NTA JEE Mains 23rd Jan 2025 Shift 2

The distance of the line $$\frac{x-2}{2}=\frac{y-6}{3}=\frac{z-3}{4}$$ from the point (1, 4, 0) along the line $$\frac{x}{1}=\frac{y-2}{2}=\frac{z+3}{3}$$ is :

Let $$A={(x,y) \in R\times R : |x+y|\geq 3}$$ and $$B={(x,y) \in R\times R : |x|+|y|\leq 3}$$. If $$C={(x,y) \in A ∩ B : x=0 \text{ or }y=0},then \sum_{(x,y)\in C}^{}|x+y|$$ is :

Let $$X=R\times R$$ Define a relation R on X as : $$(a_{1},b_{1})R(a_{2},b_{2}) \Leftrightarrow b_{1}=b_{2}$$ Statement I : R is an equivalence relation. Statement II : For some $$(a,b) \in X$$, the set $$S={(x,y) \in X : (x,y)R(a,b)}$$ represents a line parallel to y=x In the light of the above statements, choose the correct answer from the options given below :

Let $$\int_{}^{} x^{3}\sin x dx =g(x)+C$$, where is the constant of integration. If $$8(g(\frac{\pi}{2})+g'(\frac{\pi}{2}))=\alpha \pi^{3} + \beta \pi^{2} + \gamma ,\alpha ,\beta ,\gamma \in Z$$, then $$\alpha +\beta - \gamma$$ equals :

A rod of length eight units moves such that its ends A and B always lie on the lines x - y + 2=0 and y + 2 = 0. respectively. If the locus of the point P, that divides the rod AB internally in the ratio 2:1 is $$9(x^{2}+\alpha y^{2}+\beta xy+\gamma x+ 28y)-76=0$$. then $$\alpha -\beta -\gamma$$ equals to :

If the square of the shortest distance between the lines $$frac{x-2}{1}=\frac{y-1}{2}=\frac{z+3}{-3}$$ and $$\frac{x+1}{2}=\frac{y+3}{4}=\frac{x+5}{-5}\text{ is }\frac{m}{n}$$, where m, n are coprime numbers, then m + n is equals to:

$$\lim_{x \rightarrow \infty}\frac{(2x^{2}-3x+5)(3x-1)^{\frac{x}{2}}}{(3x^{2}+5x+4)\sqrt{(3x+2)^{2}}}$$ is equals to :

Let the point A divide the line segment joining the points P(−1,−1, 2) and Q(5, 5, 10) internally in the ratio $$r : 1 (r > 0)$$. If O is the origin and $$(\overrightarrow{OQ}.\overrightarrow{OA})-\frac{1}{5}|\overrightarrow{OP}.\overrightarrow{OA}|^{2}=10$$. then the value of r is :

The length of the chord of the ellipse $$\frac{x^{2}}{4}+\frac{y^{2}}{2}=1$$, whose mid-point is $$(1,\frac{1}{2})$$, is :

The system of equations $$x+y+z=6\\x+2y+5z=9,\\x+5y+\lambda z=\mu,$$ has no solution if