NTA JEE Mains 29th Jan 2025 Shift 1 - Mathematics

Let $$x_{1},x_{2},...x_{10}$$ be ten observations such that $$\sum_{i=1}^{10}(x_{i}-2)=30,\sum_{i=1}^{10}(x_{i}-\beta)^{2}=98,\beta > 2$$, and their variance is $$\frac{4}{5}$$. If $$\mu$$ and $$\sigma^{2}$$ are respectively the mean and the variance of $$2(x_{1}-1)+4\beta, 2(x_{2}-1)+4\beta,....,2(x_{10}-1)+4\beta$$, then $$\frac{\beta \mu}{\sigma^{2}}$$ is equal to :

Consider an A. P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its $$11^{th}$$ term is :

The number of solutions of the equation $$\left(\dfrac{9}{x}-\dfrac{9}{\sqrt{x}}+2\right)\left(\dfrac{2}{x}-\dfrac{7}{\sqrt{x}}+3\right)=0$$ is:

Define a relation R on the interval $$[0,\frac{\pi}{2})$$ by $$xRy$$ if and only if$$\sec^{2} x-\tan^{2} y=1$$. Then R is :

Two parabolas have the same focus (4,3) and their directrices are the x-axis and the y-axis, respectively. If these parabolas intersects at the points A and B, then $$(AB)^{2}$$ is equal to :

Let P be the set of seven digit numbers with sum of their digits equal to 11 . If the numbers in P are formed by using the digits 1,2 and 3 only, then the number of elements in the set $$P$$ is :

Let $$\overrightarrow{a}=\hat{i}+2\hat{j}+\hat{k}$$ and $$\overrightarrow{b}=2\hat{i}+7hat{j}+3\hat{k}$$. Let $$L_{1}:\overrightarrow{r}=(-\hat{i}+2\hat{j}+\hat{k})+\lambda \overrightarrow{a},\lambda \in R$$. and $$L_{2}: \overrightarrow{r}=(\hat{j}+\hat{k})+\mu \overrightarrow{b}, \mu \in R$$ be two lines. If the line $$L_{3}$$ passes through the point of intersection of $$L_{1}$$ and $$L_{2}$$, and is parallel to $$\overrightarrow{a}+\overrightarrow{b}$$, then $$L_{3}$$ passes through the point :

Let $$\overrightarrow{r}=2\hat{i}-\hat{j}+3\hat{k}, \overrightarrow{c}=3\hat{i}-5\hat{j}+\hat{k}$$ and $$\overrightarrow{c}$$ be a vector such that $$\overrightarrow{c} \times \overrightarrow{c} = \overrightarrow{c} \times \overrightarrow{b}$$ and $$(\overrightarrow{a}+\overrightarrow{c}).(\overrightarrow{b}.\overrightarrow{c})=168$$. Then the maximum value of $$|\overrightarrow{c}|^{2}$$ is :

The integral $$80\int_{0}^{\frac{\pi}{4}}\left(\frac{\sin \theta + \cos \theta}{9+16\sin 2\theta}\right)d\theta$$ is equaol to :

Let the ellipse $$E_{1}:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,a \gt b$$ and $$E_{2}:\frac{x^{2}}{A^{2}}+\frac{y^{2}}{B^{2}}=1,A \lt B$$ have same eccentricity $$\frac{1}{\sqrt{3}}$$. Let the product of their lengths of latus rectums be $$\frac{32}{\sqrt{3}}$$, and the distance between the foci of $$E_{1}$$ be 4. If $$E_{1}$$ and $$E_{2}$$ meet at $$A,B,C$$ and $$D,$$ then the area of the quadrilateral $$ABCD$$ equals:

Let $$A = [a_{ij}] = \begin{bmatrix}\log_{5}{128} & \log_{4}5 \\\log_{5}8 & \log_{4}25 \end{bmatrix}$$. If $$A_{ij}$$ is the cofactor of $$a_{ij},C_{jk} = \sum_{k=1}^{2}a_{ik}A_{ik},1 \leq i,j \leq 2$$,and $$C = [C_{ij}],$$ then $$8|C|$$ is equal to :

Let $$|z_{1}-8-2i| \leq 1$$ and $$|z_{2}-2+6i| \leq 2,z_{1},z_{2} \in C$$. Then the minimum value of $$|z_{1}-z_{2}|$$ is :

Let $$L_{1}: \frac{x-1}{1}=\frac{y-2}{-1}=\frac{z-1}{2}$$ and $$L_{2}: \frac{x+1}{-1}=\frac{y-2}{2}=\frac{z}{1}$$ be two lines. Let $$L_{3}$$ be a line passing through the point $$(\alpha ,\beta ,\gamma)$$ and be perpendicular to both $$L_{1}$$ and $$L_{2}$$. If $$L_{3}$$ intersects $$L_{1}$$, then $$|5\alpha -11\beta -8\gamma|$$ equals:

Let M and m respectively be the maximum and the minimum value of
$$f(x) =\begin{vmatrix}\mathbf{1+\sin^{2}x} & \mathbf{\cos^{2}x} & \mathbf{4\sin 4x} \\\mathbf{\sin^{2}x} &\mathbf{1+\cos^{2}x} & \mathbf{4\sin 4x} \\\mathbf{\sin^{2}x} &\mathbf{\cos^{2}x} & \mathbf{1+4\sin 4x}\end{vmatrix}$$, $$x \in R$$ then $$M^{4}-m^{4}$$ is equal to :

Let $$ABCD$$ be a triangle formed by the lines $$7x − 6y + 3 = 0, x + 2y − 31 = 0$$ and $$9x − 2y − 19 = 0.$$ Let the point $$(h,k)$$ be the image of the centroid of $$\triangle ABC$$ in the line $$3x + 6y − 53 = 0.$$ Then $$h^{2}+k^{2}+hk$$ is equal to :

The value of $$\lim_{n\rightarrow \infty}\left(\sum_{k=1}^{n}\frac{k^{3}+6k^{2}+11k+5}{(k+3)!}\right)$$ is:

The least value of n for which the number of integral terms in the Binomial expansion of $$(\sqrt[3]{7}+\sqrt[12]{11})^{n}$$ is 183, is :

Let $$y = y(x)$$ be the solution of the differential equation $$\cos x(\log_{e}(\cos x))^{2}dy + (\sin x-3y\sin x\log_{e}(\cos x))dx=0,x \in (0,\frac{\pi}{2})$$. if $$y\left(\frac{\pi}{4}\right) = \frac{-1}{\log_{e}2}$$, then $$y\left(\frac{\pi}{4}\right)$$ is equal to :

Let the line $$x + y = 1$$ meet the circle $$x^{2}+y^{2}=4$$ at the points A and B . If the line perpendicular to AB and passing through the mid point of the chord AB intersects the circle at $$C$$ and $$D$$, then the area of the quadrilateral ADBC is equal to :

Let the area of the region $$\left\{(x,y): 2y \leq x^{2}+3,y+|x| \leq 3,y \geq |x-1|\right\}$$ be A.Then 6 A is equal to :

Let $$S = \left\{x : \cos^{-1} x = \pi + \sin^{-1} x+\sin^{-1}(2x+1)\right\}$$. Then $$\sum_{x \in S}^{}(2x-1)^{2}$$ is equal to_________.

Let $$F : \left(0,\infty\right)\rightarrow R$$ be a twice differentiable function. If for some $$a \neq 0,\int_{0}^{1}f(\lambda x)d\lambda = af(x),f(1)=1$$ and $$f(16)=\frac{1}{8}$$, then $$16-f'\left(\frac{1}{16}\right)$$ is equal to _______.

The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is _______.

Let $$S = \left\{m \in Z : A^{m^{2}}+A^{m} = 3I - A^{-6}\right\}$$, where $$ A =\begin{bmatrix}2 & -1 \\1 & 0 \end{bmatrix}$$. Then n(S) is equal to ______.

Let [t] be the greatest integer less than or equal to t. Then the least value of $$p \in N$$ for which $$\lim_{x\rightarrow 0^{+}}\left(x([\frac{1}{x}]+[\frac{2}{x}]+...+[\frac{p}{x}])-x^{2}([\frac{1}{x^{2}}]+[\frac{2^{2}}{x^{2}}]+...+[\frac{9^{2}}{x^{2}}])\right) \geq 1$$ is equal to_______.