NTA JEE Main 11th April 2023 Shift 2 - Mathematics

The number of points, where the curve $$f(x) = e^{8x} - e^{6x} - 3e^{4x} - e^{2x} + 1$$, $$x \in \mathbb{R}$$ cuts $$x$$-axis, is equal to

For $$a \in \mathbb{C}$$, let $$A = \{z \in \mathbb{C}: \text{Re}(a + \bar{z}) > \text{Im}(\bar{a} + z)\}$$ and $$B = \{z \in \mathbb{C}: \text{Re}(a + \bar{z}) < \text{Im}(\bar{a} + z)\}$$. Then among the two statements:
$$(S1)$$: If $$\text{Re}(a), \text{Im}(a) > 0$$, then the set $$A$$ contains all the real numbers
$$(S2)$$: If $$\text{Re}(a), \text{Im}(a) < 0$$, then the set $$B$$ contains all the real numbers,

If the letters of the word MATHS are permuted and all possible words so formed are arranged as in a dictionary with serial numbers, then the serial number of the word THAMS is

Let $$a, b, c$$ and $$d$$ be positive real numbers such that $$a + b + c + d = 11$$. If the maximum value of $$a^5b^3c^2d$$ is $$3750\beta$$, then the value of $$\beta$$ is

For $$k \in \mathbb{N}$$, if the sum of the series $$1 + \frac{4}{k} + \frac{8}{k^2} + \frac{13}{k^3} + \frac{19}{k^4} + \ldots$$ is 10, then the value of $$k$$ is

If the 1011$$^{th}$$ term from the end in the binomial expansion of $$\left(\frac{4x}{5} - \frac{5}{2x}\right)^{2022}$$ is 1024 times 1011$$^{th}$$ term from the beginning, then $$32|x|$$ is equal to

The sum of the coefficients of three consecutive terms in the binomial expansion of $$(1 + x)^{n+2}$$, which are in the ratio 1 : 3 : 5, is equal to

The converse of $$((-p) \wedge q) \Rightarrow r$$ is

Let the mean of 6 observations 1, 2, 4, 5, $$x$$ and $$y$$ be 5 and their variance be 10. Then their mean deviation about the mean is equal to

The angle of elevation of the top $$P$$ of a tower from the feet of one person standing due south of the tower is 45$$^\circ$$ and from the feet of another person standing due west of the tower is 30$$^\circ$$. If the height of the tower is 5 meters, then the distance (in meters) between the two persons is equal to

Let $$A = \{1, 3, 4, 6, 9\}$$ and $$B = \{2, 4, 5, 8, 10\}$$. Let $$R$$ be a relation defined on $$A \times B$$ such that $$R = \{(a_1, b_1), (a_2, b_2): a_1 \leq b_2 \text{ and } b_1 \leq a_2\}$$. Then the number of elements in the set $$R$$ is

If the system of linear equations
$$7x + 11y + \alpha z = 13$$
$$5x + 4y + 7z = \beta$$
$$175x + 194y + 57z = 361$$
has infinitely many solutions, then $$\alpha + \beta + 2$$ is equal to

If $$\begin{vmatrix} x+1 & x & x \\ x & x+\lambda & x \\ x & x & x+\lambda^2 \end{vmatrix} = \frac{9}{8}(103x + 81)$$, then $$\lambda$$, $$\frac{\lambda}{3}$$ are the roots of the equation

The domain of the function $$f(x) = \frac{1}{\sqrt{[x]^2 - 3[x] - 10}}$$ is (where $$[x]$$ denotes the greatest integer less than or equal to $$x$$)

Let $$f$$ and $$g$$ be two functions defined by $$f(x) = \begin{cases} x + 1, & x < 0 \\ |x - 1|, & x \geq 0 \end{cases}$$ and $$g(x) = \begin{cases} x + 1, & x < 0 \\ 1, & x \geq 0 \end{cases}$$. Then $$(g \circ f)(x)$$ is

Let the function $$f: [0, 2] \to \mathbb{R}$$ be defined as $$f(x) = \begin{cases} e^{\min\{x^2, x-[x]\}}, & x \in [0, 1) \\ e^{[x - \log_e x]}, & x \in [1, 2] \end{cases}$$, where $$[t]$$ denotes the greatest integer less than or equal to $$t$$. Then the value of the integral $$\int_0^2 xf(x)dx$$ is

Let $$y = y(x)$$ be the solution of the differential equation $$\frac{dy}{dx} + \frac{5}{x(x^5+1)}y = \frac{(x^5+1)^2}{x^7}$$, $$x > 0$$. If $$y(1) = 2$$, then $$y(2)$$ is equal to

If four distinct points with position vectors $$\vec{a}, \vec{b}, \vec{c}$$ and $$\vec{d}$$ are coplanar, then $$[\vec{a}\vec{b}\vec{c}]$$ is equal to

Let $$P$$ be the plane passing through the points $$(5, 3, 0)$$, $$(13, 3, -2)$$ and $$(1, 6, 2)$$. For $$\alpha \in \mathbb{N}$$, if the distance of the points $$A(3, 4, \alpha)$$ and $$B(2, \alpha, a)$$ from the plane $$P$$ are 2 and 3 respectively, then the positive value of a is

Let $$S = \{z \in \mathbb{C} - \{i, 2i\}: \frac{z^2 + 8iz - 15}{z^2 - 3iz - 2} \in \mathbb{R}\}$$. $$\alpha - \frac{13}{11}i \in S$$, $$\alpha \in \mathbb{R} - \{0\}$$, then $$242\alpha^2$$ is equal to _______

If the line $$l_1: 3y - 2x = 3$$ is the angular bisector of the lines $$l_2: x - y + 1 = 0$$ and $$l_3: \alpha x + \beta y + 17 = 0$$, then $$\alpha^2 + \beta^2 - \alpha - \beta$$ is equal to _______

If the radius of the largest circle with centre (2, 0) inscribed in the ellipse $$x^2 + 4y^2 = 36$$ is $$r$$, then $$12r^2$$ is equal to _______

Let the tangent to the parabola $$y^2 = 12x$$ at the point $$(3, \alpha)$$ be perpendicular to the line $$2x + 2y = 3$$. Then the square of distance of the point $$(6, -4)$$ from the normal to the hyperbola $$\alpha^2x^2 - 9y^2 = 9\alpha^2$$ at its point $$(\alpha - 1, \alpha + 2)$$ is equal to _______

Let $$A = \{1, 2, 3, 4, 5\}$$ and $$B = \{1, 2, 3, 4, 5, 6\}$$. Then the number of functions $$f: A \to B$$ satisfying $$f(1) + f(2) = f(4) - 1$$ is equal to _______

If $$f: \mathbb{R} \to \mathbb{R}$$ be a continuous function satisfying $$\int_0^{\frac{\pi}{2}} f(\sin 2x) \sin x \, dx + \alpha \int_0^{\frac{\pi}{4}} f(\cos 2x) \cos x \, dx = 0$$, then the value of $$\alpha$$ is _______

If $$A$$ is the area in the first quadrant enclosed by the curve $$C: 2x^2 - y + 1 = 0$$, the tangent to $$C$$ at the point $$(1, 3)$$ and the line $$x + y = 1$$, then the value of $$60A$$ is _______

Let $$\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$$ and $$\vec{b} = \hat{i} + \hat{j} - \hat{k}$$. If $$\vec{c}$$ is a vector such that $$\vec{a} \cdot \vec{c} = 11$$, $$\vec{b} \cdot (\vec{a} \times \vec{c}) = 27$$ and $$\vec{b} \cdot \vec{c} = -\sqrt{3}|\vec{b}|$$, then $$|\vec{a} \times \vec{c}|^2$$ is equal to _______

Let the line passing through the points $$P(2, -1, 2)$$ and $$Q(5, 3, 4)$$ meet the plane $$x - y + z = 4$$ at the point $$R$$. Then the distance of the point $$R$$ from the plane $$x + 2y + 3z + 2 = 0$$ measured parallel to the line $$\frac{x-7}{2} = \frac{y+3}{2} = \frac{z-2}{1}$$ is _______

Let the line $$L: x = \frac{1-y}{-2} = \frac{z-3}{\lambda}$$, $$\lambda \in \mathbb{R}$$ meet the plane $$P: x + 2y + 3z = 4$$ at the point $$(\alpha, \beta, \gamma)$$. If the angle between the line $$L$$ and the plane $$P$$ is $$\cos^{-1}\left(\sqrt{\frac{5}{14}}\right)$$, then $$\alpha + 2\beta + 6\gamma$$ is equal to _______

Let the probability of getting head for a biased coin be $$\frac{1}{4}$$. It is tossed repeatedly until a head appears. Let $$N$$ be the number of tosses required. If the probability that the equation $$64x^2 + 5Nx + 1 = 0$$ has no real root is $$\frac{p}{q}$$, where $$p$$ and $$q$$ are co-prime, then $$q - p$$ is equal to _______