NTA JEE Main 31st January 2023 Shift 1 - Mathematics

The number of real roots of the equation $$\sqrt{x^2 - 4x + 3} + \sqrt{x^2 - 9} = \sqrt{4x^2 - 14x + 6}$$, is:

For all $$z \in C$$ on the curve $$C_1$$: $$|z| = 4$$, let the locus of the point $$z + \dfrac{1}{z}$$ be the curve $$C_2$$. Then

If the sum and product of four positive consecutive terms of a G.P., are 126 and 1296, respectively, then the sum of common ratios of all such GPs is

Let a circle $$C_1$$ be obtained on rolling the circle $$x^2 + y^2 - 4x - 6y + 11 = 0$$ upwards 4 units on the tangent T to it at the point (3, 2). Let $$C_2$$ be the image of $$C_1$$ in T. Let $$A$$ and $$B$$ be the centers of circles $$C_1$$ and $$C_2$$ respectively, and $$M$$ and $$N$$ be respectively the feet of perpendiculars drawn from $$A$$ and $$B$$ on the x-axis. Then the area of the trapezium AMNB is:

If the maximum distance of normal to the ellipse $$\dfrac{x^2}{4} + \dfrac{y^2}{b^2} = 1, b < 2$$, from the origin is 1, then the eccentricity of the ellipse is:

Consider:
S1: $$p \Rightarrow q \lor p \land \sim q$$ is a tautology.
S2: $$\sim p \Rightarrow \sim q \land \sim p \lor q$$ is a contradiction.
Then

Let $$R$$ be a relation on $$N \times N$$ defined by $$a, b R c, d$$ if and only if $$abd - c = bca - d$$. Then $$R$$ is

Let $$A = \begin{matrix} 1 & 0 & 0 \\ 0 & 4 & -1 \\ 0 & 12 & -3 \end{matrix}$$. Then the sum of the diagonal elements of the matrix $$A + I^{11}$$ is equal to:

For the system of linear equations
$$x + y + z = 6$$
$$\alpha x + \beta y + 7z = 3$$
$$x + 2y + 3z = 14$$
which of the following is NOT true?

If $$\sin^{-1}\dfrac{\alpha}{17} + \cos^{-1}\dfrac{4}{5} - \tan^{-1}\dfrac{77}{36} = 0$$, $$0 < \alpha < 13$$, then $$\sin^{-1}\sin\alpha + \cos^{-1}\cos\alpha$$ is equal to

Let $$y = fx$$ represent a parabola with focus $$(-\dfrac{1}{2}, 0)$$ and directrix $$y = -\dfrac{1}{2}$$. Then
$$S = \{x \in \mathbb{R}: \tan^{-1}(\sqrt{fx}) + \sin^{-1}(\sqrt{fx+1}) = \dfrac{\pi}{2}\}$$:

If the domain of the function $$f(x) = \dfrac{x}{1 + x^2}$$, where $$x$$ is greatest integer $$\le x$$, is $$[2, 6)$$, then its range is

Let $$y = f x = \sin^3\dfrac{\pi}{3}\cos\dfrac{\pi}{3\sqrt{2}} - 4x^3 + 5x^2 + 1^{\frac{3}{2}}$$. Then, at $$x = 1$$,

A wire of length 20 m is to be cut into two pieces. A piece of length $$\ell_1$$ is bent to make a square of area $$A_1$$ and the other piece of length $$\ell_2$$ is made into a circle of area $$A_2$$. If $$2A_1 + 3A_2$$ is minimum then $$\pi\ell_1 : \ell_2$$ is equal to:

Let $$\alpha \in (0, 1)$$ and $$\beta = \log_e(1 - \alpha)$$. Let $$P_n(x) = x + \dfrac{x^2}{2} + \dfrac{x^3}{3} + \ldots + \dfrac{x^n}{n}$$, $$x \in (0, 1)$$. Then the integral $$\int_0^{\alpha} \dfrac{t^{50}}{1-t} dt$$ is equal to

The value of $$\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \dfrac{2 + 3\sin x}{\sin x(1 + \cos x)} dx$$ is equal to

Let a differentiable function $$f$$ satisfy $$f(x) + \int_3^x \dfrac{f(t)}{t} dt = \sqrt{x+1}$$, $$x \ge 3$$. Then $$12f(8)$$ is equal to:

Let $$\vec{a} = 2\hat{i} + \hat{j} + \hat{k}$$, and $$\vec{b}$$ and $$\vec{c}$$ be two nonzero vectors such that $$\vec{a} + \vec{b} + \vec{c} = \vec{a} + \vec{b} - \vec{c}$$ and $$\vec{b} \cdot \vec{c} = 0$$. Consider the following two statements:
A: $$\vec{a} + \lambda\vec{c} \ge \vec{a}$$ for all $$\lambda \in \mathbb{R}$$.
B: $$\vec{a}$$ and $$\vec{c}$$ are always parallel.

Let the shortest distance between the lines L: $$\dfrac{x-5}{-2} = \dfrac{y-\lambda}{0} = \dfrac{z+\lambda}{1}$$, $$\lambda \ge 0$$ and L$$_1$$: $$x+1 = y-1 = 4-z$$ be $$2\sqrt{6}$$. If $$(\alpha, \beta, \gamma)$$ lies on L, then which of the following is NOT possible?

A bag contains 6 balls. Two balls are drawn from it at random and both are found to be black. The probability that the bag contains at least 5 black balls is

Let 5 digit numbers be constructed using the digits 0, 2, 3, 4, 7, 9 with repetition allowed, and are arranged in ascending order with serial numbers. Then the serial number of the number 42923 is ______.

Let $$a_1, a_2, \ldots, a_n$$ be in A.P. If $$a_5 = 2a_7$$ and $$a_{11} = 18$$, then $$12\left(\dfrac{1}{\sqrt{a_{10}} + \sqrt{a_{11}}} + \dfrac{1}{\sqrt{a_{11}} + \sqrt{a_{12}}} + \ldots + \dfrac{1}{\sqrt{a_{17}} + \sqrt{a_{18}}}\right)$$ is equal to ______.

Number of 4-digit numbers that are less than or equal to 2800 and either divisible by 3 or by 11, is equal to ______.

Let $$\alpha > 0$$, be the smallest number such that the expansion of $$x^{\frac{2}{3}} + \dfrac{2}{x^3}^{30}$$ has a term $$\beta x^{-\alpha}$$, $$\beta \in N$$. Then $$\alpha$$ is equal to ______.

The remainder on dividing $$5^{99}$$ by 11 is ______.

If the variance of the frequency distribution

is 3, then $$\alpha$$ is equal to ______.

Let for $$x \in R$$, $$f(x) = \dfrac{x+x}{2}$$ and $$g(x) = \begin{cases} x, & x < 0 \\ x^2, & x \ge 0 \end{cases}$$. Then area bounded by the curve $$y = f \circ g(x)$$ and the lines $$y = 0, 2y - x = 15$$ is equal to ______.

Let $$\vec{a}$$ and $$\vec{b}$$ be two vectors such that $$\vec{a} = \sqrt{14}$$, $$\vec{b} = \sqrt{6}$$ and $$\vec{a} \times \vec{b} = \sqrt{48}$$. Then $$(\vec{a} \cdot \vec{b})^2$$ is equal to ______.

Let the line $$L$$: $$\dfrac{x-1}{2} = \dfrac{y+1}{-1} = \dfrac{z-3}{1}$$ intersect the plane $$2x + y + 3z = 16$$ at the point $$P$$. Let the point $$Q$$ be the foot of perpendicular from the point $$R(1, -1, -3)$$ on the line $$L$$. If $$\alpha$$ is the area of triangle $$PQR$$, then $$\alpha^2$$ is equal to ______.

Let $$\theta$$ be the angle between the planes $$P_1 = \vec{r} \cdot (\hat{i} + \hat{j} + 2\hat{k}) = 9$$ and $$P_2 = \vec{r} \cdot (2\hat{i} - \hat{j} + \hat{k}) = 15$$. Let L be the line that meets $$P_2$$ at the point (4, -2, 5) and makes angle $$\theta$$ with the normal of $$P_2$$. If $$\alpha$$ is the angle between L and $$P_2$$ then $$\tan^2\theta \cot^2\alpha$$ is equal to ______.