Limits, Continuity & Differentiability · JEE Main previous year questions

2018 5 questions

1. Q82018 · 16 Apr 2018 · Shift 1

   ${\lim }_{x\to 0}\frac{(27+x{)}^{\frac{1}{3}}-3}{9-(27+x{)}^{\frac{2}{3}}}$ equals.

   1. A.$\frac{1}{3}$
   2. B.$-\frac{1}{3}$
   3. C.$-\frac{1}{6}$
   4. D.$\frac{1}{6}$
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   Answer: C

2. Q92018 · 16 Apr 2018 · Shift 1

   If the function f defined as $f(x)=\frac{1}{x}-\frac{k-1}{e^{2x}-1},x\ne 0$, is continuous at $x=0$, then the ordered pair (k, f(0)) is equal to :

   1. A.(3, 2)
   2. B.(3, 1)
   3. C.(2, 1)
   4. D.$\left(\frac{1}{3},2\right)$
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   Answer: B

3. Q92018 · 15 Apr 2018 · Shift 1

   If $f(x)=\left|\begin{array}{ccc}\cos x& x& 1\\ 2\sin x& x^2& 2x\\ \tan x& x& 1\end{array}\right|$, then ${\lim }_{x\to 0}\frac{f^{\prime }(x)}{x}$

   1. A.does not exist.
   2. B.exists and is equal to 2.
   3. C.exists and is equal to 0.
   4. D.exists and is equal to $-2$.
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   Answer: D

4. Q82018 · 8 Apr 2018 · Shift 1

   For each $t\in R$, let [t] be the greatest integer less than or equal to t. Then ${\lim }_{x\to 0^{+}}x\left(\left[\frac{1}{x}\right]+\left[\frac{2}{x}\right]+\dots \cdots +\left[\frac{15}{x}\right]\right)$

   1. A.does not exist in R
   2. B.is equal to 0
   3. C.is equal to 15
   4. D.is equal to 120
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   Answer: D

5. Q102018 · 8 Apr 2018 · Shift 1

   Let $\mathrm{S}=\{t\in R:f(x)=|x-\pi |\cdot (e^{|x|}-1)\sin |x|\text{ is not differentiable at t}\}$, then the set S is equal to

   1. A.$\{0,\pi \}$
   2. B.$\varphi$ (an empty set)
   3. C.$\{0\}$
   4. D.$\{\pi \}$
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   Answer: B