NTA JEE Mains 5th April 2024 Shift 1 - Mathematics

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 :

Let $$f(x) = x^5 + 2x^3 + 3x + 1, x \in \mathbb{R}$$, and $$g(x)$$ be a function such that $$g(f(x)) = x$$ for all $$x \in \mathbb{R}$$. Then $$\frac{g(7)}{g'(7)}$$ is equal to :

If the function $$f(x) = \frac{\sin 3x + \alpha \sin x - \beta \cos 3x}{x^3}, x \in \mathbb{R}$$, is continuous at $$x = 0$$, then $$f(0)$$ is equal to :

Let a rectangle $$ABCD$$ of sides 2 and 4 be inscribed in another rectangle $$PQRS$$ such that the vertices of the rectangle $$ABCD$$ lie on the sides of the rectangle $$PQRS$$. Let $$a$$ and $$b$$ be the sides of the rectangle $$PQRS$$ when its area is maximum. Then $$(a + b)^2$$ is equal to :

For the function $$f(x) = \sin x + 3x - \frac{2}{\pi}(x^2 + x)$$, where $$x \in [0, \frac{\pi}{2}]$$, consider the following two statements : (I) $$f$$ is increasing in $$(0, \frac{\pi}{2})$$. (II) $$f'$$ is decreasing in $$(0, \frac{\pi}{2})$$. Between the above two statements,

The value of $$\int_{-\pi}^{\pi} \frac{2y(1+\sin y)}{1+\cos^2 y} dy$$ is :

The integral $$\int_0^{\pi/4} \frac{136\sin x}{3\sin x + 5\cos x} dx$$ is equal to :

If $$y = y(x)$$ is the solution of the differential equation $$\frac{dy}{dx} + 2y = \sin(2x), y(0) = \frac{3}{4}$$, then $$y\left(\frac{\pi}{8}\right)$$ is equal to:

If the line $$\frac{2-x}{3} = \frac{3y-2}{4\lambda+1} = 4-z$$ makes a right angle with the line $$\frac{x+3}{3\mu} = \frac{1-2y}{6} = \frac{5-z}{7}$$, then $$4\lambda + 9\mu$$ is equal to :

Let $$d$$ be the distance of the point of intersection of the lines $$\frac{x+6}{3} = \frac{y}{2} = \frac{z+1}{1}$$ and $$\frac{x-7}{4} = \frac{y-9}{3} = \frac{z-4}{2}$$ from the point $$(7, 8, 9)$$. Then $$d^2 + 6$$ is equal to :

The coefficients $$a, b, c$$ in the quadratic equation $$ax^2 + bx + c = 0$$ are chosen from the set $$\{1, 2, 3, 4, 5, 6, 7, 8\}$$. The probability of this equation having repeated roots is :

The number of ways of getting a sum 16 on throwing a dice four times is ______

Let $$a_1, a_2, a_3, \ldots$$ be in an arithmetic progression of positive terms. Let $$A_k = a_1^2 - a_2^2 + a_3^2 - a_4^2 + \ldots + a_{2k-1}^2 - a_{2k}^2$$. If $$A_3 = -153, A_5 = -435$$ and $$a_1^2 + a_2^2 + a_3^2 = 66$$, then $$a_{17} - A_7$$ is equal to ______

If the constant term in the expansion of $$\left(1 + 2x - 3x^3\right)\left(\frac{3}{2}x^2 - \frac{1}{3x}\right)^9$$ is $$p$$, then $$108p$$ is equal to ______

Suppose $$AB$$ is a focal chord of the parabola $$y^2 = 12x$$ of length $$l$$ and slope $$m < \sqrt{3}$$. If the distance of the chord $$AB$$ from the origin is $$d$$, then $$ld^2$$ is equal to ______

Let $$f$$ be a differentiable function in the interval $$(0, \infty)$$ such that $$f(1) = 1$$ and $$\lim_{t \to x} \frac{t^2 f(x) - x^2 f(t)}{t - x} = 1$$ for each $$x > 0$$. Then $$2f(2) + 3f(3)$$ is equal to ______

From a lot of 10 items, which include 3 defective items, a sample of 5 items is drawn at random. Let the random variable $$X$$ denote the number of defective items in the sample. If the variance of $$X$$ is $$\sigma^2$$, then $$96\sigma^2$$ is equal to ______

The number of distinct real roots of the equation $$|x||x + 2| - 5|x + 1| - 1 = 0$$ is ______

If $$S = \{a \in \mathbb{R} : |2a - 1| = 3[a] + 2\{a\}\}$$, where $$[t]$$ denotes the greatest integer less than or equal to $$t$$ and $$\{t\}$$ represents the fractional part of $$t$$, then $$72\sum_{a \in S} a$$ is equal to ______

The area of the region enclosed by the parabolas $$y = x^2 - 5x$$ and $$y = 7x - x^2$$ is ______

Let $$\vec{a} = \hat{i} - 3\hat{j} + 7\hat{k}, \vec{b} = 2\hat{i} - \hat{j} + \hat{k}$$ and $$\vec{c}$$ be a vector such that $$(\vec{a} + 2\vec{b}) \times \vec{c} = 3(\vec{c} \times \vec{a})$$. If $$\vec{a} \cdot \vec{c} = 130$$, then $$\vec{b} \cdot \vec{c}$$ is equal to ______