NTA JEE Mains 28th Jan 2025 Shift 1

Let $$f(x) = \left\{\begin{array}{l l}3x & \quad {x<0}\\min\left\{1+x+[x],x+2[x]\right\}, & \quad {0\leq x\leq 2}\\ 5, & \quad {x>2,} \end{array}\right.$$ where [.] denotes greatest integer function. If $$\alpha$$ and $$\beta$$ are the number of points, where f is not continuous and is not differentiable, respectively, then $$\alpha +\beta$$ equals_________

Let M denote the set of all real matrices of order $$3\times 3$$ and let$$S=\left\{-3,-2,-1,1,2\right\}$$. Let
$$S_{1}=\left\{A=[a_{ij}] \in M : A=A^{T}\text{ and }a_{ij} \in S,\forall i,j\right\},$$
$$S_{2}=\left\{A=[a_{ij}] \in M : A=-A^{T}\text{ and }a_{ij} \in S,\forall i,j\right\},$$
$$S_{3}=\left\{A=[a_{ij}] \in M : a_{11}+a_{22}+a_{33}=0\text{ and }a_{ij} \in S,\forall i,j\right\},$$
If $$n(S_{1}\cup_{2} US_{3})=125\alpha$$, then $$alpha$$ equals___________

If $$\alpha =1+\sum_{r=1}^{6}(-3)^{r-1}$$ $$^{12}C_{2r-1}$$, then the distance of the point $$(12,\sqrt{3})$$ from the line $$\alpha x-\sqrt{3}y+1$$ is________

Let $$E_{1}:\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$$ be an ellipse. Ellipses $$E_{1}$$'s are constructed such that their centres and eccentricities are same as that of $$E_{1}$$, and the length of minor axis of $$E_{i}$$ is the length of major axis of $$E_{i+1}(i \geq 1)$$. If $$A_{i}$$ is the area of the ellipse $$E_{i}$$ then $$\frac{5}{\pi}\left(\sum_{i=1}^{\infty}A_{i}\right)$$, is equal to

Let $$\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k},\overrightarrow{b}=2\hat{i}+2\hat{j}+\hat{k}$$ and $$\overrightarrow{d}=\overrightarrow{a}\times \overrightarrow{b}$$. If$$\overrightarrow{c}$$ is a vector such that $$\overrightarrow{a}. \overrightarrow{c}=|\overrightarrow{c}|,|\overrightarrow{c}-2\overrightarrow{a}|^{2}=8$$ and the angle between $$\overrightarrow{d}$$ and $$\overrightarrow{c}$$ is $$\frac{\pi}{4}$$, then $$|10-3\overrightarrow{b}.\overrightarrow{c}|+|\overrightarrow{d}\times \overrightarrow{c}|^{2}$$ is equal to

Three infinitely long wires with linear charge density $$\lambda$$ are placed along the x-axis, y-axis and z-axis respectively. Which of the following denotes an equipotential surface?

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R Assertion A:A sound wave has higher speed in solids than gases. Reason R: Gases have higher value of Bulk modulus than solids. In the light of the above statements, choose the correct answer from the options given below

Two capacitors $$C_{1}$$ and $$C_{2}$$ are connected in parallel to a battery. Charge-time graph is shown below for the two capacitors. The energy stored with them are $$U_{1}$$ and $$U_{2}$$, respectively. Which of the given statements is true?

A thin prism $$P_{1}$$ with angle $$4^{\circ}$$ made of glass having refractive index 1.54 , is combined with another thin prism $$P_{2}$$ made of glass having refractive index 1.72 to get dispersion without deviation. The angle of the prism $$P_{2}$$ in degrees is

A particle of mass 'm' and charge 'q'is fastened to one end 'A' of a massless string having equilibrium length l, whose other end is fixed at point 'O'. The whole system is placed on a frictionless horizontal plane and is initially at rest. If uniform electric field is switched on along the direction as shown in figure, then the speed of the particle when it crosses the x -axis is