TEST FOR CLASS X (Mathematics I Term )

Class 10 – Mathematics

Maximum Marks: 40 Time Allowed: 90 minutes

General Instructions:-
1. Section A consists of 43 questions of 1 mark each. Attempt any 40 questions.
2. There is no negative marking.

1. The pairs of equations 9x + 3y + 12 = 0 and 18x + 6y + 26 = 0 have

(a) Unique solution

(b) Exactly two solutions

(d) No solution

2. A fraction becomes 1/3 when 1 is subtracted from the numerator and it becomes 1/4 when 8 is added to its denominator. The fraction obtained is:

(a) 3/12

(b) 4/12

(d) 7/12

3. The angles of cyclic quadrilaterals ABCD are: A = (6x+10), B=(5x)°, C = (x+y)° and D=(3y-10)°. The value of x and y is:

(a) x=20° and y = 10°

(b) x=20° and y = 30°

(d) x=15° and y=15°

4. A pair of linear equations which has a unique solution x = 2, y = -3 is

(a) x + y = -1; 2x – 3y = -5

(b) 2x + 5y = -11; 4x + 10y = -22

(d) x – 4y – 14 = 0; 5x – y – 13 = 0

5. Two numbers are in the ratio 5 : 6. If 8 is subtracted from each of the numbers, the ratio becomes 4 : 5. Then the numbers are:

(A) 40, 42

(B) 42, 48

(D) 44, 50

Mathematics

6. The sum of a two digit number is 8. The number obtained by reversing the digits exceeds the number by 18. Then the given number is :

(a) 53
(b) 35
(c) 26
(d) 62

7. The probability of event equal to zero is called;

(a) Unsure event

(b) Sure Event

(d) Independent event

8. If two dice are thrown in the air, the probability of getting sum as 3 will be

(a) 2/18

(b) 3/18

(d) 1/36

9. A card is drawn from a deck of 52 cards. The event E is that card is not an ace of hearts. The number of outcomes favourable to E is

(a) 1

(b) 13

(d) 51

Mathematics

10. Cards marked with numbers 2 to 101 are placed in a box and mixed thoroughly. One card is drawn from this box randomly, then the probability that the number on card is a perfect square.

(a) 9/100

(b) 1/10

(d) 19/100

11. A game consists of tossing a one rupee coin 3 times and noting its outcome each time. Aryan wins if all the tosses give the same result i.e. three heads or three tails and loses otherwise. Then the probability that Aryan will lose the game.

(a) 3/4

(b) 1/2

(d) 1/4

Mathematics

12. D and E are the midpoints of side AB and AC of a triangle ABC, respectively and BC = 6 cm. If DE || BC, then the length (in cm) of DE is:

(a) 2.5

(b) 3

(d) 6

13. Corresponding sides of two similar triangles are in the ratio of 2:3. If the area of small triangle is 48 sq.cm, then the area of large triangle is:

(a) 230 sq.cm.

(b) 106 sq.cm

(d) 108 sq.cm

14. The height of an equilateral triangle of side 5 cm is:

(a) 4.33 cm

(b) 3.9 cm

(d) 4 cm

15. In triangle ABC, ∠BAC = 90° and AD ⊥ BC. Then

(A) BD . CD = BC²

(B) AB . AC = BC²

(D) AB . AC = AD²

Mathematics

16. In triangles ABC and DEF, ∠B = ∠E, ∠F = ∠C and AB = 3 DE. Then, the two triangles are

(a) congruent but not similar

(b) similar but not congruent

(d) congruent as well as similar

17. In a square of side 10 cm, its diagonal = …

(a) 15 cm

(b) 10√2 cm

(d) 12 cm

18. If the sum of the areas of two circles with radii R₁ and R₂ is equal to the area of a circle of radius R, then

(a) R₁ + R₂ = R

(b) R₁² + R₂² = R²

(d) R₁² + R₂² < R²

19. A steel wire when bent in the form of a square encloses an area of 121 cm². If the same wire is bent in the form of a circle, then the circumference of the circle is

(a) 88 cm
(b) 44 cm
(c) 22 cm
(d) 11 cm

Mathematics

20. Area of the largest triangle that can be inscribed in a semi-circle of radius r units, in square units is:

(a) r²

(b) 1/2r²

(d) √2r²

21. If a circular grass lawn of 35 m in radius has a path 7 m wide running around it on the outside, then the area of the path is

Option A : 1450 sq cm

Option B : 1576 sq cm

Option C : 1694 sq cm

Option D : 3368 sq cm

22.Two circles touch externally. The sum of their areas is 58 cm² and the distance between their centres is 10 cm. Find the radii of the two circles
(A) 6 cm, 4 cm
(B) 7cm, 3cm
(C) 9 cm , 1 cm
(D) 8 cm ,2 cm

23. AOBC is a rectangle whose three vertices are vertices A (0, 3), O (0, 0) and B (5, 0). The length of its diagonal is

(A) 5

(B) 3

(D) 4

24.If sin θ + sin² θ = 1, then cos² θ + cos⁴ θ = ..

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

25. If sec A + tan A = x, then sec A =

26. If a pole 6m high casts a shadow $2\sqrt{3}$ m long on the ground, then the sun’s elevation is

(A) 60°

(B) 45°

(D)90°

27. If ∆ABC is right angled at C, then the value of cos(A+B) is

(a) 0

(b) 1

(d) $\frac{\sqrt{3}}{2}$

28. If sin A + sin²A = 1, then the value of the expression (cos²A + cos⁴A) is

(a) 1

(b) $\frac{1}{2}$

(d) 3

Mathematics

Case-1. ( Q 29 TO Q 33 )

Mahesh works as a manager in a hotel. He has to arrange seats in hall for a function. A hall has a certain number of chairs. Guests want to sit in different groups like in pairs, triplets, quadruplets, fives and sixes etc. When Mahesh arranges chairs in such pattern like in 2’s, 3’s, 4’s 5’s and 6’s then 1, 2, 3, 4 and 5 chairs are left respectively. But when he arranges in 11’s, no chair will be left.

29. In the hall, how many chairs are available?

(a) 407

(b) 143

(d) 209

30. If one chair is removed, which arrangements are possible now?

(a) 2

(b) 3

(c) 4

(d) 5

31. If one chair is added to the total number of chairs, how many chairs will be left when arranged in 11’s.

(a) 1

(b) 2

(c) 3

(d) 4

32. How many chairs will be left in original arrangement if same number of chairs will be arranged in 7’s?

(a) 0

(b) 1

(c) 2

(d) 3

33. How many chairs will be left in original arrangement if same number of chairs will be arranged in 9’s?

(a) 8

(b) 1

(d) 3

Mathematics

Case Study 2( Q 34 TO 38. )

2. Indian Army is the third biggest military contingent in the World next to USA and China. However, there are many firsts that make Indian army stand out in the world, making us all Indians very proud. Knowing them, will help you celebrate Republic day with greater vigour and gratitude. On 71th republic day Parade in Delhi Captian RS Meel is planing for parade of following two group:

(a) First group of Army contingent of 624 members behind an army band of 32 members.

(b) Second group of CRPF troops with 468 soldiers behind the 228 members of bikers.

These two groups are to march in the same number of columns. This sequence of soldiers is followed by different states Jhanki which are showing the culture of the respective states.

34. What is the maximum number of columns in which the army troop can march?

(a) 8

(b) 16

(c) 4

(d) 32

35. What is the maximum number of columns in which the CRPF troop can march?

(a) 4

(b) 8

(c) 12

(d) 16

36. What is the maximum number of columns in which total army troop and CRPF troop together can march past?

(a) 2

(b) 4

(d) 8

37. What should be subtracted with the numbers of CRPF soldiers and the number of bikers so that their maximum number of column is equal to the maximum number of column of army troop?

(a) 4 Soldiers and 4 Bikers

(b) 4 Soldiers and 2 Bikers

(d) 2 Soldiers and 2 Bikers

Mathematics

38. What should be added with the numbers of CRPF soldiers and the number of bikers so that their maximum number of column is equal to the maximum number of column of army troop?

(a) 4 Soldiers and 4 Bikers

(b) 12 Soldiers and 12 Bikers

(d) 12 Soldiers and 6 Bikers

( Q 39. TO Q 43 )

For the box to satisfy certain requirements, its length must be three meter greater than the width, and its height must be two meter less than the width.

39. If width is taken as x , which of the following polynomial represent volume of box ?

(a) x² – 5x – 6

(b) x³ + x² − 6x

(d) x² + x − 6

40. Which of the following polynomial represent the area of paper sheet used to make box ?

(a) x² – 5x – 6

(b) 6x² + 4x − 12

(d) 6x² + 3x − 4

41. If it must have a volume of 18 unit, what must be its length ?

(a) 6 unit

(b) 3 unit

(d) 2 unit

42. At a volume of 18 cubic unit, what must be its height ?

(a) 1 unit

(b) 3 unit

(d) 4 unit

43. If box is made of a paper sheet which cost is 100 rs per square unit, what is the cost of paper?

(a) Rs 5400

(b) Rs 10800

(d) Rs 3400

Mathematics

1. The pairs of equations 9x + 3y + 12 = 0 and 18x + 6y + 26 = 0 have

(a) Unique solution

(b) Exactly two solutions

(d) No solution

Answer: 1 (d) No solution

2. A fraction becomes 1/3 when 1 is subtracted from the numerator and it becomes 1/4 when 8 is added to its denominator. The fraction obtained is:

(a) 3/12

(b) 4/12

(d) 7/12

Answer: 2 (c) 5/12

3. The angles of cyclic quadrilaterals ABCD are: A = (6x+10), B=(5x)°, C = (x+y)° and D=(3y-10)°. The value of x and y is:

(a) x=20° and y = 10°

(b) x=20° and y = 30°

(d) x=15° and y=15°

Answer: 3 (b) x=20° and y = 30°

4. A pair of linear equations which has a unique solution x = 2, y = -3 is

(a) x + y = -1; 2x – 3y = -5

(b) 2x + 5y = -11; 4x + 10y = -22

(d) x – 4y – 14 = 0; 5x – y – 13 = 0

Answer: 4(b) 2x + 5y = -11; 4x + 10y = -22

5. Two numbers are in the ratio 5 : 6. If 8 is subtracted from each of the numbers, the ratio becomes 4 : 5. Then the numbers are:

(A) 40, 42

(B) 42, 48

(D) 44, 50

Answer:5 (C)

6. The sum of a two digit number is 8. The number obtained by reversing the digits exceeds the number by 18. Then the given number is :

(a) 53
(b) 35
(c) 26
(d) 62

Answer :6(b) 35

Let, the digit at units’ place = x and digit at ten’s place = y => x + y = 8 and lOx + y = 18 + (10y + x) No. = 35.

7. The probability of event equal to zero is called;

(a) Unsure event

(b) Sure Event

(d) Independent event

Answer:7 (c) Impossible even

8. If two dice are thrown in the air, the probability of getting sum as 3 will be

(a) 2/18

(b) 3/18

(d) 1/36

Answer: 8(c) 1/18

9. A card is drawn from a deck of 52 cards. The event E is that card is not an ace of hearts. The number of outcomes favourable to E is

(a) 1

(b) 13

(d) 51

Answer: 9(d) 51

10. Cards marked with numbers 2 to 101 are placed in a box and mixed thoroughly. One card is drawn from this box randomly, then the probability that the number on card is a perfect square.

(a) 9/100

(b) 1/10

(d) 19/100

Answer: 10(b) 1/10

(a) 3/4

(b) 1/2

(d) 1/4

Answer: 11(a) 3/4

12. D and E are the midpoints of side AB and AC of a triangle ABC, respectively and BC = 6 cm. If DE || BC, then the length (in cm) of DE is:

(a) 2.5

(b) 3

(d) 6

Answer: 12(b) 3

13. Corresponding sides of two similar triangles are in the ratio of 2:3. If the area of small triangle is 48 sq.cm, then the area of large triangle is:

(a) 230 sq.cm.

(b) 106 sq.cm

(d) 108 sq.cm

Answer: 13(d) 108 sq.cm

14. The height of an equilateral triangle of side 5 cm is:

(a) 4.33 cm

(b) 3.9 cm

(d) 4 cm

Answer: 14(a) 4.33 cm

15. In triangle ABC, ∠BAC = 90° and AD ⊥ BC. Then

(A) BD . CD = BC²

(B) AB . AC = BC²

(D) AB . AC = AD²

Answer: 15(c) BD . CD = AD²

16. In triangles ABC and DEF, ∠B = ∠E, ∠F = ∠C and AB = 3 DE. Then, the two triangles are

(a) congruent but not similar

(b) similar but not congruent

(d) congruent as well as similar

Answer: 16(b) similar but not congruent

17. In a square of side 10 cm, its diagonal = …

(a) 15 cm

(b) 10√2 cm

(d) 12 cm

Answer: 17(b) 10√2 cm

18. If the sum of the areas of two circles with radii R₁ and R₂ is equal to the area of a circle of radius R, then

(a) R₁ + R₂ = R

(b) R₁² + R₂² = R²

(d) R₁² + R₂² < R²

Answer: 18(b) R₁² + R₂² = R²

19. A steel wire when bent in the form of a square encloses an area of 121 cm². If the same wire is bent in the form of a circle, then the circumference of the circle is

(a) 88 cm
(b) 44 cm
(c) 22 cm
(d) 11 cm

Answer: 19(b) 44 cm

20. Area of the largest triangle that can be inscribed in a semi-circle of radius r units, in square units is:

(a) r²

(b) 1/2r²

(d) √2r²

Answer: 20(a) r²

21. If a circular grass lawn of 35 m in radius has a path 7 m wide running around it on the outside, then the area of the path is

Option A : 1450 sq cm

Option B : 1576 sq cm

Option C : 1694 sq cm

Option D : 3368 sq cm

Answer 21:- Option(C) 1694 sq cm

23. AOBC is a rectangle whose three vertices are vertices A (0, 3), O (0, 0) and B (5, 0). The length of its diagonal is

(A) 5

(B) 3

(D) 4

Answer: 23(C)

24.If sin θ + sin² θ = 1, then cos² θ + cos⁴ θ = ..

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

Answer :- 24(c)

25. If sec A + tan A = x, then sec A =

Answer :- 25(d)

26. If a pole 6m high casts a shadow $2\sqrt{3}$ m long on the ground, then the sun’s elevation is

(A) 60°

(B) 45°

(D)90°

Answer:26 (A)

27. If ∆ABC is right angled at C, then the value of cos(A+B) is

(a) 0

(b) 1

(d) $\frac{\sqrt{3}}{2}$

Answer: 27(a)

28. If sin A + sin²A = 1, then the value of the expression (cos²A + cos⁴A) is

(a) 1

(b) $\frac{1}{2}$

(d) 3

Answer: 28(a)

Case-1. ( Q 29 TO Q 33 )

29. In the hall, how many chairs are available?

(a) 407

(b) 143

(d) 209

Ans 29.: By dividing all the options by 2, 3, 4, 5, 6 and 11, we will get that 539 is the only option which leaves remainder 1, 2, 3, 4, 5, 0 respectively.

Thus (c) is correct option.

30. If one chair is removed, which arrangements are possible now?

(a) 2

(b) 3

(c) 4

(d) 5

Ans 30:-After removing 1 chair, we are left with 538 chairs. On arranging chairs in pair of 3’s, 4’s, 5’s, 6’s, 11’s ; 1, 2, 3 ,4, 10 chairs are left. So, only pair of 2 chairs is possible now.

Thus (a) is correct option.

31. If one chair is added to the total number of chairs, how many chairs will be left when arranged in 11’s.

(a) 1

(b) 2

(c) 3

(d) 4

Ans 31:-539 chairs are already arranged in pair of 11’s .On adding 1 extra chair, that 1 chair will be left only.

Thus (a) is correct option.

32. How many chairs will be left in original arrangement if same number of chairs will be arranged in 7’s?

(a) 0

(b) 1

(c) 2

(d) 3

Ans 32:-539 is divisible by 7 and remainder is zero, so arranging chairs in pair of 7’s, no chair will be left.

Thus (a) is correct option.

33. How many chairs will be left in original arrangement if same number of chairs will be arranged in 9’s?

(a) 8

(b) 1

(d) 3

Ans 33:-539 is divisible by 9 and remainder is 8, so arranging chairs in pair of 9’s, 8 chair will be left.

Thus (a) is correct option.

Case Study 2( Q 34 TO 38. )

(a) First group of Army contingent of 624 members behind an army band of 32 members.

(b) Second group of CRPF troops with 468 soldiers behind the 228 members of bikers.

These two groups are to march in the same number of columns. This sequence of soldiers is followed by different states Jhanki which are showing the culture of the respective states.

34. What is the maximum number of columns in which the army troop can march?

(a) 8

(b) 16

(c) 4

(d) 32

Ans 34:-We will find the HCF (624, 32) = 16

Thus (b) is correct option.

35. What is the maximum number of columns in which the CRPF troop can march?

(a) 4

(b) 8

(c) 12

(d) 16

Ans 35- We will find the HCF (228, 468) = 12.

Thus (c) is correct option.

36. What is the maximum number of columns in which total army troop and CRPF troop together can march past?

(a) 2

(b) 4

(d) 8

Ans 36- HCF(624, 32, 228, 468) = 4

Alternatively we can find,

HCF (16, 12) = 4

Thus (b) is correct option.

37. What should be subtracted with the numbers of CRPF soldiers and the number of bikers so that their maximum number of column is equal to the maximum number of column of army troop?

(a) 4 Soldiers and 4 Bikers

(b) 4 Soldiers and 2 Bikers

(d) 2 Soldiers and 2 Bikers

Ans 37-Maximum number of column of army troop is 16. But 228 and 468 are not divisible by 16. If we subtract 4 from 228 and 468, both(224 and 464) are divisible by 16.

Thus (a) is correct option.

38. What should be added with the numbers of CRPF soldiers and the number of bikers so that their maximum number of column is equal to the maximum number of column of army troop?

(a) 4 Soldiers and 4 Bikers

(b) 12 Soldiers and 12 Bikers

(d) 12 Soldiers and 6 Bikers

Ans 38:-Maximum number of column of army troop is 16. But 228 and 468 are not divisible by 16. If we add 12 from 228 and 468, both(240 and 480) are divisible by 16.

Thus (b) is correct option.

( Q 39. TO Q 43 )

For the box to satisfy certain requirements, its length must be three meter greater than the width, and its height must be two meter less than the width.

39. If width is taken as x , which of the following polynomial represent volume of box ?

(a) x² – 5x – 6

(b) x³ + x² − 6x

(d) x² + x − 6

Ans 39. V(x) = x(x + 3) (x − 2) = x(x² + x − 6) = x³ + x² − 6x

Thus (d) is correct option.

40. Which of the following polynomial represent the area of paper sheet used to make box ?

(a) x² – 5x – 6

(b) 6x² + 4x − 12

(d) 6x² + 3x − 4

Ans 40. S(x) = 2^LW+WH + HLh
= 2[x(x + 3) + (x + 3) (x − 2) + x(x − 2)]
= 2[x² + 3x + x2 + x − 6 + x² − 2x]
= 2(3x² + 2x − 6) = 6×2 + 4x − 12

Thus (b) is correct option.

41. If it must have a volume of 18 unit, what must be its length ?

(a) 6 unit

(b) 3 unit

(d) 2 unit

Ans41. We have

V^xh = x³+ x²− 6

18 = x³+ x²− 6x
x³+ x²− 6x − 18 = 0
x³− 3x²+ 4x²− 12x + 6x − 18 = 0
x² (x−3)+ 4x(x − 3)+ 6(x − 3) = 0
(x − 3)^x²+ 4x + 6h = 0
Thus width is 3 unit.
Length = x + 3 = 6 m
Thus (a) is correct option.

42. At a volume of 18 cubic unit, what must be its height ?

(a) 1 unit

(b) 3 unit

(d) 4 unit

Ans42. Height = x − 2 = 3 − 2 = 1m

Thus (a) is correct option.

43. If box is made of a paper sheet which cost is 100 rs per square unit, what is the cost of paper?

(a) Rs 5400

(b) Rs 10800

(d) Rs 3400

Ans 43. S(x) = 6x² + 4x − 12 = 6 # 3 # 3 + 4 # 3 − 12 = 54
S = 2(LW+WH + HLh
= 2(6x³ + 3×1 + 1×6)
= 2x(18 + 3 + 6) = 2×27 = 54 m2
C = 100#54 = 5400 `
Thus (a) is correct option.

TEST FOR CLASS X (Mathematics I Term )

6. The sum of a two digit number is 8. The number obtained by reversing the digits exceeds the number by 18. Then the given number is :

6. The sum of a two digit number is 8. The number obtained by reversing the digits exceeds the number by 18. Then the given number is :

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