NTA JEE Mains 23rd Jan 2025 Shift 1

Let $$f(x)=\log_{e}x$$ and $$g(x)=\frac{x^{4}-2x^{3}+3x^{2}-2x+2}{2x^{2}-2x+1}$$. Then the domain of $$f \circ g$$ is

If the system of equations $$(\lambda-1)x+(\lambda-4)y+\lambda z=5 \\\lambda x+(\lambda-1)y+(\lambda-4)z=7 \\ (\lambda+1)x+(\lambda+2)y-(\lambda+2)z=9$$ has infinitely many solutions, then $$\lambda^{2}+\lambda$$ is equal to

The number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the vowels never come together, is

Let $$R=\left\{(1,2),(2,3),(3,3)\right\}$$ be a relation defined on the set $$\left\{1,2,3,4\right\}$$. Then the minimum number of elements, needed to be added in R so that R becomes an equivalence relation, is:

Let the area of a $$\triangle PQR$$ with vertices P(5,4), Q(-2,4) and R(a,b) be 35 square units. If its orthocenter and centroid are $$O(2,\frac{14}{5})$$ and C(c,d) respectively, then c+2d is equal to

The value of $$\int_{e^{2}}^{e^{4}} \frac{1}{x}\left(\frac{e^{((log_{e}x)^{2}+1)^{-1}}}{e^{((log_{e}x)^{2}+1)^{-1}}+e^{((6-\log_{e}x)^{2}+1)^{-1}}}\right)dx$$ is

Let $$\mid\frac{\overline{z}-i}{2\overline{z}+i}\mid=\frac{1}{3}, z\in C$$, be the equation of a circle with center at C. If the area of the triangle, whose vertices are at the points (0,0), C and $$(\alpha,0)$$ is 11 square units, then $$\alpha^{2}$$ equals:

The value of $$\left(\sin 70^{\circ}\right)\left(\cot 10^{\circ}\cot 70^{\circ}-1\right)$$ is

Let $$I(x)=\int_{}^{} \frac{dx}{(x-11)^{\frac{11}{13}}(x+15)^{\frac{15}{13}}}$$. If $$I(37)-I(24)=\frac{1}{4}\left(\frac{1}{b^{\frac{1}{13}}}-\frac{1}{c^{\frac{1}{13}}}\right),b,c\in N$$, then 3(b+c) is equal to

If $$\frac{\pi}{2}\leq x\leq \frac{3\pi}{4}$$, then $$\cos^{-1}\left(\frac{12}{13}\cos x+\frac{5}{13}\sin x\right)$$ is equal to