Mathematics For Bio Technologists

Set No. 1

Code No: R05012304

I B.Tech Supplimentary Examinations, Aug/Sep 2007 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

(x3 cos 3x)

(b) If u = x3 y-y 2 x+x3 , find i. ii.

∂2u ∂x2 ∂2u ∂y 2

[8+8]

2. Evaluate the following integrals. R√ (a) 1 − sin xdx R cos xdx (b) sin x(1+sin x) √

(c)

Re

x log xdx

[5+5+6]

1

1 + a2 − b 2 2ab 3. (a) Show that 2b (b) Reduce the matrix 0 1  Where A = 4 0 2 1

2ab −2b 1 − a2 + b 2 2a −2a 1 − a2 − b 2

= (1 + a2 + b2 )3

A to its normal form. 2 −2 2 6  and hence find the rank 3 1

[8+8]

4. (a) Form equation by eliminating the arbitrary constant √ the differential 1/y sin x + e = c. (b) Solve the differential equation: dr + (2r cot θ + sin 2θ) dθ = 0.

[6+10]

5. (a) Solve the differential equation: (D2 − 5D + 6)y = ex sinx.

(b) Radium decomposes at a rate proportional to the amount present at that time. If a fraction p of the original amount disappears in 1 year how much time will remain at the end of 21 years. [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 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. 1

Code No: R05012304

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

Rπ

Sinxdx by Simpson?s

0

1 rd 3

rule. Verify your answer with integration.  t R ¯ where L{f(t)}= f (s). f (u)du = f (s) 8. (a) Prove that L s

[8+8]

0

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

2 of 2

h

s+3 (s2 +6s + 13)2

i

[8+8]

Set No. 2

Code No: R05012304

I B.Tech Supplimentary Examinations, Aug/Sep 2007 MATHEMATICS FOR BIOTECHNOLOGISTS (Bio-Technology) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) If A=[1,3,5,7,9,11], B=[2,3,4,5] and C= [7,8,9] then prove that (A ∪ B) ∩ C = A ∪ (B ∪ C) (b) Find the length of tangent for y 2 =4ax at (a,2a)

[8+8]

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

R1

dx (1+x2 )2

[6+5+5]

−1

3. (a) Solve the following system of equations x+2y-z=3, 3x-y+2z=1, 2x-2y+3z=2 using Cramer’s rule. (b)  Find 0  1   3 1

the rank of the  matrix by reducing it to the normal form. 1 −3 −1 0 1 1   1 0 2  1 −2 0

[8+8]

4. (a) Obtain the differential equation of the coaxial circles of the system x2 +y2 +2ax + c2 = 0 where c is a constant and a is a variable parameter. dy = (b) Solve the differential equation: dx

x−y cos x

1 + sin x

[6+10]

5. (a) Solve the differential equation: (D2 − 5D + 6)y = ex sinx. (b) Radium decomposes at a rate proportional to the amount present at that time. If a fraction p of the original amount disappears in 1 year how much time will remain at the end of 21 years. [10+6] 6. (a) Find a real root of x3 -4x-9=0 using bisection method (b) Solve the system of equations by Gauss Seidel method. 5x + 2y + z = 12 x + 4y + 2z = 15 x + 2y + 5z = 20

[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. 1 of 2

Set No. 2

Code No: R05012304

(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 i. Simpson’s 13 rd rule ii. Simpson’s 83 th rule

[8+8]

8. Using Laplace transforms solve the differential equation that x(0)=2, x(0)= -1 at t = 0 ⋆⋆⋆⋆⋆

2 of 2

d2 x dt2

− 2 dx + x = et , given dt [16]

Set No. 3

Code No: R05012304

I B.Tech Supplimentary Examinations, Aug/Sep 2007 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)

R (x2 +x)dx x2 +x+2

dx (16−5x2 )

(b)

R

(c)

Π/4 R

(1+sin 2x)dx (cos x+sin x)

[5+5+6]

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 y sin x = c + x. (b) Solve the differential equation:

dy dx

+

y cos x+sin y+y sin x+x cos y+x

= 0.

[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 x3 -2x2 -4=0 using Iterative method (b) Solve the system of equations by Gauss Seidel method. 3x - y + 2z = 3 x + 3y + 2z = -1 x + 2y + 5z = 1

[8+8]

7. (a) Find y(25), Given that y20 =24, y24 =32, y28 =35, y32 =40 using Gauss forward difference formula R2 2 [8+8] (b) Evaluate e−x dx using Simpson?s rule.Taking h = 0.25. 0

1 of 2

Set No. 3

Code No: R05012304 8. (a) Evaluate

R∞

t3 e−t Sint dt using the Laplace transforms.

0

(b) hFind thei inverse Laplace Tranformations of 1 s2 (s+42 )

⋆⋆⋆⋆⋆

2 of 2

[8+8]

Set No. 4

Code No: R05012304

I B.Tech Supplimentary Examinations, Aug/Sep 2007 MATHEMATICS FOR BIOTECHNOLOGISTS (Bio-Technology) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) If A=[1,3,5,7,9,11], B=[2,3,4,5] and C= [7,8,9] then prove that (A ∪ B) ∩ C = A ∪ (B ∪ C) (b) Find the length of tangent for y 2 =4ax at (a,2a)

[8+8]

2. Evaluate the following integrals. R dx (a) √3−5x R (b) (1 + x2 ) ex dx (c)

Π/2 R

x2 sin xdx

[5+6+5]

0

−2a 3. (a) Show that a + b a + c

a+b − 2b b+c

a + b b + c = 4 (a + b (b + c) (c + a)) − 2c

(b) Find non P and Q such that PAQ is in the normal form   singular matrices 1 1 2  3  [8+8] for A = 1 2 0 −1 −1

4. (a) Form the differential equation by eliminating the arbitrary constant sin−1 x + sin−1 y = c. (b) Solve the differential equation: (ey + y cos xy ) dx + ( x ey + x cos xy) dy = 0

[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) Prove that Newton’s method has a quadratic convergence (b) Solve the system of equations by Gauss Seidel method 7x + y + 4z = 5, 2x + 3y + 8z =9, x + 8y + 4z = 2

[8+8]

7. (a) Using Gauss backward interpolation formula find y (50o 42′ ) given that x 50 51 52 53 54 y = tan x 1.1918 1.2349 1.2799 1.327 1.3764 (b) Using trapezoidal rule,approximately calculate the value of with 1 of 2

R3 1

dx/

p

(1 + x)

Set No. 4

Code No: R05012304 i. four intervals and ii. six intervals.

[8+8]

8. Solve the differential equation (D2 - 3D + 2)y = e y(0) = - 3y′ (0) = 5 using Laplace transforms. ⋆⋆⋆⋆⋆

2 of 2

2x

. given that [16]