1
Numerical

### JEE Main 2021 (Online) 27th August Evening Shift

Let S = {1, 2, 3, 4, 5, 6, 9}. Then the number of elements in the set T = {A $\subseteq$ S : A $\ne$ $\phi$ and the sum of all the elements of A is not a multiple of 3} is _______________.

## Explanation

3n type $\to$ 3, 6, 9 = P

3n $-$ 1 type $\to$ 2, 5 = Q

3n $-$ 2 type $\to$ 1, 4 = R

number of subset of S containing one element which are not divisible by 3 = ${}^2$C1 + ${}^2$C1 = 4

number of subset of S containing two numbers whose some is not divisible by 3

= ${}^3$C1 $\times$ ${}^2$C1 + ${}^3$C1 $\times$ ${}^2$C1 + ${}^2$C2 + ${}^2$C2 = 14

number of subsets containing 3 elements whose sum is not divisible by 3

= ${}^3$C2 $\times$ ${}^4$C1 + (${}^2$C2 $\times$ ${}^2$C1)2 + ${}^3$C1(${}^2$C2 + ${}^2$C2) = 22

number of subsets containing 4 elements whose sum is not divisible by 3

= ${}^3$C3 $\times$ ${}^4$C1 + ${}^3$C2(${}^2$C2 + ${}^2$C2) + (${}^3$C1${}^2$C1 $\times$ ${}^2$C2)2

= 4 + 6 + 12 = 22

number of subsets of S containing 5 elements whose sum is not divisible by 3.

= ${}^3$C3(${}^2$C2 + ${}^2$C2) + (${}^3$C2${}^2$C1 $\times$ ${}^2$C2) $\times$ 2 = 2 + 12 = 14

number of subsets of S containing 6 elements whose sum is not divisible by 3 = 4

$\Rightarrow$ Total subsets of Set A whose sum of digits is not divisible by 3 = 4 + 14 + 22 + 22 + 14 + 4 = 80.
2
Numerical

### JEE Main 2021 (Online) 27th August Evening Shift

Let z1 and z2 be two complex numbers such that $\arg ({z_1} - {z_2}) = {\pi \over 4}$ and z1, z2 satisfy the equation | z $-$ 3 | = Re(z). Then the imaginary part of z1 + z2 is equal to ___________.

## Explanation

Let z1 = x1 + iy ; z2 = x2 + iy2

z1 $-$ z2 = (x1 $-$ x2) + i(y1 $-$ y2)

$\therefore$ $\arg ({z_1} - {z_2}) = {\pi \over 4}$ $\Rightarrow$ ${\tan ^{ - 1}}\left( {{{{y_1} - {y_2}} \over {{x_1} - {x_2}}}} \right) = {\pi \over 4}$

${y_1} - {y_2} = {x_1} - {x_2}$ ....... (1)

$|{z_1} - 3|\, = {\mathop{\rm Re}\nolimits} ({z_1}) \Rightarrow {({x_1} - 3)^2} + {y_1}^2 = {x_1}^2$ .... (2)

$|{z_2} - 3|\, = {\mathop{\rm Re}\nolimits} ({z_2}) \Rightarrow {({x_2} - 3)^2} + {y_2}^2 = {x_2}^2$ .... (3)

sub (2) & (3)

${({x_1} - 3)^2} - {({x_2} - 3)^2} + {y_1}^2 - {y_2}^2 = {x_1}^2 - {x_2}^2$

$({x_1} - {x_2})({x_1} + {x_2} - 6) + ({y_1} - {y_2})({y_1} + {y_2})$

$= ({x_1} - {x_2})({x_1} + {x_2})$

${x_1} + {x_2} - 6 + {y_1} + {y_2} = {x_1} + {x_2} \Rightarrow {y_1} + {y_2} = 6$
3
Numerical

### JEE Main 2021 (Online) 27th August Evening Shift

The probability distribution of random variable X is given by :

X 1 2 3 4 5
P(X) K 2K 2K 3K K

Let p = P(1 < X < 4 | X < 3). If 5p = $\lambda$K, then $\lambda$ equal to ___________.

## Explanation

$\sum {P(X) = 1 \Rightarrow k + 2k + 3} k + k = 1$

$\Rightarrow k = {1 \over 9}$

Now, $p = P\left( {{{kx < 4} \over {X < 3}}} \right) = {{P(X = 2)} \over {P(X < 3)}} = {{{{2k} \over {9k}}} \over {{k \over {9k}} + {{2k} \over {9k}}}} = {2 \over 3}$

$\Rightarrow p = {2 \over 3}$

Now, $5p = \lambda k$

$\Rightarrow (5)\left( {{2 \over 3}} \right) = \lambda (1/9)$

$\Rightarrow \lambda = 30$
4
Numerical

### JEE Main 2021 (Online) 27th August Evening Shift

Let S be the sum of all solutions (in radians) of the equation ${\sin ^4}\theta + {\cos ^4}\theta - \sin \theta \cos \theta = 0$ in [0, 4$\pi$]. Then ${{8S} \over \pi }$ is equal to ____________.

## Explanation

Given equation

${\sin ^4}\theta + {\cos ^4}\theta - \sin \theta \cos \theta = 0$

$\Rightarrow 1 - {\sin ^2}\theta {\cos ^2}\theta - \sin \theta \cos \theta = 0$

$\Rightarrow 2 - {(\sin 2\theta )^2} - \sin 2\theta = 0$

$\Rightarrow {(\sin 2\theta )^2} + (\sin 2\theta ) - 2 = 0$

$\Rightarrow (\sin 2\theta + 2)(\sin 2\theta - 1) = 0$

$\Rightarrow \sin 2\theta = 1$ or $\sin 2\theta = - 2$ (Not Possible)

$\Rightarrow 2\theta = {\pi \over 2},{{5\pi } \over 2},{{9\pi } \over 2},{{13\pi } \over 2}$

$\Rightarrow \theta = {\pi \over 4},{{5\pi } \over 4},{{9\pi } \over 4},{{13\pi } \over 4}$

$\Rightarrow S = {\pi \over 4} + {{5\pi } \over 4} + {{9\pi } \over 4} + {{13\pi } \over 4} = 7\pi$

$\Rightarrow {{8S} \over \pi } = {{8 \times 7\pi } \over \pi } = 56.00$