Match the following COLUMN-I COLUMN-IIa)y=x2+x+1x2−x+1∀x∈R then y∈p)(−∞,1]∪[4,∞)b)y=xx2−5x+9∀x∈R then y∈q)[−113,13]c)y=13x+1+1x+1−1(3x+1)(x+1)∀x∈R then y∈r)[−111,1]d)y=x+22x2+3x+6∀x∈R then y∈s)[13,3]

a) y=x2+x+1x2−x+1 (y−1)x2−(y+1)x+(y−1)=0,x∈Rb2−4ac≥0−3y2+10y−3≥0⇒3y2−10y+3≤0(3y−1)(y−3)≤0⇒y∈[13,3] b) Similarly y∈[−111,1]c) y∈(−∞,1]∪[4,∞)d) y∈[−113,13]

Match the following COLUMN-I COLUMN-IIa)y=x2+x+1x2−x+1∀x∈R then y∈p)(−∞,1]∪[4,∞)b)y=xx2−5x+9∀x∈R then y∈q)[−113,13]c)y=13x+1+1x+1−1(3x+1)(x+1)∀x∈R then y∈r)[−111,1]d)y=x+22x2+3x+6∀x∈R then y∈s)[13,3]

a) y=x2+x+1x2−x+1 (y−1)x2−(y+1)x+(y−1)=0,x∈Rb2−4ac≥0−3y2+10y−3≥0⇒3y2−10y+3≤0(3y−1)(y−3)≤0⇒y∈[13,3] b) Similarly y∈[−111,1]c) y∈(−∞,1]∪[4,∞)d) y∈[−113,13]

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Match the following

COLUMN-I

COLUMN-II
a) $\begin{array}{l} y = \frac{x^{2} + x + 1}{x^{2} - x + 1} \\ \forall x \in R then y \in \end{array}$ p) $(- \infty, 1] \cup [4, \infty)$
b) $\begin{array}{l} y = \frac{x}{x^{2} - 5 x + 9} \\ \forall x \in R then y \in \end{array}$ q) $[\frac{- 1}{13}, \frac{1}{3}]$
c) $\begin{array}{l} y = \frac{1}{3 x + 1} + \frac{1}{x + 1} - \frac{1}{(3 x + 1) (x + 1)} \\ \forall x \in R then y \in \end{array}$ r) $[\frac{- 1}{11}, 1]$
d) $\begin{array}{l} y = \frac{x + 2}{2 x^{2} + 3 x + 6} \\ \forall x \in R then y \in \end{array}$ s) $[\frac{1}{3}, 3]$

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$a \to s, b \to r, c \to p, d \to q$

$a \to s, b \to r, c \to q, d \to p$

$a \to r, b \to s, c \to q, d \to p$

$a \to q, b \to p, c \to r, d \to s$

answer is A.

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Detailed Solution

a) $y = \frac{x^{2} + x + 1}{x^{2} - x + 1}$

   $\begin{array}{l} (y - 1) x^{2} - (y + 1) x + (y - 1) = 0, x \in R \\ b^{2} - 4 a c \geq 0 \\ - 3 y^{2} + 10 y - 3 \geq 0 \Rightarrow 3 y^{2} - 10 y + 3 \leq 0 \\ (3 y - 1) (y - 3) \leq 0 \Rightarrow y \in [\frac{1}{3}, 3] \end{array}$

b) Similarly    $y \in [\frac{- 1}{11}, 1]$
c)    $y \in (- \infty, 1] \cup [4, \infty)$
d) $y \in [\frac{- 1}{13}, \frac{1}{3}]$

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	COLUMN-I		COLUMN-II
a)	$\begin{array}{l} y = \frac{x^{2} + x + 1}{x^{2} - x + 1} \\ \forall x \in R then y \in \end{array}$	p)	$(- \infty, 1] \cup [4, \infty)$
b)	$\begin{array}{l} y = \frac{x}{x^{2} - 5 x + 9} \\ \forall x \in R then y \in \end{array}$	q)	$[\frac{- 1}{13}, \frac{1}{3}]$
c)	$\begin{array}{l} y = \frac{1}{3 x + 1} + \frac{1}{x + 1} - \frac{1}{(3 x + 1) (x + 1)} \\ \forall x \in R then y \in \end{array}$	r)	$[\frac{- 1}{11}, 1]$
d)	$\begin{array}{l} y = \frac{x + 2}{2 x^{2} + 3 x + 6} \\ \forall x \in R then y \in \end{array}$	s)	$[\frac{1}{3}, 3]$

Match the following COLUMN-I COLUMN-IIa)y=x2+x+1x2−x+1∀x∈R then y∈p)(−∞,1]∪[4,∞)b)y=xx2−5x+9∀x∈R then y∈q)[−113,13]c)y=13x+1+1x+1−1(3x+1)(x+1)∀x∈R then y∈r)[−111,1]d)y=x+22x2+3x+6∀x∈R then y∈s)[13,3]

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