A wire of length $20 m$ is to be cut into two pieces A piece of length $l_1$ is bent to make a square of area $A_1$ and the other piece of length $l_2$ is made into a circle of area $A_2$ If $2 A_1+3 A_2$ is minimum then $\left(\pi l_1\right): l_2$ is equal to:

If the coefficient of x⁷ in ($ax^2 +\frac{ 1}{2bx}^{11}$) and x^-7 in ($ax - \frac{1}{3bx^2}^{11}$), are euual then

Let the line L pass through the point (0, 1, 2), intersect the line $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and be parallel to the plane 2x + y – 3z = 4. Then the distance of the point P(l, –9, 2) from the line L is

The foot of perpendicular from the origin $O$ to a plane $P$ which meets the co-ordinate axes at the points $A , B , C$ is $(2, a , 4), a \in N$ If the volume of the tetrahedron $OABC$ is 144 unit $^3$, then which of the following points is NOT on $P$ ?

If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are three non-zero vectors and $\hat{\mathbf{n}}$ is a unit vector perpendicular to $\mathbf{c}$ such that: $\mathbf{a} = \alpha \mathbf{b} - \hat{\mathbf{n}}, \, (\alpha \neq 0)$ and $ \mathbf{b} \cdot \mathbf{c} = 12 $, then $ \mathbf{c} \times (\mathbf{a} \times \mathbf{b}) $ is equal to:

The negation of the statement
(p v q) ∧ (q v(∼r)) is

All words, with or without meaning, are made using all the letters of the word MONDAY. These words are written as in a dictionary with serial numbers. The serial number of the word MONDAY is

Two dice are thrown independently Let$A$ be the event that the number appeared on the $1^{\text {st }}$ die is less than the number appeared on the $2^{\text {nd }}$ die, $B$ be the event that the number appeared on the $1^{\text {st }}$ die is even and that on the second die is odd, and $C$ be the event that the number appeared on the $1^{\text {st }}$ die is odd and that on the $2^{\text {nd }}$ is even Then :

$\lim\limits_{x\rightarrow0}\left(\left(\frac{1-cos^2(3x)}{cos^3(4x)}\right)\left(\frac{sin^3(4x)}{(log_e(2x+1))^5}\right)\right)$is equal to

The statement ~[p v (~(p ∧ q))] is equivalent to

Let x = x(y) be the solution of the differential equation 2(y+2) log_e (y+2)dx+(x+4-2log_e(y+2))dy = 0, y >-1 with x (e⁴-2)=1. Then x(e⁹-2) is equal to