Mathematics I TU Board 2024 question paper

Solve the inequality $3+2x-x^2\ge 0$.

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

If a, b, c, and d are in G.P., prove that $a^2-b^2,b^2-c^2,c^2-d^2$ are also in G.P.

Prove that $$\left|\begin{array}{ccc}1+x& 1& 1\\ 1& 1+y& 1\\ 1& 1& 1+z\end{array}\right|=xyz\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)$$

Find the equation of the ellipse whose latus rectum is 5 and the eccentricity is $\frac{1}{\sqrt{2}}$.

If $\vec{a}=\sqrt{3}\hat{i}+\hat{j}$ and $\vec{a}\times \vec{b}=(1,2,2)$, find the angle between $\vec{a}$ and $\vec{b}$.

How many numbers of three different digits less than 500 can be formed from the integers 1, 2, 3, 4, 5, and 6?

Prove that $$\frac{3+4i}{1-i}+\frac{3-4i}{1+i}$$ is a real number. b. If $x^2+y^2=11xy$, prove that $$\log {\left(\frac{x-y}{3}\right)}^2=\frac{1}{2}(\log x+\log y)$$

a.Find the Maclaurin series of the function $f(x)=\cos x$. b.Take any matrix of order 3 × 3 and express it as a sum of symmetric and skew-symmetric matrix.

a.Find the equation of a hyperbola in standard form having focus (-2, 0) and Directrix $x=-\frac{1}{2}$. b.In an examination paper on mathematics, 20 questions are set. In how many different ways you can choose 18 questions to answer?