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CBSE Class 10 Maths Basic (241) sample paper 2026–27, with marking scheme

A full practice paper in the 2026–27 board pattern, 80 marks for 3 hours. 38 questions in Sections A to E, with more direct questions than Standard: MCQs, assertion–reason, 2-, 3- and 5-mark questions and 3 case studies. Paper AW-10MB-BP-70241; it follows the board pattern but is not an official CBSE paper.

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General instructions

  1. This paper has 38 questions. All questions are compulsory.
  2. It is in five sections: A, B, C, D and E.
  3. Section A has 20 multiple-choice questions (18 MCQs and 2 assertion–reason questions) of 1 mark each.
  4. Section B has 5 very short answer questions of 2 marks each.
  5. Section C has 6 short answer questions of 3 marks each.
  6. Section D has 4 long answer questions of 5 marks each.
  7. Section E has 3 case-based questions of 4 marks each, with sub-parts of 1, 1 and 2 marks.
  8. Some questions have an internal choice (marked OR): answer only one of them.
  9. Draw neat figures wherever needed. Take π = 22/7 wherever needed, unless the question says otherwise.
  10. Use of a calculator is not allowed.

The paper

Section A: 20 questions of 1 mark each

  1. Q1. 1 mark

    The LCM of 100 and 140 is:

    1. 20
    2. 240
    3. 14000
    4. 700
  2. Q2. 1 mark

    The product of the zeroes of the polynomial x² + 4x − 45 is:

    1. 45
    2. −4
    3. −46
    4. −45
  3. Q3. 1 mark

    The solution of 4x − 6y = 8 and 3x + 6y = −36 is:

    1. (4, −4)
    2. (−3, −4)
    3. (−4, −4)
    4. (−4, 4)
  4. Q4. 1 mark

    If x + y = 27 and x − y = −13, then (x, y) is:

    1. (27, −13)
    2. (7, 20)
    3. (20, 7)
    4. (7, −20)
  5. Q5. 1 mark

    Which of these is a quadratic equation?

    1. 9x + 2 = 0
    2. (x − 3)² = x² + 2
    3. x³ + 9x + 2 = 0
    4. 5x² + 9x + 2 = 0
  6. Q6. 1 mark

    The sum of the first 14 multiples of 3 is:

    1. 315
    2. 588
    3. 42
    4. 105
  7. Q7. 1 mark

    The sum of the first 29 terms of 16, 14, 12, … is:

    1. −348
    2. 406
    3. −390
    4. −696
  8. Q8. 1 mark

    In △ABC, DE ∥ BC with D on AB and E on AC. If AD = 20 cm, DB = 28 cm and AE = 5 cm, then EC is:

    1. 12 cm
    2. 7 cm
    3. 8 cm
    4. 25/7 cm
  9. Q9. 1 mark

    △ABC ~ △DEF. AB = 3 cm, DE = 12 cm and the perimeter of △ABC is 11 cm. The perimeter of △DEF is:

    1. 46 cm
    2. 11 cm
    3. 44 cm
    4. 55 cm
  10. Q10. 1 mark

    The midpoint of (4, 6) and (6, 14) is:

    1. (2, 8)
    2. (5, 10)
    3. (10, 5)
    4. (10, 20)
  11. Q11. 1 mark

    cosec A equals:

    1. cos A / sin A
    2. 1 / cos A
    3. 1 / tan A
    4. 1 / sin A
  12. Q12. 1 mark

    If cosec A = 13/5, then tan A equals:

    1. 12/5
    2. 5/12
    3. 12/13
    4. 5/13
  13. Q13. 1 mark

    The tangent from a point P to a circle of radius 8 cm is 6 cm long. The distance of P from the centre is:

    1. 2 cm
    2. 11 cm
    3. 14 cm
    4. 10 cm
  14. Q14. 1 mark

    A line that meets a circle at two points is called a:

    1. diameter
    2. tangent
    3. secant
    4. radius
  15. Q15. 1 mark

    A solid is a cylinder of radius 7 cm and height 19 cm with a cone of the same radius and height 18 cm on top. Its volume (π = 22/7) is:

    1. 4,850 cm³
    2. 2,926 cm³
    3. 5,698 cm³
    4. 3,850 cm³
  16. Q16. 1 mark

    For the data below, the modal class is: Class: 80–100, 100–120, 120–140, 140–160, 160–180, 180–200 Frequency: 19, 17, 10, 17, 12, 12

    1. 100–120
    2. 120–140
    3. 80–100
    4. 140–160
  17. Q17. 1 mark

    The class mark of the class 175–200 is:

    1. 25
    2. 200
    3. 212.5
    4. 187.5
  18. Q18. 1 mark

    If P(E) = 0.1, then P(not E) is:

    1. 1
    2. 0.8
    3. 0.9
    4. 0.1
  19. Q19. 1 mark

    Assertion (A): When a die is thrown, the probability of getting a number greater than 4 is 1/3. Reason (R): The probability of an impossible event is 0.

    1. Both A and R are true, and R is the correct explanation of A.
    2. Both A and R are true, but R is not the correct explanation of A.
    3. A is true, but R is false.
    4. A is false, but R is true.
  20. Q20. 1 mark

    Assertion (A): In △ABC, DE ∥ BC with AD = 15 cm, DB = 9 cm and AE = 15 cm, so EC = 10 cm. Reason (R): A line drawn parallel to one side of a triangle divides the other two sides in the same ratio.

    1. Both A and R are true, and R is the correct explanation of A.
    2. Both A and R are true, but R is not the correct explanation of A.
    3. A is true, but R is false.
    4. A is false, but R is true.

Section B: 5 questions of 2 marks each

  1. Q21. 2 marks

    Find the HCF and LCM of 6 and 9, and check that HCF × LCM = 6 × 9.

    OR

    Express 66 as a product of its prime factors.

  2. Q22. 2 marks

    One zero of 2x² + 2x − 12 is 2. Find the other zero.

  3. Q23. 2 marks

    Evaluate: (3 tan² 30° + 4 cos² 45°) ÷ (3 tan² 45°)

  4. Q24. 2 marks

    Tangents PA and PB are drawn from an external point P to a circle with centre O. If ∠AOB = 100°, find ∠APB.

  5. Q25. 2 marks

    Find y if the distance between (5, 8) and (−4, y) is 15 units.

    OR

    Find the perimeter of the triangle with vertices (4, 3), (9, 3) and (4, 15).

Section C: 6 questions of 3 marks each

  1. Q26. 3 marks

    Prove that √3 is irrational. Hence show that 7 − 8√3 is irrational.

  2. Q27. 3 marks

    The sum of the digits of a two-digit number is 15. The number obtained by reversing the order of its digits is 27 less than the original number. Find the number.

  3. Q28. 3 marks

    PA and PB are tangents drawn from an external point P to a circle with centre O and radius 120 cm. If PA = 27 cm, find OP and the perimeter of the quadrilateral OAPB.

    OR

    TG is a diameter of a circle with centre O. Prove that the tangents drawn at T and G are parallel. A third tangent touches the circle at D and meets the tangents at T and G at W and B respectively. If TW = 15 cm and GB = 16 cm, find WB.

  4. Q29. 3 marks

    Show that the points P(−7, −3), Q(−4, −6) and R(−7, −9) are the vertices of a right isosceles triangle.

  5. Q30. 3 marks

    If cot θ = 5/12, where θ is an acute angle, find the value of (sec θ + tan θ)/(sec θ − tan θ).

  6. Q31. 3 marks

    The following table gives the marks obtained by the students of a class in a test. The mean of the data is 25. Find the missing frequency p. Class interval (marks) and frequency (number of students): 0–10: 7, 10–20: 13, 20–30: 5, 30–40: p, 40–50: 12.

    OR

    The following table gives the weights (in kg) of the students of a class. Find the mean using the direct method. Class interval (weight in kg) and frequency (number of students): 30–35: 10, 35–40: 4, 40–45: 3, 45–50: 18, 50–55: 5.

Section D: 4 questions of 5 marks each

  1. Q32. 5 marks

    Kabir, who is 0.9 m tall, walks away from the base of a lamp-post in a straight line at a speed of 1.2 m/s. The lamp is 4.5 m above the ground. (a) Find the length of Kabir's shadow after 8 seconds. (b) How far from the base of the lamp-post will Kabir be when the shadow is 1.9 m long?

  2. Q33. 5 marks

    Find two consecutive even positive integers, the sum of whose squares is 3700.

    OR

    A father is 23 years older than his son. 3 years ago, the product of their ages (in years) was 78. Find their present ages.

  3. Q34. 5 marks

    Isha makes a solid paperweight in the shape of a cone standing on a hemisphere of the same radius. The radius is 28 cm, the height of the cone is 21 cm and its slant height is 35 cm. Find (i) the total surface area of the paperweight and (ii) its volume. (Use π = 22/7.)

  4. Q35. 5 marks

    From a point on a bridge across a river, the angles of depression of the two banks, on opposite sides of the river, are 45° and 30°. The point is 48 m above the water. Find the width of the river, and how far the point is horizontally from each bank. (Take √3 = 1.73.)

    OR

    The angles of elevation of the top of a chimney from two points on the ground, on the same side of it and in a straight line with its foot, are 30° and 60°. The two points are 46 m apart. Find the height of the chimney and the distance of the nearer point from its foot. (Take √3 = 1.73.)

Section E: 3 case-based questions of 4 marks each

  1. Q36. 4 marks

    Sharing a round treat

    At Ananya's birthday party, a round cake of diameter 21 cm is cut from the centre into 5 equal slices. Each slice is a sector of a circle, and its curved edge is the icing border. Asha wants to work out the size of each slice. (Use π = 22/7.)

    1. What is the angle at the centre of one slice?

      1 mark

    2. Find the area of the top of one slice.

      1 mark

    3. Find the perimeter of the top of one slice.

      2 marks

      OR

      The guests eat 3 slices. Find the area of the top of the round cake that is left.

  2. Q37. 4 marks

    Seats in the school hall

    The school assembly hall has 25 rows of seats. The first row has 12 seats, and each row has 2 more seats than the row in front of it. The school is getting the assembly hall ready for its annual day. The numbers of seats in the rows form an arithmetic progression.

    1. How many seats are there in row 19?

      1 mark

    2. Which row has 32 seats?

      1 mark

    3. Find the total number of seats in the assembly hall.

      2 marks

      OR

      The first 8 rows are kept for parents. How many seats are kept for parents?

  3. Q38. 4 marks

    Checking for defects

    A workshop moulds 240 water bottles in a day. The supervisor checks them and finds that 200 are good, 37 have minor defects and the rest have major defects. Kabir, a customer, will buy only a good one, while a shopkeeper will buy one that has no major defect. One of the day's water bottles is picked at random.

    1. Find the probability that Kabir will buy it.

      1 mark

    2. Find the probability that the shopkeeper will buy it.

      1 mark

    3. Find the probability that it is not good.

      2 marks

      OR

      Find the probability that neither Kabir nor the shopkeeper will buy it.

Marking scheme

The answers and the marks for each step or point, as an examiner marks them. Open a section to check a question.

Section A: 20 questions of 1 mark each: answers and marks
  1. Q1

    (d) 700 (LCM = 20 × 5 × 7 = 700)

  2. Q2

    (d) −45 (product of zeroes = c/a = −45)

  3. Q3

    (c) (−4, −4) (solve: x = −4, y = −4)

  4. Q4

    (b) (7, 20) (adding: 2x = 14, x = 7; y = 20)

  5. Q5

    (d) 5x² + 9x + 2 = 0 (the highest power of x is 2, and the x² term does not cancel)

  6. Q6

    (a) 315 (3(1 + 2 + … + 14) = 3 × 105)

  7. Q7

    (a) −348 (29/2 × (2 × 16 + 28 × (−2)) = −348)

  8. Q8

    (b) 7 cm (AD/DB = AE/EC: 20/28 = 5/EC, so EC = 7)

  9. Q9

    (c) 44 cm (perimeters are in the ratio of corresponding sides: 11 × 4/1 = 44)

  10. Q10

    (b) (5, 10) (((4 + 6)/2, (6 + 14)/2))

  11. Q11

    (d) 1 / sin A

  12. Q12

    (b) 5/12 (cosec A = 13/5 gives sides 5 : 12 : 13; tan A = 5/12)

  13. Q13

    (d) 10 cm (OP = √(8² + 6²) = 10)

  14. Q14

    (c) secant

  15. Q15

    (d) 3,850 cm³ (πr²h₁ + (1/3)πr²h₂ = 2,926 + 924)

  16. Q16

    (c) 80–100 (the highest frequency is 19)

  17. Q17

    (d) 187.5 ((175 + 200) ÷ 2 = 187.5)

  18. Q18

    (c) 0.9 (1 − 0.1 = 0.9)

  19. Q19

    (b) Both A and R are true, but R is not the correct explanation of A. (A is true: 2 of the 6 outcomes are favourable. R is true but does not explain A.)

  20. Q20

    (d) A is false, but R is true. (A is false: EC = AE × DB/AD = 9, not 10. R is true.)

Section B: 5 questions of 2 marks each: answers and marks
  1. Q21
    1. 6 = 2 × 3, 9 = 3² (2)
    Q21 OR
    1. divide by primes in turn: 2 × 3 × 11 (2)
  2. Q22
    1. sum of zeroes = −b/a = −1, so the other zero = −1 − 2 = −3 (2)
  3. Q23
    1. 3 ÷ 3 = 1 (2)
  4. Q24
    1. OAPB has two right angles, so ∠APB = 180° − 100° = 80° (2)
  5. Q25
    1. (−9)² + (y − 8)² = 225, so y − 8 = ±12 (2)
    Q25 OR
    1. sides 5, 12 and √(5² + 12²) = 13 (2)
Section C: 6 questions of 3 marks each: answers and marks
  1. Q26
    1. Assuming √3 = r/s (r, s coprime) and getting 3s² = r², so 3 divides r (1)
    2. Showing 3 also divides s, a contradiction, so √3 is irrational (1)
    3. Writing √3 = (7 − x)/8 and concluding that 7 − 8√3 is irrational (1)
  2. Q27
    1. Let the tens digit be x and the units digit be y: x + y = 15 and (10x + y) − (10y + x) = 27, that is, x − y = 3 (1)
    2. Adding and subtracting the equations: x = 9, y = 6 (1)
    3. The number is 96 (1)
  3. Q28
    1. OA ⟂ PA (radius ⟂ tangent), so △OAP is right-angled at A (1)
    2. OP = 123 cm (1)
    3. PB = PA, perimeter = 294 cm (1)
    Q28 OR
    1. Both tangents are ⟂ TG (radius ⟂ tangent) (1)
    2. Lines perpendicular to the same line are parallel (1)
    3. WB = 31 cm using equal tangents (1)
  4. Q29
    1. PQ² = 18, QR² = 18, PR² = 36 (1)
    2. PQ = QR = 3√2, so the triangle is isosceles (1)
    3. PQ² + QR² = 36 = PR², so by the converse of Pythagoras' theorem it is right-angled at Q (1)
  5. Q30
    1. cot θ = 5/12: take the adjacent side = 5k and the opposite side = 12k; then the hypotenuse = √(5² + 12²)k = 13k, so sin θ = 12/13, cos θ = 5/13, tan θ = 12/5 (1)
    2. Substituting: (13/5 + 12/5)/(13/5 − 12/5) (1)
    3. = 25 (1)
  6. Q31
    1. Class marks xᵢ: 5, 15, 25, 35, 45; Σfᵢ = 37 + p, Σfᵢxᵢ = 895 + 35p (1)
    2. (895 + 35p)/(37 + p) = 25 (1)
    3. Solving, p = 3 (1)
    Q31 OR
    1. Class marks xᵢ: 32.5, 37.5, 42.5, 47.5, 52.5; fᵢxᵢ: 325, 150, 127.5, 855, 262.5 (1)
    2. Σfᵢ = 40, Σfᵢxᵢ = 1,720 (1)
    3. Mean = Σfᵢxᵢ/Σfᵢ = 1,720/40 = 43 (1)
Section D: 4 questions of 5 marks each: answers and marks
  1. Q32
    1. Similar triangles identified (AA), with a correct ratio (1)
    2. Distance walked = 9.6 m and the proportion set up (1)
    3. Shadow = 2.4 m (1)
    4. Proportion for (b) (1)
    5. Distance = 7.6 m (1)
  2. Q33
    1. Forming the equation: Let the integers be x and x + 2: x² + (x + 2)² = 3700, that is, 2x² + 4x − 3696 = 0 (2)
    2. Solving: x = 42 or x = −44 (2)
    3. Rejecting x = −44: the integers are 42 and 44 (1)
    Q33 OR
    1. Forming the equation: Let the son's present age be x years; the father's is x + 23. Then (x − 3)(x + 20) = 78, so x² + 17x − 138 = 0 (2)
    2. Solving: x = 6 or x = −23 (2)
    3. Rejecting x = −23: son 6 years, father 29 years (1)
  3. Q34
    1. Curved surface area of the hemisphere = 2πr² = 2 × 22/7 × 28² = 4,928 cm² (1)
    2. Curved surface area of the cone = πrl = 22/7 × 28 × 35 = 3,080 cm² (1)
    3. (i) Total surface area = 4,928 + 3,080 = 8,008 cm² (1)
    4. Volume = (1/3)πr²h + (2/3)πr³ = 17,248 + 45,994 2/3 (1)
    5. (ii) = 63,242 2/3 ≈ 63,242.67 cm³ (1)
  4. Q35
    1. Angles of depression equal the angles of elevation from the banks (alternate angles) (1)
    2. First distance = 48 m (1)
    3. Second distance = 48√3 m (1)
    4. Width = 48(√3 + 1) m (1)
    5. ≈ 131.0 m (√3 = 1.73) (1)
    Q35 OR
    1. Correct figure described and the two tan equations formed (1)
    2. Equations correct with tan 30° = 1/√3, tan 45° = 1, tan 60° = √3 (1)
    3. Solving for the unknown distance (1)
    4. Height = 23√3 m (1)
    5. ≈ 39.8 m (√3 = 1.73) (1)
Section E: 3 case-based questions of 4 marks each: answers and marks
  1. Q36 (i)
    1. 360° ÷ 5 — 72° (1)
    Q36 (ii)
    1. Area = θ/360 × πr² = 72/360 × 22/7 × 10.5 × 10.5 — 69.3 cm² (1)
    Q36 (iii)
    1. Arc = 72/360 × 2 × 22/7 × 10.5 = 13.2 cm (1)
    2. Perimeter = 13.2 + 2 × 10.5 = 34.2 cm (1)
    Q36 (iii) OR
    1. Slices left = 5 − 3 = 2 (1)
    2. Area left = 2 × 69.3 = 138.6 cm² (1)
  2. Q37 (i)
    1. a₁₉ = a + (n − 1)d = 12 + 18 × 2 — 48 seats (1)
    Q37 (ii)
    1. 12 + (n − 1) × 2 = 32 gives n − 1 = 10 — Row 11 (1)
    Q37 (iii)
    1. S = n/2 [2a + (n − 1)d] = 25/2 [2 × 12 + 24 × 2] (1)
    2. = 900 seats (1)
    Q37 (iii) OR
    1. S = 8/2 [2 × 12 + 7 × 2] (1)
    2. = 152 seats (1)
  3. Q38 (i)
    1. P = 200/240 = 5/6 — 5/6 (1)
    Q38 (ii)
    1. Good or minor defect = 200 + 37 = 237; P = 237/240 = 79/80 — 79/80 (1)
    Q38 (iii)
    1. Not good = 37 + 3 = 40 (1)
    2. P = 40/240 = 1/6 (1)
    Q38 (iii) OR
    1. Neither buys only when it has a major defect: 240 − 200 − 37 = 3 (1)
    2. P = 3/240 = 1/80 (1)

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