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

Answer: (b)

Here m and n are the roots of the equation

$x^2$ + ax + b = 0.

m + n = - a .......(i)

mn = b ......(ii)

Also m $m^2 and n^2$ are the roots of the equation of

$x^2$ - cx + d = 0.

$m^2 + n^2$ = c...(iii)

$m^2 n^2$ = d .....(iv)

by squaring Eq. (i) both sides, we get

$m^2 + n^2 + 2mn = a^2$ [from Eqs. (i) and (ii)]

⇒ c + 2b = $a^2 ⇒ c = a^2 - 2b$

⇒ 2b - $a^2$ = -c

Therefore, Statement 1 is incorrect.

From Eq. (ii)

$m^2 n^2 = b^2 ⇒ b^2$ = d

Therefore, Statement 2 is correct.

Answer: (b)

We know that two equation

$a_1 x^2 + b_1 x + c_1$ = 0

$a_2 x^2 + b_2 + c_2$ = 0

have common root when

$(c_1 a_2 - a_1 c_2)^2 = (b_1 c_2 - c_1 b_2) (a_1 b_2 - b_1 a_2)$

So, for $x^2 + 5x + 6 = 0 and x^2 + kx = 1 = 0$

we have $(5)^2$ = (5 - 6x) (x - 5)

⇒ 25 = $-6x^2$ + 35x - 25

⇒ $6x^2$ - 35x + 50 = 0

⇒ x = $5/2 or {10}/3$

Answer: (b)

Given equation,

$2x^2$ - 3x - 4 = 0

For a reciprocal roots, we replace x by $1/x$, we get

$2(1/x)^2 - 3(1/x) - 4 = 0$

⇒ $-4x^2$ - 3x + 2 = 0

⇒ $4x^2$ + 3x - 2 = 0

Answer: (b)

Given equation,

$√{{2x}/{3 - x}} - √{{3 - x}/{2x}} = 3/2$

Let $√{{2x}/{3 - x}}$ = a

∴ a - $1/a = 3/2$

⇒ $2(a^2 - 1)$ = 3a

⇒ $2a^2$ - 3a - 2 = 0

⇒ $2a^2$ - 4a + a - 2 = 0

⇒ 2a(a - 2) + 1(a - 2) = 0

⇒ (2a + 1) (a - 2) = 0

If a - 2 = 0

Now, put a = 2

⇒ $√{{2x}/{3 - x}}$ = 2

Squaring both sides, then we get

⇒ 2x = 4(3 - x)

⇒ 6x = 12 ⇒ x = 2

If 2a + 1 = 0,

⇒ a = -$1/2, a ≠ {-1}/2$

x = 2 is the root of equation.

Answer: (d)

Note: Let the quardatic equation be $ax^2$ + b + c = 0.

To find roots of this equation quickly, we find two factors of 'b' such that their sum is equal to b and their product is equal to the product of the coefficient of x2 and the constant term 'c'.

Let two such factors be α and β.

The α + β = b and α β = ca

In the second step, we divide these factors by the coefficient of $x^2$ ,

ie be 'a'.

In the next step, we change the signs of the outcome. These are the

roots of the equation.