Question: Calculus : Jun 2014 - First Year Engineering (Semester 1) | Gujarat Technological University (GTU)
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## Calculus - Jun 2014

### 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.
1 (a) (i) The value of $$\lim_{x\rightarrow 0}x \sin \dfrac {1}{x}\ is$$ (A) 1
(B) π
(C) 0
(D) ∞
(1 marks)
1 (a) (ii) $$y=\log (\sin x )\ then \ value \ of \ \dfrac {d^2y}{dx^2}=\$$
(A) sec2 x
(B) -cosec2 x
(C) -cosec xcotx
(D) secxtanx
(1 marks)
1 (a) (iii) Curve: x=t2-1, y=t2-t. For which value of t tangents of the curve are parallel to the x-axis? $$(A) \ t=0 \\ (B)\ t=\dfrac {1} {\sqrt{3}} \\ (C)\ t=\dfrac {1}{2}\\ (D) \ t=-\dfrac {1}{\sqrt{3}}$$(1 marks) 1 (a) (iv) $$\int^{1}_{-1}x|x| dx= .......$$ (A) 2 (B) 1 (C) 0 (D) None of these(1 marks) 1 (a) (v) \begin {align*} If \ f(x) &=4-2x,x<1 \\ &=6x-4, x \ge 1 \ then \ find \ \lim_{x\rightarrow 1} f(x) \end{align*} (A) -2 (B) 2 (C) 0 (D) doesn't exist.(1 marks) 1 (a) (vi) If f:R → R, f(x) = 3x+2, then find f-1
(A) Not possible
(B) 3x-2
(C) 2x-3
$$(D) \ \dfrac {x-2}{3}$$
(1 marks)
1 (a) (vii) What is the solution for following equations ?
2x+3y-5=0, 4x+6y-7=0
(A) No solution
(B) Infinitely many solution
(C) unique solution
(D) { (0,0), (-5, -7) }
(1 marks)
1 (b) (i) $$\lim_{n\to \infty}\dfrac {1-n^2}{\sum n}=$$ (A) -2 (B) 1 (C) 2 (D) doesn't exist(1 marks) 1 (b) (ii) $$\int \dfrac {1}{\sqrt{x}}\cos \sqrt{x}dx=....+c \$$A) \ \dfrac {2}{3}\cos^{\frac {3}{2}}x \\ (B) \ 2 \sqrt{x} \\ (C) 2 \sin \sqrt{x} \\ (D) \ \dfrac {2}{3} \cos \sqrt{x} $$(1 marks) 1 (b) (iii) Find the area of the curve y=x2 + 1 bounded by x-axis and lines x=1 and x=2.$$ (A) \ \dfrac {3}{10} \\ (B) \ \dfrac {10}{3} \\ (C) 6 \\ (D) \ \dfrac {1}{6} $$(1 marks) 1 (b) (iv) f(x) = [x], x ∈ R then f(x) is (A) Continuous for all real numbers. (B) Continuous for all integers (C) discontinuous for all integers (D) none of these (1 marks) 1 (b) (v) Find the vertical asymptotes of the curve$$ y=\dfrac {2x^z}{x^2-1} $$(A) x=1 (B) x=-1 (C) x= ± 1 (D) y=2 (1 marks) 1 (b) (vi) f(x) =x3-3x2+3x-100 is ....... function. (A) Increasing (B) decreasing (C) constant (D) none (1 marks) 1 (b) (vii) Find the value of c using Lagrange's mean value theorem for f(x)=ex, x ∈ [0,1] (A) log 1 (B) log c (C) log (1-e) (D) log (e-1) (1 marks) 2 (a) (i)$$ Evaluate \ \lim_{x\rightarrow 0}\dfrac {e^x+e^{-x}-x^2-2}{\sin^2 x-x^2} $$(2 marks) 2 (a) (ii) Test for convergence the series$$ \dfrac {1}{1.2}+ \dfrac {1}{2.3}+ \dfrac {1}{3.4}+ ..... $$(2 marks) 2 (a) (iii) Expand log sin x in powers of (x-2)(3 marks) 2 (b) (i) Test for convergence the series$$ 2+\dfrac {3}{2}x+\dfrac {4}{3}x^2+\dfrac {5}{4}x^3+...... $$(4 marks) 2 (b) (ii) Trace the curve r=a sin 3θ(3 marks) 3 (a) (i)$$ (i) \ Evaluate \ \lim_{x\rightarrow 0}(a^x+x)^{\frac {1}{x}} $$(ii) Test the convergence of$$ \sum^{\infty}_{n=0} \dfrac {3^{2n}}{2^{3n}} $$(4 marks) 3 (a) (ii) Trace the curve x3 + y3 = 3axy.(3 marks) 3 (b) (i) Test the convergence of$$ \sum^{\infty}_{n=1} \dfrac {[(n+1)x]^n}{n^n+1} $$(4 marks) 3 (b) (ii) Evaluate improper integral$$ \int^{3}_0 \dfrac {1}{\sqrt{3-x}}dx $$(3 marks) 4 (a) (i)$$ (i)\ If \ u=ln \left ( \dfrac {x^7+y^7+z^7}{x+y+z}\right ) \ show \ that \ x\dfrac {\partial u}{\partial x}+ y\dfrac {\partial u}{\partial y}+z\dfrac {\partial u}{\partial z}=6$$(ii) State modified Euler's theorem and find the degree of homogeneous function$$ u(x,y)=\dfrac {1}{x^2}+\dfrac {1}{xy}+\dfrac {\log x-\log y}{x^2} $$(4 marks) 4 (a) (ii) z=f(x,y), xeu + e-v, y=e-u - ev then prove that$$ \dfrac {\partial z}{\partial z}-\dfrac {\partial z}{\partial v}=x\dfrac {\partial z}{\partial x} -y\dfrac {\partial u}{\partial y} $$(3 marks) 4 (b) (i)$$ If \ u=e^{3xyz} \ show \ that \ \dfrac {\partial^3 u}{\partial x \partial y \partial z}= (3+27xyz+27x^2y^2z^2)e^{3xyz}$$(4 marks) 4 (b) (ii)$$ \begin {align*} show \ that \ f(x,y)&=\dfrac {xy}{\sqrt{x^2+y^2}}; \ &(x,y)e (0,0) \\ &=0, \ &(x,y)=(0,0) \end{align*} $$is continuous at the origin.(3 marks) 5 (a) (i) Find the extreme values of the function f(x,y) = x3+y3-3x-12y+20. (4 marks) 5 (a) (ii) Show that the surface z=xy-2 and x2+y2+z2 =3 have the same tangent plane at (1,1,1).(3 marks) 5 (b) (i) Find the minimum value of x2 y z3 subject to the condition 2x+y+3z=a using Lagrange's method of undetermined multipliers.(4 marks) 5 (b) (ii) Expand sin xy in powers of (x-1) and (y- π/2) upto second degree terms.(3 marks) 6 (a) (i) Sketch the region of integration, reverse the order of integration and evaluate$$ \int^{2}_0\int^{4-x^2}_0 \dfrac {xe^{2y}}{4-y}dydx. $$(4 marks) 6 (a) (ii) Evaluate the integral$$ \int^{\frac {\pi}{2}}_0\int^{1-\sin\theta}_0 r^2 \cos\theta drd\theta $$(3 marks) 6 (b) (i) Evaluate the integral$$ \int^2_0\int^2_1\int^{yz}_0 xyz \ dxdydx. $$(4 marks) 6 (b) (ii) Evaluate the integral$$ \int^a_0\int^{\sqrt{a^2-y^2}}_0 y^2\sqrt{x^2+y^2}dydx.$$by changing into polar co-ordinates(3 marks) 7 (a) (i) Find the volume bounded by the cylinder x2 + y2 = 4 and the planes y+z=3, z=0.(4 marks) 7 (a) (ii) Evaluate ∬R x2+y2 dA by changing the variables ,where R is the region lying in first quadrant and bounded by the hyperbola x2 - y2=1, x2 - y2=9, xy=2 and xy=4. (3 marks) 7 (b) (i) The region between the curve y=√x, 0≤x≤4 and the x-axis is revolved about the x-axis to generate a solid. Find its volume.(4 marks) 7 (b) (ii) Test the convergence of the series$$ \sum^{\infty}_{n=1}\dfrac {(-1)^{n+1}}{\log n+1}(3 marks)