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ICSE Class X Sample / Model Paper 2026 : Mathematics

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ICSE 2026 EXAMINATION SPECIMEN QUESTION PAPER MATHEMATICS Maximum Marks: 80 Time allowed: Three hours 1. Answers to this Paper must be written on the paper provided separately. 2. You will not be allowed to write during first 15 minutes. 3. This time is to be spent in reading the question paper. 4. The time given at the head of this Paper is the time allowed for writing the answers. 5. Attempt all questions from Section A and any four questions from Section B. 6. All working, including rough work, must be clearly shown, and must be done on the same sheet as the rest of the answer. 7. Omission of essential working will result in loss of marks. 8. The intended marks for questions or parts of questions are given in brackets [ ]. 9. Mathematical tables are provided. Instruction for the Supervising Examiner Kindly read aloud the Instructions given above to all the candidates present in the Examination Hall. T26 511 SPECIMEN 1 of 13 NOTE: The Specimen Question Paper in the subject provides a realistic format of the Board Examination Question Paper and should be used as a practice tool. The questions for the Board Examination can be set from any part of the syllabus. However, the format of the Board Examination Question Paper will remain the same as that of the Specimen Question Paper. SECTION A (40 Marks) (Attempt all questions from this Section.) Question 1 Choose the correct answers to the questions from the given options. [15] (Do not copy the question, write the correct answers only.) (i) ( 2) and ( + 2) are the factors of 3 + 2 4 4. The third factor of the given polynomial is: ( ) ( 1) ( ) ( 4) ( ) ( + 1) (iii) ( ) ( + 4) [Analyze] 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 (d) 98 cm2 & Analysis] T26 511 SPECIMEN 2 of 13 [Understanding (ii) Radha deposited 400 per month in a recurring deposit account for 18 months. The qualifying sum of money for the calculation of interest is (iv) (a) 3,600 (b) 7,200 (c) 68,400 (d) 1,36,800 [Application] 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) (b) (c) (d) (v) 3 2 3 3 2 [Analysis & 6 2 Evaluation] Assertion (A): If + = = then ab =1 Reason (R): 2 2 = 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). (d) Both (A) and (R) are true, but (R) is not the correct explanation of (A). T26 511 SPECIMEN [Analysis & Evaluation] 3 of 13 (vi) 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. (vii) (a) (A) is true, (R) is false. (b) (A) is false, (R) is true. (c) Both (A) and (R) are true and (R) is the correct explanation of (A). (d) Both (A) and (R) are true, but (R) is not the correct explanation of (A). [Analysis] 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 [Understanding (d) 980 & Evaluation] (viii) A bag contains 3 red and 2 blue marbles. A marble is drawn at random. The probability of drawing a black marble is (a) 0 (b) (c) (ix) (d) 1 5 2 5 3 5 [Application] 3 If matrix A=[ 1 2] and matrix B= , then matrix AB is equal to: 4 (a) (b) (c) (d) [ 3] [8] [5] 1 2 3 4 T26 511 SPECIMEN [Analysis] 4 of 13 (x) 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. (xi) 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 [Application & (d) a = 10, b = 5, c = 15 Evaluation] 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: (xii) (a) 216.00 (b) 172.80 (c) 108.00 [Analysis & (d) 86.40 Evaluation] The sum of money required to buy 50, 40 shares at 38.50 is: (a) 1,920 (b) 1,924 (c) 1,925 (d) 1,952 [Application] (xiii) The roots of quadratic equation x2 1 = 0 are: (a) 0, 0 (b) 1, 1 (c) -1, -1 [Analysis & (d) +1, -1 Evaluation] T26 511 SPECIMEN 5 of 13 (xiv) Which of the following equations represents a line equally inclined to the axes? (xv) (a) 2x 3y +7 = 0 (b) x y=7 (c) x=7 [Analysis & (d) y = 7 Evaluation] Given, + 2 3 + 3 and x is a prime number. The solution set for x is: (a) (b) (c) {1} (d) {0, 1} {0} [Understanding & Analysis] Question 2 (i) While factorizing a given polynomial, using remainder & factor theorem, [4] 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) (ii) Give a valid reason for your answer. [Analysis & Also, factorize the given polynomial completely. Application] P is a point on the x- axis which divides the line joining A (- 6, 2) and [4] B (9, - 4). Find: (a) the ratio in which P divides the line segment AB. (b) the coordinates of the point P. [Analysis & (c) equation of a line parallel to AB and passing through (-3, -2). Evaluation] T26 511 SPECIMEN 6 of 13 (iii) In the given figure, AC is the diameter of the circle with centre O. [4] CD is parallel to BE. AOB = 80 and ACE = 20 . [Analysis & Evaluation] Calculate: (a) (b) (c) BEC BCD CED Question 3 (i) -11, -7, -3, .,49, 53 are the terms of a progression. [4] Answer the following: (a) What is the type of progression? (b) How many terms are there in all? [Analysis & (c) Calculate the value of middle most term. Evaluation] T26 511 SPECIMEN 7 of 13 (ii) In the diagram given below, a tilted right circular cylindrical vessel with [4] 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 = ). 6 cm 7 cm [Application & Evaluation] (iii) Use a ruler and compass to answer this question. (a) [5] Construct a circle of radius 4.5cm and draw a chord AB of length 6.5 cm. (b) At A, construct CAB=75 , where C lies on the circumference of the circle. (c) Construct the locus of all points equidistant from A and B. (d) Construct the locus of all points equidistant from CA and BA. (e) Mark the point of intersection of the two loci as P. Measure and write down the length of CP. [Analysis & Understanding] SECTION B (40 Marks) (Attempt any four questions from this Section.) Question 4 (i) Ms. Kaur invested 8,000 in buying 100 shares of a company paying 6% [3] 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. (b) number of shares purchased by her using the total proceeds. T26 511 SPECIMEN Application & Evaluation] 8 of 13 (ii) Solve the following inequation, write the solution set, and represent it on [3] the real number line. 5x 21 < 5 7 3 6 37 + x, x R. [Evaluation] (iii) Prove the following trigonometry identity: [4] [Application & (sin + cos ) (cosec sec ) = cosec .sec 2 tan Analysis] Question 5 (i) In the given figure (not drawn to scale) chords AD and BC intersect at P, [3] where AB = 9 cm, PB = 3 cm and PD = 2 cm. A P B (ii) (a) Prove that APB ~ CPD. (b) Find the length of CD. (c) Find area APB : area CPD. C D [Application & Evaluation] Mr. Sam has a recurring deposit account and deposits 600 per month for [3] 2 years. If he gets 15,600 at the time of maturity, find the rate of interest [Application & earned by him. T26 511 SPECIMEN Evaluation] 9 of 13 (iii) Using step-deviation method, find mean for the following frequency [4] distribution: Class 0 15 15 30 30 45 45 60 60 75 75 90 Frequency 3 4 7 6 8 2 [Application & Evaluation] Question 6 (i) Find the coordinates of the centroid P of the ABC, whose vertices are [3] A( 1, 3), B(3, 1) and C(0, 0). Hence, find the equation of a line passing [Analysis & through P and parallel to AB. (ii) Evaluation] In the given figure, the parallelogram ABCD circumscribe a circle, touching [3] circle at P, Q, R and S. [Analysis & Application] (a) Prove that: AB = BC (b) What special name can be given to the parallelogram ABCD? (iii) The following bill shows the GST rate and the marked price of articles: [4] Rajdhani Departmental Store S. No. Item Marked Discount Price Rate of 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. T26 511 SPECIMEN [Application & Evaluation] 10 of 13 Question 7 (i) Draw the necessary diagram for this question. [5] 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. Give your answer correct to the nearest meter. Application & (Use Mathematical Tables for this Question) (ii) [Understanding, Evaluation] The marks of 200 students in a test were recorded as follows: [5] Marks % 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 50 - 60 60 - 70 70 - 80 80 - 90 90 - 100 No. of students 5 7 11 20 40 52 36 15 9 5 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. (c) number of students who did not pass, if the pass percentage was 35. [Application, Analysis & Evaluation] Question 8 (i) A box containing cards numbered between 0 and 100 are shuffled and a card [3] is picked at random. Find the probability of getting a card which is: (ii) (a) divisible by 6. (b) not divisible by 6. [Application & Evaluation] If x, y and z are in continued proportion, prove that: 2 . 2 T26 511 SPECIMEN + 2 . 2 + 2 . 2 = 1 1 1 + 3 + 3 3 [3] [Application & Analysis] 11 of 13 (iii) A manufacturing company prepares spherical ball bearings, each of radius [4] 7 mm and mass 4 gm. These ball bearings are packed into boxes. Each box can have a maximum of 2156 cm of ball bearings. Find the: (a) maximum number of ball bearings that each box can have. (b) mass of each box of ball bearings in kg. (Use = Application & Evaluation] 22 7 [Analysis, ) Question 9 (i) Study the graph given below and answer the following: [3] y x (a) Number of batsmen who scored 500 to 700 runs (b) Modal class interval (c) The value of mode T26 511 SPECIMEN [Analysis & Evaluation] 12 of 13 (ii) An Arithmetic Progression (A.P.) has 3 as its first term. The sum of the first [3] 8 terms is twice the sum of the first 5 terms. Find the common difference of [Analysis, Application & the A.P. Evaluation] (iii) The roots of equation (q r) x2 + (r p) x + (p - q) = 0 are equal. Prove that: 2q = p + r, that is, p, q & r are in A.P. [4] [Application & Analysis] Question 10 (i) The sum of the squares of three consecutive even numbers is 596. Find the [3] [Analysis, numbers. Application & Evaluation] (ii) Given matrix, X = 1 1 1 0 = , prove that 2 = 4 + 5 . 8 3 0 1 (iii) Use a graph sheet for this question. Take 1 cm = 1 unit along both the x and [3] [Application & Evaluation] [4] 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. T26 511 SPECIMEN [Understanding] 13 of 13 ICSE 2026 SPECIMEN DRAFT MARKING SCHEME MATHEMATICS Question 1 (i) (c) (x+1) .[15] (ii) (c) 68,400 (iii) (c) 80 cm2 (iv) (c) (v) (c) Both (A) and (R) is true and (R) is the correct reason for (A). (vi) (a) (A) is true, (R) is false. (vii) (a) 20 (viii) (ix) (a) 0 (x) (d) a = 10, b = 5, c = 15 (xi) (d) 86.40 (xii) (c) 1925 (xiii) (d) +1, -1 (xiv) (b) x y = 7 (xv) (a) 3 2 (c) [5] Question 2 (i) ( ) = + + [4] 1 1 3 1 2 1 = 2 + 7 + 2 3 0 2 2 2 2 (2 + 1) ( ). 1 1 3 1 2 1 = 2 + 7 + 2 3 = 0 2 2 2 2 (2 1) ( ) T26 511 - SPECIMEN Page 1 of 9 2 + 4 + 3 2 1 + + 2 3 2 8 2 + 2 8 2 4 6 3 6 3 ( ) = ( )( 2 + 4 + 3) (ii) (a) = 0 4 +2 (b) + Question 3 ( ) ( ) (c) (ii) 4 2 9+6 2 = 6 15 2 = 5 1 100 = 50 2 ( ) (a) = 180 80 = 100 = (c) (i) = 0, 4 = 2 : = 1: 2 + 2 = 5 ( + 3) 2 + 5 = 16 (b) [4] 9 1 + 2 ( 6) = 1 3 ( 1, 0) x= (c) = (iii) ( ) = ( )( + )( + ) [4] = + + = 40 + 20 + 50 = 110 = 180 110 50 = 20 [4] . . = 53, + ( 1) = 53 11 + ( 1)4 = 53 = 17 17+1 Middle term = 2 = 9 9 = + 8 = 11 + 8 4 = 21 1 7 (1 + 6), = 2 2 2 = + 2 ( + 2 ) = T26 511 - SPECIMEN = 22 7 7 7 + 2 = 115.5 2 7 2 2 2 [4] Page 2 of 9 (iii) [5] (a) = 4.9 . Question 4 (i) (a) . = 8000 80 SECTION B [3] = 100 6 100 100 = 600 100 (b) = 75 100 = 7500 = = 8100 . = (ii) 8100 = 300 27 5 3 6 3 + , 7 7 5 5 3 5 21 < 6 6 3 + 7 7 7 5 5 24 5 < 6 + 21 + 6 7 7 7 35 5 5 7 24 + 42 < 15 7 7 7 30 < 105 2 18 5 21 < < 3.5 T26 511 - SPECIMEN : 9 < 7 , 2 [3] 9 Page 3 of 9 (iii) Question 5 (i) [4] = ( + )( ) 1 1 = ( + ) = ( + ) . 2 2 1 2 2 1 2 2 = = = . . . . = . 2 = (a) , = ( ) (b) (c) (ii) [3] = ( ) ~ ( ) 3 = 2 = 6 ( ) 2 9 = 2 = 4 9 4 600 24 25 1 = 150 2 100 12 = 15600 = [3] 600 24 + 150 = 15600 (iii) lass 150 = 15600 14400 = = / 7.5 15 30 22.5 30 45 37.5 -1 7 -7 45 60 52.5 0 6 0 60 75 67.5 1 8 8 75 90 82.5 2 2 4 30 -12 Question 6 3 (b) = , 1 (3) 3 ( 1) 3 4 -9 -8 = 4 4 2 2 = 3 , 3 = 1 , T26 511 - SPECIMEN -2 3 [4] 12 = 52.5 + 15 = 52.5 6 = 46.50 30 1+3+0 3+( 1)+0 (a) P -3 0 15 = + (i) 1200 = 8% 150 [3] = 1 2 2 = 1 3 + 3 = 4 3 3 Page 4 of 9 (ii) ( ) [3] = , = , = = ( . ) , ( + ) + ( + ) = ( + ) + ( + ) + = + 2 = 2 = (b) Rhombus (iii) S. No. 1. 1200 (1 ) 2. 3. Total 286 1100 12% 12 1100 = 132 100 450 12% 12 450 = 54 100 286 500 [4] 1836 Grand total 5% 5 286 = 14.30 100 200.30 2036.30 Question 7 (i) [5] A 50 30 C , 30 80 B = tan 30 50 D 80 1 = = 80 3 = 80 1.7321 3 , = tan 50 = 40 = 0.8391 = 80 0.8391 80 = 80 1.732 + 80 0.839 = 80(1.7321 + 0.8391) 80(2.5712) = 205.696 = 206 T26 511 - SPECIMEN Page 5 of 9 (ii) Marks (%) 5 5 20 30 11 23 30 40 20 43 40 50 40 83 50 60 52 135 60 70 36 171 70 80 15 186 80 90 09 195 90 100 05 200 0 10 10 20 7 [5] (a) = 53 1 (b) 65% = 46 2 (c) = 31 2 12 Scale: x-axis, 2cm = 10% marks y-axis, 2cm= 20 students Question 8 (i) (a) {6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96} ( 6) = 16 99 (b) ( 6) = 1 (ii) T26 511 - SPECIMEN 16 99 = [3] 83 99 = 2 = [3] Page 6 of 9 = (iii) 3 + 3 + 3 3 3 3 = 3 3+ 3 3+ 3 3 3 3 3 1 1 1 1 1 = 3 + 6 + 3 = 3 + 3 + 3 = ( ) (b) Question 9 (i) 3 + 3 + 3 + + = 2 . 2 2 . 2 2 . 2 2 . 2 2 . = 2156 4 3 3 = 2156 [4] 3 4 22 7 3 7 10 2156 3 7 10 10 10 = 1500 4 22 7 7 7 = 4 1500 = 6 = [3] ( ) 5 (b) 400 500 (c) Mode = 430 runs y x (ii) = 3, T26 511 - SPECIMEN 8 5 8 = 2 5 [2 3 + (8 1) ] = 2 [2 3 + (5 1) ] 2 2 3 4[6 + 7 ] = 5[6 + 4 ] 24 + 28 = 30 + 20 = 4 [3] Page 7 of 9 (iii) = , = = [4] , 2 = 4 ( )2 = 4 ( )( ) 2 + 2 2 = 4[ 2 + ) 2 + 2 2 + 4 = 4[ 2 + ] ( + )2 = 4[ ( + ) 2 ] ( + )2 4 ( + ) + 4 2 = 0 ( + ) = 2 4 + 4 2 = 0 ( 2 )2 = 0 2 = 0 + = 2 Question 10 (i) [3] ( 2), ( + 2) ( 2)2 + 2 + ( + 2)2 = 596 3 2 = 588 2 = 196 = 14 The required numbers are 12, 14 & 16 (ii) = 1 2 = 8 1 1 + (1) (8) (8) 1 + 3 (8) 1 1 1 3 8 3 1 (1) + (1) 3 (8) (1) + 3 3 1+8 1+3 = 8 + 24 8 + 9 2 = 1 4 = 4 8 4 + 5 = 4 32 9 32 4 17 1 4 = 3 32 4 12 4 9 4 5 0 + = 12 32 17 0 5 2 = 4 + 5 , T26 511 - SPECIMEN [3] Page 8 of 9 (iii) (a) Square [4] [ T26 511 - SPECIMEN Page 9 of 9

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