NTA JEE Main 20th July 2021 Shift 2

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: