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ICSE Class X Prelims 2026 : Mathematics (Auxilium Convent School (ACS), Sevoke Road, Siliguri)

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AUXILIUM CONVENT SCHOOL Term - IExamination 2025- 2026 MATHEMATICS Class - X (Three hours) Answer to this Paper must be wrillen on the paper provided separutely. You will not be allowed to write during the first 15 minutes. This time is to be spent in reading the question puper. Thetime given at the head of this Paper is the time allowed for writing the answers. Attempt al questions from Section Aand 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 resultin loss of marks. The intended marks for questions or parts of questionsare given in brackets ||. Mathematicaltables are provided. SECTION -A 140 Marks) Attempt all the questions fromthis section Question 1. Choose the correct options for each of the following questions: |1x15=15] i) Inan AP, if Kth term is 5k + 1. Then, the sum of first n terms is (a) (5n +7) (b)(n +7) (d) (7n+5) (c)(n +5) i) The number that must be added to each of thenumber 6, 15, 20 and 43 to make them proportional is (a) 0 ii) IfA=| (b) 1 P (c) 2 is such that A = I then (d) 3 (a) 1 + a + By = 0 (b) 1 - a? + By = 0 (c) 1- a'- By = 0 (d) 1 +a'-By = 0 iv) If (x-2) is a factor of the polynomial x + 2x2- 13x + k, then k is: (a) -10 (b)10 (c)-26 (d) 26 v) If sin 0- cos0=0, then the value of sin0 + cost0 is (b) /2 (a) /4 (d) 1 vi) Which of the following equation represent a line passing through origin? (a) 3x 2y +5 =0 (b) 2x-3y = 0 (d) y=-6 (c) x = 5 vii) What number should be added to the polynomial p(*) = 2x-3x - 8x so that it leaves a remainder 10 when divided by 2x + 1? (b) 10 (a) 13 (c) 7 (d) 3 viii) a, b, Cand d are in proportion. Statement I: a + b, a - b, c + dand c- d are in proportion. Statement II: ad = bc (a) Both Statements are true and Statement II is the correct explanation of assertion. (b) Both Statements are true and Statement II is not the correct explanation of assertion. (c) Statement I is true, but Statement IIis false. (d) Statement I is false and Statement II is true ix) The sum of three consecutive terns of an AP is 15, and their product is 105, then the common difference is (d) +4 (b) 2 (c) 3 (a) 1 x) The top of two poles of height 20 m and 14 m are connected by a wire. If the wire makes an angle of 30 with the horizontal, then the length of the wire is (b) 10 m (c) 8 m (d) 6 m (a) 12 m xi) If M= [1 2]and N= (a) 1 2 (b) 2 x 1 he order of MN is (c) 2 x 2 (d) 1 x 1 0-bcos 0 wii) Ifb tan = a, the value ofStn a sin 0+b cos 0 a'+b2 atb (b) (c) Lxii) The 8 term from the end of the sequence 3, 6, 12, ... a2+b2 (b) 48 (a) 96 10th term is (d) 12 (c) 24 Lxiv)Two opposite vertices of arectangle are (1,3) and (5,1). Ifthe rest two vertices lie on the line y-x + A =0,then Ais equal to (b) 1 (a) -1 (d) 3 (c) 2 Lxr)Assertion (A): If a <b, c<0, then> C C Reason (R): If both sides are divided by the same negative quantity, then the inequality is reversed. (a) Both assertion and reason are true and reason is the correct explanation of assertion. (b) Both assertion and reason are true and reason is not the correct explanation of assertion. (c) Assertion is true, but reason is false. (d) Assertion is false and reason is true. Question 2. t A manufacturer has 460 litres of a 9% acid solution, How many litres of a 3 % acid solution must be added to it so that the acid content in the resulting mixture be more than 5% but less than 7%? [4) ii) Let Abe a matrix satisfying A =14 -7] a) State the order of A. b) FindA. [4] ii)Consider the AABC with vertices A(1, 4), B(2, -3) and C(-1, -2) as shown in the given figure. AD is the median and AM is the altitude through A. A(1, 4) B(2, - 3) D C(- 1,-2) Based on the above information answer the following questions. a) Find the slope of BC. b) Find the equation of the through A, parallel to BC. c) Find the equation of right bisector of side BC. [4| Question 3. i) If x = V2a+1+v2a-1 V2a+1-V20-7 Prove that: x - 4ax + 1 =0 [4] ii) The angle of elevation of ajet fighter point Aon ground is 60 . After flying 10seconds, the angle changes to 30 , If the jet is flying at a speed of 648 km/hour, find the constant height at which the jet is flying. ii) Pollution--A Major Problem: One of the major serious problems that the world is facing today is the environmental pollution. Common types of pollution include light, noise, water and air pollution. In a school, students thoughts of planting trees in and around the school to reduce noise pollution and air pollution. Condition I: It was decided that the number of trees that each section of each class willplant be the same as the class in which they are studying,e.g. asection of class I willplant Itree asection of class IIwill plant 2 trees and so on a section of class XII will plant 12 trees. Condition II: ltwas decided that the number of tres that each section of each class willplant be the double of the class in which they are studying, e.g. a section of class Iwill plant 2 trees, a section of class II will plant 4 trees and so on a section of class XII will plant 24 trees. Refer to Condition I (a)Find the AP formed by sequence i.e. number of plants by students. (b) If there are two sections of each class, how many trees will be planted by the students? (c) If there are three sections of each class, how many trees willbe planted by the students? Refer to Condition II (d) Find the number of plants planted by students of class X. (e) If there are five sections of each class, how many trees illbe planted by the students? SECTION - B [40 Marks] Attempt any four questions from this section Ouestion 4. fp= 4xy x+y find the value of p+2x p-2x + p+2y by using componendo and dividendo. p-2y i Prove that: 2(sin + cos 0) - 3(sint 0 + cos 0) +1 = 0. ) Eachside of an equilateral triangle is 24cm. The mid-points of its sides are joined to form another [3] [3] triangle. This process is going continuously infinite. A 24 cm 12 cm 24 cm 712 cm 12 cm P 24 cm Based on the above information answer the following questions. a) Find the side of the 5 triangle (in cm). b) Find the sum of perimeter of first 6triangle is (in cm). c) Find the sum of the areas of first 7 triangles. [4] Question 5. ) IfA =| find the value of 54 - 7A+ 13. -1 [3] Based on the given information, find the value of' T [3| [4] i) Consider T, = sin"9 + cos"0. iiy Find the image of the point P (-3, 1) in the line 2x - 3y = 4. Question 6. i) Solve the following inequation and represent the solution set on a number line: S*42 <x+4 <+5; xe R. [3| 5 ii) If polynomials x*-px +x+6 and 2x*-x -(p+ 3)x-6 when divided byx- 3, leaves the same remainder, then find the value of p. [3| ii) If a, b, c and d are in continued proportion, prove that: (b-c) + (c- a) + (d- b) = (a- d)2. 14| Question 7. Yr2+3 =lis find the values of x, y,z. 3) JiY The angle of elevation of the top of an unfinished tower at a distance of 75 mfrom its base is 30 . How much higher must the tower be raised so that the angle of elevation of its top at the same point may be [3| 60 ? [Take v3 = 1.732] i) The sum of three numbers in G.P. is 56. If we subtract 1,7,21 from these numbers respectively, we obtain an A.P. Find the numbers. [4| Question 8. i) Find the sum of first n terms of the series 0.7 +0.77 +0.777+... (3] ii) Show that the lines 2x + 5y = 1,x-3y = 6and x+ 5y +2 =0 are concurrent. [3] iii) Drawbridge: Adrawbridge is a bridge that can be moved in order to stop or allow passage across l. Modern drawbridges are often built across large, busy waterways. They can be lifted to allow large ships topass or lowered to allow land vehicles or pedestrians to cross. Adrawbridge is 60 meter long when stretched across a river. As shown in the figure, the two sections of the bridge can be rotated upward through an angle of 30 . a) If the water level is 5 metres below the closed bridge, find the height h between the end of a section and the water level when the bridge is fully open. b) How far apart are the ends of the two sections when the bridge is fully opened, as shown in the figure? [4] Question 9. i) Find the equation of a line which has equal intercepts on both the axes and passing through the point (4,7. [3| ii) Factorise: 6x + 17x2+ 4x - 12 by using factor theorem. [3| i i) If (x +1) and (x-2) are factors ofx* + (a + 1)x - (b- 2)x 6. Find the value of aand b. And then factorise the given expression. (4] Question 10. i) Which term of the sequence 20, 19;, 4 18, 17 ..... is the first negative term? [3) ii) Find all pairs of consecutive odd natural number, both of which are larger than 10, such that their sum is less than 40. [3| ii) Prove 4 (1+sin 8-cos 6\ that:sin @+cos e/ 1-cos 8 [4| 1+cos 0 The End Best of Luck

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