Application of Derivatives PYQs Set 2 (PDF)

Set 2 of 10 · Questions 31-60 of 274

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Application of Derivatives · Set 2 (Q31-60)
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31. $At x = 0, f (x) = cos x - 1 + \frac{x ^{2}}{2} - \frac{x ^{3}}{3}$ (1) $has a minimum value$ (2) $has a maximum value$ (3) $has no extremum value$ (4) $is not defined$ 32. $The slope of the tangent to the curve x = t^{2} + 3 t - 8, y = 2 t^{2} - 2 t - 5 at the point (2, - 1) is$ (1) $\frac{22}{7}$ (2) $\frac{6}{7}$ (3) $\frac{7}{6}$ (4) $\frac{- 6}{7}$ 33. $The equation of normal to the curve y = log_{e} x at the point P (1, 0) is$ (1) $2 x + y = 2$ (2) $x - 2 y = 1$ (3) $x - y = 1$ (4) $x + y = 1$ 34. $The number of critical points of the function f (x) = (x - 2)^{2/3} (2 x + 1) is$ (1) $1$ (2) $2$ (3) $0$ (4) $3$ 35. $The co-ordinates of the point on the x + y = 6 at which the tangent is equally inclined to the axes is$ (1) $(4, 4)$ (2) $(9, 9)$ (3) $(1, 1)$ (4) $(6, 6)$ 36. $Define f (x) = \frac{1}{2} [\| sin x \| + sin x], 0 < x \leq 2 π . Then, f is$ (1) $increasing in (\frac{π}{2}, \frac{3 π}{2})$ (2) $decreasing in (0, \frac{π}{2}) and increasing in (\frac{π}{2}, π)$ (3) $increasing in (0, \frac{π}{2}) and decreasing in (\frac{π}{2}, π)$ (4) $increasing in (0, \frac{π}{4}) and decreasing in (\frac{π}{4}, π)$ 37. $The volume of sphere is increasing at the rate of 1200 cucm / s . The rate of increase in its surface area when the radius is 10 cm is$ (1) $120 sq cm / s$ (2) $240 sq cm / s$ (3) $200 sq cm / s$ (4) $100 sq cm / s$ 38. $The slope of the tangent of the curve (\frac{x}{31})^{n} + (\frac{y}{1209})^{n} = 2 at (31,1209)..............$ (1) $-39$ (2) $39$ (3) $\frac{1}{39}$ (4) $\frac{- 1}{39}$ 39. $The interval in which the function f (x) = x^{x}, x > 0, is strictly increasing is$ (1) $(0, \frac{1}{e})$ (2) $(0, \infty)$ (3) $((\frac{1}{e}, \infty))_{V}$ (4) $(\frac{1}{e ^{2}}, 1)$ 40. $Let the curve be represented by x = 2 (cos t + t sin t), y = 2 (sin t - t cos t) . Then normal at any point ' f' of the curve is at a distance of units from the origin.$ (1) $1$ (2) $0$ (3) $2$ (4) $4$ 41. $Consider the following three statements for the function f : (0, \infty) \to R defined by f (x) = ∣ lo g_{e} x ∣ - ∣ x - 1∣ : (I) f is differentiable at all x > 0 . (II) f is increasing in (0, 1) . (III) f is decreasing in (1, \infty) . Then.$ (1) $All (I), (II) and (III) are TRUE.$ (2) $Only (II) and (III) are TRUE.$ (3) $Only (I) is TRUE.$ (4) $Only (I) and (III) are TRUE.$ 42. $The slope of the tangent to the curve f (x) = tan^{- 1} (sin x) at x = π is$ (1) $1$ (2) $0$ (3) $-1$ (4) $-2$ 43. $If the function f (x) = (\frac{1}{x})^{2 x}; x > 0 attains the maximum value at x = \frac{1}{e} then :$ (1) $e^{π} < π^{e}$ (2) $e^{π} > π^{e}$ (3) $(2 e)^{π} > π^{(2 e)}$ (4) $e^{2 π} < (2 π)^{e}$ 44. $If y = 4 x - 5 is a tangent to the curve y^{2} = p x^{3} + q at (2, 3), then p - q is$ (1) $-5$ (2) $5$ (3) $9$ (4) $-9$ 45. $Let f (x) = x cos^{- 1} (- sin \| x \|), x \in [- \frac{π}{2}, \frac{π}{2}] then which of the following is true?$ (1) $f^{’} (0) = - \frac{π}{2}$ (2) $f^{’} (x) is not defined at x = 0$ (3) $f'(x) in increasing in (\frac{- π}{2}, 0) and f'(x) in decreasing in (0, \frac{π}{2})$ (4) $f'(x) in decreasing in (\frac{- π}{2}, 0) and f'(x) in increasing in (0, \frac{π}{2})$ 46. $If the function f : [- 1, 1] \to R defined by f (x) = (2^{x} + 1, for x \in [- 1, 0) 1, for x = 0 2^{x} - 1, for x \in (0, 1]) then, in [- 1, 1], f (x) has$ (1) $a maximum$ (2) $a minimum$ (3) $both maximum and minimum$ (4) $neither maximum nor minimum$ 47. $The point on the curve y^{2} = x, the tangent at which makes an angle 45^{\circ} with X -axis is$ (1) $(\frac{1}{4}, \frac{1}{2})$ (2) $(\frac{1}{2}, \frac{1}{4})$ (3) $(\frac{1}{2}, \frac{- 1}{2})$ (4) $(\frac{1}{2}, \frac{1}{2})$ 48. $A spherical balloon is filled with 4500 π cubic metres of helium gas. If a leak in the balloon causes the gas to escape at the rate of 72 π cubic metres per minute, then the rate (in metres per minute) at which the radius of the balloon decreases 49 minutes after the leakage began is$ (1) $2/9$ (2) $9/2$ (3) $9/7$ (4) $7/9$ 49. $The locus of the point on the curve y = sin x where the tangent drawn at that point always passes through the point (0, π) is$ (1) $x = y - π$ (2) $sin x + cos y + 1 = 0$ (3) $x^{2} (1 - y^{2}) = (y - π)^{2}$ (4) $x^{2} + (y - π)^{2} = 0$ 50. $A particle moves along the curve \frac{x ^{2}}{16} + \frac{y ^{2}}{4} = 1 . When the rate of change of abscissa is 4 times that of its ordinate, then the quadrant in which the particle lies is$ (1) $III or IV$ (2) $I or III$ (3) $II or III$ (4) $II or IV$ 51. $The slope of the normal to the curve y = \frac{x}{x ^{2} + 1} at x = - 4 is$ (1) $\frac{- 289}{15}$ (2) $\frac{- 15}{16}$ (3) $\frac{289}{15}$ (4) $\frac{15}{16}$ 52. $Which of the following points lies on the tangent to the curve x^{4} e^{y} + 2 y + 1 = 3 at the point (1, 0) ?$ (1) $(2, 2)$ (2) $(- 2, 4)$ (3) $(2, 6)$ (4) $(- 2, 6)$ 53. $If the line y = - 4 x + b is tangent to the curve y = \frac{1}{x}, then b equals$ (1) $\pm 4$ (2) $\pm 2$ (3) $\pm 1$ (4) $\pm 8$ 54. $In a bank, the principal increases continuously at the rate of 6 % per year. Then the time required to double ₹ 6000 is ........... (in years)$ (1) $\frac{50}{3} log 2$ (2) $\frac{50}{3} log 6$ (3) $\frac{50}{3} log 3$ (4) $\frac{50}{3} log 12$ 55. $Let S be the set of all function f : [0, 1] \to R which are continuous on [0, 1] and differentiable on (0, 1) . Then for every f in S, there exists a c \in (0, 1) depending on f, such that:$ (1) $\| f (c) - f (1) \| < (1 - c) ∣ f^{’} (c)$ (2) $\| f (c) - f (1) \| < \| f^{’} (c) \|$ (3) $\frac{f ( 1 ) - f ( c )}{1 - c} = f^{’} (c)$ (4) $None of these$ 56. $For which of the following curves, the line x + 3 y = 23 is the tangent at the point (\frac{3 3}{2}, \frac{1}{2}) ?$ (1) $x^{2} + 9 y^{2} = 9$ (2) $2 x^{2} - 18 y^{2} = 9$ (3) $y^{2} = \frac{1}{6 3} x$ (4) $x^{2} + y^{2} = 7$ 57. $Find the positive value of a for which the equality 2 α + β = 8 holds, where α and β are the points of maximum and minimum, respectively, of the function f (x) = 2 x^{3} - 9 a x^{2} + 12 a^{2} x + 1$ (1) $0$ (2) $2$ (3) $1$ (4) $\frac{1}{4}$ 58. $Let (2 α, α) be the largest interval in which the function f (t) = \frac{\| t + 1 \|}{t ^{2}}, t < 0, is strictly decreasing. Then the local maximum value of the function g (x) = 2 log_{e} (x - 2) + α x^{2} + 4 x - α, x > 2, is____.$ 59. $Let a_{1}, a_{2} \dots a_{3} be roots of the equation x^{7} + 3 x^{5} - 13 x^{3} - 15 x = 0 and (a_{1}) \geq (a_{2}) \geq \dots (a_{7}) Then, a_{1} a_{2} - a_{3} a_{4} + a_{5} a_{6} is equal to __________$ 60. $Let f : R \to R be a differentiable function such that f^{’} (x) + f (x) = \sum f (t) dt . If f (0) = e^{- 2}, then 2 f (0) - f (2) is equal _______$