NTA JEE Mains 29th Jan 2025 Shift 2

Let $$f(x)=\int_{0}^{t}t(t^{2}-9t+20)dt$$, $$1 \le x \le 5$$. If the range of f $$[\alpha, \beta]$$ is, then $$[\alpha + \beta]$$ equals :

Let $$\widehat{a}$$ be a unit vector perpendicular to the vectors $$\overrightarrow{b}=\widehat{i}-2\widehat{j}+3\widehat{k}$$ and $$\overrightarrow{c}=2\widehat{i}+3\widehat{j}-\widehat{k}$$, and makes an angle of $$\cos^{-1}(-\frac{1}{3})$$ with the vector $$\widehat{i}+\widehat{j}+\widehat{k}$$ . If $$\widehat{a}$$ makes an angle of $$\frac{\pi}{3}$$ with the vector $$\widehat{i}+\alpha\widehat{j}+\widehat{k}$$ , then the value of $$\alpha$$ is :

If for the solution curve y = f(x) of the differential equation $$\frac{dy}{dx}+(\tan x)y = \frac{2+\sec x}{(1+2\sec x)^{2}}$$, $$x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right), \quad f\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{10}$$, then $$f\left(\frac{\pi}{4}\right)$$ is equal to :

Let P be the foot of the perpendicular from the point (1,2,2) on the line L: $$\frac{x-1}{1}=\frac{y+1}{-1}=\frac{z-2}{2}.$$ Let the line $$\vec{r}=(-\hat{i}+\hat{j}-2\hat{k})+\lambda(\hat{i}-\hat{j}+\hat{k}), \quad \lambda \in \mathbb{R},$$ intersect the line L at Q. Then $$2(PQ)^{2}$$ is equal to:

Let A = $$[a_{ij}]$$ be a matrix of order $$3 \times 3$$, with $$a_{ij}$$ = $$(\sqrt{2})^{i+j}$$. If the sum of all the elements in the third row of $$A^{2}$$ is $$\alpha + \beta\sqrt{2}, \quad \alpha,\beta \in \mathbb{Z}$$, then $$\alpha + \beta$$ is equal to:

Let the line x + y = 1 meet the axes of x and y at A and B, respectively. A right angled triangle AMN is inscribed in the triangle OAB , where O is the origin and the points M and N lie on the lines OB and AB, respectively. If the area of the triangle AMN is $$\frac{4}{9}$$ of the area of the triangle OAB and AN : NB = $$\lambda$$:1 , then the sum of all possible value(s) of is $$\lambda$$ :

If all the words with or without meaning made using all the letters of the word "KANPUR" are arranged as in a dictionary, then the word at $$440^{th}$$ position in this arrangement, is :

If the set of all $$a \in \mathbb{R}$$, for which the equation $$2x^2 + (a-5)x + 15 = 3a$$ has no real root, is the interval $$(\alpha,\beta)$$ and $$X=\{x \in \mathbb{Z} : \alpha < x < \beta\}$$, then $$\sum_{x \in X}^{}x^{2}$$ is equal to :

Let $$A =[a_{ij}]$$ be a 2$$\times$$2 matrix such that $$a_{ij} \in \left\{0,1\right\}$$ for all i and j . Let the random variable X denote the possible values of the determinant of the matrix A . Then, the variance of x is :

Let the function f(x) = $$(x^{2}+1) |x^{2}-ax+2|+\cos|x|$$ be not differentiable at the two points x = $$\alpha$$ = 2 and $$x= \beta$$. Then the distance of the point $$(\alpha , \beta)$$ from the line $$12x+5y+10=0$$ is equal to :