List of top Mathematics Questions asked in JEE Main

The solution of the differential equation
\(\frac{dy}{dx} - \frac{y+3x}{\log_e (y+3x)} + 3 = 0\)
is:
(where c is a constant of integration)

The area (in sq. units) of the largest rectangle ABCD whose vertices A and B lie on the x-axis and vertices C and D lie on the parabola, \(y = x^2 - 1\) below the x-axis, is :

If the equation of a plane P, passing through the intersection of the planes, \(x+4y-z+7=0\) and \(3x+y+5z=8\) is \(ax+by+6z=15\) for some \(a,b\in\mathbf{R}\), then the distance of the point (3, 2, \(-1\)) from the plane P is ______.

If the system of equations
\(x-2y+3z=9\)
\(2x+y+z=b\)
\(x-7y+az=24\),
has infinitely many solutions, then \(a-b\) is equal to ______.

The value of \(\sum_{r=0}^{20}{}^{50-r}C_6\) is equal to :

Let \((2x^2+3x+4)^{10}=\sum_{r=0}^{20}a_rx^r\). Then \(\frac{a_7}{a_{13}}\) is equal to ______.

Let P(3, 3) be a point on the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\). If the normal to it at P intersects the x-axis at (9, 0) and e is its eccentricity, then the ordered pair \((a^2, e^2)\) is equal to :

A triangle ABC lying in the first quadrant has two vertices as A(1, 2) and B(3, 1). If \(\angle BAC=90^\circ\), and \(\mathrm{ar}(\Delta ABC)=5\sqrt{5}\) sq. units, then the abscissa of the vertex C is :

Suppose a differentiable function \(f(x)\) satisfies the identity \(f(x+y)=f(x)+f(y)+xy^2+x^2y\), for all real \(x\) and \(y\). If \(\lim_{x\to 0}\frac{f(x)}{x}=1\), then \(f'(3)\) is equal to ______.

Let \([t]\) denote the greatest integer \(\le t\). Then the equation in \(x\), \([x]^2+2[x+2]-7=0\) has :

If \((a+\sqrt{2}\,b\cos x)(a-\sqrt{2}\,b\cos y)=a^2-b^2\), where \(a \gt b \gt 0\), then \(\frac{dx}{dy}\) at \(\left(\frac{\pi}{4},\frac{\pi}{4}\right)\) is :

Let \(u=\frac{2z+i}{z-ki},\ z=x+iy\) and \(k>0\). If the curve represented by \(\mathrm{Re}(u)+\mathrm{Im}(u)=1\) intersects the \(y\)-axis at the points P and Q where \(PQ=5\), then the value of \(k\) is :

If \(1+(1-2^2\cdot 1)+(1-4^2\cdot 3)+(1-6^2\cdot 5)+\ldots+(1-20^2\cdot 19)=\alpha-220\beta\), then an ordered pair \((\alpha,\beta)\) is equal to :

Let \(x_0\) be the point of local maxima of \(f(x)=\vec{a}\cdot(\vec{b}\times\vec{c})\), where \(\vec{a}=x\hat{i}-2\hat{j}+3\hat{k}\), \(\vec{b}=-2\hat{i}+x\hat{j}-\hat{k}\) and \(\vec{c}=7\hat{i}-2\hat{j}+x\hat{k}\). Then the value of \(\vec{a}\cdot\vec{b}+\vec{b}\cdot\vec{c}+\vec{c}\cdot\vec{a}\) at \(x=x_0\) is :

If \(A=\begin{bmatrix}\cos\theta & i\sin\theta\\ i\sin\theta & \cos\theta\end{bmatrix},\ \left(\theta=\frac{\pi}{24}\right)\) and \(A^5=\begin{bmatrix}a & b\\ c & d\end{bmatrix}\), where \(i=\sqrt{-1}\), then which one of the following is not true?

A survey shows that 63% of the people in a city read newspaper A whereas 76% read newspaper B. If \(x\)% of the people read both the newspapers, then a possible value of \(x\) can be :

Two vertical poles \(AB=15\) m and \(CD=10\) m are standing apart on a horizontal ground with points A and C on the ground. If P is the point of intersection of BC and AD, then the height of P (in m) above the line AC is :

Let \(y=y(x)\) be the solution of the differential equation, \(xy^{\prime}-y=x^2(x\cos x+\sin x),\ x>0\). If \(y(\pi)=\pi\), then \(y^{\prime\prime}\left(\frac{\pi}{2}\right)+y\left(\frac{\pi}{2}\right)\) is equal to :

Let \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\ (a>b)\) be a given ellipse, length of whose latus rectum is 10. If its eccentricity is the maximum value of the function, \(\phi(t)=\frac{5}{12}+t-t^2\), then \(a^2+b^2\) is equal to :

The integral \(\int\left(\frac{x}{x\sin x+\cos x}\right)^2dx\) is equal to (where C is a constant of integration) :

Let \(f(x)=\int\frac{\sqrt{x}}{(1+x)^2}dx\ (x\ge 0)\). Then \(f(3)-f(1)\) is equal to :

Let \(\alpha\) and \(\beta\) be the roots of \(x^2-3x+p=0\) and \(\gamma\) and \(\delta\) be the roots of \(x^2-6x+q=0\). If \(\alpha,\beta,\gamma,\delta\) form a geometric progression. Then ratio \((2q+p):(2q-p)\) is :

Let \(f(x)=|x-2|\) and \(g(x)=f(f(x)),\ x\in[0,4]\). Then \(\int_0^3 \left(g(x)-f(x)\right)dx\) is equal to :

The mean and variance of 8 observations are 10 and 13.5, respectively. If 6 of these observations are 5, 7, 10, 12, 14, 15, then the absolute difference of the remaining two observations is :

If the tangent to the curve, \(y = e^x\) at a point \((c, e^c)\) and the normal to the parabola, \(y^2 = 4x\) at the point (1, 2) intersect at the same point on the x-axis, then the value of c is ______ .