Continuity And Differentiability Test

Continuity And Differentiability

This is Continuity and Differentiability Test-05 for CBSE class 12 Maths.. There are 15 questions in this test with each question having around four answer choices.

Questions & Answers

1
If f (x) = \(ta{n^{ - 1}}\) x and g (x) = \(ta{n^{ - 1}}\)\(\left( {\frac{{x + 1}}{{1 - x}}} \right),\) then
  • A
    f (x) = g (x)
  • B
    \({D_f} = {D_g}\)
  • C
    none of these
  • D
    f ‘ (x) = g ‘ (x)
    Correct
2
Derivative of log | x | w.r.t. | x | is
  • A
    \( \pm \frac{1}{x}\)
  • B
    none of these
  • C
    \(\frac{1}{{\left| x \right|}}\)
    Correct
  • D
    \(\frac{1}{x}\)
3
If \(\sqrt x + \sqrt y = \sqrt {a,} \) then \({\left( {\frac{{{d^2}y}}{{d{x^2}}}} \right)_{x = a}}\) is equal to
  • A
    none of these
  • B
    \(\frac{1}{a}\)
  • C
    \(\frac{1}{{2a}}\)
    Correct
  • D
    a
4
If \({y^2} = a{x^2} + b,\) then \(\frac{{{d^2}y}}{{d{x^2}}}\) is equal to
  • A
    \(\frac{{ab}}{{{y^2}}}\)
  • B
    \(\frac{{ab}}{{{y^3}}}\)
    Correct
  • C
    none of these
  • D
    \(\frac{{ab}}{{{x^3}}}\)
5
If \(x = a\;{t^2},y = 2\;a\;t,then\frac{{{d^2}y}}{{d{x^2}}}\) is equal to
  • A
    \(\frac{1}{{\;{t^2}}}\)
  • B
    \( - \frac{1}{{2a\;{t^3}}}\)
    Correct
  • C
    0
  • D
    none of these
6
If f (x) \( = {x^2}g(x)\) and g (x) is twice differentiable then f’’’ (x) is equal to
  • A
    \({x^2}g''(x) + 2\;x\;g'(x) + 2\;g(x)\)
  • B
    \({x^2}g''(x) + 4\;x\;g'(x) + 2\;g(x)\)
    Correct
  • C
    none of these
  • D
    2 g ’’ (x)
7
If \(y = a\;{e^{mx}} + b\;{e^{ - mx}},\) then \({y_2}\) is equal to
  • A
    \(m{y_1}\)
  • B
    none of these
  • C
    \({m^2}y\)
    Correct
  • D
    \( - {m^2}y\)
8
If y = a sin mx + b cos m x, then \(\frac{{{d^2}y}}{{d{x^2}}}\) is equal to
  • A
    \( - {m^2}y\)
    Correct
  • B
    none of these
  • C
    \({m^2}y\)
  • D
    \(m{y_1}\)
9
If \(y = \frac{{\log x}}{x},\;then\frac{{{d^2}y}}{{d{x^2}}} = \)
  • A
    \(\frac{{2\log x - 3}}{{{x^4}}}\)
  • B
    none of these
  • C
    \(\frac{{2\log x - 3}}{{{x^3}}}\)
    Correct
  • D
    \(\frac{{3 - 2\log x}}{{{x^3}}}\)
10
If f is derivable at x = a , then \(\mathop {Lt}\limits_{x \to a} \;\;\frac{{xf(a) - af(x)}}{{x - a}}\) is equal to
  • A
    f ‘ (a)
  • B
    a f ‘ 9a) – f (a)
  • C
    \(f(a) - a\;f'(a)\)
    Correct
  • D
    none of these
11
The derivative of an even function is
  • A
    an even function
  • B
    negative
  • C
    none of these.
  • D
    an odd function
    Correct
12
The derivate of an odd function is
  • A
    an even function
    Correct
  • B
    none of these
  • C
    negative
  • D
    an odd function
13
If \(y = \frac{x}{2}\sqrt {{x^2} + 1} + \frac{1}{2}\log (x + \sqrt {{x^2} + 1} ),\) then \(\frac{{dy}}{{dx}}\) is equal to
  • A
    \(2\sqrt {{x^2} + 1} \)
  • B
    \(\sqrt {{x^2} + 1} \)
    Correct
  • C
    \(\frac{1}{{\sqrt {{x^2} + 1} }}\)
  • D
    none of these
14
If \(y = x\sqrt {1 - {x^2}} + {\sin ^{ - 1}}x,\)then \(\frac{{dy}}{{dx}}\) is equal to
  • A
    \(\sqrt {1 - {x^2}} \)
  • B
    \(\frac{1}{{\sqrt {1 - {x^2}} }}\)
  • C
    none of these
  • D
    \(2\sqrt {1 - {x^2}} \)
    Correct
15
\(\frac{{{d^4}}}{{d{x^4}}}({\sin ^3}x)\) is equal to
  • A
    \(\frac{3}{4}\cos x - \frac{{{3^4}\cos 3x}}{4}\)
  • B
    none of these
  • C
    \(\frac{3}{4}\sin x - \frac{{{3^4}\cos 3x}}{4}\)
  • D
    \(\frac{{3\sin x - {3^4}\sin 3x}}{4}\)
    Correct