NTA JEE Main 28th June 2022 Shift 2

Sum of squares of modulus of all the complex numbers $$z$$ satisfying $$\bar{z} = iz^2 + z^2 - z$$ is equal to

Let for $$n = 1, 2, \ldots, 50$$, $$S_n$$ be the sum of the infinite geometric progression whose first term is $$n^2$$ and whose common ratio is $$\frac{1}{(n+1)^2}$$. Then the value of $$\frac{1}{26} + \sum_{n=1}^{50} \left(S_n + \frac{2}{n+1} - n - 1\right)$$ is equal to

If one of the diameters of the circle $$x^2 + y^2 - 2\sqrt{2}x - 6\sqrt{2}y + 14 = 0$$ is a chord of the circle $$(x - 2\sqrt{2})^2 + (y - 2\sqrt{2})^2 = r^2$$, then the value of $$r^2$$ is equal to

If $$\lim_{x \to 1} \left(\frac{\sin(3x^2 - 4x + 1) - x^2 + 1}{2x^3 - 7x^2 + ax + b}\right) = -2$$, then the value of $$(a - b)$$ is equal to

The maximum number of compound propositions, out of $$p \vee r \vee s$$, $$p \vee r \vee \sim s$$, $$p \vee \sim q \vee s$$, $$\sim p \vee \sim r \vee s$$, $$\sim p \vee \sim r \vee \sim s$$, $$\sim p \vee q \vee \sim s$$, $$q \vee r \vee \sim s$$, $$q \vee \sim r \vee \sim s$$, $$\sim p \vee \sim q \vee \sim s$$ that can be made simultaneously true by an assignment of the truth values to $$p, q, r$$ and $$s$$, is equal to

Suppose a class has 7 students. The average marks of these students in the mathematics examination is 62, and their variance is 20. A student fails in the examination if he/she gets less than 50 marks, then in worst case, the number of students can fail is

Let $$A = \begin{pmatrix} 1+i & 1 \\ -i & 0 \end{pmatrix}$$ where $$i = \sqrt{-1}$$. Then, the number of elements in the set $$\{n \in \{1, 2, \ldots, 100\} : A^n = A\}$$ is

If the system of linear equations
$$2x - 3y = \gamma + 5$$
$$\alpha x + 5y = \beta + 1$$,
where $$\alpha, \beta, \gamma \in \mathbf{R}$$ has infinitely many solutions, then the value of $$|9\alpha + 3\beta + 5\gamma|$$ is equal to

Let $$S = \{1, 2, 3, 4\}$$. Then the number of elements in the set $$\{f : S \times S \to S : f$$ is onto and $$f(a,b) = f(b,a) \geq a \forall (a,b) \in S \times S\}$$ is

Let the image of the point $$P(1, 2, 3)$$ in the line $$L : \frac{x-6}{3} = \frac{y-1}{2} = \frac{z-2}{3}$$ be $$Q$$. Let $$R(\alpha, \beta, \gamma)$$ be a point that divides internally the line segment $$PQ$$ in the ratio 1 : 3. Then the value of $$22(\alpha + \beta + \gamma)$$ is equal to