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Let $$a \neq b$$ be two non-zero real numbers. Then the number of elements in the set $$X = \{z \in C : Re(az^2 + bz) = a$$ and $$Re(bz^2 + az) = b\}$$ is equal to
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All the letters of the word PUBLIC are written in all possible orders and these words are written as in a dictionary with serial numbers. Then the serial number of the word PUBLIC is
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If $$gcd(m, n) = 1$$ and $$1^2 - 2^2 + 3^2 - 4^2 + \ldots + (2021)^2 - (2022)^2 + (2023)^2 = 1012m^2n$$ then $$m^2 - n^2$$ is equal to
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If the coefficients of $$x^7$$ in $$\left(ax^2 + \dfrac{1}{2bx}\right)^{11}$$ and $$x^{-7}$$ in $$\left(ax - \dfrac{1}{3bx^2}\right)^{11}$$ are equal, then
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Among the statements:
(S1): $$2023^{2022} - 1999^{2022}$$ is divisible by 8.
(S2): $$13(13)^n - 11n - 13$$ is divisible by 144 for infinitely many $$n \in \mathbb{N}$$
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If the tangents at the points $$P$$ and $$Q$$ on the circle $$x^2 + y^2 - 2x + y = 5$$ meet at the point $$R\left(\dfrac{9}{4}, 2\right)$$, then the area of the triangle $$PQR$$ is
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$$\lim_{n \to \infty} \left\{\left(2^{1/2} - 2^{1/4}\right)\left(2^{1/2} - 2^{1/8}\right) \cdots \left(2^{1/2} - 2^{1/(2n+1)}\right)\right\}$$ is equal to
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Among the statements
(S1): $$(p \Rightarrow q) \lor ((\sim p) \wedge q)$$ is a tautology
(S2): $$(q \Rightarrow p) \Rightarrow ((\sim p) \wedge q)$$ is a contradiction
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In a group of 100 persons 75 speak English and 40 speak Hindi. Each person speaks at least one of the two languages. If the number of persons who speak only English is $$\alpha$$ and the number of persons who speaks only Hindi is $$\beta$$, then the eccentricity of the ellipse $$25(\beta^2 x^2 + \alpha^2 y^2) = \alpha^2\beta^2$$ is
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Let $$P$$ be a square matrix such that $$P^2 = I - P$$. For $$\alpha, \beta, \gamma, \delta \in \mathbb{N}$$, if $$P^\alpha + P^\beta = \gamma I - 29P$$ and $$P^\alpha - P^\beta = \delta I - 13P$$, then $$\alpha + \beta + \gamma - \delta$$ is equal to
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For the system of equations
$$x + y + z = 6$$
$$x + 2y + \alpha z = 10$$
$$x + 3y + 5z = \beta$$, which one of the following is NOT true?
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Let the sets $$A$$ and $$B$$ denote the domain and range respectively of the function $$f(x) = \dfrac{1}{\sqrt{[x] - x}}$$, where $$[x]$$ denotes the smallest integer greater than or equal to $$x$$. Then among the statements
(S1): $$A \cap B = (1, \infty) - \mathbb{N}$$ and
(S2): $$A \cup B = (1, \infty)$$
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Let $$f(x)$$ be a function satisfying $$f(x) + f(\pi - x) = \pi^2$$, $$\forall x \in \mathbb{R}$$. Then $$\int_0^\pi f(x) \sin x \, dx$$ is equal to
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The area bounded by the curves $$y = |x-1| + |x-2|$$ and $$y = 3$$ is equal to
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If the solution curve $$f(x, y) = 0$$ of the differential equation $$(1 + \log_e x)\dfrac{dx}{dy} - x\log_e x = e^y$$, $$x > 0$$, passes through the points (1, 0) and $$(a, 2)$$, then $$a^a$$ is equal to
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Let the vectors $$\vec{a}, \vec{b}, \vec{c}$$ represent three coterminous edges of a parallelopiped of volume $$V$$. Then the volume of the parallelopiped, whose coterminous edges are represented by $$\vec{a}, \vec{b}+\vec{c}$$ and $$\vec{a}+2\vec{b}+3\vec{c}$$ is equal to
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The sum of all values of $$\alpha$$, for which the points whose position vectors are $$\hat{i} - 2\hat{j} + 3\hat{k}$$, $$2\hat{i} - 3\hat{j} + 4\hat{k}$$, $$(\alpha+1)\hat{i} + 2\hat{k}$$ and $$9\hat{i} + (\alpha-8)\hat{j} + 6\hat{k}$$ are coplanar, is equal to
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Let the line $$L$$ pass through the point (0, 1, 2), intersect the line $$\dfrac{x-1}{2} = \dfrac{y-2}{3} = \dfrac{z-3}{4}$$ and be parallel to the plane $$2x + y - 3z = 4$$. Then the distance of the point $$P(1, -9, 2)$$ from the line $$L$$ is
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A plane $$P$$ contains the line of intersection of the plane $$\vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 6$$ and $$\vec{r} \cdot (2\hat{i} + 3\hat{j} + 4\hat{k}) = -5$$. If $$P$$ passes through the point (0, 2, -2), then the square of distance of the point (12, 12, 18) from the plane P is
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Three dice are rolled. If the probability of getting different numbers on the three dice is $$\dfrac{p}{q}$$, where $$p$$ and $$q$$ are co-prime, then $$q - p$$ is equal to
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For $$\alpha, \beta, z \in C$$ and $$\lambda > 1$$, if $$\sqrt{\lambda - 1}$$ is the radius of the circle $$|z - \alpha|^2 + |z - \beta|^2 = 2\lambda$$, then $$|\alpha - \beta|$$ is equal to ______.
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The number of 4-letter words, with or without meaning, each consisting of 2 vowels and 2 consonants, which can be formed from the letters of the word UNIVERSE without repetition is ______.
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If $$(20)^{19} + 2(21)(20)^{18} + 3(21)^2(20)^{17} + \ldots + 20(21)^{19} = k(20)^{19}$$, then $$k$$ is equal to ______.
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The value of $$\tan 9° - \tan 27° - \tan 63° + \tan 81°$$ is ______.
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Let the eccentricity of an ellipse $$\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$$ is reciprocal to that of the hyperbola $$2x^2 - 2y^2 = 1$$. If the ellipse intersects the hyperbola at right angles, then square of length of the latus-rectum of the ellipse is ______.
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If the mean and variance of the frequency distribution
$$x_i$$ 2, 4, 6, 8, 10, 12, 14, 16
$$f_i$$ 4, 4, $$\alpha$$, 15, 8, $$\beta$$, 4, 5
are 9 and 15.08 respectively, then the value of $$\alpha^2 + \beta^2 - \alpha\beta$$ is ______.
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Let a curve $$y = f(x)$$, $$x \in (0, \infty)$$ pass through the points $$P\left(1, \dfrac{3}{2}\right)$$ and $$Q\left(a, \dfrac{1}{2}\right)$$. If the tangent at any point $$R(b, f(b))$$ to the given curve cuts the y-axis at the point $$S(0, c)$$ such that $$bc = 3$$, then $$(PQ)^2$$ is equal to
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The number of points, where the curve $$y = x^5 - 20x^3 + 50x + 2$$ crosses the x-axis, is ______.
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Let $$f(x) = \dfrac{x}{(1+x^n)^{1/n}}$$, $$x \in \mathbb{R} - \{-1\}$$, $$n \in \mathbb{N}$$, $$n > 2$$. If $$f^n(x) = (f \circ f \circ f \ldots$$ upto n times$$(x)$$, then $$\lim_{n \to \infty} \int_0^1 x^{n-2}(f^n(x))dx$$ is equal to ______.
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If the lines $$\dfrac{x-1}{2} = \dfrac{2-y}{3} = \dfrac{z-3}{\alpha}$$ and $$\dfrac{x-4}{5} = \dfrac{y-1}{2} = \dfrac{z}{\beta}$$ intersect, then the magnitude of the minimum value of $$8\alpha\beta$$ is ______.
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