Application of Derivatives PYQs Set 8 (PDF)

Set 8 of 10 · Questions 211-240 of 274

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Application of Derivatives · Set 8 (Q211-240)
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211. $Let f(x) = 2 co s^{- 1} x + 4 co t^{- 1} x - 3 x^{2} - 2x + 10, x \in [-1, 1]. If [a, b] is the range of the function then 4a -b is equal to:$ (1) $11$ (2) $11- π$ (3) $11+ π$ (4) $15- π$ 212. $Let f (x) be a polynomial function such that f(x) + f'(x) + f "(x) = x^{5} + 64. Then, the value of lim_{x \to 1} \frac{f ( x )}{x - 1}$ (1) $-15$ (2) $- 60$ (3) $60$ (4) $15$ 213. $The sum of all local minimum values of the function f (x) = ⎩ ⎨ ⎧ 1 - 2 x, \frac{1}{3} (7 + 2∣ x ∣), \frac{11}{18} (x - 4) (x - 5), x < - 1 - 1 \leq x \leq 2 x > 2 is$ (1) $\frac{157}{72}$ (2) $\frac{131}{72}$ (3) $\frac{171}{72}$ (4) $\frac{167}{72}$ 214. $Let f (x) = 4 cos^{3} x + 33 cos^{2} x - 10 . The number of points of local maxima of f in interval (0, 2 π) is$ (1) $3$ (2) $4$ (3) $1$ (4) $2$ 215. $Let f : R \to (0, \infty) and g : R \to R be twice differentiable functions such that f " and g" are continuous functions on R. Suppose f '(2) = g(2) = 0, f "(2) \neq 0 and g'(2) \neq 0, If lim_{x \to 2} \frac{f ( x ) g ( x )}{f ^{'} ( x ) g ^{'} ( x )} = 1, then$ (1) $f has a local minimum at x = 2$ (2) $f has a local maximum at x = 2$ (3) $f "(2) > f(2)$ (4) $f(x) - f "(x) = 0 for at least one x \in R$ 216. $the tangent at the point (x_{1}, y_{1}) on the curve y = x^{3} + 3 x^{2} + 5 passes through the origin, then (x_{1}, y_{1}) does NOT lie on the curve :$ (1) $x^{2} + \frac{y ^{2}}{81} = 2$ (2) $\frac{y ^{2}}{9} - x^{2} = 8$ (3) $y = 4 x^{2} + 5$ (4) $\frac{x}{3} - y^{2} = 2$ 217. $If f : R \to R is a differentiable function such that f'(x) > 2f(x) for all x \in R, and f(0) = 1, the$ (1) $f(x) > e2x in (0, \infty)$ (2) $f'(x) < e2x in (0, \infty)$ (3) $f(x) is increasing in (0, \infty)$ (4) $f(x) is decreasing in (0, \infty)$ 218. $Let a tangent be drawn to the ellipse \frac{x ^{2}}{27} + y^{2} = 1 at (33 cos θ, sin θ) where θ \in (0, \frac{π}{2}) . Then the value of θ such that the sum of intercepts on axes made by this tangent is minimum is equal to:$ (1) $\frac{π}{8}$ (2) $\frac{π}{4}$ (3) $\frac{π}{6}$ (4) $\frac{π}{3}$ 219. $If f : R \to R is a differentiable function such that f'(x) > 2f(x) for all x \in R, and f(0) = 1, then$ (1) $f(x) > e2x in (0, \infty)$ (2) $f'(x) < e2x in (0, \infty)$ (3) $f(x) is increasing in (0, \infty)$ (4) $f(x) is decreasing in (0, \infty)$ 220. $Let f_{(} x_{)} = 3 s i n^{4} x + 10 s i n^{3} x + 6 s i n^{2} x - 3, x \in [- \frac{π}{6}, \frac{π}{2}] . Then, f is:$ (1) $increasing in$ (2) $decreasing in$ (3) $increasing in$ (4) $decreasing in$ 221. $Let a rectangle ABCD of sides 2 and 4 be inscribed in another rectangle PQRS such that the vertices of the rectangle ABCD lie on the sides of the rectangle PQRS . Let a and b be the sides of the rectangle PQRS when its area is maximum. Then (a + b)^{2} is equal to :$ (1) $72$ (2) $60$ (3) $64$ (4) $80$ 222. $Let f : R \to R be a function defined by f (x) = \| \| x + 2 \| - 2 \| x \| \| . If m is the number of points of local minima and n is the number of points of local maxima of f, then m + n is$ (1) $3$ (2) $4$ (3) $2$ (4) $5$ 223. $Let a, b \in R be such that the function f given by f(x) = λn \|x\| + bx2 + ax, x \neq 0 has extreme values at x = - 1 and x = 2. Statement-1 : f has local maximum at x = - 1 and at x = 2. Statement-2 : a = \frac{1}{2} and b = \frac{- 1}{4} .$ (1) $Statement-1 is false, Statement-2 is true.$ (2) $Statement-1 is true, statement-2 is true; statement-2 is a correct explanation for Statement-1.$ (3) $Statement-1 is true, statement-2 is true; statement-2 is not a correct explanation for Statement-1.$ (4) $Statement-1 is true, statement-2 is false.$ 224. $A square piece of tin of side 30 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. If the volume of the box is maximum, then its surface area (in c m^{2}) is equal to$ (1) $800$ (2) $675$ (3) $1025$ (4) $900$ 225. $Let B \equiv (0, 3) and C \equiv (4, 0) . The point A is moving on the line y = 2 x at the rate of 2 units/second. The area of △ ABC is increasing at the rate of$ (1) $\frac{11}{5} (units)^{2} / sec$ (2) $\frac{11}{5} (units)^{2} / sec$ (3) $\frac{13}{5} (units))^{2} / sec$ (4) $\frac{13}{5} (units)^{2} / sec$ 226. $Let a function f : R \to R be defined as : f (x) = {\int_{0}^{x} (5 - ∣ t - 3 ∣) d t, x^{2} + b x, x > 4 x \leq 4 where b \in R . If / is continuous at x = 4, then which of the following statements is NOT true ?$ (1) $f is not differentiable at x = 4$ (2) $f'(3) + f'(5) = \frac{35}{4}$ (3) $f is increasing in$ (4) $f has a local minima at x = \frac{1}{8}$ 227. $A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm, the ice-cream melts at the rate of 81 cm^{3} / min and the thickness of the ice-cream layer decreases at the rate of \frac{1}{4 π} cm / min . The surface area (in cm^{2}) of the chocolate ball (without the ice-cream layer) is:$ (1) $196 π$ (2) $256 π$ (3) $225 π$ (4) $128 π$ 228. $The sum of the absolute minimum and the absolute maximum values of the function f(x) = \|3x - x^{2} + 2\| - x in the interval [-1,2] is:$ (1) $\frac{17 + 3}{2}$ (2) $\frac{17 + 5}{2}$ (3) $5$ (4) $\frac{9 - 17}{2}$ 229. $The lengths of the sides of a triangle are 10 + x^{2}, 10 + x^{2} and 20 - 2 x^{2} . If for x = k, the area of the triangle is maximum, then 3 k^{2} is equal to :$ (1) $5$ (2) $8$ (3) $10$ (4) $12$ 230. $Let f be a function defined by - f(x) = {\frac{t a n x}{x}, 1, x \neq = 0 x = 0 Statement - 1 : x = 0 is point of minima of f Statement - 2 : f '(0) = 0.$ (1) $Statement-1 is true, statement-2 is true; statement-2 is a correct explanation for statement-1.$ (2) $Statement-1 is true, statement-2 is true; statement-2 is not a correct explanation for statement-1$ (3) $Statement-1 is true, statement-2 is false.$ (4) $Statement-1 is false, statement-2 is true.$ 231. $The function f (x) = 2 x + 3 (x)^{\frac{2}{3}}, x \in R, has$ (1) $exactly one point of local minima and no point of local maxima$ (2) $exactly one point of local maxima and no point of local minima$ (3) $exactly one point of local maxima and exactly one point of local minima$ (4) $exactly two points of local maxima and exactly one point of local minima$ 232. $Let S be the set of all the natural numbers, for which the line \frac{x}{a} + \frac{y}{b} = 2 is a tangent to the curve (\frac{x}{a})^{n} + (\frac{y}{b})^{n} = 2 a. at the point (a, b), ab \neq = 0. Then:$ (1) $S = φ$ (2) $n(S) = 1$ (3) $S={2k:k \in N}$ (4) $S = N$ 233. $If the minimum value of f (x) = \frac{5 x ^{2}}{2} + \frac{α}{x ^{5}}, x > 0, is 14, then the value of α is equal to :$ (1) $32$ (2) $64$ (3) $128$ (4) $256$ 234. $f(x) = 4 l o g_{e} (x - 1) - 2 x^{2} + 4x +5, x > 1, which one of the following is NOT correct ?$ (1) $f is increasing in (1, 2) and decreasing in (2, \infty)$ (2) $f(x)= -1 has exactly two solutions$ (3) $f (e) -f'' (2) < 0$ (4) $f(x) = 0 has a root in the interval (e, e +1)$ 235. $Let the maximum and minimum values of (8 x - x^{2} - 12 - 4)^{2} + (x - 7)^{2}, x \in R be M and m , respectively. Then M^{2} - m^{2} is equal to_____$ 236. $A water tank has the shape of a right circular cone with axis vertical and vertex downwards. Its semi- vertical angle is t a n^{- 1} \frac{3}{4} . Water is poured in it at a constant rate of 6 cubic meter per hour. The rate (in square meter per hour), at which the wet curved surface area of the tank is increasing, when the depth of water in the tank is 4 meters, is________.$ 237. $Let a curve y = f (x), x \in (0, \infty) pass through the points P (1, \frac{3}{2}) and Q (α, \frac{1}{2}) . If the tangent at any point R (b, f (b)) to the given curve cuts the y-axis at the point S (0, c) such that b c = 3, then (P Q)^{2} is equal to_____.$ 238. $Let A (4, - 2), B (1, 1) and C (9, - 3) be the vertices of a triangle ABC. Then the maximum area of the parallelogram AFDE, formed with vertices D, E and F on the sides BC, CA and AB of the triangle ABC respectively, is _______.$ 239. $Let f : R \to R be a thrice differentiable function such that f (0) = 0, f (1) = 1, f (2) = - 1, f (3) = 2 and f (4) = - 2 . Then, the minimum number of zeros of (3 f^{’} f^{’ ’} + f f^{’ ’ ’}) (x) is______$ 240. $If the tangent to the curve y = x^{3} - x^{2} + x at the point (a, b) is also tangent to the curve y = 5 x^{2} + 2x - 25 at the point (2, -1), then \|2a + 9b\| is equal to ___________ .$