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 \(\overline{A \cup B}\).

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

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.

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

Solve the inequality \(|3x - 1| \le 7\)

In a group of students 18 read marketing, 22 read statistics and 16 read economics, 6 read marketing only, 9 read statistics only, 5 read marketing and statistics only and 5 read statistics and economics only.

a. How many read all the subjects?

b. How many read marketing and economics only

c. How many read economics only ?

d. How many students are there altogether ?

Represent all the sets in venn diagram.

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

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

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

Express the complex number \(\frac{2-\sqrt{-25}}{1-\sqrt{-16}}\) into the form a + ib.

(a) Find the square roots of \(4 + 3i\).

(b) If \(\omega\) is an imaginary cube roots of unity then prove or disprove that

\((1 + \omega - \omega^2)^3 - (1 - \omega + \omega^2)^3 = 0\)

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

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 \(f(13)\).

Evaluate: \(\lim_{x \to 2} \frac{x^5-32}{x-2}\)

A function \(f(x)\) is defined by \(f(x) = \begin{cases} \frac{2x^2-18}{x-3} & \text{for } x \ne 3 \\ k & \text{for } x = 3 \end{cases}\)

Find the value of k so that \(f(x)\) is continuous at \(x = 3\)

Prove that: \(\lim_{x \to \infty} \sqrt{x}\left(\sqrt{x+a} - \sqrt{x}\right) = \frac{a}{2}\)

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

Find the derivatives of:

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

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

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.

Find \(\frac{dy}{dx}\) of \(y = \frac{1}{\sqrt{3x+1}}\)

Find the derivative of:

(a) \(y = \frac{1}{\sqrt[3]{6x^5-4x^3+7}}\)

(b) \(y = \frac{e^{2x}}{\log x}\)

If the demand function of monopolist is given \(P = 100 - Q\) and its cost function is given by \(C(Q) = 100 + 10Q + 4Q^2\).

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. Also, determine the intervals at which profit function is increasing or decreasing.

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

Evaluate the integrals:

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

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

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.

Find the area bounded by the curve \(y = 3x^2 - 2\), x-axis and the ordinates at \(x = 1\) and \(x = 2\).

Evaluate the integrals:

(a) \(\int \frac{x^2}{x+1}\,dx\)

(b) \(\int \frac{(1+\log x)^3}{x}\,dx\)

The demand and supply functions of a good under perfect competition are given by \(P_d = 240 - 0.6\,Q_d\) and \(P_s = 80 + 0.2Q_s\) respectively, where P and Q denote price and quantity. Find the consumer's surplus, producer's surplus and total surplus.

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

Solve the differential equation.

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

Solve the differential equation: \(\frac{dy}{dx} = 5y\)

Solve the differential equation: \((x^2 - 1)\frac{dy}{dx} + 2xy = 1\)

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 \(2\vec{a} + 3\vec{b}\).

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:

Find a vector of length 3 having direction same as the direction of vector \(\vec{a} = 5\vec{i} - 4\vec{j}\)

If \(\vec{a} = \vec{i} - 2\vec{j} + \vec{k}\) and \(\vec{b} = 2\vec{i} + 3\vec{j} - \vec{k}\) are pair of vectors then

(a) Find \(\vec{a} \times \vec{b}\)

(b) Find a unit vector perpendicular to \(\vec{a}\) and \(\vec{b}\)

(c) Show that \(\vec{a} \times \vec{b}\) is perpendicular to the vectors \(\vec{a}\) and \(\vec{b}\)

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

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

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

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

If \(P = \begin{bmatrix} -1 & 0 & 2 \\ 4 & 0 & -5 \end{bmatrix}\) and \(Q = \begin{bmatrix} 0 & 1 \\ -2 & 2 \\ 0 & 3 \end{bmatrix}\), find \(5PQ\)

Evaluate: \(\begin{vmatrix} 2 & -3 & 4 \\ -1 & 2 & 3 \\ 2 & 3 & 1 \end{vmatrix}\)

A transport company has three types trucks A, B and C which are designed to carry three different sizes of boxes P, Q and R per load as shown.

How many trucks of each type should be used to carry exactly 18 boxes of size P, 18 boxes of size Q and 21 boxes of size R? (use cramer's rule or inverse matrix method)

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.

The following table shows the yearly population of the city.

Obtain the equation of the line by least square method. Also estimate the population of the city in 2023 and 2024.