CBSE Class 10 Mathematics Sample Paper 2026 Set 2 (with Solutions) — Free PDF
This sample paper follows the latest CBSE curriculum with a mix of arithmetic, algebra, geometry, and statistics. Solve all sections carefully to master ke
TL;DR: This sample paper follows the latest CBSE curriculum with a mix of arithmetic, algebra, geometry, and statistics. Solve all sections carefully to mast…
Written & reviewed by the Syllab.in Academic Team (CBSE/NCERT subject experts) · Updated
This sample paper follows the latest CBSE curriculum with a mix of arithmetic, algebra, geometry, and statistics. Solve all sections carefully to master key concepts for your board exam. This set is longer than the board paper: 110 marks against the 80-mark CBSE pattern, so treat it as extended practice rather than a timed board rehearsal.
Board: CBSE · Class: Class 10 · Subject: Mathematics · Total: 110 marks · Time: 4 hours
Section A: Very Short Answer (1 mark each) — 20 marks
Answer all 20 questions in 1-2 sentences.
Q1. Find the HCF of 36 and 48. (1 mark)
Answer: HCF(36, 48) = 12
Q2. If tan(A) = 3/4, find sin(A). (1 mark)
Answer: sin(A) = 3/5
Q3. Write the formula for the sum of first n natural numbers. (1 mark)
Answer: Sum = n(n+1)/2
Q4. A quadratic polynomial has roots 2 and -3. Write the polynomial. (1 mark)
Answer: x^2 + x - 6
Q5. Find the distance between points (0, 0) and (3, 4). (1 mark)
Answer: Distance = 5
Q6. If sin(theta) = 1/2, find cos(theta) where theta is acute. (1 mark)
Answer: cos(theta) = sqrt(3)/2
Q7. Write the general term of AP: 5, 8, 11, 14, ... (1 mark)
Answer: a_n = 3n + 2
Q8. Find the LCM of 12 and 18. (1 mark)
Answer: LCM = 36
Q9. If a circle has radius 7 cm, find its circumference. (1 mark)
Answer: Circumference = 14π cm or 44 cm
Q10. Solve: 2x + 3 = 11 (1 mark)
Answer: x = 4
Q11. Find the median of 2, 5, 8, 9, 12. (1 mark)
Answer: Median = 8
Q12. What is the probability of getting a head when a coin is tossed? (1 mark)
Answer: Probability = 1/2
Q13. If base = 10 cm and height = 5 cm, find the area of a triangle. (1 mark)
Answer: Area = 25 cm^2
Q14. Find the 10th term of GP: 2, 6, 18, 54, ... (1 mark)
Answer: a_10 = 39366
Q15. Write the equation of line passing through (1, 2) with slope 3. (1 mark)
Answer: y = 3x - 1
Q16. If cos(A) = 4/5, find tan(A). (1 mark)
Answer: tan(A) = 3/4
Q17. Find the volume of a cube with side 5 cm. (1 mark)
Answer: Volume = 125 cm^3
Q18. Express 0.25 as a fraction. (1 mark)
Answer: 1/4
Q19. If angle in a semicircle is 90 degrees, what is the angle subtended at the center for a quarter circle? (1 mark)
Answer: 90 degrees
Q20. Find the range of 15, 20, 25, 10, 30. (1 mark)
Answer: Range = 20
Section B: Short Answer (2 marks each) — 30 marks
Answer any 10 questions from the 15 given, showing working.
Q1. Prove that sqrt(2) is irrational. (2 marks)
Answer: Assume sqrt(2) = p/q where gcd(p,q)=1. Then 2q^2 = p^2, so p is even, p=2k. Then 2q^2=4k^2, so q is even. This contradicts gcd(p,q)=1. Therefore sqrt(2) is irrational.
Q2. Solve the pair of equations: x + y = 7 and x - y = 3. (2 marks)
Answer: Adding: 2x = 10, so x = 5. Subtracting: 2y = 4, so y = 2. Solution: x=5, y=2
Q3. Find the sum of first 20 terms of AP: 3, 7, 11, 15, ... (2 marks)
Answer: a=3, d=4, n=20. S_20 = 20/2(2*3 + 19*4) = 10(6+76) = 820
Q4. Two circles have radii 5 cm and 3 cm. Find the distance between centers if they touch externally. (2 marks)
Answer: When circles touch externally, distance between centers = r1 + r2 = 5 + 3 = 8 cm
Q5. If sin(A) = 12/13, find cos(A) and tan(A) where A is acute. (2 marks)
Answer: cos(A) = 5/13, tan(A) = 12/5
Q6. Find the coordinates of the midpoint of the line segment joining (2, 3) and (6, 7). (2 marks)
Answer: Midpoint = ((2+6)/2, (3+7)/2) = (4, 5)
Q7. A number is 5 more than another. Their product is 84. Find the numbers. (2 marks)
Answer: Let numbers be x and x+5. x(x+5)=84. x^2+5x-84=0. (x+12)(x-7)=0. x=7 or x=-12. Numbers are 7 and 12 or -12 and -7.
Q8. Find the area of a circle with radius 7 cm. (2 marks)
Answer: Area = πr^2 = 49π cm^2 or approximately 154 cm^2
Q9. In a triangle ABC, if angle A = 60 degrees and angle B = 75 degrees, find angle C. (2 marks)
Answer: Sum of angles in a triangle = 180 degrees. angle C = 180 - 60 - 75 = 45 degrees
Q10. Find the mode of the data: 2, 3, 3, 5, 5, 5, 7, 9. (2 marks)
Answer: Mode = 5 (appears 3 times)
Q11. If a number is divisible by 2 and 3, is it divisible by 6? Why? (2 marks)
Answer: Yes, because 2 and 3 are coprime (gcd=1). By the property of divisibility, if a number is divisible by two coprime numbers, it is divisible by their product.
Q12. Construct a triangle with sides 4 cm, 5 cm, and 6 cm. (2 marks)
Answer: Using compass and ruler: Draw base 5 cm. From one end mark arc of 4 cm. From other end mark arc of 6 cm. Join intersection point to both ends.
Q13. Find the sum of angles in a polygon with 8 sides. (2 marks)
Answer: Sum = (n-2)*180 = (8-2)*180 = 6*180 = 1080 degrees
Q14. If 3 is a root of x^2 + kx - 12 = 0, find k. (2 marks)
Answer: Substitute x=3: 9 + 3k - 12 = 0. 3k = 3. k = 1
Q15. A train travels 300 km in 5 hours. Find its average speed. (2 marks)
Answer: Average speed = distance/time = 300/5 = 60 km/h
Section C: Long Answer (4 marks each) — 60 marks
Answer any 10 questions from the 15 given. Show full working and reasoning.
Q1. An arithmetic progression has first term a and common difference d. Find the sum of first n terms and derive it. (4 marks)
Answer: Let S_n = sum of first n terms. S_n = a + (a+d) + (a+2d) + ... + (a+(n-1)d). Writing in reverse: S_n = (a+(n-1)d) + ... + (a+d) + a. Adding: 2S_n = n(2a+(n-1)d). S_n = n/2(2a+(n-1)d) or n/2(first_term + last_term).
Q2. Two circles with centers O1 and O2 and radii r1=5 cm and r2=3 cm are such that the distance between centers is 4 cm. Prove that they intersect. (4 marks)
Answer: For two circles to intersect, distance d must satisfy |r1-r2| < d < r1+r2. Here |5-3| = 2 and 5+3 = 8. Since 2 < 4 < 8, the circles intersect at two points.
Q3. Prove that angles in the same segment of a circle are equal. (4 marks)
Answer: Let circle have center O and points A, B, C, D on the circle in the same segment. Angles ACB and ADB both subtend the same arc AB. By the inscribed angle theorem, both angles equal half the central angle AOB. Therefore angle ACB = angle ADB.
Q4. A shopkeeper bought items at Rs 50 per unit and sold at Rs 75 per unit. After selling 100 units, what is the profit percentage? (4 marks)
Answer: Cost price = 50*100 = 5000. Selling price = 75*100 = 7500. Profit = 7500 - 5000 = 2500. Profit % = (2500/5000)*100 = 50%
Q5. A quadrilateral ABCD has vertices A(1,2), B(4,2), C(4,5), D(1,5). Prove it is a rectangle and find its area. (4 marks)
Answer: Distance AB = 3, BC = 3, CD = 3, DA = 3. All sides equal. Diagonals AC and BD both = sqrt(18) = 3sqrt(2). Equal diagonals and equal sides mean it's a square. Area = 3*3 = 9 square units.
Q6. If tan(A) + cot(A) = 2, find tan(A) and cot(A). (4 marks)
Answer: Let tan(A) = x. Then cot(A) = 1/x. x + 1/x = 2. x^2 - 2x + 1 = 0. (x-1)^2 = 0. x = 1. Therefore tan(A) = 1 and cot(A) = 1. So A = 45 degrees.
Q7. Find the area of a triangle with vertices at A(0,0), B(4,0), and C(2,3). (4 marks)
Answer: Area = (1/2)|x1(y2-y3) + x2(y3-y1) + x3(y1-y2)| = (1/2)|0(0-3) + 4(3-0) + 2(0-0)| = (1/2)|0 + 12 + 0| = 6 square units
Q8. Solve the quadratic equation 2x^2 - 5x + 2 = 0 using the quadratic formula. (4 marks)
Answer: x = (5 ± sqrt(25-16))/4 = (5 ± sqrt(9))/4 = (5 ± 3)/4. x = 8/4 = 2 or x = 2/4 = 1/2. Solutions: x = 2 or x = 1/2
Q9. A cone has height 12 cm and base radius 5 cm. Find its curved surface area and volume. (4 marks)
Answer: Slant height l = sqrt(h^2 + r^2) = sqrt(144 + 25) = sqrt(169) = 13 cm. Curved surface area = πrl = 65π cm^2. Volume = (1/3)πr^2h = (1/3)π*25*12 = 100π cm^3
Q10. In a data set of 10 numbers, the mean is 25. If one number 15 is replaced by 35, find the new mean. (4 marks)
Answer: Original sum = 25*10 = 250. New sum = 250 - 15 + 35 = 270. New mean = 270/10 = 27
Q11. Prove that the sum of opposite angles in a cyclic quadrilateral is 180 degrees. (4 marks)
Answer: In cyclic quadrilateral ABCD inscribed in a circle, angle A is inscribed in arc BCD and angle C is inscribed in arc BAD. These arcs together form the complete circle (360 degrees). By inscribed angle theorem, angle A = (arc BCD)/2 and angle C = (arc BAD)/2. angle A + angle C = ((arc BCD) + (arc BAD))/2 = 360/2 = 180 degrees.
Q12. A ladder of length 13 meters leans against a wall. The foot of the ladder is 5 meters away from the wall. How high does the ladder reach on the wall? (4 marks)
Answer: Using Pythagoras theorem: h^2 + 5^2 = 13^2. h^2 + 25 = 169. h^2 = 144. h = 12 meters
Q13. Find the standard deviation of the data: 2, 4, 6, 8, 10. (4 marks)
Answer: Mean = 6. Variance = ((2-6)^2 + (4-6)^2 + (6-6)^2 + (8-6)^2 + (10-6)^2)/5 = (16+4+0+4+16)/5 = 40/5 = 8. Standard deviation = sqrt(8) = 2sqrt(2)
Q14. Two variables x and y have a linear relationship y = 3x + 2. If x increases by 5, how much does y increase? (4 marks)
Answer: The rate of change of y with respect to x is 3 (the slope). When x increases by 5, y increases by 3*5 = 15.
Q15. Construct an angle of 60 degrees using compass and ruler. Then bisect it to get 30 degrees. (4 marks)
Answer: First construct 60 degrees: Draw a line, mark a point, draw an arc of any radius to intersect the line at two points, then from intersection points draw arcs of same radius to find a point forming 60 degrees. To bisect, place compass on the vertex, draw an arc, from the intersection points on the sides of 60 degrees arc, draw two arcs intersecting at a point. The line through this point and vertex bisects the angle to create 30 degrees.
FAQs
What is the best way to prepare for CBSE Class 10 Maths exam?
Practice sample papers regularly, focus on understanding concepts rather than rote learning, solve previous years papers, and identify weak areas for targeted revision.
How much time should I allocate for each section during the exam?
Allocate about 45 minutes for Section A (20 questions), 60 minutes for Section B (10 questions), and 75 minutes for Section C (10 questions), leaving 30 minutes for review.
Are calculators allowed in CBSE Class 10 Maths?
No, calculators are not allowed. You must perform all calculations by hand using mathematical methods and formulas.
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