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ICSE Class X Board Examination

ICSE Mathematics — Specimen Question Paper 2026

Official CISCE specimen question paper published ahead of the ICSE Class X Board Examination 2026.

Specimen2026Subject code 511
Open source paper at cisce.org (opens in a new tab)
Board
ICSE Class X Board Examination
Class
Class 10
Subject
Mathematics
Year
2026
Total marks
80 marks
Duration
3 hours

Verification and rights

Published by CISCE. Khojo Papers links to the original document rather than reproducing it.

ExamVault links to the paper published on cisce.org. The document is CISCE copyright and is not reproduced here.

Questions in this paper

Section A

  1. 1((i))Choose the correct answers to the questions from the given options. [15] (Do not copy the question, write the correct answers only.)
    (x - 2) and (x + 2) are the factors of x 3 + x 2 - 4x - 4. The third
    factor of the given polynomial is:
  2. 1((iii))Choose the correct answers to the questions from the given options. [15] (Do not copy the question, write the correct answers only.)
    In the figure given below, AC is a diameter of the circle.
    AP = 3 cm and PB = 4 cm and QP ⊥ AB.
    If the area of ∆APQ is18 cm2, then the area of shaded portion QPBC is:
    (a) 32 cm2
    (b) 49 cm2
    (c) 80 cm2
  3. 1((ii))Choose the correct answers to the questions from the given options. [15] (Do not copy the question, write the correct answers only.)
    Radha deposited ₹400 per month in a recurring deposit account for 18
    months.
    The qualifying sum of money for the calculation of interest is∶
  4. 1((iv))Choose the correct answers to the questions from the given options. [15] (Do not copy the question, write the correct answers only.)
    In the given diagram, the radius of the circle with centre O is 3 cm. PA
    and PB are the tangents to the circle which are at right angle to each other.
    The length of OP is:
    (a) c
    √2
    (b) 3 c
    (c) 3√2 c
  5. 1((v))Choose the correct answers to the questions from the given options. [15] (Do not copy the question, write the correct answers only.)
    Assertion (A): If s + t = a a s - t = b then ab =1
    Reason (R): s 2 θ - t2 θ = 1
    (a) (A) is true and (R) is false.
    (b) (A) is false and (R) is true.
    (c) Both (A) and (R) are true and (R) is the correct explanation of (A).
  6. 1((vi))Choose the correct answers to the questions from the given options. [15] (Do not copy the question, write the correct answers only.)
    A solid sphere is cut into two identical hemispheres.
    Assertion (A): The total volume of two hemispheres is equal to the volume
    of the original sphere.
    Reason (R): The total surface area of two hemispheres together is equal
    to the surface area of the original sphere.
  7. 1((vii))Choose the correct answers to the questions from the given options. [15] (Do not copy the question, write the correct answers only.)
    Given that the sum of the squares of the first seven natural numbers is
    140, then their mean is:
    (a) 20
    (b) 70
    (c) 280
  8. 1((x))Choose the correct answers to the questions from the given options. [15] (Do not copy the question, write the correct answers only.)
    A mixture of paint is prepared by mixing 2 parts of red pigments with 5
    parts of the base. Using the given information in the following table, find
    the values of a, b & c to get the required mixture of paint.
    Parts of red pigment 2 4 b 6
    Parts of base 5 a 12.5 c
    (a) a = 10, b = 10, c = 10
    (b) a = 5, b = 2, c = 5
    (c) a = 10, b = 5, c = 10
  9. 1((xi))Choose the correct answers to the questions from the given options. [15] (Do not copy the question, write the correct answers only.)
    An article which is marked at ₹ 1,200 is available at a discount of 20%
    and the rate of GST is 18%. The amount of SGST is:
    (a) ₹ 216.00
    (b) ₹ 172.80
    (c) ₹ 108.00
  10. 1((xii))Choose the correct answers to the questions from the given options. [15] (Do not copy the question, write the correct answers only.)
    The sum of money required to buy 50, ₹ 40 shares at ₹ 38.50 is:
  11. 1((xiii))Choose the correct answers to the questions from the given options. [15] (Do not copy the question, write the correct answers only.)
    The roots of quadratic equation x2 – 1 = 0 are:
    (a) 0, 0
    (b) 1, 1
    (c) -1, -1
  12. 1((xv))Choose the correct answers to the questions from the given options. [15] (Do not copy the question, write the correct answers only.)
    Given, x + 2 ≤ 3 + 3 and x is a prime number. The solution set for x is:
  13. 2((i))While factorizing a given polynomial, using remainder & factor theorem,
    a student finds that (2x + 1) is a factor of 2x3 + 7x2 + 2x – 3.
    (a) Is the student’s solution correct stating that (2x + 1) is a factor of the
    given polynomial?
    (b) Give a valid reason for your answer.    4 marks
  14. 2((ii))P is a point on the x- axis which divides the line joining A (- 6, 2) and
    B (9, - 4). Find:
    (a) the ratio in which P divides the line segment AB.
    (b) the coordinates of the point P.    4 marks
  15. 2((iii))In the given figure, AC is the diameter of the circle with centre O.
    CD is parallel to BE.
    ∠AOB = 800 and ∠ACE = 200.
    Calculate:
    (a) ∠ BEC
    (b) ∠ BCD
    (c) ∠CED    4 marks
  16. 3((i))-11, -7, -3, ………….,49, 53 are the terms of a progression.
    Answer the following:
    (a) What is the type of progression?
    (b) How many terms are there in all?    4 marks
  17. 3((ii))In the diagram given below, a tilted right circular cylindrical vessel with
    base diameter 7 cm contains a liquid. When placed vertically, the height of
    the liquid in the vessel is the mean of two heights shown in the diagram.
    Find the area of wet surface, when the cylinder is placed vertically on a
    horizontal surface.
    (Use π= 7 ).
    6 cm    4 marks
  18. 3((iii))Use a ruler and compass to answer this question.5 marks

Section B

  1. 4((i))Ms. Kaur invested ₹ 8,000 in buying ₹100 shares of a company paying 6%
    dividend at ₹ 80. After a year, she sold these shares at ₹75 each and invested
    the proceeds including the dividend received during the first year in buying
    ₹ 20 shares, paying 15% dividend at ₹ 27 each. Find the:
    (a) dividend received by her during the first year. Application &
    (b) number of shares purchased by her using the total proceeds. Evaluation]    3 marks
  2. 4((ii))Solve the following inequation, write the solution set, and represent it on
    the real number line.
    5x 3
    5x – 21 < 7 – 6 ≤ –37 + x, x ∈ R.    3 marks
  3. 4((iii))Prove the following trigonometry identity:
    (sinθ + cosθ) (cosecθ – secθ) = cosecθ.secθ – 2 tanθ    4 marks
  4. 5((i))In the given figure (not drawn to scale) chords AD and BC intersect at P,
    where AB = 9 cm, PB = 3 cm and PD = 2 cm.
    A
    P C
    B D
    (a) Prove that ∆APB ~ ∆CPD.
    (b) Find the length of CD.    3 marks
  5. 5((ii))Mr. Sam has a recurring deposit account and deposits ₹ 600 per month for
    2 years. If he gets ₹ 15,600 at the time of maturity, find the rate of interest    3 marks
  6. 5((iii))Using step-deviation method, find mean for the following frequency
    distribution:
    Class 0 – 15 15 – 30 30 – 45 45 – 60 60 – 75 75 – 90
    Frequency 3 4 7 6 8 2    4 marks
  7. 6((i))Find the coordinates of the centroid P of the ∆ABC, whose vertices are
    A(–1, 3), B(3, –1) and C(0, 0). Hence, find the equation of a line passing    3 marks
  8. 6((ii))In the given figure, the parallelogram ABCD circumscribe a circle, touching
    circle at P, Q, R and S.
    (b) What special name can be given to the parallelogram ABCD?    3 marks
  9. 6((iii))The following bill shows the GST rate and the marked price of articles:
    Rajdhani Departmental Store
    S. No. Item Marked Discount Rate of
    Price GST
    (a) Dry fruits (1 kg) ₹ 1200 ₹100 12%
    (b) Packed Wheat flour (5kg) ₹ 286 Nil 5%
    (c) Bakery products ₹ 500 10% 12%
    Find the total amount to be paid (including GST) for the above bill.    4 marks
  10. 7((i))Draw the necessary diagram for this question.
    A man on the top of a lighthouse observes the angle of depression of two
    ships on the opposite sides of the lighthouse as 30° and 50° respectively. If
    the height of the lighthouse is 80m, find the distance between the two ships.    5 marks
  11. 7((ii))The marks of 200 students in a test were recorded as follows:
    Marks % 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 50 - 60 60 - 70 70 - 80 80 - 90 90 - 100
    No. of
    5 7 11 20 40 52 36 15 9 5
    students
    Using a graph sheet draw ogive for the given data and use it to find the:
    (a) median.
    (b) number of students who obtained more than 65% marks. Analysis &
    (c) number of students who did not pass, if the pass percentage was 35. Evaluation]    5 marks
  12. 8((i))A box containing cards numbered between 0 and 100 are shuffled and a card
    is picked at random. Find the probability of getting a card which is:
    (a) divisible by 6.    3 marks
  13. 8((ii))If x, y and z are in continued proportion, prove that:
    x y z 1 1 1    3 marks
  14. 8((iii))A manufacturing company prepares spherical ball bearings, each of radius
    7 mm and mass 4 gm. These ball bearings are packed into boxes. Each box
    can have a maximum of 2156 cm3 of ball bearings. Find the:
    (a) maximum number of ball bearings that each box can have.    4 marks
  15. 9((i))Study the graph given below and answer the following:
    y
    x
    (a) Number of batsmen who scored 500 to 700 runs
    (c) The value of mode    3 marks
  16. 9((ii))An Arithmetic Progression (A.P.) has 3 as its first term. The sum of the first
    8 terms is twice the sum of the first 5 terms. Find the common difference of    3 marks
  17. 9((iii))The roots of equation (q – r) x2 + (r – p) x + (p - q) = 0 are equal.4 marks
  18. 10((i))The sum of the squares of three consecutive even numbers is 596. Find the
    numbers.    3 marks
  19. 10((ii))1 1 1 0
    Given matrix, X = � � a I = � �, prove that X 2 = 4X + 5I.
    8 3 0 1    3 marks
  20. 10((iii))Use a graph sheet for this question. Take 1 cm = 1 unit along both the x and
    y axis. Plot ABCDE, where A (4, 0), B (4, 2), C (2, 2), D (2,4) and E (0,4).
    (a) Reflect the points A, B, C and D on the y-axis and name them as F, G,
    H and I respectively.
    (b) Join the points A, B, C, D, E, I, H, G and F in order. Reflect the figure
    ABCDEIHGF on the x-axis and name it as AMNPQRSTF.
    (c) Give the geometrical name of the closed figure AEFQ.
    ICSE 2026 SPECIMEN    4 marks