1
Code No: R5102305
I B.Tech (R05) Supplementary Examinations, June 2009 MATHEMATICS FOR BIOTECHNOLOGISTS (Bio-Technology) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? 1. (a) Find
d dx
(b) If u =
£
sin−1 (2x + 3) + log x
x3 +x2 y , x+y
¤
verify Euler’s Theorem.
[8+8]
2. Evaluate the following integrals. R (a) ex (tan x − log cos x) dx R√ (b) 1 + sec xdx (c)
Π/4 R
sec3 xdx
[5+5+6]
0
3. (a) Solve the following system of equations x-y+3z=5, 4x+2y-z=0, -x+3y+3=5. Using Cramer’s rule. (b) Find 1 3 5 4
the 0 −2 −2 −2
rank −5 1 −9 −4
of the matrix by reducing it to the echelon form 6 2 14 8
[8+8]
4. (a) Form the differential equation by eliminating the constant x2 +y 2 -2ay=a2 . (b) Solve the differential equation
dy dx
(x2 + y 3 + xy) = 1.
[6+10]
5. (a) Solve the differential equation (D2 +2D+1) y=x cos x. (b) If the air is maintained at 15o C and the temperature of the body cools from 70o to 40o in 10 minutes, find the temperature after 30 minutes.. [10+6] 6. (a) Find a real root of x+logx-2=0 using Newton Raphson method (b) Solve the system of equations by Gauss Seidel method 6x - y - z = 19, 3x + 4y + z = 26, x + 2y + 6z = 22
[8+8]
7. (a) Find f(2.5) using Newtons forward formula from the following table x y
0 1 0 1
2 3 16 81
4 5 256 625
(b) Derive the formula to evaluate
6 1296 Rb a
ydx using trapezoidal rule.
[8+8]
8. (a) Find the Laplace Transformation of the following functions: t e−t sin 2t . 00
0
(b) Using Laplace transform, solve y +2y +5y = e−t sin t, given that 0 y(0) = 0, y (0) = 1. ?????
[8+8]
2
Code No: R5102305
I B.Tech (R05) Supplementary Examinations, June 2009 MATHEMATICS FOR BIOTECHNOLOGISTS (Bio-Technology) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? 1. (a) If A=[1,2,3,4,5,6,7,8], B=[2,3,4,6,8] C=[4,5,6], then find (A ∩ B) ∪ C d(x3 .ex ) (b) Find dx
[8+8]
2. Evaluate the following integrals. R 2xdx (a) √sin 1+cos2 x R (x+1)dx (b) x(x+1)(x+2) (c)
R1
2
xe−x dx
[5+6+5]
0
¯ ¯ Cos2 0 CosθSinθ ¯ ¯ 3. If f(θ) = ¯ Cosθ sin θ Sin2 θ ¯ ¯ Sinθ − Cosθ then show that f(Π/6)=1
¯ − Sinθ¯¯ ¯ Cosθ ¯ ¯ ¯ 0 [16]
4. (a) Form the differential equation by eliminating the constant x2 +y 2 -2ay=a2 . (b) Solve the differential equation
dy dx
(x2 + y 3 + xy) = 1.
[6+10]
5. (a) Solve the differential equation (D2 +4)y = ex +sin2x. (b) A body kept in air with temperature 25o C cools from 140C to 80o in 20minutes. Find when the body cools down to0 35o C. [10+6] 6. (a) Find a real root of xsinx+cosx=0 by Newton Raphson method (b) Solve the system of equations by Jacobi’s method 9.9x - 1.5y + 2.6z = 0 .4x + 13.6y - 4.2z = 8.2 .7x + .4y + 7.1z = - 1.3 7. (a) Given that y0 =1, y1 =0, y2 =5, y3 =2, y4 =57, find y
[8+8] ¡1¢ 2
using Newton’s forward formula
dy (b) Find dx at x=7.5 from the following table x 7.47 7.48 7.49 7.5 7.51 7.52 7.53 y .193 .195 .198 .201 .203 .206 .208
8. (a) Prove that L [
£1 t
[8+8]
¤ R∞ f (t) = f (s) ds where L [f(t) ] = f (s) s
(b) Find the inverse Laplace Transformation of ?????
3(s2 −2)2 2 s5
[8+8]
3
Code No: R5102305
I B.Tech (R05) Supplementary Examinations, June 2009 MATHEMATICS FOR BIOTECHNOLOGISTS (Bio-Technology) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? 1. (a) If A = [x/2 < x ≤ 4] , B = [x/1 < x < 3] C = [x/5 ≤ x < 8], x is an integer. Then find (A ∪ B) ∪ C ¡ log x ¢ d (b) Find dx [8+8] x3 2. Evaluate the following integrals. (a) (b) (c)
R (x2 +x)dx R
x2 +x+2 dx (16−5x2 )
Π/4 R
(1+sin 2x)dx (cos x+sin x)
[5+5+6]
¯ ¯ a2 bc ac + c2 ¯ 2 2 ¯ b ac 3. Show that ¯ a + ab ¯ ac b2 + bc c2
¯ ¯ ¯ ¯ = 4a2 b2 c2 ¯ ¯
[16]
4. (a) Form the differential equation by eliminating the arbitrary constant tan x tan y = c. (b) Solve the differential equation:
dy dx
+
2x y (1+x2 )
=
1 given (1+x2 )2
y = 0 when x = 1. [6+10]
5. (a) Solve the differential equation: (D3 − 4D2 − D + 4)y = e3x cos 2x. (b) Find the orthogonal trajectories of the coaxial circles x2 + y2 + 2λy + c =2, λ being a parameters.
[10+6]
6. (a) Find a real root of the equation x3 - x - 4 = 0 by bisection method (b) Solve the system of equations by Jacobis method. 4x + y + 3z = 17 x + 5y + z = 14 2x - y + 8z = 12
[8+8]
7. (a) Find f(3.8) given x = 1,2,3,4, f(x) = 6, -3, 6, 2, -6. Using Gauss forward difference method. (b) Find f(3.4) from the following table using Newton’s forward formula x y
3 31
4 5 69 131
6 223
[8+8]
8. (a) Find L[t.e−t sin2t] h −1
(b) Find L
S 2 +2S−4 (S 2 +9)(S−5)
i
[8+8] ?????
4
Code No: R5102305
I B.Tech (R05) Supplementary Examinations, June 2009 MATHEMATICS FOR BIOTECHNOLOGISTS (Bio-Technology) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? 1. (a) Prove that A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) (b) If U = tan−1
x y
+ sin−1 xy , then prove that
∂2u ∂x∂y
=
∂2u ∂y∂x
[8+8]
2. Evaluate the following integrals. (a) (b) (c)
R R
cos(tan−1 )dx 1+x2 (x+2)dx (x2 +2x+3)
R4 √ x x2 − 1dx
[6+5+5]
1
3. (a) Solve the system of equations x+y-z=1, 2x-3y+4z=3, x+3y-z=3 by Matrix Inversion method. (b) Find the eigen values and the corresponding eigen vectors of the matrix. " # −2 2 −3 2 1 −6 −1 −2 0
[8+8]
4. (a) Find the differential equation of all parabolas having the axis as the axis and (a,0) as the focus. (b) Solve the differential equation
x2 dy dx
= ey − x.
[6+10]
5. (a) Solve the differential equation: (D2 −2D−3)y = x3 e−3x . Obtain the orthogonal trajectories of the family : rn = an cos nθ [10+6] 6. (a) Find a real root of the equation x3 -9x+1=0 by bisection method (b) Solve the following equations by Jacobi’s method 6x - y - z = 2 x + 3y + z = 0 x + 2y + 4z = 8 7. (a) Given that y0 =1, y1 =0, y2 =5, y3 =2, y4 =57, find y dy (b) Find dx at x=7.5 from the following table x 7.47 7.48 7.49 7.5 7.51 7.52 7.53 y .193 .195 .198 .201 .203 .206 .208
[8+8] ¡1¢ 2
using Newton’s forward formula
[8+8]
8. (a) Find L [t sinh t e−t ] (b) EvaluateL−1 log
(s2 +4)
[8+8]
(s2 +9)
?????