Class 10 Boards Mathematics Practice Questions

26 free Mathematics MCQs written in the Class 10 Boards exam pattern, each with a step-by-step solution. Tap an option for instant feedback — no sign-up, no ads.

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✍️ These are original practice questions written in the Class 10 Boards pattern for the Class 10 board exams (CBSE pattern) — not questions from any official past paper.

Practice questions with solutions

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Real Numbers

Q1. For two positive integers, HCF × LCM equals:

  1. The sum of the two numbers
  2. The product of the two numbers
  3. The larger number
  4. The difference of the numbers
Show answer

B. The product of the two numbers — For any two positive integers a and b, HCF(a,b) × LCM(a,b) = a × b.

Quadratic Equations

Q2. The roots of the quadratic equation x² − 5x + 6 = 0 are:

  1. 2 and 3
  2. 1 and 6
  3. −2 and −3
  4. 5 and 6
Show answer

A. 2 and 3 — x² − 5x + 6 = (x−2)(x−3) = 0, so x = 2 or x = 3. They add to 5 and multiply to 6, matching the equation.

Coordinate Geometry

Q3. The distance between the points (0, 0) and (3, 4) is:

  1. 5
  2. 7
  3. 12
  4. 25
Show answer

A. 5 — Distance = √((3−0)² + (4−0)²) = √(9 + 16) = √25 = 5.

Trigonometry

Q4. The value of sin 30° is:

  1. 1
  2. 1/2
  3. √3/2
  4. 0
Show answer

B. 1/2 — sin 30° = 1/2. (cos 30° = √3/2 and tan 30° = 1/√3.)

Arithmetic Progressions

Q5. In an AP with first term a and common difference d, the nth term is:

  1. a + nd
  2. a + (n − 1)d
  3. a − (n − 1)d
  4. na + d
Show answer

B. a + (n − 1)d — The nth term of an arithmetic progression is aₙ = a + (n − 1)d.

Probability

Q6. The probability of getting a head when a fair coin is tossed once is:

  1. 1
  2. 0
  3. 1/2
  4. 2
Show answer

C. 1/2 — There are 2 equally likely outcomes (head, tail), so P(head) = 1/2.

Circles

Q7. The area of a circle of radius 7 cm (take π = 22/7) is:

  1. 44 cm²
  2. 154 cm²
  3. 22 cm²
  4. 49 cm²
Show answer

B. 154 cm² — Area = πr² = (22/7) × 7 × 7 = 22 × 7 = 154 cm².

Quadratic Equations

Q8. The discriminant of the quadratic equation ax² + bx + c = 0 is:

  1. b² − 4ac
  2. b² + 4ac
  3. 2a + b
  4. √(b² − 4ac)
Show answer

A. b² − 4ac — The discriminant is D = b² − 4ac. It decides the nature of the roots: real & distinct (D>0), equal (D=0), or no real roots (D<0).

Trigonometry

Q9. The value of tan 45° is:

  1. 0
  2. 1
  3. √3
  4. 1/2
Show answer

B. 1 — tan 45° = 1, because sin 45° = cos 45° = 1/√2, so their ratio is 1.

Polynomials

Q10. The sum of the zeroes of the polynomial x² − 7x + 10 is:

  1. 7
  2. 10
  3. −7
  4. 3
Show answer

A. 7 — For ax² + bx + c, the sum of zeroes = −b/a = −(−7)/1 = 7. (The zeroes are 2 and 5.)

Real Numbers

Q11. The HCF of two numbers is 9 and their LCM is 90. If one of the numbers is 18, the other number is:

  1. 40
  2. 50
  3. 45
  4. 9
Show answer

C. 45 — For two numbers, product = HCF × LCM = 9 × 90 = 810. The other number = 810 ÷ 18 = 45.

Real Numbers

Q12. Which of the following rational numbers has a terminating decimal expansion?

  1. 3/8
  2. 5/6
  3. 7/15
  4. 11/12
Show answer

A. 3/8 — A fraction in lowest terms terminates only if the denominator has prime factors 2 and/or 5. Here 8 = 2³, so 3/8 terminates; the others have factors 3.

Polynomials

Q13. The product of the zeroes of the quadratic polynomial 2x² − 5x − 3 is:

  1. 3/2
  2. 5/2
  3. -5/2
  4. -3/2
Show answer

D. -3/2 — For ax² + bx + c, the product of zeroes = c/a = -3/2.

Polynomials

Q14. A quadratic polynomial whose zeroes are 3 and −2 is:

  1. x² + x − 6
  2. x² − x + 6
  3. x² − x − 6
  4. x² + 5x + 6
Show answer

C. x² − x − 6 — Sum of zeroes = 3 + (−2) = 1, product = 3 × (−2) = −6. The polynomial is x² − (sum)x + product = x² − x − 6.

Linear Equations

Q15. For what value of k does the pair of equations 2x + 3y = 7 and 4x + 6y = k have infinitely many solutions?

  1. 7
  2. 10
  3. 28
  4. 14
Show answer

D. 14 — Infinitely many solutions require a1/a2 = b1/b2 = c1/c2, i.e. 2/4 = 3/6 = 7/k. So 1/2 = 7/k gives k = 14.

Quadratic Equations

Q16. The nature of the roots of the quadratic equation x² + x + 1 = 0 is:

  1. Two distinct real roots
  2. Two equal real roots
  3. No real roots
  4. Exactly one real root
Show answer

C. No real roots — Discriminant = b² − 4ac = 1 − 4 = −3 < 0, so the equation has no real roots.

Quadratic Equations

Q17. For what positive value of k does the equation x² − kx + 9 = 0 have two equal roots?

  1. 3
  2. 6
  3. 9
  4. 12
Show answer

B. 6 — Equal roots need discriminant = 0: k² − 4(9) = 0, so k² = 36 and the positive value is k = 6.

Arithmetic Progressions

Q18. The sum of the first 10 terms of the AP 2, 4, 6, 8, ... is:

  1. 90
  2. 100
  3. 120
  4. 110
Show answer

D. 110 — S_n = n/2[2a + (n−1)d] = 10/2[2(2) + 9(2)] = 5[4 + 18] = 5 × 22 = 110.

Arithmetic Progressions

Q19. In an AP, the 3rd term is 12 and the 7th term is 24. The common difference is:

  1. 3
  2. 2
  3. 4
  4. 6
Show answer

A. 3 — a + 2d = 12 and a + 6d = 24. Subtracting gives 4d = 12, so d = 3.

Triangles

Q20. The areas of two similar triangles are in the ratio 9 : 16. The ratio of their corresponding sides is:

  1. 9 : 16
  2. 3 : 4
  3. 81 : 256
  4. 3 : 16
Show answer

B. 3 : 4 — In similar triangles, the ratio of areas equals the square of the ratio of corresponding sides. So sides are in ratio √9 : √16 = 3 : 4.

Triangles

Q21. In triangle ABC, DE is drawn parallel to BC with D on AB and E on AC. If AD = 2 cm, DB = 3 cm and AE = 4 cm, then EC equals:

  1. 3 cm
  2. 5 cm
  3. 8 cm
  4. 6 cm
Show answer

D. 6 cm — By the Basic Proportionality Theorem, AD/DB = AE/EC. So 2/3 = 4/EC, giving EC = 6 cm.

Coordinate Geometry

Q22. The point that divides the line segment joining (1, 2) and (7, 8) internally in the ratio 1 : 2 is:

  1. (3, 4)
  2. (5, 6)
  3. (4, 5)
  4. (3, 3)
Show answer

A. (3, 4) — Section formula: x = (1·7 + 2·1)/3 = 9/3 = 3, y = (1·8 + 2·2)/3 = 12/3 = 4. The point is (3, 4).

Trigonometry

Q23. If θ is an acute angle and sin θ = cos θ, then θ equals:

  1. 30°
  2. 45°
  3. 60°
  4. 90°
Show answer

B. 45° — sin θ = cos θ gives tan θ = 1, and the acute angle with tangent 1 is 45°.

Heights And Distances

Q24. When the angle of elevation of the Sun is 45°, a vertical pole casts a shadow 15 m long on the ground. The height of the pole is:

  1. 7.5 m
  2. 15 m
  3. 15√3 m
  4. 30 m
Show answer

B. 15 m — tan 45° = height/shadow = 1, so height = shadow length = 15 m.

Circles

Q25. The length of the tangent drawn from a point 13 cm away from the centre of a circle of radius 5 cm is:

  1. 8 cm
  2. 12 cm
  3. 18 cm
  4. √194 cm
Show answer

B. 12 cm — The tangent is perpendicular to the radius, so tangent length = √(13² − 5²) = √(169 − 25) = √144 = 12 cm.

Statistics

Q26. The empirical relationship between the mean, median and mode of a frequency distribution is:

  1. Mode = 3 Mean − 2 Median
  2. Mode = 3 Median − 2 Mean
  3. Mode = 2 Median − 3 Mean
  4. Mode = Median − Mean
Show answer

B. Mode = 3 Median − 2 Mean — The standard empirical relation is Mode = 3 Median − 2 Mean.

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