Continuity And Differentiability Test

Continuity And Differentiability

This is Continuity and Differentiability Test-04 for CBSE class 12 Maths.. There are 15 questions in this test with each question having around four answer choices.

Questions & Answers

1
\(\mathop {Lt}\limits_{x \to 0} \;\;\;\;\frac{{x{{.2}^x} - x}}{{1 - \cos x}}\) is equal to
  • A
    \(\frac{1}{2}\)
  • B
    \(\frac{1}{2}\log 2\)
  • C
    log 2
  • D
    2 log 2
    Correct
2
\(\mathop {Lt}\limits_{x \to 0} \;\;\;\frac{{x({e^{\sin }}^x - 1}}{{1 - \cos x}}\) is equal to
  • A
    2
    Correct
  • B
    \(\frac{1}{2}\)
  • C
    0
  • D
    1
3
\(\mathop {Lt}\limits_{x \to \infty } \;\;\;{\left( {1 + \frac{3}{x}} \right)^x}\) is equal to
  • A
    \({e^3}\)
    Correct
  • B
    none of these
  • C
    \({e^{1/3}}\)
  • D
    3 e
4
\(\mathop {Lt}\limits_{x \to \infty } \;\;{\left( {\;\frac{{2x + 3}}{{2x + 1}}} \right)^x}\)is equal to
  • A
    \({e^{1/2}}\)
  • B
    \({e^2}\)
    Correct
  • C
    e
  • D
    none of these
5
If \(y = {e^{2x + 3\log x}}\), then \(\frac{{dy}}{{dx}}\) is equal to
  • A
    \({x^2}{e^{2x}}(2x + 3)\)
    Correct
  • B
    none of these
  • C
    \({x^2}{e^{2x}}(2 + 3x)\)
  • D
    \({x^3}{e^{2x}}\)
6
If \(y = \frac{{{e^x} - {e^{ - x}}}}{{{e^x} + {e^{ - x}}}},\) then \(\frac{{dy}}{{dx}}\) is equal to
  • A
    \(1 - {y^2}\)
  • B
    \(1 + {y^2}\)
    Correct
  • C
    none of these
  • D
    \({y^2} + 1\)
7
\(\frac{d}{{dx}}(\log |x|)\)is equal to \(\left( {x \ne 0} \right)\)
  • A
    \(\frac{1}{x}\)
    Correct
  • B
    \( \pm \frac{1}{x}\)
  • C
    \(\frac{1}{x}or - \frac{1}{x}\)
  • D
    \(\frac{1}{{\left| x \right|}}\)
8
If \({x^p}\;{y^q}\;\)= \({(x + y)^{p + q}}\), then \(\frac{{dy}}{{dx}}\) is equal to
  • A
    \(\frac{y}{{x + y}}\)
  • B
    none of these
  • C
    \(\frac{y}{x}\)
    Correct
  • D
    \(\frac{x}{y}\)
9
If f(x) be any function which assumes only positive values and f’ (x) exists then f’ (x) is equal to
  • A
    \(f(x)\frac{d}{{dx}}\left\{ {\log (f(x))} \right\}\)
    Correct
  • B
    \(f(x)\frac{d}{{dx}}\left\{ {{e^{\log }}(f(x))} \right\}\)
  • C
    \(f(x)\frac{d}{{dx}}\left( {{e^{f(x)}}} \right)\)
  • D
    none of these
10
\(\frac{d}{{dx}}\left( {\log \left| {\tan \frac{x}{2}} \right|} \right)\)is equal to
  • A
    cosec x
    Correct
  • B
    \(\frac{2}{{\sin x}}\)
  • C
    none of these
  • D
    \(\frac{1}{{\tan \frac{x}{2}}}\)
11
The differential coefficient of log (| log x |) w.r.t. log x is
  • A
    \(\frac{1}{{\log x}}\)
    Correct
  • B
    none of these
  • C
    \(\frac{1}{{x\log x}}\)
  • D
    \(\frac{1}{{x\left| {\log x} \right|}}\)
12
If x = a \(co{s^3}\;t,\;y = a\;si{n^3}\) t then \(\frac{{dy}}{{dx}}\) is equal to
  • A
    cosec t
  • B
    cos t
  • C
    – tan t.
    Correct
  • D
    cot t
13
Derivative of \({\sin ^3}x{\text{ }}w.r.t.{\cos ^3}x{\text{ }}is\)
  • A
    – tan x
    Correct
  • B
    \({\tan ^3}x\)
  • C
    cot x.
  • D
    tan x
14
Derivative of cos (sin x) w.r.t. sin x is
  • A
    – sin (sin x)cosx
    Correct
  • B
    \({\sec ^2}\left( {\frac{x}{{\sqrt {{x^2} + 1} }}} \right)\)
  • C
    – sin (sin x)
  • D
    none of these
15
If \(y = {\sin ^{ - 1}}xandz = {\cos ^{ - 1}}\sqrt {1 - {x^2}} ,\) then \(\frac{{dy}}{{dz}}\) is equal to (0 < x < 1)
  • A
    \(\sqrt {1 - {x^2}} \)
  • B
    \(\frac{1}{{\sqrt {1 - {x^2}} }}\)
  • C
    1
    Correct
  • D
    \({\cos ^{ - 1}}x\)