BCA 1st Semester
Math I Board Question Paper 2019

In a class of 100 students, 40 failed in mathematics, 70 in English, and 20 in both subjects. Find:
a) How many students passed in both subjects?
b) How many passed in Mathematics only?
c) How many failed in mathematics only?

Find the domain and range of the function \( f(x) = \frac{2x + 1}{3 - x} \).

Find the Maclaurin series of the function \( f(x) = \sin x \).

Prove that: \[ \begin{vmatrix} x & x^2 \\ 1 & y \\ z & z^2 \end{vmatrix} = (x - y)(y - z)(z - x) \]

Find a unit vector perpendicular to the plane containing points \( P(1, -1, 0) \), \( Q(2, 1, -1) \), and \( R(-1, 1, 2) \).

In how many ways can the letters of the word "Sunday" be arranged?
How many arrangements begin with S?
How many begin with S and do not end with \( y \)?

a) Define Conic section. Find the coordinates of vertices, eccentricity, and foci of the ellipse: \[ 9x^2 + 4y^2 - 18x - 16y - 11 = 0 \]
b) If \( T: \mathbb{R}^2 \rightarrow \mathbb{R}^3 \) is defined by \( T(x_1, x_2) = (x_1 + x_2, x_2, x_1) \), find the matrix associated with \( T \).

a) Define irrational number. Prove that \( \sqrt{2} \) is irrational.
b) For functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) (\( f(x) = 2x + 1 \)) and \( g: \mathbb{R} \rightarrow \mathbb{R} \) (\( g(x) = x^2 - 2 \)):
Find \( f \circ g \) and \( g \circ f \), and verify \( f \circ g \neq g \circ f \).

a) If \( A \), \( G \), and \( H \) are the arithmetic, geometric, and harmonic means between two unequal positive numbers, prove \( A > G > H \).
b) State the relation between permutation and combination of \( n \) objects taken \( r \) at a time. A committee of 5 is formed from 6 boys and 5 girls. How many ways include at least one boy and one girl?