List of top Mathematics Questions asked in JEE Main

For \(p \gt 0\), a vector
\(\vec{v}_2 = 2\hat{i} + (p+1)\hat{j}\)
is obtained by rotating the vector
\(\vec{v}_1 = \sqrt{3}\,p\hat{i} + \hat{j}\)
by an angle \(\theta\) about origin in counter clockwise direction. If
\(\tan\theta = \frac{(\alpha\sqrt{3} - 2)}{(4\sqrt{3} + 3)}\)
, then the value of \(\alpha\) is equal to ______________.

For \(k \in N\), let
\(\frac{1}{\alpha(\alpha+1)(\alpha+2)\ldots\ldots\ldots(\alpha+20)} = \sum\limits_{K=0}^{20} \frac{A_k}{\alpha + k}\)
, where
\(\alpha \gt 0\)
. Then the value of
\(100\left(\frac{A_{14} + A_{15}}{A_{13}}\right)^2\)
is equal to ______________.

Consider a triangle having vertices A\((-2, 3)\), B\((1, 9)\) and C\((3, 8)\). If a line L passing through the circum-centre of triangle ABC, bisects line BC, and intersects y-axis at point
\(\left(0, \frac{\alpha}{2}\right)\)
, then the value of real number \(\alpha\) is ______________.

Let a function \(g : [0, 4] \to R\) be defined as
\(g(x) = \begin{cases} \max\limits_{0 \le t \le x} \{t^3 - 6t^2 + 9t - 3),\quad 0 \le x \le 3 \\ 4 - x,\quad 3 \lt x \le 4 \end{cases}\)
, then the number of points in the interval (0, 4) where g(x) is NOT differentiable, is ______________.

If
\(\lim_{x \to 0} \frac{\alpha x e^{x} - \beta \log_e(1+x) + \gamma x^2 e^{-x}}{x \sin^2 x} = 10\), \(\alpha, \beta, \gamma \in R\)
, then the value of \(\alpha + \beta + \gamma\) is ______________.

The number of solutions of the equation
\(\log_{(x+1)}(2x^2 + 7x + 5) + \log_{(2x+5)}(x+1)^2 - 4 = 0\)
, \(x \gt 0\), is :

Let
\(A = \{a_{ij}\}\)
be a \(3 \times 3\) matrix,
where
\(a_{ij} = \begin{cases} (-1)^{j-i}\ \text{if } i \lt j, \\ 2\ \text{if } i = j, \\ (-1)^{i+j}\ \text{if } i \gt j \end{cases}\)
then \(\det(3\,\mathrm{Adj}(2A^{-1}))\) is equal to ______________.

Let \(r_1\) and \(r_2\) be the radii of the largest and smallest circles, respectively, which pass through the point \((-4,1)\) and having their centres on the circumference of the circle \(x^2+y^2+2x+4y-4=0\). If \(\frac{r_1}{r_2}=a+b\sqrt2\), then \(a+b\) is equal to :

If sum of the first 21 terms of the series \(\log_{9^{1/2}} x + \log_{9^{1/3}} x + \log_{9^{1/4}} x + \ldots\ldots\), where \(x \gt 0\) is 504, then x is equal to

Let in a right angled triangle, the smallest angle be \(\theta\). If a triangle formed by taking the reciprocal of its sides is also a right angled triangle, then \(\sin\theta\) is equal to :

Consider the line L given by the equation
\(\frac{x-3}{2} = \frac{y-1}{1} = \frac{z-2}{1}\).
Let Q be the mirror image of the point \((2, 3, -1)\) with respect to L. Let a plane P be such that it passes through Q, and the line L is perpendicular to P. Then which of the following points is on the plane P?

Consider the following three statements :

(A) If 3 + 3 = 7 then 4 + 3 = 8

(B) If 5 + 3 = 8 then earth is flat.

(C) If both (A) and (B) are true then 5 + 6 = 17.

Then, which of the following statements is correct?

For the natural numbers m, n, if \((1-y)^m(1+y)^n=1+a_1y+a_2y^2+\ldots.+a_{m+n}y^{m+n}\) and \(a_1=a_2=10\), then the value of \((m+n)\) is equal to :

In a triangle ABC, if \(\left|\overrightarrow{BC}\right|=3\), \(\left|\overrightarrow{CA}\right|=5\) and \(\left|\overrightarrow{BA}\right|=7\), then the projection of the vector \(\overrightarrow{BA}\) on \(\overrightarrow{BC}\) is equal to :

Let \(y=y(x)\) satisfies the equation
\(\frac{dy}{dx}-|A|=0\)
for all \(x\gt0\), where
\(A=\begin{bmatrix}y&\sin x&1\\0&-1&1\\2&0&\frac1x\end{bmatrix}\).
If \(y(\pi)=\pi+2\), then the value of \(y\left(\frac\pi2\right)\) is :

Let P be a variable point on the parabola \(y=4x^2+1\). Then, the locus of the mid-point of the point P and the foot of the perpendicular drawn from the point P to the line \(y=x\) is :

Let \(g(t)=\int_{-\pi/2}^{\pi/2}\cos\left(\frac\pi4t+f(x)\right)dx\), where \(f(x)=\log_e\left(x+\sqrt{x^2+1}\right),\ x\in R\). Then which one of the following is correct?

The value of \(k\in R\), for which the following system of linear equations
\(3x-y+4z=3,\)
\(x+2y-3z=-2,\)
\(6x+5y+kz=-3,\)
has infinitely many solutions, is :

Let A, B and C be three events such that the probability that exactly one of A and B occurs is \((1-k)\), the probability that exactly one of B and C occurs is \((1-2k)\), the probability that exactly one of C and A occurs is \((1-k)\) and the probability of all A, B and C occur simultaneously is \(k^2\), where \(0\lt k\lt1\). Then the probability that at least one of A, B and C occur is :

Let \(f:R-\left\{\frac{\alpha}{6}\right\}\to R\) be defined by \(f(x)=\frac{5x+3}{6x-\alpha}\). Then the value of \(\alpha\) for which \((f\circ f)(x)=x\), for all \(x\in R-\left\{\frac{\alpha}{6}\right\}\), is :

If the real part of the complex number \((1-\cos\theta+2i\sin\theta)^{-1}\) is \(\frac15\) for \(\theta\in(0,\pi)\), then the value of the integral \(\int_0^{\theta}\sin x\,dx\) is equal to:

If \(f:R\to R\) is given by \(f(x)=x+1\), then the value of
\(\lim_{n\to\infty}\frac{1}{n}\left[f(0)+f\left(\frac{5}{n}\right)+f\left(\frac{10}{n}\right)+\ldots\ldots+f\left(\frac{5(n-1)}{n}\right)\right]\)
is :

The sum of all the local minimum values of the twice differentiable function \(f:R\to R\) defined by
\(f(x)=x^3-3x^2-\frac{3f^{\prime\prime}(2)}{2}x+f^{\prime\prime}(1)\)
is :

The lines \(x=ay-1=z-2\) and \(x=3y-2=bz-2\), \((ab\ne0)\) are coplanar, if :

If \([x]\) denotes the greatest integer less than or equal to \(x\), then the value of the integral \(\int_{-\pi/2}^{\pi/2}[[x]-\sin x]\,dx\) is equal to :