1

### JEE Main 2019 (Online) 10th January Evening Slot

The positive value of $\lambda$ for which the co-efficient of x2 in the expression x2 ${\left( {\sqrt x + {\lambda \over {{x^2}}}} \right)^{10}}$ is 720, is -
A
4
B
$2\sqrt 2$
C
3
D
$\sqrt 5$

## Explanation

${x^2}\left( {{}^{10}{C_r}{{\left( {\sqrt x } \right)}^{10 - r}}{{\left( {{\lambda \over {{x^2}}}} \right)}^r}} \right)$

${x^2}\left[ {{}^{10}{C_r}{{\left( x \right)}^{{{10 - r} \over 2}}}{{\left( \lambda \right)}^r}{{\left( x \right)}^{ - 2r}}} \right]$

${x^2}\left[ {{}^{10}{C_r}{\lambda ^r}{x^{{{10 - r} \over 2}}}} \right]$

$\therefore$  r = 2

Hence, ${}^{10}{C_2}{\lambda ^2} = 720$

${\lambda ^2} = 16$

$\lambda = \pm 4$
2

### JEE Main 2019 (Online) 11th January Morning Slot

The value of r for which 20Cr 20C0 + 20Cr$-$1 20C1 + 20Cr$-$2 20C2 + . . . . .+ 20C0 20Cr  is maximum, is
A
20
B
15
C
10
D
11

## Explanation

Given sum = coefficient of xr in the expansion of

(1 + x)20(1 + x)20,

Which is equal to 40Cr

It is maximum when r = 20
3

### JEE Main 2019 (Online) 11th January Morning Slot

The sum of the real values of x for which the middle term in the binomial expansion of ${\left( {{{{x^3}} \over 3} + {3 \over x}} \right)^8}$ equals 5670 is :
A
0
B
8
C
6
D
4

## Explanation

${T_5} = {}^8{C_4}{{{x^{12}}} \over {81}} \times {{81} \over {{x^4}}} = 5670$

$\Rightarrow 70{x^8} = 5670$

$\Rightarrow x = \pm \sqrt 3$
4

### JEE Main 2019 (Online) 11th January Evening Slot

Let (x + 10)50 + (x $-$ 10)50 = a0 + a1x + a2x2 + . . . . + a50x50, for all x $\in$ R; then ${{{a_2}} \over {{a_0}}}$ is equal to
A
12.25
B
12.75
C
12.00
D
12.50

## Explanation

(10 + x)50 + (10 $-$ x)50

$\Rightarrow$  a2 = 2.50C2 1048, a0 = 2.1050

${{{a_2}} \over {{a_0}}} = {{^{50}{C_2}} \over {{{10}^2}}} = 12.25$