Limits, Continuity & Differentiability · JEE Main previous year questions

2012 5 questions

1. Q82012 · 26 May 2012 · Shift 1

   ${\lim }_{x\to 0}\frac{\sin (\pi {\cos }^{2}x)}{x^2}$ equals

   1. A.$-\pi$
   2. B.1
   3. C.$-1$
   4. D.$\pi$
   Show answer

   Answer: D

2. Q132012 · 26 May 2012 · Shift 1

   If $f(x)=a|\sin x|+b{e}^{|x|}+c|x{|}^3$, where $a,b,c\in R$, is differentiable at $x=0$, then

   1. A.$a=0,b\text{ and }c\text{ are any real numbers}$
   2. B.$c=0,a=0,b\text{ is any real number}$
   3. C.$b=0,c=0,a\text{ is any real number}$
   4. D.$a=0,b=0,c\text{ is any real number}$
   Show answer

   Answer: D

3. Q142012 · 26 May 2012 · Shift 1

   Let $f:(-\infty ,\infty )\to (-\infty ,\infty )$ be defined by $f(x)=x^3+1$ Statement 1: The function $f$ has a local extremum at $x=0$ Statement 2: The function $f$ is continuous and differentiable on $(-\infty ,\infty )$ and $f^{\prime }(0)=0$

   1. A.Statement 1 is true, Statement 2 is false.
   2. B.Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.
   3. C.Statement 1 is true, Statement 2 is true, Statement 2 is not the correct explanation for Statement 1.
   4. D.Statement 1 is false, Statement 2 is true.
   Show answer

   Answer: D

4. Q122012 · 12 May 2012 · Shift 1

   Statement 1: A function $f:R\to R$ is continuous at $x_0$ if and only if ${\lim }_{x\to x_0}f(x)$ exists and ${\lim }_{x\to x_0}f(x)=f(x_0)$. Statement 2: A function $f:R\to R$ is discontinuous at $x_0$ if and only if, ${\lim }_{x\to x_0}f(x)$ exists and ${\lim }_{x\to x_0}f(x)\ne f(x_0)$.

   1. A.Statement 1 is true, Statement 2 is true; Statement 2 is not a correct explanation of Statement 1.
   2. B.Statement 1 is false, Statement 2 is true.
   3. C.Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation of Statement 1.
   4. D.Statement 1 is true, Statement 2 is false.
   Show answer

   Answer: D

5. Q82012 · 7 May 2012 · Shift 1

   ${\lim }_{x\to 0}\left(\frac{x-\sin x}{x}\right)\sin \left(\frac{1}{x}\right)$

   1. A.equals 1
   2. B.equals 0
   3. C.does not exist
   4. D.equals $-1$
   Show answer

   Answer: B