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Let \( y = y(x) \) be the solution of the differential equation \( x \, \frac{dy}{dx} - \sin 2y = x^3 (2 - x^3) \cos^2 y, \, x \neq 0 \). If \( y(2) = 0 \), then \( \tan (y(1)) \) is equal to:

The mean and variance of \( 10 \) observations are \( 9 \) and \( 34.2 \), respectively. If \( 8 \) of these observations are \( 2, 3, 5, 10, 11, 13, 15, 21 \), then the mean deviation about the median of all the \( 10 \) observations is:

The value of \( \displaystyle \lim_{x \to 0} \frac{\log_e \!\big( \sec(e{x}) \cdot \sec(e^{2}x) \cdots \sec(e^{10}x) \big)}{e^{2} - e^{2 \cos x}} \) is equal to:

The value of \( \displaystyle \sum_{k=1}^{\infty} (-1)^{k+1} \, \left( \frac{k(k+1)}{k!} \right) \) is:

Let \( f \) be a polynomial function such that \( f(x^{2}+1)=x^{4}+5x^{2}+2 \), for all \( x \in \mathbb{R} \). Then \( \displaystyle \int_{0}^{3} f(x) \, dx \) is equal to:

Let \( y = x \) be the equation of a chord of the circle \( C_{1} \) (in the closed half-plane \( x \ge 0 \)) of diameter \( 10 \) passing through the origin. Let \( C_{2} \) be another circle described on the given chord as its diameter. If the equation of the chord of the circle \( C_{2} \), which passes through the point \( (2, 3) \) and is farthest from the center of \( C_{2} \), is \( x + a y + b = 0 \), then \( a - b \) is equal to:

For three unit vectors \( \vec{a}, \vec{b}, \vec{c} \) satisfying \( \lvert \vec{a}-\vec{b} \rvert^{2} + \lvert \vec{b}-\vec{c} \rvert^{2} + \lvert \vec{c}-\vec{a} \rvert^{2} = 9 \) and \( \lvert 2\vec{a} + k\vec{b} + k\vec{c} \rvert = 3 \), the positive value of \( k \) is:

If \( \dfrac{\tan(A-B)}{\tan A} + \dfrac{\sin^{2} C}{\sin^{2} A} = 1, \ A,B,C \in \left(0, \dfrac{\pi}{2}\right), \) then:

If the distances of the point \( (1,2,a) \) from the line \( \dfrac{x-1}{1} = \dfrac{y}{2} = \dfrac{z-1}{1} \) along the lines \( L_{1}: \ \dfrac{x-1}{3} = \dfrac{y-2}{4} = \dfrac{z-a}{b} \) and \( L_{2}: \ \dfrac{x-1}{1} = \dfrac{y-2}{4} = \dfrac{z-a}{c} \) are equal, then \( a+b+c \) is equal to:

Let \( z \) be a complex number such that \( \lvert z-6 \rvert = 5 \) and \( \lvert z + 2 - 6i \rvert = 5 \). Then the value of \( z^{3} + 3z^{2} - 15z + 141 \) is equal to:

If $\int \left( \frac{1 - 5\cos^2 x}{\sin^5 x \cos^2 x} \right) dx = f(x) + C$, where $C$ is the constant of integration, then $f\left(\frac{\pi}{6}\right) - f\left(\frac{\pi}{4}\right)$ is equal to:

Let \( S = \left\{ x^{3} + a x^{2} + b x + c : a,b,c \in \mathbb{N} \ \text{and} \ a,b,c \leq 20 \right\} \) be a set of polynomials. Then the number of polynomials in \( S \), which are divisible by \( x^{2} + 2 \), is:

The common difference of the A.P.: \( a_1, a_2, \ldots, a_m \) is \( 13 \) more than the common difference of the A.P.: \( b_1, b_2, \ldots, b_n \). If \( b_{31} = -277 \), \( b_{43} = -385 \) and \( a_{78} = 327 \), then \( a_1 \) is equal to:

Let \( A, B \) and \( C \) be three \( 2 \times 2 \) matrices with real entries such that \( B = (I + A)^{-1} \) and \( A + C = I \).

If \( g(x) = 3x^2 + 2x - 3 \), \( f(0) = -3 \) and \( 4g(f(x)) = 3x^2 - 32x + 72 \), then \( f(g(2)) \) is equal to:

A bag contains $10$ balls out of which $k$ are red and $(10 - k)$ are black, where $0 \leq k \leq 10$. If three balls are drawn at random without replacement and all of them are found to be black, then the probability that the bag contains $1$ red and $9$ black balls is:

Let \( ABC \) be an equilateral triangle with orthocenter at the origin and the side \( BC \) on the line \( x + 2\sqrt{2}\,y = 4 \). If the co-ordinates of the vertex \( A \) are \( (\alpha, \beta) \) then the greatest integer less than or equal to \( \lvert \alpha + \sqrt{2}\beta \rvert \) is:

The area of the region \( R = \{ (x, y) : xy \le 8,\, 1 \le y \le x^2,\, x \ge 0 \} \) is:

If \( \alpha, \beta \), where \( \alpha < \beta \), are the roots of the equation \( \lambda x^2 - (\lambda + 3)x + 3 = 0 \) such that \( \frac{1}{\alpha} - \frac{1}{\beta} = \frac{1}{3} \), then the sum of all possible values of \( \lambda \) is:

Let \( S = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \). Let \( x \) be the number of \( 9 \)-digit numbers formed using the digits of the set \( S \) such that only one digit is repeated and it is repeated exactly twice. Let \( y \) be the number of \( 9 \)-digit numbers formed using the digits of the set \( S \) such that only two digits are repeated and each of these is repeated exactly twice. Then,

Let \( PQR \) be a triangle such that \( \overrightarrow{PQ} = -2\hat{i} - \hat{j} + 2\hat{k} \) and \( \overrightarrow{PR} = a\hat{i} + b\hat{j} - 4\hat{k},\ a, b \in \mathbb{Z} \). Let \( S \) be the point on \( QR \), which is equidistant from the lines \( PQ \) and \( PR \). If \( |\overrightarrow{PR}| = 9 \) and \( \overrightarrow{PS} = \hat{i} - 7\hat{j} + 2\hat{k} \), then the value of \( 3a - 4b \) is ________.

For some \( \theta \in \left(0, \frac{\pi}{2}\right) \), let the eccentricity and the length of the latus rectum of the hyperbola \( x^2 - y^2 \sec^2 \theta = 8 \) be \( e_1 \) and \( l_1 \), respectively, and let eccentricity and the length of the latus rectum of the ellipse \( x^2 \sec^2 \theta + y^2 = 6 \) be \( e_2 \) and \( l_2 \), respectively. If \( e_1^2 = e_2^2 \left(\sec^2 \theta + 1\right) \), then \( \left|\dfrac{l_1 l_2}{e_1 e_2}\right| \tan^2 \theta \) is equal to __________.

In a G.P., if the product of the first three terms is \( 27 \) and the set of all possible values for the sum of its first three terms is \( \mathbb{R} - (a, b) \), then \( a^2 + b^2 \) is equal to __________.

The value of is -----------.

If \( k = \tan\left( \frac{\pi}{4} + \frac{1}{2}\cos^{-1}\left( \frac{2}{3} \right) \right) + \tan\left( \frac{1}{2}\sin^{-1}\left( \frac{2}{3} \right) \right) \), then the number of solutions of the equation\( \sin^{-1}(kx - 1) = \sin^{-1} x - \cos^{-1} x \) is ____________.

JEE MAIN 2026 28TH Jan Morning Shift

Mathematics Physics Chemistry

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The value of is -----------.