NTA JEE Mains 22nd Jan 2025 Shift 2 - Mathematics

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

In a group of 3 girls and 4 boys, there are two boys $$B_{1}\text{ and }B_{2}$$. 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 $$B_{1}\text{ and }B_{2}$$ are not adjacent to each other, is :

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

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

If and are two events such that P(A ∩ 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 $$\frac{P(\overline{A} \cup \overline{B})}{P(\overline{A} \cap \overline{B}}$$ is:

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

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

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

Let $$P(4, 4\sqrt{3})$$be a point on the parabola $$y^{2}=4ax$$ and 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 :

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

If $$\lim_{x \rightarrow \infty}((\frac{e}{1-e})(\frac{1}{e}-\frac{x}{1+x}))^{x}=\alpha$$ then the value of $$\frac{\log_{e}^{\alpha}}{1+\log_{e}^{\alpha}}$$ equals :

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

Suppose that the number of terms in an A.P is $$2k, k \in N$$. 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 :

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

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

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

Let $$\alpha_{\theta}$$ anf $$\beta_{\theta}$$ 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 sum of all values of $$\theta \in [0,2\pi]$$ satisfying $$2\sin^{2}\theta =\cos2\theta \text{ and }2\cos^{2}\theta =3\sin\theta$$ is

Let the curve $$z(1+i)+\overline{z}(1-i)=4,z \in C$$,divide the region $$|z-3|\leq 1$$ into two parts of areas $$\alpha$$ and $$\beta$$. Then $$|\alpha - \beta |$$ 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 $$2sqrt{3}$$. 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:

If $$\sum_{r=1}^{30}\f\frac{r^{2}({}^{30}C_{r})^{2}}{{}^{30}C_{r-1}}=\alpha \t\times 2^{29}$$, then $$\alpha$$ 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 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 equals to_______.

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