R05012304-mathematics-for-biotechnologists

Set No. 1

Code No: R05012304

I B.Tech Supplimentary Examinations, Aug/Sep 2008 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 [8+8] (b) Find dx x3 2. Evaluate the following integrals. R x3 dx (a) 2x+1 R sin 3xdx (b) cos 4x cos x (c)

Π/2 R 0

dx 4+5 cos x

[5+5+6]

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  [8+8] −1 −2 0 4. (a) Find the differential equation of the family of cardiods r = a ( 1 + cos θ). (b) Solve the differential equation:

dy dx

+

y x log x

=

sin 2x log x

[6+10]

5. (a) Solve the differential equation: y′′ + 4y′ + 20y = 23 sint - 15cost, y(0) = 0, y′ (0) = -1. (b) The rate at which the population of the city increases at any time is proportional to the population at that time. If there were 130000 people in 1950 and 160000 in 1980 . What is the anticipated population in 2010? [10+6] 6. (a) Find a real root of xex =3 using Newton Raphson method (b) Derive the formula to find a root of the equation f(x)=0 if the initial values are a and b using Regula falsi method. [8+8] 7. (a) Find y(32) if y(10)=35.3, y(15)=32, y(20)=29.2, y(25)=26.1, y(30)=22.32, y(32)=20.5 √ (b) Evaluate 1 + x3 taking h = .1 using i. Simpson’s 31 rd rule ii. Trapezoidal rule.

[8+8] 1 of 2

Set No. 1

Code No: R05012304

8. (a) Find the Laplace Transformation of the following functions: t e−t sin 2t . ′′

′

(b) Using Laplace transform, solve y +2y +5y = e−t sin t, given that ′ y(0) = 0, y (0) = 1. ⋆⋆⋆⋆⋆

2 of 2

[8+8]

Set No. 2

Code No: R05012304

I B.Tech Supplimentary Examinations, Aug/Sep 2008 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 =

[e2x log (2x + 5) + ex sin 2x]

1 (x2 +y 2 +z 2 )

then prove that

∂2u ∂x2

+

∂2u ∂y 2

+

∂2u ∂z 2

=

2 (x2 +y 2 +z 2 )2

[8+8]

2. Evaluate the following integrals. (a)

R

(b)

R

(c)

cos(tan−1 )dx 1+x2 (x+2)dx (x2 +2x+3)

R4 √ x x2 − 1dx

[6+5+5]

1

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 arbitrary constant xy = x log x – x + c. (b) Solve the differential equation:

dy dx

−

2y x

=

5x2

(2 + x)(3 − 2x)

[6+10]

5. (a) Solve the differential equation: (D3 + 1)y = cos(2x - 1). (b) Find the orthogonal trajectories of the family of the parabolas y2 = 4ax. [10+6] 6. (a) Find a real root of xex -cosx=0 using Newton Raphson method (b) Solve the system of equations by Gauss seidel method -4x - y + z = 2, x - 5y + 3z = 5, 2x + 4y - 8z = 7

[8+8]

7. (a) The pressure of a gas at temperatures 0, 5, 20, 31, is 115.2, 131.8, 215.9, 391.8 respectively. Using Lagrange interpolation find pressure at temperature 25. (b) The velocity V of a particle at a distances from a point on its path is given by the following table. s(ft) 0 10 20 30 40 50 60 vf(t/s) 47 58 64 65 61 52 38 Estimate the time taken to travel 60ft using 1 of 2

Set No. 2

Code No: R05012304 i. Simpson’s 13 rd rule ii. Simpson’s 38 th rule

[8+8]

8. (a) Find L [ t e3t Sin 2t ]  s+2  (b) Find L−1 s2 −4s+13

[8+8] ⋆⋆⋆⋆⋆

2 of 2

Set No. 3

Code No: R05012304

I B.Tech Supplimentary Examinations, Aug/Sep 2008 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 =

[e2x log (2x + 5) + ex sin 2x]

1 (x2 +y 2 +z 2 )

then prove that

∂2u ∂x2

+

∂2u ∂y 2

+

∂2u ∂z 2

=

2 (x2 +y 2 +z 2 )2

[8+8]

2. Evaluate the following integrals. R dx (a) (1+ex )(1+e −x ) R dx (b) x√1+x2 (c)

Π/2 R

x2 sin x

[6+5+5]

0

3. (a) Solve the system of equations x+y+z=1, 2x+2y+3z=6, x+4y+9z=3. Using Cramer’s rule. (b)  For what value 4 4 −3 1  1 1 −1 0   k 2 2 2 9 9 k 3

of  K the matrix  has rank 3. 

[8+8]

4. (a) Solve the differential equation: (x+1)y1 – y = e3x (x+1)2 (b) Obtain the differential equations of the families of circles touching x – axis at the origin. [10+6] 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 the equation ex =1+2x by bisection method (b) Solve the system of equations by Jacobi?s method. 3x - y + z = 2 x +4y + z = 5 x + y - 6z = 8 7. (a) Find y(1.6) using Newton’s interpolation formula x 1 1.4 1.8 2.2 y 3.49 4.82 5.96 6.5 1 of 2

[8+8]

Set No. 3

Code No: R05012304

(b) By dividing the range into ten equal parts, evaluate 1 rd 3

Rπ

Sinxdx by Simpson?s

0

rule. Verify your answer with integration.

[8+8]

  R∞ 8. (a) Prove that L [ 1t f (t) = f (s) ds where L [f(t) ] = f (s) s

(b) Find the inverse Laplace Transformation of ⋆⋆⋆⋆⋆

2 of 2

3(s2 −2)2 2 s5

[8+8]

Set No. 4

Code No: R05012304

I B.Tech Supplimentary Examinations, Aug/Sep 2008 MATHEMATICS FOR BIOTECHNOLOGISTS (Bio-Technology) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) Prove that i. (A ∩ B)′ = A′ ∪ B′ ii. (A ∪ B)′ = A′ ∩ B′ (b) Find

dy dx

if y = em sin

−1

x

[8+8]

2. Evaluate the following integrals. R x)dx (a) (sin√x+cos 1+sin 2x R dx (b) 5+4x−2x 2 (c)

RΠ

x2 sin xdx

[5+6+5]

−Π

a2 bc ac + c2 2 b2 ac 3. Show that a + ab 2 ac b + bc c2

= 4a2 b2 c2

[16]

4. (a) Form the differential equation by eliminating the arbitrary constant sin−1 (xy) + 4x = c. (b) Solve the differential equation: (1 + y 2 )dx = ( tan−1 y – x ) dy.

[6+10]

5. (a) Solve the differential equation: (D2 + 5D + 6)y = ex . (b) In a chemical reaction a given substance is being converted into another at a rate proportional to the amount of substance unconverted. If 1/5 th of the original amount has been transformed in 4 minutes how much time will be required to transform one half. [10+6] 6. (a) Find a root of ex sinx=1 using Newton Raphson’s method (b) Solve the system of equations by jacobi’s mehod x + 8y + 2z = 3, -x + 4y + 8z = 6, 5x + 3y - z = 3

[8+8]

7. (a) Use Lagrange’s formula to calculate f(3) from the following table x : 0 1 2 4 5 6 f(x) : 1 14 15 5 6 19 (b) Evaluate

R1 0

dx 1+x

taking h= .25 using Simpson’s

1 of 2

1 3

rd taking h=.125

[8+8]

Set No. 4

Code No: R05012304

8. (a) Find the Laplace Transformations of the following functions e−3t (2cos5t – 3sin5t) 
 (b) Find L − 1 log s+1 s−1 ⋆⋆⋆⋆⋆

2 of 2

[8+8]