NTA JEE Mains 8th April Shift 2 2026

The sum of squares of all the real solutions of the equation $$\log_{(x+1)}(2x^2 + 5x + 3) = 4 - \log_{(2x+3)}(x^2 + 2x + 1)$$ is equal to __________.

If $$\int_{\pi/6}^{\pi/4} \left(\cot\left(x - \frac{\pi}{3}\right)\cot\left(x + \frac{\pi}{3}\right) + 1\right) dx = \alpha \log_e(\sqrt{3} - 1)$$, then $$9\alpha^2$$ is equal to __________.

Let a line $$L_1$$ pass through the origin and be perpendicular to the lines
$$L_2 : \vec{r} = (3+t)\hat{i} + (2t-1)\hat{j} + (2t+4)\hat{k}$$ and
$$L_3 : \vec{r} = (3+2s)\hat{i} + (3+2s)\hat{j} + (2+s)\hat{k}$$, $$t, s \in \mathbf{R}$$.
If $$(a, b, c)$$, $$a \in \mathbb{Z}$$, is the point on $$L_3$$ at a distance of $$\sqrt{17}$$ from the point of intersection of $$L_1$$ and $$L_2$$, then $$(a + b + c)^2$$ is equal to __________.

Consider the circle $$C : x^2 + y^2 - 6x - 8y - 11 = 0$$. Let a variable chord AB of the circle C subtend a right angle at the origin. If the locus of the foot of the perpendicular drawn from the origin on the chord AB is the circle $$x^2 + y^2 - \alpha x - \beta y - \gamma = 0$$, then $$\alpha + \beta + 2\gamma$$ is equal to __________.

Let $$f$$ be a polynomial function such that $$\log_2(f(x)) = \left\lfloor \log_2\left(2 + \frac{2}{3} + \frac{2}{9} + \ldots \infty\right) \right\rfloor \cdot \log_3\left(1 + \frac{f(x)}{f\left(\frac{1}{x}\right)}\right)$$, $$x > 0$$ and $$f(6) = 37$$. Then $$\sum_{n=1}^{10} f(n)$$ is equal to __________.

A new unit ($$\alpha$$) of length is chosen such that it is equal to the speed of light in vacuum. What is the distance between Venus and Earth in terms of $$\alpha$$ units if light takes 6 min. 40 s to cover this distance?

Consider the equation $$H = \frac{x^p \epsilon^q E^r}{t^s}$$
Where $$H$$ = magnetic field; $$E$$ = electric field, $$\epsilon$$ = permittivity, $$x$$ = distance, $$t$$ = time. The values of $$p, q, r$$ and $$s$$ respectively are :

A car moving with a speed of 54 km/h takes a turn of radius 20 m. A simple pendulum is suspended from the ceiling of the car. Determine the angle made by the string of the pendulum with the vertical during the turning. (Take $$g = 10$$ m/s$$^2$$)

A gas balloon is going up with a constant velocity of 10 m/s. When this balloon reached a height of 75 m, a stone is dropped from it and balloon keeps moving up with the same velocity. The height of the balloon when the stone hits the ground is __________ m. (Take $$g = 10$$ m/s$$^2$$)

A thin biconvex lens is prepared from the glass ($$\mu = 1.5$$) both curved surfaces of which have equal radii of 20 cm each. Left side surface of the lens is silvered from outside to make it reflecting. To have the position of image and object at the same place, the object should be placed, from the lens at a distance of __________ cm.