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Let \( \vec{a} \) and \( \vec{b} \) be two-unit vectors such that the angle between them is \( \frac{\pi}{3} \). If \( \lambda \vec{a} + 2\vec{b} \) and \( 3\vec{a} - \lambda \vec{b} \) are perpendicular to each other, then the number of values of \( \lambda \) in \( [-1, 3] \) is:

Let \( A = \{1, 2, 3, 4\} \) and \( B = \{1, 4, 9, 16\} \). Then the number of many-one functions \( f : A \to B \) such that \( 1 \in f(A) \) is equal to:

Let \( \alpha, \beta, \gamma \) and \( \delta \) be the coefficients of \( x^{7}, x^{5}, x^{3} \) and \( x \) respectively in the expansion of \( \left(x + \sqrt{x^{3} - 1}\right)^{5} + \left(x - \sqrt{x^{3} - 1}\right)^{5} ,\; x > 1 \). If \( u \) and \( v \) satisfy the equations \( \alpha u + \beta v = 18, \gamma u + \delta v = 20 \), then \( u + v \) equals:

Let \( E : \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1,\, a > b \) and \( H : \frac{x^{2}}{A^{2}} - \frac{y^{2}}{B^{2}} = 1 \). Let the distance between the foci of \( E \) and the foci of \( H \) be \( 2\sqrt{3} \). If \( a - A = 2 \), and the ratio of the eccentricities of \( E \) and \( H \) is \( \frac{1}{3} \), then the sum of the lengths of their latus rectums is equal to:

Let a line pass through two distinct points \( P(-2,-1,3) \) and \( Q \), and be parallel to the vector \( 3\hat{i} + 2\hat{j} + 2\hat{k} \). If the distance of the point \( Q \) from the point \( R(1,3,3) \) is \( 5 \), then the square of the area of \( \triangle PQR \) is equal to:

The area of the region enclosed by the curves \( y = x^{2} - 4x + 4 \) and \( y^{2} = 16 - 8x \) is:

Let \(\displaystyle f(x) = \int_{0}^{x^{2}} \frac{t^{2}-8t+15}{e^{t}} \, dt , \; x \in \mathbb{R} \). Then the numbers of local maximum and local minimum points of \( f \), respectively, are:

Suppose that the number of terms in an A.P. is \( 2k , \; k \in \mathbb{N} \). If the sum of all odd terms of the A.P. is \( 40 \), the sum of all even terms is \( 55 \) and the last term of the A.P. exceeds the first term by \( 27 \), then \( k \) is equal to:

If \( A \) and \( B \) are two events such that \( P(A \cap B)=0.1 \), and \( P(A|B) \) and \( P(B|A) \) are the roots of the equation \( 12x^{2}-7x+1=0 \), then the value of \(\displaystyle \frac{P(\bar{A}\cup \bar{B})}{P(\bar{A}\cap \bar{B})} \) is:

Let \( \alpha_{\theta} \) and \( \beta_{\theta} \) be the distinct roots of \( 2x^{2}+(\cos \theta)x-1=0 , \; \theta \in (0,2\pi) \). If \( m \) and \( M \) are the minimum and the maximum values of \( \alpha_{\theta}^{4}+\beta_{\theta}^{4} \), then \( 16{(M+m)} \) equals:

The perpendicular distance, of the line \(\displaystyle \frac{x-1}{2}=\frac{y+2}{-1}=\frac{z+3}{2} \) from the point \( P(2,-10,1) \), is:

Let the curve \( z(1+i)+\bar{z}(1-i)=4 , \; z \in \mathbb{C} \), divide the region \( |z-3|\le 1 \) into two parts of areas \( \alpha \) and \( \beta \). Then \( |\alpha-\beta| \) equals:

Let \( P(4, 4\sqrt{3}) \) be a point on the parabola \( y^2 = 4ax \) and PQ be a focal chord of the parabola. If M and N are the foot of perpendiculars drawn from P and Q respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to:

In a group of \( 3 \) girls and \( 4 \) boys, there are two boys \( B_1 \) and \( B_2 \). The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but \( B_1 \) and \( B_2 \) are not adjacent to each other, is:

If the system of linear equations:
\( x + y + 2z = 6 \),
\( 2x + 3y + az = a + 1 \),
\( -x - 3y + bz = 2b \),
where \( a, b \in \mathbb{R} \), has infinitely many solutions, then \( 7a + 3b \) is equal to:

For a \( 3 \times 3 \) matrix \( M \), let \( \text{trace}(M) \) denote the sum of all the diagonal elements of \( M \). Let \( A \) be a \( 3 \times 3 \) matrix such that \( |A| = \frac{1}{2} \) and \( \text{trace}(A) = 3 \). If \( B = \operatorname{adj}(\operatorname{adj}(2A)) \), then the value of \( |B| + \text{trace}(B) \) equals:

The sum of all values of \( \theta \in [0, 2\pi] \) satisfying \( 2\sin^2\theta = \cos 2\theta \) and \( 2\cos^2\theta = 3\sin\theta \) is:

If \( \displaystyle \int e^x \left( \frac{x\sin^{-1}x}{\sqrt{1-x^2}} + \frac{\sin^{-1}x}{(1-x^2)^{3/2}} + \frac{x}{1-x^2} \right) dx = g(x) + C \), where \( C \) is the constant of integration, then \(\displaystyle g\left(\frac{1}{2}\right) \) equals:

If \(\displaystyle\lim_{x \to \infty} \left( \left( \frac{e}{1 - e} \right) \left( \frac{1}{e} - \frac{x}{1 + x} \right) \right)^x = \alpha \), then the value of \(\displaystyle {\frac{\log_{e}\alpha}{1+\log_{e}\alpha}} \) equals:

If \( x=f(y) \) is the solution of the differential equation \(\displaystyle (1+y^{2})+(x-2e^{\tan^{-1}y})\frac{dy}{dx}=0, \; y \in \left(-\frac{\pi}{2},\frac{\pi}{2}\right) \) with \( f(0)=1 \), then \(\displaystyle f\!\left(\frac{1}{\sqrt{3}}\right) \) is equal to:

If \(\displaystyle \sum_{r=1}^{30} \frac{r^{2}\left({}^{30}C_{r}\right)^{2}}{{}^{30}C_{r-1}}=\alpha \times 2^{29} \), then \( \alpha \) is equal to ________________.

Let \( A(6,8) \), \( B(10\cos\alpha,-10\sin\alpha) \) and \( C(-10\sin\alpha,10\cos\alpha) \), be the vertices of a triangle. If \( L(a,9) \) and \( G(h,k) \) be its orthocenter and centroid respectively, then \( (5a-3h+6k+100\sin 2\alpha )\) is equal to ____________.

Let the distance between two parallel lines be \( 5 \) units and a point \( P \) lie between the lines at a unit distance from one of them. An equilateral triangle \( PQR \) is formed such that \( Q \) lies on one of the parallel lines, while \( R \) lies on the other. Then \( (QR)^{2} \) is equal to ____________.

Let \( A=\{1,2,3\} \). The number of relations on \( A \), containing \( (1,2) \) and \( (2,3) \), which are reflexive and transitive but not symmetric, is ____________.

Let \( y = f(x) \) be the solution of the differential equation \(\displaystyle \frac{dy}{dx} + \frac{xy}{x^2 - 1} = \frac{x^6 + 4x}{\sqrt{1 - x^2}},\ -1 < x < 1 \) such that \( f(0) = 0 \). If \(\displaystyle 6 \int_{-1/2}^{1/2} f(x)\, dx = 2\pi - \alpha \) then \( \alpha^2 \) is equal to __________.

JEE Main 2025 22nd Jan Evening Shift

Mathematics Physics Chemistry

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The area of the region enclosed by the curves \( y = x^{2} - 4x + 4 \) and \( y^{2} = 16 - 8x \) is: