List of top Questions asked in JEE Main- 2021

Let [t] denote the greatest integer \(\le t\). The number of points where the function \(f(x)=[x]\left|x^2-1\right|+\sin\left(\frac{\pi}{[x]+3}\right)-[x+1],\ x\in(-2,2)\) is not continuous is _____________.

A cube is placed inside an electric field, \(\vec{E}=150y^2\hat{j}\). The side of the cube is 0.5 m and is placed in the field as shown in the given figure. The charge inside the cube is :

All the arrangements, with or without meaning, of the word FARMER are written excluding any word that has two R appearing together. The arrangements are listed serially in the alphabetic order as in the English dictionary. Then the serial number of the word FARMER in this list is ___________.

If the sum of the coefficients in the expansion of \((x+y)^n\) is 4096, then the greatest coefficient in the expansion is _____________.

Let \(\theta\) be the acute angle between the tangents to the ellipse \(\frac{x^2}{9}+\frac{y^2}{1}=1\) and the circle \(x^2+y^2=3\) at their point of intersection in the first quadrant. Then \(\tan\theta\) is equal to :

Let f(x) be a polynomial of degree 3 such that \(f(k)=-\frac{2}{k}\) for k = 2, 3, 4, 5. Then the value of \(52-10f(10)\) is equal to :

Let the points of intersections of the lines \(x-y+1=0\), \(x-2y+3=0\) and \(2x-5y+11=0\) are the mid points of the sides of a triangle \(\Delta ABC\). Then, the area of the \(\Delta ABC\) is ______________.

If for the complex numbers z satisfying \(|z-2-2i|\le1\), the maximum value of \(|3iz+6|\) is attained at \(a+ib\), then \(a+b\) is equal to ______________.

Let \(f(x)=x^6+2x^4+x^3+2x+3,\ x\in R\). Then the natural number n for which \(\lim_{x\to1}\frac{x^nf(1)-f(x)}{x-1}=44\) is __________.

Let X be a random variable with distribution.

x	\(-2\)	\(-1\)	3	4	6
P(X = x)	\(\frac15\)	a	\(\frac13\)	\(\frac15\)	b

If the mean of X is 2.3 and variance of X is \(\sigma^2\), then \(100\sigma^2\) is equal to :

Which of the following is equivalent to the Boolean expression \(p\wedge\sim q\) ?

Let the acute angle bisector of the two planes \(x-2y-2z+1=0\) and \(2x-3y-6z+1=0\) be the plane \(P\). Then which of the following points lies on \(P\)?

If n is the number of solutions of the equation
\(2\cos x\left(4\sin\left(\frac{\pi}{4}+x\right)\sin\left(\frac{\pi}{4}-x\right)-1\right)=1,\; x\in[0,\pi]\)
and S is the sum of all these solutions, then the ordered pair (n, S) is :

Let \(S_n=1\cdot(n-1)+2\cdot(n-2)+3\cdot(n-3)+\ldots+(n-1)\cdot 1,\ n\ge 4\).
The sum \(\sum_{n=4}^{\infty}\left(\frac{2S_n}{n!}-\frac{1}{(n-2)!}\right)\) is equal to :

Consider the parabola with vertex \(\left(\frac{1}{2},\frac{3}{4}\right)\) and the directrix \(y=\frac{1}{2}\). Let P be the point where the parabola meets the line \(x=-\frac{1}{2}\). If the normal to the parabola at P intersects the parabola again at the point Q, then \((PQ)^2\) is equal to :

Let \(a_1,a_2,\ldots\ldots\ldots,a_{21}\) be an AP such that \(\sum_{n=1}^{20}\frac{1}{a_na_{n+1}}=\frac49\). If the sum of this AP is 189, then \(a_6a_{16}\) is equal to :

The function \(f(x)\), that satisfies the condition \(f(x)=x+\int_0^{\pi/2}\sin x\cdot\cos y\,f(y)\,dy\), is :

The distance of line \(3y-2z-1=0=3x-z+4\) from the point \((2,-1,6)\) is :

Let \(P_1,P_2,\ldots\ldots,P_{15}\) be 15 points on a circle. The number of distinct triangles formed by points \(P_i,P_j,P_k\) such that \(i+j+k\ne15\), is :

The numbers of pairs \((a,b)\) of real numbers, such that whenever \(\alpha\) is a root of the equation \(x^2+ax+b=0\), \(\alpha^2-2\) is also a root of this equation, is :

The range of the function,
\(f(x)=\log_{\sqrt5}\left(3+\cos\left(\frac{3\pi}{4}+x\right)+\cos\left(\frac{\pi}{4}+x\right)+\cos\left(\frac{\pi}{4}-x\right)-\cos\left(\frac{3\pi}{4}-x\right)\right)\)
is :

The function \(f(x)=x^3-6x^2+ax+b\) is such that \(f(2)=f(4)=0\). Consider two statements :
Statement 1 : there exists \(x_1, x_2\in(2,4)\), \(x_1 \lt x_2\), such that \(f'(x_1)=-1\) and \(f'(x_2)=0\).
Statement 2 : there exists \(x_3, x_4\in(2,4)\), \(x_3 \lt x_4\), such that \(f\) is decreasing in \((2,x_4)\), increasing in \((x_4,4)\) and \(2f'(x_3)=\sqrt{3}\,f(x_4)\).
Then

Consider the system of linear equations
\(-x + y + 2z = 0\)
\(3x - ay + 5z = 1\)
\(2x - 2y - az = 7\)
Let \(S_1\) be the set of all \(a \in \mathbb{R}\) for which the system is inconsistent and \(S_2\) be the set of all \(a \in \mathbb{R}\) for which the system has infinitely many solutions. If \(n(S_1)\) and \(n(S_2)\) denote the number of elements in \(S_1\) and \(S_2\) respectively, then

The area, enclosed by the curves \(y=\sin x+\cos x\) and \(y=|\cos x-\sin x|\) and the lines \(x=0,\ x=\frac{\pi}{2}\), is :

Let \(J_{n,m}=\int_0^{\frac12}\frac{x^n}{x^m-1}\,dx\), \(\forall\ n \gt m\) and \(n,m\in\mathbb{N}\). Consider a matrix \(A=[a_{ij}]_{3\times3}\) where \(a_{ij}=\begin{cases}J_{6+i,3}-J_{i+3,3}, & i\le j\\ 0, & i \gt j\end{cases}\). Then \(\left|\operatorname{adj}A^{-1}\right|\) is :