CBSE Class 10 Mathematics Practice Paper 2 (2026-27)

A full-length Class 10 Mathematics practice paper in the CBSE 2026-27 pattern, with its answer key. Free to read and to save as a PDF.

Time: 3 hours | Maximum Marks: 80

General Instructions

1. This question paper consists of 5 sections—A, B, C, D and E.

2. Section A has 18 questions of 1 mark each.

3. Section B has 5 questions of 2 marks each.

4. Section C has 6 questions of 3 marks each.

5. Section D has 4 questions of 5 marks each.

6. Section E has 3 case-based questions of 4 marks each.

7. There is no overall choice in any section.

8. However, internal choices have been provided in 2 questions of 2 marks each, 2 questions of 3 marks each, 2 questions of 5 marks each and in the 2-mark sub-part of each case-based question.

AMultiple Choice Questions

The decimal representation of the number

\frac{3}{2^{2} \times 5^{1}}

will terminate after:(a) 1 decimal place(b) 2 decimal places(c) 3 decimal places(d) 4 decimal places

α

and

β

are the zeroes of the quadratic polynomial

p (x) = x^{2} + 7 x + 10

, then the value of

\frac{1}{α} + \frac{1}{β}

is:(a) 7/10(b) -7/10(c) 10/7(d) -10/7

The pair of linear equations

x + y = 5

and

2 x + 2 y = 10

has:(a) Exactly one solution(b) No solution(c) Infinitely many solutions(d) Exactly two solutions

The value of k for which the quadratic equation

2 x^{2} + k x + 2 = 0

has real and equal roots is:(a) 4(b) -4(c)

\pm 4

(d)

\pm 2

The first term of an AP is

a = 5

and the common difference is

d = 2

. The 10th term is:(a) 19(b) 23(c) 21(d) 25

△ A B C

, if

D E ∥ B C

, then

\frac{A D}{A B} = \frac{A E}{?}

(a) AC(b) EC(c) DC(d) BC

The distance of the point P(4, 3) from the x-axis is:(a) 3 units(b) 4 units(c) 5 units(d) 7 units

\tan θ = \frac{1}{\sqrt{3}}

, then the value of

\sin θ

is:(a) 1/2(b)

\sqrt{3} / 2

From the top of a cliff 20 m high, the angle of depression of a boat sailing directly towards it is 30°. The distance of the boat from the foot of the cliff is:(a)

10 \sqrt{3}

m(b) 20 m(c)

20 \sqrt{3}

m(d) 40 m

Two tangents PA and PB are drawn to a circle with centre O from an external point P. If

∠ A P O = 30^{\circ}

, then

∠ P A B

is:(a)

30^{\circ}

(b)

60^{\circ}

(c)

90^{\circ}

(d)

120^{\circ}

The area of a sector with angle 60° and radius 6 cm is:(a)

6 π

cm²(b)

3 π

cm²(c)

12 π

cm²(d)

36 π

cm²

A solid is in the form of a cylinder with hemispherical ends. If the radius of the cylinder and hemisphere is 3 cm and the total height of the solid is 13 cm, then the volume of the solid is:(a)

126 π

cm³(b)

135 π

cm³(c)

144 π

cm³(d)

150 π

cm³

The median of the data 3, 5, 7, 9, 11 is:(a) 3(b) 5(c) 7(d) 9

If the sum of the frequencies is 20, then the sum of the deviations of the data from the mean is:(a) 0(b) 20(c)

\overset{ˉ}{x}

(d)

\overset{ˉ}{x} \times 20

The mode of the following frequency distribution is:

Class interval	Frequency
10-20	5
20-30	8
30-40	12
40-50	7
50-60	3
(a) 30-40
(b) 12
(c) 35
(d) 32.5

P (E) = \frac{2}{3}

, then the probability of 'not E' is:(a) 1/3(b) 2/3(c) 3/2(d) 1

A die is thrown once. The probability of getting a prime number less than 4 is:(a) 1/6(b) 1/3(c) 1/2(d) 2/3

A card is drawn from a well-shuffled deck of 52 cards. The probability of drawing a black king is:(a) 1/13(b) 1/26(c) 3/26(d) 1/52

Assertion-Reason Questions:

Assertion (A): The number

6^{n}

, where n is a natural number, can never end with the digit 0.Reason (R): For a number to end with the digit 0, its prime factorization must contain both 2 and 5.

Assertion (A): The pair of linear equations

x + y = 5

and

2 x + 2 y = 10

is consistent.Reason (R): A pair of linear equations

a_{1} x + b_{1} y + c_{1} = 0

and

a_{2} x + b_{2} y + c_{2} = 0

is consistent if

\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}

\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}

BShort Answer Questions

The sum of the first n terms of an AP is given by

S_{n} = 2 n^{2} + 3 n

. Find the common difference of this AP.

(A) Solve the following pair of linear equations by the substitution method:

3 x - y = 3

9 x - 3 y = 9

OR
(B) Solve the following pair of linear equations by the elimination method:

x + y = 5

2 x - 3 y = 4

△ A B C

, if

A B ∥ D E

, where D is a point on AB and E is a point on AC, and

A D = x - 1

D B = 4

A E = x

and

E C = 6

. Find the value of x.

(A) A chord of a circle of radius 15 cm subtends an angle of 60° at the centre. Find the area of the corresponding minor segment. (Use

π = 3.14

and

\sqrt{3} = 1.73

)OR(B) Find the area of a sector of a circle with radius 6 cm if the angle of the sector is 60°.

In the given figure, PQRS is a cyclic quadrilateral and its diagonals intersect at T. Prove that PT

\cdot

TR = QT

\cdot

TS.

CShort Answer Questions

(A) Prove that

3 + 2 \sqrt{5}

is an irrational number, given that

\sqrt{5}

is irrational.OR(B) State true or false for each of the following statements and justify in each case:(i)

7 \times 11 \times 13 + 13

is a composite number.(ii)

7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5

is a composite number.

Solve the following pair of linear equations:

\frac{x}{2} + \frac{2 y}{3} = - 1

x - \frac{y}{3} = 3

(A) Find the value of k for which the quadratic equation

k x^{2} - 6 k x + 3 (k + 9) = 0

has real and equal roots.OR(B) The sum of the ages of a father and his son is 45 years. Five years ago, the father's age was three times the age of his son. Find their present ages.

△ A B C

∠ A B C = 90^{\circ}

and BD

⊥

AC. If

A D = 4

cm and

C D = 3

cm, find AB and BC.

A solid toy is in the form of a cone mounted on a hemisphere of the same radius. The height of the cone is 4 cm and the radius of the base is 3 cm. Find the volume of the toy.

(A) A die is thrown once. Find the probability of getting:(i) A prime number less than 4.(ii) An even number greater than 4.OR(B) Two different coins are tossed simultaneously. Find the probability of getting:(i) Exactly one head.(ii) At least one tail.

DLong Answer Questions

(A) The sum of the first n terms of an AP is given by

S_{n} = 3 n^{2} + 5 n

. Find the nth term of this AP. Hence, find the 20th term.OR(B) The sum of the first 15 terms of an AP is 450 and the sum of its first 20 terms is 760. Find the first term and the common difference of the AP.

(A) In the given figure, ABCD is a quadrilateral with diagonals AC and BD intersecting at O. If

\frac{A O}{O C} = \frac{B O}{O D}

, prove that

△ A O B \sim △ C O D

. Also, show that

△ A O D \sim △ B O C

.OR(B) In

△ A B C

∠ A B C = 90^{\circ}

. A point D is on AB such that

C D

bisects

∠ A C B

. If

A C = 10

cm and

B C = 5

cm, then find

A D

and

D B

(A) A building has been made in the shape of a cylinder with two hemispheres attached at each end. If the height of the cylindrical part is 10 m and the radius of the base is 3.5 m, find the volume of the building. What is the cost of painting the outer surface of the building at Rs 20 per m²? (Use

π = \frac{22}{7}

)OR(B) A solid is in the form of a cylinder with hemispherical ends. If the total length of the solid is 90 cm and the diameter of each hemisphere is 42 cm, find the volume of the solid.

The following frequency distribution shows the ages of 100 teachers in a school:

Age (in years)	Number of teachers
25 - 35	14
35 - 45	26
45 - 55	20
55 - 65	14
65 - 75	26
Find the median age of the teachers. If the number of teachers in the age group 45-55 is missing, and the median is 43.5 years, find the missing frequency.

ECase-based Questions

Case Study: A farmer has a field in the shape of a quadrilateral ABCD. The farmer wants to grow wheat in triangle ABE and rice in the remaining area, where E is a point on the diagonal BD. The farmer knows the heights and areas of triangles involved.(i) If ABCD is a parallelogram and E is any point on diagonal BD, then the area of

△ A B E

is related to the area of

△ A D E

. What is this relation? (1 mark)(ii) If ABCD is a parallelogram and E is the midpoint of BD, then the area of

△ A B E

is what fraction of the area of

△ A B D

? (1 mark)(iii) (A) In a

△ A B C

, D and E are points on sides AB and AC respectively such that AD = 4 cm, BD = 6 cm, AE = 5 cm and EC = 7.5 cm. Show that

△ A D E \sim △ A B C

.OR(B) In a

△ A B C

∠ A B C = 90^{\circ}

. A point D is on AB such that

C D

bisects

∠ A C B

. If

A C = 10

cm and

B C = 5

cm, find

A D

and

D B

. 2marks

Case Study: An architect is designing a multi-story building. For structural integrity, the building's foundation must be on a stable, level ground. The design includes a rectangular base with a height of 15 m. On top of this, a conical roof is to be placed. The diameter of the base of the cone is equal to the width of the rectangular base of the building, which is 8 m. The height of the conical part is 6 m.(i) What is the radius of the base of the conical roof? (1 mark)(ii) Find the volume of the cylindrical part of the building. (1 mark)(iii) (A) Calculate the volume of the conical roof.OR(B) Calculate the total volume of the building. 2marks

Case Study: A survey was conducted in a locality about the number of hours children spent playing in the evening. The data collected is given below:

Hours of Play	Number of Children
0 - 2	5
2 - 4	15
4 - 6	25
6 - 8	10
8 - 10	5
(i) What is the modal class of the distribution? (1 mark)
(ii) Calculate the mean number of hours children spent playing. (1 mark)
(iii) (A) Find the median number of hours children spent playing.
OR
(B) If the total number of children surveyed was 60, and the number of children playing for 2-4 hours was missing, find the missing frequency. (2 marks)

Answer Key

(b) 2

(b) 5

(c)

\pm 4

(b) 23

(a) AC

(a) 3 units

(a) 1/2

(c)

20 \sqrt{3}

(a)

\sqrt{119}

(a)

6 π

cm²

(a)

126 π

cm³

(b)

\overset{ˉ}{x} + a

(a) 30-40

(a) 1/3

(b) 1/3

(b) 3/26

Assertion: True, Reason: True. Reason is the correct explanation for the assertion.

(A) Infinitely many solutions (B) x=11/5, y=14/5

x = 5

(A)

67.5

cm² (B)

6 π

cm²

Proof by congruent triangles OAP and OBP.

(A) Proof by contradiction. (B) (i) True, (ii) True.

No solution.

(A)

x = a \pm b

(B)

λ = - 4

, other root = -3

A B = \frac{16}{3}

cm,

B C = \sqrt{A B^{2} + A C^{2}} = \sqrt{(\frac{16}{3})^{2} + 7^{2}}

(Mistake in question, likely needs similarity). Assuming similar triangles,

A B = 6.4

cm,

B C = 8

cm.

94.08

cm³

(A) (i) 1/3, (ii) 1/6 (B) (i) 3/4, (ii) 3/4

(A)

a_{n} = 6 n - 1

a_{20} = 119

(B)

a = - 15

d = 30

(A) Proof using properties of similar triangles and AA similarity. (B)

A D = 20 / 3

cm,

D B = 10 / 3

(A) Volume =

539

m³, Cost = Rs 26950 (B)

107730

cm³

Median = 49.2 years. Missing frequency = 6.

(i) Area(

△ A B E

) = Area(

△ A D E

) (ii) 1/2 (iii) (A) Proof using SAS similarity. (B)

A D = 20 / 3

cm,

D B = 10 / 3

cm.

(i) 4 m (ii)

160 π

m³ (iii) (A)

24 π

m³ (B)

(160 + 24) π

m³ =

184 π

m³

(i) 4-6 hours (ii) 4.9 hours (iii) (A) 4.8 hours (B) Missing frequency = 15.

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Class 10 Mathematics — Practice Paper 2

Answer Key