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Let $$A = \left\{z \in C : \left|\frac{z+1}{z-1}\right| < 1\right\}$$ and $$B = \left\{z \in C : \arg\left(\frac{z-1}{z+1}\right) = \frac{2\pi}{3}\right\}$$. Then $$A \cap B$$ is
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The remainder when $$(2021)^{2023}$$ is divided by $$7$$ is
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Let $$R$$ be the point $$(3, 7)$$ and let $$P$$ and $$Q$$ be two points on the line $$x + y = 5$$ such that $$PQR$$ is an equilateral triangle. Then the area of $$\triangle PQR$$ is
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Let $$C$$ be a circle passing through the points $$A(2,-1)$$ and $$B(3,4)$$. The line segment $$AB$$ is not a diameter of $$C$$. If $$r$$ is the radius of $$C$$ and its centre lies on the circle $$(x-5)^2 + (y-1)^2 = \frac{13}{2}$$, then $$r^2$$ is equal to
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Let the normal at the point $$P$$ on the parabola $$y^2 = 6x$$ pass through the point $$(5, -8)$$. If the tangent at $$P$$ to the parabola intersects its directrix at the point $$Q$$, then the ordinate of the point $$Q$$ is
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$$\lim_{x \to \frac{1}{\sqrt{2}}} \frac{\sin(\cos^{-1} x) - x}{1 - \tan(\cos^{-1} x)}$$ is equal to
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Let $$\Delta, \nabla \in \{\wedge, \vee\}$$ be such that $$p\nabla q \to ((p\Delta q)\nabla r)$$ is a tautology. Then $$(p\nabla q) \Delta r$$ is logically equivalent to
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The mean of the numbers $$a, b, 8, 5, 10$$ is $$6$$ and their variance is $$6.8$$. If $$M$$ is the mean deviation of the numbers about the mean, then $$25M$$ is equal to
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Let $$A$$ be a $$3 \times 3$$ invertible matrix. If $$|\text{adj}(24A)| = |\text{adj}(3 \text{ adj}(2A))|$$, then $$|A|^2$$ is equal to
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The ordered pair $$(a, b)$$, for which the system of linear equations
$$3x - 2y + z = b$$
$$5x - 8y + 9z = 3$$
$$2x + y + az = -1$$
has no solution, is
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Let $$f(x) = \frac{x-1}{x+1}, x \in R - \{0, -1, 1\}$$. If $$f^{n+1}(x) = f(f^n(x))$$ for all $$n \in N$$, then $$f^6(6) + f^7(7)$$ is equal to
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$$f, g : R \to R$$ be two real valued functions defined as $$f(x) = \begin{cases} -|x+3| & x < 0 \\ e^x & x \geq 0 \end{cases}$$ and
$$g(x) = \begin{cases} x^2 + k_1 x & x < 0 \\ 4x + k_2 & x \geq 0 \end{cases}$$, where $$k_1$$ and $$k_2$$ are real constants. If $$gof$$ is differentiable at $$x = 0$$, then $$gof(-4) + gof(4)$$ is equal to
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The sum of the absolute minimum and the absolute maximum values of the function
$$f(x) = |3x - x^2 + 2| - x$$ in the interval $$[-1, 2]$$ is
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Let $$S$$ be the set of all the natural numbers, for which the line $$\frac{x}{a} + \frac{y}{b} = 2$$ is a tangent to the curve $$\left(\frac{x}{a}\right)^n + \left(\frac{y}{b}\right)^n = 2$$ at the point $$(a, b), ab \neq 0$$. Then
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Let $$f(x) = 2\cos^{-1}x + 4\cot^{-1}x - 3x^2 - 2x + 10, x \in [-1, 1]$$. If $$[a, b]$$ is the range of the function, then $$4a - b$$ is equal to
The area bounded by the curve $$y = |x^2 - 9|$$ and the line $$y = 3$$ is
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If $$\vec{a} \cdot \vec{b} = 1, \vec{b} \cdot \vec{c} = 2$$ and $$\vec{c} \cdot \vec{a} = 3$$, then the value of $$\left[\vec{a} \times (\vec{b} \times \vec{c}), \vec{b} \times (\vec{c} \times \vec{a}), \vec{c} \times (\vec{b} \times \vec{a})\right]$$ is
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If the two lines $$l_1 : \frac{x-2}{3} = \frac{y+1}{-2}, z = 2$$ and $$l_2 : \frac{x-1}{1} = \frac{2y+3}{\alpha} = \frac{z+5}{2}$$ are perpendicular, then an angle between the lines $$l_2$$ and $$l_3 : \frac{1-x}{3} = \frac{2y-1}{-4} = \frac{z}{4}$$ is
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Let the plane $$2x + 3y + z + 20 = 0$$ be rotated through a right angle about its line of intersection with the plane $$x - 3y + 5z = 8$$. If the mirror image of the point $$(2, -\frac{1}{2}, 2)$$ in the rotated plane is $$B(a, b, c)$$, then
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Let a biased coin be tossed 5 times. If the probability of getting 4 heads is equal to the probability of getting 5 heads, then the probability of getting atmost two heads is
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The sum of the cubes of all the roots of the equation $$x^4 - 3x^3 - 2x^2 + 3x + 1 = 0$$ is ______
There are ten boys $$B_1, B_2, \ldots, B_{10}$$ and five girls $$G_1, G_2, \ldots G_5$$ in a class. Then the number of ways of forming a group consisting of three boys and three girls, if both $$B_1$$ and $$B_2$$ together should not be the members of a group, is ______
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Let $$A = \sum_{i=1}^{10}\sum_{j=1}^{10} \min\{i, j\}$$ and $$B = \sum_{i=1}^{10}\sum_{j=1}^{10} \max\{i, j\}$$. Then $$A + B$$ is equal to ______
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If $$\sin^2(10°)\sin(20°)\sin(40°)\sin(50°)\sin(70°) = \alpha - \frac{1}{16}\sin(10°)$$, then $$16 + \alpha^{-1}$$ is equal to ______
Let the common tangents to the curves $$4(x^2 + y^2) = 9$$ and $$y^2 = 4x$$ intersect at the point $$Q$$. Let an ellipse, centered at the origin $$O$$, has lengths of semi-minor and semi-major axes equal to $$OQ$$ and $$6$$, respectively. If $$e$$ and $$l$$ respectively denote the eccentricity and the length of the latus rectum of this ellipse, then $$\frac{l}{e^2}$$ is equal to ______
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Let $$A = \{n \in N : H.C.F.(n, 45) = 1\}$$ and let $$B = \{2k : k \in \{1, 2, \ldots, 100\}\}$$. Then the sum of all the elements of $$A \cap B$$ is ______
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Let $$f(x) = \max\{|x+1|, |x+2|, \ldots, |x+5|\}$$. Then $$\int_{-6}^{0} f(x)dx$$ is equal to ______
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The value of the integral $$\frac{48}{\pi^4}\int_0^{\pi}\left(\frac{3\pi x^2}{2} - x^3\right)\frac{\sin x}{1 + \cos^2 x}dx$$ is equal to ______
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Let the solution curve $$y = y(x)$$ of the differential equation $$(4 + x^2)dy - 2x(x^2 + 3y + 4)dx = 0$$ pass through the origin. Then $$y(2)$$ is equal to ______
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Let $$S = (0, 2\pi) - \left\{\frac{\pi}{2}, \frac{3\pi}{4}, \frac{3\pi}{2}, \frac{7\pi}{4}\right\}$$. Let $$y = y(x), x \in S$$, be the solution curve of the differential equation $$\frac{dy}{dx} = \frac{1}{1 + \sin 2x}, y\left(\frac{\pi}{4}\right) = \frac{1}{2}$$. If the sum of abscissas of all the points of intersection of the curve $$y = y(x)$$ with the curve $$y = \sqrt{2}\sin x$$ is $$\frac{k\pi}{12}$$, then $$k$$ is equal to ______
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