Application of Derivatives PYQs Set 6 (PDF)

Set 6 of 10 · Questions 151-180 of 274

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Application of Derivatives · Set 6 (Q151-180)
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151. $A spherical raindrop evaporates at a rate proportional to its surface area. If originally its radius is 3 mm and 1 hour later it reduces to 2 mm, then the expression for the radius R of the raindrop at any time t is$ (1) $6 R = t + 2$ (2) $R (t + 2) = 6$ (3) $R = 6 (t + 2)$ (4) $none of these$ 152. $The function f (x) = 4 sin^{3} x - 6 sin^{2} x + 12 sin x +100 strictly$ (1) $decreasing in (\frac{- π}{2}, \frac{π}{2})$ (2) $increasing in (π, \frac{3 π}{2})$ (3) $decreasing in (0, \frac{π}{2})$ (4) $decreasing in (\frac{π}{2}, π)$ 153. $Let x_{0} be the point of local minima of f (x) = a \cdot (b \times c) where a = x i - 2 j + 3 k, b = - 2 i + x j - k, c = 7 i - 2 j + x k, then the value of a \cdot b at x = x_{0} is$ (1) $15$ (2) $-15$ (3) $12$ (4) $-12$ 154. $Let f : R \to R be defined as f (x) = ⎩ ⎨ ⎧ - 55 x, 2 x^{3} - 3 x^{2} - 120 x, 2 x^{3} - 3 x^{2} - 36 x - 336, if x < - 5 if - 5 \leq x \leq 4 if x > 4 Let A = {x \in R : f is increasing} . Then A is equal to :$ (1) $(- 5, - 4) \cup (4, \infty)$ (2) $(- 5, \infty)$ (3) $(- \infty, - 5) \cup (4, \infty)$ (4) $(- \infty, - 5) \cup (- 4, \infty)$ 155. $The shortest distance between the curves y^{2} = 8 x and x^{2} + y^{2} + 12 y + 35 = 0 is:$ (1) $23 - 1$ (2) $22 - 1$ (3) $32 - 1$ (4) $2$ 156. $The equation of the normal to the curve y = (1 + x)^{2 y} + cos^{2} (sin^{- 1} x) at x = 0 is:$ (1) $y = 4 x + 2$ (2) $y + 4 x = 2$ (3) $x + 4 y = 8$ (4) $2 y + x = 4$ 157. $The tangent to a given curve y = f (x) is perpendicular to the x -axis if$ (1) $\frac{dx}{dy} = 1$ (2) $\frac{dy}{dx} = 0$ (3) $\frac{dy}{dx} = 1$ (4) $\frac{dx}{dy} = 0$ 158. $If y = 2 x is a tangent to the curve y^{2} = a x^{3} + b at (1, 2), then (a, b) =$ (1) $(8, 4)$ (2) $(\frac{2}{3}, 1)$ (3) $(\frac{8}{3}, \frac{4}{3})$ (4) $(\frac{8}{3}, \frac{2}{3})$ 159. $If the tangent to the curve y = x + sin y at a point (a, b) is parallel to the line joining (0, \frac{3}{2}) an (\frac{1}{2}, 2) then:$ (1) $b = a$ (2) $b = \frac{π}{2} + a$ (3) $\| a + b \| = 1$ (4) $\| b - a \| = 1$ 160. $The tangent and the normal drawn to the curve y = x^{2} - x + 4 at P (1, 4) cut the X -axis at A and B respectively. If the length of the subtangent drawn to the curve at P is equal to the length of the subnormal, then the area of the triangle PAB in sq. units is$ (1) $4$ (2) $32$ (3) $8$ (4) $16$ 161. $The equation of the normal at t = \frac{π}{2} to the curve x = 2 sin t, y = 2 cost is$ (1) $x = 2$ (2) $y = 0$ (3) $y = 2 x + 3$ (4) $y = 3$ 162. $If the curves y = x^{3} - 3 x^{2} - 8 x - 4 and y = 3 x^{2} + 7 x + 4 touch each other at a point P then the equation of common tangent at P is$ (1) $x - y + 1 = 0$ (2) $2 x - y + 1 = 0$ (3) $x + y + 1 = 0$ (4) $2 x + y + 1 = 0$ 163. $The equation of the tangent to the curve x^{2} + y - 7 = 4 x at the point (1, 10) is$ (1) $y = 2 x + 8$ (2) $y = x + 8$ (3) $y = - 2 x - 14$ (4) $y = x - 4$ 164. $At present, a firm is manufacturing 2000 items. It is estimated that the rate of change of production P w.r.t. additional number of workers x is given by \frac{dP}{dx} = 100 - 12 x . If the firm employs 25 more workers, then the new level of production of items is$ (1) $3000$ (2) $3500$ (3) $4500$ (4) $2500$ 165. $A sphere of 10 cm radius has a uniform thickness of ice around it. Ice is melting at rate 50 cm^{3} / min when thickness is 5 cm then rate of change of thickness$ (1) $\frac{1}{36 π}$ (2) $\frac{1}{18 π}$ (3) $\frac{1}{9 π}$ (4) $\frac{1}{12 π}$ 166. $The normal to the curve y = f (x) at the points (3, 4) makes an angle \frac{3 π}{4} with positive x -axis then f^{’} (3) =$ (1) $3$ (2) $2$ (3) $1$ (4) $4$ 167. $If (a^{2} - 1) x + ay + (3 - a) = 0 is a normal to the curve xy = 1, then the interval in which 'a' lies is$ (1) $[- 1, 1] \cup [2, \infty)$ (2) $(- \infty, - 1] \cup (0, 1]$ (3) $[- 1, 1) \cup (1, \infty)$ (4) $(1, \infty)$ 168. $If ax + by = 1 is a normal to the parabola y^{2} = 4 px then the condition is$ (1) $4 ab = a^{2} + b^{2}$ (2) $4 pab + a b^{3} = a^{2} b^{2}$ (3) $pa^{3} = b^{2} - 2 pab^{2}$ (4) $pa^{2} + 4 pa = a + b$ 169. $The equation of the normal to the curve y (1 + x^{2}) = 2 - x where the tangent crosses x -axis is$ (1) $5 x - y - 10 = 0$ (2) $x - 5 y - 10 = 0$ (3) $5 x + y + 10 = 0$ (4) $x + 5 y + 10 = 0$ 170. $The approximation value of y = (1.01)^{3} + 2 (1.01)^{\frac{3}{2}} + 5 is$ (1) $8.06$ (2) $8.04$ (3) $8.02$ (4) $8.16$ 171. $If the slope of the tangent of the curve y = a x^{3} + bx + 4 at (2, 14) is 21 , then the values of a and b respectively$ (1) $2, - 3$ (2) $3, - 2$ (3) $- 3, - 2$ (4) $2,3$ 172. $The function f (x) = log (1 + x) - \frac{2 x}{2 + x} is increasing on$ (1) $(- \infty, \infty)$ (2) $(- 1, \infty)$ (3) $(\infty, - 1)$ (4) $(- \infty, 0)$ 173. $The equation of the normal to the curve 4 x^{2} + 9 y^{2} = 36 at the point P (\frac{7 π}{4}) is$ (1) $2 x - 3 y - 62 = 0$ (2) $2 x + 3 y = 0$ (3) $32 x + 22 y - 5 = 0$ (4) $32 x - 22 y - 13 = 0$ 174. $If the function f (x) = 2 x^{3} - 9 a x^{2} + 12 a^{2} x + 1, where a > 0, attains its local maximum and local minimum values at p and q, respectively, such that p^{2} = q, then f (3) is equal to:$ (1) $55$ (2) $37$ (3) $10$ (4) $23$ 175. $If the water is being poured at the rate 36 m^{3} / sec in a cylindrical vessel of base radius 3 m, then the rate at which water level is rising, is$ (1) $4 π m / sec$ (2) $3/ π m / sec$ (3) $4/ π m / sec$ (4) $π /4 m / sec$ 176. $If the line y = 4 x - 5 touches to the curve y^{2} = a x^{3} + b at the point (2, 3), then 7 a + 2 b =$ (1) $0$ (2) $1$ (3) $-1$ (4) $2$ 177. $Let g (x) = 3 f (\frac{x}{3}) + f (3 - x) and f^{’ ’} (x) > 0 for all x \in (0, 3) . If g is decreasing in (0, α) and increasing in (α, 3), then 8 α is$ (1) $24$ (2) $0$ (3) $18$ (4) $20$ 178. $A circular sector of perimeter 60 metre with maximum area is to be constructed. The radius of the circular arc in metre must be$ (1) $20$ (2) $5$ (3) $15$ (4) $10$ 179. $If A = (x /9 x \geq x^{2} + 20) and f : A \to R is defined by f (x) = 2 x^{3} - 15 x^{2} + 36 x - 48, then the maximum value of f (x) is$ (1) $-20$ (2) $7$ (3) $20$ (4) $-16$ 180. $If is ‘R’ the least value of ‘a’ such that the function f(x)= x^{2} +ax+1 is increasing on [1,2] and 'S' is the greatest value of ‘a’ such that the function f(x)= x^{2} +ax+1 is decreasing on [1,2], then the value of \|R-S\| is.$