NTA JEE Main 24th June 2022 Shift 2

The sum of all real roots of equation $$\left(e^{2x} - 4\right)\left(6e^{2x} - 5e^x + 1\right) = 0$$ is

Let $$x, y > 0$$. If $$x^3 y^2 = 2^{15}$$, then the least value of $$3x + 2y$$ is

The number of solutions of the equation $$\cos\left(x + \frac{\pi}{3}\right) \cos\left(\frac{\pi}{3} - x\right) = \frac{1}{4}\cos^2 2x, x \in [-3\pi, 3\pi]$$ is:

Let the area of the triangle with vertices $$A(1, \alpha)$$, $$B(\alpha, 0)$$ and $$C(0, \alpha)$$ be $$4$$ sq. units. If the points $$(\alpha, -\alpha)$$, $$(-\alpha, \alpha)$$ and $$(\alpha^2, \beta)$$ are collinear, then $$\beta$$ is equal to

A particle is moving in the $$xy$$-plane along a curve $$C$$ passing through the point $$(3, 3)$$. The tangent to the curve $$C$$ at the point $$P$$ meets the $$x$$-axis at $$Q$$. If the $$y$$-axis bisects the segment $$PQ$$, then $$C$$ is a parabola with

Let the maximum area of the triangle that can be inscribed in the ellipse $$\frac{x^2}{a^2} + \frac{y^2}{4} = 1, a > 2$$, having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel to the $$y$$-axis, be $$6\sqrt{3}$$. Then the eccentricity of the ellipse is:

Consider the following statements:
$$A$$: Rishi is a judge.
$$B$$: Rishi is honest.
$$C$$: Rishi is not arrogant.
The negation of the statement "if Rishi is a judge and he is not arrogant, then he is honest" is

Let the system of linear equations
$$x + y + az = 2$$
$$3x + y + z = 4$$
$$x + 2z = 1$$
have a unique solution $$(x^*, y^*, z^*)$$. If $$((a, x^*), (y^*, \alpha)$$ and $$(x^*, -y^*)$$ are collinear points, then the sum of absolute values of all possible values of $$\alpha$$ is:

Let $$x \times y = x^2 + y^3$$ and $$(x \times 1) \times 1 = x \times (1 \times 1)$$. Then a value of $$2\sin^{-1}\left(\frac{x^4 + x^2 - 2}{x^4 + x^2 + 2}\right)$$ is

Let $$f(x) = \begin{cases} \frac{\sin(x-|x|)}{x-|x|}, & x \in (-2, -1) \\ \max(2x, 3[|x|]), & |x| < 1 \\ 1, & \text{otherwise} \end{cases}$$
where $$[t]$$ denotes greatest integer $$\leq t$$. If $$m$$ is the number of points where $$f$$ is not continuous and $$n$$ is the number of points where $$f$$ is not differentiable, the ordered pair $$(m, n)$$ is: