BITM 1st Semester
Basic Mathematics Board Question Paper 2022

Give brief answer of the following questions.
a) Find \(\gamma(A-B)\), where \(A = \begin{pmatrix} 3 & 2 & 6 \\ 4 & 5 & 7 \end{pmatrix}\) and \(B = \begin{pmatrix} 1 & 3 & 6 \\ -3 & 2 & 5 \end{pmatrix}\).
b) Find the value of determinant:
\[ \begin{vmatrix} 1 & 2 & 2 \\ 2 & 3 & 2 \\ 3 & 4 & 3 \end{vmatrix} \]
c) Evaluate: \( \lim_{x \to \infty} \frac{7x^{2+5}x-2}{2x^{3+5}x-5} \)

d) Find the area of curve bounded by x-axis and ordinates of y=3x² from x₁=1 and x₂=2.
e) Solve the following differential equation: \() \frac{dy}{dx} = 3x + 1 \)

f) If \(A = \{a, b, c, d, e, f, g, h\}, B = \{e, f, g, h, i, j, k\}\) then find A ∪ B and A ∩ B.

g) Express the following complex number into polar form \(Z = 1 + i\sqrt{3}\)

h) Rewrite the following absolute value sign -5 ≤ x ≤ 11

i) Show that the following pair of vectors are orthogonal −5⁴ + 4j⁷ -k, 5⁴ + 7j⁷ + 3k where i, j and k are the vectors.

j) Find the derivative of \(y = 3x² + e^x - \frac{1}{x} + logx\)

(a) find the square roots of complex number \(z = 1 + i\sqrt{3}\)

(b) Express the following complex number in the form of a + ib and find the modules \(z = \frac{3-\sqrt{-25}}{2-\sqrt{-16}}\)

\(if(x) = \frac{2ax+b}{x-1}\), \(\lim_{x \to \infty} f(x) = -3\) and \(\lim_{x \to \infty} f(x) = 4\), prove that \(f(2) = 11\).

A function f(x) is defined as: f(x) = \( \begin{cases} x² + 2 & for x < 3 \\ 9 & for x = 3 \\ 4x - 1 & for x > 3 \end{cases} \) is the function continuous at x=3? If not, how can you make it continuous?

Find \(\frac{dy}{dx}\) of the following function: (a) \(y = \frac{1}{\sqrt{25x-3}-\sqrt{2x-5}}\) (b) \(x³ - y³ = a³\)

Evaluate the following integrals:
(a) \(\int_{1+x}^{x} dx\)
(b) \(\int_{0}^{1} f(5x + 3)\sqrt{2x + 1} dx\)

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

Solve the following linear differential equation: \(x\frac{dy}{dx}+y=x^{2}\)

In a survey of 100 students of a campus, the number of students who read various newspaper were found to be as follows:

The following table shows the yearly income of a family:

There are three foods X, Y and Z in a restaurant. A packet of food X contains 1 unit of calcium, 2 units of protein and 3 units of carbohydrate. A packet food Y contains 2 units of calcium, 3 units of protein and 1 unit of carbohydrate. A packet food Z contains 3 units of calcium, 1 unit of protein and 2 unit of carbohydrate. If the price of foods X,Y and Z are Rs 27, Rs 16 and Rs 19 respectively. Find the price per unit of calcium, protein and carbohydrate. ( Use Cramer’s rule or Inverse matrix method).

The demand and supply function under perfect competition are pd =20-5Q and PS = 4+3Q respectively, where p and Q denote price and quantity. Find the consumer’s surplus, producer’s surplus and total surplus.

A silver company product x quintals of silver per week and it’s total cost (Rs) is expressed as: \(C(x) = 480x - 15x^2 + \frac{1}{3}x^3\)
a. Find the minimum value of the marginal cost.
b. Find the minimum value of average cost.
c. Find the output at which marginal cost is equal to average cost.
d. Show that the marginal cost and average cost are equal at the minimum average cost.