Limits And Derivatives CBSE Questions & Answers

Limits And Derivatives

This is Mathematics Class 11 Limits and Derivatives CBSE Questions & Answers. There are 15 questions in this test with each question having around four answer choices.

Questions & Answers

1
The derivative of \({\sec ^{ - 1}}\left( {{1 \over {2{x^2} - 1}}} \right)\) with respect to \(\sqrt {1 - {x^2}} at\;x = {1 \over 2}\) is
  • A
    - 2
  • B
    1
  • C
    4
    Correct
  • D
    2
2
\({d \over {dx}}\left( {{x \over 2}\sqrt {{x^2} + {a^2}} + {{{a^2}} \over 2}\log \left( {x + \sqrt {{x^2} + {a^2}} } \right)} \right)\) is equal to
  • A
    \(\sqrt {{x^2} + {a^2}} \)
    Correct
  • B
    none of these
  • C
    \({1 \over {x + \sqrt {{x^2} + {a^2}} }}\)
  • D
    \({1 \over {\sqrt {{x^2} + {a^2}} }}\)
3
\({d \over {dx}}\left( {x\sqrt {{a^2} - {x^2}} + {a^2}{{\sin }^{ - 1}}\left( {{x \over a}} \right)} \right)\) is equal to
  • A
    \(2\sqrt {{a^2} - {x^2}} \)
    Correct
  • B
    \(\sqrt {{a^2} - {x^2}} \)
  • C
    none of these
  • D
    \({1 \over {\log x}}\)
4
\({d \over {dx}}({\sin ^{ - 1}}(\sqrt {1 - {x^2})} )\)is equal to
  • A
    \( - {x \over {\sqrt {1 - {x^2}} }}for\;0 < \left| x \right| < 1\)
  • B
    none of these
  • C
    \( - {1 \over {\sqrt {1 - {x^2}} }}for\;\left| x \right| < 1\)
  • D
    \( - {x \over {\left| x \right|\sqrt {1 - {x^2}} }}for0 < \left| x \right| < 1\)
    Correct
5
\({d \over {dx}}({\cos ^{ - 1}}(\sqrt {1 - {x^2})} )\) is equal to
  • A
    \({x \over {\left| x \right|\sqrt {1 - {x^2}} }}for\;0 < \left| x \right| < 1\)
    Correct
  • B
    none of these
  • C
    \({1 \over {\sqrt {1 - {x^2}} }}for\;0 < \left| x \right| < 1\)
  • D
    \({1 \over {\sqrt {1 - {x^2}} }}for\;\left| x \right| < 1\)
6
\({d \over {dx}}({\sec ^{ - 1}}x)\) is equal to
  • A
    none of these
  • B
    \({{ - 1} \over {x\sqrt {{x^2} - 1} }}for\left| x \right| > 1\)
  • C
    \({1 \over {\left| x \right|\sqrt {{x^2} - 1} }}for\left| x \right| > 1\)
    Correct
  • D
    \({1 \over {x\sqrt {{x^2} - 1} }}for\left| x \right| > 1\)
7
\({d \over {dx}}(\cos e{c^{ - 1}}x)\) is equal to
  • A
    \({{ - 1} \over {x\sqrt {{x^2} - 1} }}for\left| x \right| > 1\)
  • B
    none of these
  • C
    \({{ - 1} \over {\left| x \right|\sqrt {{x^2} - 1} }}for\left| x \right| > 1\)
    Correct
  • D
    \({1 \over {x\sqrt {{x^2} - 1} }}for\left| x \right| > 1\)
8
\({d \over {dx}}({\sin ^{ - 1}}(1 - x))\) is equal to
  • A
    \({1 \over {\sqrt {2x - {x^2}} }}\)
  • B
    \({1 \over {\sqrt {{x^2} - 2x} }}\)
  • C
    \({{ - 1} \over {\sqrt {2x - {x^2}} }}\)
    Correct
  • D
    none of these
9
\({{{d^2}} \over {d{x^2}}}({\cos ^{ - 1}}(1 - x))\) is equal to
  • A
    \({{1 - x} \over {{{(2x - {x^2})}^{3/2}}}}\)
  • B
    \({{x - 1} \over {{{(2x - {x^2})}^{3/2}}}}\)
    Correct
  • C
    \({1 \over {2{{(2x - {x^2})}^{3/2}}}}\)
  • D
    none of these
10
If y = \(\sqrt {x + \sqrt {x + \sqrt {x + ... + to\;\infty } } } \) then \({{dy} \over {dx}} = \)
  • A
    \({x \over {y + 1}}\)
  • B
    \({1 \over {2y - 1}}\)
    Correct
  • C
    \(\sqrt {{x \over {y + 1}}} \)
  • D
    \({1 \over {2y + 1}}\)
11
\(\left\{ \begin{gathered} x\;\;\;\;\;\;,\;\;0 \leq x < 1 \\ 3 - x\;,\;\;\;1 \leq x \leq 2 \\ \end{gathered} \right.,then\;at\) x = 1, f (x) is
  • A
    continuous but not derivable
  • B
    continuous
  • C
    not continuous
    Correct
  • D
    continuous on left and discontinuous on right.
12
\(f(x) = \left\{ \begin{gathered} {x^3},\;\;\left| x \right| \leq 1 \\ x\;\;,\;\;\left| x \right| > 1 \\ \end{gathered} \right.then\;f(x)\;is\)
  • A
    not continuous at -1 and 1
  • B
    not derivable at – 1 and 1
    Correct
  • C
    derivable at all x {tex} \in{tex} R
  • D
    none of these
13
If x = f(t) and y = g (t), then \({{{d^2}y} \over {d{x^2}}}\) is qual to
  • A
    none of these
  • B
    \({{g''(t)} \over {f''(t)}}\)
  • C
    \({{g''(t)f'(t) - g'(t)f''(t)} \over {{{(f'(t))}^3}}}\)
    Correct
  • D
    \({{g''(t)f'(t) - g'(t)f''(t)} \over {{{(f'(t))}^2}}}\)
14
\({\text{f }}\left( {\text{x}} \right){\text{ }} = {\text{ }}\left| {{\text{ }}\left[ {\text{x}} \right]{\text{ x }}} \right|{\text{ in }}--{\text{ 1}}\)\( \le \) x \( \le \) 2 is
  • A
    differentiable at x = 0
  • B
    discontinuous at x = 0
  • C
    continuous at x = 2
  • D
    continuous at x = 0
    Correct
15
\({d \over {dx}}\left( {{{\tan }^{ - 1}}\left( {{{\sqrt x - \sqrt a } \over {1 + \sqrt {x\;a} }}} \right)} \right),x,a > 0,is\) s
  • A
    \({\tan ^1}\sqrt x - {\tan ^{ - 1}}\sqrt a \)
  • B
    \({1 \over {1 + x}} - {1 \over {1 + a}}\)
  • C
    \({1 \over {1 + x}}\)
  • D
    \({1 \over {2\sqrt x (1 + x)}}\)
    Correct