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ICSE Class X Prelims 2026 : Mathematics (The Doon School, Dehradun)

8 pages, 41 questions, 17 questions with responses, 17 total responses,

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PRE BOARD - I NOVEMBER 2025 Name of the Candidate : Sch. No./House: Form Subject Paper Initials of Master(s) A-ICSE Mathematics 1 (Theory) ANC/CSG/RLR/SRT/MKS Resource Booklet / Inserts / Data Booklet (if Any) Mathematical Tables Day & Date Session Duration No. of Students Maximum Marks Monday, 17th Nov. 2025 1st Three Hours 100 80 INSTRUCTIONS TO CANDIDATES Answers to this Paper must be written on the paper provided separately. You will not be allowed to write for 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 and graph papers are to be provided by the school. This paper consists of 8 printed pages. 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. [15] (Do not copy the questions, write the correct answers only.) (i) An article 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 (d) 86.40 (ii) (iii) If x 2 + kx + 6 = ( x 2 )( x 3) for all values of x , then the value of k is (a) 5 (b) 3 (c) 2 (d) 5 In a recurring deposit account, Mohit deposited 5,000 per month for one year and at maturity gets 67,500; the total interest earned is (a) 60,000 (b) 67,500 (c) 52,500 (d) 7,500 (iv) (v) If mx 2 nx + 8 has x 2 as a factor, then (a) 2m n = 4 (b) 2m + n = 4 (c) 2n + m = 4 (d) n 2m = 4 If a number is randomly chosen from the numbers 1, 2, 3, 4, , 25, then the probability of the number to be prime is (a) (b) (c) (d) 7 25 9 25 11 25 13 25 2 (vi) The product AB of two matrices A and B is possible if (a) A and B have the same number of rows. (b) A and B have the same number of columns. (c) The number of columns of A is equal to the number of rows of B. (d) The number of rows of A is equal to the number of columns of B. (vii) In a circle with radius R, the shortest distance between two parallel tangents is equal to (a) R (b) 2R (c) 2 R (d) R (viii) If sec tan = k , then the value of sec + tan is: 1 (a) 1 k (b) 1 k (c) 1+ k 1 k A solid sphere is cut into two identical hemispheres. (d) (ix) 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. (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. (x) Assertion (A): y = x + 4 and y = 3x + 5 are two intersecting lines. Reason (R): The inclinations of both the given lines are not equal. (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). (xi) The mean of numbers in A.P. 2, 4, 6, 8, , 40 is 40 + 2 (a) 2 20 (b) ( 2 + 40 ) 2 10 (c) ( 2 + 40 ) 2 (d) 840 3 Turn Over (xii) The sum invested to purchase 15 shares of a company of nominal value 75 available at a discount of 20% is (a) 60 (b) 90 (c) 1350 (d) 900 (xiii) A point is equidistant from the sides of an obtuse angle triangle. The point is called: (a) Circumcentre of the triangle (b) Incentre of the triangle (c) Centroid of the triangle (d) Orthocentre of the triangle (xiv) The solution set for the inequation 2 x + 4 14, x W (a) 1, 2, 3, 4, 5 (xv) (b) 0, 1, (c) 1, (d) 0, 1, 2, 3, 4, 5 2, 3, 4 2, 3, 4 Assertion (A): If two triangles are congruent, then they are also similar. Reason (R): All congruent triangles are similar but similar triangles may not be congruent. (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). Question 2 (i) Solve the quadratic equation x 2 3 ( x + 3) = 0 ; give your answer correct to two significant figures. (ii) [4] Richard has a recurring deposit account in a post office for 3 years at 7.5% p.a. simple interest. If he gets 8325 as interest at the time of maturity, find [4] (a) the monthly instalment (b) the amount of maturity (iii) Three vertices of a parallelogram ABCD taken in order are A ( 3, 6 ) , B ( 5,10 ) and C ( 3, 2 ) . Find: (a) the coordinates of the fourth vertex D (b) the equation of side AB of parallelogram ABCD 4 [4] Question 3 (i) (ii) The sum of the first three terms of an Arithmetic Progression is 42 and the product of the first and the third terms is 52. Find the first term and the common difference. [4] Prove the following identity: [4] sec2 + cosec2 = tan + cot (iii) Use ruler and compass to answer this question. Construct ABC = 90 , where AB = 6 cm, BC = 8 cm. [5] (a) Construct the locus of points equidistant from B and C. (b) Construct the locus of points equidistant from A and B. (c) Mark the point which satisfies both the conditions (a) and (b) as O. Construct the locus of points keeping a fixed distance OA from the fixed point O. (d) Construct the locus of points which are equidistant from BA and BC. SECTION B (40 Marks) (Attempt any four questions from this Section.) Question 4 (i) Solve the following inequation, write the solution set and represent it on the real number line. 5 x 21 (ii) 5x 3 6 3 + x, x R 7 7 [3] A tree breaks due to storm and the broken part bends (without being detached) so that the top of the tree touches the ground making an angle of 30 with it. The distance between the foot of the (iii) tree to the point where the top touches the ground is 8 m. Find the height of the tree. [3] In the given figure, ABC is a right-angled triangle with BAC = 90 and ADC = 90 . [4] (a) Prove that ADB CDA . (b) If BD = 18 cm and CD = 8 cm, find AD. (c) Find the ratio of the area of ADB to the area of CDA . Question 5 (i) A box contains four cards numbered from 1 to 4. A card is drawn from the box, its number is noted and put back in the box. Now, one more card is drawn. Write down the sample space of this random experiment. Find the probability of getting [3] (a) The product of numbers equal to 4. (b) The sum of number less than or equal to 6. 5 Turn Over (ii) A man has a choice to invest in hundred-rupee shares of two firms at 120 or at 132. The first firm pays a dividend of 5% per annum and the second firm pays a dividend of 6% per annum. [3] (a) Find which firm is giving a better return. (b) If a man invests 26,400 with each firm, how much will be the difference between the annual returns from the two firms. (iii) In the adjoining figure, TP and TQ are two tangents [4] to the circle with centre O, touching at A and C respectively. If BCQ = 55 and BAP = 60 , find: (a) OBA and OBC (b) AOC (c) ATC Question 6 (i) The figure shows the vertical cross section of an ice cream [5] cone consisting of a cone surmounted by a hemisphere. The radius of the hemisphere is 3.5 cm and the height of the cone is 10.5 cm. The outer shell ABCDFE is shaded and is not filled with ice cream. If AE = DC = 0.5 cm, AB EF and BC FD , calculate: (a) the volume of the ice cream in the ice cream cone (the unshaded portion including the hemisphere) in cm3. (b) the volume of the outer shell (the shaded portion) in cm 3. Give your answers correct to the nearest cm3. (ii) The daily wages of 80 workers in a project are given below: Wages (in ) No. of workers [5] 400-450 450-500 500-550 550-600 600-650 650-700 700-750 2 6 12 18 24 13 5 6 Use a graph paper to draw an ogive for the above distribution. (Use a scale of 2 cm = 50 on x-axis and 2 cm = 10 workers on y-axis). Use your ogive to estimate: (a) The median wage of the workers. (b) The lower quartile wage of the workers. (c) The number of workers who earn more than 625 daily. Question 7 (i) If a, b, c are in continued proportion, prove that ( a + b + c )( a b + c ) = a 2 + b 2 + c 2 [3] (ii) 2 3 x 7 Find x and y if = 1 0 y 2 [3] (iii) There are three positive numbers in a Geometric Progression 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. [4] Question 8 (i) Using the factor theorem, show that ( x 2 ) is a factor of x3 + x 2 4 x 4 . Hence, factorize the polynomial completely. (ii) [3] In what ratio does the x-axis divide the line segment joining the points ( 4, 6 ) and ( 1, 7 ) ? Also find the coordinates of the point of division. (iii) [3] Use graph paper for this question. (take 2 cm = 1 unit on both x and y-axis). ABCD is a quadrilateral whose vertices are A ( 2, 2 ) , B ( 2, 2 ) , C ( 0, 1) and D ( 0, 1) . [4] (a) Reflect quadrilateral ABCD on the y-axis and name it as A ' B ' CD . (b) Write down the coordinates of A ' and B ' . (c) Name two points which are invariant under the above reflection. (d) Name the polygon A ' B ' CD . Question 9 (i) (ii) 1 cosec 2 sec 2 , find the value of [3] cosec 2 + sec 2 5 The mean of the following frequency distribution is 57.6 and the sum of all the frequencies is 50. If tan = Find the values of p and q . (iii) [3] Class Interval 0-20 20-40 40-60 60-80 80-100 100-120 Frequency 7 p 12 q 8 5 Using ruler and compasses only, construct a ABC such that BC = 5 cm, AB = 6.5 cm and ABC = 120 . (a) Construct circumcircle of [4] ABC . (b) Construct a cyclic quadrilateral ABCD such that D is equidistant from AB and BC. 7 Turn Over Question 10 (i) x 4 + 1 17 = , using properties of proportion, find the value of x . If 2 x2 8 (ii) The price of a bicycle is 3136 inclusive of tax (under GST) at the rate of 12% on its listed price. [3] A buyer asks for a discount on the listed price so that after charging GST, the selling price becomes equal to the listed price. Find the amount of discount which the seller has to allow for the deal. (iii) [3] A bus covers a distance of 240 km at a uniform speed. Due to heavy rain, its speed gets reduced by 10 km/h and as such it takes two hours longer to cover the total distance. Assuming the uniform speed to be x km/h, form an equation and solve it to find the value of x . 8 [4]

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