First Year Engineering (Semester 2)
Total marks: 80
Total time: 3 Hours
INSTRUCTIONS
(1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Draw neat diagrams wherever necessary.
1.a.
Evaluate $\int^\infty_0 \frac {e^{-x^2}}{\sqrt x} \ dx$
(3 marks)
12650
1.b.
Solve $(D^3 + 1)^2 y = 0$
(3 marks)
12651
1.c.
Solve the ODE $( y + \frac{1}{3} y^3 + \frac{1}{2} x^2 ) dx + (x + x y^2) dy = 0$
(3 marks)
12653
1.d.
Use Taylor’s series method to find a solution of $\frac{dy}{dx} = 1 + y^2, y(0) = 0$
at x = 0.1 taking h = 0.1 correct to three decimal value.
(3 marks)
12654
1.e.
Given $\int^x_0 \frac{dx}{x^2 + a^2} = \frac{1}{a} tan^-1 (\frac{x}{a})$, using DUIS find the value of $\int^x_0 \frac{dx}{(x^2 + a^2)^2}$
(4 marks)
12655
1.f.
Find the perimeter of the curve $r = a (1- cos\theta$)
(4 marks)
12656
2.a.
Solve $(D^3 + D^2 + D + 1) y = sin^2 x$
(6 marks)
12657
2.b.
Change the order of integration $\int^a_0 \int^{x+3a}_{\sqrt{a^2-x^2}} f (x,y) \ dx \ dy$
(6 marks)
12658
2.c.
Evaluate $\int \int_R \frac{2x \ y^5}{\sqrt{1+ x^2\ y^2 – y^4}} \ dx \ dy$, where R is a triangle whose vertices are (0, 0), (1,1), (0,1).
(8 marks)
12659
3.a.
Find the volume enclosed by the cylinder $y^2 = x \ and\ y = x^2$
Cut off by the planes z = 0, x + y + z =2
(6 marks)
12660
3.b.
Using modified Euler’s method, find an appropriate value of y at x = 0.2 in two step taking h = 0.1 and using iteration, given that $\frac{dy}{dx} = x + 3y, y = 1 \ when \ x = 0.$
(6 marks)
12661
3.c.
Solve $(1+x)^2 \frac{d^2y}{dx^2} + (1+x) \frac{dy}{dx} + y = 4 \ cos log (1 + x) $
(8 marks)
12662
4.a.
Show that $\int^\alpha_0 \sqrt\frac{x^3}{a^3 – x^3} dx$
$= \frac{a\sqrt\pi \sqrt{5/6}}{\sqrt{1/3}}$
(6 marks)
12663
4.b.
Solve $(D^2 + 2)y = e^x \ cos x + x^2 e^{3x}$
(6 marks)
12664
4.c.
Use polar co-ordinates to evaluate $\int \int \frac{(x^2+y^2)^2}{x^2 \ y^2} dx \ dy$ over the area
Common to the circle $x^2 + y^2 = ax$ and $x^2 + y^2 = by, a \gt b \gt 0$
(8 marks)
12665
5.a.
Solve $y \ dx + x (1 – 3 x^2 y^2) dy = 0$
(6 marks)
12666
5.b.
Find the mass of a lamina in the form of an eclipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} - 1$, if the density at any point varies as the product of the distance from the azes of the ellipse.
(6 marks)
12667
5.c.
Compute the value of $\int^{\pi/2}_0 \sqrt{sinx + cosx} \ dx$ using
(i) Trapezoidal rule
(ii) Simpson’s $(1/3)^{rd}$ rule
(iii) Simpson’s $(3/8)^{th}$ rule by dividing into six subintervals
(8 marks)
12668
6.a.
Evaluate $\int\int\int_v x^2 dx \ dy \ dz$ over the volume bounded by the planes
$x = 0, y = 0, z = 0 \ and \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$
(6 marks)
12669
6.b.
Change the order of integration and evaluate $\int^2_0 \int^2_\sqrt{2y} \frac{x^2}{\sqrt{x^4 – 4 y^2}} dx \ dy$
(6 marks)
12670
6.c.
Solve by the method of variation of parameters $\frac{d^2y}{dx^2} – 6 \frac{dy}{dx} + 9y = \frac{e^{3x}}{x^2}$
(8 marks)
12671