DAV INSTITUTIONS SOUTH ZONE – D SUB CODE:(041)

PRE - BOARD EXAMINATION2024-2025

CLASS : X MARKS: 80M
SUBJECT: MATHEMATICS TIME :3Hrs
General Instructions:
1. All Questions are compulsory
2. Qno.1 to Qno.20 carry 1 mark each.
3. Qno.21 to Qno.25 carry 2 marks each.
4. Qno.26 to Qno.31 carry 3 marks each.
5. Qno.32 to Qno.35 carry 5 marks each.
6. Qno.36 to Qno.38 Case-Based Questions carry 4 marks each.
SECTION- A ( 20 X 1 = 20M )
1. If two positive integers a and b are written as a = x²y² and b = xy³ where x and y are prime
numbers, then the LCM (a, b) is
a) xy b) xy ² c) x³y² d) x²y³
2. A Quadratic polynomial whose one zero is 5 and product of zeroes is 0, is
a) x² - 5 b) x²-5x c) 5x² + 1 d) x²+5x
3. From an external point P, tangents PA and PB are drawn to a circle with
centre O. If CD is tangent to a circle at point E and PA=14cm, then
perimeter of ΔPCD =
a) 28 cm b) 27 cm c) 26 cm d) 25 cm
4. For what value of k, the following pair of equations represent
coincident lines ? x - 2y =3 and -3x + ky = - 9
a) -6 b) -3 c) 3 d) 6
5. The roots of the equation x2 + 3x - 10 = 0 are:
a) 2, -5 b) -2, 5 c) 2, 5 d) -2, -5
6. If k, 2k-1 and 2k+1 3 are 3 consecutive terms of an A .P, then the value of k is
a)2 b) 3 c) -3 d) 5
7. In the given figure, AD = 1.28 cm, DB = 2.56 cm, AE = 0.64 cm,
if DE || BC, then EC is
a) 0.64cm b) 1.32cm c) 2.56cm d) 1.28 cm

8. Three vertices of a parallelogram ABCD are A (1, 4), B (-2, 3), C (5, 8). The ordinate of
the fourth vertex D is
a)8 b) 9 c) 7 d) 6

9. In ΔABC, ∠C = 90°, then tan A + tan B is

𝑏2 𝑎2 𝑐2
a) b) a + b c) d)
𝑎𝑐 𝑏𝑐 𝑎𝑏
5
10. O is centre of circle, the area of a sector OAPB is 18 of the area of
the circle, then the angle of the sector is
a) 100° b) 36° c) 18° d) 50°
11. The curved surface area of a cylinder of height 14 cm is 88 cm², then diameter of
the cylinder is
a)8.5 cm b) 1 cm c) 1.5 cm d) 2 cm
12. A bag contains 3 red, 4 black and 1 white ball, a ball is drawn from the bag, the probability
of drawing a green ball is
a) 1/8 b) 1/2 c) 0 d) 3/8
13.If mean = (3 median - mode) k, then the value of k is
a) 1 b) 2 c) 1/2 d) 3/2

1
14. In figure, PA and PB are tangents to the circle having centre C and PA ⊥ PB.
If radius of circle is 4 cm, then length of each tangent is
a) 4 cm b) 3 cm c) 8 cm d) 16 cm
15. If α and β are the roots of the equation x² + 5x + 6 = 0, then the equation whose
roots are α + 1, β + 1 is
a) x²+5x–5=0 b) x²-2 = 0 c) x²+3x +3=0 d) x²+3x +2 = 0

16. Sec θ when expressed in terms of cot θ is

1+cot²θ √1+𝑐𝑜𝑡 2 𝜃 (1−𝑐𝑜𝑡 2 𝜃)
a) b)√(1 + 𝑐𝑜𝑡 2 𝜃 c) d) √
cot θ cot 𝜃 cot θ
17. Which of the following is not a measure of central tendency?
a) Mean b) Median c) Class interval d) Mode

18. If O is the centre of a circle and chord PQ makes an angle 50° with
the tangent PR at the point of contact P, then the angle subtended by
the chord at the centre is
a) 130° b) 100° c) 50° d) 30°

In the question number 19 and 20, a statement of Assertion (A) is followed by a statement of
Reason (R). Choose the correct option.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of
assertion (A).
(b) Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation of
assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true

19. Assertion (A) : The probability that a leap year has 53 Sundays is 2/7.
Reason (R) : The probability that a non-leap year has 53 Sundays is 5/7.

20.Assertion (A) : Total surface area of the cylinder having radius of the base 14 cm and
height 30 cm is 3872 cm².
Reason (R) : If r be the radius and h be the height of the cylinder, then
total surface area = (2πrh + 2πr²)

SECTION B ( 5 X 2M = 10M)

21.The L.C.M. of two nos. is 9 times their H.C.F. and the sum of the L.C.M. and H.C.F. is 500.
Find the H.C.F. of the numbers.
22. If the points are (10, 5), (8, 4), and (6, 6) are the mid points of the sides of triangle, find its
vertices. (OR)
If (1,2), (4,y), (x,6) and (3,5) are the vertices of a parallelogram taken in order, find x and y.
23. The line segment AB joining the points A (3, -4) and B (1, 2) is trisected at the points
P (p, -2) and Q (5, q). Find the values of p and q.
24. Two dice are thrown simultaneously. What is the probability that
(i) 4 will come up on at least once?
(ii) neither prime nor composite on one die even number on other die?
𝟓𝐜𝐨𝐬 𝟐 𝟔𝟎+𝟒𝐬𝐞𝐜 𝟐 𝟑𝟎−𝐭𝐚𝐧𝟐 𝟒𝟓
25. Evaluate (OR) Prove that 𝑠𝑖𝑛2 ∅ + 𝑐𝑜𝑠 2 ∅ = 1
𝐬𝐢𝐧𝟐 𝟑𝟎+𝐜𝐨𝐬 𝟐 𝟑𝟎

2
 SECTION C (6X3M=18M)

26. Which term of the A.P. 3, 15, 27, 39, ... will be 120 more than its 21st term?
(OR)
How many terms of the AP 17, 15,13,…. must be added to get the sum 72? Explain the
double answer.
27. Find the largest possible integer that divides 125, 162, and 259 leaving reminder 5, 6, and 7
respectively.
(OR)
Prove that √5 is an irrational number.
28. Show that the points A (3, 5), B (6, 0), C (1, -3), D (-2, 2) are the vertices of a square.
29. If α, β are zeroes of quadratic polynomial 5x² + 5x + 1, find the values of
(i) α² + β² (ii) α⁻¹ + β⁻¹
1−𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃
30. Prove that 𝑐𝑜𝑠𝜃 + 1−𝑠𝑖𝑛𝜃 = 2 𝑠𝑒𝑐𝜃
31. There are two identical solid cubical boxes of side 7 cm. From the top face of the first cube a
hemisphere of diameter equal to the side of the cube is scooped out. This hemisphere is
inverted and placed on the top of the second cube's surface to form a dome. Find the volume
of each new solid formed.

SECTION D (4x5M=20M)

32. Calculate the mean of the following grouped data:

C I 10-30 30-50 50-70 70-90 90-110 110-130

F 4 10 18 6 10 2

(OR)

Median of the following data is 35 ; ∑ 𝑓𝑖 = 70 , find the values of p , q

CI 0-10 10-20 20-30 30-40 40-50 50-60

F 6 p 14 12 q 10

33. Solve the following system of equations graphically:

3x - 2y - 1 = 0, 2x - 3y + 6 = 0.

34. State and prove Basic Proportionality Theorem.

Using the above theorem, in the given figure, ∠𝐴𝑃𝑄 = ∠𝐴𝐵𝐶, AP = x,
PB = x-2, AQ = x + 2 and QC = x - 1, find the value of x

35. A straight highway leads to the foot of a tower. A man standing on the top
of the 75 m high tower observes two cars at angles of depression of 30°
and 60°, which are approaching the foot of the tower. If one car is exactly behind the other
on the same side of the tower, find the distance between the two cars. (use √3 = 1.73)
(OR)

The angle of elevation of a cloud from a point 60 m above the surface of the water of a lake is
30° and the angle of depression of its shadow in water of lake is 60°. Find the height of the
cloud from the surface of water.

3
 SECTION E (3x4M=12M)

36. Ashok and Harish are very close friends. They decided to go on a long drive with their
families in separate cars. Ashok's car travels at a speed of x km/h, while Harish was driving
the car at a speed of 5 km/h faster than Ashok's car. Ashok took 4 hours more than Harish to
complete the journey of 400 km.

Based on the above information, answer the following questions:

i) Find the distance covered by Harish's car in two hours (in terms of x). (1m)
ii) Make a quadratic equation describing the given situation. (2m)
iii) Find the speed of Ashok's car (in km/h) (1m)
OR
Find the speed of Harish's car (in km/h)

37. In a maths class, the teacher draws two circles that touch each other externally at point K
with centres A and B and radii 5cm and 4cm respectively as shown in the figure.

Based on the above information, answer the following questions.

i) The value of PK (1m)
ii) The value of PA (2m)
iii) The measure of∠𝐾𝑇𝑌 (1m)
OR
Find the distance between A and B

38. A stable owner has four horses. He usually ties these horses with 7 m long rope to pegs at
each corner of a square shaped grass field of 20 m length, to graze in his farm. But tying
with rope sometimes results in injuries to his horses, so he decided to build fence around
the area so that each horse can graze.
Based on the above information, answer the following
questions:
i) Find the area of the square shaped grass field. (1m)
ii) Find the area of the total field in which these horses can
graze if the length of the rope of each horse is 7m? (2m)
iii) What is the area of the field that is left un grazed, if the
length of the rope of each horse is 7m? (1m)

(OR)
Find the area of any one sector

*****