List of top Mathematics Questions asked in JEE Advanced

Consider a triangle \(Δ\) whose two sides lie on the x-axis and the line \(x + y + 1 = 0\). If the orthocenter of \(Δ\) is (1, 1), then the equation of the circle passing through the vertices of the triangle \(Δ\) is;

Let $O$ be the origin and $\overrightarrow{ OA }=2 \hat{ i }+2 \hat{ j }+\hat{ k }, \overrightarrow{ OB }=\hat{ i }-2 \hat{ j }+2 \hat{ k }$ and $\overrightarrow{ OC }=\frac{1}{2}(\overrightarrow{ OB }-\lambda \overrightarrow{ OA })$ for some $\lambda \(>\) 0$. If $|\overrightarrow{ OB } \times \overrightarrow{ OC }|=\frac{9}{2}$, then which of the following statements is(are) TRUE?

Let $E , F$ and $G$ be three events having probabilities
$P(E)=\frac{1}{8}, P(F)=\frac{1}{6}$ and $P(G)=\frac{1}{4}$, and let $P(E \cap F \cap G)=\frac{1}{10}$.
For any event $H$, if $H ^{ C }$ denotes its complement, then which of the following statements is(are) TRUE?

For any $3 \times 3$ matrix $M$, let $|M|$ denote the determinant of $M$. Let $I$ be the $3 \times 3$ identity matrix Let $E$ and $F$ be two $3 \times 3$ matrices such that $( I - EF )$ is invertible. If $G =( I - EF )^{-1}$, then which of the following statements is(are) TRUE?

Consider three sets E₁ = {1, 2, 3}, F₁ = {1, 3, 4} and G₁ = {2, 3, 4, 5}. Two elements are chosen at random, without replacement, from the set E₁, and let S₁ denote the set of these chosen elements. Let E₂ = E₁ – S₁ and F₂ = F₁ ⋃ S₁. Now two elements are chosen at random, without replacement, from the set F₂ and let S₂ denote the set of these chosen elements.
Let G₂ = G₁ ⋃ S₂. Finally, two elements are chosen at random, without replacement from the set G₂ and let S₃ denote the set of these chosen elements. Let E₃ = E₂ ⋃ S₃. Given that E₁ = E₃, let p be the conditional probability of the event S₁ = {1, 2}. Then the value of p is;

For any complex number $w=c+$ id, let $\arg (w) \in(-\pi, \pi]$, where $i=\sqrt{-1}$ Let $\alpha$ and $\beta$ be real numbers such that for all complex numbers $z=x+$ iy satisfying $\arg \left(\frac{z+\alpha}{z+\beta}\right)=\frac{\pi}{4}$, then ordered pair $(x, y)$ lies on the circle
$x^{2}+y^{2}+5 x-3 y+4=0$
Then which of the following statements is(are) TRUE?

For any real number $x$, let $[x]$ denote the largest integer less than or equal to $x$ If
$I=\int\limits_{0}^{10}\left[\sqrt{\frac{10 x}{x+1}}\right] d x,$
then the value of $9I$ is _____

Let $E$ denote the parabola $y ^{2}=8 x$ Let $P =(-2,4)$ and let $Q$ and $Q ^{\prime}$ be two distinct points on $E$ such that the lines $P Q$ and $P Q^{\prime}$ are tangents to $E$ Let $F$ be the focus of $E$. Then which of the following statements is(are) TRUE?

Let $E$ be the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$ For any three distinct points $P, Q$ and $Q'$ on $E$, let $M(P, Q)$ be the midpoint of the line segment joining $P$ and $Q$, and $M \left( P , Q '\right)$ be the mid-point of the line segment joining $P$ and $Q'$. Then the maximum possible value of the distance between $M ( P , Q )$ and $M \left( P , Q '\right)$, as $P , Q$ and $Q'$ vary on $E$, is _____

Consider a triangle $P Q R$ having sides of lengths $p, q$ and $r$ opposite to the angles $P, Q$ and $R$, respectively. Then which of the following statements is(are) TRUE?

A number is chosen at random from the set $\{1,2,3, \ldots , 2000\} $ Let $p$ be the probability that the chosen number is a multiple of $3$ or a multiple of $7$. Then the value of $500\, p$ is_____

For any real numbers $\alpha$ and $\beta$, let $y _{\alpha, \beta}( x ), x \in R$, be the solution of the differential equation
$\frac{d y}{d x}+\alpha y=x e^{\beta x}, y(1)=1$
Let $S=\left\{y_{\alpha, \beta}(x): \alpha, \beta \in R\right\}$. Then which of the following functions belong(s) the set $S$ ?

For $x \in R$, the number of real roots of the equation $3 x^{2}-4\left|x^{2}-1\right|+x-1=0$ is ______

Let $f:\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \rightarrow R$ be a continuous function such that
$f(0)=1$ and $\int\limits_{0}^{\frac{\pi}{3}} f(t) d t=0$
Then which of the following statements is(are) TRUE?

Let $\Psi_{1}:[0, \infty) \rightarrow R, \Psi_{2}:[0, \infty) \rightarrow R, f:[0, \infty) \rightarrow R$ and $g:[0, \infty) \rightarrow R$ be functions such that $f (0)= g (0)=0$,
$\psi_{1}(x)=e^{-x}+x, x \geq 0, $ 
$\psi_{2}(x)=x^{2}-2 x-2 e^{-x}+2, x \geq 0,$
\(f(x)=\int\limits_{-x}^{x}\left(|t|-t^{2}\right) e^{-t^{2}} d t\),\(x > 0\) 
and \(g(x)=\int\limits_{0}^{x^{2}} \sqrt{t} e^{-t} d t, \)\(x > 0\) .

Let 
$M=\left\{(x, y) \in R \times R: x^{2}+y^{2} \leq r^{2}\right\}$, 
where \(r\)\(>\)\(0\). Consider the geometric progression $a_{n}=\frac{1}{2^{n-1}}, n=1,2,3, \ldots $ Let $S_{0}=0$ and for $n \geq 1$, let $S_{n}$ denote the sum of the first $n$ terms of this progression. For $n \geq 1$, let $C_{n}$ denote the circle with center $\left(S_{n-1}, 0\right)$ and radius $a_{n}$, and $D_{n}$ denote the circle with center $\left(S_{n-1}, S_{n-1}\right)$ and radius $a_{n}$.

Let $f_{1}:(0, \infty) \rightarrow R$ and $f_{2}:(0, \infty) \rightarrow R$ be defined by
\(f_{1}(x)=\int\limits_{0}^{x} \displaystyle\prod_{j=1}^{21}(t-j)^{j} d t,\) x\(>\)0 
and $f _{2}( x )=98( x -1)^{50}-600( x -1)^{49}+2450, x >0$, 
where, for any positive integer $n$ and real numbers $a _{1}, a _{2}$, $\ldots , a_{n}, \displaystyle\prod_{i=1}^{n} a_{i}$ denotes the product of $a_{1}, a_{2}, \ldots , a_{n} $. Let $m_{i}$ and $n_{i}$, respectively, denote the number of points of local minima and the number of points of local maxima of function $f _{ i }, i =1,2$, in the interval $(0$, $\infty$ )

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 region R = {(x,y)∈ R×R : x ≥ 0 and y² ≤ 4 – x. Let F be the family of all circles that are contained in R and have centres on the x-axis. Let C be the circle that has the largest radius among the circles in F. Let (α, β) be a point where circle C meets the curve y² = 4 – x.

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.

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