BCA 1st Semester
Math I Board Question Paper 2022

In a certain village in Nepal, all the people speak Nepali or Tharu or both languages. If 90% speak Nepali and 20% speak Tharu, how many people speak:
i) Nepali language only
ii) Tharu language only
iii) both languages

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

Define a function. Show that the function \( f: \mathbb{R} \rightarrow \mathbb{R} \) defined by \( f(x) = 3x + 5 \) is bijective.

If \( A \) is the A.M. and \( H \) is the H.M. between two numbers \( a \) and \( b \), show that: \[ \frac{a - A}{a - H} \times \frac{b - A}{b - H} = \frac{A}{H} \]

Define matrix. If \( A = \begin{pmatrix} 2 & 0 \\ 1 & 3 \end{pmatrix} \) and \( B = \begin{pmatrix} -2 & 1 \\ 3 & 2 \end{pmatrix} \), show that: \( (AB)^T = B^T A^T \).

Prove that: \[ \begin{vmatrix} a & b & c \\ a^2 & b^2 & c^2 \end{vmatrix} = (a - b)(b - c)(c - a) \]

a) A bag contains 8 red balls and 5 blue balls. In how many ways can 3 red balls and 4 blue balls be drawn?
b) Find the volume of the parallelepiped whose concurrent edges are represented by the vectors \( \vec{i} - 2\vec{j} + 3\vec{k} \), \( -3\vec{i} + 4\vec{j} - 5\vec{k} \), and \( \vec{i} + 2\vec{j} - 3\vec{k} \).

a) Find the Taylor Series expansion of \( f(x) = x^3 - 2x + 4 \) at \( a = 2 \).
b) In how many ways can the letters of the word 'CALCULUS' be arranged so that the two C's do not come together?

Define exponential and logarithmic functions. If \( f(x) = \log \frac{1 - x}{1 + x} \) for \( -1 < x < 1 \), show that: \[ f\left(\frac{2ab}{1 + a^2 b^2}\right) = 2f(ab) \quad \text{where } |ab| < 1. \]