CBSE Class 10 · Maths · board exam 2027
CBSE Class 10 Maths Question Paper 2026
The board’s 2026 question paper, set 30/1/1 — all 38 questions as printed; 20 objective answers from the board’s own key; full model answers for 3 written questions, how the answer begins for 8 more. Each question names its chapter, where that chapter’s most-asked questions are.
Full marks 80 · Time 3 h
Section A · Objective questions (20)
The HCF of 960 and 432 is :
- (A) 48 ✓
- (B) 54
- (C) 72
- (D) 36
Answer (the board’s key): (A)
The natural number 2 is :
- (A) a prime number ✓
- (B) a composite number
- (C) prime as well as composite
- (D) neither prime nor composite
Answer (the board’s key): (A)
For any natural number n, 6^n ends with the digit :
- (A) 0
- (B) 6 ✓
- (C) 3
- (D) 2
Answer (the board’s key): (B)
The graph of y = f(x) is given. The number of zeroes of f(x) is :
- (A) 0 ✓
- (B) 1
- (C) 2
- (D) 4
Answer (the board’s key): (A)
If a pair of linear equations in two variables is represented by two coincident lines, then the pair of equations has :
- (A) a unique solution
- (B) two solutions
- (C) no solution
- (D) an infinite number of solutions ✓
Answer (the board’s key): (D)
The common difference of the AP : sqrt(2), 2sqrt(2), 3sqrt(2), 4sqrt(2), ..... is :
- (A) sqrt(2) ✓
- (B) 1
- (C) 2sqrt(2)
- (D) -sqrt(2)
Answer (the board’s key): (A)
If triangle ABC and triangle DEF are similar, where 2 AB = DE and BC = 8 cm, then EF is equal to :
- (A) 4 cm
- (B) 8 cm
- (C) 12 cm
- (D) 16 cm ✓
Answer (the board’s key): (D)
The mid-point of the line segment joining the points (5, - 4) and (6, 4) lies on :
- (A) x-axis ✓
- (B) y-axis
- (C) origin
- (D) neither x-axis nor y-axis
Answer (the board’s key): (A)
If sin theta = a/b, then cos theta is equal to :
- (A) b / sqrt(b^2 - a^2)
- (B) b / a
- (C) sqrt(b^2 - a^2) / b ✓
- (D) a / sqrt(b^2 - a^2)
Answer (the board’s key): (C)
If cos A = 1/2, then the value of sin^2 A + 2 cos^2 A is :
- (A) 3/2
- (B) 5/4 ✓
- (C) -1
- (D) 1/2
Answer (the board’s key): (B)
A car is moving away from the base of a 30 m high tower. The angle of elevation of the top of the tower from the car at an instant, when the car is 10sqrt(3) m away from the base of the tower, is :
- (A) 30°
- (B) 45°
- (C) 90°
- (D) 60° ✓
Answer (the board’s key): (D)
If two tangents TP and TQ are drawn to a circle with centre O from an external point T such that angle POQ = 120 degrees, then angle PTQ is equal to :
- (A) 60 degrees ✓
- (B) 70 degrees
- (C) 80 degrees
- (D) 90 degrees
Answer (the board’s key): (A)
In the given figure, PA is a tangent from an external point P to a circle with centre O. If angle POB = 125 degrees, then angle APO is equal to :
- (A) 25 degrees
- (B) 65 degrees
- (C) 90 degrees
- (D) 35 degrees ✓
Answer (the board’s key): (D)
The length of the arc of a sector of a circle with radius 21 cm and central angle 60 degrees is :
- (A) 22 cm ✓
- (B) 44 cm
- (C) 88 cm
- (D) 11 cm
Answer (the board’s key): (A)
The minute hand of a clock is 7 cm long. The angle swept by it between 7:00 a.m. and 8:10 a.m. is :
- (A) (35/4) degrees
- (B) (35/2) degrees
- (C) 35 degrees ✓
- (D) 70 degrees
Answer (the board’s key): (C)
The total surface area of a solid hemisphere of diameter '2d' is :
- (A) 3 pi d^2 ✓
- (B) 2 pi d^2
- (C) 1/2 pi d^2
- (D) 3/4 pi d^2
Answer (the board’s key): (A)
If the mean and mode of a data are 12 and 21 respectively, then its median is :
- (A) 6
- (B) 13.5
- (C) 15 ✓
- (D) 14
Answer (the board’s key): (C)
A die is thrown once. Probability of getting a number other than 3 is :
- (A) 1/6
- (B) 3/6
- (C) 5/6 ✓
- (D) 1
Answer (the board’s key): (C)
Assertion (A) : The probability that a leap year has 53 Mondays is 2/7. Reason (R) : The probability that a non-leap year has 53 Mondays is 5/7.
- (A) Both, Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
- (B) Both, Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
- (C) Assertion (A) is true, but Reason (R) is false. ✓
- (D) Assertion (A) is false, but Reason (R) is true.
Answer (the board’s key): (C)
Assertion (A) : The polynomial p(y) = y^2 + 4y + 3 has two zeroes. Reason (R) : A quadratic polynomial can have at most two zeroes.
- (A) Both, Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
- (B) Both, Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A). ✓
- (C) Assertion (A) is true, but Reason (R) is false.
- (D) Assertion (A) is false, but Reason (R) is true.
Answer (the board’s key): (B)
Section B · Written questions (18)
If alpha, beta are the zeroes of the polynomial p(x) = x^2 - 3x - 1, then find the value of 1/alpha + 1/beta.
Medha Ma’am
AskMedha’s AI teacher
Model answer · Zeroes and coefficients relationship
- 1.Given p(x) = x² – 3x – 1, with zeroes α and β. Sum of zeroes = α + β = –b/a = –(–3)/1 = 3 Product of zeroes = αβ = c/a = –1/1 = –1 Here, α + β = 3 and αβ = –1.1 mark
- 2.1/α + 1/β = (α + β)/αβ = 3/(–1)0.5 marks
- 3.= –3 So, 1/α + 1/β = –3.0.5 marks
- Write both values first: "α + β = 3 and αβ = –1", using sum of zeroes = –b/a and product of zeroes = c/a.
- Show the step "1/α + 1/β = (α + β)/αβ" before you put in the numbers.
- Keep the minus sign of αβ = –1, so that the final answer is –3.
Marks split as in the board’s published marking scheme
(A) In triangle ABC, DE || BC. If AD = x, DB = x - 2, AE = x + 2 and EC = x - 1, then find the value of x. OR (B) In the figure given above, triangle ABC ~ triangle XYZ, then find the values of x and y.
The coordinates of the centre of a circle are (x - 7, 2x). Find the value(s) of 'x', if the circle passes through the point (- 9, 11) and has radius 5*sqrt(2) units.
(A) If tan theta = 24/7, then find the value of sin theta + cos theta. OR (B) If cot theta = 7/8, then find the value of (1 + sin theta)(1 - sin theta) / (1 + cos theta)(1 - cos theta).
Two concentric circles are of radii 5 cm and 4 cm. Find the length of the chord of the larger circle which touches the smaller circle.
Medha Ma’am
AskMedha’s AI teacher
Model answer · tangent, concentric circles chord
- 1.Given: two concentric circles with centre O, where the larger circle has radius OA = 5 cm and the smaller circle has radius OM = 4 cm. AB is a chord of the larger circle that touches the smaller circle at the point M (figure drawn alongside).0.5 marks
- 2.AB is a tangent to the smaller circle at M, and OM is its radius. The tangent to a circle is perpendicular to the radius through the point of contact, so OM ⊥ AB. In right triangle OMA, by the Pythagoras Theorem: AM = √(OA² − OM²) = √(5² − 4²) = √(25 − 16) = √9 = 3 cm1 mark
- 3.The perpendicular from the centre bisects the chord, so AM = MB. AB = 2 × AM = 2 × 3 = 6 cm So the length of the chord is 6 cm.0.5 marks
Diagram: Draw two concentric circles with centre O. Draw chord AB of the larger circle so that it touches the smaller circle at M. Join OM and OA, mark the right angle at M, and write OA = 5 cm and OM = 4 cm.
- Write the reason for "OM ⊥ AB": the tangent to a circle is perpendicular to the radius through the point of contact.
- Give the reason for doubling AM: the perpendicular from the centre bisects the chord.
- On the figure, label O, A, B and M, mark the right angle at M, and write both radii.
Marks split as in the board’s published marking scheme
Prove that sqrt(3) is an irrational number.
Medha Ma’am
AskMedha’s AI teacher
Model answer · Proving irrationality of roots
- 1.Let us assume, to the contrary, that √3 is a rational number. Then √3 = p/q, where p and q are integers, q ≠ 0, and p and q are coprime.0.5 marks
- 2.Squaring on both sides and rearranging, we get 3q² = p² So p² is divisible by 3. Since 3 is a prime, if 3 divides p², then 3 divides p. So p is divisible by 3. ----- (i)1 mark
- 3.Let p = 3a, where a is some integer. Substituting for p, we get 3q² = 9a², that is, q² = 3a² So q² is divisible by 3, and therefore q is divisible by 3. ----- (ii)1 mark
- 4.From (i) and (ii), p and q have at least 3 as a common factor. But this contradicts the fact that p and q are coprime. This contradiction has arisen because of our incorrect assumption that √3 is rational. Therefore, √3 is an irrational number.0.5 marks
- In the first line, write that p and q are integers, q ≠ 0 and "p and q are coprime".
- Write both steps, "p² is divisible by 3" and then "p is divisible by 3", and do the same for q² and q.
- End with the contradiction, "this contradicts the fact that p and q are coprime", and then the conclusion that √3 is an irrational number.
Marks split as in the board’s published marking scheme
Find the ratio in which the x-axis divides the line segment joining the points (- 6, 5) and (- 4, - 1). Also, find the point of intersection.
How the model answer begins
1. Given: A(–6, 5) and B(–4, –1). Let the x-axis meet AB at P, with coordinates of P be (x, 0), and let P divide the line segment AB in the ratio k : 1.
The full answer from Medha Ma’am →(A) If x = h + a cos theta, y = k + b sin theta, then prove that : ((x - h)/a)^2 + ((y - k)/b)^2 = 1 OR (B) Prove that : tan A / (1 + sec A) - tan A / (1 - sec A) = 2 cosec A
How the model answer begins
1. Solving (A). Given: x = h + a cos θ and y = k + b sin θ. From the first equation, subtract h from both sides: x − h = a cos θ Divide both sides by a: (x − h)/a = cos θ ... (1)
The full answer from Medha Ma’am →(A) In the given figure, triangle ABC is a right triangle in which angle B = 90 degree, AB = 4 cm and BC = 3 cm. Find the radius of the circle inscribed in the triangle ABC. OR (B) In the given figure, if a circle touches the side QR of triangle PQR at S and extended sides PQ and PR at M and N respectively, then prove that : PM = 1/2 (PQ + QR + PR)
A solid is in the form of a cylinder with hemispherical ends. The total height of the solid is 20 cm and the diameter of the cylinder is 7 cm. Find the total volume of the solid. (Use pi = 22/7)
How the model answer begins
1. r = 7/2 = 3.5 cm; height of the cylinder h = 20 − 2 × 3.5 = 13 cm.
The full answer from Medha Ma’am →31. Two dice of different colours are thrown at the same time. Write down all the possible outcomes. What is the probability that : (i) same number appears on both the dice ? (ii) different number appears on both the dice ?
How the model answer begins
1. When two dice of different colours are thrown at the same time, each die can show any one of the numbers 1, 2, 3, 4, 5, 6. In each ordered pair, the first number is the number on the first die and the second number is…
The full answer from Medha Ma’am →Find the coordinates of the vertices of the triangle formed by the graph, whose sides are represented by the equations 2x - 3y + 6 = 0; 2x + 3y - 18 = 0 and x = 0. Also find the area of the triangle.
How the model answer begins
1. Line 1: 2x - 3y + 6 = 0, so y = (2x + 6)/3. Table of two or more solutions: x: 0, 3, -3 y: 2, 4, 0 Plot the points A(0, 2), B(3, 4) and (-3, 0) on graph paper. Join them to get the line of 2x - 3y + 6 = 0.
The full answer from Medha Ma’am →33. (A) A faster train takes one hour less than a slower train for a journey of 200 km. If the speed of the slower train is 10 km/hr less than that of the faster train, find the speeds of the two trains. OR (B) The sum of the areas of two squares is 640 m^2. If the difference in their perimeters is 64 m, find the sides of the two squares.
How the model answer begins
1. The speed of the faster train is 50 km/h and the speed of the slower train is 40 km/h. The working is shown below. Given: the journey is 200 km, the faster train takes 1 hour less than the slower train, and the slower…
The full answer from Medha Ma’am →(A) State and prove Basic Proportionality Theorem. OR (B) In the given figure, CM and RN are the medians of triangle ABC and triangle PQR respectively. If triangle ABC ~ triangle PQR, then prove that : (i) triangle AMC ~ triangle PNR (ii) triangle CMB ~ triangle RNQ
How the model answer begins
1. Basic Proportionality Theorem (Thales Theorem): If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
The full answer from Medha Ma’am →The mean of the following frequency distribution is 35. If the sum of frequencies is 25, find the values of x and y : Class: 0-10, 10-20, 20-30, 30-40, 40-50, 50-60, 60-70 Frequency: 1, x, 5, 7, y, 3, 1
How the model answer begins
1. Given: the mean of the distribution is 35 and the sum of frequencies is 25. We find the mean by the direct method, x̄ = Σf_i x_i / Σf_i, where x_i is the class mark of each class. The table of class marks (x_i) and…
The full answer from Medha Ma’am →36. In a potato race, a bucket is placed at the starting point, which is 5 m from the first potato. The other potatoes are arranged 3 m apart in a straight line, with a total of 10 potatoes, as shown in the figure : A competitor starts from the bucket, picks up the nearest potato, runs back to the bucket to drop it in, then returns to pick up the next potato. This process continues until all the potatoes are in the bucket. Based on the above information, answer the following questions : (i) What is the distance covered to pick up the first potato and drop it in bucket ? (ii) What is the distance covered to pick up the second potato and drop it in bucket ? (iii) (a) What is the total distance the competitor has to run ? OR (iii) (b) If average speed of competitor is 5 m/s, then find the average time taken by competitor to put all the potatoes in the bucket.
Radio towers are used for transmitting a range of communication services including radio and television. The tower will either act as an antenna itself or support one or more antennas on its structure. On a similar concept, a radio station tower was built in two sections ‘A’ and ‘B’. Tower is supported by wires from a point ‘O’ (as shown in figure). Distance between the base of the tower and point ‘O’ is 6 m. From point ‘O’, the angle of elevation of the top of the section ‘B’ is 30° and the angle of elevation of the top of section ‘A’ is 60°. Based on the above information, answer the following questions : (i) Find the length of the wire from the point ‘O’ to the top of section ‘B’. (ii) Find the length of the wire from the point ‘O’ to the top of section ‘A’. (iii) (a) Find the distance AB. OR (iii) (b) Find the area of Δ OPB.
A brooch is crafted from silver wire in the shape of a circle with a diameter of 35 cm. The wire is also used to create 5 diameters, dividing the circle into 10 equal sectors as shown in figure. Based on the above information, answer the following questions : (i) What is the radius of circle ? (ii) What is the circumference of the brooch ? (iii) (a) What is the total length of silver wire required ? OR (iii) (b) What is the area of each sector of the brooch ?
The same preparation, with Medha Ma’am — on your phone
Every question of every chapter with its model answer, practice questions in the board’s own shapes, 60 minutes of class a day, written answers marked, a test every Sunday and a WhatsApp report to parents.
Medha Ma’am is an AI teacher, not a live tutor. Every answer is checked twice; a person takes responsibility.