List of top Mathematics Questions asked in JEE Advanced

Let $g _{ i }:\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow R , i =1,2$ and $f :\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow R$ be functions such that
$g_{1}(x)=1, g_{2}(x)=|4 x-\pi|$ and $f(x)=\sin ^{2} x$, for all $x \in\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] $ ,
Define $S_{i}=\int\limits_{\frac{\pi}{8}}^{\frac{3 \pi}{8}} f(x) \cdot g_{i}(x) d x, i=1,2$

Consider the lines L₁ and L₂ defined by
\(L_1: x\sqrt{2} + y - 1 = 0\)and \(L_2: x\sqrt{2} - y + 1 = 0\)
For a fixed constant λ, let C be the locus of a point P such that the product of the distance of P from L₁ and the distance of P from L₂ is λ². The line \(y = 2x + 1\) meets C at two points R and S, where the distance between R and S is \(\sqrt{270}.\)
Let the perpendicular bisector of RS meet C at two distinct points R’ and S’. Let D be the square of the distance between R’ and S’.

Let α, β and γ be real numbers such that the system of linear equations
\(x + 2y + 3z = α\)
\(4x + 5y + 6z = β\)
\(7x + 8y + 9z = γ – 1 \)
is consistent. Let\( |M|\) represent the determinant of the matrix.
\(M = \begin{bmatrix} \alpha & 2 & \gamma &\\ \beta & 1 & 0 & \\-1& 0&1 \end{bmatrix}\)
Let P be the plane containing all those \((α, β, γ)\) for which the above system of linear equations is consistent, and D be the square of the distance of the point (0, 1, 0) from the plane P.

Three numbers are chosen at random, one after another with replacement, from the set S = {1, 2, 3, …, 100}. Let p₁ be the probability that the maximum of chosen numbers is at least 81 and p₂ be the probability that the minimum of chosen numbers is at most 40.

Let $a$ and $b$ be positive real numbers. Suppose $\overrightarrow{P Q}=a \hat{i}+b \hat{j}$ and $\overrightarrow{P S}=a \hat{i}-b \hat{j}$ are adjacent sides of a parallelogram $PQRS$. Let $\vec{ u }$ and $\vec{ v }$ be the projection vectors of $\vec{ w }=\hat{ i }+\hat{ j }$ along $\overrightarrow{ PQ }$ and $\overrightarrow{ PS }$, respectively. If $|\vec{u}|+|\vec{v}|=|\vec{w}|$ and if the area of the parallelogram $PQRS$ is $8$, then which of the following statements is/are TRUE?

Let $f: R \rightarrow R$ be a differentiable function such that its derivative $f^{\prime}$ is continuous and $f(\pi)=-6$. If $F$ : $[0$, $\pi] \rightarrow R$ is defined by $F(x)=\int\limits_{0}^{x} f(t) d t$, and if
$\int\limits_{0}^{\pi}\left(f^{\prime}(x)+F(x)\right) \cos x d x=2$
then the value of $f (0)$ is _____

Let $M$ be a $3 \times 3$ invertible matrix with real entries and let $I$ denote the $3 \times 3$ identity matrix. If $M^{-1}=\text{adj}(\text{adj} M)$, then which of the following statements is/are ALWAYS TRUE?

Let \(a_{1}, a_{2}, a_{3}, \ldots\) be a sequence of positive integers in arithmetic progression with common difference \(2\) Also, let \(b_{1}, b_{2}, b_{3}, \ldots\) be a sequence of positive integers in geometric progression with common ratio \(2\) .If \(a_{1}=b_{1}=c\), then the number of all possible values of \(c\), for which the equality \(2\left(a_{1}+a_{2}+\ldots+a_{n}\right)=b_{1}+b_{2}+\ldots +b_{n}\) holds for some positive integer \(n\), is

If the function $f: R \rightarrow R$ is defined by $f(x)=|x|(x-\sin x)$, then which of the following statements is TRUE?

The trace of a square matrix is defined to be the sum of its diagonal entries. If $A$ is a $2 \times 2$ matrix such that the trace of $A$ is $3$ and the trace of $A^{3}$ is $-18$, then the value of the determinant of $A$ is ______

The value of the limit
$\displaystyle\lim _{x \rightarrow \frac{\pi}{2}} \frac{4 \sqrt{2}(\sin 3 x+\sin x)}{\left(2 \sin 2 x \sin \frac{3 x}{2}+\cos \frac{5 x}{2}\right)-\left(\sqrt{2}+\sqrt{2} \cos 2 x+\cos \frac{3 x}{2}\right)}$
is _______

In a hotel, four rooms are available Six persons are to be accommodated in these four rooms in such a way that each of these rooms contains at least one person and at most two persons. Then the number of all possible ways in which this can be done is ______

Let $a, b$ and $\lambda$ be positive real numbers. Suppose $P$ is an end point of the latus rectum of the parabola $y^{2}=4 \lambda x$, and suppose the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ passes through the point $P$. If the tangents to the parabola and the ellipse at the point $P$ are perpendicular to each other, then the eccentricity of the ellipse is

Let the function $f: R \rightarrow R$ be defined by $f(x)=x^{3}-x^{2}+(x-1) \sin x$ and let $g: R \rightarrow R$ be an arbitrary function. Let $f g: R \rightarrow R$ be the product function defined by $(f, g)(x)=f(x) g(x)$. Then which of the following statements is/are TRUE?

Suppose $a, b$ denote the distinct real roots of the quadratic polynomial $x^{2}+20 x-2020$ and suppose $c, d$ denote the distinct complex roots of the quadratic polynomial $x^{2}-20 x+2020$. Then the value of $a c(a-c)+a d(a-d)+b c(b-c)+b d(b-d)$ is

Let \(m\) be the minimum possible value of \(\log _{3}\left(3^{y_{1}}+3^{y_{2}}+3^{y_{3}}\right)\), where \(y_{1}, y_{2}, y_{3}\) are real numbers for which \(y_{1}+y_{2}+y_{3}=9\) .Let \(M\) be the maximum possible value of \(\left(\log _{3} x_{1}+\log _{3} x_{2}+\log _{3} x_{3}\right)\), where \(x_{1}, x_{2}, x_{3}\) are positive real numbers for which \(x_{1}+x_{2}+x_{3}=9\) .Then the value of \(\log _{2}\left(m^{3}\right)+\log _{3}\left(M^{2}\right)\) is_______

Let $S$ be the set of all complex numbers $z$ satisfying $\left|z^{2}+z+1\right|=1$. Then which of the following statements is/are TRUE?

For a complex number $z$, let $Re(z)$ denote the real part of $z$. Let $S$ be the set of all complex numbers $z$ satisfying $z^{4}-|z|^{4}=4 i z^{2}$, where $i=\sqrt{-1}$. Then the minimum possible value of $\left|z_{1}-z_{2}\right|^{2}$, where $z_{1}, z_{2} \in S$ with \(Re\left( z _{1}\right)\)\(>\)0 and \(Re\left( z _{2}\right)\) \(<\) 0, is ______

Let the function $f: R \rightarrow R$ and $g: R \rightarrow R$ be defined by
$f(x)=e^{x-1}-e^{-|x-1|}$ and $g(x)=\frac{1}{2}\left(e^{x-1}+e^{1-x}\right)$.
Then the area of the region in the first quadrant bounded by the curves $y=f(x), y=g(x)$ and $x=0$ is

In a triangle \(P Q R\), let \(\vec{a}=\overrightarrow{Q R}, \vec{b}=\overrightarrow{R P}\) and \(\vec{c}=\overrightarrow{P Q}\) If
 \(|\vec{a}|=3,|\vec{b}|=4\) and \(\frac{\vec{a} \cdot(\vec{c}-\vec{b})}{\vec{c} \cdot(\vec{a}-\vec{b})}=\frac{|\vec{a}|}{|\vec{a}|+|\vec{b}|}\) then the value of \(|\vec{a} \times \vec{b}|^{2}\) is

Let $x, y$ and $z$ be positive real numbers Suppose $x, y$ and $z$ are the lengths of the sides of a triangle opposite to its angles $X, Y$ and $Z$, respectively If $\tan \frac{x}{2}+\tan \frac{z}{2}=\frac{2 y}{x+y+z}$ then which of the following statements is/are TRUE?

Let \(f:[0,2] \rightarrow R\) be the function defined by
\(f(x)=(3-\sin (2 \pi x)) \sin \left(\pi x-\frac{\pi}{4}\right)-\sin \left(3 \pi x+\frac{\pi}{4}\right)\) If \(\alpha, \beta \in[0,2]\) are such that \(\{x \in[0,2]: f(x) \geq 0\}=[\alpha, \beta]\), then the value of \(\beta-\alpha\) is ______

Which of the following inequalities is/are TRUE?

Let $L_{1}$ and $L_{2}$ be the following straight lines
$L_{1}: \frac{x-1}{1}=\frac{y}{-1}=\frac{z-1}{3}$ and $L_{2}: \frac{x-1}{-3}-\frac{y}{-1}=\frac{z-1}{1} $
Suppose thestraight line
$L: \frac{x-\alpha}{1}=\frac{y-1}{m}=\frac{z-\gamma}{-2}$
lies in the plane containing $L_{1}$ and $L_{2}$, and passes through thepoint of intersection of $L_{1}$ and $L_{2}$. If the line $L$ bisects the acute angle between the lines $L_{1}$ and $L_{2}$, then which of the following statements is/are TRUE?

Let the function $f:[0,1] \rightarrow R$ be defined by
$f(x)=\frac{4^{x}}{4^{x}+2}$.
Then the value of
$f\left(\frac{1}{40}\right)+f\left(\frac{2}{40}\right)+f\left(\frac{3}{40}\right)+\ldots+f\left(\frac{39}{40}\right)-f\left(\frac{1}{2}\right)$
is ____