NTA JEE Mains 29th Jan 2025 Shift 1

Let $$A = [a_{ij}] = \begin{bmatrix}\log_{e}{128} & \log_{4}5 \\\log_{5}8 & \log_{4}25 \end{bmatrix}$$. 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 :