Class X - Mathematics (Basic) Practice Paper - Answers

Subject: Mathematics (Basic)

Class X

Time Allowed: 3 hours

Max 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 02 marks each.

4. Section C has 6 questions carrying 03 marks each.

5. Section D has 4 questions carrying 05 marks each.

6. Section E has 3 case based integrated units of assessment (04 marks each) with sub-parts of the values of 1, 1 and 2 marks each respectively.

7. All Questions are compulsory. However, an internal choice in 2 Qs of 5 marks, 2 Qs of 3 marks and 2 Questions of 2 marks has been provided. An internal choice has been provided in the 2marks questions of Section E

8. Draw neat figures wherever required. Take π = 22/7 wherever required if not stated.

SECTION A

Section A consists of 20 questions of 1 mark each.

1. If α and β are the zeroes of the polynomial 2x² – 13x + 6, then α + β is equal to

(a) -3

(b) 3

(d) -13/2

View Answer

Ans. (c)

2. The point on the x -axis which is equidistant from the points A (- 2, 3) and B (5, 4) is

(a) (0, 2)

(b) (2, 0)

(d) (- 2, 0)

View Answer

Ans. (b)

3. The value of k for which the system of linear equations x + 2y = 3, 5x + ky + 7 = 0 is inconsistent is

(a) -14/3

(b) 2/5

(d) 10

View Answer

Ans. (d)

4. The roots of the quadratic equation x² – 0.04 = 0 are

(a) ±0.2

(b) ±0.02

(d) 2

View Answer

Ans. (a)

5. A tree casts a shadow 15 m long on the level of ground, when the angle of elevation of the sun is 45c. The height of a tree is

(a) 10 m

(b) 14 m

(d) 15 m

View Answer

Ans. (d)

6. A sector is cut from a circular sheet of radius 100 cm, the angle of the sector being 240c. If another circle of the area same as the sector is formed, then radius of the new circle is

(a) 79.5 cm

(b) 81.5 cm

(d) 88.5 cm

View Answer

Ans. (b)

7. The quadratic equation 2x² – √5 x + 1 = 0 has

(a) two distinct real roots

(b) two equal real roots

(d) more than 2 real roots

View Answer

Ans. (c)

8. If the common difference of an AP is 5, then what is a a₁₈ – a₁₃?

(a) 5

(b) 20

(d) 30

View Answer

Ans. (c)

9. Δ ABC is an equilateral triangle with each side of length 2p. If AD ⊥ BC then the value of AD is

(a) √3

(b) √3 p

(d) 4p

View Answer

Ans. (b)

10. In figure, O is the centre of circle. PQ is a chord and PT is tangent at P which makes an angle of 50° with ∠POQ is

(a) 130°

(b) 90°

(d) 75°

View Answer

Ans. (c)

11. If cos (α + β) = 0, then sin (α – β) can be reduced to

(a) cosβ

(b) cos 2β

(d) sin 2α

View Answer

Ans. (b)

12. The base radii of a cone and a cylinder are equal. If their curved surface areas are also equal, then the ratio of the slant height of the cone to the height of the cylinder is

(a) 2 : 1

(b) 1 : 2

(d) 3 : 1

View Answer

Ans. (a)

13. In a frequency distribution, the mid value of a class is 10 and the width of the class is 6. The lower limit of the class is

(a) 6

(b) 7

(d) 12

View Answer

Ans. (b)

14. If a card is selected from a deck of 52 cards, then the probability of its being a red face card is

(a) 3/26

(b) 3/13

(d) ½

View Answer

Ans. (a)

15. The point P on x -axis equidistant from the points A (-1, 0) and B (5, 0) is

(a) (2, 0)

(b) (0, 2)

(d) (-3, 5)

View Answer

Ans. (a)

16. If one zero of the polynomial (3x² + 8x + k) is the reciprocal of the other, then value of k is

(a) 3

(b) -3

(d) -1/3

View Answer

Ans. (a)

17. If 3x + 4y : x + 2y = 9:4, then3x + 5y : 3x – y is equal to

(a) 4 : 1

(b) 1 : 4

(d) 1 : 7

View Answer

Ans. (c)

18. The distance between the points (a cos θ + b sin θ, 0), and (0, a sin θ – b cos θ) is

(a) a²+ b²

(b) a²– b²

(c) √ (a²+ b²)

(d) √ (a²– b²)

View Answer

Ans. (c)

19. Assertion: Sum of first 10 terms of the arithmetic progression -0.5, -1.0, -1.5, ………. is 31.

Reason: Sum of n terms of an AP is given as S_n = n/2[2a + (n – 1) d] where a is first term and d common difference.

(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).

(d) Assertion (A) is false but reason (R) is true

View Answer

Ans. (d)

20. Assertion: When a positive integer a is divided by 3, the values of remainder can be 0, 1 or 2.

Reason: According to Euclid’s Division Lemma a = bq + r, where 0 ≤ r < b and r is an integer.

(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).

(d) Assertion (A) is false but reason (R) is true

View Answer

Ans. (a)

Section – B

Section B consists of 5 questions of 2 marks each.

21. ABCD is a trapezium in which AB || CD and its diagonals intersect each other at the point O. Show that, AO/BO =CO/DO

22. In given figure, AB is the diameter of a circle with centre O and AT is a tangent. If ∠AOQ = 58c, find ∠ATQ.

View Answer

Ans.

23. Find the value of cos 2θ, if 2 sin 2θ = √3.

View Answer

Ans.

24. Find the mean of the following distribution:

View Answer

Ans.

Find the mean of the following data:

View Answer

Ans.

25. Show that 5√6 is an irrational number

Write a rational number between √2 and √3.

Section – C

Section C consists of 6 questions of 3 marks each.

26.

View Answer

Ans.

27. If 1 + 3 sin² θ = 3 sin θ cos θ, prove that tan θ = 1 or ½.

View Answer

28. A horse is tethered to one corner of a rectangular field of dimensions 70 m x 52 m, by a rope of length 21 m. How much area of the field can it graze?

View Answer

Ans.

In the given figure, a chord AB of the circle with centre O and radius 10 cm, that subtends a right angle at the centre of the circle. Find the area of the minor segment AQBP. Hence find the area of major segment ALBQA. (Use π = 3.14)

View Answer

Ans.

29. Find the mode of the following frequency distribution:

View Answer

Ans.

30. Find the ratio in which the segment joining the points (1, -3) and (4, 5) is divided by x -axis? Also find the coordinates of this point on x -axis.

View Answer

The vertices of ΔABC are A (6, -2), B (0, -6) and C (4, 8). Find the co-ordinates of mid-points of AB, BC and AC.

View Answer

31. Write the smallest number which is divisible by both 306 and 657.

View Answer

Ans.

Section – D

Section D consists of 4 questions of 5 marks each

32. Determine graphically the coordinates of the vertices of triangle, the equations of whose sides are given by 2y – x = 8, 5y – x = 14 and y – 2x = 1.

View Answer

Ans.

Draw the graphs of the equations x – y + 1 = 0 and 3x + 2y – 12 = 0. Determine the co-ordinates of the vertices of the triangle formed by these lines and the X-axis and shade the triangular region.

View Answer

Ans.

33. Two tangents PA and PB are drawn from an external point P to a circle with centre O, such that ∠APB = ∠x and ∠AOB = ∠y. Prove that opposite angles are supplementary.

View Answer

Ans.

34. The person standing on the bank of river observes that the angle of elevation of the top of a tree standing on opposite bank is 60°. When he moves 30 m away from the bank, he finds the angle of elevation to be 30°. Find the height of tree and width of the river.

View Answer

Ans.

As observed from the top of a 100 m high light house from the sea-level, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the light house, find the distance between the two ships [Use √3 = 1.732] View Answer

Ans.

35. A hemispherical depression is cut from one face of a cubical block, such that diameter l of hemisphere is equal to the edge of cube. Find the surface area of the remaining solid.

View Answer

Ans.

Section – E

Case study based questions are compulsory.

36. Maximum Profit: A kitchen utensils manufacturer can produce up to 200 utensils per day. The profit made from the sale of these utensils can be modelled by the function P(x) = -0.5x + 175x – 330, where P(x) is the profit in Rupees, and x is the number of utensils made and sold. Based on this model,

(i) Find the y -intercept and explain what it means in this context.

View Answer

Ans.

(ii) Find the x -intercepts and explain what they mean in this context.

View Answer

Ans.

(iii) How many utensils should be sold to maximize profit?

View Answer

Ans.

What is the maximum profit?

View Answer

Ans. The maximum profit is Rs 12012.5

37. Tania is very intelligent in math’s. She always try to relate the concept of math’s in daily life. One day she plans to cross a river and want to know how far it is to the other side. She takes measurements on her side of the river and make the drawing as shown below.