GTU Calculus - December 2013 Exam Question Paper

GTU First Year Engineering (Semester 1)
Calculus
December 2013

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

Answer the following questions by choosing the most appropriate answer:

1 (a) (i)

W h a t i s \int \frac{x - 3}{x} d x ? (a) 1 - 3 l n x + C (b) x - 3 l n x + C (c) 1 + \frac{3}{x^{2}} + C (d) \frac{x^{2} - 3 x}{x^{2}} + C

$What \ is \ \int \dfrac {x-3}{x}dx? \\(a) \ 1-3 \ ln \ x+C \\ (b) x-3 \ ln \ x+C \\ (c) \ 1+\dfrac {3}{x^{2}}+C \\ (d) \dfrac {x^2-3x}{x^2}+C$

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1 (a) (ii)

i f \tan y + x^{3} = y^{2} + 1 a n d \frac{d x}{d t} = - 2, w h a t i s t h e v a l u e o f \frac{d y}{d t} a t t h e p o i n t (1, 0)

$if \ \tan y+x^3=y^2+1 \ and \ \dfrac {dx}{dt}=-2, \\ what \ is \ the \ value \ of \dfrac {dy}{dt} \ at \ the \ point \ (1,0)$
(A) -6
(B) -2.5
(C) 0
(D) 6

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1 (a) (iii) The values of x for which the graphs of y=x and y² =4x intersect are
(A) 4 and 4 (B) -4 and 4 (C) 0 and 4 (D) 0 and -4

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1 (a) (iv) The value of the limit

lim_{x \to 0} \frac{\tan x}{x} i s

$\lim_{x\rightarrow 0}\dfrac {\tan x}{x} \ is$
(A) 0
(B) 1
(C) π
(D) ∞

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1 (a) (v)

i f y = \frac{e^{x} - e^{- x}}{e^{x} + e^{- x}}

$if \ y=\dfrac {e^x-e^{-x}}{e^x+e^{-x}}$ then the derivative of the function y w.r.t. x is

(A) 0 (B) 1 (C) \frac{2}{(e^{x} + e^{- x})^{2}} (D) \frac{4}{(e^{x} + e^{- x})^{2}}

$(A)\ 0 \\ (B) \ 1 \\ (C) \ \dfrac {2}{(e^x+e^{-x})^2} \\ (D) \ \dfrac {4}{(e^x+e^{-x})^2}$

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1 (a) (vi)

i f y = l n (\sqrt{2} x)

$if \ y=ln (\sqrt{2}\ x)$ the derivative of the function y w.r.t. x is

(A) \frac{\sqrt{2}}{x} (B) \frac{1}{\sqrt{2} x} (C) \frac{1}{2 x} (D) \frac{1}{x}

$(A) \ \dfrac {\sqrt{2}}{x} \\ (B) \ \dfrac {1}{\sqrt{2} \ x} \\ (C) \ \dfrac {1}{2x} \\ (D) \ \dfrac {1}{x}$

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1 (a) (vii) The sum of the squares of two positive numbers is 200; their minimum product is
(A) 200 (B) 25 √7 (C) 28 (D) none of these

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Answer the following questions by choosing the most appropriate answer

1 (b) (i) The value of the integral

\int \frac{\sin \sqrt{x}}{\sqrt{x}} d x i s (A) - 2 \cos^{1 / 2} x + C (B) - \cos \sqrt{x} + C (C) - 2 \cos \sqrt{x} + C (D) \frac{1}{2} \cos \sqrt{x} + C

$\int \dfrac {\sin \sqrt{x}}{\sqrt{x}}dx \ is \\ (A) \ -2\cos^{1/2}x+C \\ (B) \ -\cos \sqrt{x}+C \\ (C) \ -2 \cos \sqrt{x}+C \\ (D) \ \dfrac {1}{2}\cos\sqrt{x}+C$

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1 (b) (ii) The value of the limit

lim_{x \to 0} \frac{| x |}{x} i s

$\lim_{x\rightarrow 0}\dfrac {|x|}{x} \ is$
(A) 0 (B) 1
(C) π (D) ∞

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1 (b) (iii) The value of the integral

\int \frac{l n y}{y^{2}} i s

$\int \dfrac {ln \ y} {y^2} \ is$

(A) \frac{1}{y} (1 - l n y) + C (B) \frac{1}{2 y} l n^{2} y + C (C) \frac{1}{3 y^{3}} (4 l n y + 1) (D) - \frac{1}{y} (l n y + 1) + C

$(A)\ \dfrac {1}{y}(1-ln \ y)+C \\ (B) \ \dfrac {1}{2y}ln^2 \ y+C \\ (C) \ \dfrac {1}{3y^3}(4\ ln\ y+1)\\ (D)\ - \dfrac {1}{y}(ln \ y+1)+C$ For

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1 (b) (iv) f(x) = x² + 2x -1, 0 < x < 1 the value of C is mean value theorem is
(A) /2 (B) 0 (C) 1 (D) 1/3

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1 (b) (v) The total area bounded by x-axis and x y sin = is equal to (a) 4 (b) 1 (c) -1 (d) 0

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1 (b) (vi)

i f A = [\begin{matrix} - 5 & - 3 \\ 2 & 1 \end{matrix}] t h e n A^{- 1} i s : (A) [\begin{matrix} - 5 & - 3 \\ 2 & - 1 \end{matrix}] (B) [\begin{matrix} - 1 & - 3 \\ 2 & 5 \end{matrix}] (C) [\begin{matrix} 1 & 3 \\ - 2 & - 5 \end{matrix}] (D) [\begin{matrix} 5 & 3 \\ - 2 & 1 \end{matrix}]

$if \ A=\begin{bmatrix} -5&-3 \\2 &1 \end{bmatrix} \ then \ A^{-1}\ is: \\ (A) \ \begin{bmatrix}-5 &-3 \\2 &-1 \end{bmatrix} \\ (B) \ \begin{bmatrix} -1&-3 \\2 &5 \end{bmatrix} \\ (C) \ \begin{bmatrix} 1&3 \\-2 &-5 \end{bmatrix} \\ (D) \ \begin{bmatrix} 5&3 \\-2 &1 \end{bmatrix}$

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1 (b) (vii) The function 2x³ + 3x² - 12x + 7 is decreasing in
(A) [-2,1] (B) -[-2,1] (C) [0,2] (D) [1,3]

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2 (a) (i) Test the convergence of the series

\frac{1.2}{3^{2} {.4}^{2}} + \frac{3.4}{5^{2} {.6}^{2}} + \frac{5.6}{7^{2} {.8}^{2}} + . . . .

$\dfrac {1.2}{3^2.4^2}+ \dfrac {3.4}{5^2.6^2}+\dfrac {5.6}{7^2.8^2}+....$

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2 (a) (ii) Find value of xfor which the given series

\frac{1}{2 \sqrt{1}} + \frac{x^{2}}{3 \sqrt{2}} + \frac{x^{4}}{4 \sqrt{3}} + \frac{x^{6}}{5 \sqrt{4}} + . . . . c o n v e r g e s .

$\dfrac {1}{2\sqrt{1}}+ \dfrac {x^2}{3\sqrt{2}}+ \dfrac {x^4}{4\sqrt{3}}+ \dfrac {x^6}{5\sqrt{4}} + .... \ converges.$

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2 (b) (i) Determine convergence or divergence of series

\sum_{n = 1}^{\infty} \frac{(2 n^{2} - 1)^{\frac{1}{3}}}{(3 n^{3} + 2 n + 5)^{\frac{1}{4}}}

$\sum^{\infty}_{n=1}\dfrac {(2n^2-1)^{\frac {1}{3}}}{(3n^3+2n+5)^{\frac {1}{4}}}$

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2 (b) (ii) Determine absolute or conditional convergence of the series

\sum_{n = 1}^{\infty} (- 1)^{n} \cdot \frac{n^{2}}{n^{3} + 1}

$\sum^{\infty}_{n=1}(-1)^n \cdot \dfrac {n^2}{n^3+1}$

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3 (a) (i) Find the expansion of

\tan (x + \frac{π}{4})

$\tan \left ( x+\dfrac {\pi}{4} \right )$ in ascending powers of x upto terms in x⁴ and find approximately the value of tan 43°.

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3 (a) (ii)

P r o v e t h a t : \tan^{- 1} (\frac{\sqrt{1 + x^{2}} - 1}{x}) = \frac{1}{2} (x - \frac{x^{3}}{3} + \frac{x^{5}}{5} - \frac{x^{7}}{7} + . . . .)

$Prove\ that: \ \tan^{-1}\left ( \dfrac {\sqrt{1+x^2}-1}{x} \right )=\dfrac {1}{2} \left ( x-\dfrac {x^3}{3}+ \dfrac {x^5}{5}-\dfrac {x^7}{7}+.... \right )$

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3 (b) (i)

E v a l u a t e (i) lim_{x \to 0} (\cos x)^{\cot x} (i i) lim_{x \to \infty} (a^{\frac{1}{x}} - 1) x

$Evaluate \\ (i) \ \lim_{x\rightarrow 0}(\cos x)^{\cot x} \\ (ii) \ \lim_{x\rightarrow \infty}\left ( a^{\frac {1}{x}}-1 \right )x$

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

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4 (a) (i) Evaluate the iterated integral

\int_{0}^{1} \int_{x}^{1} \sin (y^{2}) d y d x

$\int^{1}_{0}\int^{1}_{x}\sin (y^2)dy \ dx$

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4 (a) (ii) Evaluate the integral

\int_{0}^{4 a} \int_{\frac{y^{2}}{4 a}}^{y} \frac{x^{2} - y^{2}}{x^{2} + y^{2}} d x d y

$\int^{4a}_{0}\int^{y}_{\frac {y^2}{4a}}\dfrac {x^2-y^2}{x^2+y^2}dx \ dy$ by transforming into Polar coordinates.

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4 (b) (i) Evaluate the triple integral

\int_{0}^{1} \int_{0}^{π} \int_{0}^{π} y \sin z d x d y d z .

$\int^1_0 \int^\pi_0 \int^\pi_0 y \sin z \ dx \ dy \ dz.$

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4 (b) (ii) Find the area common to both of the circles r=cos θ and r=sin θ

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5 (a) (i) Determine the set of points at which the given function is continuous

f (x, y) = {\begin{matrix} \frac{3 x^{2} y}{x^{2} + y^{2}} & i f & (x, y) e (0, 0) \\ 0 & i f & (x, y) = (0, 0) \end{matrix}

$f(x,y)=\left\{\begin{matrix}\dfrac {3x^2 y}{x^2+y^2} & if &(x,y)e (0,0) \\ 0 & if &(x,y)=(0,0)\end{matrix}\right.$

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5 (a) (ii) If z=x²y+3xy⁴, where x= sin 2t and y=cos t, find

\frac{d x}{d t}

$\dfrac {dx}{dt}$ when t=0

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5 (b) (i)

I f u = \sin^{- 1} \frac{x + y}{\sqrt{x} + \sqrt{y}} t h e n p r o v e t h a t : (A) 2 x \frac{\partial u}{\partial x} + 2 x \frac{\partial u}{\partial y} = \tan u (B) x^{2} \frac{\partial^{2} u}{\partial x^{2}} + 2 x y \frac{\partial^{2} u}{\partial x \partial y} + y^{2} \frac{\partial^{2} u}{\partial y^{2}} = \frac{1}{4} (\tan^{- 3} u - \tan u)

$If \ u=\sin^{-1}\dfrac {x+y}{\sqrt{x}+\sqrt{y}} \ then \ prove \ that: \\ (A) \ 2x\dfrac {\partial u}{\partial x}+2x\dfrac {\partial u}{\partial y}=\tan u \\ (B) \ x^2 \dfrac {\partial^2 u}{\partial x^2}+2xy \dfrac {\partial^2u}{\partial x \partial y}+y^2 \dfrac {\partial^2 u}{\partial y^2 }= \dfrac {1}{4} (\tan^{-3}u-\tan u)$

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5 (b) (ii)

I f z = x + y^{x}, p r o v e t h a t \frac{\partial^{2} z}{\partial x \partial y} = \frac{\partial^{2} z}{\partial y \partial x}

$If \ z=x+y^x, \ prove \ that \ \dfrac {\partial^2 z}{\partial x \partial y}=\dfrac {\partial^2 z} {\partial y \partial x}$

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6 (a) (i) Find the volume of the solid obtained by rotating the region enclosed by the curves y=x and y=x² about the x-axis.

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6 (a) (ii) Trace the witch of agnessi xy²=4a² (a-x)

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6 (b) (i) A rectangular box without a lid is to be made from 12m² of cardboard. Find the maximum volume of such a box.

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6 (b) (ii) Find the equations of the tangent plane and normal line at the point (-z, 1, -3) to the ellipsoid

\frac{x^{2}}{4} + y^{2} + \frac{x^{2}}{9} = 3.

$\dfrac {x^2}{4}+y^2+\dfrac {x^2}{9}=3.$

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7 (a) (i)

I f u = f (\frac{x}{y}, \frac{y}{x}, \frac{z}{x}), p r o v e t h a t x + \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} + z \frac{\partial u}{\partial z} = 0

$If \ u=f\left ( \dfrac {x}{y}, \dfrac {y}{x}, \dfrac {z}{x} \right ), \ prove \ that \ x+\dfrac {\partial u}{\partial x}+ y\dfrac {\partial u}{\partial y}+ z \dfrac {\partial u}{\partial z}=0$

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7 (a) (ii)

E v a l u a t e t h e l i m i t : lim_{x \to 1} (1 x^{2})^{\frac{1}{\log (1 - x)}}

$Evaluate \ the \ limit: \ \lim_{x\rightarrow 1}(1x^2)^{\frac{1}{\log(1-x)}}$

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7 (b) (i)

E v a l u a t e \int_{0}^{a} \int_{y}^{a} \frac{x}{x^{2} + y^{2}} d x d y

$Evaluate \int^{a}_{0}\int^a_y \dfrac {x}{x^2+y^2}dxdy$ by transforming into polar coordinates.

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7 (b) (iii) Find the interval of convergence of the series

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

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

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