NTA JEE Main 29th July 2022 Shift 1

If $$z = 2 + 3i$$, then $$z^5 + \bar{z}^5$$ is equal to:

If $$\frac{1}{(20-a)(40-a)} + \frac{1}{(40-a)(60-a)} + \ldots + \frac{1}{(180-a)(200-a)} = \frac{1}{256}$$, then the maximum value of $$a$$ is

Let the circumcentre of a triangle with vertices $$A(a, 3)$$, $$B(b, 5)$$ and $$C(a, b)$$, $$ab > 0$$ be $$P(1, 1)$$. If the line AP intersects the line BC at the point $$Q(k_1, k_2)$$, then $$k_1 + k_2$$ is equal to

Let a line L pass through the point of intersection of the lines $$bx + 10y - 8 = 0$$ and $$2x - 3y = 0$$, $$b \in \mathbb{R} - \{\frac{4}{3}\}$$. If the line L also passes through the point (1, 1) and touches the circle $$17(x^2 + y^2) = 16$$, then the eccentricity of the ellipse $$\frac{x^2}{5} + \frac{y^2}{b^2} = 1$$ is

Let the focal chord of the parabola $$P: y^2 = 4x$$ along the line $$L: y = mx + c, m > 0$$ meet the parabola at the points M and N. Let the line L be a tangent to the hyperbola $$H: x^2 - y^2 = 4$$. If O is the vertex of P and F is the focus of H on the positive x-axis, then the area of the quadrilateral OMFN is

If $$\lim_{x \to 0} \frac{\alpha e^x + \beta e^{-x} + \gamma \sin x}{x \sin^2 x} = \frac{2}{3}$$, where $$\alpha, \beta, \gamma \in \mathbb{R}$$, then which of the following is NOT correct?

The statement $$(p \wedge q) \Rightarrow (p \wedge r)$$ is equivalent to

The angle of elevation of the top of a tower from a point A due north of it is $$\alpha$$ and from a point B at a distance of 9 units due west of A is $$\cos^{-1}\left(\frac{3}{\sqrt{13}}\right)$$. If the distance of the point B from the tower is 15 units, then $$\cot\alpha$$ is equal to

Let R be a relation from the set $$\{1, 2, 3, \ldots, 60\}$$ to itself such that $$R = \{(a, b) : b = pq$$, where $$p, q \geq 3$$ are prime numbers$$\}$$. Then, the number of elements in R is

Let A and B be two $$3 \times 3$$ non-zero real matrices such that AB is a zero matrix. Then