NTA JEE Main 24th June 2022 Shift 1

In an examination, there are $$5$$ multiple choice questions with $$3$$ choices, out of which exactly one is correct. There are $$3$$ marks for each correct answer, $$-2$$ marks for each wrong answer and $$0$$ mark if the question is not attempted. Then, the number of ways a student appearing in the examination gets $$5$$ marks is ______

Let $$A\left(\frac{3}{\sqrt{a}}, \sqrt{a}\right), a > 0$$, be a fixed point in the $$xy$$-plane. The image of $$A$$ in $$y$$-axis be $$B$$ and the image of $$B$$ in $$x$$-axis be $$C$$. If $$D(3\cos\theta, a\sin\theta)$$, is a point in the fourth quadrant such that the maximum area of $$\triangle ACD$$ is $$12$$ square units, then $$a$$ is equal to ______

If two tangents drawn from a point $$(\alpha, \beta)$$ lying on the ellipse $$25x^2 + 4y^2 = 1$$ to the parabola $$y^2 = 4x$$ are such that the slope of one tangent is four times the other, then the value of $$(10\alpha + 5)^2 + (16\beta^2 + 50)^2$$ equals ______

The number of one-one functions $$f : \{a, b, c, d\} \to \{0, 1, 2, \ldots, 10\}$$ such that $$2f(a) - f(b) + 3f(c) + f(d) = 0$$ is ______

The number of points where the function
$$f(x) = \begin{cases} |2x^2 - 3x - 7| & \text{if } x \leqslant -1 \\ [4x^2 - 1] & \text{if } -1 < x < 1 \\ |x+1| + |x-2| & \text{if } x \geqslant 1 \end{cases}$$
where $$[t]$$ denotes the greatest integer $$\leqslant t$$, is discontinuous is ______

If $$f(\theta) = \sin\theta + \int_{-\pi/2}^{\pi/2} (\sin\theta + t\cos\theta) \cdot f(t)\,dt$$, then $$\left|\int_0^{\pi/2} f(\theta)\,d\theta\right|$$ is ______

Let $$\underset{0 \leqslant x \leqslant 2}{\text{Max}}\left\{\frac{9-x^2}{5-x}\right\} = \alpha$$ and $$\underset{0 \leqslant x \leqslant 2}{\text{Min}}\left\{\frac{9-x^2}{5-x}\right\} = \beta$$. If $$\int_{\beta - 8/3}^{2\alpha - 1} \text{Max}\left\{\frac{9-x^2}{5-x}, x\right\}dx = \alpha_1 + \alpha_2 \log_e\left(\frac{8}{15}\right)$$, then $$\alpha_1 + \alpha_2$$ is equal to ______

Let $$S$$ be the region bounded by the curves $$y = x^3$$ and $$y^2 = x$$. The curve $$y = 2|x|$$ divides $$S$$ into two regions of areas $$R_1$$ and $$R_2$$. If $$ |R_1, R_2 | = R_2$$, then $$\frac{R_2}{R_1}$$ is equal to ______

Let a line having direction ratios $$1, -4, 2$$ intersect the lines $$\frac{x-7}{3} = \frac{y-1}{-1} = \frac{z+2}{1}$$ and $$\frac{x}{2} = \frac{y-7}{3} = \frac{z}{1}$$ at the points $$A$$ and $$B$$. Then $$(AB)^2$$ is equal to ______

If the shortest distance between the lines $$\vec{r} = (-\hat{i} + 3\hat{k}) + \lambda(\hat{i} - a\hat{j})$$ and $$\vec{r} = (-\hat{j} + 2\hat{k}) + \mu(\hat{i} - \hat{j} + \hat{k})$$ is $$\sqrt{\frac{2}{3}}$$, then the integral value of $$a$$ is equal to ______