The number of integral values of a for which | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) For $x^2 - (a - 1)x + 3 = 0$ to have both roots positive:$1$. Discriminant $D_1 \ge 0 \Rightarrow (a - 1)^2 - 12 \ge 0 \Rightarrow (a - 1)^2 \ge 1

Vedclass · Accepted Answer

$1$

Vedclass · Answer

(D) For $x^2 - (a - 1)x + 3 = 0$ to have both roots positive:$1$. Discriminant $D_1 \ge 0 \Rightarrow (a - 1)^2 - 12 \ge 0 \Rightarrow (a - 1)^2 \ge 12 \Rightarrow a - 1 \ge 2\sqrt{3}$ or $a - 1 \le -2\sqrt{3}$. Since $2\sqrt{3} \approx 3.46$,$a \ge 4.46$ or $a \le -2.46$.$2$. Sum of roots $> 0 \Rightarrow a - 1 > 0 \Rightarrow a > 1$.$3$. Product of roots $> 0 \Rightarrow 3 > 0$ (always true).Combining these,$a \ge 4.46$.For $x^2 + 3x + 6 - a = 0$ to have both roots negative:$1$. Discriminant $D_2 \ge 0 \Rightarrow 3^2 - 4(6 - a) \ge 0 \Rightarrow 9 - 24 + 4a \ge 0 \Rightarrow 4a \ge 15 \Rightarrow a \ge 3.75$.$2$. Sum of roots $$3$. Product of roots $> 0 \Rightarrow 6 - a > 0 \Rightarrow a Combining these,$3.75 \le a Intersection of both conditions: $a \ge 4.46$ and $3.75 \le a The integral values of $a$ are $5$. Thus,there is only $1$ such value.

The number of integral values of $a$ for which $x^2 - (a - 1)x + 3 = 0$ has both roots positive and $x^2 + 3x + 6 - a = 0$ has both roots negative is

Similar Questions

The area of a rectangle is the same as that of a circle of radius $\frac{\sqrt{35}}{\sqrt{11}} \text{ cm}$. If the length of the rectangle exceeds its breadth by $3 \text{ cm}$,find the dimensions of the rectangle.

If $\alpha$ and $\beta$ are two roots of the equation $x^{2}-64x+256=0$,then the value of $\left(\frac{\alpha^{3}}{\beta^{5}}\right)^{\frac{1}{8}}+\left(\frac{\beta^{3}}{\alpha^{5}}\right)^{\frac{1}{8}}$ is

If the roots of the given equation $(2k + 1)x^2 - (7k + 3)x + k + 2 = 0$ are reciprocal to each other,then the value of $k$ will be

Let one root of the equation $ax^{2} + bx + c = 0$ be $\alpha$. Then the other root is $2\alpha$. Find the relation between $a, b,$ and $c$.

Solve the given two equations and select the correct answer from the given options.
$I.$ $(441)^{\frac{1}{2}} x^{2} - 111 = (15)^{2}$
$II.$ $\sqrt{121} y^{2} + (6)^{3} = 260$

Vedclass Test Series

Exam Paper Generator

Online Exam Module