BITM 1st Semester
Basic Mathematics Board Question Paper 2023

Brief Answer Questions:
a. If \( A = \{1, 2, 3, 4, 5\}, B = \{3, 4, 5, 6, 7\} \) and \( C = \{1, 3, 5, 7\} \), find \((A \cap B \cap C)\).

b. Express the complex number \(-1 + i\sqrt{3}\) into polar form.

c. Rewrite \(-4 \leq x \leq -1\) by using the modulus sign.

d. Evaluate: \(\lim_{x \to \infty} \frac{5x^3 + 3x + 7}{2x^3 + 7x + 9} \)

e. Find a unit vector perpendicular to each of the vectors \(\vec{a} = \vec{i} + 3\vec{j} + 2\vec{k}\) and \(\vec{b} = 2\vec{i} - 4\vec{j} + \vec{k}\)

f. Find the derivative of \( y = e^{2x} \)

g. If \( A = \begin{bmatrix} 2 & -4 \\ 4 & 1 \end{bmatrix} \) and \( B = \begin{bmatrix} 3 & 6 \\ 5 & 2 \end{bmatrix} \), find \( 5(A + B) \).

h. Find the area bounded by the line \( y = 2x + 3 \), the x-axis and the ordinates at \( x = 2 \) and \( x = 4 \)

i. Solve the different equation: \( \frac{dy}{dx} = 3x^2 \)

j. Find the value of determinant: \( \begin{vmatrix} 1 & 3 & 2 \\ 3 & 4 & 1 \\ 2 & 5 & 1 \end{vmatrix} \)

(a) If \(\sqrt{a - ib} = x - iy\), prove that \(\sqrt{a + ib} = x + iy\)
(b) Find the square roots of \( 7 - 24i \)

A function \( f(x) \) is defined as follows:
\[ f(x) = \begin{cases} 2x + 5 & for \quad x < 3 \\ 3x + 2 & for \quad x = 3 \\ 2x^2 - 7 & for \quad x > 3 \end{cases} \]
Find \( \lim_{x \to 3} f(x) \) if it exists. Discuss the continuity of the function \( f(x) \) at \( x = 3 \).

Evaluate: \[ \lim_{x \to 2} \frac{x - \sqrt{8 - x^2}}{\sqrt{x^2 + 12} - 4} \]

Find the derivatives of:
(a) \( y = \frac{1}{\sqrt{2x + 3} - \sqrt{2x - 3}} \)
(b) \( x^3 + y^3 = 27 \)

Evaluate the integrals:
(a) \( \int x^2 e^x dx \)
(b) \( \int x^2 \cdot logx \, dx \)

Prove or disprove that the vectors \( \vec{a} - 2\vec{b} + 3\vec{c}, -2\vec{d} + 3\vec{b} - 4\vec{c} \) and \( \vec{a} - 3\vec{b} + 5\vec{c} \) are coplanar.

Solve the differential equation: \( (1 + x^2) \frac{dy}{dx} + 2xy = 4x^2 \)

A survey of 500 students who read various newspapers produced the following information:
280 read Kathmandu Post, 190 read Rising Nepal, 110 read Himalayan Times, 75 read Kathmandu Post and Rising Nepal, 50 read Rising Nepal and Himalayan Times, 45 read Kathmandu Post and Himalayan Times. If 55 students read none of the newspapers, find how many of them read
a. All three newspapers
b. Two newspapers only
c. One newspaper only
Represent all the sets in venn diagram.

The following table shows annual profits in thousand rupees in an industrial concern.

Solve the following equations by using determinant or matrix method.
\[ 2x + 5y + 7z = 12 \quad x + 2y - z = 0 \quad x + y + z = 9 \]

The demand and supply functions for a good are
\[ P_d = 50 - 2Q_d \text{ and } P_s = 14 + 4Q_s \]
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 at equilibrium.
c. Also, find total surplus.

The total cost and demand function for a company are
\[ TC = \frac{1}{3}Q^3 - 15Q^2 + 480Q + 750 \text{ and } \]
\[ P = 536 - 2Q \text{ respectively.} \]
a. Find the revenue function and profit function.
b. Determine the level of output Q for which profit is maximized.
c. Find the maximum value of marginal profit.
d. Find the maximum revenue.