List of top Mathematics Questions asked in JEE Advanced

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

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

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?

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 ______

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

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 ____

Let $b$ be a nonzero real number. Suppose $f : R \rightarrow R$ is a differentiable function such that $f(0)=1$. If the derivative $f^{\prime}$ of $f$ satisfies the equation
$f^{\prime}(x)=\frac{f(x)}{b^{2}+x^{2}}$
for all $x \in R$, then which of the following statements is/are TRUE?

Let the functions $f :(-1,1) \rightarrow R$ and $g :(-1,1) \rightarrow(-1,1)$ be defined by
$f(x)=|2 x-1|+|2 x+1|$ and $g(x)=x-[x] \text {, }$
where $[ x ]$ denotes the greatest integer less than or equal to $x$. Let fog : $(-1,1) \rightarrow R$ be the composite function defined by $(f o g)(x)=f(g(x))$. Suppose $c$ is the number of points in the interval $(-1,1)$ at which fog is NOT continuous, and suppose $d$ is the number of points in the interval $(-1,1)$ at which fog is NOT differentiable. Then the value of $c + d$ is ____

For a polynomial $g(x)$ with real coefficients, let $m_{g}$ denote the number of distinct real roots of $g(x)$. Suppose $S$ in the set of polynomials with real coefficients defined by
$S =\left\{\left( x ^{2}-1\right)^{2}\left( a _{0}+ a _{1} x + a _{2} x ^{2}+ a _{3} x ^{3}\right): a _{0}, a _{1}, a _{2}, a _{3} \in R \right\} $
For a polynomial $f$, let $f'$ and $f"$ denote its first and second order derivatives, respectively. Then the minimum possible value of $\left(m_{f'}+m_{f''}\right)$, where $f \in S$, is

Let $C_{1}$ and $C_{2}$ be two biased coins such that the probabilities of getting head in a single toss are $\frac{2}{3}$ and $\frac{1}{3}$, respectively. Suppose $\alpha$ is the number of heads that appear when $C_{1}$ is tossed twice, independently, and suppose $\beta$ is the number of heads that appear when $C_{2}$ is tossed twice, independently. Then the probability that the roots of the quadratic polynomial $x^{2}-\alpha x+\beta$ are real and equal, is

Consider all rectangles lying in the region
 \(\{(x, y) \in R \times R: 0 \leq x \leq \frac{\pi}{2}\ and\ 0 \leq y \leq 2 \sin (2 x)\}\) 
and having one side on the $x$-axis. The area of the rectangle which has the maximum perimeter among all such rectangles, is

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 ______

For nonnegative integers $s$ and $r$, 
let\(\begin{pmatrix}s \\ r\end{pmatrix}=\begin{cases}\frac{s !}{r !(s-r) !} & \text { if } r \leq s \\ 0 & \text { if } r>s\end{cases}\).
For positive integers $m$ and $n$,
let $g(m, n)-\displaystyle\sum_{p=0}^{m+n} \frac{f(m, n, p)}{\begin{pmatrix}n+p \\ p\end{pmatrix}}$ 
where for any nonnegative integer $p$, 
$f(m, n, p)=\displaystyle\sum_{i=0}^{p}\begin{pmatrix}m \\ i\end{pmatrix}\begin{pmatrix}n+i \\ p\end{pmatrix}\begin{pmatrix}p+n \\ p-i\end{pmatrix}$.
Then which of the following statements is/are TRUE?

Let $\alpha, \beta, \gamma, \delta$ be real numbers such that $\alpha^{2}+\beta^{2}+\gamma^{2} \neq 0$ and $\alpha+\gamma=1$. Suppose the point $(3,2,-1)$ is the mirror image of the point $(1,0,-1)$ with respect to the plane $\alpha x+\beta y+\gamma z=\delta$. Then which of the following statements is/are TRUE?

Let $a$ and $b$ be positive real numbers such that \(a >1\) and \(b < a\). Let $P$ be a point in the first quadrant that lies on the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$. Suppose the tangent to the hyperbola at P passes through the point $(1,0)$, and suppose the normal to the hyperbola at $P$ cuts off equal intercepts on the coordinate axes. Let $\Delta$ denote the area of the triangle formed by the tangent at $P$, the normal at $P$ and the $x$-axis. If e denotes the eccentricity of the hyperbola, then which of the following statements is/are TRUE?

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?