Mathematics TU Board 2080 question paper

Section A Long Answer Questions. Attempt any TWO questions.

(a) Define gradient of vector function of $f(x,y,z)$ and find the derivative of $f(x,y,z)=x^3-x{y}^2-z^2$ at $p_0(1,1,0)$ in the direction of $\vec{V}=2\vec{i}-\vec{j}+6\vec{k}$. (b) Define Volume of the solid and find the volume of the solid generated by revolving the region bounded by $y=\sqrt{x}$ and the line $y=1$, $x=4$ about the line $y=1$.

Evaluate (a) ${\int }_{\pi /4}^{\pi /2}\frac{dx}{1-\sin x}$

(b) $\int x^2\sin xdx$

(c) Solve the differential equation $\frac{dy}{dx}-\frac{3}{x}y=x$, $x>0$.

(a) State Rolle's Theorem and show that $x^3+3x+1=0$ has exactly one real solution. (b) Find the area of the region enclosed by the parabola $y=2-x^2$ and the line $y=-x$.

Section B Short Answer Questions. Attempt any EIGHT questions.

Define absolute value function and sketch the graph of absolute value.

Find the limit of $\underset{h\to \infty }{\lim }\frac{\sqrt{6h+25}-5}{h}$.

State integral test and apply it to test the convergence of the series $\sum _{n=1}^{\infty }\frac{1}{n^2+1}$.

Find the Taylor's Series generated by $f(x)=\frac{1}{x}$ at $a=2$. Where, if anywhere, does the series converge to $\frac{1}{x}$?

Define implicit differentiation and find the slope of the circle $x^2+y^2=25$ at the point $(3,-4)$.

Define partial derivative and find the value of $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ at the point $(4,-5)$ if $f(x,y)=x^2+3xy+y-1$.

Evaluate (a) $\int (\sin 2x\cos x+\cos 2x\sin x)dx$

(b) ${\int }_{\pi /4}^{\pi /2}\frac{dx}{1-\sin x}$

Determine the concavity of $y=3+\sin x$ on $[0,2\pi ]$.

Test for convergence of the series $\sum _{n=1}^{\infty }{\left(\frac{1}{n+1}\right)}^n$.