JEE Main 2022 (Online) 24th June Morning Shift | Application of Derivatives Question 91 | Mathematics

Let $$\lambda x - 2y = \mu $$ be a tangent to the hyperbola $${a^2}{x^2} - {y^2} = {b^2}$$. Then $${\left( {{\lambda \over a}} \right)^2} - {\left( {{\mu \over b}} \right)^2}$$ is equal to :

$$-$$2

$$-$$4

JEE Main 2021 (Online) 1st September Evening Shift

MCQ (Single Correct Answer)

-1

The function $$f(x) = {x^3} - 6{x^2} + ax + b$$ is such that $$f(2) = f(4) = 0$$. Consider two statements :

Statement 1 : there exists x₁, x₂ $$\in$$(2, 4), x₁ < x₂, such that f'(x₁) = $$-$$1 and f'(x₂) = 0.

Statement 2 : there exists x₃, x₄ $$\in$$ (2, 4), x₃ < x₄, such that f is decreasing in (2, x₄), increasing in (x₄, 4) and $$2f'({x_3}) = \sqrt 3 f({x_4})$$.

Then

both Statement 1 and Statement 2 are true

Statement 1 is false and Statement 2 is true

both Statement 1 and Statement 2 are false

Statement 1 is true and Statement 2 is false

JEE Main 2021 (Online) 31st August Morning Shift

MCQ (Single Correct Answer)

-1

The number of real roots of the equation

$${e^{4x}} + 2{e^{3x}} - {e^x} - 6 = 0$$ is :

JEE Main 2021 (Online) 27th August Evening Shift

MCQ (Single Correct Answer)

-1

A box open from top is made from a rectangular sheet of dimension a $$\times$$ b by cutting squares each of side x from each of the four corners and folding up the flaps. If the volume of the box is maximum, then x is equal to :

$${{a + b - \sqrt {{a^2} + {b^2} - ab} } \over {12}}$$

$${{a + b - \sqrt {{a^2} + {b^2} + ab} } \over 6}$$

$${{a + b - \sqrt {{a^2} + {b^2} - ab} } \over 6}$$

$${{a + b + \sqrt {{a^2} + {b^2} + ab} } \over 6}$$

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