NTA JEE Mains 5th April 2024 Shift 1

Consider the following two statements : Statement I : For any two non-zero complex numbers $$z_1, z_2$$, $$(|z_1| + |z_2|)\left|\frac{z_1}{|z_1|} + \frac{z_2}{|z_2|}\right| \leq 2(|z_1| + |z_2|)$$. Statement II : If $$x, y, z$$ are three distinct complex numbers and $$a, b, c$$ are three positive real numbers such that $$\frac{a}{|y-z|} = \frac{b}{|z-x|} = \frac{c}{|x-y|}$$, then $$\frac{a^2}{y-z} + \frac{b^2}{z-x} + \frac{c^2}{x-y} = 1$$. Between the above two statements,

If $$\frac{1}{\sqrt{1}+\sqrt{2}} + \frac{1}{\sqrt{2}+\sqrt{3}} + \ldots + \frac{1}{\sqrt{99}+\sqrt{100}} = m$$ and $$\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \ldots + \frac{1}{99 \cdot 100} = n$$, then the point $$(m, n)$$ lies on the line

Suppose $$\theta \in [0, \frac{\pi}{4}]$$ is a solution of $$4\cos\theta - 3\sin\theta = 1$$. Then $$\cos\theta$$ is equal to :

Let two straight lines drawn from the origin $$O$$ intersect the line $$3x + 4y = 12$$ at the points $$P$$ and $$Q$$ such that $$\triangle OPQ$$ is an isosceles triangle and $$\angle POQ = 90°$$. If $$l = OP^2 + PQ^2 + QO^2$$, then the greatest integer less than or equal to $$l$$ is :

If $$A(1, -1, 2), B(5, 7, -6), C(3, 4, -10)$$ and $$D(-1, -4, -2)$$ are the vertices of a quadrilateral $$ABCD$$, then its area is :

Let a circle $$C$$ of radius 1 and closer to the origin be such that the lines passing through the point $$(3, 2)$$ and parallel to the coordinate axes touch it. Then the shortest distance of the circle $$C$$ from the point $$(5, 5)$$ is :

Let the line $$2x + 3y - k = 0, k > 0$$, intersect the $$x$$-axis and $$y$$-axis at the points $$A$$ and $$B$$, respectively. If the equation of the circle having the line segment $$AB$$ as a diameter is $$x^2 + y^2 - 3x - 2y = 0$$ and the length of the latus rectum of the ellipse $$x^2 + 9y^2 = k^2$$ is $$\frac{m}{n}$$, where $$m$$ and $$n$$ are coprime, then $$2m + n$$ is equal to

Let $$A$$ and $$B$$ be two square matrices of order 3 such that $$|A| = 3$$ and $$|B| = 2$$. Then $$|A^T A(\text{adj}(2A))^{-1}(\text{adj}(4B))(\text{adj}(AB))^{-1}AA^T|$$ is equal to :

If the system of equations $$11x + y + \lambda z = -5$$, $$2x + 3y + 5z = 3$$, $$8x - 19y - 39z = \mu$$ has infinitely many solutions, then $$\lambda^4 - \mu$$ is equal to :

Let $$A = \{1, 3, 7, 9, 11\}$$ and $$B = \{2, 4, 5, 7, 8, 10, 12\}$$. Then the total number of one-one maps $$f : A \to B$$, such that $$f(1) + f(3) = 14$$, is :