List of top Questions asked in JEE Main

If tangents are drawn to the ellipse \(x^2 + 2y^2 = 2\) at all points on the ellipse other than its four vertices then the mid points of the tangents intercepted between the coordinate axes lie on the curve :

The value of \(r\) for which \({}^{20}C_r\,{}^{20}C_0 + {}^{20}C_{r-1}\,{}^{20}C_1 + {}^{20}C_{r-2}\,{}^{20}C_2 + \ldots + {}^{20}C_0\,{}^{20}C_r\) is maximum, is

The direction ratios of normal to the plane through the points \((0, -1, 0)\) and \((0, 0, 1)\) and making an angle \(\frac{\pi}{4}\) with the plane \(y - z + 5 = 0\) are :

The plane containing the line \(\frac{x-3}{2} = \frac{y+2}{-1} = \frac{z-1}{3}\) and also containing its projection on the plane \(2x + 3y - z = 5\), contains which one of the following points ?

If \(q\) is false and \(p \wedge q \leftrightarrow r\) is true, then which one of the following statements is a tautology ?

If \(x\log_e(\log_e x) - x^2 + y^2 = 4\ (y \gt 0)\), then \(\frac{dy}{dx}\) at \(x = e\) is equal to :

Let \(f_k(x) = \frac{1}{k}\left(\sin^k x + \cos^k x\right)\) for \(k = 1, 2, 3, \dots\) Then for all \(x \in \mathbb{R}\), the value of \(f_4(x) - f_6(x)\) is equal to

Equation of a common tangent to the parabola \(y^2 = 4x\) and the hyperbola \(xy = 2\) is :

Let \(\vec{a} = \hat{i} + 2\hat{j} + 4\hat{k}\), \(\vec{b} = \hat{i} + \lambda\hat{j} + 4\hat{k}\) and \(\vec{c} = 2\hat{i} + 4\hat{j} + (\lambda^2 - 1)\hat{k}\) be coplanar vectors. Then the non-zero vector \(\vec{a} \times \vec{c}\) is :

Let \(\left(-2 - \frac{1}{3}i\right)^3 = \frac{x + iy}{27}\ (i = \sqrt{-1})\), where x and y are real numbers, then \(y - x\) equals :

Let \(f(x) = \begin{cases} -1, & -2 \le x \lt 0 \\ x^2 - 1, & 0 \le x \le 2 \end{cases}\) and \(g(x) = |f(x)| + f(|x|)\). Then, in the interval \((-2, 2)\), g is :

Let \([x]\) denote the greatest integer less than or equal to \(x\). Then \(\lim_{x \to 0} \frac{\tan(\pi \sin^2 x) + \left(|x| - \sin(x[x])\right)^2}{x^2}\)

The sum of the real values of \(x\) for which the middle term in the binomial expansion of \(\left(\frac{x^3}{3} + \frac{3}{x}\right)^8\) equals 5670 is :

If \(y(x)\) is the solution of the differential equation \(\frac{dy}{dx} + \left(\frac{2x+1}{x}\right)y = e^{-2x},\ x \gt 0\), where \(y(1) = \frac{1}{2}e^{-2}\), then

Let \(f : \mathbb{R} \to \mathbb{R}\) be defined by \(f(x) = \frac{x}{1+x^2},\ x \in \mathbb{R}\). Then the range of f is :

The maximum value of the function \(f(x) = 3x^3 - 18x^2 + 27x - 40\) on the set \(S = \{x \in \mathbb{R} : x^2 + 30 \le 11x\}\) is :

The sum of an infinite geometric series with positive terms is 3 and the sum of the cubes of its terms is \(\frac{27}{19}\). Then the common ratio of this series is :

A square is inscribed in the circle \(x^2 + y^2 - 6x + 8y - 103 = 0\) with its sides parallel to the coordinate axes. Then the distance of the vertex of this square which is nearest to the origin is :

The straight line \(x + 2y = 1\) meets the coordinate axes at A and B. A circle is drawn through A, B and the origin. Then the sum of perpendicular distances from A and B on the tangent to the circle at the origin is :

If one real root of the quadratic equation \(81x^2 + kx + 256 = 0\) is cube of the other root, then a value of \(k\) is

The area (in sq. units) of the region bounded by the curve \(x^2 = 4y\) and the straight line \(x = 4y - 2\) is :

The reaction that is NOT involved in the ozone layer depletion mechanism in the stratosphere is :

Among the following reactions of hydrogen with halogens, the one that requires a catalyst is :

The electrolytes usually used in the electroplating of gold and silver, respectively are :

The major product of the following reaction is