List of top Mathematics Questions asked in JEE Advanced

For \(x ∈ R\), let \(tan^{-1} (x) ∈ \) (\(-\frac{\pi}{2},\frac{\pi}{2}\)). Then the minimum value of function \(f: R \rightarrow R\) defined by \(f(x) =\) \(\int_{0}^{x\,tan^{-1}x}\frac{e^{(t-cos\,t)}}{1+t^{2023}}dt\)  is
Let M=(aij), i,j∈{1,2,3}, be the 3×3 matrix such that aij=1 if j+1 is divisible by i,otherwise aij=0. Then which of the following statements is(are) true? 
Consider an obtuse-angled triangle ABC in which the difference between the largest and the smallest angle is \(\frac{\pi}{2}\) and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius 1.Let a be the area of the triangle ABC. Then the value of (64a)2 is
Let S be the set of all twice differentiable functions f from R to R such that \(\frac{d^2f}{dx^2}(x)>0\) for all x∈(-1,1). For f∈S, let Xf be the number of points x∈(-1,1) for which f(x)=x.Then which of the following statements is(are) true? 
Consider an experiment of tossing a coin repeatedly until the outcomes of two consecutive tosses are the same. If the probability of a random toss resulting in a head is \(\frac{1}{3}\), then the probability that the experiment stops with the head is

Let the position vectors of the points P, Q, R and S be
\(\vec{a}=\hat{i}+2\hat{j}-5\hat{k}\), \(\vec{b}=3\hat{i}+6\hat{j}+3\hat{k}\), \(\vec{c}=\frac{17}{5}\hat{i}+\frac{16}{5}\hat{j}+7\hat{k}\) and \(\vec{d}=2\hat{i}+\hat{j}+\hat{k}\)
respectively. Then which of the following statements is true?

Let α,β, and γ be real numbers. Consider the following system of linear equations:
x + 2y + z = 7
x + αz = 11
2x - 3y + βz = γ
Match each entry in List I to the correct entries in List II

List IList II
(P)If β=\(\frac{1}{2}\)(7α - 3) and \(\gamma\)=28, then the system has(1)a unique solution
(Q)If β=\(\frac{1}{2}\)(7α - 3) and \(\gamma\)\(\neq\)28, then the system has(2)no solution
(R)If β\(\neq\)\(\frac{1}{2}\)(7α - 3)  where \(\alpha\)=1 and \(\gamma\)\(\neq\)28, then the system has(3)infinitely many solutions
(S)If β\(\neq\)\(\frac{1}{2}\)(7α - 3)  where \(\alpha\)=1 and \(\gamma\)=28, then the system has(4)x = 11, y = - 2 and z = 0 as a solution
  (5)x = -15 , y = 4 and z = 0 as a solution
Let a and b be two nonzero real numbers. If the coefficient of x5 in the expansion of (\(ax^2+(\frac{70}{27bx})^4\) is equal to the coefficient of x-5 in the expansion of \((ax-\frac{1}{bx^2})^7\). then the value of 2b is
Let A1, A2, A3, A4,........, A8 be the vertices of the regular octagons that lie on the circle of radius 2. Let p be a point on the circle and let PAi denote the distance between the point P and Ai for i = 1,2,3,....,8. If P varies over the circle, then the maximum value of the product is PA1.Pa2..........PA8, is 
Let \(X\) be the set of all five-digit numbers formed using \(1,2,2,2,2,4,4,0\). for example \(22240\) is in \(X\) while \(02244\) and \(44422\) are not in \(X\). Suppose that each element of \(X\) has an equal chance of being chosen. Let \(p\) be the conditional probability that a component chosen at random is a multiple of \(20 \) given that it is a multiple of \(5\). Then the numerous of \(38p\) is equal to

Consider the 6 x 6 square in the figure. Let A1, A2, ........, A49 be the points of intersections (dots in the picture) in some order. We say that Ai and Aj are friends if they are adjacent along a row or a column. Assume that each point Ai has an equal chance of being chosen. Two distinct points are chosen randomly out of the points A1, A2, ........, A49. Let p be the probability that they are friends. Then the value of 7p is

Let n ≥ 2 be a natural number and f:[0,1)]\(\rightarrow\) R be the function defined by
\(f(x) \begin{cases} n(1-2nx)\ \ \ \ \ \ \ \ \ \text{if}\ 0\le x\le\frac{1}{2n} \\ 2n((2nx-1)\ \ \ \  \ \text{if}\ \frac{1}{2n}\le x\le \frac{3}{4n}\\4n(1-nx)\ \ \ \ \ \ \ \ \ \text{if}\ \frac{3}{4n}\le x\le\frac{1}{n} \\ \frac{n}{n-1}(nx-1)\ \ \ \ \ \ \ \text{if}\ \frac{1}{n}\le x\le1 \end{cases}\)
If n is such that the area of the region bounded by the curves x = 0, x = 1, y = 0 and y = f(x) is 4, then the maximum value of the function f is

If $y(x)$ is the solution of the differential equation $x d y-\left(y^2-4 y\right) d x=0 \text { for } x>0, y(1)=2,$ and the slope of the curve $y=y(x)$ is never zero, then the value of $10 y(\sqrt{2})$ is ____.

Let A1, A2, A3, A4,........, A8 be the vertices of the regular octagons that lie on the circle of radius 2. Let p be a point on the circle and let PAi denote the distance between the point P and Ai for i = 1,2,3,....,8. If P varies over the circle, then the maximum value of the product is PA1.PA2..........PA8, is 

Let l1 and l2 be the lines r1 = λ(\(\hat{i}+\hat{j}+\hat{k}\)) and r2 = (\(\hat{j}-\hat{k}\)) + μ (\(\hat{i}+\hat{k}\)), respectively. Let X be the set of all the planes H containing line l1. For a plane H, let d (H) denote the smallest possible distance between the points of l2 and H. Let H0 be a plane in X for which d (H0) is the maximum value of d (H ) as H varies over all planes in X . Match each entry in List-I to the correct entries in List-II.
 List-IList-II
(P)The value of d (H0) is(1)\(\sqrt3\)
(Q)The distance of the point (0,1,2) from H0 is(2)\(\frac{1}{\sqrt3}\)
(R)The distance of origin from H0 is(3)0
(S)The distance of origin from the point of intersection of planes y = z, x = 1, and H0 is(4)\(\sqrt2\)
  (5)\(\frac{1}{\sqrt2}\)
 The correct option is: 
Consider the 6 x 6 square in the figure. Let A1, A2, ........, A49 be the points of intersections (dots in the picture) in some order. We say that Ai and Aj are friends if they are adjacent along a row or a column. Assume that each point Ai has an equal chance of being chosen.  Let i p be the probability that a randomly chosen point has i many friends, i = 0,1,2,3,4. Let X be a random variable such that for i = 0,1,2,3,4, the probability P(X = i) =pi. Then the value of 7E(X) is
Consider the 6 x 6 square

Consider the 6 x 6 square in the figure. Let A1, A2, ........, A49 be the points of intersections (dots in the picture) in some order. We say that Ai and Aj are friends if they are adjacent along a row or a column. Assume that each point Ai has an equal chance of being chosen. Two distinct points are chosen randomly out of the points A1, A2, ........, A49. Let p be the probability that they are friends. Then the value of 7p is

Consider the 6 x 6 square in the figure. Let A1, A2, ........, A49 be the points of intersections (dots in the picture) in some order. We say that Ai and Aj are friends if they are adjacent along a row or a column. Assume that each point Ai has an equal chance of being chosen.  Let i p be the probability that a randomly chosen point has i many friends, i = 0,1,2,3,4. Let X be a random variable such that for i = 0,1,2,3,4, the probability P(X = i) =pi. Then the value of 7E(X) is

Consider the 6 x 6 square in the figure. Let A1, A2, ........, A49 be the points of intersections (dots in the picture) in some order. We say that Ai and Aj are friends if they are adjacent along a row or a column. Assume that each point Ai has an equal chance of being chosen.
Consider the 6 x 6 square
Consider an obtuse-angled triangle ABC in which the difference between the largest and the smallest angle is \(\frac{\pi}{2}\) and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius 1.
Let f:(0,1)→R be the function defined as f(x)=[4x](x-\(\frac{1}{4}\))2(x-\(\frac{1}{2}\)), where [x] denotes the greatest integer less than or equal to x . Then which of the following statements is(are) true?
Let C1 be the circle of radius 1 with the center at the origin. Let C2 be the circle of radius r with center at the point A = (4,1), where 1 < r < 3 . Two distinct common tangents PQ and ST of C1 and C2 are drawn. The tangent PQ touches C1 at P and C2 at Q. The tangent ST touches C1 at S and C2 at T. Midpoints of the line segments PQ and ST are joined to form a line that meets the x-axis at a point B. If AB = √5, then the value of r2 is : 
Consider an obtuse-angled triangle ABC in which the difference between the largest and the smallest angle is \(\frac{\pi}{2}\) and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius 1.Then the inradius of the triangle ABC is
Let R=\(\begin{pmatrix}a&3&b\\c&2&d\\0&5&0\end{pmatrix}\): a,b,c,d ∈ {0,3,5,7,11,13,17,19}. Then the number of invertible matrices in R is
Let P be the plane √3x+2y+3z =16 and let and let S = {\(\alpha \hat{i}+\beta \hat{j}+\gamma\hat{k}:\alpha^2+\beta^2+\gamma^2=1\) and the distance of (α, β, γ) from the plane P is \(\frac{7}{2}\) }. Let u, v, and w be three distinct vectors in s such that |\(\vec{u}-\vec{v}\)| = |\(\vec{v}-\vec{w}\)| = |\(\vec{w}-\vec{u}\)|. Let V be the volume of the parallelepiped determined by vectors \(\vec{u},\vec{v},\vec{w}\). Then the value of \(\frac{80}{\sqrt3}\)V is