NIMCET 2024 — Mathematics PYQ

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Question

NIMCET 2024 — Mathematics PYQ

NIMCET | Mathematics | 2024

For what values of $λ$ does the equation $6 x^{2} - x y + λ y^{2} = 0$ represent two perpendicular lines and two lines inclined at an angle of $\frac{π}{4}$ .

Choose the correct answer:

Explanation

1. For Perpendicular Lines:

Condition: $a + b = 0$

In $6x^2 - xy + \lambda y^2 = 0$ , we have $a = 6$ and $b = \lambda$ .

6 + \lambda = 0

\lambda = -6

2. For Lines Inclined at $\pi/4$ :

Formula: $\tan \theta = \left| \frac{2\sqrt{h^2 - ab}}{a + b} \right|$

Given $\theta = \pi/4 \implies \tan(\pi/4) = 1$ .

Values: $a = 6, b = \lambda, h = -1/2$ .

1 = \left| \frac{2\sqrt{(-1/2)^2 - 6\lambda}}{6 + \lambda} \right|

Squaring both sides:

1 = \frac{4(1/4 - 6\lambda)}{(6 + \lambda)^2}

(6 + \lambda)^2 = 1 - 24\lambda

36 + 12\lambda + \lambda^2 = 1 - 24\lambda

\lambda^2 + 36\lambda + 35 = 0

Factoring the quadratic:

(\lambda + 35)(\lambda + 1) = 0

\lambda = -35 \text{ or } \lambda = -1

Correct Answer:

$\lambda = -6$ for perpendicular lines and $\lambda = -35, -1$ for the angle $\pi/4$ .

Choose the correct answer:

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