Class 11Mathematics4-2.Quadratic Equations and InequationsSolution of quadratic equations and Nature of roots
EasyFind the number of real solutions for the equation $\left( \frac{5}{7} \right)^x = -x^2 + 2x - 3$.
- A$2$
- B$0$
- C$1$
- DNone of these
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Let $\alpha$ and $\beta$ be the roots of the quadratic equation $a x^2+b x+c=0$. Observe the lists given below:
List-$I$ List-$II$ $(i)$ $\alpha = \beta$ $(A)$ $(ac^2)^{1/3} + (a^2c)^{1/3} + b = 0$ $(ii)$ $\alpha = 2\beta$ $(B)$ $2b^2 = 9ac$ $(iii)$ $\alpha = 3\beta$ $(C)$ $b^2 = 6ac$ $(iv)$ $\alpha = \beta^2$ $(D)$ $3b^2 = 16ac$ $(E)$ $b^2 = 4ac$ $(F)$ $(ac^2)^{1/3} + (a^2c)^{1/3} = b$
The correct match of List-$I$ from List-$II$ is:
| List-$I$ | List-$II$ |
| $(i)$ $\alpha = \beta$ | $(A)$ $(ac^2)^{1/3} + (a^2c)^{1/3} + b = 0$ |
| $(ii)$ $\alpha = 2\beta$ | $(B)$ $2b^2 = 9ac$ |
| $(iii)$ $\alpha = 3\beta$ | $(C)$ $b^2 = 6ac$ |
| $(iv)$ $\alpha = \beta^2$ | $(D)$ $3b^2 = 16ac$ |
| $(E)$ $b^2 = 4ac$ | |
| $(F)$ $(ac^2)^{1/3} + (a^2c)^{1/3} = b$ |
The correct match of List-$I$ from List-$II$ is:
The equation $\sqrt{x + 1} - \sqrt{x - 1} = \sqrt{4x - 1}$ has
Two positive distinct numbers $a$ and $b$ each differ from their reciprocal by $1$. The value of $a + b$ is
What is the number of real solutions to the equation $(5 + 2\sqrt{6})^{x^2 - 3} + (5 - 2\sqrt{6})^{x^2 - 3} = 10$?
Medium
View SolutionIf the roots of the equation $\frac{\alpha}{x - \alpha} + \frac{\beta}{x - \beta} = 1$ are equal in magnitude but opposite in sign,then $\alpha + \beta =$
Easy
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