NTA JEE Mains 21st Jan 2026 Shift 1

$$6\int_{0}^{\pi} |(\sin3x+\sin2x+\sin x)|dx$$ is equal to _________.

Let $$a_{1}=1$$ and for $$n\geq1,a_{n+1}=\frac{1}{2}a_{n}+\frac{n^{2}-2n-1}{n^{2}(n+1)^2}$$. Then $$\left|\sum_{ n=1}^{ \infty}\left( a_n - \frac{2}{n^2}\right)\right|$$ is equal to ______.

Let S= {(m, n) :m, n $$\epsilon$$ {1, 2, 3, .... , 50}}. lf the number of elements (m, n) in S such that $$6^m+9^n$$ is a multiple of 5 is p and the number of elements (m, n) in S such that m + n is a square of a prime number is q, then p +q is equal to ________.

Let $$f:R\rightarrow R$$ be a twice differentiable function such that the quadratic equation $$f(x)m^{2}-2 f'(x)m+ f''(x)=0$$ in m, has two equal roots for every $$x \epsilon R$$. If $$ f(0)=1,f'(0)=2$$, and $$(\alpha,\beta)$$ is the largest interval in which the function $$f(\log_{e}{x-x})$$ is increasing, then $$\alpha+\beta$$ is equal to ________.

For some $$\alpha,\beta\epsilon R$$, let $$A=\begin{bmatrix}\alpha &  2 \\ 1 &  2 \end{bmatrix}\text{ and }B=\begin{bmatrix}1 &  1 \\1 &   \beta \end{bmatrix}$$ be such that $$A^{2}-4A+2I=B^2-3B+I=0$$. Then $$(det(adj(A^3-B^3)))^2$$ is equal to _______.

Two strings (A, B) having linear densities $$\mu_{A}=2\times10^{-4}kg/m\text{ and },\mu_{B}=4\times10^{-4}kg/m$$ and lengths $$L_{A}=2.5m$$ and $$L_{B}=1.5m$$ respectively are joined. Free ends of A and B are tied to two rigid supports C and D, respectively creating a tension of 500 N in the wire. Two identical pulses, sent from C and D ends, take time $$t_{1}\text{ and } t_{2}$$, respectively, to reach the joint. The ratio $$t_{1}/ t_{2}$$ is :

A gas based geyser heats water flowing at the rate of 5.0 litres per minute from 27°C to 87°C. The rate of consumption of the gas is ___ g/s. (take heat of combustion of $$gas=5.0\times10^{4}J/g$$) specific heat capacity of water =4200 J/ kg.°C

A 1 m long metal rod AB completes the circuit as shown in figure. The area of circuit is perpendicular to the magnetic field of 0.10 T. lf the resistance of the total circuit is 2Ω then the force needed to move the rod towards right with constant speed (v) of 1.5 m/s is ___ N

Consider a modified Bernoulli equation.
$$\left( P + \frac{A}{B t^2} \right) + \rho g (h + B t) + \frac{1}{2}\,\rho V^2 = \text{constant}$$
If t has tile dimension of time then the dimensions of A and B are ___.___ respectively.

In a double slit experiment the distance between the slits is 0.1 cm and the screen is placed at 50 cm from the slits plane. When one slit is covered with a transparent sheet having thickness t and refractive index n(=1.5), the central fringe shifts by 0.2 cm. The value of t is____ cm