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Continuity and Differentiability

Multiple Choice Questions

201.
If x = $\frac{1 - t^{2}}{1 + t^{2}}$ and y = $\frac{2 at}{1 + t^{2}}$ , then $\frac{dy}{dx}$ is equal to

$\frac{a (1 - t^{2})}{2 t}$

$\frac{a (t^{2} - 1)}{2 t}$

$\frac{a (t^{2} + 1)}{2 t}$

$\frac{a (t^{2} - 1)}{t}$

$\frac{a (t^{2} - 1)}{2 t}$

$Given x = \frac{1 - t^{2}}{1 + t^{2}} and y = \frac{2 at}{1 + t^{2}} On differentiating w . r . t . t, respectively, we get \frac{dx}{dt} = \frac{(1 + t^{2}) (0 - 2 t) - (1 - t^{2}) (0 + 2 t)}{{(1 + t^{2})}^{2}} = \frac{- 4 t}{{(1 + t^{2})}^{2}} and \frac{dy}{dt} = \frac{(1 + t^{2}) 2 a - 2 at (2 t)}{{(1 + t^{2})}^{2}} = \frac{2 a (1 - t^{2})}{{(1 + t^{2})}^{2}} ∴ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{a (1 - t^{2})}{- 2 t} \Rightarrow \frac{dy}{dx} = \frac{a (t^{2} - 1)}{2 t}$

202.

The value of $∆ \log (f (x)) + ∆^{2} (3^{x})$ is

$\log [1 + \frac{∆ f (x)}{f (x)}] + 4 . 3^{x}$
$\log [1 + \frac{∆ f (x)}{f (x)}] + 3^{x}$
$\log [\frac{∆ f (x)}{1 + f (x)}] + 4 . 3^{x}$
$\log [\frac{∆ f (x)}{1 + f (x)}] + 3^{x}$

203.

If x²y⁵ = (x + y)⁷, then $\frac{d^{2} y}{{dx}^{2}}$ is equal to

y/x²
x/y
1
0

204.

If x = $x = \sec (θ), y = \tan (θ)$ , then the value of $\frac{d^{2} y}{{dx}^{2}}$ at $θ = \frac{π}{4}$ is

205.

If x = f(t) and y =g(t), then the value of $\frac{d^{2} y}{{dx}^{2}}$ is

$\frac{f' (t) g'' (t) - g' (t) f'' (t)}{{\{f' (t)\}}^{3}}$
$\frac{f' (t) g'' (t) - g' (t) f'' (t)}{{\{f' (t)\}}^{2}}$
$\frac{' (t) f'' (t) - g'' (t) f' (t)}{{\{f' (t)\}}^{2}}$
$\frac{g' (t) f'' (t) - g'' (t) f' (t)}{{\{f' (t)\}}^{3}}$

206.

The value of a and b such that the function

$f (x) = \{- 2 \sin (x), - π \leq x \leq - \frac{π}{2} asin (x) + b, - \frac{π}{2} < x < \frac{π}{2} \cos (x), \frac{π}{2} \leq x \leq π\}$

is continuous in $[- π, π]$ , are

- 1, 0
1, 0
1, 1
- 1, 1

207.

If g is the inverse of f and f'(x) = $\frac{1}{1 + x^{2}}$ , then g'(x) is equal to

1 + [g(x)]²
$\frac{- 1}{1 + [g (x)]^{2}}$
$\frac{1}{2 (1 + x^{2})}$
None of these

208.

If f(x) = $\{{(1 + |\sin (x)|)}^{\frac{a}{|\sin (x)|}}, - \frac{π}{6} < x < 0 e^{\frac{\tan (2 x)}{\tan (3 x)}}, 0 < x < \frac{π}{6}\}$ is continuous at x = 0, find the values of a and b.