JEE Main 2021 (18 March Shift 1)

The value of $$3 + \cfrac{1}{4 + \cfrac{1}{3 + \cfrac{1}{4 + \cfrac{1}{3 + \ldots \infty}}}}$$ is equal to:

If the equation $$a|z|^2 +\overline{\bar{\alpha}z + \alpha\bar{z}} + d = 0$$ represents a circle where $$a, d$$ are real constants then which of the following condition is correct?

The sum of all the 4-digit distinct numbers that can be formed with the digits 1, 2, 2 and 3 is:

If $$\alpha, \beta$$ are natural numbers such that $$100^\alpha - 199\beta = (100)(100) + (99)(101) + (98)(102) + \ldots + (1)(199)$$, then the slope of the line passing through $$(\alpha, \beta)$$ and origin is:

$$\frac{1}{3^2 - 1} + \frac{1}{5^2 - 1} + \frac{1}{7^2 - 1} + \ldots + \frac{1}{(201)^2 - 1}$$ is equal to:

Let $$(1 + x + 2x^2)^{20} = a_0 + a_1 x + a_2 x^2 + \ldots + a_{40} x^{40}$$, then $$a_1 + a_3 + a_5 + \ldots + a_{37}$$ is equal to:

The solutions of the equation $$\begin{vmatrix} 1 + \sin^2 x & \sin^2 x & \sin^2 x \\ \cos^2 x & 1 + \cos^2 x & \cos^2 x \\ 4\sin 2x & 4\sin 2x & 1 + 4\sin 2x \end{vmatrix} = 0$$, $$(0 < x < \pi)$$, are:

The number of integral values of $$m$$ so that the abscissa of point of intersection of lines $$3x + 4y = 9$$ and $$y = mx + 1$$ is also an integer, is:

The equation of one of the straight lines which passes through the point $$(1, 3)$$ and makes an angles $$\tan^{-1}(\sqrt{2})$$ with the straight line, $$y + 1 = 3\sqrt{2}x$$ is:

Choose the correct statement about two circles whose equations are given below:
$$x^2 + y^2 - 10x - 10y + 41 = 0$$
$$x^2 + y^2 - 22x - 10y + 137 = 0$$