JEE Main & Advanced Mathematics Limits, Continuity and Differentiability Question Bank

done Self Evaluation Test - Continuity and Differentiability Total Questions - 100

• question_answer1)
  If \[f(x)={{x}^{\alpha }}log\text{ }x\] and \[f(0)=0\], then the value of a for which Rolle's theorem can be applied in [0, 1] is
  A)
  -2 done clear
  B)
  -1   done clear
  C)
  0 done clear
  D)
  ½ done clear
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• question_answer2)
  Let \[f\] be a function which is continuous and differentiable for all real x.  If \[f(2)=-4\] and \[f'(x)\ge 6\] for all \[x\text{ }\in [2,\text{ }4],\] then
  A)
  \[f(4)<8\] done clear
  B)
  \[f(4)\ge 8\] done clear
  C)
  \[f(4)\ge 12\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer3)
  If \[f(0)=0,f'(0)=2\], then the derivative of \[y=f(f(f(f(x)))\] at \[x=0\] is
  A)
  2 done clear
  B)
  8 done clear
  C)
  16 done clear
  D)
  4 done clear
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• question_answer4)
  Let \[f(x)=\frac{{{({{e}^{x}}-1)}^{2}}}{\sin \left( \frac{x}{a} \right)\log \left( 1+\frac{x}{4} \right)}\]  for \[x\ne 0,\] and \[f(0)=12\]. If f is continuous at \[x=0\], then the value of a is equal to
  A)
  1 done clear
  B)
  -1 done clear
  C)
  2 done clear
  D)
  3 done clear
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• question_answer5)
  If \[y=lo{{g}_{10}}x+lo{{g}_{x}}10+lo{{g}_{x}}x+lo{{g}_{10}}10\] then what is \[{{\left( \frac{dy}{dx} \right)}_{x=10}}\] equal to?
  A)
  10 done clear
  B)
  2 done clear
  C)
  1 done clear
  D)
  0 done clear
  View Solution play_arrow

• question_answer6)
  The function \[f(x)=\frac{1-\sin x+\cos x}{1+\sin x+\cos x}\] is not defined at \[x=\pi \]. The value of \[f(\pi )\] so that \[f(x)\] is continuous at \[x=\pi \] is
  A)
  \[-\frac{1}{2}\] done clear
  B)
  \[\frac{1}{2}\] done clear
  C)
  \[-1\] done clear
  D)
  \[1\] done clear
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• question_answer7)
  Which one of the following functions is differentiable for all real values of x?
  A)
  \[\frac{x}{\left| x \right|}\] done clear
  B)
  \[x\left| x \right|\] done clear
  C)
  \[\frac{1}{\left| x \right|}\] done clear
  D)
  \[\frac{1}{x}\] done clear
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• question_answer8)
  Let \[0<x<\pi \] and y(x) be given by\[(1+\sin \,x){{y}^{3}}-(\cos \,x){{y}^{2}}+2(1+\sin \,x)\] \[y-2\,\cos \,x=0\]. The derivative of y with respect to \[\tan \frac{x}{2}\] at \[x=\frac{\pi }{2}\] is:
  A)
  \[\frac{1}{2}\] done clear
  B)
  \[-\frac{1}{2}\] done clear
  C)
  \[2\] done clear
  D)
  \[-2\] done clear
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• question_answer9)
  If\[y={{\log }^{n}}x\], where \[{{\log }^{n}}\] means log log log... (repeated n time), then \[x\,log\text{ }x\text{ }log\text{ }x\text{ }lo{{g}^{2}}x\text{ }lo{{g}^{3}}x\]\[....{{\log }^{n-1}}x{{\log }^{n}}x\frac{dy}{dx}\] is equal to
  A)
  \[\log x\] done clear
  B)
  \[{{\log }^{n}}x\] done clear
  C)
  \[\frac{1}{\log \,x}\] done clear
  D)
  1 done clear
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• question_answer10)
  Let \[f(x)=g(x).\frac{{{e}^{1/x}}-{{e}^{-1/x}}}{{{e}^{1/x}}+{{e}^{-1/x}}}\], where g is a continuous function then \[\underset{x\to 0}{\mathop{\lim }}\,f(x)\] does not exist if
  A)
  g(x) is any constant function done clear
  B)
  g(x)=x done clear
  C)
  \[g(x)={{x}^{2}}\] done clear
  D)
  g(x) = x h (x), where h(x) is a polynomial. done clear
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• question_answer11)
  If \[f(x)=\left\{ \begin{align}   & \frac{x\log \cos x}{\log (1+{{x}^{2}})},x\ne 0 \\  & \,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,,x=0 \\ \end{align} \right.\text{then}\,\,\text{f(x)is}\] is
  A)
  Continuous as well as differentiable at x = 0 done clear
  B)
  Continuous but not differentiable at x = 0 done clear
  C)
  Differentiable but not continuous at x = 0 done clear
  D)
  Neither continuous nor differentiable at x = 0 done clear
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• question_answer12)
  Let \[f(x)=\left\{ \begin{align}   & {{5}^{1/x}},x<0 \\  & \lambda [x],x\ge 0 \\ \end{align} \right.\] and \[\lambda \in R\], then at x = 0
  A)
  f is discontinuous done clear
  B)
  f is continuous only, if \[\lambda =0\] done clear
  C)
  f is continuous only, whatever \[\lambda \] may be done clear
  D)
  None of these done clear
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• question_answer13)
  Consider the following statements:

  1. The function f (x) = greatest integer \[\le x,\text{ }x\in R\]is a continuous function.

  2. All trigonometric functions are continuous on R.

  Which of the statements given above is/are correct?

  A)
  1 only done clear
  B)
  2 only done clear
  C)
  Both 1 and 2 done clear
  D)
  Neither 1 nor 2 done clear
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• question_answer14)
  What is \[\underset{x\to 0}{\mathop{\lim }}\,\frac{2(1-\cos x)}{{{x}^{2}}}\] equal to?
  A)
  0 done clear
  B)
  1/2 done clear
  C)
   ¼ done clear
  D)
  1 done clear
  View Solution play_arrow

• question_answer15)
  The number of points at which the function \[f(x)=\left| x-0.5 \right|+\left| x-1 \right|+\tan x\] does not have a derivative in the interval (0, 2) is
  A)
  0 done clear
  B)
  1 done clear
  C)
  2 done clear
  D)
  3 done clear
  View Solution play_arrow

• question_answer16)
  Let \[f(x+y)=f(x)+f(y)\] and \[f(x)={{x}^{2}}g(x)\] for all\[x,\text{ }y\in R\], where g(x) is continuous function. Then f?(x) is equal to
  A)
  g'(x) done clear
  B)
  g(0) done clear
  C)
  g(0)+g'(x) done clear
  D)
  0 done clear
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• question_answer17)
  Given \[f:[-2a,2a]\to R\] is an odd function such that the left hand derivative at x = a is zero and \[f(x)=f(2a-x)\forall x\in (a,2a),\] then its left had derivative at \[x=-a\] is
  A)
  0 done clear
  B)
  a done clear
  C)
  -a done clear
  D)
  Does not exist done clear
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• question_answer18)
  If \[f(xy)=f(x).f(y)\] for all x, y and f(x) is continuous at x = 2, then f(x) is not necessarily continuous in:
  A)
  \[(-\infty ,\infty )\] done clear
  B)
  \[(0,\infty )\] done clear
  C)
  \[(-\infty ,0)\] done clear
  D)
  \[(2,\infty )\] done clear
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• question_answer19)
  Let \[y={{t}^{10}}+1\] and \[x={{t}^{8}}+1\], then \[\frac{{{d}^{2}}y}{d{{x}^{2}}}\] is equal to:
  A)
  \[\frac{5}{2}t\] done clear
  B)
  \[20{{t}^{8}}\] done clear
  C)
  \[\frac{5}{16{{t}^{6}}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer20)
  Consider the function \[f(x)=\left\{ \begin{matrix}    ax-2 & for & -2<x<-1  \\    -1 & for & -1\le x\le 1  \\    a+2{{(x-1)}^{2}} & for & 1<x<2  \\ \end{matrix} \right.\] What is the value of a for which f(x) is continuous at x =-1 and x=1?
  A)
  -1 done clear
  B)
  1 done clear
  C)
  0 done clear
  D)
  2 done clear
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• question_answer21)
  Which one of the following is correct in respect of the function \[f(x)=\frac{{{x}^{2}}}{\left| x \right|}\] for \[x\ne 0\] and f(0) = 0?
  A)
  f (x) is discontinuous every where done clear
  B)
  f (x) is continuous every where done clear
  C)
  f(x) is continuous at x = 0 only done clear
  D)
  f(x) is discontinuous at x = 0 only done clear
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• question_answer22)
  If f(x) is differentiable everywhere, then which one of the following is correct?
  A)
  \[\left| f \right|\] is differentiable everywhere done clear
  B)
  \[{{\left| f \right|}^{2}}\]is differentiable everywhere done clear
  C)
  \[f\left| f \right|\]is not differentiable at some points done clear
  D)
  None of the above done clear
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• question_answer23)
  Suppose \[f(x)\] is differentiable at \[x=1\] and \[\underset{h\to 0}{\mathop{\lim }}\,\frac{1}{h}f(1+h)=5\] then \[f'(1)\] equals
  A)
  3 done clear
  B)
  4 done clear
  C)
  5 done clear
  D)
  6 done clear
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• question_answer24)
  If \[f''(x)<0,\forall x\in (a,b),\] then \[f'(x)=0\] occurs
  A)
  Exactly once in (a, b) done clear
  B)
  At most once in (a, b) done clear
  C)
  At least once in (a, b) done clear
  D)
  None of these done clear
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• question_answer25)
  Let \[f'(x)=-1+\left| x-2 \right|,\] and \[g(x)=1-\left| x \right|;\] then the set of all points where fog is discontinuous is:
  A)
  {0, 2} done clear
  B)
  {0, 1, 2} done clear
  C)
  {0} done clear
  D)
  An empty set done clear
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• question_answer26)
  The value of p for which the function\[f(x)=\left\{ \begin{matrix}    \frac{{{({{4}^{x}}-1)}^{3}}}{\sin \frac{x}{p}\log \left[ 1+\frac{{{x}^{2}}}{3} \right]},x\ne 0  \\    12{{(log\,4)}^{3}},x=0  \\ \end{matrix} \right.\]may be continuous at \[x=0\], is
  A)
  1 done clear
  B)
  2 done clear
  C)
  3 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer27)
  If\[f(x)=\left\{ \begin{matrix}    mx+1x\le \frac{\pi }{2}  \\    \sin x+nx>\frac{\pi }{2}  \\ \end{matrix}\,\,\,\text{is}\,\,\text{continuous}\,\,\text{at} \right.\]\[x=\frac{\pi }{2}\], then which one of the following is correct?
  A)
  m = 1, n = 0 done clear
  B)
  \[m=\frac{n\pi }{2}+1\] done clear
  C)
  \[n=m\left( \frac{\pi }{2} \right)\] done clear
  D)
  \[m=n=\frac{\pi }{2}\] done clear
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• question_answer28)
  Consider the function\[f(x)=\left\{ \begin{matrix}    {{x}^{2}}, & x>2  \\    3x-2, & x\le 2  \\ \end{matrix} \right.\]. Which one of the following statements is correct in respect of the above function?
  A)
  f(x) is derivable but not continuous at x = 2. done clear
  B)
  f(x) is continuous but not derivable at x = 2. done clear
  C)
  f(x) is neither continuous nor derivable at x = 2. done clear
  D)
  f(x) is continuous as well as derivable at x = 2.   done clear
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• question_answer29)
  If the polynomial equation \[{{a}_{n}}{{x}^{n}}+{{a}_{n-1}}{{x}^{n-1}}+.....+{{a}_{2}}{{x}^{2}}+{{a}_{1}}x+{{a}_{0}}=0\],n positive integer, has two different real roots \[\alpha \]and \[\beta \], then between \[\alpha \text{ }and\text{ }\beta \], the equation \[n{{a}_{n}}{{x}^{n-1}}+(n-1){{a}_{n-1}}{{x}^{n-2}}+....+{{a}_{1}}=0\] has
  A)
  Exactly one root done clear
  B)
  At most one root done clear
  C)
  At least one root done clear
  D)
  No root done clear
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• question_answer30)
  If \[f(x)=\sqrt[3]{\frac{{{x}^{4}}}{\left| x \right|}},x\ne 0\] and f(0) = 0 is:
  A)
  Continuous for all x but not differentiable for any x done clear
  B)
  Continuous and differentiable for all x done clear
  C)
  Continuous for all x and differentiable for all \[x\ne 0\] done clear
  D)
  Continuous and differentiable for all \[x\ne 0\] done clear
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• question_answer31)
  Let \[f:[2,7]\to [0,\infty )\] be a continuous and differentiable function. Then, \[(f(7)-f(2))\frac{{{(f(7))}^{2}}+{{(f(2))}^{2}}+f(2)f(7)}{3}\] is, where \[c\in [2,7]\] [2, 7].
  A)
  \[5{{f}^{2}}(c)f'(c)\] done clear
  B)
  \[5f'(c)\] done clear
  C)
  \[f(c)f'(c)\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer32)
  If \[f(x)=\left\{ \begin{align}   & \left( {{x}^{2}}/a \right)-a,\,\,when\,\,\,x<0 \\  & 0,\,\,when\,\,\,x=a,\,\,then \\  & a-\left( {{x}^{2}}/a \right),\,\,when\,\,x>a \\ \end{align} \right.\]
  A)
  \[\underset{x\to a}{\mathop{\lim }}\,f(x)=a\] done clear
  B)
  \[f(x)\] is continuous at x = a done clear
  C)
  \[f(x)\] is discontinuous at x = a done clear
  D)
  None of these done clear
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• question_answer33)
  The number of points in (1, 3), where \[f(x)={{a}^{[{{x}^{2}}]}},a>1\], is not differentiable, where [x] denotes the integral part of x.
  A)
  5 done clear
  B)
  7 done clear
  C)
  9 done clear
  D)
  11 done clear
  View Solution play_arrow

• question_answer34)
  A value of c for which conclusion of Mean Value Theorem holds for the function \[f(x)={{\log }_{e}}x\] on the interval [1, 3] is
  A)
  \[lo{{g}_{3}}e\] done clear
  B)
  \[lo{{g}_{e}}3\] done clear
  C)
  \[2lo{{g}_{3}}e\] done clear
  D)
  \[\frac{1}{2}{{\log }_{3}}e\] done clear
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• question_answer35)
  The derivative of ln \[(x+sin\text{ }x)\] with respect to \[(x+cos\text{ }x)\] is
  A)
  \[\frac{1+\cos x}{(x+\sin x)(1-\sin x)}\] done clear
  B)
  \[\frac{1-\cos x}{(x+\sin x)(1+\sin x)}\] done clear
  C)
  \[\frac{1-\cos x}{(x-\sin x)(1+\cos x)}\] done clear
  D)
  \[\frac{1+\cos \,\,x}{(x-sin\,\,x)(1-cos\,\,x)}\] done clear
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• question_answer36)
  What is the derivative of \[{{x}^{3}}\] with respect to\[{{x}^{2}}\]?
  A)
  \[3{{x}^{2}}\] done clear
  B)
  \[\frac{3x}{2}\] done clear
  C)
  x done clear
  D)
  \[\frac{3}{2}\] done clear
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• question_answer37)
  If the derivative of the function\[f(x)=\left\{ \begin{matrix}    a{{x}^{2}}+b & x<-1  \\    b{{x}^{2}}+ax+4 & x\ge -1  \\ \end{matrix} \right.\]is everywhere continuous, then what are the values of a and b?
  A)
  a=2, b=3 done clear
  B)
  a=3, b=2 done clear
  C)
  a=-2, b=-3 done clear
  D)
  a=-3, b=-2 done clear
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• question_answer38)
  Consider the following: 1. \[\underset{x\to 0}{\mathop{\lim }}\,\frac{1}{x}\] exists. 2. \[\underset{x\to 0}{\mathop{\lim }}\,\,\,{{e}^{\frac{1}{x}}}\] does not exist. Which of the above is/are correct?
  A)
  1 only done clear
  B)
  2 only done clear
  C)
  Both 1 and 2 done clear
  D)
  Neither 1 nor 2 done clear
  View Solution play_arrow

• question_answer39)
  If the function \[f(x)=\left[ \frac{{{(x-2)}^{3}}}{a} \right]\sin (x-2)+a\,\cos (x-2),\] [.] denotes the greatest integer function is continuous and differentiable in [4, 6], then
  A)
  \[a\in [8,\,64]\] done clear
  B)
  \[a\in (0,\,8]\] done clear
  C)
  \[a\in [64,\,\infty )\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer40)
  Given \[f(x)=b({{[x]}^{2}}+[x])+1\] for \[x\ge -1\]\[=\sin (\pi (x+a))\] for \[x<-1\]where [x] denotes the integral part of x, then for what values of a, b, the function is continuous at x = -1?
  A)
  \[a=2n+(3/2);b\in R;n\in I\] done clear
  B)
  \[a=4n+2;b\in R;n\in I\] done clear
  C)
  \[a=4n+(3/2);b\in {{R}^{+1}};n\in I\] done clear
  D)
  \[a=4n+1;b\in {{R}^{+}};n\in I\] done clear
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• question_answer41)
  If \[y=\frac{1}{{{t}^{2}}+t-2}\] where \[t=\frac{1}{x-1}\], then find the number of points of discontinuities of \[y=f(x),\],
  A)
  1 done clear
  B)
  2 done clear
  C)
  3 done clear
  D)
  4 done clear
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• question_answer42)
  If \[f(x)=\left| 1-x \right|,\] then the points where \[{{\sin }^{-1}}(f\left| x \right|)\] is non-differentiable are
  A)
  \[\left\{ 0,1 \right\}\] done clear
  B)
  \[\left\{ 0,-1 \right\}\] done clear
  C)
  \[\left\{ 0,1,-1 \right\}\] done clear
  D)
  None of these done clear
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• question_answer43)
  If \[y={{\cot }^{-1}}\left[ \frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}} \right]\], where \[0<x<\frac{\pi }{2}\], then \[\frac{dy}{dx}\] is equal to
  A)
  \[\frac{1}{2}\] done clear
  B)
  2 done clear
  C)
  \[\sin x+\cos x\] done clear
  D)
  \[\sin x-\cos x\] done clear
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• question_answer44)
  A function \[f:R\to R\] is defined as \[f(x)={{x}^{2}}\] for \[x\ge 0\] and \[f(x)=-x\] for \[x<0\]. Consider the following statements in respect of the above function:

  1. The function is continuous at x = 0.

  2. The function is differentiable at x = 0.

  Which of the above statements is/are correct?

  A)
  1 only done clear
  B)
  2 only done clear
  C)
  Both 1 and 2 done clear
  D)
  Neither 1 nor 2 done clear
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• question_answer45)
  A function  f is defined as follows \[f(x)={{x}^{p}}\cos \left( \frac{1}{x} \right),x\ne 0f(0)=0\] What conditions should be imposed on p so that f may be continuous at x = 0?
  A)
  p = 0 done clear
  B)
  p > 0 done clear
  C)
  p < 0 done clear
  D)
  No value of p done clear
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• question_answer46)
  If \[f(x)=x+\frac{x}{1+x}+\frac{x}{{{(1+x)}^{2}}}+....to\,\,\infty \], then at \[x=0,f(x)\]
  A)
  Has no limit done clear
  B)
  Is discontinuous done clear
  C)
  Is continuous but not differentiable done clear
  D)
  Is differentiable done clear
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• question_answer47)
  Let f be a continuous function on R such that\[f(1/4n)=(\sin {{e}^{n}}){{e}^{-{{n}^{2}}}}+\frac{{{n}^{2}}}{{{n}^{2}}+1}\]. Then the value of f (0) is
  A)
  1 done clear
  B)
  1/2 done clear
  C)
  0 done clear
  D)
  None of these done clear
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• question_answer48)
  Suppose that \[f(0)=-3\] and \[f'(x)=\,\le 5\] for all values of x. Then, the largest value which f(2) can attain is????               
  A)
  7 done clear
  B)
  10 done clear
  C)
  2 done clear
  D)
  9 done clear
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• question_answer49)
  Let f'(x) be continuous at x = 0 and f"(0) = 4. Then value of \[\underset{x\to 0}{\mathop{\lim }}\,\frac{2f(x)-3f(2x)+f(4x)}{{{x}^{2}}}\] is
  A)
  12 done clear
  B)
  10 done clear
  C)
  6 done clear
  D)
  4 done clear
  View Solution play_arrow

• question_answer50)
  If function \[\text{f(x)=}\left\{ \begin{align}   & \text{x,if}\,\,\text{x}\,\,\text{is}\,\,\text{rational} \\  & \text{1-x,if}\,\,\text{x}\,\,\text{is}\,\,\text{irrational} \\ \end{align} \right.\text{,then}\] the number of points at which f(x) is continuous, is-
  A)
  \[\infty \] done clear
  B)
   1 done clear
  C)
  0 done clear
  D)
  None of these done clear
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• question_answer51)
  Consider the following in respect of the function \[f(x)=\left\{ \begin{matrix}    2+x, & x\ge 0  \\    2-x, & x<0  \\ \end{matrix} \right.\]

  1. \[\underset{x\to 1}{\mathop{\lim \,f}}\,(x)\] does not exist.

  2. f(x) is differentiable at x = 0

  3. f(x) is continuous at x = 0

  Which of the above statements is/are correct?

  A)
  1 only done clear
  B)
  3 only done clear
  C)
  2 and 3 only done clear
  D)
  1 and 3 only done clear
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• question_answer52)
  If \[{{x}^{a}}{{y}^{b}}={{(x-y)}^{a+b}},\] then the value of \[\frac{dy}{dx}-\frac{y}{x}\] is equal to
  A)
  \[\frac{a}{b}\] done clear
  B)
  \[\frac{b}{a}\] done clear
  C)
  1 done clear
  D)
  0 done clear
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• question_answer53)
  If \[f(x)={{(x+1)}^{\cot \,x}}\] is continuous at\[x=0\], then what is f (0) equal to?
  A)
  1 done clear
  B)
  e done clear
  C)
  \[\frac{1}{e}\] done clear
  D)
  \[{{e}^{2}}\] done clear
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• question_answer54)
  If the function \[f(x)=\left\{ \begin{align}   & \frac{k\,\,\cos \,\,x}{\pi -2x},when\,x\ne \frac{\pi }{2} \\  & 3,when\,x=\frac{\pi }{2} \\ \end{align} \right.\,\,be\]continuous at \[x=\frac{\pi }{2},\] then k =
  A)
  3 done clear
  B)
  6 done clear
  C)
  12 done clear
  D)
  None of these done clear
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• question_answer55)
  Which of the following functions is not differentiable at\[x=1\]?
  A)
  \[f(x)=({{x}^{2}}-1)\left| (x-1)(x-2) \right|\] done clear
  B)
  \[f(x)=\sin (\left| x-1 \right|)-\left| x-1 \right|\] done clear
  C)
  \[f(x)=\tan (\left| x-1 \right|)+\left| x-1 \right|\] done clear
  D)
  None of these done clear
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• question_answer56)
  If \[y=\sqrt{\log x+\sqrt{\log x+\sqrt{\log x+.....\infty }}}\], then \[\frac{dy}{dx}=\]
  A)
  \[\frac{x}{2y-1}\] done clear
  B)
  \[\frac{x}{2y+1}\] done clear
  C)
  \[\frac{1}{x(2y-1)}\] done clear
  D)
  \[\frac{1}{x(1-2y)}\] done clear
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• question_answer57)
  Which of the following function(s) has/have removable discontinuity at \[x=1\]?
  A)
  \[f(x)=\frac{1}{In\left| x \right|}\] done clear
  B)
  \[f(x)=\frac{1}{{{x}^{3}}-1}\] done clear
  C)
  \[f(x)={{2}^{{{2}^{\frac{1}{1-x}}}}}\] done clear
  D)
  \[f(x)=\frac{\sqrt{x+1}-\sqrt{2x}}{{{x}^{2}}-x}\] done clear
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• question_answer58)
  What is \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sqrt{1+x}-1}{x}\] equal to?
  A)
  0 done clear
  B)
  \[\frac{1}{2}\] done clear
  C)
  1 done clear
  D)
  \[-\frac{1}{2}\] done clear
  View Solution play_arrow

• question_answer59)
  Let \[f(x)=\left\{ \begin{matrix}    3x-4, & 0\le x\le 2  \\    2x+\ell , & 2<x\le 9  \\ \end{matrix} \right.\] If f is continuous at x = 2, then what is the value of\[\ell \]?
  A)
  0 done clear
  B)
  2 done clear
  C)
  -2 done clear
  D)
  -1 done clear
  View Solution play_arrow

• question_answer60)
  Let \[f(x)=[{{x}^{3}}-3],[x]=\]G.I.F. Then the no. of points in the interval (1, 2) where function is discontinuous is
  A)
  5 done clear
  B)
  4 done clear
  C)
  6 done clear
  D)
  3 done clear
  View Solution play_arrow

• question_answer61)
  If \[\theta \] are the points of discontinuity of \[f(x)=\underset{n\to \infty }{\mathop{\lim }}\,{{\cos }^{2n}}x\] then the value of sin \[\theta \] is
  A)
  0 done clear
  B)
  1 done clear
  C)
  -1 done clear
  D)
  ½ done clear
  View Solution play_arrow

• question_answer62)
  Let \[f:R\to R\] be a function defined by f(x) max\[\{x,\text{ }{{x}^{3}}\}\]. The set of all points where f(x) is NOT differentiable is
  A)
  {-1, 1} done clear
  B)
  {-1, 0} done clear
  C)
  {0, 1} done clear
  D)
  {-1, 0, 1} done clear
  View Solution play_arrow

• question_answer63)
  The derivative of \[{{\sin }^{-1}}\left( \frac{2x}{1+{{x}^{2}}} \right)\] with respect to \[{{\cos }^{-1}}\left[ \frac{1-{{x}^{2}}}{1+{{x}^{2}}} \right]\] is equal to:
  A)
  1 done clear
  B)
  -1 done clear
  C)
  2 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer64)
  If \[y={{(1+1/x)}^{x}}\] then \[\frac{2\sqrt{{{y}_{2}}(2)+1/8}}{(log\,3/2-1/3)}\] is equal to-
  A)
  3 done clear
  B)
  4 done clear
  C)
  1 done clear
  D)
  2 done clear
  View Solution play_arrow

• question_answer65)
  If \[f(x)=\frac{1}{1-x}\], then the points of discontinuity of the function \[f[f\{f(x)\}]\] are
  A)
  {0, -1} done clear
  B)
  {0, 1} done clear
  C)
  {1, -1} done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer66)
  What is the derivative of \[\left| x-1 \right|\] at\[x=2\]?
  A)
  -1 done clear
  B)
  0 done clear
  C)
  1 done clear
  D)
  Derivative does not exist done clear
  View Solution play_arrow

• question_answer67)
  Let \[f:R\to R\] be defined as \[f(x)=sin(\left| x \right|)\] Which one of the following is correct?
  A)
  f is not differentiable only at 0 done clear
  B)
  f is differentiable at 9 only done clear
  C)
  f is differentiable everywhere done clear
  D)
  f is non-differentiable at many points done clear
  View Solution play_arrow

• question_answer68)
  Let f be a function which is continuous and differentiable for all real x. If \[f(2)=-4\] and \[f'(x)\ge 6\] for all \[x\in [2,4],\] then
  A)
  \[f(4)<8\] done clear
  B)
  \[f(4)\ge 8\] done clear
  C)
  \[f(4)\ge 12\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer69)
  The function \[f(x)=\sin ({{\log }_{e}}\left| x \right|),x\ne 0\], and 1 if \[x=0\]
  A)
  Is continuous at \[x=0\] done clear
  B)
  Has removable discontinuity at \[x=0\] done clear
  C)
  Has jump discontinuity at \[x=0\] done clear
  D)
  Has oscillating discontinuity at \[x=0\] done clear
  View Solution play_arrow

• question_answer70)
  \[\frac{{{d}^{n}}}{d{{x}^{n}}}(log\,x)=\]
  A)
  \[\frac{(n-1)!}{{{x}^{n}}}\] done clear
  B)
  \[\frac{n!}{{{x}^{n}}}\] done clear
  C)
  \[\frac{(n-2)!}{{{x}^{n}}}\] done clear
  D)
  \[{{(-1)}^{n-1}}\frac{(n-1)!}{{{x}^{n}}}\] done clear
  View Solution play_arrow

• question_answer71)
  If \[y={{(1+1/x)}^{x}}\] then \[\frac{2\sqrt{{{y}_{2}}(2)+1/8}}{(log\,3/2-1/3)}\] is equal to-
  A)
  3 done clear
  B)
  4 done clear
  C)
  1 done clear
  D)
  2 done clear
  View Solution play_arrow

• question_answer72)
  The set of points of discontinuity of the function\[f(x)=\underset{n\to \infty }{\mathop{\lim }}\,\frac{{{(2\,\sin \,x)}^{2}}^{n}}{{{3}^{n}}-{{(2\cos \,x)}^{2n}}}\] is given by
  A)
  R done clear
  B)
  \[\left\{ n\pi \pm \frac{\pi }{3},n\in I \right\}\] done clear
  C)
  \[\left\{ n\pi \pm \frac{\pi }{6},n\in I \right\}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer73)
  Which of the following function is continuous at for all value of x?         

  (i) \[f\left( x \right)\] =sgn\[({{x}^{3}}-x)\]

  (ii)  \[f\left( x \right)\] =sgn \[(2\cos x-1)\]

  (iii) \[f\left( x \right)\] =sgn \[({{x}^{2}}-2x+3)\]

   
  A)
  Only (i) done clear
  B)
  Only (iii) done clear
  C)
  Both (ii) and (iii) done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer74)
  Which one of the following statements is correct in respect of the function\[f(x)={{x}^{3}}\sin x\]?
  A)
  f'(x) changes sign from positive to negative at x = 0 done clear
  B)
  f '(x) changes sign from negative to positive at x = 0 done clear
  C)
  Does not change sign at x = 0 done clear
  D)
  \[f''(0)\ne 0\] done clear
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• question_answer75)
  What is the set of all points, where the function \[f(x)=\frac{x}{1+\left| x \right|}\] is differentiable?
  A)
  \[(-\infty ,\infty )\] only done clear
  B)
  \[(0,\infty )\] only done clear
  C)
  \[(-\infty ,0)\cup (0,\infty )\]only done clear
  D)
  \[(-\infty ,0)\] only done clear
  View Solution play_arrow

• question_answer76)
  The number of points of non-differentiability for\[f(x)=max\,\{\left| \left| x \right|-1 \right|,1/2\}\] is
  A)
  4 done clear
  B)
  3 done clear
  C)
  2 done clear
  D)
  5 done clear
  View Solution play_arrow

• question_answer77)
  Which of the following functions have finite number of points of discontinuity? (where \[[\cdot ]\]represents greatest integer functions)
  A)
  \[tan\,x\] done clear
  B)
  \[x\text{  }\!\![\!\!\text{ }x]\] done clear
  C)
  \[\frac{\left| x \right|}{x}\] done clear
  D)
  \[\sin \,[n\pi x]\] done clear
  View Solution play_arrow

• question_answer78)
  If \[f''(x)=-f(x)\] and \[g(x)=f'(x)\] and \[F(x)={{\left( f\left( \frac{x}{2} \right) \right)}^{2}}+{{\left( g\left( \frac{x}{2} \right) \right)}^{2}}\] and given that \[F(5)=5\], then F (10) is equal to-
  A)
  5 done clear
  B)
  10 done clear
  C)
  0 done clear
  D)
  15 done clear
  View Solution play_arrow

• question_answer79)
  Let \[f:R\to R\] be a function defined by\[f(x)=min\{x+1,\left| x \right|+1\}\], Then which of the following is true?
  A)
  \[f(x)\] is differentiable everywhere done clear
  B)
  \[f(x)\] is not differentiable at x = 0 done clear
  C)
  \[f(x)\ge 1\] for all \[x\in R\] done clear
  D)
  \[f(x)\] is not differentiable at \[x=1\] done clear
  View Solution play_arrow

• question_answer80)
  The function \[f(x)=\frac{1-\sin x+\cos x}{1+\sin x+\cos x}\] is not defined at \[x=\pi \]. The value of \[f(\pi )\], so that \[f(x)\] is continuous at \[x=\pi \], is
  A)
  \[-\frac{1}{2}\] done clear
  B)
  \[\frac{1}{2}\] done clear
  C)
  \[-1\] done clear
  D)
  1 done clear
  View Solution play_arrow

• question_answer81)
  If \[{{(x-a)}^{2}}+{{(y-b)}^{2}}={{c}^{2}}\], for some c > 0, then\[\frac{{{\left[ 1+{{\left( \frac{dy}{dx} \right)}^{2}} \right]}^{\frac{3}{2}}}}{\frac{{{d}^{2}}y}{d{{x}^{2}}}}\] is
  A)
  Is a constant dependent on a done clear
  B)
  Is a constant dependent on b done clear
  C)
  Is a constant independent of a and b done clear
  D)
  0 done clear
  View Solution play_arrow

• question_answer82)
  Which one of the following is correct in respect of the function \[f(x)=\left| x \right|+{{x}^{2}}\]
  A)
  \[f(x)\] is not continuous at x = 0 done clear
  B)
  \[f(x)\] is differentiable at x = 0 done clear
  C)
  \[f(x)\] is continuous but not differentiable at x = 0 done clear
  D)
  None of the above done clear
  View Solution play_arrow

• question_answer83)
  If \[f(x)=\cos \left[ \frac{\pi }{x} \right]\cos \left( \frac{\pi }{2}(x-1) \right);\] where [x] is the greatest integer function of x, then f(x) is continuous at
  A)
  x = 0 done clear
  B)
  x = 1, 2 done clear
  C)
  x = 0, 2, 4 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer84)
  If the mean value theorem is\[f(b)-f(a)=(b-a)f'(c)\]. Then, for the function \[{{x}^{2}}-2x+3\] in \[\left[ 1,\frac{3}{2} \right]\] the value of c is
  A)
  6/5 done clear
  B)
  5/4 done clear
  C)
  4/3 done clear
  D)
  7/6 done clear
  View Solution play_arrow

• question_answer85)
  Which of the following is correct for \[f(x)=\left\{ \begin{matrix}    (x-e){{2}^{-{{2}^{\left( \frac{1}{(e-x)} \right)}},}} & x\ne e\,\,at\,\,x=e  \\    0, & x=e  \\ \end{matrix} \right.\]
  A)
  f(x) is discontinuous at x = e done clear
  B)
  f(x) is differentiable at x = e done clear
  C)
  f(x) is non-differentiable at x = e done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer86)
  The number of points at which the function \[f(x)=\frac{1}{{{\log }_{e}}\left| x \right|}\] is discontinuous, is
  A)
  1 done clear
  B)
  2 done clear
  C)
  3 done clear
  D)
  4 done clear
  View Solution play_arrow

• question_answer87)
  If \[y={{\tan }^{-1}}\left( \frac{{{2}^{x}}}{1+{{2}^{2x+1}}} \right),\]then \[\frac{dy}{dx}at\,x=0\] is
  A)
  \[\frac{3}{5}\log \,2\] done clear
  B)
  \[\frac{2}{5}\log \,2\] done clear
  C)
  \[-\frac{3}{2}\log \,2\] done clear
  D)
  \[\log \,2\left( \frac{-1}{10} \right)\] done clear
  View Solution play_arrow

• question_answer88)
  If \[f(x)={{\log }_{x}}(In\,\,x)\], then at \[x=e,f'(x)\]equals-
  A)
  0 done clear
  B)
  1 done clear
  C)
  e done clear
  D)
  1/e done clear
  View Solution play_arrow

• question_answer89)
  If \[s=\sqrt{{{t}^{2}}+1}\], then \[\frac{{{d}^{2}}s}{d{{t}^{2}}}\] is equal to
  A)
  \[\frac{1}{s}\] done clear
  B)
  \[\frac{1}{{{s}^{2}}}\] done clear
  C)
  \[\frac{1}{{{s}^{3}}}\] done clear
  D)
  \[\frac{1}{{{s}^{4}}}\] done clear
  View Solution play_arrow

• question_answer90)
  What is the derivative of\[{{\tan }^{-1}}\left( \frac{\sqrt{1+{{x}^{2}}}-1}{x} \right)\] with respect to \[{{\tan }^{-1}}x\]?
  A)
  0 done clear
  B)
  \[\frac{1}{2}\] done clear
  C)
  1 done clear
  D)
  x done clear
  View Solution play_arrow

• question_answer91)
  If the function\[f(x)=\frac{x(x-2)}{{{x}^{2}}-4},x\ne \pm 2\]is continuous at\[x=2\], then what is \[f(2)\] equal to?
  A)
  0 done clear
  B)
  \[\frac{1}{2}\] done clear
  C)
  1 done clear
  D)
  2 done clear
  View Solution play_arrow

• question_answer92)
  If the functions \[f(x)\] and \[g(x)\] are continuous in [a, b] and differentiable in (a, b), then the equation \[\left| \begin{matrix}    f(a) & f(b)  \\    g(a) & g(b)  \\ \end{matrix} \right|=(b-a)\left| \begin{matrix}    f(a) & f'(x)  \\    g(a) & g'(x)  \\ \end{matrix} \right|\] has in the interval [a, b]
  A)
  At least one root done clear
  B)
  Exactly one root done clear
  C)
  At most one root done clear
  D)
  No root done clear
  View Solution play_arrow

• question_answer93)
  If \[u=f({{x}^{3}}),v=g({{x}^{2}}),f'(x)=\cos x\] and \[g'(x)=\sin x,\] then \[\frac{du}{dv}=\]
  A)
  \[\frac{1}{2}x\cos {{x}^{3}}\cos ec\,{{x}^{2}}\] done clear
  B)
  \[\frac{3}{2}x\cos {{x}^{3}}\cos ec\,{{x}^{2}}\] done clear
  C)
  \[\frac{1}{2}x\sec {{x}^{3}}\sin \,{{x}^{2}}\] done clear
  D)
  \[\frac{3}{2}x\sec {{x}^{3}}\cos ec\,{{x}^{2}}\] done clear
  View Solution play_arrow

• question_answer94)
  What is the value of k for which the following function f(x) is continuous for all x? \[f(x)=\left\{ \begin{align}   & \frac{{{x}^{3}}-3x+2}{{{(x-1)}^{2}}},for\,\,x\ne 1 \\  & k\,\,\,\,\,\,\,\,,for\,\,x=1 \\ \end{align} \right.\]
  A)
  3 done clear
  B)
  2 done clear
  C)
  1 done clear
  D)
  -1 done clear
  View Solution play_arrow

• question_answer95)
  Suppose\[f(x)={{e}^{ax}}+{{e}^{bx}}\], where\[a\ne b\], and that \[f{{\,}^{n}}(x)-2f'(x)-15f(x)=0\] for all x. Then the product ab is
  A)
  25 done clear
  B)
  9 done clear
  C)
  -15 done clear
  D)
  -9 done clear
  View Solution play_arrow

• question_answer96)
  Let \[3f(x)-2f(1/x)=x,\] then \[f'(2)\] is equal to
  A)
  \[\frac{2}{7}\] done clear
  B)
  \[\frac{1}{2}\] done clear
  C)
  \[2\] done clear
  D)
  \[\frac{7}{2}\] done clear
  View Solution play_arrow

• question_answer97)
  If \[{{I}_{n}}=\frac{{{d}^{n}}}{d{{x}^{n}}}({{x}^{n}}\log \,x)\], then \[{{I}_{n}}-n{{I}_{n-1}}=\]
  A)
  n done clear
  B)
  \[n-1\] done clear
  C)
  \[n!\] done clear
  D)
  \[\left( n-1 \right)!\] done clear
  View Solution play_arrow

• question_answer98)
  If \[\left. \begin{align}   & f(x)=sin\text{ }x,when\,\,x\,\,is\,\,rational \\  & \,\,\,\,\,\,\,\,\,\,=\,\cos x,when\,\,x\,\,is\,\,irrational \\ \end{align} \right\}\]
  A)
  Discontinuous at \[x=n\pi +\pi /4\] done clear
  B)
  Continuous at \[x=n\pi +\pi /4\] done clear
  C)
  Discontinuous at all x done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer99)
  Let \[f(x)=\left\{ \begin{align}   & \frac{1-{{\sin }^{3}}x}{3{{\cos }^{2}}x},x<\frac{\pi }{2} \\  & p,x=\frac{\pi }{2} \\  & \frac{q(1-\sin x)}{{{(\pi -2x)}^{2}}},x>\frac{\pi }{2} \\ \end{align} \right.\] If f(x) is continuous at \[x=\frac{\pi }{2},(p,q)=\]
  A)
  \[(1,4)\] done clear
  B)
  \[\left( \frac{1}{2},2 \right)\] done clear
  C)
  \[\left( \frac{1}{2},4 \right)\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer100)
  If \[y=\frac{(a-x)\sqrt{a-x}-(b-x)\sqrt{x-b}}{\sqrt{a-x}+\sqrt{x-b}}\], then \[\frac{dy}{dx}\] wherever it is defined is
  A)
  \[\frac{x+(a+b)}{\sqrt{(a-x)(x-b)}}\] done clear
  B)
  \[\frac{2x-a-b}{2\sqrt{a-x}\sqrt{x-b}}\] done clear
  C)
  \[-\frac{(a+b)}{2\sqrt{(a-x)(x-b)}}\] done clear
  D)
  \[\frac{2x+(a+b)}{2\sqrt{(a-x)(x-b)}}\] done clear
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