2024 Mathematics Past Paper PDF (OCR)

2024 III 15 0930 Seat No. Time : 3 Hours MATHEMATICS Subject Code H 4 7 5 4 Total No. Of Questions : 36 (Printed Pages : 10) Maximu...

2024 III 15 0930 Seat No. Time : 3 Hours MATHEMATICS Subject Code H 4 7 5 4 Total No. Of Questions : 36 (Printed Pages : 10) Maximum Marks : 80 INSTRUCTIONS : (i) The question paper consists of 36 questions. (ii) All questions are compulsory. (iii) Question numbers 1 to 8 are multiple choice type questions of one mark each. (iv) Question numbers 9 to 16 are very short answer type questions of one mark each. (v) Question numbers 17 to 22 are short answer type-I questions of two marks each. (vi) Question numbers 23 to 28 are short answer type-II questions of three marks each. (vii) Question numbers 29 to 34 are long answer type-I questions of four marks each. (viii) Question numbers 35 to 36 are long answer type-II questions of five marks each. (ix) There is no overall choice. However an internal choice has been provided in two questions of 4 marks each and 2 questions of 5 marks each. (x) Use of calculator is not permitted. (xi) Log tables will be supplied on request. (xii) Graph should be drawn on the answer paper only. H-4754 1 P.T.O.  3 4 d2 y dy 1. The degree of the differential equation xy is................. dx2 dx 1 2 3 4 2. If f : R R and g : R R are the two real functions defined by f ( x) 3 x2 1 and g( x) 1 x , then ( gof ) ( 2) is................... 12 28 –12 –28 3. The value of iˆ ˆj 2 ˆj 3kˆ is.................... 0 6 3 5 H-4754 2  1 1 1 4. If sin tan x cot 1 , then the value of x is................... 3 3 –3 2 1 3 5. If R is a relation in the set A 3 as R {( x, y) / x, y A and x2 y 4 is a perfect square}, then R is................ reflexive, symmetric but not transitive reflexive but neither symmetric nor transitive an equivalence relation symmetric but neither reflexive nor transitive 3 4 3 1 6. If X , then the matrix X is................. 1 2 5 0 6 5 4 4 0 5 4 2 0 5 4 2 0 5 4 2 H-4754 3 P.T.O.  3 7. The value of | x 3| dx is................ 1 1 –2 2 0 1 8. The value of x4 sin3 x dx is.................. 1 1 0 –1 2 9. Using determinants, show that the points (1, 3), (2, 2) and (0, 4) are collinear. 10. Find the slope of tangent to the curve 2y = 3 – x3 at the point (1, 1). 11. Find the distance between the two planes : x + y + 3z = 4 and 2x + 2y + 6z = 10. H-4754 4 12. Find the area of parallelogram whose adjacent sides are given by the vectors a iˆ jˆ and b 2iˆ 3 kˆ. 13. Find the principal value of cosec 1 ( 2). 14. If 1 A 2 and B 2 0 11 3 4 3 1 then find the matrix (AB)', where (AB)' is the transpose of matrix (AB). 15. The random variable X has the following probability distribution : X P(X) 0 K 1 2K 2 3K 3 4K Find P(X < 2). dy 16. If y ex y2 , then find. dx H-4754 5 P.T.O. 17. Find the angle between the following pairs of lines : r 2iˆ 5 ˆj kˆ (3iˆ 2 ˆj 6kˆ ) and r 7iˆ 6kˆ µ (2iˆ 2 ˆj kˆ ). 18. If E and F are the events of a sample space S, such that P(F) 0 , then prove that : P(E'/F) 1 P(E/F) , where E' is the complement of event E. 19. Form the differential equation of family of curves represented by A( y A)2 x3 , by eliminating arbitrary constant A. 20. Let * be a binary operation defined on set A = {1, 2, 3, 6, 12} as a * b = H.C.F {a, b}. Prepare composition table for the binary operation *. Also, compute 3* (6 * 12). 21. Prove that : 1 1 1 x y tan x tan y tan , xy 1. 1 xy 22. If A(1, –2, 3) and B(–1, –4, 3) are the given points and d 3iˆ 4 ˆj 5kˆ is a given vector, then find the scalar projection of vector AB on d. H-4754 6 23. Using properties of determinants, prove that : a b c 2a 2a 2b b c a 2b ( a b c)3. 2c 2c c a b 24. Find the general solution of the differential equation : dy x2 x2 xy 2 y2. dx 25. One third of the students in a class are boys and the rest are girls. It is known that the probability of a girl getting first class is 0.32 and that a boy getting first class is 0.22. If a student chosen at random gets first class marks in the subject, what is the probability that the chosen student is a boy ? 26. If y ( x )sin 2 x (log 3 x) x 1, dy find. dx 27. Find the equation of the plane passing through the line of intersection of planes r. (2iˆ ˆj kˆ ) 3 and r. (5iˆ 3 ˆj 4 kˆ ) 9 and parallel to the line r (iˆ 3 ˆj 5kˆ ) (2iˆ 4 ˆj 5kˆ ). 28. Find : x 5 dx. x2 3x 7 H-4754 7 P.T.O. 29. Prove that : 2a a f ( x) dx [ f ( x) f (2a x)] dx 0 0 Hence, show that : 2a a f ( x) dx 2 f ( x) dx, if f (2a x) f ( x) 0 0 0 , if f (2a x) f ( x). 30. Using integration, find the area of the smaller region bounded by the ellipse x2 y2 x y 1 and the straight line 1. 16 9 4 3 Or Using integration, find the area of the smaller region enclosed between parabola y2 = 16x and the line x – y + 3 = 0. 31. Solve the following Linear Programming Problem graphically : Minimize Z = 5x + 7y Subject to constraints : 2x + y 8 x + 2y 10 2x + 3y 24 x 0, y 0. H-4754 8 32. If x sin t , y sin( pt) prove that : d2 y dy (1 x 2 ) 2 x p2 y 0 dx dx where t is a parameter. Or If y kx Ae cos( px 4) , prove that : d2 y dy 2 2k ( p2 k2 ) y 0. dx dx 33. If the function f(x) defined by : 1 sin x f ( x) ; x A cos2 x 2 3sin 2 x B ; x 0 2 e5 x e3 x ; 0 x x is continuous on [ , ] , then find the values of A and B. 1 2 5 34. Find the inverse of matrix A 1 1 1. 2 3 1 Hence, solve the system of equations : x + 2y + 5z = 10 x – y – z = –2 2x + 3y – z = –11. H-4754 9 P.T.O. 35. The perimeter of an isosceles triangle is 200 cm. If its base is changing at the rate 5 cm/sec, then find the rate at which the altitude is changing when the base is 40 cm. Or Show that the semivertical angle of the cone of maximum volume and of 1 1 given slant height is cos. 3 36. Find : 2 cos2 x cos x dx. (sin x 2) (sin2 x 3) Or Find : 1 cos x dx. H-4754 10