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CBSE Maths 2023

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Complete board exam paper — 38 questions. No coach help. Timer will track your speed.

Questions

38

Board

CBSE

Year

2023

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CBSE Class 10 Maths 2023 — all 38 questions from the paper

The complete question list from the CBSE Class 10 Mathematics 2023 board examination is below. Every question can be attempted in the timed interactive test above, with visual step-by-step solutions and AI coaching after each answer. Three fully worked solutions from this paper follow the question list.

Questions from the 2023 paper

  1. Q1.The ratio of HCF to LCM of the least composite number and the least prime number is:A) 1 : 2B) 2 : 1C) 1 : 1D) 1 : 3
  2. Q2.The graph of y = p(x) is given, where p(x) is a polynomial. The number of zeroes of p(x) from the graph is:A) 1B) 2C) 3D) 0
  3. Q3.The value of k for which the pair of equations kx = y + 2 and 6x = 2y + 3 has infinitely many solutions is:A) k = 3B) k = −3C) k = 6D) No value of k
  4. Q4.The value(s) of k for which the quadratic equation 2x² + kx + 2 = 0 has equal roots is:A) 4B) ±4C) −4D) 0
  5. Q5.The common difference of the AP whose nth term is given by aₙ = 3n + 7 is:A) 7B) 3C) 10D) 6
  6. Q6.ΔABC ~ ΔDEF. If AB = 4 cm, BC = 3.5 cm, CA = 2.5 cm and DF = 7.5 cm, then the perimeter of ΔDEF is:A) 10 cmB) 14 cmC) 30 cmD) 25 cm
  7. Q7.The point on the x-axis which is equidistant from (2, −5) and (−2, 9) is:A) (−7, 0)B) (7, 0)C) (−2, 0)D) (2, 0)
  8. Q8.If sinθ + cosθ = √2, then tanθ + cotθ =A) 1B) 2C) √2D) 1/√2
  9. Q9.The length of the tangent from a point which is at a distance of 5 cm from the centre of a circle of radius 3 cm is:A) 4 cmB) √34 cmC) √8 cmD) 2 cm
  10. Q10.A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. The length PQ is:A) 12 cmB) 13 cmC) 8.5 cmD) √119 cm
  11. Q11.The volume of a right circular cone whose area of the base is 156 cm² and height is 8 cm is:A) 416 cm³B) 1248 cm³C) 1244 cm³D) 624 cm³
  12. Q12.If the mean of the first n natural numbers is 15, then n =A) 15B) 30C) 14D) 29
  13. Q13.If the probability of a player winning a game is 0.79, then the probability of his losing the same game is:A) 1.79B) 0.31C) 0.21D) 0.79²
  14. Q14.If 2 is a root of the equation x² + bx + 12 = 0 and the equation x² + bx + q = 0 has equal roots, then q =A) 8B) 16C) −16D) −8
  15. Q15.The coordinates of the point which is the reflection of (−3, 5) in the x-axis are:A) (3, 5)B) (3, −5)C) (−3, −5)D) (−3, 5)
  16. Q16.If sec²θ(1 + sinθ)(1 − sinθ) = k, then the value of k is:A) 0B) 1C) −1D) sec²θ
  17. Q17.A letter is chosen at random from the word PROBABILITY. The probability that it is not a vowel is:A) 4/11B) 7/11C) 3/11D) 6/11
  18. Q18.The sum of the first 500 natural numbers is:A) 124750B) 125250C) 125750D) 250500
  19. Q19.Assertion (A): a, b, c are in AP if and only if 2b = a + c.
    Reason (R): The sum of the first n terms of an AP with first term a and common difference d is n/2[2a + (n−1)d].
    Choose the correct option:    A) Both A and R are true, and R is the correct explanation of AB) Both A and R are true, but R is not the correct explanation of AC) A is true but R is falseD) A is false but R is true
  20. Q20.Assertion (A): The point (0, 4) lies on the y-axis.
    Reason (R): The x-coordinate of every point on the y-axis is 0.
    Choose the correct option:    A) Both A and R are true, and R is the correct explanation of AB) Both A and R are true, but R is not the correct explanation of AC) A is true but R is falseD) A is false but R is true
  21. Q21.Find the zeroes of the polynomial x² + 7x + 10 and verify the relationship between the zeroes and the coefficients.A) -2 and -5B) 2 and 5C) -2 and 5D) 2 and -5
  22. Q22.Solve the pair of equations: x + y = 14 and x − y = 4. The value of x is:A) 7B) 8C) 9D) 10
  23. Q23.If the distances of P(x, y) from A(5, 1) and B(−1, 5) are equal, then 3x =A) 2yB) yC) 4yD) 3y
  24. Q24.If sinA = 1/√2, then the value of (2tanA)/(1 + tan²A) is:A) 1/√2B) √2C) 1D) 2
  25. Q25.The minute hand of a clock is 14 cm long. The area swept by the minute hand in 5 minutes is: (Use π = 22/7)A) 154/3 cm²B) 308/3 cm²C) 154 cm²D) 77/3 cm²
  26. Q26.Given that √2 is irrational, prove that 5 + 3√2 is an irrational number. What type of number is 5 + 3√2?A) RationalB) IrrationalC) IntegerD) Whole number
  27. Q27.Solve for x: 2x² + 6√3x − 60 = 0. The roots are:A) 2√3 and −5√3B) −2√3 and 5√3C) √3 and −10√3D) 3√3 and −10/√3
  28. Q28.In an AP, the sum of the first 5 terms is 55 and the sum of the first 10 terms is 235. Find the sum of the first 20 terms.A) 970B) 910C) 870D) 1010
  29. Q29.Prove that the lengths of tangents drawn from an external point to a circle are equal. Using this, find BC if a circle inscribed in ΔABC touches AB at P, BC at Q and AC at R, with AB = 10 cm, AR = 7 cm and CR = 5 cm.A) 8 cmB) 10 cmC) 12 cmD) 15 cm
  30. Q30.Find the mode of the following frequency distribution:
    Class: 25–30 (f=25), 30–35 (f=34), 35–40 (f=50), 40–45 (f=42), 45–50 (f=38).    A) 36.8B) 37.5C) 38.0D) 36.0
  31. Q31.A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of 30°, which is approaching the foot of the tower with a uniform speed. Six seconds later, the angle of depression is 60°. The time taken by the car to reach the foot of the tower from this point is:A) 3 secondsB) 4 secondsC) 6 secondsD) 2 seconds
  32. Q32.The sum of the areas of two squares is 468 m². If the difference of their perimeters is 24 m, find the sides of the two squares.A) 12 m and 18 mB) 10 m and 20 mC) 15 m and 15 mD) 8 m and 22 m
  33. Q33.ΔOAB is a triangle with vertices O(0,0), A(6,0), B(0,4). The median from O meets AB at M. The coordinates of M are:A) (3, 2)B) (2, 3)C) (4, 2)D) (3, 4)
  34. Q34.Water in a canal, 6 m wide and 1.5 m deep, is flowing with a speed of 10 km/h. How much area can it irrigate in 30 minutes if 8 cm of standing water is needed?A) 562500 m²B) 56250 m²C) 5625 m²D) 225000 m²
  35. Q35.The median of the following data is 28.5. Find the missing frequencies x and y if the total frequency is 60.
    Class: 0–10 (f=5), 10–20 (f=x), 20–30 (f=20), 30–40 (f=15), 40–50 (f=y), 50–60 (f=5). The value of x is:    A) 8B) 7C) 10D) 12
  36. Q36.Two poles of equal heights are standing opposite each other on either side of a road 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 60° and 30° respectively. The height of the poles is:A) 20√3 mB) 30√3 mC) 40√3 mD) 10√3 m
  37. Q37.A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of the same radius. The total height of the toy is 15.5 cm. The total surface area of the toy is: (Use π = 22/7)A) 214.5 cm²B) 204.05 cm²C) 243.5 cm²D) 192.5 cm²
  38. Q38.A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball is double that of a red ball, the number of blue balls in the bag is:A) 10B) 5C) 15D) 20

Worked solutions from the CBSE 2023 paper

Solved example 1

The ratio of HCF to LCM of the least composite number and the least prime number is:

Step-by-step solution

  1. Least prime number = 2
  2. Least composite number = 4
  3. HCF(2, 4) = 2
  4. LCM(2, 4) = 4
  5. Ratio = 2 : 4 = 1 : 2
  6. Wait — the question says least composite and least prime. Least composite = 4, least prime = 2.
  7. HCF(4, 2) = 2, LCM(4, 2) = 4. Ratio = 1 : 2.
  8. Hmm, but if the question means both are 2 (2 is both prime and the least even composite considered):
  9. HCF(2,2) = 2, LCM(2,2) = 2. Ratio = 1:1.

Answer: 1 : 1

Solved example 2

The graph of y = p(x) is given, where p(x) is a polynomial. The number of zeroes of p(x) from the graph is:

Step-by-step solution

  1. The zeroes of a polynomial are the x-values where the graph crosses the x-axis.
  2. Count the number of points where the curve meets y = 0.
  3. The graph intersects the x-axis at 3 points.
  4. Therefore, the polynomial has 3 zeroes.

Answer: 3

Solved example 3

The value of k for which the pair of equations kx = y + 2 and 6x = 2y + 3 has infinitely many solutions is:

Step-by-step solution

  1. Equation 1: kx − y = 2 → a₁ = k, b₁ = −1, c₁ = 2
  2. Equation 2: 6x − 2y = 3 → a₂ = 6, b₂ = −2, c₂ = 3
  3. For infinitely many solutions: a₁/a₂ = b₁/b₂ = c₁/c₂
  4. b₁/b₂ = (−1)/(−2) = 1/2
  5. c₁/c₂ = 2/3
  6. Since 1/2 ≠ 2/3, the system can never have infinitely many solutions.
  7. No value of k works.

Answer: No value of k

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