Let $25^x+25^{-x},\ \dfrac{\alpha}{3},\ 20^{1+x}+20^{1-x}$, where $x,\alpha\in\mathbb{R}$, be the first three terms of an A.P. of increasing terms. For the least value of $\alpha$, the sum of its first $10$ terms is _____

Let \[ f(x)= \begin{cases} 3x, & x<0,\\ 1+x+[x], & 0\le x\le 2,\\ 5, & x>2, \end{cases} \] where $[x]$ denotes the greatest integer function. If $\alpha$ and $\beta$ are the number of points in $\mathbb{R}$ where $f$ is not continuous and is not differentiable respectively, then $\alpha+\beta$ equals:

Let the value of $p$, such that the sum of the squares of the roots of the quadratic equation \[ x^2+(7-p)x+4=p \] has the least value, be $\alpha$, and the corresponding roots be $\beta$ and $\gamma$. Then $\alpha^3+\beta^3+\gamma^3$ equals \underline{\hspace{2cm}.}

The area of the region \[ \{(x,y): \sin x \le y \le \sqrt{\pi^2-x^2}\} \] is:

Let \[ f(x)=\int\left(\frac{1}{\log_e x}-\frac{2}{(\log_e x)^3}\right)\,dx. \] If $f(e)=2e$, then $f(e^2)$ is equal to:

Let \[ f(x)= \begin{vmatrix} -\cos x & \tan x & 3\sin x\\ 1 & -3x & 2x^2\\ x^3 & x & x^2 \end{vmatrix}. \] Then the value of \[ \lim_{x\to 0}\frac{(1+x)f(x)-3x\sin x}{x^3} \] is:

Two persons $A$ and $B$ alternately throw a pair of dice. $A$ wins if he throws a sum of $4$ before $B$ throws a sum of $9$, and $B$ wins if he throws a sum of $9$ before $A$ throws a sum of $4$. The probability that $A$ wins if $B$ makes the first throw is:

The number of integral terms in the binomial expansion of \[ \left(11^{\frac{1}{2}} + 17^{\frac{1}{8}}\right)^{1024} \] is:

The number of integral values of $n$, for which the equation \[ 3\cos x + 5\sin x = 2n + 1 \] has a solution, is:

If all the words, with or without meaning, made using all the letters of the word ``RANCHI'' are arranged as in a dictionary, then the word at $560^{\text{th}}$ position is:

If the system of linear equations : \[ x+y+z=4,\quad x+2y+3z=6,\quad 4x+5y+\lambda z=\mu \] has more than one solution, then the value of $ \lambda+\mu $ is equal to:

Let $ b_1 = 3, b_2, b_3, \ldots $ be a geometric progression of increasing positive numbers. Let \[ \sum_{n=1}^{20} b_{3n} = 4\sum_{n=1}^{20} b_{3n-2}. \] Then, the sum of the first ten terms of the G.P. is:

The given graph illustrates:

If $ \sqrt{1 - x^2} + \sqrt{1 - y^2} = a(x - y) $, then prove that $ \frac{dy}{dx} = \frac{\sqrt{1 - y^2}}{\sqrt{1 - x^2}} $.

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