ICSE 2025 Mathematics Specimen Question Paper PDF

ICSE 2025 EXAMINATION SPECIMEN QUESTION PAPER MATHEMATICS Maximum Marks: 80 Time allowed: Two and half hours Three hours Answers to this Paper must be writ...

ICSE 2025 EXAMINATION SPECIMEN QUESTION PAPER MATHEMATICS Maximum Marks: 80 Time allowed: Two and half hours Three hours Answers to this Paper must be written on the paper provided separately. You will not be allowed to write during first 15 minutes. This time is to be spent in reading the question paper. The time given at the head of this Paper is the time allowed for writing the answers. Attempt all questions from Section A and any four questions from Section B. All working, including rough work, must be clearly shown, and must be done on the same sheet as the rest of the answer. Omission of essential working will result in loss of marks. The intended marks for questions or parts of questions are given in brackets [ ] 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. T25 511 – SPECIMEN 1 of 12  SECTION A (Attempt all questions from this Section.) Question 1 Choose the correct answers to the questions from the given options. (Do not copy the question, write the correct answers only.) (i) A polynomial in ‘x’ is divided by (x − a) and for (𝑥𝑥 − 𝑎𝑎) to be a factor of this polynomial, the remainder should be: (a) − 𝑎𝑎 (b) 0 (c) a (d) 2𝑎𝑎 [Analyze] (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∶ (a) ₹ 3600 (b) ₹ 7200 (c) ₹ 68,400 (d) ₹ 1,36,800 [Application] (iii) In the adjoining figure, 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 [Understanding & Analysis] T25 511 – SPECIMEN 2 of 12 (iv) In the adjoining diagram, O is the centre of the circle and PT is a tangent. The value of x is: (a) 20° (b) 40° (c) 55° (d) 70° [Application] (v) In the adjoining diagram the length of PR is: (a) 3√3 cm R (b) 6√3 cm 9 cm (c) 9√3 cm 60° (d) 18 cm Q P [Application] (vi) A solid sphere is cut into two identical hemispheres. Statement 1: The total volume of two hemispheres is equal to the volume of the original sphere. Statement 2: The total surface area of two hemispheres together is equal to the surface area of the original sphere. Which of the following is valid? (a) Both the statements are true. (b) Both the statements are false. (c) Statement 1 is true, and Statement 2 is false. (d) Statement 1 is false, and Statement 2 is true. [Analysis] (vii) 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 & Evaluation] (d) 980 T25 511 – SPECIMEN 3 of 12 (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 1 (b) 5 2 (c) 5 3 (d) [Application] 5 (ix) −1 4 If A = [3 −2] and B = 2 0 Assertion (A): Product AB of the two matrices A and B is possible. Reason (R): Number of columns of matrix A is equal to number of rows in matrix B. (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 incorrect reason for A. [Analysis] (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. 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] T25 511 – SPECIMEN 4 of 12 (xi) An article which is marked at ₹ 1200 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 [Analysis & (d) ₹ 86.40 Evaluation] (xii) The sum of money required to buy 50, ₹ 40 shares at ₹ 38.50 is: (a) ₹1920 (b) ₹1924 (c) ₹1925 (d) ₹1952 [Application] (xiii) The roots of quadratic equation x2 – 1 = 0 are: (a) 0 (b) 1 (c) -1 [Analysis & (d) ±1 Evaluation] (xiv) Which of the following equation represents a line equally inclined to the axes? (a) 2x – 3y +7 = 0 (b) x–y=7 (c) x=7 [Analysis & (d) y = –7 Evaluation] 𝑥𝑥 (xv) Given, 𝑥𝑥 + 2 ≤ 3 + 3 and x is a prime number. The solution set for x is: (a) ∅ (b) {0} (c) {1} [Understanding & Analysis] (d) {0, 1} T25 511 – SPECIMEN 5 of 12 Question 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. [Analysis & Also, factorize the given polynomial completely. Application] (ii) A line segment joining P (2, –3) and Q (0, –1) is cut by the x-axis at the point R. A line AB cuts the y axis at T(0,6) and is perpendicular to PQ at S. Find the: (a) equation of line PQ (b) equation of line AB [Analysis & (c) coordinates of points R and S. Evaluation] (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 [Analysis & Evaluation] Question 3 (i) In a Geometric Progression (G.P.) the first term is 24 and the fifth term is 8. Find the ninth term of the G.P. [Analysis & Evaluation] T25 511 – SPECIMEN 6 of 12 (ii) In the adjoining diagram, 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 6 cm diagram. Find the area of wet surface, when 7 cm the cylinder is placed vertically on a horizontal [Application & surface. (Use π= 𝟐𝟐𝟐𝟐 ). Evaluation] 𝟕𝟕 (iii) Study the graph and answer each of the following: (a) Write the coordinates of points A, B, C & D. (b) Given that, point C is the image of point A. Name and write the equation of the line of reflection. (c) Write the coordinates of the image of the point D under reflection in y-axis. (d) What is the name given to a point whose image is the point itself? (e) On joining the points A, B, C, D and A in order, a figure is formed. Name the closed figure. [Analyze & Application] T25 511 – SPECIMEN 7 of 12  SECTION B (Attempt any four questions from this Section.) Question 4 (i) A man buys 250, ten-rupee shares each at ₹ 12.50. If the rate of dividend is 7%, find the: (a) dividend he receives annually. [Application & (b) percentage return on his investment. Evaluation] (ii) Solve the following inequation, write the solution set and represent it on the real number line. 5𝑥𝑥 3 5x – 21 < – 6 ≤ –3 + x, x ∈ R. 7 7 [Evaluation] (iii) Prove the following trigonometry identity: (sinθ + cosθ) (cosecθ – secθ) = cosecθ.secθ – 2 tanθ [Application & Analysis] Question 5 A (i) In the given figure (drawn not to scale) chords AD and BC intersect at P, P C where AB = 9 cm, PB = 3 cm and PD = 2 cm. (a) Prove that ∆APB ~ ∆CPD. B D (b) Find the length of CD. [Application & Evaluation] (c) Find area ∆APB : area ∆CPD. (ii) Mr. Sameer has a recurring deposit account and deposits ₹ 600 per month for 2 years. If he gets ₹ 15600 at the time of maturity, find the rate of interest [Application & earned by him. Evaluation] T25 511 – SPECIMEN 8 of 12 (ii) 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 [Application & Evaluation] Question 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 [Analysis & through P and parallel to AB. Evaluation] (ii) In the given figure PT is a tangent to the circle. Chord BA produced meets the tangent PT at P. Given PT=20cm and PA= 16cm. T (a) Prove ΔPTB ~ΔPAT (b) Find the length of AB. [Analysis & P Evaluation] B A (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% [Application & Evaluation] Find the total amount to be paid (including GST) for the above bill. T25 511 – SPECIMEN 9 of 12 Question 7 (i) A vertical tower standing on a horizontal plane is surmounted by a vertical flagstaff. At a point 100m away from the foot of the tower, the angle of elevation of the top and bottom of the flagstaff are 54o and 42o respectively. Find the height of the flagstaff. [Application & Give your answer correct to nearest metre. Evaluation] (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 graph sheet draw ogive for the given data and use it to find the, (a) median, [Application, (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] Question 8 (i) In a TV show, a contestant opts for video call a friend life line to get an answer from three of his friends, named Amar, Akbar & Anthony. The question which he asks from one of his friends has four options. Find the probability that: (a) Akbar is chosen for the call. [Analysis & (b) Akbar couldn't give the correct answer. Evaluation] (ii) If x, y and z are in continued proportion, Prove that: 𝑥𝑥 𝑦𝑦 𝑧𝑧 1 1 1 [Application & + + = + + 𝑦𝑦 2. 𝑧𝑧 2 𝑧𝑧 2. 𝑥𝑥 2 𝑥𝑥 2. 𝑦𝑦 2 𝑥𝑥 3 𝑦𝑦 3 𝑧𝑧 3 Analysis] T25 511 – SPECIMEN 10 of 12 (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 maximum of 2156 cm³ of ball bearings. Find the: (a) maximum number of ball bearings that each box can have. [Analysis, (b) mass of each box of ball bearings in kg. Application & (use π = 22 ) Evaluation] 7 Question 9 (i) The table given below shows the runs scored by a cricket team during the overs of a match. 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 20 – 30 37 30 – 40 45 40 – 50 40 50 - 60 60 60 – 70 51 70 - 80 35 Use graph sheet for this question. Take 2 cm = 10 overs along one axis and 2 cm = 10 runs along the other axis. (a) Draw a histogram representing the above distribution. [Application & (b) Estimate the modal runs scored. Evaluation] (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 [Analysis, the A.P. Application & 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. [Application & Analysis] T25 511 – SPECIMEN 11 of 12 Question 10 (i) A car travels a distance of 72 km at a certain average speed of x km per hour and then travels a distance of 81 km at an average speed of 6 km per hour [Analysis, more than its original average speed. If it takes 3 hours to complete the total Application & journey then form a quadratic equation and solve it to find its original average Evaluation] speed. (ii) 1 1 1 0 Given matrix, X = 𝑎𝑎𝑎𝑎𝑎𝑎 𝐼𝐼 = , prove that 𝑋𝑋 2 = 4𝑋𝑋 + 5𝐼𝐼 8 3 0 1 [Application & Evaluation] (iii) Use ruler and compasses for the following question taking a scale of 10 m=1 cm. A park in a city is bounded by straight fences AB, BC, CD and DA. Given that AB = 50 m, BC = 63 m, ∠ABC = 75°. D is a point equidistant from the fences AB and BC. If ∠BAD = 90°, construct the outline of the park ABCD. Also locate a point P on the line BD for the flag post which is equidistant from [Analysis & the corners of the park A and B. Creativity] T25 511 – SPECIMEN 12 of 12