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Find the equation of the plane passing through the line of intersection of the planes $x + 2y +3z - 4 = 0$ and $2x + y - z + 5 = 0$ and perpendicular to the plane $5x + 3y - 6z + 8 = 0.$

A square piece of tin of side 18cm is to be made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible.

Evaluate the following integrals:
$\int\limits^{\frac{3}{2}}_0\big|\text{x}\cos\pi\text{x}\big|\text{dx}$

Discuss the continuity and differentiability of the function $\text{f (x) = |x| + x - 1|}$ in the interval (-1, 2).

A biased die is such that $\text{P}(4)=\frac{1}{10}$ and other scores being equally likely. The die is tossed twice. If X is the ‘number of fours seen’, find the variance of the random variable X.

Evaluate the following integrals:
$\int\limits_{0}^{\frac{\pi}{2}}\frac{1}{5+4\sin\text{x}}\text{ dx}$

Solve the Linear Programming Problem graphically:
Minimize Z = 3x + 5y such that $x + 3y \geq 3,x + y \geq 2,x,y \geq 0$.

Suppose 10,000 tickets are sold in a lottery each for Rs. 1. First prize is of Rs. 3000 and the second prize is of Rs. 2000. There are three third prizes of Rs. 500 each. If you buy one ticket, what is your expectation.

Integrate the function in Exercise:$\frac{1}{\big(\text{x}^{2}+1\big)\big(\text{x}^{2}+4\big)}$

If the axes are rectangular and p is the point (2, 3, -1), find the equation of the plane throught p at right angles to OP.