NTA JEE Main 28th June 2022 Shift 1

The number of real solutions of the equation $$e^{4x} + 4e^{3x} - 58e^{2x} + 4e^x + 1 = 0$$ is ______

The number of elements in the set $$\{z = a + ib \in C : a, b \in \mathbb{Z}$$ and $$1 < |z - 3 + 2i| < 4\}$$ is ______

The number of positive integers $$k$$ such that the constant term in the binomial expansion of $$\left(2x^3 + \frac{3}{x^k}\right)^{12}, x \neq 0$$ is $$2^8 \cdot l$$, where $$l$$ is an odd integer, is ______

A ray of light passing through the point $$P(2, 3)$$ reflects on the $$X$$-axis at point $$A$$ and the reflected ray passes through the point $$Q(5, 4)$$. Let $$R$$ be the point that divides the line segment $$AQ$$ internally into the ratio $$2 : 1$$. Let the co-ordinates of the foot of the perpendicular $$M$$ from $$R$$ on the bisector of the angle $$PAQ$$ be $$(\alpha, \beta)$$. Then, the value of $$7\alpha + 3\beta$$ is equal to ______

Let the lines $$y + 2x = \sqrt{11} + 7\sqrt{7}$$ and $$2y + x = 2\sqrt{11} + 6\sqrt{7}$$ be normal to a circle $$C : (x-h)^2 + (y-k)^2 = r^2$$. If the line $$\sqrt{11}y - 3x = \frac{5\sqrt{77}}{3} + 11$$ is tangent to the circle $$C$$, then the value of $$(5h - 8k)^2 + 5r^2$$ is equal to ______

The mean and standard deviation of 15 observations are found to be $$8$$ and $$3$$ respectively. On rechecking it was found that, in the observations, $$20$$ was misread as $$5$$. Then, the correct variance is equal to ______

Let $$R_1$$ and $$R_2$$ be relations on the set $$\{1, 2, \ldots, 50\}$$ such that $$R_1 = \{(p, p^n) : p$$ is a prime and $$n \geq 0$$ is an integer$$\}$$ and $$R_2 = \{(p, p^n) : p$$ is a prime and $$n = 0$$ or $$1\}$$. Then, the number of elements in $$R_1 - R_2$$ is ______

Let $$A = \{1, a_1, a_2, \ldots a_{18}, 77\}$$ be a set of integers with $$1 < a_1 < a_2 < \ldots < a_{18} < 77$$. Let the set $$A + A = \{x + y : x, y \in A\}$$ contain exactly $$39$$ elements. Then, the value of $$a_1 + a_2 + \ldots + a_{18}$$ is equal to ______

Let $$l$$ be a line which is normal to the curve $$y = 2x^2 + x + 2$$ at a point $$P$$ on the curve. If the point $$Q(6, 4)$$ lies on the line $$l$$ and $$O$$ is origin, then the area of the triangle $$OPQ$$ is equal to ______

If $$\vec{a} = 2\hat{i} + \hat{j} + 3\hat{k}, \vec{b} = 3\hat{i} + 3\hat{j} + \hat{k}$$ and $$\vec{c} = c_1\hat{i} + c_2\hat{j} + c_3\hat{k}$$ are coplanar vectors and $$\vec{a} \cdot \vec{c} = 5, \vec{b} \perp \vec{c}$$, then $$122(c_1 + c_2 + c_3)$$ is equal to ______