Mathematics I TU Board 2019 question paper

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: $$\left|\begin{array}{cc}x& x^2\\ 1& y\\ z& z^2\end{array}\right|=(x-y)(y-z)(z-x)$$

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 $y$?

If $X+iy=\sqrt{\frac{1+i}{1-i}}$, show that $x^2+y^2=1$.

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 $T:{\mathbb{R}}^2\to {\mathbb{R}}^3$ 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 $f:\mathbb{R}\to \mathbb{R}$ ($f(x)=2x+1$) and $g:\mathbb{R}\to \mathbb{R}$ ($g(x)=x^2-2$): Find $f\circ g$ and $g\circ f$, and verify $f\circ g\ne g\circ f$.

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?