List I has four entries and nbsp-List II has five entries- Each entry of nbsp-List I is to be matched with one or more than one entries of nbsp-List II- nbsp- - amp-List I amp- amp- List II- -A- amp- The vertices of a triangle are -1-a- amp- -P- amp-a-b-13- amp-2-b- and -c-2-3- If the centroid of- amp- the triangle lies on the x-axis- then- nbsp- -B- amp-If a-b-c -a b c- are the roots of amp-Q- amp-ab-12- nbsp- amp- x-3-6x-2-11x-6-0- then- nbsp-C- amp-If a-b-c lie between 2 and 18 such that amp-R- amp-a-b-5-8- amp- -i-a-b-c-25- amp- -ii-2-a-b are in A-P- amp- -iii-b-c-18 are in G-P- then- nbsp- nbsp- -D- amp- If a-b-c are in G-P- with common ratio amp-S- amp-a-b-8- amp- r-r 1- such that abc-216 and- amp- ab-bc-ca-156- then- amp- nbsp- amp-T- nbsp- amp- a-b-3- nbsp- - Which of the following is the only CORRECT combination-C-P- -Q-C-P- -R-D-R- -S-D-Q- -S- nbsp-

The correct option is D (D)→(Q), (S) nbsp; nbsp;(C) a,b,c are three numbers between 2 and 18 a+b+c=25 ⋯(1) 2,a,b are in A.P. ⇒ 2a=2+b ⋯(2) b,c,18 a ...

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Byju's Answer

Standard XII

Mathematics

Centroid

List I has fo...

Question

$List I$ has four entries and $List II$ has five entries. Each entry of $List I$ is to be matched with one or more than one entries of $List II .$

$\begin{matrix} List I & List II (A) & The vertices of a triangle are (1, a), & (P) & a + b = 13 (2, b) and (c^{2}, - 3) . If the centroid of the triangle lies on the x -axis, then (B) & If a, b, c (a < b < c) are the roots of & (Q) & a b = 12 x^{3} - 6 x^{2} + 11 x - 6 = 0, then (C) & If a, b, c lie between 2 and 18 such that & (R) & (a, b) = (- 5, 8) (i) a + b + c = 25 (ii) 2, a, b are in A.P. (iii) b, c, 18 are in G.P., then (D) & If a, b, c are in G.P. with common ratio & (S) & a + b = 8 r (r > 1) such that a b c = 216 and a b + b c + c a = 156, then (T) & a + b = 3 \end{matrix}$

Which of the following is the only CORRECT combination?

(C) \to (P), (Q)

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(C) \to (P), (R)

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(D) \to (R), (S)

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(D) \to (Q), (S)

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Solution

The correct option is D $(D) \to (Q), (S)$
$(C)$
$a, b, c$ are three numbers between $2$ and $18$
$a + b + c = 25 \dots (1)$
$2, a, b$ are in A.P.
$\Rightarrow 2 a = 2 + b \dots (2)$
$b, c, 18$ are in G.P.
$\Rightarrow c^{2} = 18 b \dots (3)$

From $(1), (2)$ and $(3),$
$\frac{1}{2} (2 + \frac{c^{2}}{18}) + \frac{c^{2}}{18} + c = 25 \Rightarrow c = 12$
$∴ b = \frac{c^{2}}{18} = 8 \Rightarrow a = 5$
$(C) \to (P)$

$(D)$
$a, b, c$ are in G.P., common ratio $= r (r > 1)$
Let $a = \frac{b}{r}, c = b r$
$a b c = 216 \Rightarrow b^{3} = 216 \Rightarrow b = 6$

$a b + b c + c a = 156$
$\Rightarrow b^{2} (\frac{1}{r} + r + 1) = 156 \Rightarrow 3 r^{2} - 10 r + 3 = 0$
$\Rightarrow r = \frac{1}{3}$ or $3$
But $r > 1. ∴ r = 3$
$\Rightarrow a = 2, c = 18$
$(D) \to (Q), (S)$

Suggest Corrections

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Which of the following is the only CORRECT combination?

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