Match the statements of Column I with values of Column II.

COLUMN - I COLUMN - II
a. The least positive integral values of $λ$ for which
$(λ - 2) x^{2} + 8 x + (λ + 4) > 0$ for all real x is p. 3
b. The equation $x^{2} + 2 (a^{2} + 1) x + (a^{2} - 14 a + 48) = 0$ possesses
roots of opposite signs then x value of ‘a’ can be q. 5
c. If the equation ${ax}^{2} + 2 bx + 4 c = 16$ has no real roots
and $a + c > b + 4$ then integral value of c can be equal to r. 7
d. If N be the number of solution of the equation
$|x^{2} - x - 6| = x + 2$ then the value of N is s. 12
t. 20

Question

Match the statements of Column I with values of Column II.

COLUMN - I COLUMN - II
a. The least positive integral values of $λ$ for which
$(λ - 2) x^{2} + 8 x + (λ + 4) > 0$ for all real x is p. 3
b. The equation $x^{2} + 2 (a^{2} + 1) x + (a^{2} - 14 a + 48) = 0$ possesses
roots of opposite signs then x value of ‘a’ can be q. 5
c. If the equation ${ax}^{2} + 2 bx + 4 c = 16$ has no real roots
and $a + c > b + 4$ then integral value of c can be equal to r. 7
d. If N be the number of solution of the equation
$|x^{2} - x - 6| = x + 2$ then the value of N is s. 12
t. 20

Accepted Answer

A) λ>2,64−4(λ−2)(λ+4)<0λ<−6 or λ>4The least positive integral value of λ is 5B) Roots are of opposite signs ⇒a2−14a+48<0⇒(a=6)(a−8)<0 so a can be 7The equation is x2+100x−1=0discriminant = =D=1002+4>0 .Roots are realC) Let f(x)=ax2+2bx+4c−16 ,Clearly f(−2)=4a−4b+4c−16=4(a−b+c−4)>0=f(x)>0,∀x∈R,⇒f(0)>0⇒4c−16>0⇒c>4D) ∵x2−x−6=x+2⇒|(x−3)(x+2)|=x+2⇒|x−3||x+2|=x+2(x−3)(x+2)=x+2,x<−2−⇒(x−3)(x+2)=x+2,−2≤x<3(x−3)(x+2)=x+2,x>3x=4,x<−2x=−2,2−2≤x<3x=4,x<3, Hence, x=−2,2,4 N=3

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	COLUMN - I		COLUMN - II
a.	The least positive integral values of $λ$ for which $(λ - 2) x^{2} + 8 x + (λ + 4) > 0$ for all real x is	p.	3
b.	The equation $x^{2} + 2 (a^{2} + 1) x + (a^{2} - 14 a + 48) = 0$ possesses roots of opposite signs then x value of ‘a’ can be	q.	5
c.	If the equation ${ax}^{2} + 2 bx + 4 c = 16$ has no real roots and $a + c > b + 4$ then integral value of c can be equal to	r.	7
d.	If N be the number of solution of the equation $\|x^{2} - x - 6\| = x + 2$ then the value of N is	s.	12
		t.	20