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ICSE Class X Prelims 2026 : Mathematics

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FIRST TRIAL ASSESSMENT DECEMBER - 2025 CLASS: X (ICSE) MATHEMATICS _________________________________________________________________________ Maximum Marks: 80 Time allowed: Three hours Answers to this Paper must be written on the paper provided separately. You will not be allowed to write during the 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 work will result in loss of marks. The intended marks for questions or parts of questions are given in brackets [ ] Mathematical tables are provided. ____________________________________________________________________________________ SECTION A (Attempt all questions from this Section.) Question 1 Choose the correct answer to the questions from the given options. 2 (i) The given quadratic equation 3 + 7 + 2 = 0 has: (a) two equal real roots. (b) two distinct real roots. (c) more than two real roots. (d) no real roots. (ii) Mr. Anuj deposits 500 per month for 18 months in a recurring deposit account at a certain rate. If he earns 570 as interest at the time of maturity, then his [15] 2 matured amount is: (a) (500 18 + 570) (b) (500 19 + 570) (c) (500 18 19 + 570) (d) (500 9 19 + 570) (iii) The equation of the line passing through origin and parallel to the line 3x+4y+7 = 0 is: (a) 3x+4y+5 =0 (b) 4x-3y-5 = 0 (c) 4x-3y = 0 (d) 3x+4y = 0 0 (iv) In the given diagram, chords AC and BC are equal. If ACD = 120 then AEC is: 0 (a) 30 0 (b) 60 0 (c) 90 0 (d) 120 (v) Which of the following cannot be the probability of any event? (a) 5 4 (b) 0.25 (c) 1 33 (d) 67% (vi) If 3 Which of the following operations is possible? (a) A-B (b) A+B (c) AB (d) BA (vii) The marked price of an article is 1375. If the CGST is charged at a rate of 4%, then the price of the article including GST is: (a) 55 (b) 110 (c) 1430 (d) 1485 (viii) In the given diagram ABC ~ EFG. If ABC = EFG = 60 , then the length of the side FG is: (a) 15 cm (b) 20 cm (c) 25 cm (d) 30 cm (ix) Statement 1: The point which is equidistant from three non-collinear points D,E and F is the circumcentre of the DEF. Statement 2: The incentre of a triangle is the point where the bisector of the angles intersect. (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. 2 4 2 (x) Assertion(A): If + = 1 + = 1 2 2 Reason (R): 1 = (a) (A) is true, (R) is false. (b) (A) is false, (R) is true. 4 (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) (xi) If the volume of two spheres is in the ratio 27 64, then the ratio of their radii is: (a) 3 4 (b) 4 : 3 (c) 9 16 (d) 16 : 9 (xii) A man invested in a company paying 12% dividend on its share. If the percentage return on his investment is 10%, then the shares are: (a) at par (b) below par (c) above par (d) cannot be determined (xiii) The table shows the values of x and y, where x is proportional to y. What are the values of M and N? (a) M = 4, N = 9 (b) M = 9, N = 3 (c) M = 9, N = 4 (d) M = 12, N =0 (xiv) Given, + 2 3 + 3 and x is a prime number. The solution set for x is: (a) (b) {0} (c) {1} (d) { 0,1} (xv) A right angle triangle shaped piece of hard board is rotated completely about its hypotenuse, as shown in the diagram. The solid so formed is always: 1. a single cone 2. a double cone Which of the statements is valid? (a) only 1. (b) only 2. (c) both 1. and 2. (d) neither 1. nor 2. Question 2: (i) Mrs. Rao deposited 250 per month in a recurring deposit account for a period of 3 years. She received 10,110 at the time of maturity. Find: (a) the rate of interest. [4] 5 (b) how much more interest Mrs. Rao will receive if she had deposited 50 more per month at the same rate of interest and for the same time. (ii) Find: [4] (iii) In ABC, ABC = 90 , AB = 20 cm, AC = 25 cm , DE is perpendicular to AC such that DEA = 90 and DE = 3 cm as shown in the given figure. [4] (a) (sin + cosec ) (b) (cos + sec ) 2 2 Using the above results prove the following trigonometry identity. 2 2 2 2 (sin + cosec ) + (cos + sec ) = 7 + ta + co (a) Prove that ABC ~ AED. (b) Find the lengths of BC, AD and AE. (c) If BCED represents a plot of land on a map whose 2 actual area on ground is 576 , then find the scale factor of the map. Question 3: (i)Using ruler and compass only construct ABC = 60 , AB = 6 cm and BC = 5 cm. (a) construct the locus of points equidistant from AB and BC. (b) construct the locus of points equidistant from A and B. (c) Mark the point which satisfies both the conditions (a) and (b) as P. Hence, construct a circle with centre P and passing through A and B. (ii) If If a, b and c are in continued proportion, then prove that (iii) In unit y-axis, 2 cm = 1 unit [4] [4] the given graph ABCD is a parallelogram. Scale: x-axis, 2 cm = 1 [5] 6 Using the graph, answer the following: (a) write down the coordinates of A, B, C and D. (b) calculate the coordinates of P , the point of intersection of the diagonals AC and BD. (c) find the slope of sides CB and DA and verify that they represent parallel lines. (d) find the equation of the diagonal AC. SECTION B (Answer any Four from this Section) Question 4: (i) Solve the following inequation, write the solution set and represent it on the real number line. (ii) There are three positive numbers in a Geometric Progression (G.P.) such that: their product is 3375 and the result of the product of the first and second number added to the product of the second and third number is 750. Find the numbers (iii) If A 2 2 find . = . Question 5: (i) The following bill shows the GST rates and the marked price of articles. [3] [4] [3] [3] 7 Find the total amount to be paid for the above bill. 3 2 (ii) It is given that (x - 2 ) is a factor of polynomial 2 7 + 2 Find: (a) the value of k . (b) hence, factorize the resulting polynomial completely. (iii) A solid wooden capsule is shown in Figure 1. The capsule is formed of a cylindrical block and two hemispheres. Find the sum of total surface area of the three parts as shown in Figure 2. Given,the radius of the capsule is 3.5 cm and the length of the cylindrical block is 14 cm. (Use = 22 7 [3] [4] ) Question 6: (i) Use a graph paper for this question taking 2 cm = 1 unit along both axes. [5] (a) Plot A(1, 3), B(1, 2) and C(3, 0). (b) Reflect A and B on the x-axis and name their images as E and D, respectively . Write down their coordinates. (c) Reflect A and B through the origin and name their images as F and G respectively. (d) Reflect A, B and C on the y-axis and name their images as J, I and H respectively. (e) Join all the points A, B, C, D, E, F, G, H, I and J in order and name the closed figure so formed. 8 (ii) In the given diagram, AB is a vertical tower 100 m away from the foot of a 30 storied building CD. The angles of depression from the point C and E, (E being the midpoint 0 [5] 0 of CD), are 35 and 14 respectively. (Use mathematical table for the required values rounded off correct to two places of decimals only) Find the height of the: (a) tower AB (b) building CD Question 7: (i) Use a graph paper for this question. (Take 2 cm = 10 Marks along one axis and 2 cm = 10 students along another axis). Draw a Histogram for the following distribution which gives the marks obtained by 164 students in a particular class and hence find the Mode. [3] [3] (iii) Mr. and Mrs. Das were travelling by car from Delhi to Kasauli for a holiday. Distance between Delhi and Kasauli is approximately 350 km (via NH 152D). Due to heavy rain they had to slow down. The average speed of the car was reduced by 20 km/hr and the time of the journey increased by 2 hours. Find: (a) the original speed of the car. (b) with the reduced speed, the number of hours they took to reach their destination. [4] (ii) A man bought 200 shares of a company at 25% premium. If he received a return of 5% on his investment. Find the: (a) market value (b) dividend percent declared (c) number of shares purchased, if annual dividend is 1000. 9 Question 8: (i) For the given frequency distribution, find the: (a) mean, to the nearest whole number (b) median [3] [4] [3] Question 9: (i) A hollow sphere of external diameter 10 cm and internal diameter 6 cm is melted and made into a solid right circular cone of height 8 cm. Find the radius of the cone [3] (ii) The line segment joining A(2,-3) and B(-3, 2) is intercepted by the -axis at the point M and the y axis at the point N. PQ is perpendicular to AB produced at R and meets the y- axis at a distance of 6 units from the origin O, as shown in the diagram, at S. Find the: (a) coordinates of M and N [4] (ii) In the given figure, O is the centre of the circle and AB is a tangent to the circle at B. 0 If PQB=55 (a) find the value of the angles x, y and z. (b) prove that RB is parallel to PQ. (iii) (ii) Solve the Quadratic Equation 2 7 + 2 2 = 0 ( ) so formed. [Use = (b) coordinates of S (c) slope of AB. 22 7 ) 10 (d) equation of line PQ. (iii) Ms. Sushmita went to a fair and participated in a game. The game consisted of a box having number cards with numbers from 01 to 30. The three prizes were as per the given table:Find the probability of winning a: [3] (a) Wall Clock (b) Water Bottle (c) Purse Question 10: (i) ABCD is a cyclic quadrilateral in which BC = CD and EF is a tangent at A. CBD = 43 and ADB = 62 . Find: [4] (a) ADC (b) ABD (c) FAD (ii) Use a graph sheet for this section.The daily wages of 120 workers working at a site are [6] given below Take 2cm = 50 and 2cm = 20 workers along x - axis and y - axis respectively to draw an ogive curve and hence estimate: 11 (a) Median Wages (b) The inter-Quartlie range of wages (c) Percentage of workers whose daily wage is above 475

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