Match the elements of the list I, which contains inequalities, with the elements of list II, which has solution sets.
List-I
(Inequality) List-II
(Solution set)
A) $x^{2} - 4 x + 3 > 0$ 1) $(3, 4)$
B) $x^{2} - 5 x + 6 \leq 0$ 2) $(- 1, 1) \cup (2, 4)$
C) $x^{2} + 6 x - 27 > 0, - x^{2} + 3 x + 4 > 0$
3) $(- \infty, 1) \cup (3, \infty)$
D) $x^{2} - 3 x - 4 < 0, x^{2} - 3 x + 2 > 0$ 4) $[3, 4]$
5) $[2, 3]$
Then the correct order for A B C D, is

1 2 3 4

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3 5 1 2

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1 3 5 4

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3 2 4 1

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Solution

The correct option is C 3 5 1 2
A: $x^{2} - 4 x + 3 > 0$
$\Rightarrow (x - 1) (x - 3) > 0$
So, $x \in (- \infty, 1) \cup (3, \infty)$ $\to (3)$

B: $x^{2} - 5 x + 6 \leq 0$
$\Rightarrow (x - 2) (x - 3) \leq 0$
So, $x \in [2, 3]$ $\to (5)$

C: $x^{2} + 6 x - 27 > 0$ and $- x^{2} + 3 x + 4 > 0$
$\Rightarrow (x + 9) (x - 3) > 0$ and $(x - 4) (x + 1) < 0$
So, $x \in (- \infty, - 9) \cup (3, \infty)$ and $x \in (- 1, 4)$
Taking intersection, we get
$x \in (3, 4)$ $\to (1)$

D: $x^{2} - 3 x - 4 < 0$ and $x^{2} - 3 x + 2 > 0$
$\Rightarrow (x - 4) (x + 1) < 0$ and $(x - 1) (x - 2) > 0$
So, $x \in (- 1, 4)$ and $x \in (- \infty, 1) \cup (2, \infty)$
Taking intersection, we get
$x \in (- 1, 1) \cup (2, 4)$ $\to (2)$

$∴$ The final order for A B C D is 3 5 1 2.

Hence, option B.

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Similar questions

Question 10
Which of the following equations has no real roots?
(A) $x^{2} - 4 x + 3 \sqrt{2} = 0$
(B) $x^{2} + 4 x - 3 \sqrt{2} = 0$
(C) $x^{2} - 4 x - 3 \sqrt{2} = 0$
(D) $3 x^{2} - 4 \sqrt{3} x + 4 = 0$

Q. The solution set of

x^{2} - 16 \leq 0

and

x^{2} - 9 \geq 0

Q. Observe the following lists, In List-I there are ellipses and in List -II their foci
are given
List -I List -II
A

\frac{x^{2}}{16} + \frac{y^{2}}{12} = 1

(0, \pm 5)

\frac{x^{2}}{11} + \frac{y^{2}}{36} = 1

(\pm 2, 0)

\frac{x^{2}}{7} + \frac{y^{2}}{16} = 1

(0, \pm \sqrt{5})

\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1

(0, \pm 3)

(\pm \sqrt{5}, 0)

The correct match for List-I from List-II is

Q. Match the equations in List 1 with the solutions in List 2

	List I		List II
A.	${log}_{0.25} {(x^{2} + 2 x - 8)}^{2}$ $- {log}_{0.5} (10 + 3 x - x^{2}) = 1$	1.	${- 3}$
B.	${log}_{2} (x^{2} + 7)$ $= 5 + {log}_{2} x$ $- \frac{6}{{log}_{2} (x + \frac{7}{x})}$	2.	${\frac{1}{4}}$
C.	${log}_{(1 - 2 x)} (6 x^{2} - 5 x + 1)$ $- {log}_{(1 - 3 x)} (4 x^{2} - 4 x + 1) = 2$	3.	${\frac{1}{6} (\sqrt{313} - 1), \frac{1}{2} (\sqrt{73} - 7)}$
D.	${log}_{10} (1 {+ x}^{2} - 2 x) + 1 - {log}_{10} (1 + x^{2}) =$ $2 {log}_{10} (1 - x)$	4.	${1, 7}$

Q. Find the discriminant of the following quadratic equations and discuss the nature of the roots .
1.

6 x^{2} - 13 x + 6 = 0

\sqrt{6} x^{2} - 5 x + \sqrt{6} = 0

24 x^{2} - 17 x + 3 = 0

x^{2} + 2 x + 4 = 0

x^{2} + x + 1 = 0

x^{2} - 3 \sqrt{3} x - 30 = 0

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List-I (Inequality)	List-II (Solution set)
A) $x^{2} - 4 x + 3 > 0$	1) $(3, 4)$
B) $x^{2} - 5 x + 6 \leq 0$	2) $(- 1, 1) \cup (2, 4)$
C) $x^{2} + 6 x - 27 > 0, - x^{2} + 3 x + 4 > 0$	3) $(- \infty, 1) \cup (3, \infty)$
D) $x^{2} - 3 x - 4 < 0, x^{2} - 3 x + 2 > 0$	4) $[3, 4]$
	5) $[2, 3]$