NTA JEE Main 31st August 2021 Shift 2 - Mathematics

The sum of the roots of the equation, $$x+1-2\log_2\left(3+2^x\right)+2\log_4\left(10-2^{-x}\right)=0$$, is:

The number of solutions of the equation $$32^{\tan^2 x} + 32^{\sec^2 x} = 81$$, $$0 \leq x \leq \frac{\pi}{4}$$ is:

If $$z$$ is a complex number such that $$\frac{z-i}{z-1}$$ is purely imaginary, then the minimum value of $$|z - (3 + 3i)|$$ is:

Let $$a_1, a_2, a_3, \ldots$$ be an A.P. If $$\frac{a_1 + a_2 + \ldots + a_{10}}{a_1 + a_2 + \ldots + a_p} = \frac{100}{p^2}$$, $$p \neq 10$$, then $$\frac{a_{11}}{a_{10}}$$ is equal to:

Let $$A$$ be the set of all points $$\alpha, \beta$$ such that the area of triangle formed by the points $$(5, 6)$$, $$(3, 2)$$ and $$(\alpha, \beta)$$ is 12 square units. Then the least possible length of a line segment joining the origin to a point in $$A$$, is:

The locus of mid-points of the line segments joining -3, -5 and the points on the ellipse $$\frac{x^2}{4} + \frac{y^2}{9} = 1$$ is:

If $$\alpha = \lim_{x \to \pi/4} \frac{\tan^3 x - \tan x}{\cos x + \frac{\pi}{4}}$$ and $$\beta = \lim_{x \to 0} \cos x^{\cot x}$$ are the roots of the equation, $$ax^2 + bx - 4 = 0$$, then the ordered pair $$a, b$$ is:

Negation of the statement $$(p \vee r) \Rightarrow (q \vee r)$$ is:

The mean and variance of 7 observations are 8 and 16 respectively. If two observations are 6 and 8, then the variance of the remaining 5 observations is:

If $$\alpha + \beta + \gamma = 2\pi$$, then the system of equations
$$x + \cos\gamma y + \cos\beta z = 0$$
$$\cos\gamma x + y + \cos\alpha z = 0$$
$$\cos\beta x + \cos\alpha y + z = 0$$
has:

Let $$f : N \rightarrow N$$ be a function such that $$f(m+n) = f(m) + f(n)$$ for every $$m, n \in N$$. If $$f(6) = 18$$ then $$f(2) \cdot f(3)$$ is equal to:

The domain of the function, $$f(x) = \sin^{-1}\frac{3x^2+x-1}{(x-1)^2} + \cos^{-1}\frac{x-1}{x+1}$$ is:

An angle of intersection of the curves, $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ and $$x^2 + y^2 = ab$$, $$a \gt b$$, is:

Let $$f$$ be any continuous function on $$[0, 2]$$ and twice differentiable on $$(0, 2)$$. If $$f(0) = 0$$, $$f(1) = 1$$ and $$f(2) = 2$$, then:

If $$x$$ is the greatest integer $$\leq x$$, then $$\pi^2 \int_0^2 \sin\frac{\pi x}{2} x - x^{[x]} dx$$ is equal to:

If $$y\frac{dy}{dx} = x\frac{y^2}{x^2} + \frac{\phi\frac{y^2}{x^2}}{\phi'\frac{y^2}{x^2}}$$, $$x > 0$$, $$\phi > 0$$, and $$y(1) = -1$$, then $$\phi\frac{y^2}{4}$$ is equal to:

If $$\frac{dy}{dx} = \frac{2^x y + 2^y \cdot 2^x}{2^x + 2^x + y\log_e 2}$$, $$y(0) = 0$$, then for $$y = 1$$, the value of $$x$$ lies in the interval:

Let $$\vec{a}$$, $$\vec{b}$$, $$\vec{c}$$ be three vectors mutually perpendicular to each other and have same magnitude. If a vector $$\vec{r}$$ satisfies $$\vec{a} \times \{\vec{r} - \vec{b} \times \vec{a}\} + \vec{b} \times \{\vec{r} - \vec{c} \times \vec{b}\} + \vec{c} \times \{\vec{r} - \vec{a} \times \vec{c}\} = \vec{0}$$, then $$\vec{r}$$ is equal to:

The distance of the point $$(-1, 2, -2)$$ from the line of intersection of the planes $$2x + 3y + 2z = 0$$ and $$x - 2y + z = 0$$ is:

Let $$S = \{1, 2, 3, 4, 5, 6\}$$. Then the probability that a randomly chosen onto function $$g$$ from $$S$$ to $$S$$ satisfies $$g(3) = 2g(1)$$ is:

The number of 4-digit numbers which are neither multiple of 7 nor multiple of 3 is _________.

If $$S = \frac{7}{5} + \frac{9}{5^2} + \frac{13}{5^3} + \frac{19}{5^4} + \ldots$$, then $$160 S$$ is equal to _________.

If the coefficient of $$a^7 b^8$$ in the expansion of $$(a + 2b + 4ab)^{10}$$ is $$K \cdot 2^{16}$$, then $$K$$ is equal to _________.

Let $$B$$ be the centre of the circle $$x^2 + y^2 - 2x + 4y + 1 = 0$$. Let the tangents at two points $$P$$ and $$Q$$ on the circle intersect at the point $$A(3, 1)$$. Then $$8 \cdot \frac{\text{area } \triangle APQ}{\text{area } \triangle BPQ}$$ is equal to _________.

A tangent line $$L$$ is drawn at the point $$(2, -4)$$ on the parabola $$y^2 = 8x$$. If the line $$L$$ is also tangent to the circle $$x^2 + y^2 = a$$, then $$a$$ is equal to _________.

The number of elements in the set $$\{A = \begin{pmatrix} a & b \\ 0 & d \end{pmatrix} : a, b, d \in \{-1, 0, 1\}$$ and $$(I - A)^3 = I - A^3\}$$, where $$I$$ is $$2 \times 2$$ identity matrix, is _________.

Let $$f(x)$$ be a cubic polynomial with $$f(1) = -10$$, $$f(-1) = 6$$, and has a local minima at $$x = 1$$, and $$f'(x)$$ has a local minima at $$x = -1$$. Then $$f(3)$$ is equal to _________.

If $$\int \frac{\sin x}{\sin^3 x + \cos^3 x} dx = \alpha\log_e |1 + \tan x| + \beta\log_e|1 - \tan x + \tan^2 x| + \gamma\tan^{-1}\frac{2\tan x - 1}{\sqrt{3}} + C$$, when $$C$$ is constant of integration, then the value of $$18\left(\alpha+\beta+\gamma^2\right)$$ is _________.

If the line $$y = mx$$ bisects the area enclosed by the lines $$x = 0$$, $$y = 0$$, $$x = \frac{3}{2}$$ and the curve $$y = 1 + 4x - x^2$$, then $$12m$$ is equal to _________.

Suppose the line $$\frac{x-2}{\alpha} = \frac{y-2}{-5} = \frac{z+2}{2}$$ lies on the plane $$x + 3y - 2z + \beta = 0$$. Then $$(\alpha + \beta)$$ is equal to _________.