JEE Main & Advanced Mathematics Differentiation Question Bank

done Mock Test - Limits and Derivatives Total Questions - 25

• question_answer1)
  If \[f(x)=0\]is a quadratic equation such that \[f(-\pi )=f(\pi )=0\]and \[f\left( \frac{\pi }{2} \right)=-\frac{3{{\pi }^{2}}}{4}\], then \[\underset{x\to -\pi }{\mathop{\lim }}\,\frac{f(x)}{\sin (sinx)}\]is equal to
  A)
  0                                  done clear
  B)
   \[\pi \] done clear
  C)
   \[2\pi \]             done clear
  D)
  none of these done clear
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• question_answer2)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{1}{x}{{\cos }^{-1}}\left( \frac{1-{{x}^{2}}}{1+{{x}^{2}}} \right)\]is equal to
  A)
  1                      done clear
  B)
  0 done clear
  C)
  2                     done clear
  D)
  none of these done clear
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• question_answer3)
  The value of \[\underset{x\to 2}{\mathop{\lim }}\,\frac{\sqrt{1+\sqrt{2+x}}-\sqrt{3}}{x-2}\] is
  A)
  \[\frac{1}{8\sqrt{3}}\]   done clear
  B)
  \[\frac{1}{4\sqrt{3}}\] done clear
  C)
  0                     done clear
  D)
  None of these done clear
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• question_answer4)
  If \[\underset{x\to \infty }{\mathop{\lim }}\,\left\{ \frac{{{x}^{3}}+1}{{{x}^{2}}+1}-(ax+b) \right\}=2\], then
  A)
  \[a=1,\text{ }b=1\]       done clear
  B)
  \[a=1,\text{ }b=2\] done clear
  C)
  \[a=1,\text{ }v=-2\]      done clear
  D)
  None of these done clear
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• question_answer5)
  \[\underset{n\to \infty }{\mathop{\lim }}\,\sum\limits_{x=1}^{20}{{{\cos }^{2n}}(x-10)}\]is equal to
  A)
  0                      done clear
  B)
  1 done clear
  C)
  19                    done clear
  D)
  20 done clear
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• question_answer6)
  The value of \[\underset{m\to \infty }{\mathop{\lim }}\,{{\left( \cos \frac{x}{m} \right)}^{m}}\]is
  A)
  1                      done clear
  B)
  \[e\] done clear
  C)
  \[{{e}^{-1}}\]             done clear
  D)
  none of these done clear
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• question_answer7)
  The value of \[\underset{x\to 2}{\mathop{\lim }}\,\frac{{{2}^{x}}+{{2}^{3-x}}-6}{\sqrt{{{2}^{-x}}}-{{2}^{1-x}}}\]is
  A)
  16                     done clear
  B)
                                      8 done clear
  C)
  4                     done clear
  D)
  2 done clear
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• question_answer8)
  \[\underset{n\to \infty }{\mathop{\lim }}\,{{\left( \frac{{{n}^{2}}-n+1}{{{n}^{2}}-n-1} \right)}^{n(n-1)}}\]is equal to
  A)
  \[e\]                  done clear
  B)
  \[{{e}^{2}}\] done clear
  C)
  \[{{e}^{-1}}\]             done clear
  D)
  1 done clear
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• question_answer9)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin ({{x}^{2}})}{in(cos(2{{x}^{2}}-x))}\]is equal to
  A)
  2                      done clear
  B)
  -2 done clear
  C)
  1                     done clear
  D)
  -1 done clear
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• question_answer10)
  \[\underset{x\to \infty }{\mathop{\lim }}\,{{\left( \frac{{{x}^{2}}+5x+3}{{{x}^{2}}+x+3} \right)}^{1/x}}\]is equal to
  A)
  \[{{e}^{4}}\]              done clear
  B)
  \[{{e}^{2}}\] done clear
  C)
  \[{{e}^{3}}\]              done clear
  D)
  1 done clear
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• question_answer11)
  If \[y={{x}^{({{x}^{x}})}}\], then \[\frac{dy}{dx}\]is
  A)
   \[y\left[ {{x}^{x}}(logex)logx+{{x}^{x}} \right]\] done clear
  B)
   \[y\left[ {{x}^{x}}(logex)logx+x \right]\] done clear
  C)
   \[y\left[ {{x}^{x}}(logex)logx+{{x}^{x-1}} \right]\] done clear
  D)
   \[y\left[ {{x}^{x}}(lo{{g}_{e}}x)logx+{{x}^{x-1}} \right]\] done clear
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• question_answer12)
  If \[y={{\left( x+\sqrt{{{x}^{2}}+{{a}^{2}}} \right)}^{n}}\], then \[\frac{dy}{dx}\]is
  A)
  \[\frac{ny}{\sqrt{{{x}^{2}}+{{a}^{2}}}}\]    done clear
  B)
  \[-\frac{ny}{\sqrt{{{x}^{2}}+{{a}^{2}}}}\] done clear
  C)
  \[\frac{nx}{\sqrt{{{x}^{2}}+{{a}^{2}}}}\]    done clear
  D)
  \[-\frac{nx}{\sqrt{{{x}^{2}}+{{a}^{2}}}}\] done clear
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• question_answer13)
  If \[y=1+x+\frac{{{x}^{2}}}{2!}+\frac{{{x}^{3}}}{3!}+...+\frac{{{x}^{n}}}{n!}\], then \[\frac{dy}{dx}\]is equal to
  A)
  y                      done clear
  B)
  \[y+\frac{{{x}^{n}}}{n!}\] done clear
  C)
  \[y-\frac{{{x}^{n}}}{n!}\]       done clear
  D)
  \[y-1-\frac{{{x}^{n}}}{n!}\] done clear
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• question_answer14)
  If \[y=\sqrt{\log x+\sqrt{\log x+\sqrt{\log x+...\infty },}}\]then \[\frac{dy}{dx}\]is
  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_answer15)
  If \[y=a{{e}^{mx}}+b{{e}^{-mx}}\], then \[\frac{{{d}^{2}}y}{d{{x}^{2}}}-{{m}^{2}}y\]is equal to
  A)
  \[{{m}^{2}}(a{{e}^{mx}}-b{{e}^{-mx}})\] done clear
  B)
  1 done clear
  C)
  0                     done clear
  D)
  none of these done clear
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• question_answer16)
  If \[y={{\tan }^{-1}}\sqrt{\frac{x+1}{x-1}}\]. Then \[\frac{dy}{dx}\]is
  A)
  \[\frac{-1}{2\left| x \right|\sqrt{{{x}^{2}}-1}}\] done clear
  B)
  \[\frac{-1}{2x\sqrt{{{x}^{2}}-1}}\] done clear
  C)
  \[\frac{1}{2x\sqrt{{{x}^{2}}-1}}\]       done clear
  D)
  none of these done clear
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• question_answer17)
  If then \[\frac{dy}{dx}\] is
  A)
  \[\frac{2xy}{2y-{{x}^{2}}}\]   done clear
  B)
  \[\frac{xy}{y+{{x}^{2}}}\] done clear
  C)
  \[\frac{xy}{y-{{x}^{2}}}\] done clear
  D)
  \[\frac{2xy}{2+{{\frac{x}{y}}^{2}}}\] done clear
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• question_answer18)
  If \[\sqrt{1-{{x}^{2}}}+\sqrt{1-{{y}^{2}}}=a(x-y)\], then \[\frac{dy}{dx}\]=
  A)
   \[\sqrt{\frac{1-{{x}^{2}}}{1-{{y}^{2}}}}\]                done clear
  B)
   \[\sqrt{\frac{1-{{y}^{2}}}{1-{{x}^{2}}}}\] done clear
  C)
   \[\sqrt{\frac{{{x}^{2}}-1}{1-{{y}^{2}}}}\]                done clear
  D)
   \[\sqrt{\frac{{{y}^{2}}-1}{1-{{x}^{2}}}}\] done clear
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• question_answer19)
  Let y be an implicit function of x defined by \[{{x}^{2x}}-2{{x}^{x}}\cot y-1=0\]. Then y'(1) equals
  A)
  -1                     done clear
  B)
  1 done clear
  C)
  \[log\text{ }2\]                done clear
  D)
  \[-log\text{ }2\] done clear
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• question_answer20)
  If \[{{x}^{m}}{{y}^{n}}={{(x+y)}^{m+n}}\], then \[dy/dx\]is equal to
  A)
  \[\frac{y}{x}\]              done clear
  B)
  \[\frac{x+y}{xy}\] done clear
  C)
  \[xy\]                done clear
  D)
  \[\frac{x}{y}\] done clear
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• question_answer21)
  If\[f(x)=\frac{2}{x-3},g(x)=\frac{x-3}{x+4},\]and\[h(x)=-\frac{2(2x+1)}{{{x}^{2}}+x-12}\], then \[\underset{x\to 3}{\mathop{\lim }}\,[f(x)+g(x)+h(x)]\]is __________.
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• question_answer22)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{(1-cos2x)(3+cosx)}{x\tan 4x}\] is equal to _________.
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• question_answer23)
  if\[f(0)=0,f'(0)=2\], then the derivative of
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• question_answer24)
  Suppose the function \[f(x)-f(2x)\]has the derivative 5 at x=1 and derivative 7 at x = 2. The derivative of the function \[f(x)-f(4x)\]at x=1has the value equal to __________.
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• question_answer25)
  If \[f(x+y)=f(x)f(y)\forall x,y\]and\[f(5)=2,f'(0)=3\], then f'(5) is equal to _______.
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