Mathematics I TU Board 2020 question paper

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: i) How many people like all three drinks? ii) How many people like only one drink?

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

If $H$ is the harmonic mean between $a$ and $b$, prove that: $$\frac{1}{H-a}+\frac{1}{H-b}=\frac{1}{a+b}$$

Define singular and non-singular matrix. Find the inverse of the matrix $A$: $$A=\left[\begin{array}{ccc}1& -2& -1\\ 2& 1& 1\\ 3& -5& 8\end{array}\right]$$

Find the focus, vertex, equation of axis, equation of directrix, and length of latus rectum of the ellipse: $$4x^2+9y^2=36$$

If $\theta$ is the angle between two unit vectors $\vec{a}$ and $\vec{b}$, show that: $$\left|\vec{a}-\vec{b}\right|=\sin \frac{\theta }{2}$$

How many numbers of at least three different digits can be formed using the digits 1, 2, 3, 4, 5, 6?

Prove by vector method: $\cos (A+B)=\cos A\cdot \cos B-\sin A\cdot \sin B$.

Find the equation of the circle passing through the points (1, 2), (3, 1), and (-3, -1).

Let $f:\mathbb{R}\to \mathbb{R}$ and $g:\mathbb{R}\to \mathbb{R}$ be defined by $f(x)=2x+3$ and $g(x)=x^2-1$. Find: i) $f\circ g$ ii) $f+g$ iii) $g\circ f$ iv) $f\circ (f\circ g)$