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ISC Class XI Sample / Model Paper 2026 : Mathematics : half yearly exam

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Raghvendra Prajapati
Vikas Bharti School, Gorakhpur

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Vikas Bharti School S(SCI) Form Time: 3 hours Gorakhpur, U.P. Half Yearly Examination 2024 Class: 11th Subject: Mathematics Max. Marks: 80 GENERAL INSTRUCTIONS1. 2. 3. 4. 5. 6. 7. The question paper consists of three sections A, B and C. Candidates are required to attempt all questions from Section A and all questions either from Section B or Section C Section A Internal choice has been provided in two questions of two marks each, two questions of 4 marks each and two questions of 6 marks each. Section B Internal choice has been provided in one question of two marks and one question of four marks. Section C Internal choice has been provided in one question of 2 marks and one question of 4 marks. All working, including rough work should be done on the same sheet as, and adjacent to the rest of the answer The intended marks for questions or parts of questions are given in brackets [ ]. Section A (65 marks) 1. In sub-parts (i) to (x) choose the correct option and in sub-parts (xi) to (xv), answer the questions as instructed. (i) {x: x is the capital of India} is a/an [1] a) empty set b) singleton set c) infinite set d) none of these (ii) The domain of f(x) = | x - 3 | includes [1] a) all real numbers greater than 3 b) all real numbers smaller than 3 c) all real numbers less than or equal to 3 d) all real numbers (iii) The value of tan 19 /3 is. [1] a) 1/ 3 b) 3 c) 3 d) (iv) The real values of x and y if (x - iy) (3 + 5i) is the conjugate of ( -6 - 24i ) are [1] a) x = 5 and y = 3 b) x = 3 and y = -3 c) x = -3 and y = 3 d) x = 3 and y = 5 (v) The value of 7! - 5! is [1] a) 4920 b) 4290 c) 4092 d) 4029 (vi) The number of terms in the expansion of (x + a)51 - (x - a)51 after simplification [1] a) 50 b) 52 c) 26 d) 25 (vii) Eighteenth term of the sequence 9, 5, 1, -3,... is [1] a) -57 b) -58 c) -59 d) -60 (viii) If the distance between the points (a, -2) and (5,1) is 5 units, find the value(s) of a [1] a) 1 b) 9 c) 1 and 9 d) -1 and -9 In the following questions, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct answer out of the following choices. a) Both A and R are true and R is the correct explanation of A. b) Both A and R are true but R is not the correct explanation of A. c) A is true but R is false. d) A is false but R is true. (ix) Assertion(A): The number of ways in which 4 prizes can be distributed among 5 students, when no student gets more than one prize is 4! [1] Reason (R): Number of ways of arranging r objects in n positions is nPr . (x) Assertion (A): Equation of a line with slope 2 and intersecting the x-axis at a distance of 3 units to the left of origin is 2x y 6 = 0. [1] Reason (R): Equation of a line passing through the point (x1, y1) and with slope m is given by: (y y1) = m(x x1) (xi) Find the distance between P(x1, y1) and Q(x2, y2) when PQ is parallel to the Y-axis. [1] (xii) Find the number of terms in the expansion of (1 + x2)4 . [1] (xiii) Find the sum of 24 terms of an AP 1, 3, 5, 7 [1] (xiv) If sin x = , find the value of cos 2x. [1] (xv) Write the equation of the line through origin and parallel to the line 4x - 7y + 13= 0. [1] 2. Draw the graph of the real function f(x) = x3 and find its range. [2] 3. The large hand of a big clock is 35 cm long. How many cm does its tip move in 9 minutes? [2] 4. Find the number of non-zero integral solutions of the equation | 1 - i |x = 2x . [2] 5. Divide 39 into 2 parts whose product is 338. [2] 6. (i) Prove that sin x + sin 3x + sin 5x + sin 7x = 4 cosx.cos2x.sin4x. [2] OR (ii) Find the value of cos ( 1710 ). 7. Find the domain and range of the function f(x) = (x - 2)/(3 - x). [4] 8. If the sums of n terms of two arithmetic progression are in the ratio (3n + 8) : (7n + 15), find the ratio of their 12th terms. [4] 9. (i) Prove the following using principle of mathematical induction. [4] 32n+2 - 8n - 9 is divisible by 8. OR (ii) Prove by using principle of mathematical induction (2n + 7) < (n + 3)2 . 10.(i) Find the equation of a circle passing through the points (4, 1) and (6, 5) and whose centre lies on the line 4x + y = 16. [4]. OR' (ii) Find the equation of the circle which passes through the centre of the circle x2 + y2 + 8x + 10y - 7 = 0 and is concentric with the circle 2x2 + 2y2 - 8x - 12y - 9 = 0 11. If sec x + tan x = 1.5, find the value of sec x, tan x, cos x and sin x. In which quadrant does x lie? [6] 12. (i) If a and b are the roots of x2 - 3x + p = 0 and c, d are the roots of x2 - 12x + q = 0, where a, b, c and d form a GP, then prove that (q +p)/(q - p) = 17/15. [6] OR' (ii) The sum of three numbers in GP is 56. If we subtract 1, 7, 21 from these numbers in that order we obtain an arithmetic progression, then find the numbers. 13. A solution of 8% boric acid is to be diluted by adding 2% boric acid solution to it. The resulting mixture is to be more than 4% but less than 6% boric acid. If we have 640 L of 8% solution, then how many litres of the 2% solutions will have to be added? [6] 14. If one diagonal of a square is along the line 8x - 15y = 0 and one of its vertex is at (1, 2), then find the equation of sides of the square passing this vertex. [6] Section B (15 marks) 15. In sub-parts (i) and (ii) choose the correct option and in sub-parts (iii) to (v), answer the questions as instructed. [5] (i) If a parabola has the origin as its focus and the line x = 2 as the directrix, then the co-ordinates of the vertex of the parabola are a) (0, 1) b) (1, 0) c) (0, -1) d) (-1, 0) (ii) The vertex of the parabola y2 + 8x - 2y + 17 = 0 a) (1, -2) b) (-2, 1) c) (1, 2) d) (2, -1) (iii) Find the coordinates of the focus of the conic represented by 5x2 = -12y (iv) Find the equation of the parabola with focus (6, 0) and directrix x = -6. (v) If a parabola has the origin as its focus and the line x = 2 as the directrix, then find the coordinates of the vertex of the parabola. 16. (i) Find the length of latus-rectum of the parabola x2 = 16 OR' (ii) Find the focus, directrix and eccentricity of the conic represented by the equation y2 = 8 3 x. [2] 17 (i) Show that the equation y2 - 8y - x + 19 = 0 represents a parabola. Find its vertex, focus, directrix and axis. [4] OR' (ii) Find the equation to the parabola whose vertex is at (-2, 2) and focus at (-6, 6). 18. The towers of a suspension bridge, hung in the form of a parabola, have their tops 30 metres above the roadway and are 200 metres apart. If the cable is 5 metres above the roadway at the centre of the bridge, find the length of the vertical supporting cable 30 metres from the centre. [4] Section C(15 Marks) 19. In sub-parts (i) and (ii) choose the correct option and in sub-parts (iii) to (v), answer the questions as instructed. [5] (i) For the data: 28, 17, 12, 25, 26, 19, 13, 27, 21, 16, the third quartile is a) 25.5 b) 26 c) 26.25 d) 26.75 (ii) If the median and mean of a moderately asymmetrical frequency distribution are 72 and 74 respectively, then the mode is a) 68 b) 76 c) 75 d) 70 (iii) Two samples of sizes 50 and 100 are given. The means of these samples respectively are 56 and 50. Find the mean of size 150 by combining the two samples. (iv) The daily earnings (in rupees) of 11 workers in a factory are 16, 11, 3, 7, 5, 28, 9, 31, 28, 43, 15. Find the median. If another worker with earning rupees 17 is also included, what would the median be? (v) Find the lower quartile for the data: 9, 5, 7, 11, 13, 17, 15 20. (i) If the mode of the following series is 66, then find the value of f. [2] Class Interval 0-15 15-30 30-35 45-60 60-75 75-90 Frequency 3 5 10 12 16 f OR' (ii) Calculate the median from the following data. Expenditure 40-59 60-79 80-99 100-119 120-139 Number of families 50 250 500 100 50 21. An analysis of daily wages of casual laborers in two firms A and B, belonging to the same industry, gives the following result: [4] No. of workers Average daily wages(in rupees) S.D. Firm A Firm B 50 113 6.5 60 120 8.2 Find the mean and S.D. of wages of all casual laborers in the two firms taken together. 22. (i) The heights( to nearest cm) of 60 students of a certain school are given in the following frequency distribution table: [4] Heights(in cm) 151 152 153 154 155 156 157 No. of students 6 4 11 9 16 12 2 Find the (i) median (ii) lower quartile (iii) upper quartile (iv) inter quartile range OR' (ii) Find the values of a and b from the following data: Variable 10-20 20-30 30-40 40-50 50-60 Frequency 5 a 15 b 7 Given that mode = 37 and total frequency is 47.

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