Continuity & Differentiability PYQs Set 13 (PDF)

Set 13 of 13 · Questions 361-371 of 371

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Continuity & Differentiability · Set 13 (Q361-371)
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361. $Let M and m respectively denote the maximum and the minimum values of [f (θ)]^{2}, where f (θ) = a^{2} cos^{2} θ + b^{2} sin^{2} θ + a^{2} sin^{2} θ + b^{2} cos^{2} θ . Then M - m =$ (1) $a^{2} + b^{2}$ (2) $(a - b)^{2}$ (3) $a^{2} b^{2}$ (4) $(a + b)^{2}$ 362. $If f (x) = cos^{- 1} (\frac{1 - ( log x ) ^{2}}{1 + ( log x ) ^{2}}), then f^{’} (e) =$ (1) $1/ e$ (2) $2/ e^{2}$ (3) $2/ e$ (4) $1$ 363. $The maximum volume (in cubic units) of the cylinder which can be inscribed in a sphere of diameter 6 units is$ (1) $123 π$ (2) $43 π$ (3) $33 π$ (4) $83 π$ 364. $Let α, β be two roots of the quadratic equation x^{2} + ax - b = 0, b \neq = 0 . If the straight line x cos θ + y sin θ = C touches the curve (\frac{x}{α})^{n} + (\frac{y}{β})^{n} = 2 at the point (α, β), then (\frac{a}{b})^{2} + \frac{2}{b} =$ (1) $\frac{1}{2 C ^{2}}$ (2) $\frac{4}{C ^{2}}$ (3) $\frac{2}{C ^{2}}$ (4) $\frac{1}{C ^{2}}$ 365. $The maximum area of a rectangle inscribed in the circle (x + 1)^{2} + (y - 3)^{2} = 64 is$ (1) $64 sq. units$ (2) $72 sq. units$ (3) $128 sq. units$ (4) $8 sq. units$ 366. $The trigonometric function y = tan x in the II quadrant$ (1) $decreases from 0 to \infty$ (2) $increases from 0 to \infty$ (3) $decreases from - \infty to 0$ (4) $increases from - \infty to 0$ 367. $Let f : R \to R be defined as f (x) = {\frac{x ^{3}}{( 1 - c o s 2 x ) ^{2}} lo g_{e} (\frac{1 + 2 x e ^{- 2 x}}{( 1 - x e ^{- x} ) ^{2}}), α, x \neq = 0 x = 0 If f is continuous at x=0, then α is equal to:$ (1) $1$ (2) $3$ (3) $0$ (4) $2$ 368. $Let f : R -> R be defined as f (x) = x^{3} + x - 5. If g (x) is a function such that f (g(x)) = x, V x \in R, then g' (63) is equal to______.$ (1) $\frac{1}{49}$ (2) $\frac{3}{49}$ (3) $\frac{43}{49}$ (4) $\frac{91}{49}$ 369. $If y (x) = cot^{- 1} (\frac{1 + s i n x + 1 - s i n x}{1 + s i n x - 1 - s i n x}), x \in (\frac{π}{2}, π), then \frac{d y}{d x} at x = \frac{5 π}{6} is:$ (1) $- \frac{1}{2}$ (2) $-1$ (3) $\frac{1}{2}$ (4) $0$ 370. $Let the function f (x) = {\frac{l o g _{e} ( 1 + 5 x ) - l o g _{e} ( 1 + α x )}{x}; 10; if x \neq = 0 if x = 0 be continuous at x = 0. The α is equal to :$ (1) $10$ (2) $-10$ (3) $5$ (4) $-5$ 371. $If cos^{- 1} (\frac{y}{2}) = lo g_{e} (\frac{x}{5})^{5}, ∣ y ∣ < 2,then:$ (1) $x^{2} y^{''} + x y^{'} + 25 y = 0$ (2) $x^{2} y^{''} - x y^{'} - 25 y = Q$ (3) $x^{2} y^{''} - x y^{'} + 25 y = 0$ (4) $x^{2} y^{''} + x y^{'} + 25 y = Q$