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Mathematics

Let \( \vec{a}, \vec{b} \) be two vectors, and let \( P, Q \) and \( R \) be the points with position vectors \( \vec{a}, \vec{b} \) and \( \vec{a} + \vec{b} \), respectively, with respect to the origin \( O \). If \( |\vec{a} + \vec{b}| = \sqrt{21} \), \( |\vec{a} - \vec{b}| = 3 \), and \( \vec{a} \) and \( (\vec{a} - \vec{b}) \) are perpendicular to each other, then the area of the triangle \( OPR \) is:

Let \( T \) be the tangent to the parabola \( y^{2}=16x \) at the point \( (64,\,32) \). Let \( L \) be the tangent to the same parabola at another point \( (x_{1},y_{1}) \) on the parabola. If \( L \) and \( T \) are perpendicular to each other, then the distance between the point \( (x_{1},y_{1}) \) and the focus of the parabola, is:

Let \( y:(-\infty,\infty)\to(0,\infty) \) be the solution of the differential equation \( \frac{dy}{dx}=\frac{e^{5x}y^{3}+y^{3}}{e^{x}+e^{x}y^{4}} \), satisfying \( y(0)=\frac{1}{\sqrt{2}} \). Then the value of \( y(\log_{e}2) \) is:

The value of the definite integral \( \int_{0}^{2}\dfrac{1}{3^{x}+3}\,dx \) is:

Let \( \mathbb{R} \) denote the set of all real numbers. Consider the polynomial function \( f:\mathbb{R}\to\mathbb{R} \) defined by
\( f(x)=\dfrac{d^{10}}{dx^{10}}\!\left((x^{2}-1)^{10}\right) \), for all \( x\in\mathbb{R} \).
Here \( \dfrac{d^{10}}{dx^{10}}\!\left((x^{2}-1)^{10}\right) \) is the \( 10^{\text{th}} \) order derivative of the function \( (x^{2}-1)^{10} \).
Then which of the following statements is (are) TRUE?

Let \( a, b, c \) be positive integers in arithmetic progression such that the equation \( ax^2 + bx + c = 0 \) has only integer solutions.
Then which of the following statements is (are) TRUE?

Let \( L \) be the straight line joining the points \( P(1,\,2,\,-1) \) and \( Q(2,\,3,\,1) \). Let \( S \) be the foot of the perpendicular drawn from the point \( R(4,\,-1,\,5) \) to the line \( L \). Another line passing through \( R \) intersects \( L \) at a point \( T \) such that the point \( S \) divides the line segment \( PT \) internally in the ratio \( |PS| : |ST| = 1 : 2 \), where \( |PS| \) and \( |ST| \) are the lengths of the line segments \( PS \) and \( ST \), respectively. Then which of the following statements is (are) TRUE?

Let \( y=f(x) \) be the real valued function defined on the interval \( (0,\infty) \), satisfying \( y(1)=0 \) and the differential equation \( x\,\dfrac{dy}{dx}=y-x^3 \). Then which of the following statements is (are) TRUE?

Let \( R \) denote the set of all real numbers and let \( i = \sqrt{-1} \), Consider the matrices . Let \( a, b, c, d \) be real numbers such that .
Let $H = \{x + iy : \quad x,y \in \mathbb{R} \text{ and } y > 0\}$
Then which of the following statements is (are) TRUE?

Let \( \mathbb{N} \) denote the set of all positive integers. Consider the sets \( A = \{ 1, 2, 3, 4, 5 \} \) and \( B = \{ 1, 2, 3, 4, 5, 6, 7 \} \).
Let \( S \) be the set of all functions \( f : A \to B \) such that \( f(2) \neq 2 \) and \( f(4) \neq 4 \). Consider the set \( T = \{ f \in S : \text{ there exists a function } g : B \to \mathbb{N} \text{ such that } g(f(x)) = 2^x \text{ for all } x \in A \} \).
Then the number of elements in the set \( T \) is ________.

A bookshelf contains \( 6 \) distinct books of Mathematics and \( 5 \) distinct books of Physics. From these \( 11 \) books, \( 6 \) books are chosen at random. Let \( X \) be the absolute value of the difference between the number of Mathematics books chosen and the number of Physics books chosen. If \( \alpha \) is the mean of the random variable \( X \), then the value of \( 77\alpha \) is _________.

Consider a data consisting of \( 10 \) observations \( x_1, x_2, \ldots, x_{10} \), whose mean is \( 5 \) and variance is \( 7 \). If the mean and the variance of the first \( 8 \) observations \( x_1, x_2, \ldots, x_8 \) are \( 4 \) and \( 3.5 \), respectively, and \( x_9 < x_{10} \), then the value of \( 3x_9 + 2x_{10} \) is ________.

Consider the ellipse \( E \) given by \( \frac{x^2}{18} + \frac{y^2}{12} = 1 \). Let \( H \) be the hyperbola whose eccentricity is the reciprocal of the eccentricity of \( E \) and whose foci are the same as that of \( E \). Let \( P \) and \( Q \) be the points of intersection of \( H \) and the parabola \( \sqrt{5}\,y = x^2 \) in the first quadrant. Let \( d \) be the distance between \( P \) and \( Q \).

For a real number \( \alpha \), let \( [\alpha] \) denote the greatest integer less than or equal to \( \alpha \). For a finite set \( S \), let \( |S| \) denote the number of elements in the set \( S \).
Consider the functions \( f:(-3,3)\rightarrow(-\infty,\infty) \) and \( g:(-3,3)\rightarrow(-\infty,\infty) \) defined by \( f(x)=[x^{3}] \log_{e}\!\big(1+\sin^{2}(\pi(x-[x]))\big) \) and \( g(x)=x^{3}\sin^{2}\!\big(\pi\log_{e}(1+x-[x])\big) \).
Let \( A=\{x\in(-3,3): f \text{ is discontinuous at } x\} \) and \( B=\{x\in(-3,3): g \text{ is discontinuous at } x\} \). Then the value of \( |A|+2|B|-|A\cap B| \) is ____.

Consider the curve \( C_{1} \) given by \( y=e^{-x} \) for \( x\in[0,10\pi] \), and the curve \( C_{2} \) given by \( y=e^{-x}(\sin x+\cos x) \) for \( x\in[0,10\pi] \).
Let \( n \) be the total number of points of intersection of the curves \( C_{1} \) and \( C_{2} \).
Suppose that \( \alpha_{1},\alpha_{2},\ldots,\alpha_{n}\in[0,10\pi] \) are the \( x \)-coordinates of the points of intersection of the curves \( C_{1} \) and \( C_{2} \) such that \( \alpha_{1}<\alpha_{2}<\ldots<\alpha_{n} \).
The value of \( n \) is ____.

Consider the curve \( C_{1} \) given by \( y=e^{-x} \) for \( x\in[0,10\pi] \), and the curve \( C_{2} \) given by \( y=e^{-x}(\sin x+\cos x) \) for \( x\in[0,10\pi] \).
Let \( n \) be the total number of points of intersection of the curves \( C_{1} \) and \( C_{2} \).
Suppose that \( \alpha_{1},\alpha_{2},\ldots,\alpha_{n}\in[0,10\pi] \) are the \( x \)-coordinates of the points of intersection of the curves \( C_{1} \) and \( C_{2} \) such that \( \alpha_{1}<\alpha_{2}<\ldots<\alpha_{n} \).
Let \( \beta \) be the area of the region enclosed between the curves \( C_{1}, C_{2} \), and the lines \( x=\alpha_{1} \) and \( x=\alpha_{4} \). Then the value of \( -\frac{1}{\pi}\log_{e}\!\left(\beta-2e^{-\pi/2}\right) \) is __________.

Consider the ellipses given by \( x^{2}+4y^{2}=1 \) and \( 4x^{2}+y^{2}=1 \).
Let \( P \) be the point in the first quadrant where the given ellipse intersect. If \( \theta \) is the acute angle between the tangents to the given ellipses at the point \( P \), then the value of \( 4\tan\theta \) is __________________.

Consider the ellipses given by \( x^2 + 4y^2 = 1 \) and \( 4x^2 + y^2 = 1 \).
If \( \alpha \) is the area of the common region that lies inside both the given ellipses, then the value of \( \cot \alpha \) is ____________.

JEE Advanced 2026 Paper - 2

Physics Chemistry Mathematics

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Let \( y:(-\infty,\infty)\to(0,\infty) \) be the solution of the differential equation \( \frac{dy}{dx}=\frac{e^{5x}y^{3}+y^{3}}{e^{x}+e^{x}y^{4}} \), satisfying \( y(0)=\frac{1}{\sqrt{2}} \). Then the value of \( y(\log_{e}2) \) is: