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Mathematics

Let [.] denote the greatest integer function. If the domain of the function \( f(x)=\sin^{-1}\!\left(\frac{x+[x]}{3}\right) \) is \( [\alpha,\beta] \), then \( \alpha^{2}+\beta^{2} \) is equal to:

Let one root of the quadratic equation in \( x \):
\( (k^{2}-15k+27)x^{2}+9(k-1)x+18=0 \) be twice the other. Then the length of the latus rectum of the parabola \( y^{2}=6kx \) is equal to:

Let \( e_{1} \) and \( e_{2} \) be two distinct roots of the equation \( x^{2}-ax+2=0 \). Let the sets \( \{\, a \in \mathbb{R} : e_{1} \) and \( e_{2} \) are the eccentricities of hyperbolas \} \( = (\alpha,\beta) \), and \( \{\, a \in \mathbb{R} : e_{1} \) and \( e_{2} \) are the eccentricities of an ellipse and a hyperbola, respectively \} \( = (\gamma,\infty) \).
Then \( \alpha^{2}+\beta^{2}+\gamma^{2} \) is equal to:

Let the set of all values of \( k \in \mathbb{R} \) such that the equation \( {z}(\overline{z}+2+i)+k(2+3i)=0, \; z \in \mathbb{C} \), has at least one solution, be the interval \( [\alpha,\beta] \). Then \( 9(\alpha+\beta) \) is equal to:

The value of \( 1^{3}-2^{3}+3^{3}-\cdots+15^{3} \) is:

The sum of the first ten terms of an A.P. is \( 160 \) and the sum of the first two terms of a G.P. is \( 8 \). If the term of the A.P. is equal to the common ratio of the G.P. and the first term of the G.P. is equal to common difference of the A.P., then the sum of all possible values of the first term of the G.P. is:

The number of \( 4 \)-letter words, with or without meaning, each consisting of two vowels and two consonants that can be formed the letters of the word INCONSEQUENTIAL, without repeating any letter, is:

If the coefficients of the middle terms in the binomial expansions of \( (1+\alpha x)^{26} \) and \( (1-\alpha x)^{28} \), \( \alpha \neq 0 \) are equal, then the value of \( \alpha \) is:

A data consists of \( 20 \) observations \( x_{1}, x_{2}, \ldots, x_{20} \). If \( \sum_{i=1}^{20} (x_{i}+5)^{2} = 2500 \) and \( \sum_{i=1}^{20} (x_{i}-5)^{2} = 100 \), then the ratio of mean to standard deviation of this data is:

A bag contains \( (N + 1) \) coins – \( N \) fair coins, and one coin with ‘Head’ on both sides. A coin is selected at random and tossed. If the probability of getting ‘Head’ is \( \frac{9}{16} \), then \( N \) is equal to:

If the eccentricity \( e \) of the hyperbola \( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \), passing through \( (6, 4\sqrt{3}) \), satisfies \( 15(e^{2} + 1) = 34e \), then the length of the latus rectum of the hyperbola \( \frac{x^{2}}{b^{2}} - \frac{y^{2}}{2(a^{2} + 1)} = 1 \) is:

Let chord \( PQ \) of length \( 3\sqrt{13} \) of the parabola \( y^2 = 12x \) be such that the ordinates of points \( P \) and \( Q \) are in the ratio \( 1:2 \). If the chord \( PQ \) subtends an angle \( \alpha \) at the focus of the parabola, then \( \sin \alpha \) is equal to:

Let \( 0<\alpha<1, \; \beta=\frac{1}{3\alpha} \) and \( \tan^{-1}(1-\alpha)+\tan^{-1}(1-\beta)=\frac{\pi}{4} \). Then \( 6(\alpha+\beta) \) is equal to:

Let \( S=\{\theta \in (-2\pi,2\pi): \cos\theta+1=\sqrt{3}\sin\theta\} \). Then \( \sum_{\theta\in S}\theta \) is equal to:

Let the image of the point \( P(1,6,a) \) in the line \( L : \, \dfrac{x}{1} = \dfrac{y-1}{2} = \dfrac{z-a+1}{b}, \, b>0 \), be \( \left(\dfrac{a}{3},0,a+c\right) \). If \( S(\alpha,\beta,\gamma), \, \alpha>0 \), is the point on \( L \) such that the distance of \( S \) from the foot of perpendicular from the point \( P \) on \( L \) is \( 2\sqrt{14} \), then \( \alpha + \beta + \gamma \) is equal to:

Let a line \( L \) be perpendicular to both the lines \( L_{1} : \, \dfrac{x+1}{3} = \dfrac{y+3}{5} = \dfrac{z+5}{7} \) and \( L_{2} : \, \dfrac{x-2}{1} = \dfrac{y-4}{4} = \dfrac{z-6}{7} \). If \( \theta \) is the acute angle between the lines \( L \) and \( L_{3} : \, \dfrac{x-8/7}{2} = \dfrac{y-4/7}{1} = \dfrac{z}{2} \), then \( \theta \) is equal to:

The value of \( \displaystyle \lim_{x\to 0} \left( \dfrac{x^{2}\sin^{2}x}{x^{2}-\sin^{2}x} \right) \) is:

The value o the integral \( \int_{-\pi/4}^{\pi/4} \left( \frac{32\cos^4 x}{1+e^{\sin x}} \right) dx \) is:

The area of the region \( \{(x,y) : 0 \le y \le 6 - x, \, y^{2} \ge 4x - 3, \, x \ge 0 \} \) is:

Let \( e \) be the base of natural logarithm and let \(\alpha/\beta\) and \( g : \{1 \), \( e \), \( e^2 \), \( e^3 \} \to \{ 1 \), \( \tfrac{1}{2} \), \( \tfrac{1}{3} \), \( \tfrac{1}{4} \} \) be two bijective functions such that \( f \) is strictly decreasing and \( g \) is strictly increasing. If \( \phi(x)=\left[ f^{-1}\!\left\{\,g^{-1}\!\left( \tfrac{1}{2} \right)\!\right\}\right]^x \), then the area of the region \( R=\{(x,y): x^2 \le y \le \phi(x),\; 0 \le x \le 1\} \) is:

Let satisfy
\( A^2+\alpha(\operatorname{adj}(\operatorname{adj}(A)))+\beta(\operatorname{adj}(A)\operatorname{adj}(\operatorname{adj}(A)))\)= for some \( \alpha,\beta \in \mathbb{R} \). Then \( (\alpha-\beta)^2 \) is equal to _____.

Let the centre of the circle \( x^2+y^2+2gx+2fy+25=0 \) be in the first quadrant and lie on the line \( 2x-y=4 \). Let the area of an equilateral triangle inscribed in the circle be \( 27\sqrt{3} \). Then the square of the length of the chord of the circle on the line \( x=1 \) is ______.

If \( \vec{a}=\hat{i}+\hat{j}+\hat{k} \), \( \vec{b}=\hat{j}-\hat{k} \) and \( \vec{c} \) be three vectors such that \( \vec{a}\times\vec{c}=\vec{b} \) and \( \vec{a}\cdot\vec{c}=3 \), then \( \vec{c}\cdot(\vec{a}-2\vec{b}) \) is equal to ______.

For the function \( f(\theta)=\alpha \tan^2\theta+\beta \cot^2\theta \), and \( g(\theta)=\alpha \sin^2\theta+\beta \cos^2\theta, \ \alpha,\beta>0 \), let If the first terms of a G.P. is \( \left(\frac{\alpha}{2\beta}\right) \), its common ratio is \( \left(\frac{2\beta}{\alpha}\right) \) and the sum of its first \( 10 \) terms is \( \frac{m}{n} \), \( \gcd(m,n)=1 \), then \( m+n \) is equal to ______.

Let \( y=y(x) \) be the solution of the differential equation \( \left(x^2-x\sqrt{x^2-1}\right)\,dy+\left(y\left(x-\sqrt{x^2-1}\right)-x\right)\,dx=0,\ x\ge 1 \). If \( y(1)=1 \), then the greatest integer less than \( y(\sqrt{5}) \) is ______.

JEE Main 2026 6th April Morning Shift

Mathematics Physics Chemistry

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The value o the integral \( \int_{-\pi/4}^{\pi/4} \left( \frac{32\cos^4 x}{1+e^{\sin x}} \right) dx \) is: