BCA Semester 1 – Mathematics I – Unit 1: Set Theory and Real & Complex Number

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

If $X+iy=1+i1−i,$ show that $x2+y2=1.$

Out of 500 people, 285 like tea, 195 like coffee, 115 like lemon juice, 45 like tea and coffee, 70 like tea and juice, 50 like juice and coffee. If 50 do not like any drinks:
$\quad$i) How many people like all three drinks?
$\quad$ii) How many people like only one drink?

If $x - i y = \frac{3-2i}{3+2i}$, prove that $x^2 + y^2 = 1.$

In an examination, $58\%$ failed in Accounts, $39\%$ in English, and $25\%$ in Statistics;
$32\%$ failed in Accounts and English, $19\%$ in Accounts and Statistics,
$17\%$ in English and Statistics, and $13\%$ failed in all three subjects.
Find:
$\quad$i) What percent passed in all three subjects?
$\quad$ii) What percent failed in exactly two subjects?

If $\sqrt{x + ty} = a + ib$, find $\sqrt{x - ty}$ and $x^2 + y^2$.

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

x^2 + y^2 = 1. $

Solve the inequality: $6 + 5x - x^2 \ge 0$

a) If $A$ and $B$ are two subsets of universal set $U$ such that$n(U) = 350, \quad n(A) = 100, \quad n(B) = 150, \quad \text{and} \quad n(A \cap B) = 50,$ then find $n(A' \cap B')$.
b) If $a,b,c$ are in $A.P., b,c,d$ are in $G.P.,$ and $c,d,e$ are in $H.P.,$ then prove that $a,c,e$ are in $G.P.$

Solve the inequality: $3 + 2x - x^2 \ge 0$

Prove that

\quad\quad\quad\quad\quad\log \left( \frac{x - y}{3} \right)^2 = \frac{1}{2} (\log x + \log y) $