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ICSE Class X Board Exam 2026 : Mathematics

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Abhijeet
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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 and graph papers are to be provided by the school. Instruction for the Supervising Examiner Kindly read aloud the Instructions given above to all the candidates present in the Examination Hall. T25 511 I Copyright reserved. This paper consists of 15 printed pages and 1 blank page. Turn Over SECTION A (40 Marks) (Attempt all questions from this Section.) Question 1 Choose the correct answers to the questions from the given options. (Do not copy the questions, write the correct answers only.) (i) When polynomial + is divided by ( + ), the remainder is zero. The probability of ( + 2) to be one of the factors of the given polynomial is: (a) 0 (b) 1 3 (c) (d) (ii) 2 3 1 Assertion (A): In ABC and PQR, if BAC = QPR and Reason (R): T25 511 I ABC = PQR, then ABC ~ PQR ABC ~ PQR by SSS axiom (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 reason for (A). (d) Both (A) and (R) are true, and (R) is the incorrect reason for (A). 2 [15] (iii) The ratio of diameters of two right circular cones is 3 : 7 and that of their heights is 14 : 9, then their volumes are in ratio: (iv) (v) (a) 3:7 (b) 2:7 (c) 3:2 (d) 9 : 49 The value of p for which ( ) is a factor of + + is: (a) 5 (b) 4 (c) 5 (d) p+5 The GST of an article is reduced from 12% to 5% and due to this, the price paid for the article is cut down by 14. The original price of the article is: (vi) T25 511 I (a) 50 (b) 98 (c) 100 (d) 200 The mean of 12, 22, 33 and 44 is: (a) 24 (b) 72 (c) 144 (d) 264 3 Turn Over (vii) If > + , then: (a) (b) (c) (d) (viii) < 12 < 12 > 12 > 12 Assertion(A): If the length of shadow of a person is equal to his height, then the angle of elevation of the sun is 45 . Reason(R): (ix) (x) For any right-angled triangle, = (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 reason for (A). (d) Both (A) and (R) are true, and (R) is the incorrect reason for (A). The roots of the quadratic equation = is: (a) 0 (b) 2 (c) 0 and 2 (d) 0 and 6 The locus of a toy bird fixed at the tip of one of the blades of a rotating ceiling fan is a: T25 511 I (a) straight line (b) circle (c) semi-circular arc (d) diameter of the circle so formed 4 (xi) (xii) Percentage return on 100, 12% share of a company bought at 4% discount is: (a) 10% (b) 12% (c) 12.5% (d) 16% If matrix A of order 2 1 and matrix B of order 2 2 are added, then the order of the matrix A + B is: (xiii) (a) 2 2 (b) 2 1 (c) 1 2 (d) A + B is not possible In the adjoining diagram, PQ is a tangent at A to the circle with centre O. If OAC = 25o, then ABC is: (xiv) (a) 20o (b) 65o (c) 70o (d) 130o The line segment joining A( 7, 2) and B(3, 8) is divided by the x-axis in the ratio: T25 511 I (a) 1:4 (b) 3:7 (c) 4:1 (d) 7:3 5 Turn Over (xv) Mr. Rahul deposited 11,700 in a recurring deposit account for years. The amount deposited by him per month is: (a) 650 (b) 780 (c) 6,500 (d) 7,800 Question 2 (i) A retailer purchased an air conditioner (A.C.) for 30,000. He marked up its [4] price by 20% and then allows a discount of 10% on the marked price to a customer. If the sale is intra-state and the rate of GST is 28%, find the: (ii) (a) marked price of A.C. (b) total amount paid by the customer including GST. (c) tax collected by the central and the state governments respectively. In the adjoining diagram DOE = 46o and BGH = 113o. (a) (b) T25 511 I Find DBC and DCE. Prove that CBGH is a cyclic quadrilateral. 6 [4] (iii) The table given below shows a record of the weight in kilogram of 200 [4] students of a school. Weight (kg) Number of students 40 45 8 45 50 19 50 55 24 55 60 45 60 65 51 65 70 31 70 75 22 Draw a histogram and find the modal weight. [Take 2 cm = 5 kg along one axis and 2 cm = 5 students along the other axis] Question 3 (i) (ii) T25 511 I Prove that: [4] ( ) ( ) = ( ) ( ) + = ( )( + + ): (a) using Remainder and Factor theorem, find the value of a and b . (b) hence, factorise the polynomial + completely. 7 [4] Turn Over (iii) Scale: x-axis, 2 cm = 1 unit y-axis, 2 cm = 1 unit Using the given graph, answer the following: (a) Write down the coordinates of the points A, B, C, and E. (b) Name and write down the coordinates of the image of B under reflection in x-axis. (c) Name and write the coordinates of the image of D under reflection through the origin. (d) Which point is the image of A under reflection on the line BH? Write its coordinates. (e) T25 511 I Name the closed figure ABCDEFGH. 8 [5] SECTION B (40 Marks) (Attempt any four questions from this Section.) Question 4 (i) The sum of two numbers is 2 and the sum of their reciprocals is 2.25. Find the [3] numbers. (ii) A right circular cone of radius 20 cm has its volume 8800 cm3. Find its: (a) height (b) curved surface area [3] Give your answer to the nearest whole number. (iii) [Use = ] Construct a regular hexagon of side 4.5 cm. Hence, construct a circle [4] circumscribing the regular hexagon. Use ruler and compass for the construction. Measure and write down the radius of the circle. Question 5 (i) 164, 160, 156, 152, are in Arithmetic Progression (A.P.). Find: (a) which term is equal to 0. (b) the sum of its first 20 terms. T25 511 I 9 [3] Turn Over (ii) Solve the following quadratic equation: [3] + = Give your answer correct to two places of decimals (Use Mathematical tables, if necessary) (iii) In the adjoining figure of a circle with centre O and diameter AD, BED = 70 [4] and BC is parallel to AD. Find: (a) (b) (c) (d) BAD BOD DBC DCF C B A O F D 70 E Question 6 (i) Solve the inequation, write down the solution set and represent it on a real number line: < T25 511 I + ; 10 [3] (ii) If the 6th term of a series in Geometric Progression (G.P.) is 32 and the 9th term [3] is 256, find the: (iii) (a) first term and the common ratio. (b) sum of its first 10 terms. An ice cream cone has a diameter of 7 cm and its height is 9 cm. It is filled with [4] a scoop of spherical shaped ice cream of radius 3.5 cm. Find: (Give all answers correct to the nearest whole number) (a) on melting, is the ice cream sufficient to fill the cone completely without any wastage? (b) the volume of ice cream, if any, is in excess or less. [Use = T25 511 I ] 11 Turn Over Question 7 (i) There are some red, green and white marbles in a box. One marble is picked up at random from this box. If the probability of picking up a red marble is that of picking up a green marble is (ii) then find the: (a) probability of picking up a white marble. (b) number of green marbles, if total number of marbles is 54. (c) probability of not picking up a red marble. [3] and Mr. Anil has a recurring deposit account. He deposits a certain amount of money [3] per month for 2 years. If he received an interest whose value is the double of the deposit made per month, then find the rate of interest. (iii) If a, b, c and d are in continued proportion, prove that [4] ( + ) = ( + ) Question 8 (i) 100 shares of a company giving 10% dividend are selling at 150. Mr. Saha invests 18000 to buy these shares. He sells 80% of his shares after one year. Find: (a) the number of shares he purchased. (b) the number of shares he sold. (c) his annual income from the remaining 20% shares he still holds. T25 511 I 12 [3] (ii) Equation of a line AB is + + = . A perpendicular PQ is dropped on AB [3] from the point P(3, 2) meeting AB at Q. Find the: (iii) (a) equation of PQ. (b) coordinates of the point Q. Divide 20 into two parts such that the sum of their squares is 272. The larger of [4] two parts is square of the other. Assuming the smaller part to be x , form an equation and solve it to find the two parts. Question 9 (i) Use a graph paper for this question: [5] The Marks out of 80 obtained by 160 students in a Mathematics test were recorded as given in the table: Marks No. of Students 0 10 10 20 20 30 30 40 40 50 50 60 60 70 70 80 12 20 28 35 29 16 12 8 (Take 2 cm = 10 Marks on one axis and 2 cm = 20 students on the other axis). Draw an Ogive and use it to find the following: (a) median marks (b) upper quartile marks (c) number of students who scored above 65 marks (d) the lowest marks scored by the top 30% students. T25 511 I 13 Turn Over (ii) The angle of elevation of the top of a hill from the foot of a tower at B is 50o. The [5] angle of elevation of the top of the tower 100 m high from the foot of the hill at C is 35o. D A Hill Tower 100 m 350 500 B C Find the: (a) horizontal distance BC between the Hill and the Tower. (b) height CD of the Hill. (Take tan50 = 1.20) (c) time taken by a cyclist to cover the distance BC, cycling at 20 m/sec. Question 10 (i) Using Remainder and Factor theorem factorise the given polynomial completely. [3] (ii) + + [3] Using short-cut method, find Mean of the given frequency distribution: Class 20 - 30 30 - 40 40 - 50 50 - 60 60 - 70 70 - 80 Frequency 6 9 14 10 7 4 T25 511 I 14 (iii) Use ruler and compass for the following constructions: Construct: (a) an isosceles ABC in which AB = AC = 7 cm and BC = 6 cm. (b) the locus of points which moves such that it is 2.5 cm from the point A. (c) the locus of points equidistant from B and C. Mark point P which satisfies both the conditions mentioned in (b) and (c). (d) T25 511 a circle passing through P, B and C. 15 [4]

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