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First Term Exam (2021-22)

Class 10 - Mathematics
Sample Paper 01

Maximum Marks: 40
Time Allowed: 90 minutes

General Instructions:

1. The question paper contains three parts A, B and C.

2. Section A consists of 20 questions of 1 mark each. Attempt any 16 questions.
3. Section B consists of 20 questions of 1 mark each. Attempt any 16 questions.
4. Section C consists of 10 questions based on two Case Studies. Attempt any 8 questions.
5. There is no negative marking.

Section A
1. The least positive integer divisible by 20 and 24 is
a. 480
b. 240
c. 360
d. 120
2. One equation of a pair of dependent linear equations is – 5x + 7y = 2, then the second equation can be
a. – 10x + 14y + 4 = 0
b. – 10x – 14y + 4 = 0
c. 10x – 14y +4 = 0
d. 10x + 14y + 4 = 0
3. The zeros of the quadratic polynomial x2 + 88x + 125 are
a. both negative
b. both positive
c. both equal
d. one positive and one negative
4. The pair of equations 5x – 15y = 8 and 3x – 9y = has
a. infinitely many solutions
b. no solution
c. two solutions
d. one solution

5.
a. 6
b.
c. 5
d. 4
6. If two positive integers p and q can be expressed as p = ab2 and q = a3 b; a, b being prime numbers,
then LCM (p, q) is

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a. a3 b3
b. a3 b2
c. a2 b2
d. ab
7. A quadratic polynomial, the sum of whose zeroes is 0 and one zero is 3, is
a. x2 - 9
b. x2 + 3
c. x2 - 3
d. x2 + 9
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8. The distance between the points (sin , cos ) and (cos , – sin ) is
a. units
b. 2 units
c. units
d. units
9. If α and β are the zeros of the polynomial f(x) = x2 + px + q, then a polynomial having is its
zero is
a. qx2 + px + 1
b. x2 + qx + p
c. x2 − px + q
d. px2 + qx + 1
10. The zeroes of the quadratic polynomial x2 + 9x + 20 are
a. - 4 and 5
b. - 4 and - 5
c. 4 and 5
d. 4 and - 5
11. The probability that it will rain on a particular day is 0.76. The probability that it will not rain on that
day is
a. 0.24
b. 0.76
c. 0
d. 1
12. is
a. a non-terminating repeating decimal
b. a rational number
c. a terminating decimal
d. an irrational number
13. A circle has its centre at the origin and a point P(5, 0) lies on it. Then the point Q(8, 6) lies ________ the
circle.
a. in side
b. out side
c. on
d. None of these

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14. If the centroid of the triangle formed by the points (a, b),(b, c) and (c, a) is at the origin, then a3 + b3 + c3
=
a. 2a
b. 0
c. 3 abc
d. a + b + c
15. On dividing the polynomial x4 - 5x + 6 by 2 - x2, the remainder is
a. -5x - 10
b. 5x - 10
c. - 5x + 10
d. 5x + 10
16. is equal to
a. sin + cos
b. sin - cos
c. 0
d. 1
17. The sum of the digits of a two digit number is 9. Nine times this number is twice the number obtained
by reversing the digits, then the number is
a. 72
b. 27
c. 18
d. 81
18. Three coins are tossed simultaneously. What is the probability of getting exactly two heads?
a.
b.
c.
d.
19. If m2 - 1 is divisible by 8, then m is
a. an odd integer
b. a natural number
c. an even integer
d. a whole number
20. Three consecutive vertices of a parallelogram ABCD are A(1, 2), B(1, 0) and C(4, 0). The co – ordinates of
the fourth vertex D are
a. (– 4, 2)
b. (4, – 2)
c. (4, 2)
d. (– 4, – 2)
Section B
21. The area of the triangle formed by the lines 2x + y = 6, 2x – y + 2 = 0 and the x – axis is
a. 12 sq. units
b. 15 sq. units
c. 10 sq. units
d. 8 sq. units
22. The zeroes of the quadratic polynomial x2 + 99x + 127 are

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a. both negative
b. one positive and one negative
c. both positive
d. both equal
23. is
a. none of these
b. an integer
c. a rational number
d. an irrational number
24. The value of tan15o tan20o tan70o tan75o is
a. 2
b. 0
c. 1
d. -1
25. If x - y = 2 and then
a. x = 6, y = 4
b. x = 7, y = 5
c. x = 5, y = 3
d. x = 4, y = 2
26. If , , are the zeros of the polynomial f(x) = ax3 + bx2 + cx + d, then

a.

b.

c.

d.
27. In a ABC, , AB = 5 cm and AC = 12 cm. Also , Then AD =
a.
b.
c.
d.
28. If the coordinates of a point are (-5, 11), then its abscissa is
a. -5
b. 11
c. 5
d. -11
29. If sec + tan = x, then tan =

a.

b.

c.

d.
30. The area of the triangle formed by the lines
2x + 3y = 12 , x - y = 1 and x = 0 is
a. 6.5 sq. units

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b. 7 sq. units
c. 7.5 sq. units
d. 6 sq. units
31. 3 + is a/an:
a. natural Number
b. integer
c. irrational number
d. rational number
32. If a pole 18 m high casts a shadow 9.6 m long, then the distance of the far end of the shadow from the
top of the pole is
a. 24.8 m.
b. 20.4 m.
c. 20 m.
d. 28.4 m.
33. If (cos + sec ) = then (cos2 + sec2 ) = ?
a.
b.
c.
d.
34. Which point on x-axis is equidistant from the points A(7, 6) and B(-3, 4)?
a. (-4, 0)
b. (0, 4)
c. (0, 3)
d. (3, 0)
35. If a digit is chosen at random from the digits 1, 2, 3, 4, 5, 6, 7, 8, 9, then the probability that it is odd, is
a.
b.
c.
d.
36. A system of two linear equations in two variables is dependent consistent, if their graphs
a. do not intersect at any point
b. cut the x-axis
c. intersect only at a point
d. coincide with each other
37. The exponent of 2 in the prime factorisation of 144, is
a. 4
b. 5
c. 6
d. 3
38. If a cos + b sin = m and a sin - b cos = n, then a2 + b2 =
a. n2 - m2
b. m2 + n2
c. m2 - n2
d. m2n2

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39. 3 rotten eggs are mixed with 12 good ones. One egg is chosen at random. The probability of choosing a
rotten egg is
a.
b.
c.
d.
40. The points A(9, 0), B(9, 6), C(-9, 6) and D(-9, 0) are the vertices of a
a. rhombus
b. trapezium
c. rectangle
d. square
Section C

Question No. 41 to 45 are based on the given text. Read the text carefully and answer the
questions:

Two hotels are at the ground level on either side of a mountain. On moving a certain distance towards
the top of the mountain two huts are situated as shown in the figure. The ratio between the distance
from hotel B to hut-2 to that of hut-2 to mountain top is 3:7.

41. What is the ratio of the perimeters of the triangle formed by both hotels and mountain top to the
triangle formed by both huts and mountain top?
a. 5 : 2
b. 7 : 3
c. 3 : 10
d. 10 : 7
42. The distance between the hotel A and hut-1 is
a. 2.5 miles
b. 4.29 miles
c. 29 miles
d. 1.5 miles
43. If the horizontal distance between the hut-1 and hut-2 is 8 miles, then the distance between the two
hotels is
a. 9 miles
b. 2.4 miles

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c. 11.43 miles
d. 7 miles
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44. If the distance from mountain top to hut-1 is 5 miles more than that of distance from hotel B to
mountain top, then what is the distance between hut-2 and mountain top?
a. 5.5 miles
b. 6 miles
c. 3.5 miles
d. 4 miles
45. What is the ratio of areas of two parts formed in the complete figure?
a. 53:21
b. 49:51
c. 10:41
d. 51:33

Question No. 46 to 50 are based on the given text. Read the text carefully and answer the
questions:

Renu wants to change the design of the floor of her living room which is of dimensions 6 m 4 m and
it is covered with circular tiles of diameters 50 cm each, as shown in the figure.

46. Number of circular tiles along length of room is

a. 11
b. 14
c. 13
d. 12
47. Total number of circular tiles equals
a. 92
b. 94
c. 96
d. 90
48. Area covered by each circular tile is
a. 1960.08 cm2
b. 1980 cm2
c. 1954.28 cm2
d. 1964.28 cm2
49. Area of rectangular floor is

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a. 420000 cm2
b. 240000 cm2
c. 204000 cm2
d. None of these
50. Find the area of the floor that remains uncovered with tiles.
a. 4.6 m2
b. 3.6 m2
c. 5.142 m2
d. 5 m2

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Class 10 - Mathematics
Sample Paper 01

Solution

Section A
1. (d) 120
Explanation: Least positive integer divisible by 20 and 24 is
LCM of (20, 24).
20 = 22 5
24 = 23 3
LCM (20, 24) = 23 3 5 = 120
Thus 120 is divisible by 20 and 24.
2. (c) 10x – 14y +4 = 0
Explanation: If the equation of a pair of dependent linear equations, then
Given: a1 = -5, b1 = -5and c1 = 2.
For satisfying the condition of dependent linear equations, the values of a2, b2 and c2 should be the
multiples of the values of a1, b1 and c1.
The values would be and
The second equation can be 10x - 14y = - 4
3. (a) both negative
Explanation: Given;

Now,

There roots are

Which are both negative.
4. (a) infinitely many solutions
Explanation: Given: and Here

Since all have the same answer .

Therefore, the pair of given linear equations has infinitely many solutions.
5. (d) 4

Explanation:

= 4

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6. (b) a3 b2
Explanation: Let p = ab2 = a b b
And q = a3b = a a a b
LCM of p and q = LCM (ab2, a3b) = a b b a a = a3b2
[Since, LCM is the product of the greatest power of each prime factor involved in the number]
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7. (a) x2 - 9
Explanation: Since and are the zeros of the quadratic polynomials such that

If one of zero is 3 then




Substituting in we get


Let S and P denote the sum and product of the zeros of the polynomial respectively then
S =
S = 0


P = -9
Hence, the required polynomials is
= (x2 - Sx + P)
= (x2 - 0x - 9)
= x2 - 9
8. (a) units
Explanation: Distance between and

=
=
=

= units
9. (a) qx2 + px + 1
Explanation: Let and be the zeros of the polynomial .Then,

And = q
Let S and R denote respectively the sum and product of the zeros of a polynomial whose zeros are
and , then

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Hence, the required polynomial whose sum and product of zeros are S and R is given by

So
10. (b) - 4 and - 5
Explanation: (x2 + 9x + 20) =0 Splitting the middle term, we get
x2 + 5x + 4x + 20 = 0
= x(x + 5) + 4(x + 5) = 0
= (x + 5)(x + 4) = 0
x + 5 = 0 and x + 4 = 0
x = -5 and x = -4
11. (a) 0.24
Explanation: Given: P (It will rain on a particular day) = 0.76
P (It will not rain on a particular day) = 1 - P (It will rain particular day)
= 1 - 0.76 = 0.24
12. (d) an irrational number
Explanation: Let is a rational number.
= , where p and q are some integers and HCF(p, q) = 1 .... (1)

⇒ p2 is divisible by 2
⇒ p is divisible by 2 .... (2)
Let p = 2m, where m is some integer.
= p

⇒ q2 is divisible by 2
⇒ q is divisible by 2 .... (3)
From (2) and (3), 2 is a common factor of both p and q, which contradicts (1).
Hence, our assumption is wrong.
Thus, is an irrational number.
13. (b) out side
Explanation: Given: Coordinates of centre O (0, 0) and Radius is OP.

OP =
=
= = 5 units

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Now, OQ =
=
= = 10 units
Since OQ > OP
Therefore, point Q lies outside the circle.
14. (c) 3 abc
Explanation: Centroid of the triangle formed by the points (a, b), (b, c) and (c, a) is origin (0, 0)

a3 + b3 + c3 = 3abc
a + b + c = 0
Hence a3 + b3 + c3 = 3abc
15. (c) - 5x + 10

Explanation:

16. (a) sin + cos

Explanation: We have,

17. (c) 18
Explanation: Let unit digit = x , Tens digit = y , therefore original no will be 10y + x
Sum of digits are 9 So that x + y = 9 ... (i)
nine times this number is twice the number obtained by reversing the order of the digits 9(10y + x) =
2(10x + y)
90y + 9x = 20 x + 2y
88y - 11x = 0
Divide by 11 we get 8y - x = 0 ... (ii)
Adding equations (i) and (ii), we get
9y = 9
y = = 1
Putting this value in equation 1 we get
x + y = 9

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x + 1 = 9
x = 8
Therefore the number is 10(1) + 8 = 18
18. (c)
Explanation: When 3 coins are tossed, all possible outcomes are HHH, HHT, HTH, THH, HTT, THT, TTH,
TTT.
Their number is 8.
Favourable outcomes are HHT, HTH, TH H. Their number is 3.
P (getting exactly 2 heads) =
19. (a) an odd integer
Explanation: Let a = m2 - 1
Here m can be ever or odd.
Case I: m = Even i.e., m = 2k, where k is an integer,
a = (2k)2 - 1
a = 4k2 - 1
At k = -1, = 4 (-1)2 - 1 = 4 -1 = 3, which is not divisible by 8.
At k = 0, a = 4 (0)2 - 1 = 0 - 1 = -1, which is not divisible by 8, which is not.
Case II: m = Odd i.e., m = 2k + 1, where k is an odd integer.
a = 2k + 1
a = (2k + 1)2 - 1
a 4k2 + 4k + 1 - 1
a = 4k2 + 4k
a = 4k(k + 1)
At k = -1, a = 4(-1)(-1 + 1) = 0 which is divisible by 8.
At k = 0, a = 4(0)(0 + 1) = 4 which is divisible by 8.
At k = 1, a = 4(1)(1 + 1) = 8 which is divisible by 8.
Hence, we can conclude from the above two cases, if m is odd, then m2 - 1 is divisible by 8.
20. (c) (4, 2)
Explanation: Coordinates are given for A(1, 2), B(1, 0) and C(4, 0)
Let coordinates of D be (x, y).
Since diagonals of a parallelogram bisect each other. at point O
Therefore O is the midpoint of diagonal AC
Therefore, coordinates of O will be =
O is also the midpoint of diagonal BD

And Therefore, the required coordinates are (4, 2).

Section B
21. (d) 8 sq. units

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Explanation: Here are the two solutions of each of the given equations.
2x + y = 6,
x -1 2 3

y 8 2 0

2x - y + 2 = 0
x -1 2 3

y 0 6 8

The area bounded by the given lines and axis has been shaded in the graph.
Area of shaded region = Base Height = DC GH = = 8 sq. units
22. (a) both negative
Explanation: As the Discriminant of the given quadratic polynomial x2 + 99x + 127 is less than Zero.
Both the zeros are negative.
23. (d) an irrational number
Explanation: is an irrational number.
If it is rational, then the difference of two rational is rational.
= irrational, which is a contradiction.
Hence, , is an irrational number.
24. (c) 1
Explanation: Given: tan15o tan20o tan70o tan75o
= tan15o tan20o tan(90o - 20o) tan(90o - 15o)
= tan15o tan20o tan20o tan15o
= (tan15o cot15o) (tan20o cot20o)
= 1 1 = 1
25. (a) x = 6, y = 4

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Explanation: We have:
x - y = 2 …(i)
x + y = 10 …(ii)
Now, adding (i) and (ii) we get:
2x = 12
x =
x = 6
Putting the value of x in (ii), we get
6 + y = 10
y = 10 - 6
y = 4

26. (d)
Explanation: We have to find the value of
Given be the zeros of the polynomial f(x) = ax3 + bx2 + cx + d

=
Now

The value of
27. (b)
Explanation: In ABC , AB = 5 cm, AC = 12 cm

( Pythagoras Theorem)
=
=

Now area of

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=
and also area of

28. (a) -5
Explanation: Since x-coordinate of a point is called abscissa.
Therefore, the abscissa is -5.

29. (d)
Explanation: Given, sec + tan = x ...(i)
We know that

...(ii)
Subtracting (ii) from (i)

30. (c) 7.5 sq. units

Explanation: Graph of the equation 2x + 3y - 12 = 0
We have
2x + 3y = 12
2x = 12 - 3y

Putting y = 4
We get
Putting y = 2,
We get
Thus, we have the following table for the points:
x 0 3

y 4 2

Plotting point A(0, 4), B(3, 2) on the graph paper and drawing a line passing through them we obtain a
graph of the equation.
Graph of the equation x - y - 1
We have x - y = 1
x = 1 + y
Thus, we have the following table for the points for the line x - y = 1
x 1 0

y 0 -1

Plotting point C(1, 0) and D(0, -1) on the same graph paper drawing a line passing through them, we
obtain the graph of the line represented by the equation x - y = 1

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Clearly two lines intersect at A(3, 2).

The graph of line 2x + 3y = 12 intersect with y-axis at B(0, 4) and the graph of the line x - y = 1 intersect
with y-axis at C(0, -1)
So, the vertices of the triangle formed by the two straight lines and y-axis are A(3, 2) and B(0, 4) and C(0,
-1)
Now,
Area of [Base Height]

31. (c) irrational number

Explanation: Here, 3 is rational and is irrational.
We know that the sum of a rational and an irrational is an irrational number, therefore, is
irrational.
32. (b) 20.4 m.
Explanation: Let the height of pole be AB = 18 m
And its shadow be BC = 9.6 m
Using Pythagoras theorem,

33. (c)
Explanation: (cos + sec )2 =
34. (d) (3, 0)
Explanation: Let the required point be P(x, 0) then,

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35. (c)
Explanation: Total number of digits from 1 to 9(n) = 9
Numbers which are odd (m) = 1, 3, 5, 7, 9 = 5
Probability =
36. (d) coincide with each other
Explanation: A system of two linear equations in two variables is dependent consistent, if their graphs
coincide with each other i.e. they superimpose each other and all points in one line are also a solution
for the other line.
37. (a) 4
Explanation: Using the factor tree for prime factorisation, we have:

Therefore, 144 =

Thus, the exponent of 2 in 144 is 4.
38. (b) m2 + n2
Explanation: Given,
a cos + b sin = m
a sin - b cos = n
Now, Squaring and adding, we have;
a2 cos2 + b2 sin2 + 2ab sin cos = m2
a2 sin2 + b2 cos2 - 2ab sin cos = n2
a2 (cos2 + sin2 ) + b2 (sin2 + cos2 ) = m2 + n2 { sin2 + cos2 = 1}
a2 + 1 + b2 1 = m2 + n2
a2 + b2 = m2 + n2
Hence a2 + b2 = m2 + n2
39. (c)
Explanation: Number of possible outcomes = 3
Number of Total outcomes = 15
Required Probability =
40. (c) rectangle
Explanation: A (9, 0), B(9, 6), C( - 9, 6) and D( - 9, 0) are the given vertices.
Then,
AB2 = (9 - 9)2 + (6 - 0)2

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= (0)2 + (6)2 = 0 + 36 = 36 units

BC2 = ( - 9 - 9)2 + (6 - 6)2
= (-18)2 + (0)2 = 324 + 0 = 324 units
CD2 = (- 9 + 9)2 + (0 - 6)2 = (0)2 + (-6)2 = 0 + 3 = 36 units
DA2 = ( - 9 - 9)2 + (0 - 0)2 = (- 18)2 + (0)2 = 324 + 0 = 324 units
Therefore, we have:
AB2 = CD2 and BC2 = DA2
Now, the diagonals are:
AC2 = ( - 9 - 9)2 + (6 - 0)2 = (- 18)2 + (6)2 = 324 + 36 = 360 units
BD2 = ( - 9 - 9)2 + (0 - 6)2 = ( - 18)2 + ( - 6)2 = 324 + 36 = 360 units
Therefore,
AC2 = BD2
Hence, ABCD is a rectangle.
Section C
41. (d) 10 : 7
Explanation: Let ABC is the triangle formed by both hotels and the mountain top. CDE is the
triangle formed by both huts and mountain top.
Clearly, DE || AB and so ABC DEC [By AA-similarity criterion]

Now, required ratio = Ratio of their corresponding sides = i.e., 10 : 7

42. (c) 29 miles
Explanation: Since, DE || AB, therefore
AD = = 4.29 miles
43. (c) 11.43 miles
Explanation: Since, ABC DEC
[ Corresponding sides of similar triangles are proportional]
AB = = 11.43 miles
44. (c) 3.5 miles
Explanation: Given, DC = 5 + BC
Clealy, BC = 10 - 5 = 5 miles
Now, CE = BC = 5 = 3.5 miles
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45. (b) 49:51
Explanation: Clearly, the ratio of areas of two triangles (i.e., ABC to DEC)

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Required ratio =
46. (d) 12
Explanation: Number of circular tiles along length of room = = 12
47. (c) 96
Explanation: Number of circular tiles along breadth of room = = 8
Total number of circular tiles = 12 8 = 96
48. (d) 1964.28 cm2
Explanation: Area covered by each circular tile = (25)2
= 25 25 = 1964.28 cm2

49. (b) 240000 cm2

Explanation: Area of rectangular floor = 600 400
= 240000 cm2
50. (c) 5.142 m2
Explanation: Area of the floor that remains uncovered with tiles = Area of rectangular floor - Area of
96 circular tiles
= 240000 - 96 1964.28
= 51429.12 cm2 = 5.142 m2

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