If the coefficient of $${x^7}$$ in $${\left( {a{x^2} + {1 \over {2bx}}} \right)^{11}}$$ and $${x^{ - 7}}$$ in $${\left( {ax - {1 \over {3b{x^2}}}} \right)^{11}}$$ are equal, then :

The area bounded by the curves $$y=|x-1|+|x-2|$$ and $$y=3$$ 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 :

Let $$a \neq b$$ be two non-zero real numbers. Then the number of elements in the set $$X=\left\{z \in \mathbb{C}: \operatorname{Re}\left(a z^{2}+b z\right)=a\right.$$ and $$\left.\operatorname{Re}\left(b z^{2}+a z\right)=b\right\}$$ is equal to :

Let $$P$$ be a square matrix such that $$P^{2}=I-P$$. For $$\alpha, \beta, \gamma, \delta \in \mathbb{N}$$, if $$P^{\alpha}+P^{\beta}=\gamma I-29 P$$ and $$P^{\alpha}-P^{\beta}=\delta I-13 P$$, then $$\alpha+\beta+\gamma-\delta$$ is equal to :

Among the statements :

(S1) : $$2023^{2022}-1999^{2022}$$ is divisible by 8

(S2) : $$13(13)^{n}-12 n-13$$ is divisible by 144 for infinitely many $$n \in \mathbb{N}$$

If the solution curve $$f(x, y)=0$$ of the differential equation

$$\left(1+\log _{e} x\right) \frac{d x}{d y}-x \log _{e} x=e^{y}, x > 0$$,

passes through the points $$(1,0)$$ and $$(\alpha, 2)$$, then $$\alpha^{\alpha}$$ is equal to :

Let $$f(x)$$ be a function satisfying $$f(x)+f(\pi-x)=\pi^{2}, \forall x \in \mathbb{R}$$. Then $$\int_\limits{0}^{\pi} f(x) \sin x d x$$ is equal to :

$$\lim _\limits{n \rightarrow \infty}\left\{\left(2^{\frac{1}{2}}-2^{\frac{1}{3}}\right)\left(2^{\frac{1}{2}}-2^{\frac{1}{5}}\right) \ldots . .\left(2^{\frac{1}{2}}-2^{\frac{1}{2 n+1}}\right)\right\}$$ is equal to :

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 speak 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 :

Three dice are rolled. If the probability of getting different numbers on the three dice is $$\frac{p}{q}$$, where $$p$$ and $$q$$ are co-prime, then $$q-p$$ is equal to :

For the system of equations

$$x+y+z=6$$

$$x+2 y+\alpha z=10$$

$$x+3 y+5 z=\beta$$, which one of the following is NOT true?

Let the sets A and B denote the domain and range respectively of the function $$f(x)=\frac{1}{\sqrt{\lceil x\rceil-x}}$$, where $$\lceil x\rceil$$ denotes the smallest integer greater than or equal to $$x$$. Then among the statements

(S1) : $$A \cap B=(1, \infty)-\mathbb{N}$$ and

(S2) : $$A \cup B=(1, \infty)$$

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 ___________.

Let the eccentricity of an ellipse $$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$$ is reciprocal to that of the hyperbola $$2 x^{2}-2 y^{2}=1$$. If the ellipse intersects the hyperbola at right angles, then square of length of the latus-rectum of the ellipse is ___________.

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 __________.

The value of $$\tan 9^{\circ}-\tan 27^{\circ}-\tan 63^{\circ}+\tan 81^{\circ}$$ is __________.

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 __________.

If the lines $$\frac{x-1}{2}=\frac{2-y}{-3}=\frac{z-3}{\alpha}$$ and $$\frac{x-4}{5}=\frac{y-1}{2}=\frac{z}{\beta}$$ intersect, then the magnitude of the minimum value of $$8 \alpha \beta$$ is _____________.

If

$$(20)^{19}+2(21)(20)^{18}+3(21)^{2}(20)^{17}+\ldots+20(21)^{19}=k(20)^{19}$$,

then $$k$$ is equal to ___________.

The number of points, where the curve $$y=x^{5}-20 x^{3}+50 x+2$$ crosses the $$\mathrm{x}$$-axis, is ____________.