Let s , t , r be non-zero complex numbers and L be the set of solutions z = x + i y x , y ∈ R , i = - 1 of the equation s z + t z ¯ + r = 0 , where z ¯ = x - i y Then, which of the following statement(s) is(are) TRUE?

If L has more than one element, then L has infinitely many elements.

If s = t , then L has infinitely many elements

If L has more than one element, then L has infinitely many elements.

Let z k = cos 2 k π 10 - i sin 2 k π 10 ; k = 1 , 2 , . . . , 9 List I List II a . For each z k there exists a z j such that z k . z j = 1 p . True b . There exists a k ∈ 1 , 2 , . . . . , 9 such that z 1 z = z k has no solution z in the set of complex numbers. q . False c . 1 - z 1 1 - z 2 . . . . . 1 - z 9 10 equals r . 1 d . 1 - ∑ k = 0 9 cos 2 k π 10 equals s . 2

a → p ; b → q ; c → r ; d → s

a → q ; b → s ; c → r ; d → p

a → s ; b → r ; c → p ; d → q

a → q ; b → s ; c → p ; d → r

A → p , q , s ; B → p , q ; C → p , q , s , t ; D → q , t

A → p , q , r ; B → p , s ; C → p , q , r , s ; D → q

A → s , q , p ; B → p , t ; C → p , s , r , t ; D → q

HARD

JEE Main/Advance

IMPORTANT

Earn 100

50% studentsanswered this correctly

HARD

JEE Main/Advance

IMPORTANT

Let $z_{k} = \cos (\frac{2 k π}{10}) - i \sin (\frac{2 k π}{10}); k = 1, 2, . . ., 9$

List I	List II
$a .$ For each $z_{k}$ there exists a $z_{j}$ such that $z_{k .} z_{j} = 1$	$p .$ True
$b .$ There exists a $k \in \{1, 2, . . . ., 9\}$ such that $z_{1} z = z_{k}$ has no solution $z$ in the set of complex numbers.	$q .$ False
$c .$ $\frac{\|1 - z_{1}\| \|1 - z_{2}\| . . . . . \|1 - z_{9}\|}{10}$ equals	$r . 1$
$d .$ $1 - \sum_{k = 0}^{9} \cos (\frac{2 k π}{10})$ equals	$s . 2$

HARD

JEE Main/Advance

IMPORTANT

Match the statements given in Column I with the values given in Column II.

	Column I		Column II
A.	$\ln R^{2}$ , if the magnitude of the projection vector of the vector $α \hat{i} + β \hat{j}$ on $\sqrt{3 \hat{i}} + \hat{j}$ is $\sqrt{3}$ and if $α = 2 + \sqrt{3} β$ then possible value(s) of $\| α \|$ is/are	p	$1$
B.	Let $a$ and $b$ be real numbers such that the function $f (x) = \{\begin{cases} - 3 a x^{2} - 2, & x < 1 \\ b x + a^{2}, & x \geq 1 \end{cases}$ is differentiable for all $x \in R$ . Then, possible value(s) of $a$ is/are	q	$2$
C.	Let $ω (\neq 1)$ be a complex cube root of unity. If ${(3 - 3 ω + 2 ω^{2})}^{4 n + 3} +$ ${(2 + 3 ω - 3 ω^{2})}^{4 n + 3}$ $+ {(- 3 + 2 ω + 3 ω^{2})}^{4 n + 3} = 0,$ then the possible value(s) of $n$ is/are	r	$3$
D.	Let the harmonic mean of two positive real numbers $a$ and $b$ be $4$ . If $q$ is a positive real number such that $a, 5, q, b$ is in arithmetic progression, then the value(s) of $\| q - a \|$ is/are	s	$4$
		t	$5$