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A · all 20 · 1 mark each · 18 multiple choice and 2 assertion-reason20

B · all 5 · 2 marks each · very short answer, internal choice in 210

C · all 6 · 3 marks each · short answer, internal choice in 218

D · all 4 · 5 marks each · long answer, internal choice in 220

E · all 3 · 4 marks each · case-based, 1 + 1 + 2 marks12

Board paper total80

CBSE Mathematics Standard (041) and Basic (241) question paper design and sample papers, 2026-27: 38 questions, 80 marks, 3 hours.

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2 marks · short answer

Check whether 14ⁿ can end with the digit 0 for any natural number n.Unit I Explain why 11 × 13 × 17 + 17 is a composite number.Unit I Find the HCF and LCM of 72 and 120 by the prime factorisation method.Unit I The HCF of two numbers is 9 and their LCM is 360. If one number is 45, find the other.Unit I Given that √7 is irrational, show that 5 − 2√7 is irrational.Unit I Find the zeroes of the quadratic polynomial x² − 2x − 24 and verify the relationship between the zeroes and the coefficients.Unit II Find a quadratic polynomial whose zeroes are 3/2 and −1/3.Unit II If α and β are the zeroes of x² + 5x + 6, find the value of 1/α + 1/β.Unit II If x = −1 is a zero of the polynomial p(x) = kx² − 4x + k, find the value of k.Unit II Without solving, find whether the pair 5x − 2y = 6 and 10x − 4y = 9 is consistent or inconsistent.Unit III Find the value of k for which the pair x + ky = 4 and 2x + 6y = 8 has infinitely many solutions.Unit III Solve by substitution: x + 2y = 7 and 2x − y = 4.Unit III Solve by elimination: 2x + 5y = 1 and 2x + 3y = 3.Unit III Find the roots of 3x² − 2x − 1 = 0 by factorisation.Unit IV Find the nature of the roots of 4x² + 4√3 x + 3 = 0.Unit IV If x = −2 is a root of kx² + x − 6 = 0, find the value of k.Unit IV Check whether (x − 3)² + 5 = (x + 1)(x − 2) is a quadratic equation.Unit IV Show that x + 2, 2x + 3 and 3x + 4 form an AP for every value of x. Find the common difference.Unit V Is 0 a term of the AP 31, 28, 25, ...? Give a reason.Unit V Find the 8th term from the end of the AP 5, 9, 13, ..., 185.Unit V Find the sum of all the odd numbers between 10 and 60.Unit V Is a triangle with sides 4 cm, 6 cm and 8 cm similar to a triangle with sides 6 cm, 9 cm and 12 cm? Give a reason.Unit VI In ΔABC, DE ∥ BC with D on AB and E on AC. If AD = 1.6 cm, DB = 2.4 cm and AE = 2 cm, find AC.Unit VI In ΔPQR, S and T lie on PQ and PR. PS = 1.2 cm, PQ = 3 cm, PT = 1.6 cm and PR = 4 cm. Is ST ∥ QR? Justify your answer.Unit VI Give one counter-example to show that two quadrilaterals with equal corresponding angles need not be similar.Unit VI Find the distance between the points P(−6, 7) and Q(−1, −5).Unit VII Find the point on the y-axis that is equidistant from A(−4, 1) and B(2, 3).Unit VII Find the ratio in which the x-axis divides the segment joining (−3, −4) and (1, 2). Also find the point of division.Unit VII The centre of a circle is (2, −3) and one end of a diameter is (−1, 4). Find the other end of the diameter.Unit VII In ΔABC, right-angled at B, AB = 9 cm and AC = 15 cm. Find sin C and cos C.Unit VIII If 4 tan θ = 3, find the value of (4 sin θ − cos θ)/(4 sin θ + cos θ).Unit VIII Evaluate sin²60° + cos²45° − tan²30°.Unit VIII Prove that cot θ − tan θ = (2 cos²θ − 1)/(sin θ cos θ).Unit VIII Define the angle of elevation and the angle of depression, with one example of each.Unit IX A 12 m ladder leans against a wall and makes 60° with the ground. How high up the wall does it reach? Leave your answer in surd form.Unit IX The shadow of a tower on level ground is √3 times its height. Find the angle of elevation of the sun.Unit IX From the top of a 50 m high cliff, the angle of depression of a boat is 45°. How far is the boat from the foot of the cliff?Unit IX AB is a diameter of a circle. Show that the tangents drawn at A and B are parallel.Unit X A point P is 26 cm from the centre of a circle of radius 10 cm. Find the length of the tangent from P to the circle.Unit X PA and PB are tangents from P to a circle with centre O. If ∠AOB = 110°, find ∠APB and ∠OPA.Unit X A quadrilateral ABCD circumscribes a circle. AB = 9 cm, BC = 11 cm and CD = 8 cm. Find AD.Unit X Find the area of a sector of a circle of radius 21 cm with central angle 120°. (Use π = 22/7)Unit XI The minute hand of a clock is 21 cm long. Find the distance travelled by its tip in 10 minutes. (Use π = 22/7)Unit XI A chord of a circle of radius 7 cm subtends a right angle at the centre. Find the area of the minor segment. (Use π = 22/7)Unit XI The circumference of a circular plate is 88 cm. Find the area of a quadrant of the plate. (Use π = 22/7)Unit XI A cone has radius 8 cm and height 15 cm. Find its slant height and its curved surface area in terms of π.Unit XII Two cubes, each of volume 125 cm³, are joined end to end. Find the surface area of the cuboid formed.Unit XII A hemisphere of radius r is fixed on one face of a cube. Show that the total surface area increases by πr².Unit XII A solid is a cone of radius 6 cm and height 8 cm standing on a hemisphere of the same radius. Find its volume in terms of π.Unit XII Find the class marks of the classes 15-25 and 35-45. What is the class size?Unit XIII In the assumed mean method, a = 55, Σfᵢdᵢ = −120 and Σfᵢ = 40. Find the mean.Unit XIII For a distribution with modal class 20-30, f₁ = 14, f₀ = 9 and f₂ = 11. Find the mode.Unit XIII The mean and mode of a distribution are 36 and 33. Find the median using the empirical relation.Unit XIII Two coins are tossed together. Find the probability of getting at most one tail.Unit XIV If P(E) = 0.38, find P(not E). Can 1.38 be the probability of an event? Give a reason.Unit XIV A die is thrown once. Find the probability of getting a number that is a multiple of 2 or of 3.Unit XIV A card is drawn from a well-shuffled pack of 52 cards. Find the probability that it is a black face card.Unit XIV

3 marks · short answer

Prove that √5 is irrational.Unit I Given that √2 is irrational, prove that 3 + 4√2 is irrational.Unit I Three bells ring at intervals of 12, 18 and 30 minutes. If they ring together at 7:00 a.m., at what time will they next ring together?Unit I Prove that √3 is irrational. Hence show that 7 − 2√3 is irrational.Unit I Find the HCF and LCM of 144, 180 and 192 by prime factorisation. Show that the product of the HCF and LCM is not equal to the product of the three numbers.Unit I Find the zeroes of the quadratic polynomial 6x² − x − 2 and verify the relationship between the zeroes and the coefficients.Unit II If one zero of the polynomial x² − 6x + k is twice the other, find the zeroes and the value of k.Unit II If α and β are the zeroes of x² − 3x + 2, find a quadratic polynomial whose zeroes are 2α + 1 and 2β + 1.Unit II Solve graphically: x + y = 6 and x − y = 2. Shade the triangle formed by these lines and the y-axis.Unit III Solve 2x + 3y = 12 and 3x − 2y = 5 by elimination. Hence find p if y = px − 1.Unit III A fraction becomes 2/3 when 2 is added to both its numerator and denominator. It becomes 1/2 when 1 is subtracted from both. Find the fraction.Unit III Solve x² − 3√5 x + 10 = 0 using the quadratic formula.Unit IV Find the values of k for which the equation kx² + 4x + (k − 3) = 0 has two equal roots.Unit IV The product of the digits of a two-digit number is 14. When 45 is added to the number, its digits are interchanged. Find the number.Unit IV The 4th term of an AP is 11 and its 9th term exceeds its 6th term by 9. Find the AP and its 20th term.Unit V The sum of the first n terms of an AP is Sₙ = 2n² + 3n. Find the first term, the common difference and the 10th term.Unit V How many terms of the AP 45, 39, 33, ... must be taken so that their sum is 180? Explain why there are two answers.Unit V In ΔABC, D, E and F lie on AB, AC and BC respectively, with DE ∥ BC and EF ∥ AB. Prove that AD/DB = BF/FC.Unit VI A vertical pole 2.4 m high casts a shadow 1.6 m long. At the same time, a building casts a shadow 24 m long. Find the height of the building.Unit VI In ΔABC, P and Q lie on AB and AC with AP = 3 cm, PB = 6 cm, AQ = 4 cm and QC = 8 cm. Prove that ΔAPQ ~ ΔABC and find PQ if BC = 13.5 cm.Unit VI Check whether the points (−4, −1), (−1, 1) and (5, 5) are collinear.Unit VII Find the points of trisection of the segment joining (−6, 3) and (3, −6).Unit VII The point P(2, k) lies on the segment joining A(−2, 5) and B(4, −1). Find the ratio in which P divides AB, and the value of k.Unit VII Given sin (A + B) = 1 and cos (A − B) = √3/2. Here 0° < A + B ≤ 90° and A > B. Find A and B.Unit VIII Prove that √(sec²θ + cosec²θ) = tan θ + cot θ.Unit VIII If 3 tan θ = 4, find sin θ and cos θ, and evaluate (sin θ + cos θ)/(sec θ + cosec θ).Unit VIII The angle of elevation of the top of a tower from a point on the ground is 30°. On moving 30 m nearer the tower, it becomes 60°. Find the height of the tower. (Use √3 = 1.732)Unit IX A 4 m tall statue stands on a pedestal. From a point on the ground, the angles of elevation of the top of the statue and the top of the pedestal are 60° and 30°. Find the height of the pedestal.Unit IX A girl whose eyes are 1.2 m above the ground stands 30 m from a building. The angle of elevation of its top from her eyes is 30°. Find the height of the building. (Use √3 = 1.732)Unit IX Prove that the lengths of tangents drawn from an external point to a circle are equal.Unit X ABCD is a parallelogram whose sides all touch a circle. Prove that ABCD is a rhombus.Unit X Two concentric circles have radii 17 cm and 8 cm. Find the length of the chord of the larger circle that touches the smaller circle.Unit X Find the areas of the minor and major sectors of a circle of radius 14 cm, when the minor sector's angle is 45°. (Use π = 22/7)Unit XI A chord of a circle of radius 35 cm subtends a right angle at the centre. Find the area of the minor segment. (Use π = 22/7)Unit XI Find the perimeter and the area of a sector of a circle of radius 10.5 cm with central angle 60°. (Use π = 22/7)Unit XI A wooden toy is a cone on a hemisphere, both of radius 7 cm. The total height of the toy is 31 cm. Find its total surface area. (Use π = 22/7.)Unit XII From a solid cylinder of height 8 cm and radius 6 cm, a cone of the same height and radius is hollowed out. Find the total surface area of the remaining solid in terms of π.Unit XII A pen stand is a cuboid 12 cm × 7 cm × 6 cm. It has two cylindrical holes, each of radius 1.4 cm and depth 5 cm. Find the volume of wood in the stand. (Use π = 22/7.)Unit XII Find the mean of the following data by the step-deviation method: 0-20: 7, 20-40: 10, 40-60: 15, 60-80: 10, 80-100: 8.Unit XIII Find the mode of the following data: 10-20: 6, 20-30: 11, 30-40: 18, 40-50: 13, 50-60: 2.Unit XIII Find the median of the following data: 0-10: 4, 10-20: 6, 20-30: 12, 30-40: 10, 40-50: 8.Unit XIII Two dice are thrown together. Find the probability that (i) the sum is 6, (ii) the numbers on the two dice are different, (iii) the sum is at least 10.Unit XIV A bag has 8 red balls and some blue balls. The probability of drawing a blue ball is 3/5. Find the number of blue balls.Unit XIV Tokens numbered 1 to 40 are put in a box and one is drawn at random. Find the probability that the number is (i) a prime, (ii) a multiple of 7, (iii) a perfect square.Unit XIV

5 marks · long answer

Find the zeroes of p(x) = 4x² − 4x − 3 and verify the relationship between the zeroes and the coefficients. Write the points where the graph of y = p(x) cuts the x-axis. Also find a quadratic polynomial whose zeroes are the reciprocals of the zeroes of p(x).Unit II If α and β are the zeroes of 2x² − 7x + 3, find (i) α² + β² and (ii) a quadratic polynomial whose zeroes are α² and β².Unit II Draw the graphs of 2x − y = 2 and 4x − y = 8 on the same axes. Find the coordinates of the vertices of the triangle formed by these lines and the x-axis, and find its area.Unit III Ravi invested part of his savings at 10% and the rest at 12% simple interest per year. His total interest for one year was ₹ 1400. If the two rates had been swapped, he would have got ₹ 50 less. Find the two amounts he invested.Unit III A cyclist travels 72 km at a uniform speed. If she increased her speed by 2 km/h, she would take 3 hours less. Find her original speed.Unit IV The sum of the squares of two consecutive positive even numbers is 452. Form a quadratic equation, check the nature of its roots using the discriminant, and find the numbers using the quadratic formula.Unit IV In a stadium, the first row has 30 seats and each row has 4 more seats than the row in front of it. Find (i) the number of seats in the 20th row, (ii) the total number of seats in the first 25 rows, and (iii) which row has 110 seats.Unit V Derive the formula for the sum of the first n terms of an AP. Use it to find the sum of all three-digit numbers that are multiples of 11.Unit V State and prove the Basic Proportionality Theorem. Use it to find EC in ΔABC, where DE ∥ BC, AD = 2.5 cm, DB = 5 cm and AE = 3 cm.Unit VI A lamp is fixed 6 m above the ground on a pole. A boy 1.5 m tall stands 9 m from the foot of the pole. Find the length of his shadow. How far from the pole must he stand for his shadow to be 4 m long?Unit VI Show that A(1, 1), B(5, 3), C(4, 5) and D(0, 3) are the vertices of a rectangle. Is it a square? Give a reason.Unit VII A(−2, −1), B(1, 0) and C(4, 3) are three vertices of parallelogram ABCD. Find the fourth vertex D. Find the lengths of all four sides and say whether ABCD is a rhombus.Unit VII Prove that sin²A + cos²A = 1 for an acute angle A. Hence prove that 1 + tan²A = sec²A and 1 + cot²A = cosec²A, and state the values of A for which each identity holds.Unit VIII If sec θ + tan θ = p, prove that sin θ = (p² − 1)/(p² + 1). Hence find sin θ and cos θ when p = 2.Unit VIII A lighthouse is 120 m high. Two boats are on the same side of it, in line with its foot. From its top, their angles of depression are 45° and 60°. Find the distance between the boats. (Use √3 = 1.732)Unit IX A building is 45 m high. From its top, the angles of depression of the top and the foot of a lamp tower are 30° and 60°. Find the height of the lamp tower and the distance between the building and the lamp tower. (Use √3 = 1.732)Unit IX Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. Use it to find the length of the tangent from a point 25 cm from the centre of a circle of radius 7 cm.Unit X A circle touches the sides AB, BC and CA of ΔABC at P, Q and R. AB = 14 cm, BC = 16 cm and CA = 12 cm. Find AP, BQ and CR, and show that AP + BQ + CR is half the perimeter of the triangle.Unit X A chord of a circle of radius 7 cm subtends 120° at the centre. Find the areas of the minor and major segments. (Use π = 22/7 and √3 = 1.73)Unit XI A square park of side 42 m has a flower bed shaped as a quadrant of radius 21 m at each corner. Find the area of the park outside the flower beds, and the perimeter of one flower bed. (Use π = 22/7)Unit XI A shamiana is a cylinder of radius 2.1 m and height 3 m with a conical top of height 2.8 m. Find the canvas needed (no floor) and its cost at ₹ 120 per m². Also find the volume of air inside. (Use π = 22/7.)Unit XII A solid is a cylinder with a hemisphere at each end. Its total length is 17 cm and its diameter is 7 cm. Find its volume and its total surface area. (Use π = 22/7.)Unit XII The median of the following distribution of 40 values is 35. Find x and y: 0-10: 2, 10-20: 5, 20-30: x, 30-40: 12, 40-50: y, 50-60: 3.Unit XIII The milk (in litres) given by 50 farmers to a dairy is: 20-30: 6, 30-40: 11, 40-50: 15, 50-60: 10, 60-70: 8. Find the mean by any method and the mode. Which measure tells the most common amount?Unit XIII One card is drawn from a well-shuffled pack of 52 cards. Find the probability of getting (i) a black queen, (ii) a card that is not a face card, (iii) a diamond or an ace, (iv) neither a red card nor a queen, (v) the 7 of clubs.Unit XIV Three coins are tossed together. Find the probability of getting (i) exactly two heads, (ii) at least two heads, (iii) at most one head, (iv) no tail. Show that the answers to (ii) and (iii) add up to 1, and explain why.Unit XIV

4 marks · case-based (1 + 1 + 2)

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