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
\(\mathop {Lt}\limits_{h \to 0} \;\;\;\;{{\root 3 \of {8 + h} - 2} \over 4}\) is equal to
  • A
    \({1 \over 24}\)
  • B
    \({1 \over 12}\)
    Correct
  • C
    \({1 \over 3}\)
  • D
    none of these
2
If f (x) = \(\sqrt {1 - {x^2}} ,x \in (0,1),thenf'(x)\) is equal to
  • A
    \(\sqrt {{x^2} - 1} \)
  • B
    \({{ - x} \over {\sqrt {1 - {x^2}} }}\)
    Correct
  • C
    \(\sqrt {1 - {x^2}} \)
  • D
    \({1 \over {\sqrt {1 - {x^2}} }}\)
3
If \(y = {\sin ^{ - 1}}\) x and z = \({\cos ^{ - 1}}\sqrt {1 - {x^2}} ,\) then \({{dy} \over {dz}} = \)
  • A
    1
  • B
    \({{\left| x \right|} \over x}\)
    Correct
  • C
    - 1
  • D
    \({\tan ^{ - 1}}{x \over {\sqrt {1 - {x^2}} }}\)
4
\({d \over {dx}}\left\{ {{{\tan }^{ - 1}}\left( {{{3x - {x^3}} \over {1 - 3{x^2}}}} \right)} \right\}\) is equal to
  • A
    \({3 \over {1 + {x^2}}}\)
    Correct
  • B
    \({\sec ^2}3x\)
  • C
    \({1 \over {1 + {x^2}}}\)
  • D
    \({3 \over {1 + 9{x^2}}}\)
5
The function, \(f(x) = {(x - a)^2}\cos {1 \over {x - a}}for\;x \ne a\) and f (a) = 0, is
  • A
    continuous but not derivable at x = 0
  • B
    none of these
  • C
    not continuous at x = a
  • D
    derivable at x = a
    Correct
6
\({d \over {dx}}\left( {{{\cos }^{ - 1}}\left( {{{{x^{ - 1}} - x} \over {{x^{ - 1}} + x}}} \right)} \right)\) is equal to
  • A
    \({2 \over {1 + {x^2}}},x > 0\)
    Correct
  • B
    \({{ - 2} \over {\sqrt {1 + {x^2}} }}\)
  • C
    \({2 \over {1 + {x^2}}},x > 0\)
  • D
    none of these
7
\({d \over {dx}}\left( {{{\sin }^{ - 1}}\left( {{2 \over {{x^{ - 1}} + x}}} \right)} \right)\) is equal to
  • A
    \({2 \over {1 + {x^2}}},0 < \left| x \right| < 1\)
    Correct
  • B
    none of these
  • C
    \({2 \over {\sqrt {1 + {x^2}} }},\left| x \right| < 1\)
  • D
    \({2 \over {\sqrt {1 + {x^2}} }}\)
8
\({d \over {dx}}\left( {{{\tan }^{ - 1}}\left( {{x \over {\sqrt {1 - {x^2}} }}} \right)} \right)\) is equal to
  • A
    \({{ - 1} \over {\sqrt {1 - {x^2}} }}\)
  • B
    none of these
  • C
    \({1 \over {1 + {x^2}}}\)
  • D
    \({1 \over {\sqrt {1 - {x^2}} }}\)
    Correct
9
\({d \over {dx}}\left( {{{\tan }^{ - 1}}\left( {{2 \over {{x^{ - 1}} - x}}} \right)} \right)\) is equal to
  • A
    \({2 \over {1 + {x^2}}},x \ne 0, \pm 1\)
    Correct
  • B
    \({2 \over {1 + {x^2}}},x \ne 0, \pm 1\)
  • C
    \({{ - 2} \over {\sqrt {1 - {x^2}} }}\)
  • D
    none of these
10
If \(\sin x = {t \over {\sqrt {1 + {t^2}} }},then{{dx} \over {dt}}\) is equal to
  • A
    \({1 \over {\sqrt {1 - {t^2}} }}\)
  • B
    \({1 \over {1 + {t^2}}}\)
    Correct
  • C
    \({1 \over {{{\left( {1 + {t^2}} \right)}^{3/2}}}}\)
  • D
    none of these
11
\({d \over {dx}}({\sin ^{ - 1}}x) = {1 \over {\sqrt {1 - {x^2}} }}\) holds true for
  • A
    all real x
  • B
    all real x for which | x| > 1
  • C
    all x \(\) ( - 1, 1)
    Correct
  • D
    all x \(\) [- 1, 1]
12
Derivative of tan \(\sqrt {{x^2} + 1} \;w.r.t.\sqrt {{x^2} + 1} \) is
  • A
    \({\sec ^2}x\)
  • B
    \({\sec ^2}\sqrt {{x^2} + 1} \)
    Correct
  • C
    none of these
  • D
    \({\sec ^2}\left( {{x \over {\sqrt {{x^2} + 1} }}} \right)\)
13
If \(y = {{1 - x} \over {1 + x}}then\;{y_6} = \)
  • A
    \(\frac{{2\left| \!{\underline {\, 6 \,}} \right. }}{{{{(1 + x)}^7}}}\)
    Correct
  • B
    \(\frac{{\left| \!{\underline {\, 6 \,}} \right. }}{{{{(1 + x)}^7}}}\)
  • C
    \(\frac{{ - 2\left| \!{\underline {\, 6 \,}} \right. }}{{{{(1 + x)}^7}}}\)
  • D
    none of these
14
If y = log \((x + \sqrt {1 + {x^2})} \) then \({{{d^2}y} \over {d{x^2}}} = \)
  • A
    \({{ - x} \over {{{({x^2} + 1)}^{3/2}}}} = \)
    Correct
  • B
    \({1 \over {\sqrt {{x^2} + 1} }}\)
  • C
    \({x \over {{{({x^2} + 1)}^{3/2}}}}\)
  • D
    \({{ - 2x} \over {{{({x^2} + 1)}^{3/2}}}} = \)
15
If y = log x , then \({y_n} = \)
  • A
    \({{{{( - 1)}^n}.n!} \over {{x^{n + 1}}}}\)
  • B
    \({{{{( - 1)}^n}.n!} \over {{x^n}}}.\)
  • C
    \({{{{( - 1)}^{n - 1}}(n - 1)!} \over {{x^n}}}\)
    Correct
  • D
    \({{{{( - 1)}^n}(n - 1)!} \over {{x^n}}}\)