List of top Mathematics Questions asked in JEE Advanced

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 $f: R \rightarrow R$ 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 $O$ be the centre of the circle $x ^{2}+ y ^{2}= r ^{2}$, where \(r >\frac{\sqrt{5}}{2}\). Suppose $PQ$ is a chord of this circle and the equation of the line passing through $P$ and $Q$ is $2 x +4 y =5$. If the centre of the circumcircle of the triangle OPQ lies on the line $x+2 y=4$, then the value of $r$ is _____

Let $f : R \rightarrow R$ and $g : R \rightarrow R$ be functions satisfying
$f(x+y)=f(x)+f(y)+f(x) f(y)$ and $f(x)=x g(x)$
for all $x, y \in R $. If $\displaystyle\lim _{x \rightarrow 0} g(x)=1$, then which of the following statements is/are TRUE?

Two fair dice, each with faces numbered $1,2,3,4,5$ and $6$, are rolled together and the sum of the numbers on the faces is observed. This process is repeated till the sum is either a prime number or a perfect square. Suppose the sum turns out to be a perfect square before it turns out to be a prime number. If $p$ is the probability that this perfect square is an odd number, then the value of $14\, p$ is _____

Let the function $f :(0, \pi) \rightarrow R$ be defined by 
$f (\theta)=(\sin \theta+\cos \theta)^{2}+(\sin \theta-\cos \theta)^{4} \text { }$. 
Suppose the function f has a local minimum at $\theta$ precisely when $\theta \in\left\{\lambda_{1} \pi, \ldots, \lambda_{r} \pi\right\}$, where \(0 < \lambda_{1} < \ldots < \lambda_{r} < 1\) Then the value of $\lambda_{1}+\ldots+\lambda_{r}$ is ______

The probability that a missile hits a target successfully is $0.75$. In order to destroy the target completely, at least three successful hits are required. Then the minimum number of missiles that have to be fired so that the probability of completely destroying the target is NOT less than $0.95$, is______

Let e denote the base of the natural logarithm. The value of the real number a for which the right hand limit
$\displaystyle\lim_{x\rightarrow 0^+} \frac{(1-x)^{\frac{1}{x}} - e^{-1}}{x^a}$
is equal to a nonzero real number, is_____

Let f : R $/to$ R be given by f(x) = (x - 1)(x - 2)(x - 5). Define $F\left(x\right) = \int\limits^{x}_{0} f\left(t\right) dt, x > 0.$ Then which of the following options is/are correct?

Let $x \, \in \, R$ and let $P = \begin{bmatrix}1&1&1\\ 0&2&2\\ 0&0&3\end{bmatrix},\quad Q = \begin{bmatrix}2&x&x\\ 0&4&0\\ x&x&6\end{bmatrix}$ and $R = PQP^{-1}$. Then which of the following options is/are correct?

There are three bags $B_1, B_2$ and $B_3$. The bag $B_1$ contains 5 red and 5 green balls, $B_2$ contains 3 red and 5 green balls, and $B_3$ contains 5 red and 3 green balls. Bags $B_1, B_2$ and $B_3$ have probabilities $\frac{3}{10}, \frac{3}{10}$ and $\frac{4}{10}$ respectively of being chosen. A bag is selected at random and a ball is chosen at random from the bag. Then which of the following options is/are correct?

Let $L_1$ and $L_2$ denote the lines $\vec{r}=\hat{i}+\lambda\left(-\hat{i}+2\hat{j}+2\hat{k}\right), \lambda\,\in\,\mathbb{R}$ and $\vec{r}=\mu\left(2\hat{i}-\hat{j}+2\hat{k}\right), \mu\,\in\,\mathbb{R}$ respectively. If $L_3$ is a line which is perpendicular to both $L_1$ and $L_2$ and cuts both of them, then which of the following options describe(s) $L_3$?

Let the circles $C_{1} : x_{2} + y_{2} = 9$ and $C_{2} : \left(x ??3\right)^{2} + \left(y ??4\right)^{2} = 16$, intersect at the points X and Y. Suppose that another circle $C_{3} : \left(x ??h\right)^{2} + \left(y ??k\right)^{2} = r^{2}$ satisfies the following conditions: $\left(i\right) \quad$ centre of $C_{3}$ is collinear with the centres of $C_{1}$ and $C_{2}$, $\left(ii\right) \quad C_{1}$ and $C_{2}$ both lie inside $C_{3}$, and $\left(iii\right)\quad C_{3}$ touches $C_{1}$ at M and $C_{2}$ at N. Let the line through X and Y intersect $C_3$ at Z and W, and let a common tangent of $C_{1}$ and $C_{3}$ be a tangent to the parabola $x^{2} = 8ay.$ There are some expressions given in the List-I whose values are given in List-II below: Which of the following is the only INCORRECT combination?

Let the circles $C_{1} : x_{2} + y_{2} = 9$ and $C_{2} : \left(x ??3\right)^{2} + \left(y ??4\right)^{2} = 16$, intersect at the points X and Y. Suppose that another circle $C_{3} : \left(x ??h\right)^{2} + \left(y ??k\right)^{2} = r^{2}$ satisfies the following conditions: $\left(i\right) \quad$ centre of $C_{3}$ is collinear with the centres of $C_{1}$ and $C_{2}$, $\left(ii\right) \quad C_{1}$ and $C_{2}$ both lie inside $C_{3}$, and $\left(iii\right)\quad C_{3}$ touches $C_{1}$ at M and $C_{2}$ at N. Let the line through X and Y intersect $C_3$ at Z and W, and let a common tangent of $C_{1}$ and $C_{3}$ be a tangent to the parabola $x^{2} = 8ay.$ There are some expressions given in the List-I whose values are given in List-II below: Which of the following is the only CORRECT combination?

Let S be the set of all complex numbers z satisfying $\left|z-2+i\right|\ge\sqrt{5}.$ If the complex number $z_0$ is such that $\frac{ 1}{\left|z_{0}-1\right|}$ is the maximum of the set $\left\{\frac{ 1}{\left|z_{0}-1\right|}: z\,\in\,S\right\},$ then the principal argument of $\frac{4-z_{0}-z^{-}_{0}}{z_{0}-z^{-}_{0}+2i}$ is

A line $y = mx + 1$ intersects the circle $(x ? 3)^2 + (y + 2)^2 = 25$ at the points $P$ and $Q$ If the midpoint of the line segment $PQ$ has $x$-coordinate $-\frac{3}{5}$, then which one of the following options is correct?

Let $M=\begin{bmatrix}sin^{4}\theta &-1-sin^{2}\theta\\ 1+cos^{2}\theta&cos^{4}\theta\end{bmatrix}=\alpha I+\beta M^{-1}$ where $\alpha=\alpha\left(\theta\right)$ and $\beta=\beta\left(\theta\right)$ are real numbers, and $I$ is the 2 x 2 identity matrix. If $\alpha^*$ is the minimum of the set $\left\{\alpha\left(\theta\right):\theta\,\in[\,0,2\pi)\right\}$ and $\beta^*$ is the minimum of the set $\left\{\beta\left(\theta\right):\theta\,\in[\,0,2\pi)\right\}$, then the value of $\alpha^*+\beta^*$ is

Three lines $L_{1} : \vec{r} = \lambda\hat{i},\, \lambda\,\in\,R$ $L_{2} : \vec{r} = \hat{k} + \mu\hat{j},\,\mu\,\in\,R $ and $L_{3} : \vec{r} = \hat{i} + \hat{j} + v\hat{k}, \,v \,\in\,R $ are given. For which point(s) $Q$ on $L_{2}$ can we find a point P on $L_{1}$ and a point R on $L_{3}$ so that P, Q and R are collinear?

For non-negative integers n, let $f\left(n\right) = \frac{\displaystyle\sum^{n}_{k = 1}sin\left(\frac{k+1}{n+2}\pi\right) sin \left(\frac{k+2}{n+2}\pi\right)}{\displaystyle\sum^{n}_{k = 1} sin^{2}\left(\frac{k+1}{n+2}\pi\right)}$ Assuming $cos^{-1}x$ takes values in $\left[0, \pi\right]$, which of the following options is/are correct?

Let $S=\{1,2,3, \ldots \ldots, 9\}$. For $k=1,2, \ldots \ldots, 5$, let $N_{k}$ be the number of subsets of $S$, each containing five elements out of which exactly $k$ are odd. Then $N_{1}+N_{2}+N_{3}+N_{4}+N_{5}=$

If $f: R \rightarrow R$ is a differentiable function such that $f'(x)>2 f(x)$ for all $x \in R$, and $f(0)=1$, then

If '' > 0$ for all $

Three randomly chosen nonnegative integers $x, y$ and $z$ are found to satisfy the equation $x+y+z=10$. Then the probability that $z$ is even, is

How many $3 \times 3$ matrices $M$ with entries from $\{0,1,2\}$ are there, for which the sum of the diagonal entries of $M^{T} M$ is $5$ ?