NTA JEE Main 26th August 2021 Shift 1 - Mathematics

The equation $$\arg\left(\frac{z-1}{z+1}\right) = \frac{\pi}{4}$$ represents a circle with:

The sum of the series $$\frac{1}{x+1} + \frac{2}{x^2+1} + \frac{2^2}{x^4+1} + \ldots + \frac{2^{100}}{x^{2^{100}}+1}$$ when $$x = 2$$ is:

If the sum of an infinite GP, $$a, ar, ar^2, ar^3, \ldots$$ is 15 and the sum of the squares of its each term is 150, then the sum of $$ar^2, ar^4, ar^6, \ldots$$ is:

If $$^{20}C_r$$ is the co-efficient of $$x^r$$ in the expansion of $$(1 + x)^{20}$$, then the value of $$\sum_{r=0}^{20} r^2(^{20}C_r) $$ is equal to:

The sum of solutions of the equation $$\frac{\cos x}{1+\sin x} = |\tan 2x|$$, $$x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) - \left\{-\frac{\pi}{4}, \frac{\pi}{4}\right\}$$ is:

Let $$ABC$$ be a triangle with $$A(-3, 1)$$ and $$\angle ACB = \theta$$, $$0 < \theta < \frac{\pi}{2}$$. If the equation of the median through B is $$2x + y - 3 = 0$$ and the equation of angle bisector of C is $$7x - 4y - 1 = 0$$, then $$\tan \theta$$ is equal to:

If a line along a chord of the circle $$4x^2 + 4y^2 + 120x + 675 = 0$$, passes through the point $$(-30, 0)$$ and is tangent to the parabola $$y^2 = 30x$$, then the length of this chord is:

On the ellipse $$\frac{x^2}{8} + \frac{y^2}{4} = 1$$, let P be a point in the second quadrant such that the tangent at P to the ellipse is perpendicular to the line $$x + 2y = 0$$. Let S and S' be the foci of the ellipse and $$e$$ be its eccentricity. If A is the area of the triangle SPS', then the value of $$(5 - e^2) \cdot A$$ is

If the truth value of the Boolean expression $$((p \vee q) \wedge (q \rightarrow r) \wedge (\sim r)) \rightarrow (p \wedge q)$$ is false, then the truth values of the statements $$p$$, $$q$$, $$r$$ respectively can be:

The mean and standard deviation of 20 observations were calculated as 10 and 2.5 respectively. It was found that by mistake one data value was taken as 25 instead of 35. If $$\alpha$$ and $$\sqrt{\beta}$$ are the mean and standard deviation respectively for correct data, then $$(\alpha, \beta)$$ is:

Out of all the patients in a hospital 89% are found to be suffering from heart ailment and 98% are suffering from lungs infection. If $$K$$% of them are suffering from both ailments, then $$K$$ can not belong to the set:

If $$A = \begin{bmatrix} \frac{1}{\sqrt{5}} & \frac{2}{\sqrt{5}} \\ \frac{-2}{\sqrt{5}} & \frac{1}{\sqrt{5}} \end{bmatrix}$$, $$B = \begin{bmatrix} 1 & 0 \\ i & 1 \end{bmatrix}$$, $$i = \sqrt{-1}$$, and $$Q = A^T B A$$, then the inverse of the matrix $$AQ^{2021}A^T$$ is equal to:

Let $$\theta \in \left(0, \frac{\pi}{2}\right)$$. If the system of linear equations
$$(1 + \cos^2 \theta)x + \sin^2 \theta y + 4\sin 3\theta z = 0$$
$$\cos^2 \theta x + (1 + \sin^2 \theta)y + 4\sin 3\theta z = 0$$
$$\cos^2 \theta x + \sin^2 \theta y + (1 + 4\sin 3\theta)z = 0$$
has a non-trivial solution, then the value of $$\theta$$ is:

Let $$f(x) = \cos\left(2\tan^{-1}\sin\left(\cot^{-1}\sqrt{\frac{1-x}{x}}\right)\right)$$, $$0 \lt x \lt 1$$. Then:

The value of $$\lim_{n \to \infty} \frac{1}{n} \sum_{r=0}^{2n-1} \frac{n^2}{n^2 + 4r^2}$$ is:

The value of $$\int_{\frac{-1}{\sqrt{2}}}^{\frac{1}{\sqrt{2}}} \left(\left(\frac{x+1}{x-1}\right)^2 + \left(\frac{x-1}{x+1}\right)^2 - 2\right)^{\frac{1}{2}} dx$$ is:

Let $$y = y(x)$$ be a solution curve of the differential equation $$(y+1)\tan^2 x \, dx + \tan x \, dy + y \, dx = 0$$, $$x \in \left(0, \frac{\pi}{2}\right)$$. If $$\lim_{x \to 0^+} xy(x) = 1$$, then the value of $$y\left(\frac{\pi}{4}\right)$$ is:

Let $$\vec{a} = \hat{i} + \hat{j} + \hat{k}$$ and $$\vec{b} = \hat{j} - \hat{k}$$. If $$\vec{c}$$ is a vector such that $$\vec{a} \times \vec{c} = \vec{b}$$ and $$\vec{a} \cdot \vec{c} = 3$$, then $$\vec{a} \cdot (\vec{b} \times \vec{c})$$ is equal to:

A plane $$P$$ contains the line $$x + 2y + 3z + 1 = 0 = x - y - z - 6$$, and is perpendicular to the plane $$-2x + y + z + 8 = 0$$. Then which of the following points lies on $$P$$?

Let A and B be independent events such that P(A) = p, P(B) = 2p. The largest value of p, for which P(exactly one of A, B occurs) = $$\frac{5}{9}$$, is:

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 _________