NTA JEE Main 20th July 2021 Shift 2 - Mathematics

If the real part of the complex number $$(1 - \cos\theta + 2i\sin\theta)^{-1}$$ is $$\frac{1}{5}$$ for $$\theta \in (0, \pi)$$, then the value of the integral $$\int_0^\theta \sin x \, dx$$ is equal to:

If sum of the first 21 terms of the series $$\log_{9^{1/2}} x + \log_{9^{1/3}} x + \log_{9^{1/4}} x + \ldots$$ where $$x > 0$$ is 504, then $$x$$ is equal to:

For the natural numbers $$m$$, $$n$$, if $$(1-y)^m(1+y)^n = 1 + a_1 y + a_2 y^2 + \ldots + a_{m+n}y^{m+n}$$ and $$a_1 = a_2 = 10$$, then the value of $$(m+n)$$ is equal to:

Let $$r_1$$ and $$r_2$$ be the radii of the largest and smallest circles, respectively, which pass through the point $$(-4, 1)$$ and having their centres on the circumference of the circle $$x^2 + y^2 + 2x + 4y - 4 = 0$$. If $$\frac{r_1}{r_2} = a + b\sqrt{2}$$, then $$a + b$$ is equal to:

Let $$P$$ be a variable point on the parabola $$y = 4x^2 + 1$$. Then, the locus of the mid-point of the point $$P$$ and the foot of the perpendicular drawn from the point $$P$$ to the line $$y = x$$ is:

Consider the following three statements:
(A) If $$3 + 3 = 7$$ then $$4 + 3 = 8$$
(B) If $$5 + 3 = 8$$ then earth is flat.
(C) If both (A) and (B) are true then $$5 + 6 = 17$$.
Then, which of the following statements is correct?

If the mean and variance of six observations 7, 10, 11, 15, $$a$$, $$b$$ are 10 and $$\frac{20}{3}$$, respectively, then the value of $$|a - b|$$ is equal to:

Let in a right angled triangle, the smallest angle be $$\theta$$. If a triangle formed by taking the reciprocal of its sides is also a right angled triangle, then $$\sin \theta$$ is equal to:

The value of $$k \in R$$, for which the following system of linear equations
$$3x - y + 4z = 3$$
$$x + 2y - 3z = -2$$
$$6x + 5y + kz = -3$$
has infinitely many solutions, is:

The value of $$\tan\left(2\tan^{-1}\left(\frac{3}{5}\right) + \sin^{-1}\left(\frac{5}{13}\right)\right)$$ is equal to:

Let $$f : R - \{\frac{\alpha}{6}\} \to R$$ be defined by $$f(x) = \left(\frac{5x+3}{6x-\alpha}\right)$$. Then the value of $$\alpha$$ for which $$(f \circ f)(x) = x$$, for all $$x \in R - \{\frac{\alpha}{6}\}$$, is:

The sum of all the local minimum values of the twice differentiable function $$f : R \to R$$ defined by $$f(x) = x^3 - 3x^2 - \frac{3f''(2)}{2}x + f''(1)$$ is:

If $$[x]$$ denotes the greatest integer less than or equal to $$x$$, then the value of the integral $$\int_{-\pi/2}^{\pi/2} [x] - \sin x] dx$$ is equal to:

If $$f : R \to R$$ is given by $$f(x) = x + 1$$, then the value of
$$\lim_{n \to \infty} \frac{1}{n}\left[f(0) + f\left(\frac{5}{n}\right) + f\left(\frac{10}{n}\right) + \ldots + f\left(\frac{5(n-1)}{n}\right)\right]$$ is:

Let $$g(t) = \int_{-\pi/2}^{\pi/2} (\cos \frac{\pi}{4}t + f(x))dx$$, where $$f(x) = \log_e(x + \sqrt{x^2+1})$$, $$x \in R$$. Then which one of the following is correct?

Let $$y = y(x)$$ satisfies the equation $$\frac{dy}{dx} - |A| = 0$$, for all $$x > 0$$, where $$A = \begin{bmatrix} y & \sin x & 1 \\ 0 & -1 & 1 \\ 2 & 0 & \frac{1}{x} \end{bmatrix}$$. If $$y(\pi) = \pi + 2$$, then the value of $$y\left(\frac{\pi}{2}\right)$$ is:

In a triangle $$ABC$$, if $$|\overrightarrow{BC}| = 3$$, $$|\overrightarrow{CA}| = 5$$ and $$|\overrightarrow{BA}| = 7$$, then the projection of the vector $$\overrightarrow{BA}$$ on $$\overrightarrow{BC}$$ is equal to:

The lines $$x = ay - 1 = z - 2$$ and $$x = 3y - 2 = bz - 2$$, $$(ab \neq 0)$$ are coplanar, if:

Consider the line $$L$$ given by the equation $$\frac{x-3}{2} = \frac{y-1}{1} = \frac{z-2}{1}$$. Let $$Q$$ be the mirror image of the point $$(2, 3, -1)$$ with respect to $$L$$. Let a plane $$P$$ be such that it passes through $$Q$$, and the line $$L$$ is perpendicular to $$P$$. Then which of the following points is on the plane $$P$$?

Let $$A$$, $$B$$, $$C$$ be three events such that the probability that exactly one of $$A$$ and $$B$$ occurs is $$(1-k)$$, the probability that exactly one of $$B$$ and $$C$$ occurs is $$(1-2k)$$, the probability that exactly one of $$C$$ and $$A$$ occurs is $$(1-k)$$ and the probability of all $$A$$, $$B$$ and $$C$$ occur simultaneously is $$k^2$$, where $$0 < k < 1$$. Then the probability that at least one of $$A$$, $$B$$ and $$C$$ occurs is:

The number of solutions of the equation $$\log_{(x+1)}(2x^2 + 7x + 5) + \log_{(2x+5)}(x+1)^2 - 4 = 0$$, $$x > 0$$, is ___.

Let $$\{a_n\}_{n=1}^\infty$$ be a sequence such that $$a_1 = 1$$, $$a_2 = 1$$ and $$a_{n+2} = 2a_{n+1} + a_n$$ for all $$n \ge 1$$. Then the value of $$47\sum_{n=1}^\infty \frac{a_n}{2^{3n}}$$ is equal to ___.

For $$k \in N$$, let $$\frac{1}{\alpha(\alpha+1)(\alpha+2)\ldots(\alpha+20)} = \sum_{K=0}^{20} \frac{A_K}{\alpha+k}$$, where $$\alpha > 0$$. Then the value of $$100\left(\frac{A_{14}+A_{15}}{A_{13}}\right)^2$$ is equal to ___.

Consider a triangle having vertices $$A(-2, 3)$$, $$B(1, 9)$$ and $$C(3, 8)$$. If a line $$L$$ passing through the circumcentre of triangle $$ABC$$, bisects line $$BC$$, and intersects y-axis at point $$\left(0, \frac{\alpha}{2}\right)$$, then the value of real number $$\alpha$$ is ___.

If the point on the curve $$y^2 = 6x$$, nearest to the point $$\left(3, \frac{3}{2}\right)$$ is $$(\alpha, \beta)$$, then $$2(\alpha + \beta)$$ is equal to ___.

If $$\lim_{x \to 0} \left[\frac{\alpha x e^x - \beta \log_e(1+x) + \gamma x^2 e^{-x}}{x \sin^2 x}\right] = 10$$, $$\alpha, \beta, \gamma \in R$$, then the value of $$\alpha + \beta + \gamma$$ is ___.

Let $$A = \{a_{ij}\}$$ be a $$3 \times 3$$ matrix, where $$a_{ij} = \begin{cases} (-1)^{j-i} & \text{if } i < j \\ 2 & \text{if } i = j \\ (-1)^{i+j} & \text{if } i > j \end{cases}$$
then det$$(3 \text{Adj}(2A^{-1}))$$ is equal to ___.

Let a function $$g : [0, 4] \to R$$ be defined as
$$g(x) =\begin{cases}\max\limits_{0 \le t \le x} \{ t^3 - 6t^2 + 9t - 3 \}, & 0 \le x \le 3 \\4 - x, & 3 < x \le 4\end{cases}$$
then the number of points in the interval $$(0, 4)$$ where $$g(x)$$ is NOT differentiable, is ___.

Let a curve $$y = y(x)$$ be given by the solution of the differential equation $$\cos\left(\frac{1}{2}\cos^{-1}(e^{-x})\right)dx = \left(\sqrt{e^{2x}-1}\right)dy$$. If it intersects y-axis at $$y = -1$$, and the intersection point of the curve with x-axis is $$(\alpha, 0)$$, then $$e^\alpha$$ is equal to ___.

For $$p > 0$$, a vector $$\vec{v_2} = 2\hat{i} + (p+1)\hat{j}$$ is obtained by rotating the vector $$\vec{v_1} = \sqrt{3}p\hat{i} + \hat{j}$$ by an angle $$\theta$$ about origin in counter clockwise direction. If $$\tan\theta = \frac{(\alpha\sqrt{3}-2)}{(4\sqrt{3}+3)}$$, then the value of $$\alpha$$ is equal to ___.