NTA JEE Main 29th July 2022 Shift 1

Let $$S = \{4, 6, 9\}$$ and $$T = \{9, 10, 11, \ldots, 1000\}$$. If $$A = \{a_1 + a_2 + \ldots + a_k : k \in \mathbb{N}, a_1, a_2, a_3, \ldots, a_k \in S\}$$, then the sum of all the elements in the set $$T - A$$ is equal to _______

Let $$a_1, a_2, a_3, \ldots$$ be an A.P. If $$\sum_{r=1}^{\infty} \frac{a_r}{2^r} = 4$$, then $$4a_2$$ is equal to ______

If $$\frac{1}{2 \times 3 \times 4} + \frac{1}{3 \times 4 \times 5} + \frac{1}{4 \times 5 \times 6} + \ldots + \frac{1}{100 \times 101 \times 102} = \frac{k}{101}$$, then $$34k$$ is equal to _______

Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of $$\left(\sqrt[4]{2} + \frac{1}{\sqrt[4]{3}}\right)^n$$, in the increasing powers of $$\frac{1}{\sqrt[4]{3}}$$ be $$\sqrt[4]{6} : 1$$. If the sixth term from the beginning is $$\frac{\alpha}{\sqrt[4]{3}}$$, then $$\alpha$$ is equal to _______

Let $$S = \{\theta \in (0, 2\pi) : 7\cos^2\theta - 3\sin^2\theta - 2\cos^2(2\theta) = 2\}$$. Then the sum of roots of all the equations $$x^2 - 2(\tan^2\theta + \cot^2\theta)x + 6\sin^2\theta = 0$$, $$\theta \in S$$, is _______

Let the mirror image of a circle $$c_1: x^2 + y^2 - 2x - 6y + \alpha = 0$$ in line $$y = x + 1$$ be $$c_2: 5x^2 + 5y^2 + 10gx + 10fy + 38 = 0$$. If r is the radius of circle $$c_2$$, then $$\alpha + 6r^2$$ is equal to ______

Let the mean and the variance of 20 observations $$x_1, x_2, \ldots, x_{20}$$ be 15 and 9, respectively. For $$\alpha \in \mathbb{R}$$, if the mean of $$(x_1 + \alpha)^2, (x_2 + \alpha)^2, \ldots, (x_{20} + \alpha)^2$$ is 178, then the square of the maximum value of $$\alpha$$ is equal to _______

The number of matrices of order $$3 \times 3$$, whose entries are either 0 or 1 and the sum of all the entries is a prime number, is _______

Let p and p+2 be prime numbers and let $$\Delta = \begin{vmatrix} p! & (p+1)! & (p+2)! \\ (p+1)! & (p+2)! & (p+3)! \\ (p+2)! & (p+3)! & (p+4)! \end{vmatrix}$$
Then the sum of the maximum values of $$\alpha$$ and $$\beta$$, such that $$p^\alpha$$ and $$(p+2)^\beta$$ divide $$\Delta$$, is _______

Let a line with direction ratios $$a, -4a, -7$$ be perpendicular to the lines with direction ratios $$3, -1, 2b$$ and $$b, a, -2$$. If the point of intersection of the line $$\frac{x+1}{a^2+b^2} = \frac{y-2}{a^2-b^2} = \frac{z}{1}$$ and the plane $$x - y + z = 0$$ is $$(\alpha, \beta, \gamma)$$, then $$\alpha + \beta + \gamma$$ is equal to ________