First Year Engineering (Semester 2)
TOTAL MARKS: 80
TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Assume data if required.
(4) Figures to the right indicate full marks.
1(a) Evaluate $$\displaystyle \int_{0}^{2} x^{4}(8-x^{3})^{-1/3}dx$$
(3 marks)
1(b) Solve $\dfrac{d^{4}y}{dx^{4}}+2\dfrac{d^{2}y}{dx^{2}}+y =0$(3 marks)
1(c) Prove that E = 1+Δ = e4D
(3 marks)
1(d) Solve $[x\sqrt{x^{2}+y^{2}}-y]dx +[y\sqrt{x^{2}+y^{2}}-x]dy =0$
(3 marks)
1(e) Change to polar coordinates and evaluate $\displaystyle \int_{0}^{2a} \int_{0}^{\sqrt{2ax-x^{3}}} \dfrac{x}{\sqrt{x^{2}+y^{2}}}dy\ dx$
(4 marks)
1(f) Evaluate$ \displaystyle \int_{a}^{1}\int_{a}^{x} e^{x+y} dydx$
(4 marks)
2(a) Solve $\dfrac{dy}{dx} +x\sin 2y =x^{3}\cos^{2}y $(6 marks)
2(b) Change the order of integration and evaluate $\displaystyle \int_{0}^{a}\int_{\frac{y^{2}}{a}}^{y} \dfrac{y}{(a-x)\sqrt{ax-y^{2}}}dxdy$(6 marks)
2(c)
prove that using
$\displaystyle \int_{0}^{\infty}cis\lambda(e^{-ax}-e^{-bx})dx=\dfrac{1}{2} log\left(\dfrac{b^{2}+\lambda^{2}}{a^{2}+\lambda^{2}}\right), a>o ,b>0 $
DUIS rule
(8 marks)
3(a) Evaluate $\displaystyle \iiint\dfrac{dx\ dy\ dy}{x^{2}+y^{2}+z^{2}}$ throughout the volume of the sphere $x^{2}-y^{2} +z^{2} =a^{2}$
(6 marks)
3(b) Find the area common to the cardiods r=a(1+cosθ) and r = a(a-cosθ).
(6 marks)
3(c)
Apply the method of variation of parameters to solve $\dfrac{d^{2}y}{dx^{2}}-4\dfrac{dy}{dx}+4y =e^{2x}sec^{2}x$
(8 marks)
4(a) Find the length of one are of the cycloid x =a (θ -sinθ)and y = a(a+cos θ)
(6 marks)
4(b) Solve $\dfrac{d^{2}y}{dx^{2}} +2y =x^{2}e^{3x} + e^{x} \cos x $
(6 marks)
4(c) Appply Runge-Kutla method of fourth order to find an approximate value of y at x =1.2 if $\dfrac{dy}{dx} =x^{2} +y^{2}$, given that y = 1.5 when x=1 choosing h= 0.1(8 marks)
5(a) Solve $\left [xy^{2}-e^{\frac{1}{x^{3}}}\right]dx-yx^{2}dy =0$
(6 marks)
5(b) If y satisfies the equation $\dfrac{dy}{dx} =x^{2}y-1$ and with y=1 when x=0, using Taylor's series method for y about x=0, find y when x =0.1 and x=0.2(6 marks)
5(c) Compute the value of the definite integral$ \displaystyle \int_{-1}^{1}\dfrac{dx}{1+x^{2}}$,by using
Trapezoidal Rule
Simpson's (1/3)rd Rule
Simpson's (3/8)th Rule.
(8 marks)
6(a) A radial displacement 'u' in rotating a disc at a distance 'r' from the axis in given by $\dfrac{d^{2}u}{dr^{2}}+\dfrac{1}{r}\dfrac{du}{dr}-\dfrac{u}{r^{2}}+kr=0.$ find the displacement given u=0 when r=0 and r=a
(6 marks)
6(b) Evaluate$\displaystyle \iint x^{2} dsdy $ over the region bounded by xy = a2, x=2a, y=0 and y=x in the first quadrant.(6 marks)
6(c) Find the volume of the tetrahedron bounded by the co-ordinates plane and the plane $\dfrac{x}{2}+\dfrac{y}{3}+\dfrac{z}{4} =1$(8 marks)
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