1

JEE Main 2017 (Online) 9th April Morning Slot

If 2x = y${^{{1 \over 5}}}$ + y${^{ - {1 \over 5}}}$ and

(x2 $-$ 1) ${{{d^2}y} \over {d{x^2}}}$ + $\lambda$x ${{dy} \over {dx}}$ + ky = 0,

then $\lambda$ + k is equal to :
A
$-$ 23
B
$-$ 24
C
26
D
$-$ 26
2

JEE Main 2017 (Online) 9th April Morning Slot

The value of k for which the function

$f\left( x \right) = \left\{ {\matrix{ {{{\left( {{4 \over 5}} \right)}^{{{\tan \,4x} \over {\tan \,5x}}}}\,\,,} & {0 < x < {\pi \over 2}} \cr {k + {2 \over 5}\,\,\,,} & {x = {\pi \over 2}} \cr } } \right.$

is continuous at x = ${\pi \over 2},$ is :
A
${{17} \over {20}}$
B
${{2} \over {5}}$
C
${{3} \over {5}}$
D
$-$ ${{2} \over {5}}$
3

JEE Main 2017 (Online) 9th April Morning Slot

Let f be a polynomial function such that

f (3x) = f ' (x) . f '' (x), for all x $\in$ R. Then :
A
f (2) + f ' (2) = 28
B
f '' (2) $-$ f ' (2) = 0
C
f '' (2) $-$ f (2) = 4
D
f (2) $-$ f ' (2) + f '' (2) = 10
4

JEE Main 2018 (Offline)

For each t $\in R$, let [t] be the greatest integer less than or equal to t.

Then $\mathop {\lim }\limits_{x \to {0^ + }} x\left( {\left[ {{1 \over x}} \right] + \left[ {{2 \over x}} \right] + ..... + \left[ {{{15} \over x}} \right]} \right)$
A
does not exist in R
B
is equal to 0
C
is equal to 15
D
is equal to 120

Explanation

Given,

$\mathop {lim}\limits_{x \to {0^ + }} \,\,x\,\left( {\left[ {{1 \over x}} \right] + \left[ {{2 \over x}} \right]} \right. +$ $\left. {\,.\,.\,.\,.\,.\, + \left[ {{{15} \over x}} \right]} \right)$

as we know that

${1 \over x} = \left[ {{1 \over x}} \right] + \left\{ {{1 \over x}} \right\}$

$\Rightarrow \,\,\,\,\left[ {{1 \over x}} \right] = {1 \over x} - \left\{ {{1 \over x}} \right\}$

$= \,\,\,\,\mathop {\lim }\limits_{x \to {0^ + }} \,\,x\left[ {{1 \over x} - \left\{ {{1 \over x}} \right\} + {2 \over 2} - \left\{ {{2 \over x}} \right\} + ........{{15} \over x} - \left\{ {{{15} \over x}} \right\}} \right]$

$= \,\,\,\,\mathop {\lim }\limits_{x \to {0^ + }} \,\left[ {x.{1 \over x} + x.{2 \over x} + .....x.{{15} \over x}} \right]$

$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - \,\left[ {x.\left\{ {{1 \over 2}} \right\} + .. + x.\left\{ {{{15} \over x}} \right\}} \right]$

We know $\left\{ {{1 \over x}} \right\}$ is fractional part of ${1 \over x}.$

So, the range of $\,\left\{ {{1 \over x}} \right\}$ is $0 \le \left\{ {{1 \over x}} \right\} < 1$

So, $\mathop {lim}\limits_{x \to {0^ + }} \,\,x\,\left\{ {{1 \over x}} \right\} = 0.$ (finite no) $=0$

Similarly $\mathop {\lim }\limits_{x \to {0^ + }} x.\left\{ {{2 \over x}} \right\} = 0$

$= \,\,\,\,\left( {1 + 2 + ... + 15} \right) - \left( {0 + 0...} \right)$

$= \,\,\,\,{{15 \times 16} \over 2}$

$= \,\,\,\,120$