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12th Maths Sectionwise paper - 02 (Differentiation)

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MATHEMATICS Class II Pre-Uni Plus 2016-17 Topics Board Section-wise Test - 02 Differentiability, Differentiation and Application of derivatives Max. Marks 100 Duration Date 3 hrs 15 min 27-12-2016 Instructions 1. Do not overwrite. 2. Draw a line using pen at the end of each answer. 3. Steps adopted in solving problems should be clearly shown. 4. In case of violation of the above conditions, the answer scripts will not be considered for valuation. PART - A I. Answer the following questions [10 1 = 10] 1. Examine the continuity of the function f(x) = 2x2 1 at x = 3. 2. Find the derivative of y 1 x 3. If y = cos(2cos x). Then find 4. Find 4 3 3 x 3 2 . dy . dx dy if y = sin(xx). dx 5. If x = 4t, y 4 dy y . Then show that . t dx x 6. State Rolle s theorem. 7. Find the rate of change of the area of a circle with respect to radius r when r = 3 cm. 8. Find the slope of the tangent to the curve y = 3x4 4x at x = 4. 9. Show that the function f(x) = 3x + 17 is strictly increasing on R. 10. Find the extreme values of the function f(x) = x2. PART - B II. Answer any TEN of the following questions 2x 3; x 2 11. Find all the points of discontinuity of the function f (x) . 2x 3; x 2 12. Prove that the function f(x) = |x 1|, x R is not differentiable at x = 1. 13. If y x 14. If y 1 dy prove that x 2 xy 2 0 . x dx e x log x dy . Then find . sin 3x dx 2P1617MBSWT2 1 [10 2 = 20] dy 4 3 7x 15. If y tan 1 show that . 2 2 dx 1 16x 1 9x 2 1 12x 16. If f(x) = (1 + x) (1 + x2) (1 + x4) (1 + x8). Then find f (1) . 17. If x = cos 1(2t2 1) and y = sin 1 (4t3 3t) show that 18. If x2 + 2xy + 3y2 = 1. Then prove that dy 3 . dx 2 d2 y 2 . 2 dx (x 3y) 2 19. Prove that tangents to the curve y = x2 5x + 6 at the points (2, 0) and (3, 0) are at right angles. 20. Show that the ellipse 4x2 + 9y2 = 45 and the hyperbola x2 4y2 = 5 cut each other orthogonally at (3, 1). 21. The sides of an equilateral triangle are increasing at the rate of 2 cm/s. How fast is the area increasing when the side is 10 cm? 22. Find the range of values of x, for which the function f(x) = x2 6x + 3 is (i) strictly increasing (ii) strictly decreasing. 23. Show that the function x 1 has a local maxima and local minima and the local maximum value is less x than the local minimum value. 24. Find the maximum profit that a company can make, if the profit function is given by P(x) = 41 + 72x 18x2. PART - C III. Answer any TEN of the following questions [10 3 = 30] 25. Show that the function defined by f(x) = |1 x + |x| | where x is any real number, is a continuous function. 26. Prove that, if a function f(x) is differentiable at x = a then it is continuous at x = a. 27. If y x 2 a2 dy a x 2 log x a 2 x 2 . Then show that a2 x2 . 2 2 dx 28. If x 1 y y 1 x 0 and x y show that dy 1 . dx (1 x) 2 1 x2 1 dy 1 29. If y tan 1 . prove that x dx 2(1 x 2 ) 30. If y cos 1 x , find d2 y in terms of y alone. dx 2 31. Verify Mean Value Theorem for the function f(x) = x2 4x 3 in the interval [1, 4]. 32. The radius of a air bubble is increasing at the rate of increasing when the radius is 1 cm? 2P1617MBSWT2 2 1 cm/s. At what rate is the volume of the bubble 2 33. Find the points on the curve y = x3 at which the slope of the tangent is equal to the x-coordinate of the point. 34. Find the angle of intersection of the curves 2y2 = x3 and y2 = 32x at (8, 16). 35. Using differentials find the approximate value of 661/3. 36. Show that y log(1 x) 2x , x 1 is an increasing function of x throughout its domain. 2 x 37. Find the maximum and minimum values of f(x) = x + sin 2x in the interval [0, 2 ]. 38. Find two positive numbers x and y such that their sum is 35 and the product x2y5 is a maximum. PART - D IV. Answer any SIX of the following questions x 2 if 39. Discuss the continuity of the function f (x) x 2 if [6 5 = 30] x 1 . x 1 1 sin x 1 sin x dy 1 40. If y cot 1 . , x . Then show that dx 2 1 sin x 1 sin x 2 41. If 1 x 4 1 y 4 k(x 2 y 2 ) show that dy x 1 y 4 . dx y 1 x 4 2 x dy 3 d y 42. If y x log show that x x y . 2 dx a bx dx 43. A cone has a depth of 10 cm and the base of 5 cm radius. Water is poured into it at the rate of 3 cm/min . 2 Find the rate at which the level of water in the cone is rising when the depth is 4 cm. 44. Find the equations of tangent and normal to the curve at t . Given x = cos t, y = sin t. 4 45. Find the condition for the curves ax2 + by2 = 1 and Ax2 + By2 = 1 to intersect orthogonally. 46. Determine the value of x for which the function f(x) = sin x cos x, 0 x 2 . (i) increasing (ii) decreasing. 47. Find the points of local maxima or local minima and corresponding values for the function, f(x) = sin 2x, 0 < x < . 48. Show that of all rectangles inscribed in a given fixed circle, the square has maximum area. 2P1617MBSWT2 3 PART - E IV. Answer any ONE of the following questions 49. (a) If y (sin x) tan x (cos x)sec x , then find [1 10 = 10] dy . dx [6] k cos x 2x ; (b) Find the value of k so that the function f (x) 3; 2 is continuous at x . 2 x 2 x [4] 50. (a) A square piece of tin of side 24 cm is to be made into a box without top, by cutting a square from each corner and folding up the flaps to form a box. What should be the side of the square to be cut off so that the volume of the box is maximum? Also find this maximum volume. (b) If y cos n x cos nx show that dy n cos n 1 x sin(n 1)x . dx *** 2P1617MBSWT2 4 [6] [4]

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