$ H$m h¡ &
IÊS> g _| 10 àíZ VrZ-VrZ A§H$m| Ho$ h¢ & IÊS> X _| 8 àíZ h¢ {OZ_| go àË`oH$ 4 A§H$ H$m h¡ &
(iv) àíZ-nÌ _| H$moB© g_J« {dH$ën Zht h¡ & VWm{n 1 A§H$ dmbo 2 àíZm| _|, 2 A§H$m| dmbo
2 àíZm| _|, 3 A§H$m| dmbo 4 àíZm| _| Am¡a 4 A§H$m| dmbo 3 àíZm| _| Am§V[aH$ {dH$ën àXmZ {H$E JE
h¢ & Eogo àíZm| _| AmnH$mo {XE JE {dH$ënm| _| go Ho$db EH$ àíZ hr H$aZm h¡ &
(v) H¡$bHw$boQ>am| Ho$ à`moJ H$s AZw_{V Zht h¡ &
General Instructions :
(i) All questions are compulsory.
(ii) The question paper consists of 30 questions divided into four sections
A, B, C and D.
(iii) Section A contains 6 questions of 1 mark each. Section B contains
6 questions of 2 marks each, Section C contains 10 questions of 3 marks each
and Section D contains 8 questions of 4 marks each.
(iv) There is no overall choice. However, an internal choice has been provided in
two questions of 1 mark each, two questions of 2 marks each, four questions of
3 marks each and three questions of 4 marks each. You have to attempt only
one of the alternatives in all such questions.
(v) Use of calculators is not permitted.
IÊS> A
SECTION A
àíZ g§»`m 1 go 6 VH$ àË`oH$ àíZ 1 A§H$ H$m h¡ &
Question numbers 1 to 6 carry 1 mark each.
1. `{X _.g. (HCF) (336, 54) = 6 h¡, Vmo b.g. (LCM) (336, 54) kmV H$s{OE &
If HCF (336, 54) = 6, find LCM (336, 54).
2. {ÛKmV g_rH$aU 2x2 – 4x + 3 = 0 Ho$ _ybm| H$s àH¥${V kmV H$s{OE &
Find the nature of roots of the quadratic equation 2x2 – 4x + 3 = 0.
1 3 – a 3 – 2a
3. g_mÝVa lo‹T>r, , , ... (a 0) Ho$ {bE gmd© AÝVa kmV H$s{OE &
a 3a 3a
Find the common difference of the Arithmetic Progression (A.P.)
1 3 – a 3 – 2a
, , , ... (a 0)
a 3a 3a
4. _mZ kmV H$s{OE :
sin2 60 + 2 tan 45 – cos2 30
AWdm
30/2/1 2
3
`{X sin A = h¡, Vmo sec A n[aH${bV H$s{OE &
4
Evaluate :
sin2 60 + 2 tan 45 – cos2 30
OR
3
If sin A = , calculate sec A.
4
5. x-Aj na pñWV {~ÝXw P Ho$ {ZX©oem§H$ {b{IE Omo {~ÝXþ A(– 2, 0) VWm {~ÝXþ B(6, 0) go
g_mZ Xÿar na hmo &
Write the coordinates of a point P on x-axis which is equidistant from the
points A(– 2, 0) and B(6, 0).
6. AmH¥${V 1
_|, ABC EH$ g_{Û~mhþ {Ì^wO h¡ {OgH$m H$moU C g_H$moU h¡ VWm
AC = 4 cm h¡ & AB H$s b§~mB© kmV H$s{OE &
AmH¥${V 1
AWdm
AmH¥${V 2 _|, DE BC h¡ & ^wOm AD H$s bå~mB© kmV H$s{OE O~{H$ {X`m J`m h¡
AE = 1·8 go_r, BD = 7·2 go_r VWm CE = 5·4 go_r &
AmH¥${V 2
In Figure 1, ABC is an isosceles triangle right angled at C with
AC = 4 cm. Find the length of AB.
Figure 1
OR
30/2/1 3 P.T.O.
In Figure 2, DE BC. Find the length of side AD, given that AE = 1·8 cm,
BD = 7·2 cm and CE = 5·4 cm.
Figure 2
IÊS> ~
SECTION B
àíZ g§»`m 7 go 12 VH$ àË`oH$ àíZ Ho$ 2 A§H h¢ &
Question numbers 7 to 12 carry 2 marks each.
7. 306 VWm 657 XmoZm| go nyU©V`m {d^m{OV hmo &
dh g~go N>moQ>r g§»`m {b{IE Omo
Write the smallest number which is divisible by both 306 and 657.
.
8. x Am¡a y _| EH$ g§~§Y kmV H$s{OE Vm{H$ {~ÝXþ A(x, y), B(–4, 6) VWm C(–2, 3) gaoIr`
hm| &
AWdm
Cg {Ì^wO H$m joÌ\$b kmV H$s{OE {OgHo$ erf© (1, – 1) (– 4, 6) VWm (– 3, – 5) h¢ &
Find a relation between x and y if the points A(x, y), B(– 4, 6) and
C(– 2, 3) are collinear.
OR
Find the area of a triangle whose vertices are given as (1, – 1) (– 4, 6) and
(– 3, – 5).
9. EH$ Oma _| Ho$db Zrbo, H$mbo VWm hao H§$Mo h¢ & Bg Oma _| go `mÑÀN>`m EH$ Zrbo H§$Mo Ho$
{ZH$mbZo H$s àm{`H$Vm 1 h¡ VWm Cgr Oma _| go EH$ H$mbo H§$Mo Ho$ `mÑÀN>`m {ZH$mbZo H$s
5
1
àm{`H$Vm h¡ & `{X Oma _| 11 hao a§J Ho$ H§$Mo h¢, Vmo Oma _| Hw$b H§$Mm| H$s g§»`m kmV
4
H$s{OE &
The probability of selecting a blue marble at random from a jar that
1
contains only blue, black and green marbles is . The probability of
5 1
selecting a black marble at random from the same jar is . If the jar
4
contains 11 green marbles, find the total number of marbles in the jar.
30/2/1 4
10. k Ho$ {H$Z _mZm| ({H$g _mZ) Ho$ {bE {ZåZ g_rH$aUm| Ho$ `w½_ H$m EH$ A{ÛVr` hb h¡ :
x + 2y = 5 Am¡a 3x + ky + 15 = 0
Find the value(s) of k so that the pair of equations x + 2y = 5 and
3x + ky + 15 = 0 has a unique solution.
11. Xmo g§nyaH$ H$moUm| _| go ~‹S>o H$moU H$m _mZ N>moQ>o H$moU Ho$ _mZ go 18 A{YH$ h¡ & XmoZm| H$moUm|
Ho$ _mZ kmV H$s{OE &
AWdm
gw{_V H$s Am`w CgHo$ ~oQ>o H$s Am`w H$s VrZ JwZr h¡ & nm±M df© Ho$ ~mX, CgH$s Am`w AnZo
~oQ>o H$s Am`w H$s ‹T>mB© JwZm hmo OmEJr & Bg g_` gw{_V H$s Am`w {H$VZo df© h¡ ?
The larger of two supplementary angles exceeds the smaller by 18. Find
the angles.
OR
Sumit is 3 times as old as his son. Five years later, he shall be two and a
half times as old as his son. How old is Sumit at present ?
12. {ZåZ{b{IV ~ma§~maVm ~§Q>Z H$m ~hþbH$ kmV H$s{OE :
dJ© A§Vamb : 25 – 30 30 – 35 35 – 40 40 – 45 45 – 50 50 – 55
~ma§~maVm : 25 34 50 42 38 14
Class Interval : 25 – 30 30 – 35 35 – 40 40 – 45 45 – 50 50 – 55
Frequency : 25 34 50 42 38 14
IÊS> g
SECTION C
àíZ g§»`m 13 go 22 VH$ àË`oH$ àíZ Ho$ 3 A§H$ h¢ &
Question numbers 13 to 22 carry 3 marks each.
13. {gÕ H$s{OE {H$ 2+5 3 EH$ An[a_o` g§»`m h¡, {X`m J`m h¡ {H$ 3 EH$ An[a_o`
g§»`m h¡ &
AWdm
`ypŠbS> EoëJmo[aÏ_ Ho$ à`moJ go 2048 VWm 960 H$m _.g. (HCF) kmV H$s{OE &
Prove that 2 + 5 3 is an irrational number, given that 3 is an
irrational number.
OR
Using Euclid’s Algorithm, find the HCF of 2048 and 960.
30/2/1 5 P.T.O.
14. H$U© BC na
EH$ hr Va\$ Xmo g_H$moU {Ì^wO ABC VWm DBC ~ZmE JE h¢ & `{X AC VWm
BD EH$ Xÿgao H$mo q~Xþ P na à{VÀN>oX H$aVo h¢, Vmo {gÕ H$s{OE {H$
AP PC = BP DP.
AWdm
EH$ g_b§~ PQRS {Og_| PQ RS h¡, Ho$ {dH$U© nañna {~ÝXþ O na à{VÀN>oX H$aVo h¢ &
`{X PQ = 3RS hmo, Vmo {Ì^wOm| POQ VWm ROS Ho$ joÌ\$bm| H$m AZwnmV kmV H$s{OE &
Two right triangles ABC and DBC are drawn on the same hypotenuse BC
and on the same side of BC. If AC and BD intersect at P, prove that
AP PC = BP DP.
OR
Diagonals of a trapezium PQRS intersect each other at the point O,
PQ RS and PQ = 3RS. Find the ratio of the areas of triangles POQ and
ROS.
15. AmH¥${V 3 _|, PQ VWm RS, O Ho$ÝÐ dmbo {H$gr d¥Îm na Xmo g_m§Va ñne©-aoImE± h¢ Am¡a
ñne© {~ÝXþ C na ñním©-aoIm AB, PQ H$mo A VWm RS H$mo B na à{VÀN>oX H$aVr h¡ & {gÕ
H$s{OE {H$ AOB = 90 h¡ &
AmH¥${V 3
In Figure 3, PQ and RS are two parallel tangents to a circle with centre O
and another tangent AB with point of contact C intersecting PQ at A and
RS at B. Prove that AOB = 90.
Figure 3
16. aoIm x – 3y = 0 {~ÝXþAm| (– 2, – 5) VWm (6, 3) H$mo Omo‹S>Zo dmbo aoImI§S> H$mo {H$g AZwnmV
_| {d^m{OV H$aVr h¡ ? Bg à{VÀN>oX {~ÝXþ Ho$ {ZX©oem§H$ ^r kmV H$s{OE &
Find the ratio in which the line x – 3y = 0 divides the line segment
joining the points (– 2, – 5) and (6, 3). Find the coordinates of the point of
intersection.
30/2/1 6
17. _mZ kmV H$s{OE :
2
3 sin 43 cos 37 cosec 53
–
cos 47 tan 5 tan 25 tan 45 tan 65 tan 85
Evaluate :
2
3 sin 43 cos 37 cosec 53
–
cos 47 tan 5 tan 25 tan 45 tan 65 tan 85
18. 4 _|, EH$
AmH¥${V d¥Îm Ho$ MVwWmªe OPBQ Ho$ A§VJ©V EH$ dJ© OABC ~Zm hþAm h¡ & `{X
OA = 15 go_r h¡, Vmo N>m`m§{H$V joÌ H$m> joÌ\$b kmV H$s{OE & ( = 3·14 à`moJ H$s{OE)
AmH¥${V 4
AWdm
AmH¥${V 5 _|, 2 2 go_r ^wOm dmbm dJ© ABCD EH$ d¥Îm Ho$ A§VJ©V ~Zm hþAm h¡ &
N>m`m§{H$V joÌ H$m joÌ\$b kmV H$s{OE & ( = 3·14 à`moJ H$s{OE)
AmH¥${V 5
In Figure 4, a square OABC is inscribed in a quadrant OPBQ. If
OA = 15 cm, find the area of the shaded region. (Use = 3·14)
Figure 4
OR
30/2/1 7 P.T.O.
In Figure 5, ABCD is a square with side 2 2 cm and inscribed in a circle.
Find the area of the shaded region. (Use = 3·14)
Figure 5
19. EH$ R>mog ~obZ Ho$ AmH$ma H$m h¡ {OgHo$ XmoZm| {gao AY©JmobmH$ma h¢ & R>mog H$s Hw$b bå~mB©
20 go_r h¡ VWm ~obZ H$m ì`mg 7 go_r h¡ & R>mog H$m Hw$b Am`VZ kmV H$s{OE &
22
( = à`moJ H$s{OE)
7
A solid is in the form of a cylinder with hemispherical ends. The total
height of the solid is 20 cm and the diameter of the cylinder is 7 cm. Find
22
the total volume of the solid. (Use = )
7
20. ZrMo {X`m hþAm ~§Q>Z 100 {dÚm{W©`m| Ûmam EH$ narjm _| àmßV A§H$m§o H$mo Xem© ahm h¡ :
àmßVm§H$ : 30 – 35 35 – 40 40 – 45 45 – 50 50 – 55 55 – 60 60 – 65
{dÚm{W©`m| H$s
14 16 28 23 18 8 3
g§»`m :
{dÚm{W©`m| Ho$ _mÜ` A§H$ kmV H$s{OE &
The marks obtained by 100 students in an examination are given below :
Marks : 30 – 35 35 – 40 40 – 45 45 – 50 50 – 55 55 – 60 60 – 65
Number of
14 16 28 23 18 8 3
Students :
Find the mean marks of the students.
21. k Ho$ {H$g _mZ Ho$ {bE, ~hþnX
f(x) = 3x4 – 9x3 + x2 + 15x + k,
3x2 – 5 go nyU©V`m {d^m{OV hmoVm h¡ ?
AWdm
{ÛKmV ~hþnX 7y2 – y – 2 Ho$ eyÝ`H$ kmV H$s{OE Am¡a eyÝ`H$m| VWm JwUm§H$m| Ho$ ~rM
11
3 3
Ho$ g§~§Y H$s gË`Vm H$s Om±M H$s{OE &$
30/2/1 8
For what value of k, is the polynomial
f(x) = 3x4 – 9x3 + x2 + 15x + k
completely divisible by 3x2 – 5 ?
OR
11 2
Find the zeroes of the quadratic polynomial 7y2 – y – and verify
3 3
the relationship between the zeroes and the coefficients.
22. p Ho$ Eogo g^r _mZ {b{IE {H$ {ÛKmV g_rH$aU x2 + px + 16 = 0 Ho$ ~am~a _yb
hm| & àmá g_rH$aU Ho$ _yb kmV H$s{OE &
Write all the values of p for which the quadratic equation x2 + px + 16 = 0
has equal roots. Find the roots of the equation so obtained.
IÊS> X
SECTION D
23. `{X {H$gr {Ì^wO H$s EH$ ^wOm Ho$ g_m§Va AÝ` Xmo ^wOmAm| H$mo {^Þ -{^Þ {~ÝXþAm| na
à{VÀN>oX H$aZo Ho$ {bE EH$ aoIm ItMr OmE, Vmo {gÕ H$s{OE {H$ `o AÝ` Xmo ^wOmE± EH$ hr
AZwnmV _| {d^m{OV hmo OmVr h¢ &
If a line is drawn parallel to one side of a triangle to intersect the other
two sides in distinct points, then prove that the other two sides are
divided in the same ratio.
24. A{_V Omo {H$ EH$ g_Vb O_rZ na I‹S>m h¡, AnZo go 200 _r. Xÿa C‹S>Vo hþE njr H$m
CÞ`Z H$moU 30 nmVm h¡ & XrnH$ Omo {H$ 50 _r. D±$Mo ^dZ H$s N>V na I‹S>m h¡, Cgr njr
H$m CÞ`Z H$moU 45 nmVm h¡ & A{_V Am¡a XrnH$ njr Ho$ {dnarV {Xem _| h¢ & XrnH$ go
njr H$s Xÿar kmV H$s{OE &
Amit, standing on a horizontal plane, finds a bird flying at a distance of
200 m from him at an elevation of 30. Deepak standing on the roof of a
50 m high building, finds the angle of elevation of the same bird to be 45.
Amit and Deepak are on opposite sides of the bird. Find the distance of
the bird from Deepak.
30/2/1 9 P.T.O.
25. bmoho Ho$ EH$ R>mog I§^o _| 220 go_r D±$MmB© Ho$ EH$ ~obZ {OgHo$ AmYma H$m ì`mg 24 go_r
h¡, Ho$ D$na 60 go_r D±$MmB© H$m EH$ AÝ` ~obZ AÜ`mamo{nV h¡ {OgH$s {ÌÁ`m 8 go_r h¡ &
Bg I§^o H$m ^ma kmV H$s{OE, O~{H$ {X`m J`m h¡ {H$ 1 KZ go_r bmoho H$m bJ^J ^ma
8 J«m_ h¡ & ( = 3·14 à`moJ H$s{OE)
A solid iron pole consists of a cylinder of height 220 cm and base diameter
24 cm, which is surmounted by another cylinder of height 60 cm and
radius 8 cm. Find the mass of the pole, given that 1 cm3 of iron has
approximately 8 gm mass. (Use = 3·14)
26. 5 go_r ^wOm dmbo g_~mhþ {Ì^wO ABC H$s aMZm H$s{OE & {\$a EH$ AÝ` {Ì^wO H$s aMZm
2
H$s{OE {OgH$s ^wOmE± {XE hþE {Ì^wO ABC H$s g§JV ^wOmAm| H$s JwZr hm| &
3
AWdm
2 go_r {ÌÁ`m Ho$ d¥Îm na 5 go_r {ÌÁ`m H$m EH$ g§Ho$ÝÐr d¥Îm It{ME & ~mø d¥Îm na {bE
JE EH$ {~ÝXþ P go N>moQ>o d¥Îm na Xmo ñne©-aoImAm| PA VWm PB H$s aMZm H$s{OE &
PA H$s b§~mB© _m{nE &
Construct an equilateral ABC with each side 5 cm. Then construct
2
another triangle whose sides are times the corresponding sides of
3
ABC.
OR
Draw two concentric circles of radii 2 cm and 5 cm. Take a point P on the
outer circle and construct a pair of tangents PA and PB to the smaller
circle. Measure PA.
27. {ZåZ{b{IV ~§Q>Z H$mo "go H$_ àH$ma' Ho$ ~§Q>Z _| ~X{bE Am¡a {\$a CgH$m VmoaU It{ME :
dJ© AV§amb : 30 – 40 40 – 50 50 – 60 60 – 70 70 – 80 80 – 90 90 – 100
~ma§~maVm : 7 5 8 10 6 6 8
Change the following data into ‘less than type’ distribution and draw its
ogive :
Class
30 – 40 40 – 50 50 – 60 60 – 70 70 – 80 80 – 90 90 – 100
Interval :
Frequency : 7 5 8 10 6 6 8
30/2/1 10
28. {gÕ H$s{OE {H$ :
tan cot
1 sec cosec
1 – cot 1 – tan
AWdm
{gÕ H$s{OE {H$ :
sin sin
2
cot cosec cot – cosec
Prove that :
tan cot
1 sec cosec
1 – cot 1 – tan
OR
Prove that :
sin sin
2
cot cosec cot – cosec
29. g_m§Va lo‹T>r –7, –12, –17, –22, ... H$m H$m¡Z-gm nX –82 hmoJm ? Š`m –100 Bg g_m§Va
lo‹T>r H$m H$moB© nX hmoJm ? gH$maU CÎma ~VmBE &
AWdm
g_m§Va lo‹T>r 45, 39, 33, ... Ho$ {H$VZo nXm| H$m `moJ\$b 180 hmoJm ? Xmohao CÎma H$s
ì`m»`m H$s{OE &
Which term of the Arithmetic Progression –7, –12, –17, –22, ... will be
–82 ? Is –100 any term of the A.P. ? Give reason for your answer.
OR
How many terms of the Arithmetic Progression 45, 39, 33, ... must be
taken so that their sum is 180 ? Explain the double answer.
30. {hÝXr VWm A§J«oµOr H$s H$jm narjm _| AéU Ho$ àmá A§H$m| H$m `moJ\$b 30 h¡ & `{X CgHo$
{hÝXr _| 2 A§H$ A{YH$ hmoVo Am¡a A§J«oµOr _| 3 A§H$ H$_ hmoVo, Vmo àmá A§H$m| H$m JwUZ\$b
210 hmoVm & XmoZm| {df`m| _| CgHo$ Ûmam àmá A§H$m| H$mo kmV H$s{OE &
In a class test, the sum of Arun’s marks in Hindi and English is 30. Had
he got 2 marks more in Hindi and 3 marks less in English, the product of
the marks would have been 210. Find his marks in the two subjects.
30/2/1 11 P.T.O.
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