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Mathematics

Consider the function \( f : (0,\infty) \to (-\infty,\infty) \) given by
\( f(x) = \sqrt{x}\,\log_e(x) - x + 1 \).
Hence which one of the following statements is TRUE?

Let \( P \) be the point on the parabola \( y=x^2 \) such that the slope of the tangent to the parabola at the point \( P \) is \( 4 \). Let \( Q \) be the point in the first quadrant lying on the circle \( x^2+y^2=2 \) such that the slope of the tangent to the circle at the point \( Q \) is \( -1 \). Let \( R \) be the point in the first quadrant lying on the ellipse \( x^2+4y^2=8 \) such that the slope of the tangent to the ellipse at the point \( R \) is \( -\frac{1}{2} \). Then the radius of the circle passing through the points \( P, Q \) and \( R \) is:

Which one of the following matrices can be obtained by performing elementary row transformations on the \( 3\times 3 \) identity matrix?

Considering only the principal values of the inverse trigonometric functions, the value of
\( \cot^{-1}(\cot(-11)) + 10\sin\!\left(2\cos^{-1}\!\left(\frac{1}{\sqrt{2}}\right)\right) + 10\sin\!\left(2\tan^{-1}(2)\right) \) is:

Suppose that Box I contains \( 6 \) red balls and \( 9 \) green balls, and Box II contains \( 8 \) red balls and \( 12 \) green balls. All the balls of Box I and Box II are mixed together and a ball is chosen at random from them. Let \( E_1 \) be the event that the ball chosen belonged to Box I and let \( E_2 \) be the event that the ball chosen belonged to Box II. Let \( F_1 \) be the event that the ball chosen is red and let \( F_2 \) be the event that the ball chosen is green. Then which of the following statement is (are) TRUE?

Let \( P \) be the plane such that it contains the straight line \( \dfrac{x-1}{2} = \dfrac{y-3}{3} = \dfrac{z+2}{1} \) and is perpendicular to the plane \( x + 2y + 3z = 4 \). Let \( P_1 \) be the plane which passes through the point \( (4, 2, 2) \) and is parallel to \( P \). Then which of the following statements is (are) TRUE?

Let \( \mathbb{R} \) denote the set of all real numbers. Let \( f : \mathbb{R} \to \mathbb{R} \) be an arbitrary function and let \( g : \mathbb{R} \to \mathbb{R} \) be the function defined by \( g(x) = x f(x) \), for all \( x \in \mathbb{R} \).
Then which of the following statements is (are) TRUE?

Consider the matrix . Let \( p, q, r, s, a, b, c \) and \( d \) be integers such that .
Then which of the following statements is (are) TRUE?

Let \( S = \{ 1, 2, 3, \ldots, 10 \} \). Consider the set \( X = \{ R : R \) is an equivalence relation on the set \( S \) such that \( R \) has exactly \( 42 \) elements\( \} \). Then the number of elements in \( X \) is ____________.

Consider the function \( f : \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \rightarrow (-\infty, \infty) \) defined by
\( f(x) = (|x| + |x - 1|)\sin x + [ x\sin x ]\), where \( [x\sin x ]\) is the greatest integer less than or equal to \( x\sin x \). Let \( \alpha \) be the total number of points in the interval \( \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \) at which \( f \) is NOT continuous and let \( \beta \) be the total number of points in the interval \( \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \) at which \( f \) is NOT differentiable. Then the value of \( \alpha + \beta \) is __________.

The number of ways to distribute \( 10 \) identical red pens and \( 14 \) identical blue pens among four persons such that each person gets \( 6 \) pens, is __________.

$Let\, \alpha = \left(1 - 2\cos\left(\frac{\pi}{11}\right)\right)\left(1 - 2\cos\left(\frac{3\pi}{11}\right)\right)\left(1 - 2\cos\left(\frac{9\pi}{11}\right)\right)\left(1 - 2\cos\left(\frac{27\pi}{11}\right)\right)\left(1 - 2\cos\left(\frac{81\pi}{11}\right)\right)$
Then the value of $5 - \alpha^2$ is_______.

Match each entry in List-I to the correct entry in List-II and choose the correct option.

Match each entry in List-I to the correct entry in List-II and choose the correct option.

For real numbers \( \alpha, \beta, \gamma, \delta \) and \( \mu \), consider the matrix
.
Suppose that \( MM^{T} = I \), where \( M^{T} \) is the transpose of the matrix \( M \), and \( I \) is the \( 3 \times 3 \) identity matrix.
Let $\vec{u} = \alpha\hat{i} + \frac{1}{\sqrt{3}}\hat{j} + \gamma\hat{k}$, $\vec{v} = \frac{1}{\sqrt{2}}\hat{i} + \beta\hat{j} + \delta\hat{k}$ and $\vec{w} = -\frac{1}{\sqrt{2}}\hat{i} + \frac{1}{\sqrt{3}}\hat{j} + \mu\hat{k}$.
Match each entry in List-I to the correct entry in List-II and choose the correct option.

Match each entry in List-I to the correct entry in List-II and choose the correct option.

JEE Advanced 2026 Paper - 1

Physics Chemistry Mathematics

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Consider the matrix . Let \( p, q, r, s, a, b, c \) and \( d \) be integers such that .
Then which of the following statements is (are) TRUE?