List of top Mathematics Questions asked in JEE Main

Let \(f(x)\) be a differentiable function at \(x = a\) with \(f'(a) = 2\) and \(f(a) = 4\).
Then \(\displaystyle\lim_{x \to a} \frac{x f(a) - a f(x)}{x - a}\) equals :

Let \(m,n\in\mathbb{N}\) and \(\gcd(2,n)=1\). If \(30\binom{30}{0}+29\binom{30}{1}+\cdots+2\binom{30}{28}+1\binom{30}{29}=n\cdot2^m\), then \(n+m\) is equal to __________.
(Here \(\binom{n}{k}={}^nC_k\))

The number of integral values of 'k' for which the equation \(3\sin x+4\cos x=k+1\) has a solution, \(k\in\mathbb{R}\) is ___________.

If \(\sqrt3\,(\cos^2x)=(\sqrt3-1)\cos x+1\), the number of solutions of the given equation when \(x\in\left[0,\frac{\pi}{2}\right]\) is __________.

If \(y=y(x)\) is the solution of the equation
\(e^{\sin y}\cos y\,\frac{dy}{dx}+e^{\sin y}\cos x=\cos x\), \(y(0)=0\); then
\(1+y\left(\frac{\pi}{6}\right)+\frac{\sqrt3}{2}\,y\left(\frac{\pi}{3}\right)+\frac{1}{\sqrt2}\,y\left(\frac{\pi}{4}\right)\) is equal to ____________.

The area bounded by the lines \(y=\big|\,|x-1|-2\,\big|\) is ___________.

The difference between degree and order of a differential equation that represents the family of curves given by \(y^2=a\left(x+\frac{\sqrt{a}}{2}\right)\), \(a\gt0\) is _________.

The sum of \(162^{th}\) power of the roots of the equation \(x^3-2x^2+2x-1=0\) is ________.

If \(\vec{a}\) and \(\vec{b}\) are perpendicular, then
\(\vec{a}\times\left(\vec{a}\times\left(\vec{a}\times\left(\vec{a}\times\vec{b}\right)\right)\right)\)
is equal to :

Consider the three planes
\(P_1: 3x+15y+21z=9\),
\(P_2: x-3y-z=5\), and
\(P_3: 2x+10y+14z=5\)
Then, which one of the following is true?

If \((1,5,35)\), \((7,5,5)\), \((1,\lambda,7)\) and \((2\lambda,1,2)\) are coplanar, then the sum of all possible values of \(\lambda\) is :

The number of solutions of the equation \(\log_4(x-1)=\log_2(x-3)\) is _________.

The value of the integral \(\int_0^{\pi} |\sin 2x|\,dx\) is ___________.

The sum of the infinite series
\(1+\frac{2}{3}+\frac{7}{3^2}+\frac{12}{3^3}+\frac{17}{3^4}+\frac{22}{3^5}+\ldots\ldots\)
is equal to :

The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteria count is 1000 at initial time t = 0. The number of bacteria is increased by 20% in 2 hours. If the population of bacteria is 2000 after
\(\displaystyle\frac{k}{\log_e\left(\frac{6}{5}\right)}\)
hours, then
\(\displaystyle\left(\frac{k}{\log_e2}\right)^2\)
is equal to :

If
\(\displaystyle\frac{\sin^{-1}x}{a}=\frac{\cos^{-1}x}{b}=\frac{\tan^{-1}y}{c}\)
;
\(0\lt x\lt1\)
, then the value of
\(\cos\left(\frac{\pi c}{a+b}\right)\)
is :

The maximum slope of the curve
\(y=\frac{1}{2}x^4-5x^3+18x^2-19x\)
occurs at the point :

Let f be any function defined on R and let it satisfy the condition :
\(|f(x)-f(y)|\le|(x-y)^2|,\ \forall(x,y)\in R\)
If f(0) = 1, then :

The value of
\(\begin{vmatrix}(a+1)(a+2) & a+2 & 1\\ (a+2)(a+3) & a+3 & 1\\ (a+3)(a+4) & a+4 & 1\end{vmatrix}\)
is :

The intersection of three lines \(x-y=0\), \(x+2y=3\) and \(2x+y=6\) is a :

In the circle given below, let OA = 1 unit, OB = 13 unit and PQ \(\perp\) OB. Then, the area of the triangle PQB (in square units) is :

The number of seven digit integers with sum of the digits equal to 10 and formed by using the digits 1, 2 and 3 only is :

The value of
\(\displaystyle\lim_{h\to 0}\, 2\left\{\frac{\sqrt{3}\sin\left(\frac{\pi}{6}+h\right)-\cos\left(\frac{\pi}{6}+h\right)}{\sqrt{3}\,h\left(\sqrt{3}\cos h-\sin h\right)}\right\}\)
is :

The value of
\(\displaystyle\int_{-\pi/2}^{\pi/2}\frac{\cos^2 x}{1+3^x}\,dx\)
is :

The value of
\(\displaystyle\sum_{n=1}^{100}\int_{n-1}^{n}e^{x-[x]}\,dx\)
, where [ x ] is the greatest integer \(\le\) x, is :