Choose the correct answer from the given four options in the foll… - Vidyadip

MCQ

Choose the correct answer from the given four options in the following questions:
The zeroes of the quadratic polynomial $\text{x}^2+\text{kx}+\text{k},\text{ k}\neq0,$:

✓
Cannot both be positive.
B
Cannot both be negative
C
Are always unequal.
D
Are always equal.

✓

Answer

Correct option: A.

Cannot both be positive.

Let p(x) $=\text{x}^2+\text{kx}+\text{k},\text{ k}\neq0,$
On comparing p(x) with $ax^2 + bx + c,$ we get
a = 1, b = k and c = k
Now, $\text{x}=\frac{-\text{b}\pm\sqrt{\text{b}^2-4\text{ac}}}{2\text{a}}$ [by quadratic formula]
$=\frac{-\text{k}\pm\sqrt{\text{k}^2-4\text{k}}}{2\times1}$
$=\frac{-\text{k}\pm\sqrt{\text{k}(\text{k}-4)}}{2},\text{k}\neq0$

Here, we see that
k(k - 4) > 0
$\Rightarrow\ \text{k }\in(-\infty,0)\text{ u }(4,\infty)$
Now, we know that
In quadratic Polynomial $ax^2 + bx + c$
If a > 0, b > 0, c > 0 or a < 0, b < 0, c < 0,
Then the polynomial has always all negative zeroes.
and if a > 0, c > 0 of a < 0, c > 0, then the ploynomial has always zeroes of opposite sing
Case I:
If $\text{k}\in(-\infty, 0)\text{ i.e., k}<0$
⇒ a = 1 > 0, b,c = k < 0
So, both zeroes are of opposite sign.
Case II:
If $\text{k}\in(4,\infty)\text{ i.e., k}\leq4$
⇒ a = 1 > 0, b, c > 4
So, both zeroes are negative,
Hence, in any case zeroes of the quadratic polynomial cannot both be positive.

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