AM-GM-HM inequalities

If $\vec a$ and $\vec b$ are two vectors such that $|\vec a|=2$ and $|\vec b|=3$, then the maximum value of $3\left|\left(3\vec a+2\vec b\right)\right|+4\left|\left(3\vec a-2\vec b\right)\right|$ is :

Let $\left(2^{1-a}+2^{1+a}\right)$, $f(a)$, $\left(3^a+3^{-a}\right)$ be in A.P. and $\alpha$ be the minimum value of $f(a)$. Then the value of the integral $\displaystyle\int_{\log_e(\alpha-1)}^{\log_e(\alpha)}\frac{dx}{\left(e^{2x}-e^{-2x}\right)}$ is :

For the functions $f(\theta)=\alpha\tan^2\theta+\beta\cot^2\theta$, and $g(\theta)=\alpha\sin^2\theta+\beta\cos^2\theta$, $\alpha>\beta>0$, let $\displaystyle\min_{0<\theta<\frac\pi2}f(\theta)=\max_{0<\theta<\pi}g(\theta)$. If the first term of a G.P. is $\left(\dfrac\alpha{2\beta}\right)$, its common ratio is $\left(\dfrac{2\beta}\alpha\right)$ and the sum of its first 10 terms is $\dfrac mn$, $\gcd(m,n)=1$, then $m+n$ is equal to ________.

Let
$$\alpha = \sum_{k=1}^{\infty} \sin^{2k}\left(\frac\pi6\right).$$
Let $g : [0, 1] \to \mathbb{R}$ be the function defined by
$$g(x) = 2^{\alpha x} + 2^{\alpha(1-x)}.$$
Then, which of the following statements is/are TRUE?

Let $A\left(\dfrac3{\sqrt a},\sqrt a\right)$, $a>0$, be a fixed point in the $xy$-plane. The image of $A$ in $y$-axis be $B$ and the image of $B$ in $x$-axis be $C$. If $D(3\cos\theta,a\sin\theta)$ is a point in the fourth quadrant such that the maximum area of $\Delta ACD$ is 12 square units, then $a$ is equal to ______.

Let $x,y>0$. If $x^3y^2=2^{15}$, then the least value of $3x+2y$ is

For two positive real numbers a and b such that $\dfrac1{a^2}+\dfrac1{b^3}=4$, the minimum value of the constant term in the expansion of $\left(ax^{\frac18}+bx^{-\frac1{12}}\right)^{10}$ is :

The number of elements in the set $S=\left\{x\in\mathbb R:2\cos\left(\dfrac{x^2+x}6\right)=4^x+4^{-x}\right\}$ is :

The minimum value of α for which the equation $\dfrac4{\sin x}+\dfrac1{1-\sin x}=\alpha$ has at least one solution in $\left(0,\dfrac\pi2\right)$ is _______.

Let $m$ be the minimum possible value of $\log_3\left(3^{y_1} + 3^{y_2} + 3^{y_3}\right)$, where $y_1, y_2, y_3$ are real numbers for which $y_1 + y_2 + y_3 = 9$. Let $M$ be the maximum possible value of $(\log_3 x_1 + \log_3 x_2 + \log_3 x_3)$, where $x_1, x_2, x_3$ are positive real numbers for which $x_1 + x_2 + x_3 = 9$. Then the value of $\log_2(m^3) + \log_3(M^2)$ is ______

If the triangle $PQR$ varies, then the minimum value of

$$\cos(P + Q) + \cos(Q + R) + \cos(R + P)$$

is

If the arithmetic mean of two numbers a and b, a > b > 0, is five times their geometric mean, then $\dfrac{a+b}{a-b}$ is equal to :

If three positive numbers a, b and c are in A.P. such that abc = 8, then the minimum possible value of b is :

The least value of $\alpha\in\mathbb R$ for which $4\alpha x^2+\dfrac1x\ge1$, for all $x>0$, is

Let $b_i>1$ for $i=1,2,\ldots,101$. Suppose $\log_eb_1,\log_eb_2,\ldots,\log_eb_{101}$ are in Arithmetic Progression (A.P.) with the common difference $\log_e2$. Suppose $a_1,a_2,\ldots,a_{101}$ are in A.P. such that $a_1=b_1$ and $a_{51}=b_{51}$. If $t=b_1+b_2+\cdots+b_{51}$ and $s=a_1+a_2+\cdots+a_{51}$, then

Let x, y, z be positive real numbers such that x + y + z = 12 and $x^3y^4z^5=(0.1)(600)^3$. Then $x^3+y^3+z^3$ is equal to :

The least value of the product $xyz$ for which the determinant $\begin{vmatrix}x&1&1\\1&y&1\\1&1&z\end{vmatrix}$ is non-negative, is :

Let G be the geometric mean of two positive numbers a and b, and M be the arithmetic mean of $\dfrac1a$ and $\dfrac1b$. If $\dfrac1M:G$ is 4 : 5, then a : b can be :

The minimum value of the sum of real numbers $a^{-5}$, $a^{-4}$, $3a^{-3}$, $1$, $a^8$ and $a^{10}$ with $a>0$ is

If
$$\lim_{x\to0}\left[1+x\ln(1+b^2)\right]^{\frac1x}=2b\sin^2\theta,\quad b>0\ \text{and}\ \theta\in(-\pi,\pi],$$
then the value of $\theta$ is