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 $\int^{\infty}_0 5^{-4x^2} \ dx$
(3 marks)
12629
1.b.
Solve $\frac{dy}{dx} = xy$ with the help of Euler’s method, given that y(0) = 1 and find y when x = 0.3 (h = 0.1)
(3 marks)
12672
1.c.
Evaluate $\frac{d^4y}{dx^4} + 2 \frac{d^2y}{dx^2} + y = 0$
(3 marks)
12673
1.d.
Evaluate $\int^1_0 \sqrt{\sqrt{x} – x } dx$
(3 marks)
12674
1.e.
Solve $(1 + log xy) dx + (1 + \frac{x}{y}) dy = 0$
(4 marks)
12675
1.f.
Evaluate $\int^1_0 \int^\sqrt{1+x^2}_0 \frac{dxdy}{1 + x^2 + y^2}$
(4 marks)
12676
2.a.
Solve $xy (1+ x y^2) \frac{dy}{dx} = 1$
(6 marks)
12677
2.b.
Find the area inside the circle $r = a sin \theta$ and outside the cardioide $r = a (1 + cos \theta)$
(6 marks)
12678
2.c.
Apply Runge-kutta Method of fourth order to find an approximate value of y when x = 0.2 given that $\frac{dy}{dx} = x + y$ when y = 1 at x = 0 with step size h = 0.2
(8 marks)
12679
3.a.
Show that the length of the curve $9ay^2 = x(x-3a)^2$ is $4\sqrt3a$
(6 marks)
12680
3.b.
Change the order of the integration of $\int^1_0 \int^{1+\sqrt{1-y^2}}_{-\sqrt{2y – y^2}} f (x,y) dx \ dy$
(6 marks)
12681
3.c.
Find the volume of the paraboloid $x^2 + y^2 = 4z$ cut off by the plane z = 4
(8 marks)
12682
4.a.
Show that $\int^1_0 \frac{x-1}{log x} dx = log (a+1)$
(6 marks)
12683
4.b.
If Y satisfies the equation $\frac{dy}{dx} = x^2 y -1$ with $x_0 = 0, y_0 = 1$, using Taylor’s series method find y at x = 0.1 (take h = 0.1)
(6 marks)
12684
4.c.
Find the value of the integral $\int^1_0 \frac{x^2}{1 +x^2} dx$ using
(i) Trapezoidal rule
(ii) Simpsons $1/3^{rd}$ rule
(iii) Simpsons $3/8^{th}$ rule.
(8 marks)
12685
5.a.
Solve $(y – xy^2) dx – (x + x^2y)dy = 0$
(6 marks)
12686
5.b.
Evaluate $\int\int\int \sqrt{1 - \frac{x^2}{a^2} - \frac{y^2}{b^2} - \frac{z^2}{c^2}}$ dx dy dz over the ellipsoid $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$
(6 marks)
12687
5.c.
Evaluate $(2x + 1)^2 \frac{d^2y}{dx^2} – 2(2x +1) \frac{dy}{dx} – 12y = 6x$.
(8 marks)
12688
6.a.
A resistance of 100 ohms and inductance of 0.5 henneries are connected in series with a battery of 20 volts. Find the current at any instant if the relation between L, R, E is $L \frac{di}{dt} + Ri = E$
(6 marks)
12689
6.b.
Solve by variation parameter method $\frac{d^2y}{dx^2} + 3 \frac{dy}{dx} + 2y = e^{e^x}$
(6 marks)
12690
6.c.
Evaluate $\int\int xy (x – y)$ dx dy over the region bounded by xy = 4, y = 0, x = 1 and x = 4.
(8 marks)
12691