Quadratic Equations Model Questions Set 1 Practice Questions Answers Test with Solutions & More Shortcuts

Answer: (d)

Let &alpha, β be the roots of the equation $lx^2$ + mx + m = 0

Given ${α}/{β} = p/q$

Now α + β (sum of roots) = ${- m}/l$

and α β (product of roots) = $m/l$

Consider $√{p/q} + √{q/p} + √{m/l}$

Using (1)

= $√{{α}/{β}} = √{{β}/{α}} + √{m/l}$

= ${α + β}/{√{α β}} + √{m/l}$

= ${- m/l}/{√{m/l}} + √{m/l} = - √{m/l} + √{m/l}$ = 0

∴ option (d) is correct.

Answer: (d)

Consider $ax^2 - (a^2 + 1)$ x + a .......(1)

⇒ $ax^2 - a^2 x$ - x + a

⇒ $a^x$ (x - a) - 1 (x - a)

= (x - a) (ax - 1)

Given p and q are two linear factors of (1)

∴ p = x + a and q = ax - 1

⇒ p + q = x - a + ax - 1

= x(a + 1) - 1 (a + 1)

= (x - 1) (a + 1)

∴ Option (d) is correct.

Answer: (d)

Let the roots of equation be α and 2 α

Sum of root

- (2 α + α) = ${3a - 1}/{a^2 - 5a + 3}$

-3 α = ${3a - 1}/{a^2 - 5a + 3}$....

Squaring Both the side

$9α^2 = {(3a - 1)^2}/{(a^2 - 5a + 3)^2}$ ......(i)

Product of root

2 α × α = $2/{a^2 - 5a + 3}$

$α^2 = 1/{a^2 - 5a + 3}$

Dividing equation (i) from (ii)

${9 α^2}/{α^2} = {(3a - 1)^2}/{(a^2 - 5a + 3)^2} × {a^2 - 5a + 3}/1$

9 = ${(3a - 1)^2}/{(a^2 - 5a + 3)}$

$9a^2 - 45a + 27 = 9a^2 + 1 - 6a$

26 = 39a

a = $2/3$