Mathematics I TU Board 2023 question paper

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

Find the domain and range of the function $f(x)=\sqrt{6-x-x^2}$.

Prove that the function $f:\mathbb{R}\to \mathbb{R}$ defined by $f(x)=3x-1$ is bijective.

Expand $e^x$ about $x=0$ by using the Maclaurin series.

Find the inverse matrix of the matrix $\left(\begin{array}{ccc}1& 4& 1\\ 3& 3& -2\\ 0& -4& 1\end{array}\right)$.

There are 7 men and 3 ladies. Find the number of ways in which a committee of 6 persons can be formed if the committee should have at least one lady.

Find the equation of a parabola having vertex (+, 2) and directrix $x=4$.

a) If $A$ and $B$ be two subsets of universal set $U$ such that $n(U)=350$, $n(A)=100$, $n(B)=150$, and $n(A\cap B)=50$, then find $n(\overline{A}\cap \overline{B})$. 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.

a) Prove that $\left|\begin{array}{ccc}1& a& bc\\ 1& b& ca\\ 1& c& ab\end{array}\right|=(a-b)(b-c)(c-a)$. b) By using the vector method, prove that $\cos (A-B)=\cos A\cos B+\sin A\sin B$.

a) Define a parabola with different parts using the figure and derive the standard equation of parabola $y^2=4ax$. b) In how many ways can the letters of the word "ARRANGE" be arranged so that all the vowels are always together?