Limits, Continuity & Differentiability · JEE Main previous year questions

2005 6 questions

1. Q202005 · 28 Apr 2005 · Shift 1

   Let $f:R\to R$ be a differentiable function having $f(2)=6$, $f^{\prime }(2)=\left(\frac{1}{48}\right)$. Then ${\lim }_{x\to 2}{\int }_6^{f(x)}\frac{4t^3}{x-2}dt$ equals :

   1. A.$24$
   2. B.$36$
   3. C.$12$
   4. D.$18$
   Show answer

   Answer: D

2. Q322005 · 28 Apr 2005 · Shift 1

   Let $f(x)$ be a non-negative continuous function such that the area bounded by the curve $y=f(x)$, $x$-axis and the ordinates $x=\frac{\pi }{4}$ and $x=\beta >\frac{\pi }{4}$ is $\left(\beta \sin \beta +\frac{\pi }{4}\cos \beta +\sqrt{2}\beta \right)$. Then $f\left(\frac{\pi }{2}\right)$ is

   1. A.$\left(\frac{\pi }{4}+\sqrt{2}-1\right)$
   2. B.$\left(\frac{\pi }{4}-\sqrt{2}+1\right)$
   3. C.$\left(1-\frac{\pi }{4}-\sqrt{2}\right)$
   4. D.$\left(1-\frac{\pi }{4}+\sqrt{2}\right)$
   Show answer

   Answer: D

3. Q472005 · 28 Apr 2005 · Shift 1

   Let $\alpha$ and $\beta$ be the distinct roots of $a{x}^2+bx+c=0$, then $$\underset{x\to \alpha }{\lim }\frac{1-\cos (a{x}^2+bx+c)}{(x-\alpha {)}^2}$$ is equal to

   1. A.$\frac{a^2(\alpha -\beta {)}^2}{2}$
   2. B.$0$
   3. C.$-\frac{a^2(\alpha -\beta {)}^2}{2}$
   4. D.$\frac{(\alpha -\beta {)}^2}{2}$
   Show answer

   Answer: A

4. Q482005 · 28 Apr 2005 · Shift 1

   Suppose $f(x)$ is differentiable at $x=1$ and $$\underset{h\to 0}{\lim }\frac{1}{h}f(1+h)=5,$$ then $f^{\prime }(1)$ equals

   1. A.$3$
   2. B.$4$
   3. C.$5$
   4. D.$6$
   Show answer

   Answer: C

5. Q732005 · 28 Apr 2005 · Shift 1

   $$\underset{n\to \infty }{\lim }\left[\frac{1}{n^2}{\sec }^2\frac{1}{n^2}+\frac{2}{n^2}{\sec }^2\frac{4}{n^2}\cdots +\frac{1}{n}{\sec }^{2}1\right]$$ equals

   1. A.$\frac{1}{2}\sec 1$
   2. B.$\frac{1}{2}\mathrm{cosec}1$
   3. C.$\tan 1$
   4. D.$\frac{1}{2}\tan 1$
   Show answer

   Answer: D

6. Q742005 · 28 Apr 2005 · Shift 1

   Let $f$ be differentiable for all $x$. If $f(1)=-2$ and $f^{\prime }(x)\ge 2$ for $x\in [1,6]$, then

   1. A.$f(6)\ge 8$
   2. B.$f(6)<8$
   3. C.$f(6)<5$
   4. D.$f(6)=5$
   Show answer

   Answer: A