NTA JEE Main 27th June 2022 Shift 1 - Mathematics

The area of the polygon, whose vertices are the non-real roots of the equation $$\bar{z} = iz^2$$ is

If $$x = \sum_{n=0}^{\infty} a^n, y = \sum_{n=0}^{\infty} b^n, z = \sum_{n=0}^{\infty} c^n$$, where $$a, b, c$$ are in A.P. and $$|a| < 1, |b| < 1, |c| < 1, abc \neq 0$$, then

The value of $$\cos\left(\frac{2\pi}{7}\right) + \cos\left(\frac{4\pi}{7}\right) + \cos\left(\frac{6\pi}{7}\right)$$ is equal to

In an isosceles triangle $$ABC$$, the vertex $$A$$ is $$(6, 1)$$ and the equation of the base $$BC$$ is $$2x + y = 4$$. Let the point $$B$$ lie on the line $$x + 3y = 7$$. If $$(\alpha, \beta)$$ is the centroid of $$\triangle ABC$$, then $$15(\alpha + \beta)$$ is equal to

Let the eccentricity of an ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b$$, be $$\frac{1}{4}$$. If this ellipse passes through the point $$\left(-4\sqrt{\frac{2}{5}}, 3\right)$$, then $$a^2 + b^2$$ is equal to

Let $$a$$ be an integer such that $$\lim_{x \to 7} \frac{18 - [1-x]}{[x-3a]}$$ exists, where $$[t]$$ is greatest integer $$\leq t$$. Then $$a$$ is equal to

The boolean expression $$(\sim(p \wedge q)) \vee q$$ is equivalent to

Let the system of linear equations $$x + 2y + z = 2, \alpha x + 3y - z = \alpha, -\alpha x + y + 2z = -\alpha$$ be inconsistent. Then $$\alpha$$ is equal to

$$\sin^{-1}\left(\sin\frac{2\pi}{3}\right) + \cos^{-1}\left(\cos\frac{7\pi}{6}\right) + \tan^{-1}\left(\tan\frac{3\pi}{4}\right)$$ is equal to

If $$\cos^{-1}\left(\frac{y}{2}\right) = \log_e\left(\frac{x}{5}\right)^5, |y| < 2$$, then

The number of distinct real roots of $$x^4 - 4x + 1 = 0$$ is

The lengths of the sides of a triangle are $$10 + x^2, 10 + x^2$$ and $$20 - 2x^2$$. If for $$x = k$$, the area of the triangle is maximum, then $$3k^2$$ is equal to

$$\int \frac{(x^2+1)e^x}{(x+1)^2} dx = f(x)e^x + C$$, where $$C$$ is a constant, then $$\frac{d^3f}{dx^3}$$ at $$x = 1$$ is equal to

The value of the integral $$\int_{-2}^{2} \frac{|x^3+x|}{(e^{x|x|}+1)} dx$$ is equal to

Let $$\frac{dy}{dx} = \frac{ax - by + a}{bx + cy + a}$$, where $$a, b, c$$ are constants, represent a circle passing through the point $$(2, 5)$$. Then the shortest distance of the point $$(11, 6)$$ from this circle is

If $$\frac{dy}{dx} + \frac{2^x y(2y-1)}{2^x - 1} = 0, x, y > 0, y(1) = 1$$, then $$y(2)$$ is equal to

Let $$\vec{a} = \hat{i} + \hat{j} - \hat{k}$$ and $$\vec{c} = 2\hat{i} - 3\hat{j} + 2\hat{k}$$. Then the number of vectors $$\vec{b}$$ such that $$\vec{b} \times \vec{c} = \vec{a}$$ and $$|\vec{b}| \in \{1, 2, \ldots, 10\}$$ is

If two straight lines whose direction cosines are given by the relations $$l + m - n = 0, 3l^2 + m^2 + cnl = 0$$ are parallel, then the positive value of $$c$$ is

Five numbers $$x_1, x_2, x_3, x_4, x_5$$ are randomly selected from the numbers $$1, 2, 3, \ldots, 18$$ and are arranged in the increasing order $$(x_1 < x_2 < x_1 < x_4 < x_2)$$. The probability that $$x_2 = 7$$ and $$x_4 = 11$$ is

Let $$X$$ be a random variable having binomial distribution $$B(7, p)$$. If $$P(X = 3) = 5P(X = 4)$$, then the sum of the mean and the variance of $$X$$ is

The number of ways, 16 identical cubes, of which 11 are blue and rest are red, can be placed in a row so that between any two red cubes there should be at least 2 blue cubes, is ______

If the sum of the first ten terms of the series $$\frac{1}{5} + \frac{2}{65} + \frac{3}{325} + \frac{4}{1025} + \frac{5}{2501} + \ldots$$ is $$\frac{m}{n}$$, where $$m$$ and $$n$$ are co-prime numbers, then $$m + n$$ is equal to ______

If the coefficient of $$x^{10}$$ in the binomial expansion of $$\left(\frac{\sqrt{x}}{5^{1/4}} + \frac{\sqrt{5}}{x^{1/3}}\right)^{60}$$ is $$5^k l$$, where $$l, k \in N$$ and $$l$$ is coprime to $$5$$, then $$k$$ is equal to ______

A rectangle $$R$$ with end points of the one of its sides as $$(1, 2)$$ and $$(3, 6)$$ is inscribed in a circle. If the equation of a diameter of the circle is $$2x - y + 4 = 0$$, then the area of $$R$$ is ______

A circle of radius $$2$$ unit passes through the vertex and the focus of the parabola $$y^2 = 2x$$ and touches the parabola $$y = \left(x - \frac{1}{4}\right)^2 + \alpha$$, where $$\alpha > 0$$. Then $$(4\alpha - 8)^2$$ is equal to ______

The positive value of the determinant of the matrix $$A$$, whose $$Adj(Adj(A)) = \begin{bmatrix} 14 & 28 & -14 \\ -14 & 14 & 28 \\ 28 & -14 & 14 \end{bmatrix}$$, is ______

Let $$f : R \to R$$ be a function defined $$f(x) = \frac{2e^{2x}}{e^{2x}+e}$$. Then $$f\left(\frac{1}{100}\right) + f\left(\frac{2}{100}\right) + f\left(\frac{3}{100}\right) + \ldots + f\left(\frac{99}{100}\right)$$ is equal to ______

If the sum of all the roots of the equation $$e^{2x} - 11e^x - 45e^{-x} + \frac{81}{2} = 0$$ is $$\log_e P$$, then $$P$$ is equal to ______

Let $$A_1 = \{(x,y) : |x| \leq y^2, |x| + 2y \leq 8\}$$ and $$A_2 = \{(x,y) : |x| + |y| \leq k\}$$. If $$27$$ (Area $$A_1$$) $$= 5$$ (Area $$A_2$$), then $$k$$ is equal to ______

Let the mirror image of the point $$(a, b, c)$$ with respect to the plane $$3x - 4y + 12z + 19 = 0$$ be $$(a - 6, \beta, \gamma)$$. If $$a + b + c = 5$$, then $$7\beta - 9\gamma$$ is equal to ______