DAV INSTITUTIONS : MAHARASHTRA AND GUJARAT ZONE

(UNDER THE AEGIS OF DAVCAE, DAVCMC, NEW DELHI)

FIRST TERM EXAMINATION
2023-2024
MATHEMATICS (041) - SET: B
Class: X Time Allowed: 3 Hours
Date: 01/09/2023 Maximum Marks: 80
____________________________________________________________________________
General Instructions:
1. This question paper has 5 sections A-E.
2. Section A has 20 MCQs carrying 1 mark each.
3. Section B has 5 questions carrying 2 marks each.
4. Section C has 6 questions carrying 3 marks each.
5. Section D has 4 questions carrying 5 marks each.
6.Section E has 3 case-based questions of 4 marks with subparts of values 1,1 and 2 marks each
respectively.
7.All questions are compulsory. However, an internal choice has been provided in 2 questions of 2 marks, 2
questions of 3 marks and 2 questions of 5 marks. An internal choice has been provided in the 2 marks
questions of Section E.
…………………………………………………………………………………………………………………………………………………………………………………

SECTION- A
1. Given that LCM (91, 26) = 182, then HCF (91, 26) is
(a) 26 (b) 13 (c) 39 (d) 91
2. The sum of exponents of prime factors in the prime factorisation of 1250 is
(a) 4 (b) 6 (c) 5 (d) 10
3. If α and β are zeroes of quadratic polynomial x2 – 4x + 3, then the value of α4β3 + α3β4 is

(a) 104 (b) 108 (c) 112 (d) 125

4. If graph of a polynomial does not intersect the x-axis but intersects y-axis in one point, then
no. of zeroes of the polynomial is equal to

(a) 0 (b) 1 (c) 0 or 1 (d) none of these

5. The number of polynomials having zeroes as –2 and 5 is

(a) 1 (b) 2 (c) 3 (d) more than 3
6. The pair of equations x + 2y + 5 = 0 and – 3x – 6y + 1 = 0 has

(a) A unique solution (b) Exactly two solutions (c) Infinitely many solutions (d) No solution
7. The equation of x- axis is
(a) x = 0 (b) y = 0 (c) x = y (d) x + y = 0
8.The equation mx2 + 2x + m = 0 (m being real) has two distinct roots, if
(a) m ≠ 0 (b) m ≠ 0, 1 (c) m ≠ 1, -1 (d) m ≠ 0, 1, -1
9. If the equation 9x2 +6kx +4 = 0 has equal roots, then the value of k must be
(a) 0 (b) either 2 or 0 (c) either – 2 or 0 (d) either 2 or – 2
10. The discriminant of quadratic equation: 2x2 + 4x - 1 = 0 is
(a) 24 (b) 16 (c) 20 (d) 18
11.If the mean of x – 2, x + 2 , x – 3 and x +3 is 10, then the value of 2x - 14 is

(a) 3 (b) 4 (c) 10 (d) 6

12.Consider the following frequency distribution:

C.I 50 - 55 55 - 60 60 - 65 65 - 70 70 - 75 75 - 80
Frequency 5 3 10 8 4 6
The difference of the upper limit of modal class and lower limit of the median class is
(a) 15 (b) 5 (c) 0 (d) 25
13. The mean and median of a data are 14 and 15 respectively, then value of mode is
(a) 17 (b) 15 (c)16 (d) 14
𝑨𝑫 𝑨𝑬
14. In the given figure, 𝑫𝑩 = and ∠ADE = 70°, ∠BAC = 50°, then angle ∠BCA =
𝑬𝑪

(a) 70° (b) 50° (c) 60° (d) 80°

A

D E

B C

15. In the fig., P and Q are points on the sides AB and AC respectively of ΔABC such that
AP = 3.5 cm, PB = 7 cm, AQ = 3 cm and QC = 6 cm. I f PQ = 4.5 cm, find BC

(a) 9cm (b) 13.5cm (c) 12.5cm (d) 11.5cm

 16.If A = 60° and B = 30°, then value of sin A cos B – cos A sin B is
1 3 2 3
(a) 2 (b) 2 (c) 5 (d) 4

17. If 2 sin 2θ = then the value of cosecθ is

1
(a) 1 (b) 2 (c) 0 (d) 2

3𝑠𝑖𝑛𝐴+𝑐𝑜𝑠𝐴
18. If 3tanA = 4, then value of 3𝑠𝑖𝑛𝐴−𝑐𝑜𝑠𝐴 is
12 5 1 7
(a) (b) 3 (c) 2 (d) 5
5

DIRECTION: In the question numbers 19 and 20, a statement of assertion (A) is followed by a statement
of Reason. Choose the correct options.
a. Both A and B are true and R is the correct explanation for A.
b. Both A and B 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.

1 1
19.Statement (A): A quadratic polynomial having 2 and 3 as its zeroes is 6x2 – 5x +1.

Statement (R): Quadratic polynomial having α and β as zeroes is given by

p(x) = k[x2 – (α +β)x +αβ], where k is a nonzero constant.

20.Statement (A) : If ac ≠ 0, then atleast one of the two equations ax2 + bx + c = 0 and
ax2 + bx – c = 0 has real and distinct roots.
Statement (R) : A quadratic equation has real and distinct roots if its discriminant
is positive.
SECTION- B
21.(a) If one zero of quadratic polynomial 4x2 - 2x + (k - 4) is reciprocal of other. Find the
value of k.
OR
(b) Find a quadratic polynomial whose one zero is 7 and sum of zeroes is –18.
22. Solve for x and y : x – 5y =7 and 3x + 15y = 9

23.The difference of the squares of two numbers is 45. The square of the smaller number is
4 times the larger number. Find the numbers.
24. (a) If 4 tanA = 3, Evaluate sec2A + cosec2A
OR
1 1
(b) If sin (A – B) = 2 , cos (A + B) = 2 , find A and B.
25. Prove that: tan2 A + cot2 A + 2 = sec2 A cosec2 A

SECTION- C

26.(a) Prove 𝑡ℎ𝑎𝑡 √5 is irrational.

OR

(b) The LCM of two numbers is 14 times their HCF. The sum of HCF and LCM is 600.
If one number is 280, then find the other number.
1 1
27. If α and β are the zeroes of x2 + 7x +12, then find the value of 𝛼 + - 2αβ.
𝛽

28. 7 audio cassettes and 3 video cassettes cost ₹ 1110. 5 audio cassettes and 4 video cassettes cost
₹ 1350. Find the cost of an audio cassette and a video cassette.

29. Two years ago, Salim was thrice as old as his daughter and six years later, he will be
Four years older than twice her age. How old are they now?

30.(a) The distribution below gives the weight of 30 students of a class, find the median weight

of the students.

Weight (in 40-45 45-50 50-55 55-60 60-65 65-70 70-75

Kg)

No of 2 3 8 6 6 3 2
Students

OR

(b) A student noted the number of cars passing through a spot on a road for 100

Periods each of 3 minutes and summarised in the table given below, Find the

Mode of the data.

No. of 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80

cars

frequency 7 14 13 12 20 11 15 8
31.In the given figure 𝑃𝑄 ∥ 𝐵𝐴; 𝑃𝑅 ∥ 𝐶𝐴. If PD = 12 cm. Find BD X CD

SECTION-D
32.(a) Solve the following pair of linear equations graphically x + 3y = 6 : 2x – 3y = 12. Also, find
The area of the triangle formed by the lines representing the given equations with y-axis.
OR

(b) A man has 70 notes in all of ₹10 and ₹20 denominations. If the total worth of the notes is
₹900, find out graphically how many notes of each kind does he have?

33. The mean of the following frequency distribution is 50, but the frequencies f1 and f2 in classes
20 – 40 and 60 – 80 respectively are not known. Find the missing frequencies, if the sum of all the
frequencies are 120.

variable 0 - 20 20 - 40 40 - 60 60 - 80 80 - 100
Frequency 17 f1 32 f2 19

34. (a) ABCD is a trapezium in which ABIIDC and its diagonals intersect each other at point O.
𝐴𝑂 𝐶𝑂
using BPT show that = 𝐷𝑂 .If AO = 3x – 19, OB = x – 4, OC = x – 3 and OD = 4.
𝐵𝑂

Find x

OR

(b) Sides AB and AC and median AD of a ∆ABC are respectively proportional to sides PQ

and PR and median PM of another ∆PQR. Show that ∆ABC ~ ∆PQR.

𝑡𝑎𝑛𝜃 𝑐𝑜𝑡𝜃
35. Prove that 1−𝑐𝑜𝑡𝜃 + = 1 + secθcosecθ
𝑡𝑎𝑛𝜃−1
 SECTION-E

36. CASE STUDY 1

A Pizza Journey: Neeraj and Hrithik's Delicious Lunch

Looking to satiate their hunger, Neeraj and Hrithik made their way to a quaint pizza shop located
nearby. Little did they know that this shop had an intriguing method for determining the price of their
pizzas. Each day, the cost of a pizza is equal to 4 more than twice the total number of pizzas produced.
On that particular day, the total cost of producing pizzas amounted to 448 rupees.

1. Write the quadratic equation for the above given situation.

2. What is the nature of roots for the above situation.
3. Find the number of pizzas produced.
OR
Find the cost of each pizza.

37. CASE STUDY 2

Exploring the Realm of Mathematics:

At your esteemed school, an exciting mathematics exhibition is underway, captivating the minds of
students and teachers alike. As you manoeuvre through the various exhibits, you come across your
friend who is diligently working on a unique model of a factor tree. The intricate construction of this
model, showcasing the fundamental principles of prime factorization, has proven to be quite
challenging for your friend. Your friend turns to you, seeking your assistance in completing this
intricate task. Eager to lend a helping hand, you step up to the challenge.
 x

5 2783

y 253

11 Z

1. What will be the value of x in the factor tree?

2. What will be the value of y in the factor tree?
3. What is the product of x, y and z?
OR
Justify why 4n cannot end with digit 0?

38 CASE STUDY 3

Analyzing the Distance Covered by Existing Buses for Sustainable

Transportation Planning

In their quest to embrace sustainable transportation, the transport department of a bustling

city is eager to add electric buses to their fleet. To ensure efficiency and reliability,
they have embarked on a thorough analysis of the distance covered by the existing public
transport buses in a single day. To gather accurate and comprehensive data, a diligent
effort has been made to record the distance travelled by 50 of these buses within a
specified timeframe. This data will serve as a valuable foundation for informed decision-
making, paving the way for a greener and more eco-friendly transport system for the
city's residents.
Daily distance travelled
150-155 155-160 160-165 165-170 170-175 175-180
in km
No. of buses 12 n 10 p q 2
cf m 25 x 43 48 y

Answer the Following questions:

1. Find the value of y and m.

2. Find the value of n and p.
𝑥𝑖 − 20
3. If 𝑢𝑖 = , ∑𝑓𝑖 𝑢𝑖 = 30, ∑𝑓𝑖 = 40, then find the value of 𝑥.
̅
10

OR

If the difference of mode and median of a data is 24, then find the difference of median and mean.