The owner of a small restaurant can prepare a particular meal at a cost of Rupee 100. He estimate that if the menu price of the meal is x rupees, then the number of customers who will order that meal at that price in an evening is given by the function D(x) = 200 - x. Express his day revenue total cost and profit on this meal as a function of x.

Let A = {1, 2, 3, 4} and B = {a, b, c, d}. Give a function from A\(\rightarrow\)B for each of the following:
one-to-one but not onto.

Consider the functions: 
i) \(f(x)=x^2,\)
ii) \(f(x)={1\over 2}x^2,\)
iii) \(f(x)=2x^2\)

Solve \(-{ x }^{ 2 }+3x-2\ge 0\)

If x = -2 is one root of x³ - x²- 17x = 22, then find the other roots of the equation.

Resolve the following rational expressions into partial fractions.
\({{3x+1}\over{(x-2)(x+1)}}\)

Solve 3x² + 5x - 2 ≤ 0.

Solve the equation \(\sqrt{6-4x-x^2}=x+4\)

If \(\sin { x } =\frac { 15 }{ 17 } \) and \(\cos {y } =\frac { 12 }{ 13 } \), 0 < x < \(\frac{\pi}{2}\), 0 < y < \(\frac{\pi}{2}\), find the value of sin (x + y)

Prove that sin² (A + B) - sin² (A - B) = sin 2A sin 2B

Solve the following equation cos 2\(\theta\)=\(\frac { \sqrt { 5+1 } }{ 4 } \)

A polygon has 90 diagonals. Find the number of its sides?

In a race, 20 balls are placed in a line at intervals of 4 meters, with the first ball 24 meters away from the starting point. A contestant is required to bring the balls back to the starting place one at a time. How far would the contestant run to bring back all balls?

Find the value of k and b, if the points P(-3, 1) and Q(2, b) lie on the locus of x² - 5x + ky = 0.

Evaluate: \(\underset { x\rightarrow \infty }{ lim } \sqrt { x } \left( \sqrt { x+c } -\sqrt { x } \right) \) 

Construct a 2 \(\times\) 3 matrix whose (i, j)^th element is given by \(a_ij={\sqrt{3}\over 2}|2i-3j|(1\le i\le2,1\le j\le3)\) .

Write the general form of a 3 \(\times\) 3 skew-symmetric matrix and prove that its determinant is 0.

A tank contains 5000 litres of pure water. Brine (very salty water) that contains 30 grams of salt per litre of water is pumped into the tank at a rate of 25 litres per minute. The concentration of salt water after t minutes (in grams per litre) is\(C(t)={30t\over 200+t}\)  What happens to the concentration as \(t\rightarrow \infty?\) 

Find the derivatives of the following functions with respect to corresponding independent variables: y = (x² + 5)log(1 + x)e^-3x

Integrate the following with respect to x : \(5x^2-4+{7\over x}+{2\over \sqrt{x}}\)

Evaluate the following integrals : \(\int e^{-x^2}xdx\)

Integrate the following with respect to x : 25xe^-5x

If P(A) = 0.5, P(B) = 0.8 and P(B/A) = 0.8, find P(A/B) and P(A\(\cup \)B)

(i) The odds that the event A occurs is 5 to 7, find P(A)..
(ii) Suppose \(P(B)=\frac{2}{5},\) Express the odds that the event B occurs.