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Calculus : Dec 2013 - First Year Engineering (Semester 1) | Gujarat Technological University (GTU)

Calculus - Dec 2013

First Year Engineering (Semester 1)

TOTAL MARKS: 100
TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.
(2) Attempt any four from the remaining questions.
(3) Assume data wherever required.
(4) Figures to the right indicate full marks.


Answer the following questions by choosing the most appropriate answer:

1 (a) (i) $$ What \ is \ \int \dfrac {x-3}{x}dx? \$$a) \ 1-3 \ ln \ x+C \\ (b) x-3 \ ln \ x+C \\ (c) \ 1+\dfrac {3}{x^{2}}+C \\ (d) \dfrac {x^2-3x}{x^2}+C $$(1 marks) 1 (a) (ii) $$ if \ \tan y+x^3=y^2+1 \ and \ \dfrac {dx}{dt}=-2, \\ what \ is \ the \ value \ of \dfrac {dy}{dt} \ at \ the \ point \ (1,0)$$
(A) -6
(B) -2.5
(C) 0
(D) 6
(1 marks)
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
(1 marks)
1 (a) (iv) The value of the limit $$ \lim_{x\rightarrow 0}\dfrac {\tan x}{x} \ is $$
(A) 0
(B) 1
(C) π
(D) ∞
(1 marks)
1 (a) (v) $$ if \ y=\dfrac {e^x-e^{-x}}{e^x+e^{-x}} $$ then the derivative of the function y w.r.t. x is $$ (A)\ 0 \\ (B) \ 1 \\ (C) \ \dfrac {2}{(e^x+e^{-x})^2} \\ (D) \ \dfrac {4}{(e^x+e^{-x})^2} $$(1 marks) 1 (a) (vi) $$ if \ y=ln (\sqrt{2}\ x)$$ the derivative of the function y w.r.t. x is $$ (A) \ \dfrac {\sqrt{2}}{x} \\ (B) \ \dfrac {1}{\sqrt{2} \ x} \\ (C) \ \dfrac {1}{2x} \\ (D) \ \dfrac {1}{x} $$ (1 marks) 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
(1 marks)


Answer the following questions by choosing the most appropriate answer

1 (b) (i) The value of the integral $$ \int \dfrac {\sin \sqrt{x}}{\sqrt{x}}dx \ is \\ (A) \ -2\cos^{1/2}x+C \\ (B) \ -\cos \sqrt{x}+C \\ (C) \ -2 \cos \sqrt{x}+C \\ (D) \ \dfrac {1}{2}\cos\sqrt{x}+C $$(1 marks) 1 (b) (ii) The value of the limit $$ \lim_{x\rightarrow 0}\dfrac {|x|}{x} \ is $$
(A) 0 (B) 1
(C) π (D) ∞
(1 marks)
1 (b) (iii) The value of the integral $$ \int \dfrac {ln \ y} {y^2} \ is $$ $$ (A)\ \dfrac {1}{y}(1-ln \ y)+C \\ (B) \ \dfrac {1}{2y}ln^2 \ y+C \\ (C) \ \dfrac {1}{3y^3}(4\ ln\ y+1)\\ (D)\ - \dfrac {1}{y}(ln \ y+1)+C $$ For (1 marks) 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
(1 marks)
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(1 marks) 1 (b) (vi) $$ if \ A=\begin{bmatrix} -5&-3 \\2 &1 \end{bmatrix} \ then \ A^{-1}\ is: \\ (A) \ \begin{bmatrix}-5 &-3 \\2 &-1 \end{bmatrix} \\ (B) \ \begin{bmatrix} -1&-3 \\2 &5 \end{bmatrix} \\ (C) \ \begin{bmatrix} 1&3 \\-2 &-5 \end{bmatrix} \\ (D) \ \begin{bmatrix} 5&3 \\-2 &1 \end{bmatrix} $$(1 marks) 1 (b) (vii) The function 2x3 + 3x2 - 12x + 7 is decreasing in
(A) [-2,1] (B) -[-2,1] (C) [0,2] (D) [1,3]
(1 marks)
2 (a) (i) Test the convergence of the series $$ \dfrac {1.2}{3^2.4^2}+ \dfrac {3.4}{5^2.6^2}+\dfrac {5.6}{7^2.8^2}+.... $$(3 marks) 2 (a) (ii) Find value of xfor which the given series $$ \dfrac {1}{2\sqrt{1}}+ \dfrac {x^2}{3\sqrt{2}}+ \dfrac {x^4}{4\sqrt{3}}+ \dfrac {x^6}{5\sqrt{4}} + .... \ converges. $$ (4 marks) 2 (b) (i) Determine convergence or divergence of series $$ \sum^{\infty}_{n=1}\dfrac {(2n^2-1)^{\frac {1}{3}}}{(3n^3+2n+5)^{\frac {1}{4}}} $$(3 marks) 2 (b) (ii) Determine absolute or conditional convergence of the series $$ \sum^{\infty}_{n=1}(-1)^n \cdot \dfrac {n^2}{n^3+1} $$(4 marks) 3 (a) (i) Find the expansion of $$ \tan \left ( x+\dfrac {\pi}{4} \right ) $$ in ascending powers of x upto terms in x4 and find approximately the value of tan 43°.(4 marks) 3 (a) (ii) $$ Prove\ that: \ \tan^{-1}\left ( \dfrac {\sqrt{1+x^2}-1}{x} \right )=\dfrac {1}{2} \left ( x-\dfrac {x^3}{3}+ \dfrac {x^5}{5}-\dfrac {x^7}{7}+.... \right ) $$(3 marks) 3 (b) (i) $$ Evaluate \\ (i) \ \lim_{x\rightarrow 0}(\cos x)^{\cot x} \\ (ii) \ \lim_{x\rightarrow \infty}\left ( a^{\frac {1}{x}}-1 \right )x $$(4 marks) 3 (b) (ii) Express 5+4(x-1)2-3(x-1)3+(x-1)4 in ascending powers of x.(3 marks) 4 (a) (i) Evaluate the iterated integral $$ \int^{1}_{0}\int^{1}_{x}\sin (y^2)dy \ dx $$(3 marks) 4 (a) (ii) Evaluate the integral $$ \int^{4a}_{0}\int^{y}_{\frac {y^2}{4a}}\dfrac {x^2-y^2}{x^2+y^2}dx \ dy $$ by transforming into Polar coordinates.(4 marks) 4 (b) (i) Evaluate the triple integral $$ \int^1_0 \int^\pi_0 \int^\pi_0 y \sin z \ dx \ dy \ dz.$$(3 marks) 4 (b) (ii) Find the area common to both of the circles r=cos θ and r=sin θ(4 marks) 5 (a) (i) Determine the set of points at which the given function is continuous $$ f(x,y)=\left\{\begin{matrix}\dfrac {3x^2 y}{x^2+y^2} & if &(x,y)e (0,0) \\ 0 & if &(x,y)=(0,0)\end{matrix}\right. $$(4 marks) 5 (a) (ii) If z=x2y+3xy4, where x= sin 2t and y=cos t, find $$ \dfrac {dx}{dt} $$ when t=0(3 marks) 5 (b) (i) $$ If \ u=\sin^{-1}\dfrac {x+y}{\sqrt{x}+\sqrt{y}} \ then \ prove \ that: \\ (A) \ 2x\dfrac {\partial u}{\partial x}+2x\dfrac {\partial u}{\partial y}=\tan u \\ (B) \ x^2 \dfrac {\partial^2 u}{\partial x^2}+2xy \dfrac {\partial^2u}{\partial x \partial y}+y^2 \dfrac {\partial^2 u}{\partial y^2 }= \dfrac {1}{4} (\tan^{-3}u-\tan u)$$(4 marks) 5 (b) (ii) $$ If \ z=x+y^x, \ prove \ that \ \dfrac {\partial^2 z}{\partial x \partial y}=\dfrac {\partial^2 z} {\partial y \partial x} $$(3 marks) 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.(3 marks) 6 (a) (ii) Trace the witch of agnessi xy2=4a2 (a-x)(4 marks) 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.(4 marks) 6 (b) (ii) Find the equations of the tangent plane and normal line at the point (-z, 1, -3) to the ellipsoid $$\dfrac {x^2}{4}+y^2+\dfrac {x^2}{9}=3. $$ (3 marks) 7 (a) (i) $$ If \ u=f\left ( \dfrac {x}{y}, \dfrac {y}{x}, \dfrac {z}{x} \right ), \ prove \ that \ x+\dfrac {\partial u}{\partial x}+ y\dfrac {\partial u}{\partial y}+ z \dfrac {\partial u}{\partial z}=0 $$(4 marks) 7 (a) (ii) $$ Evaluate \ the \ limit: \ \lim_{x\rightarrow 1}(1x^2)^{\frac{1}{\log(1-x)}} $$(3 marks) 7 (b) (i) $$ Evaluate \int^{a}_{0}\int^a_y \dfrac {x}{x^2+y^2}dxdy $$ by transforming into polar coordinates. (3 marks) 7 (b) (iii) Find the interval of convergence of the series $$ \sum_{n=0}^{\infty}\dfrac {(-3)^n x^n}{\sqrt{n+1}} $$(4 marks)

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