Let $a \neq b$ be two non-zero real numbers. Then the number of elements in the set $X = \left\{ z \in C : \operatorname{Re}\left(az^{2} + bz\right) = a$ and $\operatorname{Re}\left(bz^{2} + az\right) = b\right\}$ is equal to (1) 0 (2) 1 (3) 3 (4) 2
For $\alpha, \beta, z \in \mathbb{C}$ 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 $\_\_\_\_$.
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 (1) 576 (2) 578 (3) 580 (4) 582
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 $\_\_\_\_$.
If the coefficients of $x^{7}$ in $\left(ax^{2} + \frac{1}{2bx}\right)^{11}$ and $x^{-7}$ in $\left(ax - \frac{1}{3bx^{2}}\right)^{11}$ are equal, then (1) $729ab = 32$ (2) $32ab = 729$ (3) $64ab = 243$ (4) $243ab = 64$
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 $n \in \mathbb{N}$ (1) Only $(S2)$ is correct (2) Only $(S1)$ is correct (3) Both $(S1)$ and $(S2)$ are correct (4) Both $(S1)$ and $(S2)$ are incorrect
If the tangents at the points $P$ and $Q$ on the circle $x^{2} + y^{2} - 2x + y = 5$ meet at the point $R\left(\frac{9}{4}, 2\right)$, then the area of the triangle $PQR$ is (1) $\frac{5}{4}$ (2) $\frac{13}{8}$ (3) $\frac{5}{8}$ (4) $\frac{13}{4}$
Let the eccentricity of an ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ 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 $\_\_\_\_$.
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 $25\left(\beta^{2}x^{2} + \alpha^{2}y^{2}\right) = \alpha^{2}\beta^{2}$ is (1) $\frac{\sqrt{119}}{12}$ (2) $\frac{\sqrt{117}}{12}$ (3) $\frac{3\sqrt{15}}{12}$ (4) $\frac{\sqrt{129}}{12}$
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 (1) 18 (2) 40 (3) 22 (4) 24
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? (1) System has no solution for $\alpha = 3, \beta = 24$ (2) System has a unique solution for $\alpha = -3, \beta = 14$ (3) System has infinitely many solutions for $\alpha = 3, \beta = 14$ (4) System has a unique solution for $\alpha = 3, \beta = 14$