List of top Questions asked in JEE Main

The number of unpaired electrons responsible for the paramagnetic nature of the following complex species are respectively : $ [Fe(CN)_6]^{3-}, [FeF_6]^{3-}, [CoF_6]^{3-}, [Mn(CN)_6]^{3-} $

Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R)
Assertion (A) : Refractive index of glass is higher than that of air.
Reason (R) : Optical density of a medium is directly proportionate to its mass density which results in a proportionate refractive index.
In the light of the above statements, choose the most appropriate answer from the options given below :

Let the lengths of the transverse and conjugate axes of a hyperbola in standard form be $ 2a $ and $ 2b $, respectively, and one focus and the corresponding directrix of this hyperbola be $ (-5, 0) $ and $ 5x + 9 = 0 $, respectively. If the product of the focal distances of a point $ (\alpha, 2\sqrt{5}) $ on the hyperbola is $ p $, then $ 4p $ is equal to:

If $ \int \left( \frac{1}{x} + \frac{1}{x^3} \right) \left( \sqrt[23]{3x^{-24}} + x^{-26} \right) \, dx $ is equal to $ -\frac{\alpha}{3(\alpha + 1)} \left( 3x^\beta + x^\gamma \right)^{\alpha + 1} + C, \quad x>0, $ where $ \alpha, \beta, \gamma \in \mathbb{Z} $ and $ C $ is the constant of integration, then $ \alpha + \beta + \gamma $ is equal to _______.

For $ t>-1 $, let $ \alpha_t $ and $ \beta_t $ be the roots of the equation $ \left( (t + 2)^{\frac{1}{7}} - 1 \right)x^2 + \left( (t + 2)^{\frac{1}{6}} - 1 \right)x + \left( (t + 2)^{\frac{1}{21}} - 1 \right) = 0. $ If $ \lim_{t \to 1^+} \alpha_t = a $ and $ \lim_{t \to 1^+} \beta_t = b $, then $ 72(a + b)^2 $ is equal to:

The number of real roots of the equation $ x|x-2| + 3|x-3| + 1 = 0 $ is:

If the function $ f(x) = \frac{\tan(\tan x) - \sin(\sin x)}{\tan x - \sin x} $ is continuous at $ x = 0 $, then $ f(0) $ is equal to:

Let the length of a latus rectum of an ellipse $ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $ be 10. If its eccentricity is $ e $, and the minimum value of the function $ f(t) = t^2 + t + \frac{11}{12} $, where $ t \in \mathbb{R} $, then $ a^2 + b^2 $ is equal to:

If the locus of $ z \in \mathbb{C} $, such that $ \text{Re} \left( \frac{z - 1}{2z + i} \right) + \text{Re} \left( \frac{ \bar{z} - 1}{2 \bar{z} - i} \right) = 2, $ is a circle of radius $ r $ and center $ (a, b) $, then $ \frac{15ab}{r^2} \text{ is equal to:} $

Let $ y = y(x) $ be the solution of the differential equation $ (x^2 + 1)y' - 2xy = (x^4 + 2x^2 + 1)\cos x, $ with the initial condition $ y(0) = 1 $. Then $ \int_{-3}^{3} y(x) \, dx \text{ is:} $

Let $ p $ be the number of all triangles that can be formed by joining the vertices of a regular polygon $ P $ of $ n $ sides, and $ q $ be the number of all quadrilaterals that can be formed by joining the vertices of $ P $. If $ p + q = 126 $, then the eccentricity of the ellipse $ \frac{x^2}{16} + \frac{y^2}{n} = 1 $ is:

Let a random variable X take values 0, 1, 2, 3 with $ P(X = 0) = P(X = 1) = p, \, P(X = 2) = P(X = 3), \, \text{and} \, F(X^2) = 2F(X). $ Then the value of $ 8p - 1 $ is: