List of top Mathematics Questions asked in JEE Main

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 :

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 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 :

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 :

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 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 \(P(x) = x^2 + bx + c\) 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 :

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 :

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 \(f: S \to S\) 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 :

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 :

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?

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 \(f(x)=\begin{vmatrix}\sin^2x&-2+\cos^2x&\cos2x\\2+\sin^2x&\cos^2x&\cos2x\\\sin^2x&\cos^2x&1+\cos2x\end{vmatrix},\ x\in[0,\pi]\). Then the maximum value of \(f(x)\) is equal to ______________.

Let \(A=\begin{bmatrix}1&2\\-1&4\end{bmatrix}\). If \(A^{-1}=\alpha I+\beta A,\ \alpha,\beta\in\mathbb{R}\), \(I\) is a \(2\times2\) identity matrix then \(4(\alpha-\beta)\) is equal to :

Let \(f : \mathbb{R} \to \mathbb{R}\) be defined as
\(f(x)=\begin{cases}\dfrac{\lambda\left|x^2-5x+6\right|}{\mu\left(5x-x^2-6\right)}, & x\lt 2\\[2mm] e^{\frac{\tan(x-2)}{x-[x]}}, & x\gt 2\\[2mm] \mu, & x=2\end{cases}\)
where \([x]\) is the greatest integer less than or equal to \(x\). If \(f\) is continuous at \(x=2\), then \(\lambda+\mu\) is equal to :

The solutions of the equation \(\begin{vmatrix} 1 + \sin^2 x & \sin^2 x & \sin^2 x \\ \cos^2 x & 1 + \cos^2 x & \cos^2 x \\ 4\sin 2x & 4\sin 2x & 1 + 4\sin 2x \end{vmatrix} = 0\), \((0 \lt x \lt \pi)\), are

For the four circles M, N, O and P, following four equations are given :
Circle M : \(x^2 + y^2 = 1\)
Circle N : \(x^2 + y^2 - 2x = 0\)
Circle O : \(x^2 + y^2 - 2x - 2y + 1 = 0\)
Circle P : \(x^2 + y^2 - 2y = 0\)
If the centre of circle M is joined with centre of the circle N, further center of circle N is joined with centre of the circle O, centre of circle O is joined with the centre of circle P and lastly, centre of circle P is joined with centre of circle M, then these lines form the sides of a :

If \(\int \frac{\cos x - \sin x}{\sqrt{8 - \sin 2x}}\,dx = a\sin^{-1}\left(\frac{\sin x + \cos x}{b}\right) + c\), where c is a constant of integration, then the ordered pair (a, b) is equal to :

Let \(P = \begin{bmatrix} -30 \& 20 \& 56 \\ 90 \& 140 \& 112 \\ 120 \& 60 \& 14 \end{bmatrix}\) and \(A = \begin{bmatrix} 2 \& 7 \& \omega^2 \\ -1 \& -\omega \& 1 \\ 0 \& -\omega \& -\omega + 1 \end{bmatrix}\) where \(\omega = \frac{-1 + i\sqrt3}{2}\), and \(I_3\) be the identity matrix of order 3. If the determinant of the matrix \((P^{-1}AP - I_3)^2\) is \(\alpha\omega^2\), then the value of \(\alpha\) is equal to ____________.

Let \(f : (0, 2) \to R\) be defined as \(f(x) = \log_2\left(1 + \tan\left(\frac{\pi x}{4}\right)\right)\). Then, \(\lim\limits_{n \to \infty} \frac{2}{n}\left(f\left(\frac1n\right) + f\left(\frac2n\right) + \dots + f(1)\right)\) is equal to ____________.

Let ABCD be a square of side of unit length. Let a circle \(C_1\) centered at A with unit radius is drawn. Another circle \(C_2\) which touches \(C_1\) and the lines AD and AB are tangent to it, is also drawn. Let a tangent line from the point C to the circle \(C_2\) meet the side AB at E. If the length of EB is \(\alpha + \sqrt3\,\beta\), where \(\alpha, \beta\) are integers, then \(\alpha + \beta\) is equal to ____________.

Let z and \(\omega\) be two complex numbers such that \(\omega = z\overline{z} - 2z + 2\), \(\left|\frac{z+i}{z-3i}\right| = 1\) and Re(\(\omega\)) has minimum value. Then, the minimum value of \(n \in N\) for which \(\omega^n\) is real, is equal to ____________.

If the normal to the curve \(y(x) = \int\limits_0^x (2t^2 - 15t + 10)dt\) at a point \((a, b)\) is parallel to the line \(x + 3y = -5\), \(a \gt 1\), then the value of \(|\,a + 6b\,|\) is equal to ____________.

Consider an arithmetic series and a geometric series having four initial terms from the set \(\{11, 8, 21, 16, 26, 32, 4\}\). If the last terms of these series are the maximum possible four digit numbers, then the number of common terms in these two series is equal to ____________.