Mathematics I TU Board 2022 question paper

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 $f:\mathbb{R}\to \mathbb{R}$ 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 $A=\left(\begin{array}{cc}2& 0\\ 1& 3\end{array}\right)$ and $B=\left(\begin{array}{cc}-2& 1\\ 3& 2\end{array}\right)$, show that: $(AB{)}^T=B^T{A}^T$.

Prove that: $$\left|\begin{array}{ccc}a& b& c\\ a^2& b^2& c^2\end{array}\right|=(a-b)(b-c)(c-a)$$

Find the eccentricity and foci of the ellipse: $25x^2+4y^2=100$.

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: $$f\left(\frac{2ab}{1+a^2{b}^2}\right)=2f(ab)\text{where }|ab|<1.$$