NTA JEE Mains 29th Jan 2025 Shift 2 - Mathematics

Let $$f(x)=\int_{0}^{t}t(t^{2}-9t+20)dt$$, $$1 \le x \le 5$$. If the range of $$f$$ is $$[\alpha, \beta]$$, then $$4(\alpha + \beta)$$ 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 :

Let the area enclosed between the curves $$|y|= 1-x^{2}$$ and $$x^{2}+y^{2}=1$$ be $$\alpha$$. If $$9\alpha$$ = $$\beta \pi + \gamma; \beta,\gamma$$ are integers, then the value of $$|\beta - \gamma |$$ equals.

The remainder, when $$7^{103}$$ is divided by 23 , is equal to :

If $$\alpha x+ \beta y = 109$$ is the equation of the chord of the ellipse $$\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$$, whose mid point is $$(\frac{5}{2},\frac{1}{2})$$ , then $$\alpha + \beta$$
is equal to :

If the domain of the function $$\log_{5}(18x - x^{2} - 77)$$ is $$(\alpha,\beta)$$ and the domain of the function $$\log_{(x-1)}\left(\frac{2x^{2}+3x-2}{x^{2}-3x-4}\right)$$ is $$(\gamma,\delta)$$, then $$\alpha^{2}+\beta^{2}+\gamma^{2}$$ is equal to :

Let a circle C pass through the points (4,2) and (0,2) , and its centre lie on 3x+2y+2=0. Then the length of the chord, of the circle C, whose mid-point is (1,2), is:

Let a straight line L pass through the point P(2,-1,3) and be perpendicular to the lines $$\frac{x-1}{2}=\frac{y+1}{1}=\frac{z-3}{-2}$$ and $$\frac{x-3}{1}=\frac{y-2}{3}=\frac{z+2}{4}.$$ If the line L intersects the yz-plane at the point Q , then the distance between the points P and Q is :

Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains n white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to Bag 2. A ball is then drawn randomly from Bag 2. If the probability, that the ball drawn is white, is 29/45 , then n is equal to :

Let $$\alpha, \beta (\alpha \neq \beta)$$ be the values of m , for which the equations x + y + z = 1, x + 2y + 4z = m and x + 4y + 10z = $$m^{2}$$ have infinitely many solutions. Then the value of $$\sum_{n=1}^{10} \left(n^{\alpha} + n^{\beta}\right)$$ is equal to :

Let $$S = \mathbb{N} \cup \{0\}$$. Define a relation R from S to $$\mathbb{R}$$ by : $$R = \{(x,y) : \log_{e} y = x \log_e\left(\frac{2}{5}\right),\ x \in S,\ y \in \mathbb{R}\}$$ Then, the sum of all the elements in the range of $$\mathbb{R}$$ is equal to :

If $$\sin x + \sin^2 x = 1$$, $$x \in (0, \tfrac{\pi}{2})$$ then $$(\cos^{12}x+\tan^{12}x)+3(\cos^{10}x+\tan^{10}x+\cos^{8}x+\tan^{8} x)+(\cos^{6}x+\tan^{6}x)$$
is equal to :

If $$24 \int_{0}^{\frac{\pi}{4}} \left(\sin\left|4x-\frac{\pi}{12}\right| + \left[2\sin x\right]\right) dx = 2\pi + \alpha,$$ , where $$[\cdot]$$ denotes the greatest integer function, then $$\alpha$$ is equal to _______.

Let $$a_1, a_2, \ldots, a_{2024}$$ be an Arithmetic Progression such that $$a_1 + (a_5 + a_{10} + a_{15} + \cdots + a_{2020}) + a_{2024} = 2233$$. Then $$a_1 + a_2 + a_3 + \cdots + a_{2024}$$ is equal to _______

If $$\lim_{t \to 0} \left(\int_{0}^{1} (3x+5)^t \, dx\right)^{\frac{1}{t}} = \frac{\alpha}{5e}\left(\frac{8}{5}\right)^{\frac{2}{3}}$$, then $$\alpha$$ is equal to ________

Let $$y^{2} = 12x$$ be the parabola and S be its focus. Let PQ be a focal chord of the parabola such that (SP)(SQ) = $$\frac{147}{4}$$. Let C be the circle described taking PQ as a diameter. If the equation of a circle C is $$64x^2 + 64y^2 - \alpha x - 64\sqrt{3}\,y = \beta$$, then $$\beta - \alpha$$ is equal to ________.

Let integers $$a,b \in [-3,3]$$ be such that $$a+b \neq 0$$. Then the number of all possible ordered pairs (a, b), for which $$\left|\frac{z-a}{z+b}\right| = 1$$ and $$\begin{vmatrix} z+1 &amp; \omega &amp; \omega^2 \\ \omega &amp; z+\omega^2 &amp; 1 \\ \omega^2 &amp; 1 &amp; z+\omega \end{vmatrix} = 1,\quad z \in \mathbb{C}$$, where $$\omega$$ and $$\omega^{2}$$ are the roots of $$x^{2} + x + 1 = 0$$, is equal to_________.