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Match List-I with List-II and Select the correct answer(s) in the given options
List-I List-II
P) Remainder when $x^{4} - 3 x + 5$ is divided by $x - 2$ is 1) -4
Q) Coefficient of $x^{4}$ in the product of $x^{3} - 3 x^{2} + x - 2$ and $2 x^{3} + x^{2} - 3 x + 1$ is 2) 15
R) The value of $x$ satisfying $2 x - 3 y = 4$ and $x^{2} - 3 x y + 3 y^{2} = 7$ is 3) -1
S) The sum of the solutions of $3^{2 x^{2}} - {2.3}^{x^{2} + x + 6} + 3^{2 (x + 6)} = 0$ is 4) 1
5) 5

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$P \to 2$

$Q \to 1$

$R \to 5$

$S \to 4$

answer is A, B, C, D.

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Detailed Solution

P) $f (x) = x^{4} - 3 x + 5$
Remainder is $f (2) = 15$
Q) $(x^{3} - 3 x^{2} + x - 2) (2 x^{3} + x^{2} - 3 x + 1)$
Coefficient of $x^{4}$ is -1
R)
$2 x - 3 y = 4 \Rightarrow 3 y = 2 x - 4 x^{2} - 3 x y + 3 y^{2} = 7 \Rightarrow x^{2} - 4 x - 5 = 0 x = - 1, 5$

S) ${(3^{x^{2}})}^{2} - 2.3. x^{2} {.3}^{x + 6} + {(3^{x + 6})}^{2} = 0 {(3^{x^{2}} - 3^{x + 6})}^{2} = 0 \Rightarrow x^{2} - x - 6 = 0$
Sum of solutions $= 1$

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