Application of Derivatives PYQs Set 7 (PDF)

Set 7 of 10 · Questions 181-210 of 274

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Application of Derivatives · Set 7 (Q181-210)
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181. $The shortest distance between the line x - y = 1 and the curve x^{2} = 2 y is:$ (1) $\frac{1}{2}$ (2) $0$ (3) $\frac{1}{2 2}$ (4) $\frac{1}{2}$ 182. $Let g(x)= f(x)+f(1-x) and f"(x)>0, x \in (0,1). If g is decreasing in the interval (0, α) and increasing in the interval (α, 1), then tan^{- 1} 2 α + tan^{- 1} (\frac{1}{α}) + tan^{- 1} (\frac{α + 1}{α}) is equal to$ (1) $π$ (2) $\frac{5 π}{4}$ (3) $\frac{3 π}{4}$ (4) $\frac{3 π}{2}$ 183. $The set of all real values of λ for which the function f (x) = (1 - cos^{2} x) \cdot (λ + sin x), x \in (- \frac{π}{2}, \frac{π}{2}) has exactly one maxima and exactly one minima, is:$ (1) $(- \frac{3}{2}, \frac{3}{2})$ (2) $(- \frac{1}{2}, \frac{1}{2})$ (3) $(- \frac{1}{2}, \frac{1}{2}) - 0$ (4) $(- \frac{3}{2}, \frac{3}{2}) - 0$ 184. $The number of distinct real roots of x^{4} - 4x +1 = 0 is .$ (1) $4$ (2) $2$ (3) $1$ (4) $0$ 185. $A variable line L passes through the point (3, 5) and intersects the positive coordinate axes at the points A and B. The minimum area of the triangle OAB , where O is the origin, is :$ (1) $30$ (2) $25$ (3) $40$ (4) $35$ 186. $Let f : R \to R be a function defined by f(x) = (x- 3)^{n 1} (x- 5)^{n 2}, n_{1}, n_{2} \in N. The, which of the following is NOT true?$ (1) $For n_{1} = 3, n_{2} = 4, there exists α \in (3,5) where f attains local maxima.$ (2) $For n_{1} = 4, n_{2} = 3, there exists α \in (3,5) where f attains local manima.$ (3) $For n_{1} = 3, n_{2} = 5, there exists α \in (3,5) where f attains local maxima.$ (4) $For n_{1} = 4, n_{2} = 6, there exists α \in (3,5) where f attains local maxima.$ 187. $The maximum area of a rectangle that can be inscribed in a circle of radius 2 units is (in square units)$ (1) $8$ (2) $5$ (3) $4$ (4) $8 π$ 188. $Consider a cuboid of sides 2x, 4x and 5x and a closed hemisphere of radius r. If the sum of their surface areas is a constant k, then the ratio x : r, for which the sum of their volumes is maximum, is :$ (1) $2 : 5$ (2) $19:45$ (3) $3 : 8$ (4) $19 : 15$ 189. $Water is being filled at the rate of 1 c m^{3} / sec in a right circular conical vessel (vertex downwards) of height 35 cm and diameter 14 cm. When the height of the water level is 10 cm, the rate (in cm /sec) at which the wet conical surface area of the vessel increases is$ (1) $5$ (2) $\frac{21}{5}$ (3) $\frac{26}{5}$ (4) $\frac{26}{10}$ 190. $The sum of all the local minimum values of the twice differentiable function f : R \to R defined by f (x) = x^{3} - 3 x^{2} - \frac{3 f ^{''} ( 2 )}{2} x + f^{''} (1) is :$ (1) $-22$ (2) $5$ (3) $-27$ (4) $0$ 191. $Let a > 0 . If the function f (x) = 6 x^{3} - 45 a x^{2} + 108 a^{2} x + 1 attains its local maximum and minimum values at the points x_{1} and x_{2} respectively such that x_{1} x_{2} = 54, then a + x_{1} + x_{2} is equal to :$ (1) $15$ (2) $13$ (3) $24$ (4) $18$ 192. $A rectangular sheet of fixed perimeter with sides having their lengths in the ratio 8 : 15 is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is 100, the resulting box has maximum volume. The lengths of the sides of the rectangular sheet are$ (1) $24$ (2) $32$ (3) $45$ (4) $60$ 193. $The function f(x) = 2\|x\| + \|x + 2\| - \|\|x + 2\| - 2\|x\|\| has a local minimum or a local maximum at x =$ (1) $- 2$ (2) $\frac{-2}{3}$ (3) $2$ (4) $\frac{2}{3}$ 194. $The normal to the curve y(x - 2)(x - 3) = x + 6 at the point where the curve intersects the y-axis passes through the point :$ (1) $(- \frac{1}{2}, - \frac{1}{2})$ (2) $(\frac{1}{2}, \frac{1}{2})$ (3) $(\frac{1}{2}, - \frac{1}{3})$ (4) $(\frac{1}{2}, \frac{1}{3})$ 195. $If y = y(x) is the solution of the differential equation x \frac{d y}{d x} + 2y = x e^{x}, y(l) = 0 then the local maximum value of the function z(x) = x^{2} y(x)- e^{x}, x \in R is :$ (1) $1-e$ (2) $0$ (3) $\frac{1}{2}$ (4) $\frac{4}{e} - e$ 196. $If the absolute maximum value of the function f(x) = e^{(4 x^{3} - 12 x^{2} - 180 x + 31)} (x^{2} - 2x + 7) in the interval [-3, 0] is f(α), then :$ (1) $α = 0$ (2) $α = -3$ (3) $α \in (-1,0)$ (4) $α \in (-3-1)$ 197. $The sum of the absolute maximum and absolute minimum values of the function f(x) = t a n^{- 1} (sin x - cosx) in the interval [0, π] is$ (1) $0$ (2) $tan^{- 1} (\frac{1}{2}) - \frac{π}{4}$ (3) $cos^{- 1} (\frac{1}{3}) - \frac{π}{4}$ (4) $- \frac{π}{12}$ 198. $OA and OB are two roads enclosing an angle of 120^{\circ} . X and Y start from ' O' at the same time. X travels along OA with a speed of 4 km / hour and Y travels along OB with a speed of 3 km / hour. The rate at which the shortest distance between X and Y is increasing after 1 hour is$ (1) $37 km / h$ (2) $37 km / h$ (3) $13 km / h$ (4) $13 km / h$ 199. $The sum of absolute maximum and absolute minimum values of the function f(x) = \| 2 x^{2} + 3x - 2\| + sin x cos x in the interval [0, 1] is :$ (1) $3 + \frac{s i n ( 1 ) c o s ^{2} ( 1/2 )}{2}$ (2) $3 + \frac{1}{2} (1 + 2 cos (1)) sin (1)$ (3) $5+ \frac{1}{2} (sin (1) + sin (2))$ (4) $2 + sin (\frac{1}{2}) cos (\frac{1}{2})$ 200. $The curve y(x) = a x^{3} + b x^{2} + ex + 5 touches the x-axis at the point P(-2, 0) and cuts the y-axis at the point Q, where y' is equal to 3. Then the local maximum value of y(x) is :$ (1) $\frac{27}{4}$ (2) $\frac{29}{4}$ (3) $\frac{37}{4}$ (4) $\frac{9}{2}$ 201. $The number of real solutions of x^{7} + 5 x^{3} + 3x + 1 = 0 is equal to __________.$ (1) $0$ (2) $1$ (3) $3$ (4) $5$ 202. $Let M and N be the number of points on the curve y^{5} - 9xy + 2x = 0, where the tangents to the curve are parallel to x-axis and y-axis, respectively. Then the value of M + N equals________.$ 203. $Let the set of all positive values of λ, for which the point of local minimum of the function (1 + x (λ^{2} - x^{2})) satisfies \frac{x ^{2} + x + 2}{x ^{2} + 5 x + 6} < 0, be (α, β) . Then α^{2} + β^{2} is equal to______$ 204. $Let f(x) = \|(x-1)(x^{2} -2x-3)\| + x-3, x \in R. If m and M are respectively the number of points of local minimum and local maximum of f in the interval (0,4), then m + M is equal to____$ 205. $If the set of all values of a, for which the equation 5 x^{3} - 15 x - a = 0 has three distinct real roots, is the interval (α, β), then β - 2 α is equal to .$ 206. $Let l be a line which is normal to the curve y = 2 x^{2} + x + 2 at a point P on the curve. If the point Q(6, 4) lies on the line l and O is origin, then the area of the triangle OPQ is equal to_____.$ 207. $Let A be the region enclosed by the parabola y^{2} = 2 x and the line x = 24 . Then the maximum area of the rectangle inscribed in the region A is_______$ 208. $The number of points, where the curve y = x^{5} - 20 x^{3} + 50x + 2 crosses the x-axis, is_____.$ 209. $The slope of the tangent to the curve (y - x5)2 = x(1 + x2)2 at the point (1, 3) is$ 210. $Let the quadratic curve passing through the point (-1,0) and touching the line y = x at (1,1) be y = f(x). Then the x-intercept of the normal to the curve at the point (α, α + 1) in the first quadrant is$