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Continuity and Differentiability

Multiple Choice Questions

411.

$If x^{y} = y^{x}, then x (x - ylog (x)) \frac{dy}{dx} is equal to :$

$y (y - xlog (y))$
$y (y + xlog (y))$
$x (x + y \log (x))$
$x (y - x \log (y))$

412.

$f (x) = e^{x} \sin (x), then f^{(6)} (x) = ?$

$e^{6 x} \sin (6 x)$
$- 8 e^{x} \cos (x)$
$8 e^{x} \sin (x)$
$8 e^{x} \cos (x)$

413.

$If f (x) = \{\frac{1 - \sqrt{2} \sin (x)}{π - 4 x} if x \neq \frac{π}{4} a if x = \frac{π}{4}\} is continuous at \frac{π}{4}, then a is equal to :$

4
2
1
$\frac{1}{4}$

414.

$If u = \sin^{- 1} (\frac{x^{2} + y^{2}}{x + y}) then x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} is$

sin(u)
tan(u)
cos(u)
cot(u)

415.

$If f (x, y) = \frac{\cos (x - 4 y)}{\cos (x + 4 y)}, then {\frac{\partial f}{\partial x}}_{y = \frac{x}{2}} is equal to$

416.

$y = \log ({\{\frac{1 + x}{1 - x}\}}^{\frac{1}{4}}) - \frac{1}{2} \tan^{- 1} (x), then \frac{dy}{dx} is equal to$

$\frac{x}{1 - x^{2}}$
$\frac{x^{2}}{1 - x^{4}}$
$\frac{x}{1 + x^{4}}$
$\frac{x}{1 - x^{4}}$

417.
$x = \cos (θ), y = \sin (5 θ) \Rightarrow (1 - x^{2}) \frac{d^{2} y}{{dx}^{2}} - x \frac{dy}{dx} is equal to$

- 5y

5y

25y

- 25y

- 25y

$Given, x = \cos (θ), y = \sin (5 θ) \frac{dx}{dθ} = - \sin (θ), \frac{dy}{dθ} = 5 \cos (5 θ) ∴ \frac{dy}{dx} = \frac{\frac{dy}{dθ}}{\frac{dx}{dθ}} = - \frac{5 \cos (5 θ)}{\sin (θ)} \Rightarrow \frac{d^{2} y}{{dx}^{2}} = \frac{d}{dx} (\frac{dy}{dx}) = \frac{d}{dθ} (\frac{dy}{dx}) . \frac{dθ}{dx} = \frac{d}{dθ} (\frac{- 5 \cos (5 θ)}{\sin (θ)}) \frac{1}{- \sin (θ)} = (\frac{\sin (θ) \sin (5 θ) . 25 + 5 \cos (5 θ) \cos (θ)}{\sin^{2} (θ)}) = - \frac{25 \sin (5 θ)}{\sin^{2} (θ)} - \frac{5 \cos (5 θ) \cos (θ)}{\sin^{3} (θ)} Now, (1 - x^{2}) \frac{d^{2} y}{{dx}^{2}} - x \frac{dy}{dx} = (1 - \cos^{2} (θ)) (- \frac{25 \sin (5 θ)}{\sin^{2} (θ)} - \frac{5 \cos (5 θ) \cos (θ)}{\sin^{3} (θ)}) - \cos (θ) (\frac{- 5 \cos (5 θ)}{\sin (θ)})$

$= \sin^{2} (θ) (\frac{- 25 \sin (5 θ)}{\sin^{2} (θ)} - \frac{5 \cos (5 θ) \cos (θ)}{\sin^{3} (θ)}) + \frac{5 \cos (θ) \cos (5 θ)}{\sin (θ)} = - 25 \sin (5 θ) - \frac{5 \cos (5 θ) \cos (θ)}{\sin (θ)} + \frac{5 \cos (5 θ) \cos (θ)}{\sin (θ)} = - 25 \sin (5 θ) = - 25 y$

418.

$If f : R \to R is defined by f (x) = \{\frac{\cos (3 x) - \cos (x)}{x^{2}}, for x \neq 0 λ, for x = 0\} and if f is continuous at x = 0, then λ = ?$

419.

$If f (2) = 4 and f' (2) = 1, then \lim_{x \to 2} \frac{xf (2) - 2 f (x)}{x - 2} = ?$

420.

If $y = \sin (\log_{e} (x))$ , then $x^{2} \frac{d^{2} y}{{dx}^{2}} + x \frac{dy}{dx}$ is equal to

$y = \sin (\log_{e} (x))$
$\cos (\log_{e} (x))$
y²
- y

Previous Year Papers

Test Series

Test Yourself

Multiple Choice Questions

417. x = cosθ, y = sin5θ ⇒ 1 - x2d2ydx2 - xdydx is equal to - 5y 5y 25y - 25y

417.
$x = \cos (θ), y = \sin (5 θ) \Rightarrow (1 - x^{2}) \frac{d^{2} y}{{dx}^{2}} - x \frac{dy}{dx} is equal to$

- 5y

5y

25y

- 25y