CBSE Class 10 Mathematics · Exam Practice

Practice Paper — Set 1

CBSE Class 10 Mathematics (Ganita Prakash), covering real numbers, polynomials, linear equations, quadratic equations, arithmetic progressions, coordinate geometry, trigonometry, circles, mensuration, statistics, and probability, with a full answer key and worked solutions.

🎯 Full Marks: 63 ⏱️ Time: 2½ Hours 📝 Questions: 42
SectionTopicQsMarks
AReal Numbers, Polynomials & Linear Equations1515
BQuadratic Equations, AP & Coordinate Geometry1520
CTrigonometry, Circles, Mensuration, Statistics & Probability1228
Total4263
Before you start
  • Attempt the paper first, without looking at the answer key at the bottom of the page.
  • Show your working for Sections B and C — method marks are given even if the final answer is wrong.
15 MARKS

Section A — Real Numbers, Polynomials & Linear Equations

1. Multiple Choice Questions (MCQs) (1 mark each)

1. The HCF of 12 and 18 is

A. 3B. 6C. 9D. 36

2. Which of the following is an irrational number?

A. √4B. √9C. √7D. √16

3. The zeroes of the polynomial x² − 5x + 6 are

A. 1, 6B. 2, 3C. −2, −3D. 2, −3

4. The degree of the polynomial 2x³ − 3x + 1 is

A. 1B. 2C. 3D. 0

5. The LCM of 4 and 6 is

A. 10B. 12C. 24D. 2

6. The solution of the pair of equations x + y = 5 and x − y = 1 is

A. x = 2, y = 3B. x = 3, y = 2C. x = 4, y = 1D. x = 1, y = 4

7. A quadratic polynomial can have at most how many zeroes?

A. 1B. 2C. 3D. Infinitely many

8. Which of these is a rational number?

A. πB. √2C. 22/7D. √3

9. For two positive integers a and b, HCF(a, b) × LCM(a, b) is equal to

A. a + bB. a − bC. a × bD. a ÷ b

10. If p(x) = x² − 3x + 2, then p(2) =

A. 0B. 2C. 4D. −2

2. Fill in the Blanks (1 mark each)

1. HCF(15, 20) = ______.

2. The sum of the zeroes of x² − 7x + 10 is ______.

3. If 2x + 3y = 12 and x = 3, then y = ______.

4. A quadratic polynomial has at most ______ zeroes.

5. LCM(6, 8) = ______.

20 MARKS

Section B — Quadratic Equations, AP & Coordinate Geometry

3. True or False (1 mark each)

1. x² − 4 = 0 has two real roots. (   )

2. Every quadratic equation has real roots. (   )

3. The distance between (0, 0) and (3, 4) is 5 units. (   )

4. In an AP, the common difference is found by subtracting consecutive terms. (   )

5. The midpoint of (2, 4) and (6, 8) is (4, 6). (   )

4. Match the Following (1 mark each)

Column A
1.Roots of x² − 5x + 6 = 0
2.10th term of AP: 3, 7, 11, ...
3.Distance between (1, 1) and (4, 5)
4.Sum of first 5 terms of AP: 2, 4, 6, 8, ...
5.Discriminant of x² + 2x + 1 = 0
Column B
a)2, 3
b)39
c)5
d)30
e)0

5. Very Short Answer Type Questions (2 marks each)

1. Solve: x² − 9 = 0.

2. Find the 15th term of the AP: 5, 8, 11, ...

3. Find the distance between (0, 0) and (6, 8).

4. Find the sum of the first 10 terms of the AP: 1, 2, 3, ...

5. Solve the quadratic equation x² − 6x + 9 = 0.

28 MARKS

Section C — Trigonometry, Circles, Mensuration, Statistics & Probability

6. Solve the Following (2 marks each)

1. Find the value of sin 30° + cos 60°.

2. Find the value of tan 45°.

3. A tangent to a circle touches the circle at how many points?

4. Find the circumference of a circle with radius 7 cm (use π = 22/7).

5. Find the mean of 2, 4, 6, 8, 10.

7. Do As Directed (2 marks each)

1. If cos θ = 3/5, find sin θ.

2. Find the area of a circle with radius 14 cm (use π = 22/7).

3. A die is thrown once. Find the probability of getting an even number.

4. Find the median of: 3, 5, 7, 9, 11.

5. Find the curved surface area of a cylinder with radius 7 cm and height 10 cm (use π = 22/7).

8. Think & Solve (Word Problems) (4 marks each)

1. The sum of the first n terms of an AP is given by Sₙ = 2n² + 3n. Find (a) the first term, and (b) the 5th term.

2. A tower is 30 m high. From a point on the ground, the angle of elevation to the top of the tower is 30°. Find the distance of the point from the base of the tower (use tan 30° = 1/√3).

CBSE Class 10 Mathematics · Practice Paper — Set 1 · Answer Key

Answers & Worked Solutions

Mark your own paper, then read the solution for every question you got wrong. For the multiple-choice questions, the note in pink shows the answer most students pick by mistake, and why.

Section A — Real Numbers, Polynomials & Linear Equations (15 marks)
1. Multiple Choice Questions
Q1. B. 6

12 = 2² × 3, 18 = 2 × 3². The common factors give HCF = 2 × 3 = 6.

Q2. C. √7

√4 = 2, √9 = 3, and √16 = 4 are whole numbers (rational). √7 cannot be simplified to a whole number or simple fraction — it's irrational.

Q3. B. 2, 3

Check: sum of roots = 2 + 3 = 5 (matches −(−5)/1) and product = 2 × 3 = 6 (matches 6/1) ✓.

Q4. C. 3

The degree is the highest power of x present — here it's x³ (power 3).

Q5. B. 12

Multiples of 4: 4,8,12... Multiples of 6: 6,12... The smallest common one is 12.

Q6. B. x = 3, y = 2

Check: 3 + 2 = 5 ✓ and 3 − 2 = 1 ✓.

Q7. B. 2

A quadratic (degree 2) polynomial can have at most 2 zeroes.

Q8. C. 22/7

22/7 is a ratio of two integers, so it's rational. π, √2, and √3 are all irrational.

Tempting wrong answer: π — 22/7 is often used as an approximation for π in calculations, but π itself is irrational; 22/7 is a genuinely different, rational number.

Q9. C. a × b

For any two positive integers, HCF × LCM always equals the product of the two numbers.

Q10. A. 0

p(2) = 2² − 3(2) + 2 = 4 − 6 + 2 = 0.

2. Fill in the Blanks
Q1. 5

15 = 3 × 5, 20 = 4 × 5. Common factor: 5.

Q2. 7

For x² − 7x + 10, sum of zeroes = −(coefficient of x)/(coefficient of x²) = −(−7)/1 = 7.

Q3. 2

2(3) + 3y = 12 → 6 + 3y = 12 → 3y = 6 → y = 2.

Q4. 2

Degree 2 means at most 2 zeroes.

Q5. 24

Multiples of 6: 6,12,18,24... Multiples of 8: 8,16,24... Smallest common: 24.

Section B — Quadratic Equations, AP & Coordinate Geometry (20 marks)
3. True or False
Q1. True

x² = 4 → x = 2 or x = −2, both real.

Q2. False

Some quadratic equations have no real roots (when the discriminant is negative) — for example, x² + 1 = 0.

Q3. True

Distance = √(3² + 4²) = √(9+16) = √25 = 5.

Q4. True

Common difference d = any term − the term before it.

Q5. True

Midpoint = ((2+6)/2, (4+8)/2) = (4, 6).

4. Match the Following
1 → a) 2, 3

x² − 5x + 6 = (x−2)(x−3) = 0, so x = 2 or x = 3.

2 → b) 39

a=3, d=4. 10th term = a + 9d = 3 + 36 = 39.

3 → c) 5

Distance = √((4−1)² + (5−1)²) = √(9+16) = √25 = 5.

4 → d) 30

a=2, d=2, n=5. Sum = (n/2)[2a+(n−1)d] = (5/2)[4+8] = (5/2)(12) = 30.

5 → e) 0

Discriminant = b² − 4ac = 2² − 4(1)(1) = 4 − 4 = 0.

5. Very Short Answer Type Questions
Q1. x = 3 or x = −3

x² = 9 → x = ±3.

Q2. 47

a=5, d=3. 15th term = a + 14d = 5 + 42 = 47.

Q3. 10

Distance = √(6² + 8²) = √(36+64) = √100 = 10.

Q4. 55

a=1, d=1, n=10. Sum = (10/2)[2(1)+9(1)] = 5 × 11 = 55.

Q5. x = 3 (repeated root)

x² − 6x + 9 = (x − 3)² = 0, so x = 3 (a repeated/double root).

Section C — Trigonometry, Circles, Mensuration, Statistics & Probability (28 marks)
6. Solve the Following
Q1. 1

sin 30° = 1/2, cos 60° = 1/2. Sum = 1/2 + 1/2 = 1.

Q2. 1

tan 45° = 1 (a standard trigonometric value).

Q3. Exactly one point

A tangent, by definition, touches a circle at exactly one point.

Q4. 44 cm

Circumference = 2πr = 2 × (22/7) × 7 = 44 cm.

Q5. 6

Mean = (2+4+6+8+10) ÷ 5 = 30 ÷ 5 = 6.

7. Do As Directed
Q1. 4/5
  1. Using sin²θ + cos²θ = 1: sin²θ = 1 − (3/5)² = 1 − 9/25 = 16/25.
  2. sin θ = √(16/25) = 4/5.
Q2. 616 sq cm

Area = πr² = (22/7) × 14 × 14 = 22 × 28 = 616 sq cm.

Q3. 1/2

Even numbers on a die: 2, 4, 6 — that's 3 out of 6 outcomes. P = 3/6 = 1/2.

Q4. 7

Already in order: 3, 5, 7, 9, 11. The middle (3rd) value is the median: 7.

Q5. 440 sq cm

Curved Surface Area = 2πrh = 2 × (22/7) × 7 × 10 = 440 sq cm.

8. Think & Solve (Word Problems)
Q1. (a) first term = 5 (b) 5th term = 21
  1. S₁ = 2(1)² + 3(1) = 5 — this is the first term, a₁ = 5.
  2. S₅ = 2(5)² + 3(5) = 50 + 15 = 65.
  3. S₄ = 2(4)² + 3(4) = 32 + 12 = 44.
  4. 5th term = S₅ − S₄ = 65 − 44 = 21.
Q2. 30√3 m (≈ 51.96 m)
  1. Let the distance from the base be d. Using tan 30° = height ÷ distance: 1/√3 = 30/d.
  2. d = 30 × √3 = 30√3 m (approximately 51.96 m).
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