NTA JEE Main 26th August 2021 Shift 1

The sum of all integral values of $$k$$ ($$k \neq 0$$) for which the equation $$\frac{2}{x-1} - \frac{1}{x-2} = \frac{2}{k}$$ in $$x$$ has no real roots, is _________

Let $$z = \frac{1-i\sqrt{3}}{2}$$, $$i = \sqrt{-1}$$. Then the value of $$$21 + \left(z + \frac{1}{z}\right)^3 + \left(z^2 + \frac{1}{z^2}\right)^3 + \left(z^3 + \frac{1}{z^3}\right)^3 + \ldots + \left(z^{21} + \frac{1}{z^{21}}\right)^3$$$ is _________

The number of three-digit even numbers, formed by the digits 0, 1, 3, 4, 6, 7 if the repetition of digits is not allowed, is _________

If $$^1P_1 + 2 \cdot ^2P_2 + 3 \cdot ^3P_3 + \ldots + 15 \cdot ^{15}P_{15} = ^qP_r - s$$, $$0 \leq s \leq 1$$, then $$^{q+s}C_{r-s}$$ is equal to _________

The locus of a point, which moves such that the sum of squares of its distances from the points $$(0, 0)$$, $$(1, 0)$$, $$(0, 1)$$, $$(1, 1)$$ is 18 units, is a circle of diameter $$d$$. Then $$d^2$$ is equal to _________

Let $$a, b \in R$$, $$b \neq 0$$. Defined a function, $$f(x) = \begin{cases} a\sin\frac{\pi}{2}(x-1), & \text{for } x \leq 0 \\ \frac{\tan 2x - \sin 2x}{bx^3}, & \text{for } x > 0 \end{cases}$$
If $$f$$ is continuous at $$x = 0$$, then $$10 - ab$$ is equal to _________

If $$y = y(x)$$ is an implicit function of $$x$$ such that $$\log_e(x + y) = 4xy$$, then $$\frac{d^2y}{dx^2}$$ at $$x = 0$$ is equal to _________

A wire of length 36 m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is $$k$$ (meter), then $$\left(\frac{4}{\pi} + 1\right)k$$ is equal to _________

The area of the region $$S = \{(x, y) : 3x^2 \leq 4y \leq 6x + 24\}$$ is _________

Let the line $$L$$ be the projection of the line $$\frac{x-1}{2} = \frac{y-3}{1} = \frac{z-4}{2}$$ in the plane $$x - 2y - z = 3$$. If $$d$$ is the distance of the point $$(0, 0, 6)$$ from $$L$$, then $$d^2$$ is equal to _________