Application of Derivatives PYQs Set 9 (PDF)

Set 9 of 10 · Questions 241-270 of 274

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Application of Derivatives · Set 9 (Q241-270)
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241. $The diagonal of a square is changing at the rate of 0.5 cm / sec . Then the rate of change of area when the area is 400 cm^{2} is equal to$ (1) $202 cm^{2} / sec$ (2) $102 cm^{2} / sec$ (3) $\frac{1}{10 2} cm^{2} / sec$ (4) $\frac{10}{2} cm^{2} / sec$ 242. $The function f (x) = x^{2} + 2 x - 5 is strictly increasing in the interval$ (1) $[- 1, \infty)$ (2) $(- \infty, - 1)$ (3) $(- \infty, - 1]$ (4) $(- 1, \infty)$ 243. $Let the function f (x) = 2 x^{3} + (2 p - 7) x^{2} + 3 (2 p - 9) x - 6 have a maxima for some value of x < 0 and a minima for some value of x > 0 . Then, the set of all values of p is$ (1) $(\frac{9}{2}, \infty)$ (2) $(0, \frac{9}{2})$ (3) $(- \infty, \frac{9}{2})$ (4) $(- \frac{9}{2}, \frac{9}{2})$ 244. $The number of points on the curve y = 54 x^{5} - 135 x^{4} - 70 x^{3} + 180 x^{2} + 210 x at which the normal lines are parallel to x + 90 y + 2 = 0 is$ (1) $2$ (2) $3$ (3) $4$ (4) $0$ 245. $Let x = 2 be a local minimum of the function f (x) = 2 x^{4} - 18 x^{2} + 8 x + 12, x \in (- 4, 4) . If M is the local maximum value of the function f in (- 4, 4), then M =$ (1) $126 - \frac{33}{2}$ (2) $126 - \frac{31}{2}$ (3) $186 - \frac{33}{2}$ (4) $186 - \frac{31}{2}$ 246. $Let f (x) = 1 + sin^{2} x sin^{2} x sin^{2} x cos^{2} x 1 + cos^{2} x cos^{2} x sin 2 x sin 2 x 1 + sin 2 x, x \in (\frac{π}{6}, \frac{π}{3}) . If α and β respectively are the maximum and the minimum values of f, then$ (1) $β^{2} - 2 α = \frac{19}{4}$ (2) $β^{2} + 2 α = \frac{19}{4}$ (3) $α^{2} - β^{2} = 43$ (4) $α2-β2=53$ 247. $The time period T of a simple pendulum of length l is given by T = 2 π \frac{l}{g}, where g denotes the acceleration due to gravity. If the length of the pendulum is increased by 1%, then the approximate change in its time period is$ (1) $0.5 %$ (2) $2 %$ (3) $1 %$ (4) $4 %$ 248. $The length of the subtangent at '' on the curve x = a (t + sin t), y = a (1 - cos t) is$ (1) $2 a sin \frac{t}{2}$ (2) $2 a sin^{3} (\frac{t}{2}) sec (\frac{t}{2})$ (3) $a sin t$ (4) $2 a sin (\frac{t}{2}) tan (\frac{t}{2})$ 249. $Let f : R \to R be a twice differentiable function such that f^{’ ’} (x) > 0 for all x \in R and f^{’} (a - 1) = 0, where a is a real number. Let g (x) = f (tan^{2} x - 2 tanx + a), 0 < x < \frac{π}{2} . Consider the following two statements: (I) g is increasing in (0, \frac{π}{4}) (II) g is decreasing in (\frac{π}{4}, \frac{π}{2}) . Then,$ (1) $Neither (I) nor (II) is True$ (2) $Only (I) is True$ (3) $Both (I) and (II) are True$ (4) $Only (II) is True$ 250. $The curve represented by x = t^{5} + 5 t^{3} + 20 t + 7 and y = 4 t^{3} - 3 t^{2} - 18 t + 3 is decreasing in the interval$ (1) $(- 2, - 1)$ (2) $(3/2, 2)$ (3) $(- 1, \frac{3}{2})$ 251. $The sum of the absolute maximum and minimum values of the function f (x) = ∣ x^{2} - 5 x + 6∣ - 3 x + 2 in the interval [- 1, 3] is equal to$ (1) $10$ (2) $12$ (3) $13$ (4) $24$ 252. $The number of distinct real roots of the equation x^{7} - 7 x - 2 = 0 is$ (1) $5$ (2) $7$ (3) $1$ (4) $3$ 253. $Let x = - 1 and x = 2 be the critical points of the function f (x) = x^{3} + a x^{2} + b lo g_{e} ∣ x ∣ + 1, x \neq = 0 . Let m and M respectively be the absolute minimum and the absolute maximum values of f in the interval [- 2, - \frac{1}{2}] . Then ∣ M + m ∣ is equal to (Take lo g_{e} 2 = 0.7) :$ (1) $21.1$ (2) $19.8$ (3) $22.1$ (4) $20.9$ 254. $If f(x) = cos (2 x) - cos x sin x cos (2 x) cos x sin x sin (2 x) - sin x cos x, then$ (1) $f (x) attains its minimum at x = 0$ (2) $f (x) attains its maximum at x = 0$ (3) $f^{'} (x) = 0 at more than three points in (-π, π)$ (4) $f^{'} (x) = 0 at exactly three points in (-π, π)$ 255. $I = \int_{π /4}^{π /3} (\frac{8 s i n x - s i n 2 x}{x}) d x^{.} Then$ (1) $\frac{π}{2} < I < \frac{3 π}{4}$ (2) $\frac{π}{5} < I < \frac{5 π}{12}$ (3) $\frac{5 π}{12} < I < \frac{2}{3} π$ (4) $\frac{3 π}{4} < I < π$ 256. $If f(x) = cos (2 x) - cos x sin x cos (2 x) cos x sin x sin (2 x) - sin x cos x, then [$ (1) $f (x) attains its minimum at x = 0$ (2) $f (x) attains its maximum at x = 0$ (3) $f^{'} (x) = 0 at more than three points in (-π, π)$ (4) $f^{'} (x) = 0 at exactly three points in (-π, π)$ 257. $max_{0 \leq x \leq π} {x - 2 sin x cos x + \frac{1}{3} sin 3 x} =$ (1) $\frac{π + 2 - 3 3}{6}$ (2) $π$ (3) $0$ (4) $\frac{5 π + 2 + 3 3}{6}$ 258. $If m and n respectively are the number of local maximum and local minimum points of the function f (x) = \int_{0}^{x^{2}} \frac{t ^{2} - 5 t + 4}{2 + e ^{t}} d t, then the ordered pair (m, n) is equal to$ (1) $(3, 2)$ (2) $(2, 3)$ (3) $(2, 2)$ (4) $(3, 4)$ 259. $A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is :$ (1) $\frac{22}{9 + 4 3}$ (2) $\frac{66}{9 + 4 3}$ (3) $\frac{22}{4 + 9 3}$ (4) $\frac{66}{4 + 9 3}$ 260. $The function f (x) = x e^{x (1 - x)}, x \in R, is$ (1) $increasing in (- \frac{1}{2}, 1)$ (2) $decreasing in (\frac{1}{2}, 1)$ (3) $increasing in (- 1, - \frac{1}{2})$ (4) $decreasing in (- \frac{1}{2}, \frac{1}{2})$ 261. $A container is the shape of an inverted cone. Its height is 6 m and radius is 4 m at the top. If it is filled with water at the rate of 3 m^{3} / min then the rate of change of height of water (in mt / min) when the water level is 3 m, is$ (1) $\frac{3}{4 π}$ (2) $\frac{2}{9 π}$ (3) $16 π$ (4) $2 π$ 262. $Let A (1, 2), B (4, - 4), C (2, 22) be points on the parabola y^{2} = 4 x . If α and β respectively represent the area of △ ABC and the area of the triangle formed by the tangents at A, B, C to the above parabola, then αβ =$ (1) $6$ (2) $32$ (3) $9$ (4) $62$ 263. $Let f (x) be a cubic polynomial with f (1) = - 10, f (- 1) = 6, and has a local minima at x = 1, and f^{'} (x) has a local minima at x = - 1 . Then f (3) is equal to ____.$ 264. $Let P (x) be a real polynomial of degree 3 which vanishes at x = - 3 . Let P (x) have local minima at x = 1, local maxima at x = - 1 and \int_{- 1}^{1} P (x) d x = 18, then the sum of all the coefficients of the polynomial P (x) is equal to_____$ 265. $If the normal to the curve y (x) = \int_{0}^{x} (2 t^{2} - 15 t + 10) d t at a point (a, b) is parallel to the line x + 3 y = - 5, a > 1, then the value of ∣ a + 6 b ∣ is equal to____$ 266. $If f (x) = x^{2} + g^{'} (1) x + g^{''} (2) and g (x) = f (1) x^{2} + x f^{'} (x) + f^{''} (x), then the value of f (4) - g (4) is equal to ________$ 267. $Let f : R \to R be a twice differentiable function such that the quadratic equation f (x) m^{2} - 2 f^{’} (x) m + f^{’ ’} (x) = 0 in m , has two equal roots for every x \in R . If f (0) = 1, f^{’} (0) = 2, and (α, β) is the largest interval in which the function f (log_{e} x - x) is increasing, then α + β is equal to____.$ 268. $Let f be a differentiable function satisfying f (x) = 1 - 2 x + \int_{0}^{x} e^{(x - t)} f (t) dt, x \in R and let g (x) = \int_{0}^{x} (f (t) + 2)^{15} (t - 4)^{6} (t + 12)^{17} dt, x \in R . If p and q are respectively the points of local minima and local maxima of g, then the value of \| p + q \| is equal to____.$ 269. $Let f be a function defined on R (the set of all real numbers) such that f^{'} (x) = 2010 (x - 2009) (x - 2010)^{2} (x - 2011)^{3} (x - 2012)^{4}, for all x \in R . If g is a function defined on R with values in the interval (0, \infty) such that f (x) = ln (g (x)), for all x \in R, then the number of points in R at which g has a local maximum is$ 270. $Let the area enclosed by the x-axis, and the tangent and normal drawn to the curve 4 x^{3} - 3 x y^{2} + 6 x^{2} - 5xy - 8 y^{2} + 9x + 14 = 0 at the point (-2, 3) be A. Then 8A is equal to_____$