GTU Calculus - May 2016 Exam Question Paper

GTU First Year Engineering (Semester 1)
Calculus
May 2016

Total marks: --
Total time: --

INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

Choose the most appropriate answer out of the following options given for each part of the question:

1(a)(i) The sum of the series

1 - \frac{1}{2} + \frac{1}{2^{2}} - \frac{1}{2^{3}} + \dots

$1-\dfrac{1}{2}+\dfrac{1}{2^2}-\dfrac{1}{2^3}+\cdots$ is _______
(A) 2/3 (B) 3/2 (C) 1/2 (D) None of these

1 M

1(a)(ii) The series

x + \frac{x^{3}}{3!} + \frac{x^{5}}{5!} + \dots

$x+\dfrac{x^3}{3!}+\dfrac{x^5}{5!}+\cdots$ represent expansion of _______
(A) sin x (B) cos x (C) sinh x (D) coshx

1 M

1(a)(iii) The value of the

lim_{x \to \infty} \frac{\sin x}{x}

$\lim_{x \rightarrow \infty }\dfrac{\sin x}{x}$ is _______
(A) 1 (B) -1 (C) 0 (D) None of these

1 M

1(a)(iv) The curve r=cos2θ has _______ petals.
(A) 2 (B) 4 (C) 3 (D) None of these

1 M

1(a)(v) If x=rcosθ, y=rsinθ, then

\frac{\partial (r, θ)}{\partial (x, y)}

$\dfrac{\partial (r,\theta)}{\partial (x,y)}$ = _______
(A) r (B) 1/r (C) 3 (D) None of these

1 M

1(a)(vi) If

u = x^{3} \cos (y / x), then x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y}

$u=x^3\cos\left ( y/x \right ),\text{then}\ x\dfrac{\partial u}{\partial x}+y\dfrac{\partial u}{\partial y}$ = _______
(A) u (B) 2u (C) 3u (D) None of these

1 M

1(a)(vii) The equation of a cylindrical surface x²+y²=9 becomes _______ when converted to cylindrical polar coordinates.
(A) r=9 (B) r² = 9 (C)

r = \pm 3

$r=\pm 3$ (D) r=3

1 M

1(b)(i) The series

\sum_{n = 1}^{\infty} \frac{1}{(\log n)^{n}}

$\sum ^{\infty}_{n=1}\dfrac{1}{(\log n)^n}$ is _______
(A) oscillatory (B) divergent (C) convergent (D) None of these

1 M

1(b)(ii) If in the equation of a curve, x occurs only as an even power then the curve is symmetrical about
(A) x-axis (B) y-axis (C) Origin (D) None of these

1 M

1(b)(iii) The value of the

lim_{x \to 0} (\cos x)^{\cot x}

$\lim_{x\rightarrow 0}(\cos x)^{\cot x}$ is _______
(A) 0 (B) 1 (C) &infin (D) None of these

1 M

1(b)(iv) The integral

\int_{- a}^{a} f (x) d x = 2 \int_{0}^{a} f (x) d x

$\int ^a_{-a}f(x)dx=2\int ^a_0f(x)dx$ if _______
(A) f is an odd function (B) f is neither even nor odd function (C) f is a even function (D) None of these

1 M

1(b)(v) The volume of the solid generated by revolving the region between the y - axis and the curve (x=2\sqrt{y},0\leq y\leq 4, \) about the y ' axis.
(A) 2π (B) 32π (C) 16π (D) None of these

1 M

1(b)(vi)

\int_{0}^{2} \int_{0}^{x^{2}} e^{y / x} d y d x

$\int ^2_0 \int ^{x^2}_0e^{y/x}dydx$ is equal to _______ (A) e²-1 (B) e² (C) e²+1 (D) e^-2

1 M

1(b)(vii) A point ( a , b ) is said to be a saddle point if at ( a , b )
(A) rt-S² >0 (B) rt-S² = 0 (C) rt-S² < 0 (D) rt-S² ≥ 0

1 M

2(a) Expand

(\frac{π}{4} + x)

$\left (\dfrac{\pi}{4}+x \right)$ in powers of x by using the Taylor's series. Also, find the value of tan 44°.

4 M

2(b) Evaluate:

lim_{x \to 0} {(\frac{1}{x})}^{\tan x} .

$\lim_{x\rightarrow 0}\left ( \dfrac{1}{x} \right )^{\tan x}.$

3 M

2(c)(i) Test for convergence the series

x - \frac{x^{3}}{3} + \frac{x^{5}}{5} - \dots, x > 0.

$x-\dfrac{x^3}{3}+\dfrac{x^5}{5}-\cdots,\ \ x>0.$

3 M

2(c)(ii) Trace the curve r² = a²sin 2θ.

4 M

3(a) If

θ = t^{n} e^{- \frac{r^{2}}{4 t}} then find n so that \frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} \frac{\partial θ}{\partial r}) = \frac{\partial θ}{\partial t}

$\theta=t^n e^{-\dfrac{r^2}{4t}}\text {then find n so that }\ \frac{1}{r^2}\frac{\partial }{\partial r}\left ( r^2\frac{\partial \theta }{\partial r} \right )=\frac{\partial \theta}{\partial t}$ .

4 M

3(b) Determine the continuity of the function

f (x, y) = {\begin{matrix} (x^{2} + y^{2}) \sin (\frac{1}{x^{2} + y^{2}}), & (x, y) \neq (0, 0) \\ 0, & (x, y) = (0, 0) \end{matrix} at origin .

$f(x,y)=\left\{\begin{matrix} (x^2+y^2)\sin \left ( \dfrac{1}{x^2+y^2} \right ) , & (x,y) \neq (0,0)\\ 0 , & (x,y)=(0,0) \end{matrix}\right. \ \ \ \ \text {at origin}.$

3 M

3(c)(i) If u = tan^-1(x²+2y²), show that x²u_xx + 2xy u_xy + y²u_yy = 2sin u.cos 3u

4 M

3(c)(ii) If

u = f (\frac{y - x}{x y}, \frac{z - x}{x z}), show that x^{2} \frac{\partial u}{\partial x} + y^{2} \frac{\partial u}{\partial y} + z^{2} \frac{\partial u}{\partial z} = 0

$u=f\left ( \dfrac{y-x}{xy},\dfrac{z-x}{xz} \right ), \text{show that}\ x^2\dfrac{\partial u}{\partial x}+y^2\frac{\partial u}{\partial y}+z^2\dfrac{\partial u}{\partial z}=0$

3 M

4(a) Find the extreme values of the function f(x, y) = x³+3xy²-15x²-15y²+72x.

4 M

4(b) Find the equation of the tangent plane and the normal line of the surface

\frac{x^{2}}{4} + y^{2} + \frac{z^{2}}{9} = 3 at (- 2, 1, - 3)

$\dfrac{x^2}{4}+y^2+\dfrac{z^2}{9}=3\ \text{at}\ (-2,1,-3)$ .

3 M

4(c)(i) Show that the rectangular solid of maximum volume that can be inscribed in a sphere is cube.

4 M

4(c)(ii) Expand xy²+xy+3 in powers of (x-1) and (y+2) using Taylor's series expansion.

3 M

5(a) Find the radius of convergence and interval of convergence of the series

\sum_{n = 0}^{\infty} \frac{(- 3)^{n} x^{n}}{\sqrt{n + 1}}

$\sum ^{\infty}_{n=0}\dfrac{(-3)^n x^n}{\sqrt{n+1}}$

4 M

5(b) Test the convergence of the series

\sum_{n = 1}^{\infty} \sqrt{n^{4} + 1} - \sqrt{n^{4} - 1}

$\sum ^{\infty}_{n=1}\sqrt{n^4+1}-\sqrt{n^4-1}$ .

3 M

5(c)(i) Define geometric series and show that the geometric series is :
(i) convergent if |r| < 1, (ii) divergent if r ≥ 1

4 M

5(c)(ii) Express (x-1)⁴+2(x-1)³+5(x-1)+2 in ascending power of x.

3 M

6(a) Evaluate

\iint_{R} (x + y) d y d x

$\iint _R (x+y)dy\ dx$ , where R is the region bounded by x=0, x=2, y=x, y=x+2.

4 M

6(b) Evaluate

\iint_{R} r^{3} \sin 2 θ d r d θ

$\iint _R r^3 \sin 2\theta \ dr\ d\theta$ over the area bounded in the first quadrant between the circles r=2 and r=4.

3 M

6(c)(i) Evaluate

\int_{1}^{e} \int_{1}^{\log y} \int_{1}^{e^{x}} \log z d z d x d y

$\int ^e_1 \int ^{\log y}_1 \int ^{e^x}_1 \log z\ dz\ dx\ dy$

4 M

6(c)(ii) Evaluate

lim_{x \to \frac{π}{2}} (\cos)^{\frac{π}{2} - x} .

$\lim_{x\rightarrow \frac{\pi}{2}}(\cos)^{\frac{\pi}{2}-x}.$

3 M

7(a) Find the volume of the solid generated by revolving the curve x^2/3+y^2/3=a^2/3 about the x-axis.

4 M

7(b) Find the area lying inside the circle r = a sinθ and outside the cardioid r = a (1 ' cosθ).

3 M

7(c)(i) Evaluate

\int_{0}^{1} \int_{x}^{\sqrt{2 - x^{2}}} \frac{x}{\sqrt{x^{2} + y^{2}}} d y d x

$\int _0^1 \int _x^{\sqrt{2-x^2}}\dfrac{x}{\sqrt{x^2+y^2}}dy\ dx$ by changing the order of integration.

4 M

7(c)(ii) Check the convergence of

\int_{0}^{5} \frac{1}{x^{2}} d x .

$\int ^5_0 \dfrac{1}{x^2}dx.$

3 M

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