Mathematics I TU Board 2021 question paper

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

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

A class consists of boys whose ages are in A.P. (common difference = 4 months). The youngest boy is 8 years old, and the total age of the class is 168 years. Find the number of boys.

If $A=\left(\begin{array}{cc}4& 0\\ 0& 5\end{array}\right)$, find a matrix $X$ such that $AX=\left(\begin{array}{cc}1& 2\\ 2& 4\end{array}\right)$.

Find the equation of the ellipse whose latus rectum is 3 and eccentricity is $\frac{1}{\sqrt{2}}$.

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

A committee of 5 is to be constituted from 6 boys and 5 girls. How many ways can this be done to include at least one boy and one girl?

Define a function, its domain, and range. Find the domain and range of $f(x)=\sqrt{2-x-x^2}$.

a) Three numbers in A.P. sum to 15. If 1, 3, 9 are added to them respectively, they form a G.P. Find the original numbers. b) Find the sum to $n$ terms of the series $\frac{1}{2}+\frac{2}{4}+\frac{3}{8}+\cdots$.

a) Find the angle between vectors $\mathbf{u}=4\mathbf{i}-2\mathbf{j}+\mathbf{k}$ and $\mathbf{v}=\mathbf{i}+\mathbf{j}-\mathbf{k}$. b) Find the Maclaurin series of $f(x)=\cos x$.