NTA JEE Mains 28th Jan 2025 Shift 2 - Mathematics

Let $$A = \begin{bmatrix}\frac{1}{\sqrt{2}} & -2 \\0 & 1 \end{bmatrix}$$ and $$P = \begin{bmatrix}\cos \theta & -\sin \theta \\\sin \theta & \cos \theta \end{bmatrix}$$ ,$$\theta > 0$$. If $$B = PAP^{T}, C = P^{T}B^{10}P$$ and the sum of the diagonal elements of $$C$$ is $$\frac{m}{n}$$, where $$gcd(m,n)=1,$$ m + n is :

If the components of $$\overrightarrow{a} = \alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}$$ along and perpendicular to $$\overrightarrow{b}= 3\hat{i}+\hat{j}-\hat{k}$$ respectively, are $$frac{16}{11}(3\hat{i}+\hat{j}-\hat{k})$$ and $$frac{1}{11}(-4\hat{i}-5\hat{j}-17\hat{k})$$, then $$\alpha^{2} + \beta^{2} + \gamma^{2}$$ is equals to :

Let A, B, C be three points in $$xy-plane$$, whose position vector are given by $$\sqrt{3}\hat{i}+\hat{j}, \hat{i}+\sqrt{3}\hat{j}$$ and $$a\hat{i}+ (1-a)\hat{j}$$ respectively with respect to the origin O . If the distance of the point C from the line bisecting the angle between the vectors $$\overrightarrow{OA}$$ and $$\overrightarrow{OB}$$ is $$\frac{9}{\sqrt{2}}$$, then the sum of all the possible values of $$a$$ is :

Let the coefficients of three consecutive terms $$T_{r},T_{r+1}$$ and $$T_{r+2}$$ in the binomial expansion of $$(a+b)^{12}$$ be in a G.P. and let $$p$$ be the number of all possible values of $$r$$. Let $$q$$ be the sum of all rational terms in the binomial expansion of $$(\sqrt[4]{3}+\sqrt[3]{4})^{12}$$ Then p + q is equals to :

Let $$[x]$$ denote the greatest integer less than or equal to $$x$$. Then the domain of $$f(x)=sec^{-1}(2[x]+1)$$ is:

Let S be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set S, one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is :

If $$\sum_{r=1}^{13}\left\{\frac{1}{\sin(\frac{\pi}{4}+(r-1)\frac{\pi}{6})\sin(\frac{\pi}{4}+\frac{r\pi}{6})}\right\}=a\sqrt{3}+b,a,b \in Z$$ then $$a^{2}+b^{2}$$ is equal to:

Let $$f$$ be a real valued continuous function defined on the positive real axis such that $$g(x)=\int_{0}^{x}t f(t)dt$$. If $$g(x^{3})=x^{6}+x^{7}$$, then Value of $$\sum_{r=1}^{15}f(r^{3})$$ is:

Let $$f : [0:3]\rightarrow A$$ be difined by $$f(x)=2x^{3}-15x^{2}+36x+7$$ and $$g: [0,\infty)\rightarrow B$$ be difined by $$g(x)=\frac{x^{2015}}{x^{2025}+1}$$. If both the functions are onto and $$S=\left\{x \in Z : x \in A or x \in B \right\}$$, then n(S) is equal to:

Bag $$B_{1}$$ contains 6 white and 4 blue balls, Bag $$B_{2}$$ contains 4 white and 6 blue balls, and Bag $$B_{3}$$ contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability, that the ball is drawn from Bag $$B_{2}$$, is :

Let $$f : R\rightarrow R$$ be a twice differentiable function such that $$f(2)=1$$. If $$F(x)=xf(x)$$ for all $$x \in R$$, $$\int_{0}^{2}x F'(x)dx=6$$ and $$\int_{0}^{2}x^{2}F''(x)dx=40$$, then $$F'(2)+\int_{0}^{2}F(x)dx$$ is equal to :

For positive integers $$n$$, if $$4a_{n}=(n^{2}_5n+6)$$ and $$S_{n}= \sum_{k=1}^{n}\left(\frac{1}{a_{k}}\right)$$, then the value of $$507S_{2025}$$ is :

Let $$f : R - {0} \rightarrow (-\infty , 1)$$ be a polynomial of degree 2, satisfying $$f(x)f\left(\frac{1}{x}\right)=f(x)+f\left(\frac{1}{x}\right)$$. If $$f(K)=-2K$$, then the sum of squares of all possible values of K is:

If A and B are the points of intersection of the circle $$x^{2}+y^{2}-8x=0$$ and the hyperbola $$\frac{x^{2}}{9}-\frac{y^{2}}{4}=1$$ and a point P moves on the line $$2x-3y+4=0$$, then the centroid of $$ \triangle PAB $$ lies on the line:

If $$f(x)=\int_{}^{}\frac{1}{x^{1/4}(1+x^{1/4})}dx,f(0)=-6$$, then $$f(1)$$ is equal to :

The area of the region bounded by the curves $$x(1+y^{2})=1$$ and $$y^{2}=2x$$ is:

The square of the distance of the point $$(\frac{15}{7},\frac{32}{7},7)$$ from the line $$\frac{x+1}{3}=\frac{y+3}{5}=\frac{z+5}{7}$$ in the direction of the vector $$\hat{i}+4\hat{j}+7\hat{k}$$ is :

If the midpoint of a chord of the ellipse $$\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$$ is $$(\sqrt{2},4/3)$$, and the length of the chord is $$\frac{2sqrt{\alpha}}{3}$$, then $$\alpha$$ is :

If $$\alpha + i\beta $$ and $$\gamma + i \delta $$ are the roots of $$x^{2}-(3-2i)x-(2i-2)=0,i=sqrt{-1}$$, then $$\alpha \gamma + \beta \delta $$ is equal to :

Two equal sides of an isosceles triangle are along $$−x + 2y = 4$$ and $$x + y = 4$$. If m is the slope of its third side, then the sum, of all possible distinct values of $$m$$, is:

Let A and B be the two points of intersection of the line $$y + 5 = 0$$ and the mirror image of the parabola $$y^{2}=4x$$ with respect to the line $$x + y + 4 = 0$$. If d denotes the distance between A and B , and a denotes the area of $$\triangle SAB$$ where $$S$$ is the focus of the parabola $$y^{2}=4x$$, then the value of (a + d) is

The number of natural numbers, between 212 and 999 , such that the sum of their digits is 15 , is

If $$y=y(x)$$ is the solution of the differential equation.
$$\sqrt{4 - x^2}\,\frac{dy}{dx} = \left(\left(\sin^{-1}\left(\frac{x}{2}\right)\right)^2 - y\right)\sin^{-1}\left(\frac{x}{2}\right), \quad -2 \le x \le 2,\quad y(2) = \frac{\pi^2 - 8}{4}$$, then $$y^{2}(0)$$ is equal to

The interior angles of a polygon with n sides, are in an A.P. with common difference $$6^{\circ}$$ . If the largest interior angle of the polygon is $$219^{\circ}$$, then n is equal to

Let $$f(x) = \lim_{n \to \infty} \sum_{r=0}^{n}\left(\frac{\tan\left(\frac{x}{2^{r+1}}\right) + \tan^{3}\left(\frac{x}{2^{r+1}}\right)}{1 - \tan^{2}\left(\frac{x}{2^{r+1}}\right)}\right).\quad$$ Then $$\lim_{x \to 0} \frac{e^{x} - e^{f(x)}}{x - f(x)}$$ is equal to.