BITM Basic Mathematics

If \(U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\), \(A = \{1, 2, 3, 4, 5\}\) and \(B = \{3, 4, 5, 6, 7\}\),find \(\overline{A \cup B}\).

Express \(Z = (2 + 3i)^2\) in the form of \(a + ib\).

Rewrite \(-7 \le x \le 2\) by using the modulus sign.

Evaluate: \(\lim_{x \to \infty} \frac{4x^2-5x+7}{3x^2-6x-1}\)

If \(\vec{a} = 3\vec{i} + 2\vec{j}\) and \(\vec{b} = -4\vec{i} + 3\vec{j}\) are two vectors. Find the value of \(2\vec{a} + 3\vec{b}\).

Find the derivative of \(y = x^3.e^x\)

If \(A = \begin{bmatrix} 2 & -4 \\ 3 & 1 \end{bmatrix}\), find \(3A^T\).

Find the area bounded by the line \(y = 4x + 7\), the x –axis and ordinates at \(x = 1\) and \(x = 3\).

Solve the differential equation: \(\frac{dy}{dx} = e^{-y}\)

Find the value of determinant \(A = \begin{vmatrix} 1 & 2 & 1 \\ 2 & 1 & 4 \\ 3 & 4 & 2 \end{vmatrix}\)

(a) If \(x - iy = \frac{5-6i}{5+6i}\) then prove that \(x^2 + y^2 = 1\).

(b) Find the square roots of \(-4 + 4\sqrt{3}i\).

A function \(f(x)\) is defined as follows:

\(f(x) = \begin{cases} 3x + 2 & \text{for } -2 \le x < 0 \\ 2 - 5x & \text{for } 0 \le x < 2 \\ 5 - 2x & \text{for } x \ge 2 \end{cases}\)

Examine the continuity and discontinuity of the function at \(x = 0\) and \(x = 2\).

If \(f(x) = \frac{x+6}{cx-d}\), \(\lim_{x \to 0} f(x) = -6\) and \(\lim_{x \to \infty} f(x) = \frac{1}{3}\), find the value of \(f(13)\).

Find the derivatives of:

(a) \(y = t^3 + 1\) and \(x = t^2 + 2\)

(b) \(x^2 + y^2 = 36\)

Evaluate the integrals:

(a) \(\int x(\sqrt{x+4})dx\)

(b) \(\int \left(1 - \frac{1}{x^2}\right).e^{x+\frac{1}{x}} dx\).

Prove or disprove that the vectors \(\vec{a} + 2\vec{b} - \vec{c}\), \(2\vec{a} - \vec{b} + \vec{c}\) and \(3\vec{a} + \vec{b} + \vec{c}\) are Coplanar:

Solve the differential equation.

\(\frac{dy}{dx} + 3y = e^{2x} + 12\)

A survey of 600 television viewers produced the following information:

285 watch channel A, 195 watch channel B, 115 watch channel C, 70 watch channel A and B ,50 watch channel B and C, 45 watch channel A and C. If 150 of the viewers watch none of the channels, find how many of them watch

a. at least one of the channels

b. all three channels

c. two channels only

d. one channel only

Represent all the sets in venn diagram.

The following table shows annual production of wheat ( in thousand tons) of a certain village.

a. Determine the equation of the trend line by least square method.

b. Estimate the production in the year 2023 and 2025.

Solve the following system of linear equations by using determinant or matrix method:

\(2x + 5y + 3z = 9\); \(3x + y + 2z = 3\); \(x + 2y - z = 6\)

The demand and supply functions for a good are

\(P_d = 100 - 0.5Q_d\) and \(P_s = 10 + 0.5Q_s\)

respectively, where P and Q denote price and quantity.

a. Find the equilibrium price and quantity.

b. Find the consumer's surplus and producer's surplus.

c. Also, find total surplus.

The demand function for a commodity is \(P = 15 - Q\) ,where \(P\) is the price in rupees and \(Q\) is number of outputs produced and sold. The cost function is given by \(C(Q) = -20 - 3Q + Q^2\)

a. Find the value of Q at which the revenue is maximum. Also find the maximum revenue.

b. Find the value of Q at which profit is maximum. Also find the maximum profit.

c. Find the price(P) at which profit is maximum.

d. Find the break- even point.