List of top Questions asked in JEE Main

The maximum value of
\(f(x)=\left|\begin{matrix}\sin^2 x & 1+\cos^2 x & \cos 2x\\ 1+\sin^2 x & \cos^2 x & \cos 2x\\ \sin^2 x & \cos^2 x & \sin 2x\end{matrix}\right|,\ x\in R\)
is :

Let C be the locus of the mirror image of a point on the parabola \(y^2=4x\) with respect to the line \(y=x\). Then the equation of tangent to C at \(P(2, 1)\) is :

Let \(\frac{1}{16}\), a and b be in G.P. and \(\frac{1}{a}\), \(\frac{1}{b}\), 6 be in A.P., where \(a, b>0\). Then \(72(a+b)\) is equal to ___________.

If \((x, y, z)\) be an arbitrary point lying on a plane P which passes through the points \((42, 0, 0)\), \((0, 42, 0)\) and \((0, 0, 42)\), then the value of the expression
\(3+\frac{x-11}{(y-19)^2(z-12)^2}+\frac{y-19}{(x-11)^2(z-12)^2}+\frac{z-12}{(x-11)^2(y-19)^2}-\frac{x+y+z}{14(x-11)(y-19)(z-12)}\)
is equal to :

If the points of intersections of the ellipse \(\frac{x^2}{16}+\frac{y^2}{b^2}=1\) and the circle \(x^2+y^2=4b,\ b>4\) lie on the curve \(y^2=3x^2\), then b is equal to :

Given that the inverse trigonometric functions take principal values only. Then, the number of real values of \(x\) which satisfy
\(\sin^{-1}\left(\frac{3x}{5}\right) + \sin^{-1}\left(\frac{4x}{5}\right) = \sin^{-1}x\)
is equal to :

Let \(\alpha\in\mathbb{R}\) be such that the function
\(f(x) = \begin{cases}\dfrac{\cos^{-1}(1 - \{x\}^2)\,\sin^{-1}(1 - \{x\})}{\{x\} - \{x\}^3}, & x \ne 0\\[2mm] \alpha, & x = 0\end{cases}\)
is continuous at \(x = 0\), where \(\{x\} = x - [\,x\,]\) and \([\,x\,]\) is the greatest integer less than or equal to \(x\). Then :

Let \(A(-1, 1)\), \(B(3, 4)\) and \(C(2, 0)\) be given three points.
A line \(y = mx\), \(m > 0\), intersects lines AC and BC at point P and Q respectively. Let \(A_1\) and \(A_2\) be the areas of \(\Delta ABC\) and \(\Delta PQC\) respectively, such that \(A_1 = 3A_2\), then the value of \(m\) is equal to :

Let the lengths of intercepts on x-axis and y-axis made by the circle
\(x^2 + y^2 + ax + 2ay + c = 0\), \((a < 0)\) be \(2\sqrt{2}\) and \(2\sqrt{5}\), respectively. Then the shortest distance from origin to a tangent to this circle which is perpendicular to the line \(x + 2y = 0\), is equal to :

Let A denote the event that a 6-digit integer formed by 0, 1, 2, 3, 4, 5, 6 without repetitions, be divisible by 3. Then probability of event A is equal to :

Let \(P(x) = x^2 + bx + c\) be a quadratic polynomial with real coefficients such that \(\int_0^1 P(x)\,dx = 1\) and \(P(x)\) leaves remainder 5 when it is divided by \((x - 2)\). Then the value of \(9(b + c)\) is equal to :

The least value of \(|z|\) where \(z\) is complex number which satisfies the inequality
\(\exp\left(\frac{(|z|+3)(|z|-1)}{||z|+1|}\log_e 2\right)\ge\log_{\sqrt{2}}\left|5\sqrt{7} + 9i\right|,\ i = \sqrt{-1}\), is equal to :

Consider the integral
\(I = \int_0^{10}\frac{[x]e^{[x]}}{e^{x-1}}\,dx\),
where \([x]\) denotes the greatest integer less than or equal to \(x\). Then the value of \(I\) is equal to :

Let \(C_1\) be the curve obtained by the solution of differential equation
\(2xy\frac{dy}{dx} = y^2 - x^2,\ x > 0\). Let the curve \(C_2\) be the solution of
\(\frac{2xy}{x^2 - y^2} = \frac{dy}{dx}\). If both the curves pass through \((1, 1)\), then the area enclosed by the curves \(C_1\) and \(C_2\) is equal to :

Consider a rectangle ABCD having 5, 7, 6, 9 points in the interior of the line segments AB, CD, BC, DA respectively. Let \(\alpha\) be the number of triangles having these points from different sides as vertices and \(\beta\) be the number of quadrilaterals having these points from different sides as vertices. Then \((\beta - \alpha)\) is equal to :

If the foot of the perpendicular from point \((4, 3, 8)\) on the line \(L_1: \frac{x-a}{l} = \frac{y-2}{3} = \frac{z-b}{4}\), \(l \ne 0\) is \((3, 5, 7)\), then the shortest distance between the line \(L_1\) and line \(L_2: \frac{x-2}{3} = \frac{y-4}{4} = \frac{z-5}{5}\) is equal to :

Let \(\vec{a} = \hat{i} + 2\hat{j} - 3\hat{k}\) and \(\vec{b} = 2\hat{i} - 3\hat{j} + 5\hat{k}\). If \(\vec{r}\times\vec{a} = \vec{b}\times\vec{r}\),
\(\vec{r}\cdot\left(\alpha\hat{i} + 2\hat{j} + \hat{k}\right) = 3\) and \(\vec{r}\cdot\left(2\hat{i} + 5\hat{j} - \alpha\hat{k}\right) = -1\), \(\alpha\in\mathbb{R}\), then the value of \(\alpha + |\vec{r}|^2\) is equal to :

Let \(f: S \to S\) where \(S = (0, \infty)\) be a twice differentiable function such that \(f(x+1) = xf(x)\). If \(g: S \to \mathbb{R}\) be defined as \(g(x) = \log_e f(x)\), then the value of \(|g''(5) - g''(1)|\) is equal to :

If \(y = y(x)\) is the solution of the differential equation
\(\frac{dy}{dx} + (\tan x)\,y = \sin x,\ 0 \le x \le \frac{\pi}{3}\), with \(y(0) = 0\), then \(y\left(\frac{\pi}{4}\right)\) is equal to :

Let \(f\) be a real valued function, defined on \(\mathbb{R} - \{-1, 1\}\) and given by
\(f(x) = 3\log_e\left|\frac{x-1}{x+1}\right| - \frac{2}{x-1}\).
Then in which of the following intervals, function \(f(x)\) is increasing?

Find the truth table for the function Y of A and B represented in the following figure.

Consider the above reaction and identify the Product P :

Match List - I with List - II :

	List-I (Drug)		List-II (Class of Drug)
(a)	Furacin	(i)	Antibiotic
(b)	Arsphenamine	(ii)	Tranquilizers
(c)	Dimetone	(iii)	Antiseptic
(d)	Valium	(iv)	Synthetic antihistamines

Choose the most appropriate match :

List-I	List-II
(a) MI of the rod (length L, Mass M, about an axis \(\perp\) to the rod passing through the midpoint)	(i) \(8ML^2/3\)
(b) MI of the rod (length L, Mass 2M, about an axis \(\perp\) to the rod passing through one of its end)	(ii) \(ML^2/3\)
(c) MI of the rod (length 2L, Mass M, about an axis \(\perp\) to the rod passing through its midpoint)	(iii) \(ML^2/12\)
(d) MI of the rod (Length 2L, Mass 2M, about an axis \(\perp\) to the rod passing through one of its end)	(iv) \(2ML^2/3\)

Choose the correct answer from the options given below:

The major product formed in the following reaction is :