(D) The given point $(t^2 + 2t + 5, 2t^2 + t - 2)$ lies on the line $x + y = 2$ if the coordinates satisfy the equation:$(t^2 + 2t + 5) + (2t^2 + t -

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(D) The given point $(t^2 + 2t + 5, 2t^2 + t - 2)$ lies on the line $x + y = 2$ if the coordinates satisfy the equation:$(t^2 + 2t + 5) + (2t^2 + t - 2) = 2$$3t^2 + 3t + 3 = 2$$3t^2 + 3t + 1 = 0$For this quadratic equation,the discriminant $D = b^2 - 4ac = (3)^2 - 4(3)(1) = 9 - 12 = -3$.Since the discriminant is negative $(D Therefore,the point cannot lie on the line for any real value of $t$.

Class 11 Mathematics 4-2.Quadratic Equations and Inequations Solution of quadratic equations and Nature of roots

Easy

The point $(t^2 + 2t + 5, 2t^2 + t - 2)$ lies on the line $x + y = 2$ for

A
All real values of $t$
B
Some real values of $t$
C
$t = \frac{-3 \pm \sqrt{3}}{6}$
D
None of these

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4-2.Quadratic Equations and Inequations

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Solution of quadratic equations and Nature of roots

If $ax^2 + bx + c = 0$ has real and distinct roots,$\alpha$ and $\beta$ where $\beta > \alpha$. Further,if $a > 0, b < 0$,and $c < 0$,then:

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If $a$ and $b$ are odd integers, then the roots of the equation $2ax^2 + (2a + b)x + b = 0$, where $a \neq 0$, are

MediumWBJEE 2022

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For which interval of values of $a$ will the equation $3x^2 + 2(a^2 + 1)x + (a^2 - 3a + 2) = 0$ have roots of opposite signs?

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If the expression $x^3+3x^2-9x+\lambda$ is of the form $(x-\alpha)^2(x-\beta)$,then the values of $\lambda$ are

MediumTS EAMCET 2023

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The solution of the equation $2x^3 - x^2 - 22x - 24 = 0$,given that two of the roots are in the ratio $3:4$,is:

MediumAP EAMCET 2021

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The point $(t^2 + 2t + 5, 2t^2 + t - 2)$ lies on the line $x + y = 2$ for

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