List of top Mathematics Questions asked in JEE Main

Let \(L\) be a common tangent line to the curves
\(4x^2+9y^2=36\) and \((2x)^2+(2y)^2=31\). Then the square of the slope of the line \(L\) is __________.

If \(I_{m,n}=\int_0^1x^{m-1}(1-x)^{n-1}dx\), for \(m,n\ge1\), and
\(\int_0^1\frac{x^{m-1}+x^{n-1}}{(1+x)^{m+1}}dx=\alpha I_{m,n}\), \(\alpha\in R\), then \(\alpha\) equals ___________.

If the matrix \(A=\begin{bmatrix}1&0&0\\0&2&0\\3&0&-1\end{bmatrix}\) satisfies the equation
\(A^{20}+\alpha A^{19}+\beta A=\begin{bmatrix}1&0&0\\0&4&0\\0&0&1\end{bmatrix}\)
for some real numbers \(\alpha\) and \(\beta\), then \(\beta-\alpha\) is equal to ___________.

If the arithmetic mean and geometric mean of the \(p^{th}\) and \(q^{th}\) terms of the sequence \(-16,8,-4,2,\dots\) satisfy the equation \(4x^2-9x+5=0\), then \(p+q\) is equal to __________.

Let \(\alpha\) and \(\beta\) be two real numbers such that \(\alpha+\beta=1\) and \(\alpha\beta=-1\). Let \(p_n=(\alpha)^n+(\beta)^n\), \(p_{n-1}=11\) and \(p_{n+1}=29\) for some integer \(n\ge1\). Then, the value of \(p_n^2\) is ___________.

Let \(z\) be those complex numbers which satisfy
\(|z+5|\le4\) and \(z(1+i)+\overline{z}(1-i)\ge-10\), \(i=\sqrt{-1}\).
If the maximum value of \(|z+1|^2\) is \(\alpha+\beta\sqrt2\), then the value of \((\alpha+\beta)\) is ____________.

The total number of 4-digit numbers whose greatest common divisor with 18 is 3, is _________.

If the mirror image of the point \((1,3,5)\) with respect to the plane \(4x-5y+2z=8\) is \((\alpha,\beta,\gamma)\), then \(5(\alpha+\beta+\gamma)\) equals :

Let \(F_1(A,B,C)=(A\wedge\sim B)\vee[\sim C\wedge(A\vee B)]\vee\sim A\) and
\(F_2(A,B)=(A\vee B)\vee(B\to\sim A)\) be two logical expressions. Then :

Let slope of the tangent line to a curve at any point \(P(x,y)\) be given by \(\frac{xy^2+y}{x}\). If the curve intersects the line \(x+2y=4\) at \(x=-2\), then the value of \(y\), for which the point \((3,y)\) lies on the curve, is :

The triangle of maximum area that can be inscribed in a given circle of radius 'r' is :

Let \(L\) be a line obtained from the intersection of two planes \(x + 2y + z = 6\) and \(y + 2z = 4\). If point \(P(\alpha, \beta, \gamma)\) is the foot of perpendicular from \((3, 2, 1)\) on \(L\), then the value of \(21(\alpha + \beta + \gamma)\) equals :

The sum of the series
\(\displaystyle\sum_{n=1}^{\infty}\frac{n^2+6n+10}{(2n+1)!}\)
is equal to :

A seven digit number is formed using digits \(3, 3, 4, 4, 4, 5, 5\). The probability, that number so formed is divisible by 2, is :

If vectors \(\vec{a_1} = x\hat{i} - \hat{j} + \hat{k}\) and \(\vec{a_2} = \hat{i} + y\hat{j} + z\hat{k}\) are collinear, then a possible unit vector parallel to the vector \(x\hat{i} + y\hat{j} + z\hat{k}\) is :

Let \(f(x) = \sin^{-1}x\) and \(g(x) = \frac{x^2 - x - 2}{2x^2 - x - 6}\). If \(g(2) = \displaystyle\lim_{x \to 2} g(x)\), then the domain of the function \(f \circ g\) is :

If the locus of the mid-point of the line segment from the point \((3, 2)\) to a point on the circle, \(x^2 + y^2 = 1\) is a circle of radius \(r\), then \(r\) is equal to :

Let \(f : R \to R\) be defined as
\(f(x) = \begin{cases} 2\sin\left(-\frac{\pi x}{2}\right), & \text{if } x \lt -1 \\ |ax^2 + x + b|, & \text{if } -1 \le x \le 1 \\ \sin(\pi x), & \text{if } x \gt 1 \end{cases}\)
If \(f(x)\) is continuous on \(R\), then \(a + b\) equals :

A natural number has prime factorization given by \(n = 2^x 3^y 5^z\), where \(y\) and \(z\) are such that \(y + z = 5\) and \(y^{-1} + z^{-1} = \frac{5}{6}\), \(y \gt z\). Then the number of odd divisions of \(n\), including 1, is :

Let \(A = \{1, 2, 3, \dots, 10\}\) and \(f : A \to A\) be defined as
\(f(k) = \begin{cases} k + 1 & \text{if } k \text{ is odd} \\ k & \text{if } k \text{ is even} \end{cases}\)
Then the number of possible functions \(g : A \to A\) such that \(g \circ f = f\) is :

Let \(f(x) = \displaystyle\int_0^x e^t f(t)\,dt + e^x\) be a differentiable function for all \(x \in R\). Then \(f(x)\) equals :

For \(x \gt 0\), if \(f(x) = \displaystyle\int_1^x \frac{\log_e t}{(1 + t)}\,dt\), then \(f(e) + f\left(\frac{1}{e}\right)\) is equal to :

If \(0 \lt a, b \lt 1\), and \(\tan^{-1}a + \tan^{-1}b = \frac{\pi}{4}\), then the value of
\((a + b) - \left(\frac{a^2 + b^2}{2}\right) + \left(\frac{a^3 + b^3}{3}\right) - \left(\frac{a^4 + b^4}{4}\right) + \dots\)
is :

Consider the following system of equations :
\(x + 2y - 3z = a\)
\(2x + 6y - 11z = b\)
\(x - 2y + 7z = c\),
where \(a\), \(b\) and \(c\) are real constants. Then the system of equations :

Let \(A(1, 4)\) and \(B(1, -5)\) be two points. Let \(P\) be a point on the circle \((x - 1)^2 + (y - 1)^2 = 1\) such that \((PA)^2 + (PB)^2\) have maximum value, then the points, \(P\), \(A\) and \(B\) lie on :