beginarray -c- c-c- c - hhine-List-I-List-II- IIVhlineP-are vertices of an equilateral triangle-1-0- - IF a complex number-z satisfying - z -2- i - - leq 1- - 1Q-then the maximum distance of - origin from 4- i 2- z is- 2 - - 1- 211 hlineR-Then the number of distinct ordered pairslalpha- beta - 3 - - 2- 2lendarray A- P-3-Q-2-R-4-S-1B- P -4- Q -3- R -2- S -1C- P-2-Q-3-R-1- S-4D- P-2-Q-3-R-4-S-1

The correct option is D P 2,Q 3,R 4,S 1 P : Point lies on x^2 + y^2 = x ∴ centre is (1/2,0) a =1/2, b = 0 Q : |4+2i iz|=|z 2+i+3i| ≤ 1 + 3 = 4 R :xy = 25! = 2 ...

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Byju's Answer

Standard XII

Mathematics

Relation between Roots and Coefficients for Quadratic

beginarray |c...

Question

$\begin{matrix} L i s t - I & L i s t - I I P (t) = (\frac{1}{t^{2} + 1}, \frac{t}{t^{2} + 1}) . I f P (α), B (β) . C (γ) P . & a r e v e r t i c e s o f a n e q u i l a t e r a l t r i a n g l e & 1. & 0 (α, β, γ > 0) a n d i t s c e n t r o i d i s (a, b) t h e n 2 a + b = I F a c o m p l e x n u m b e r! z s a t i s f y i n g | z - 2 + i | \leq 1, Q . & t h e n t h e m a x i m u m d i s t a n c e o f o r i g i n f r o m 4 + i (2 - z) i s & 2. & 1^{2} C o n s i d e r t h e c u r v e x y = 25! s u c h t h a t (α, β) i s a p o i n t o n t h e c u r v e . R . & T h e n t h e n u m b e r o f d i s t i n c t o r d e r e d p a i r s (α, β) & 3. & 2^{2} s u c h t h a t H C F (α, β) = 1 i s 2^{k} t h e n k = S . & I f 2^{(1 + c o s π x)} l o g_{2}^{5} + 2^{x^{2} - 1} + 2^{2 (1 - | x |)} = 3, t h e n s u m o f t h e r o o t s i s & 4. & 3^{2} \end{matrix}$

P-4,Q-3,R-2, S-1

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P-2,Q-3,R-1, S-4

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P-2,Q-3,R-4,S-1

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P-3,Q-2,R-4,S-1

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Solution

The correct option is C
P-2,Q-3,R-4,S-1

$P : P o i n t l i e s o n x^{2} + y^{2} = x$

$∴ c e n t r e i s (\frac{1}{2}, 0)$

$a = \frac{1}{2}$ , $b = 0$

$Q : | 4 + 2 i - i z | = | z - 2 + i + 3 i |$ $\leq 1 + 3 = 4$

$R : x y = 25!$ $= 2^{22} \cdot 3^{10} 5^{6} 7^{3} 11^{2} 13^{1} 17^{1} 19^{1} 23^{1}$

$∴ 2^{9}$

$S : 1 + c o s π x = 0$ , $x^{2} - 1 = 0$ $| x | = 1 \Rightarrow x = \pm 1$ .

$s u m = 0$

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Similar questions

$\begin{matrix} L i s t - I & L i s t - I I P . & I f_{α} = \frac{π}{7} t h e n \frac{1}{c o s α} + \frac{2 c o s α}{c o s 2 α} = & 1. & 2 Q . & u n d e r s e t x \to \infty L t [(x - 1) (x - 2) (x + 3) (x + 5) (x + 10)]^{\frac{1}{5}} - x = & 2. & 3 R . & \int_{- 2}^{2} \frac{3 x^{2} d x}{1 + e^{x}} = & 3. & 4 S . & L e t f (x) = x^{3} + x^{2} + 2 x - 1. T h e m i n i m u m i n t e g r a l v a l u e o f x i f x s a t i s f i e s f (f (x)) > f (2 x + 1) i s & 4. & 8 \end{matrix}$

List-II gives distance between pair of points given in List-I. Match them correctly.
$\begin{matrix} L i s t - I & L i s t I I (P) & (- 5, 7), (- 1, 3) & (1) & 5 (Q) & (5, 6), (1, 3) & (2) & \sqrt{8} (R) & (\sqrt{3} + 1, 1), (0, \sqrt{3}) & (3) & \sqrt{6} (S) & (0, 0), (- \sqrt{3}, \sqrt{3}) & (4) & 4 \sqrt{2} \end{matrix}$

Q. Let A and B be two independent events such that

P (A) = \frac{1}{3}

and

P (B) = \frac{1}{4}

\begin{matrix} L i s t I & L i s t I I P . & P (A \cup B) i s e q u a l t o & 1. & \frac{1}{2} Q . & P (\frac{A}{A \cup B}) i s e q u a l t o & 2. & \frac{2}{3} R . & P (\frac{B}{A^{'} \cap B^{'}}) i s e q u a l t o & 2. & \frac{1}{3} S . & P (\frac{A}{B}) i s e q u a l t o & 4. & 0 \end{matrix}

Q. Match the electronic configurations of the elements given in List - I with their correct characteristic(s) (i.e., properties for given configuration) given in List - II.

\begin{matrix} List-I & List - II P. & 1 s^{2} & 1. & Element shows highest negative oxidation state Q. & 1 s^{2} 2 s^{2} 2 p^{5} & 2. & Element shows highest first ionisation enthalpy R. & 1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{5} & 3. & Element shows highest electronegativity on Pauling scale S. & 1 s^{2} 2 s^{2} 2 p^{2} & 4. & Element shows maximum electron gain enthalpy (most exothermic) \end{matrix}

Q. List-II gives distance between pair of points given in List-I, match them correctly.

\begin{matrix} L i s t - I & L i s t I I (P) & (- 5, 7), (- 1, 3) & (1) & 5 (Q) & (5, 6), (1, 3) & (2) & \sqrt{8} (R) & (\sqrt{3} + 1, 1), (0, \sqrt{3}) & (3) & \sqrt{6} (S) & (0, 0), (- \sqrt{3}, \sqrt{3}) & (4) & 4 \sqrt{2} \end{matrix}

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The correct option is C P-2,Q-3,R-4,S-1 P:Point lies on x2+y2=x ∴ centre is (12,0) a=12, b=0 Q:|4+2i−iz|=|z−2+i+3i| ≤1+3=4 R:xy=25! =222⋅31056 73 112 131 171 191 231 ∴29 S:1+cosπx=0, x2–1=0 |x|=1⇒x=±1. sum=0