NTA JEE Main 1st February 2023 Shift 2

If the term without $$x$$ in the expansion of $$\left(x^{\frac{2}{3}} + \frac{\alpha}{x^3}\right)^{22}$$ is $$7315$$, then $$|\alpha|$$ is equal to ______.

Let the sixth term in the binomial expansion of $$\left(\sqrt{2^{\log_2(10-3^x)}} + \sqrt[5]{2^{(x-2)\log_2 3}}\right)^m$$ powers of $$2^{(x-2)\log_2 3}$$, be $$21$$. If the binomial coefficients of the second, third and fourth terms in the expansion are respectively the first, third and fifth terms of an A.P., then the sum of the squares of all possible values of $$x$$ is ______.

If the $$x$$-intercept of a focal chord of the parabola $$y^2 = 8x + 4y + 4$$ is $$3$$, then the length of this chord is equal to ______.

The line $$x = 8$$ is the directrix of the ellipse $$E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ with the corresponding focus $$(2, 0)$$. If the tangent to $$E$$ at the point $$P$$ in the first quadrant passes through the point $$(0, 4\sqrt{3})$$ and intersects the $$x$$-axis at $$Q$$, then $$(3PQ)^2$$ is equal to ______.

The value of the integral $$\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{x + \frac{\pi}{4}}{2 - \cos 2x} dx$$ is:

If $$\int_0^{\pi} \frac{5^{\cos x}(1+\cos x \cos 3x + \cos^2 x + \cos^3 x \cos 3x) dx}{1+5^{\cos x}} = \frac{k\pi}{16}$$, then $$k$$ is equal to ______.

Let $$\vec{a} = 2\hat{i} - 7\hat{j} + 5\hat{k}$$, $$\vec{b} = \hat{i} + \hat{k}$$ and $$\vec{c} = \hat{i} + 2\hat{j} - 3\hat{k}$$ be three given vectors. If $$\vec{r}$$ is a vector such that $$\vec{r} \times \vec{a} = \vec{c} \times \vec{a}$$ and $$\vec{r} \cdot \vec{b} = 0$$, then $$|\vec{r}|$$ is equal to:

Let the plane $$P$$ pass through the intersection of the planes $$2x + 3y - z = 2$$ and $$x + 2y + 3z = 6$$, and be perpendicular to the plane $$2x + y - z + 1 = 0$$. If $$d$$ is the distance of $$P$$ from the point $$(-7, 1, 1)$$, then $$d^2$$ is equal to:

Let $$\alpha x + \beta y + \gamma z = 1$$ be the equation of a plane passing through the point $$(3, -2, 5)$$ and perpendicular to the line joining the points $$(1, 2, 3)$$ and $$(-2, 3, 5)$$. Then the value of $$\alpha \beta y$$ is equal to ______.

The point of intersection $$C$$ of the plane $$8x + y + 2z = 0$$ and the line joining the points $$A(-3, -6, 1)$$ and $$B(2, 4, -3)$$ divides the line segment $$AB$$ internally in the ratio $$k : 1$$. If $$a, b, c$$ ($$|a|, |b|, |c|$$ are coprime) are the direction ratios of the perpendicular from the point $$C$$ on the line $$\frac{1-x}{1} = \frac{y+4}{2} = \frac{z+2}{3}$$, then $$|a + b + c|$$ is equal to ______.