# Gujarat Technological University 2010-1st Sem A.M.Ae.S.I Aeronautical Engineering -Maths 1 - Question Paper

GUJARAT TECHNOLOGICAL UNIVERSITY

B.E. Sem-I Examination January 2010

Seat No. Enrolment No.

Subject code: 110008 Subject Name: Mathematics - I

Date: 11 / 01 /2010 Time: 11.00 am - 02.00 pm

Total Marks: 70

2. Make suitable assumptions wherever necessary.

3. Figures to the right indicate full marks.

Q1. (a) (i) Find the value of k so that the function given below is continuous at a given 02 point x=2.

2^{x}+^{2} -16

^{f} (x) =

4 ^{x} -16 k , x 2

^{(ii)} State Sandwich theorem and using it find lim g(x) if 3 - x^{3} < g(x) < 3sec x ^{02}

x-0

for all x.

(b) (i) If f (x) and g(x) are continuous functions for 0 < x < 1, could 02

f (x)/g (x) possibly be discontinuous at a point in the interval [0,1]? Give

reasons for your answer.

(ii) If f : (a, b) is differentiable at c e (a, b), then show that 02

f (c + h) - f (c - h) . '

lim - exists and equals j (c).Is the converse true?

h0+ 2h

x

e -1

< 1, for x > 0. 03

x

v

(c) (i) Using Mean Value Theorem, Prove 0 < log

(ii) For what values of a, m and b does the function 03

[3 , x = 0

- x^{2} + 3x + a, 0 < x < 1

^{f} (x) = <

mx + b ,1 < x < 2 satisfy the hypothesis of the Mean Value Theorem on the interval [0,2].

Q2. (a) (i) Find the area of the region between the x-axis and the graph of 02

f (x) x x 2x, 1 < x < 2.

x^{2}

^{(ii) Using Fundamental Theorem of Calculus find =1 cos tdt. 02}

1

(iii) Evaluate the integral J-.

02

0 ^{x} + ^{1}

(b) (i) Find the absolute maximum and minimum values of the function on the given 03 interval f (t) |t - 5, 4 < t < 7 .

The geometric mean of two positive numbers a and b is the number 4ab . ^{02} Show that the value of c in the conclusion of the Mean Value Theorem for

f (x) = ^{1} on an interval of positive numbers [a, b] is c = yfab. x

(b) (i) Test the convergence or divergence of the following series (ANY TWO) 04

ro n i ro ro

a. I ^{b}. I -nr c. I n!^{(}x - 4f

n=0 3 n=1e n=0

b l

(ii) Using Riemann Sum show that J x dx = (b^{2} - a^{2})

^{2} ~^{21} 03

a

Q3. (a) Suppose that w = f (x, y), x = g(r, 5) and y = h(r, 5) then write the chain rule 05

dw dw dw dw

for and . Also evaluate and in terms of r and 5 if

dr d5 dr d5

w = x + 2y + z^{2}, x = y_{s} , y = r^{2} + ln 5, z = 2r.

ro b

(b) (i) Show that J f (x) dx may not equal to lim J f (x) dx. 02

b ro

-ro -b

^{(ii) -1 fx2 + y2 'I , . du du l . 03}

If u = tan ^{1}

show that x--+ y = sin 2u

dx dy 2

^{x -} y

x

(c) Find the length of the curve y = Jyjcos2t dt from x = 0 to x = . 04

0

OR

Q3. (a) Let w = f (x, y, z )be a function of three independent variables, write the 05

formal definition of the partial derivative for at (x_{0}, y_{0}, z_{0}). Using this

definition find at (l, 2,3) for f (x, y, z) = x^{2}yz^{2}.

(b) (i) Show that 03

2xy

2 2> ^{(}x y^{)} *^{(0,0)}

^{f (}x y ^{)} =

x ^{2} + y ^{2}

0 , (x, y) = (0,0)

is continuous at every point except at the origin.

^{(ii)} Find ^{d}dt f ^{w} = xy + z, x = cos t, y = sin t, z = t. ^{02}

(c) Find the volume of the solid generated by revolving the region bounded by 04

y = 4x and the lines y = 2 and x = 0 about the line y = 2.

Find the equations for tangent plane and normal line at the point (l, 1, l) on the 03

2 2 2 surface x + y + z = 3.

(b)

^{(c)}

Q4. (a) (b)

(c)

Q5. (a)

(b)

(c)

Q5. (a)

(b)

(c)

Find the area of the region that lies inside the cardioid r = 1 + cos 3 and 04

outside the circle r = 1.

OR

1 4z 2n \

Evaluate J J J(r2 cos2 3 + z2 )r d3 dr dz . 04

0 0 0

22

Integrate f (x, y) = ^{ln(x} + ^{y}_l over the region 1 < x^{2} + y^{2} < e by changing

^{(i) Vx2 + y2 04}

to polar coordinates

^{(ii)} Find the derivative of f (x, y, z) = x^{3} - xy^{2} - z at Po (1,1,0) in the direction ^{02}

of v = 2 i - 3 j + 6 k .

Find the volume of the prism whose base is the triangle in xy - plane 04

bounded by the x - axis and the line y = x and x = 1 and whose top lies in the plane z = f (x, y) = 3 - x - y..

05

Integrate f (x, y, z) = x + -Jy - z^{2} over the path C = Q u C2 from (0,0,0) to (1,1,1) with

Cj : r (t) = ti +1^{2} j, 0 < t < 1

C_{2} : r (t) = i + j + tk, 0 < t < 1

State Greens theorem and also evaluate the integral J (6y + x)dx + (y + 2 x )dy ^{05}

C

where C : The circle (x - 2)^{2} + (y - 3)^{2} = 4.

Trace the curve r2 = a2 cos 23. 04

OR

Use Greens theorem to evaluate the integral j(y2 dx + x2 dy) where 05

C

C : The triangle bounded by x = 0, x + y = 1, y = 0.

^{ 2 }

Find the flux of F = yz j + z k outward through the surface S cut from the 05

22

cylinder y + z = 1, z > 0, by the planes x = 0 and x = 1.

Use Stokes theorem to evaluate J F. dr if 04

C

F = (x + y) i + (2x - z) j + (y + z) k and C is the boundary of the triangle (2,0,0), (0,3,0) and (0,0,6).

3

GUJARAT TECHNOLOGICAL UNIVERSITY

B.E. Sem-I Examination January 2010

Seat No. Enrolment No.

Subject code: 110008 Subject Name: Mathematics - I

Date: 11 / 01 /2010 Time: 11.00 am - 02.00 pm

Total Marks: 70

2. Make suitable assumptions wherever necessary.

3. Figures to the right indicate full marks.

Q1. (a) (i) Find the value of k so that the function given below is continuous at a given 02 point x=2.

2^{x}+^{2} -16

^{f} (x) =

4 ^{x} -16 k , x 2

^{(ii)} State Sandwich theorem and using it find lim g(x) if 3 - x^{3} < g(x) < 3sec x ^{02}

x-0

for all x.

(b) (i) If f (x) and g(x) are continuous functions for 0 < x < 1, could 02

f (x)/g (x) possibly be discontinuous at a point in the interval [0,1]? Give

reasons for your answer.

(ii) If f : (a, b) is differentiable at c e (a, b), then show that 02

f (c + h) - f (c - h) . '

lim - exists and equals j (c).Is the converse true?

h0+ 2h

x

e -1

< 1, for x > 0. 03

x

v

(c) (i) Using Mean Value Theorem, Prove 0 < log

(ii) For what values of a, m and b does the function 03

[3 , x = 0

- x^{2} + 3x + a, 0 < x < 1

^{f} (x) = <

mx + b ,1 < x < 2 satisfy the hypothesis of the Mean Value Theorem on the interval [0,2].

Q2. (a) (i) Find the area of the region between the x-axis and the graph of 02

f (x) x x 2x, 1 < x < 2.

x^{2}

^{(ii) Using Fundamental Theorem of Calculus find =1 cos tdt. 02}

1

(iii) Evaluate the integral J-.

02

0 ^{x} + ^{1}

(b) (i) Find the absolute maximum and minimum values of the function on the given 03 interval f (t) |t - 5, 4 < t < 7 .

The geometric mean of two positive numbers a and b is the number 4ab . ^{02} Show that the value of c in the conclusion of the Mean Value Theorem for

f (x) = ^{1} on an interval of positive numbers [a, b] is c = yfab. x

(b) (i) Test the convergence or divergence of the following series (ANY TWO) 04

ro n i ro ro

a. I ^{b}. I -nr c. I n!^{(}x - 4f

n=0 3 n=1e n=0

b l

(ii) Using Riemann Sum show that J x dx = (b^{2} - a^{2})

^{2} ~^{21} 03

a

Q3. (a) Suppose that w = f (x, y), x = g(r, 5) and y = h(r, 5) then write the chain rule 05

dw dw dw dw

for and . Also evaluate and in terms of r and 5 if

dr d5 dr d5

w = x + 2y + z^{2}, x = y_{s} , y = r^{2} + ln 5, z = 2r.

ro b

(b) (i) Show that J f (x) dx may not equal to lim J f (x) dx. 02

b ro

-ro -b

^{(ii) -1 fx2 + y2 'I , . du du l . 03}

If u = tan ^{1}

show that x--+ y = sin 2u

dx dy 2

^{x -} y

x

(c) Find the length of the curve y = Jyjcos2t dt from x = 0 to x = . 04

0

OR

Q3. (a) Let w = f (x, y, z )be a function of three independent variables, write the 05

formal definition of the partial derivative for at (x_{0}, y_{0}, z_{0}). Using this

definition find at (l, 2,3) for f (x, y, z) = x^{2}yz^{2}.

(b) (i) Show that 03

2xy

2 2> ^{(}x y^{)} *^{(0,0)}

^{f (}x y ^{)} =

x ^{2} + y ^{2}

0 , (x, y) = (0,0)

is continuous at every point except at the origin.

^{(ii)} Find ^{d}dt f ^{w} = xy + z, x = cos t, y = sin t, z = t. ^{02}

(c) Find the volume of the solid generated by revolving the region bounded by 04

y = 4x and the lines y = 2 and x = 0 about the line y = 2.

Find the equations for tangent plane and normal line at the point (l, 1, l) on the 03

2 2 2 surface x + y + z = 3.

(b)

^{(c)}

Q4. (a) (b)

(c)

Q5. (a)

(b)

(c)

Q5. (a)

(b)

(c)

Find the area of the region that lies inside the cardioid r = 1 + cos 3 and 04

outside the circle r = 1.

OR

1 4z 2n \

Evaluate J J J(r2 cos2 3 + z2 )r d3 dr dz . 04

0 0 0

22

Integrate f (x, y) = ^{ln(x} + ^{y}_l over the region 1 < x^{2} + y^{2} < e by changing

^{(i) Vx2 + y2 04}

to polar coordinates

^{(ii)} Find the derivative of f (x, y, z) = x^{3} - xy^{2} - z at Po (1,1,0) in the direction ^{02}

of v = 2 i - 3 j + 6 k .

Find the volume of the prism whose base is the triangle in xy - plane 04

bounded by the x - axis and the line y = x and x = 1 and whose top lies in the plane z = f (x, y) = 3 - x - y..

05

Integrate f (x, y, z) = x + -Jy - z^{2} over the path C = Q u C2 from (0,0,0) to (1,1,1) with

Cj : r (t) = ti +1^{2} j, 0 < t < 1

C_{2} : r (t) = i + j + tk, 0 < t < 1

State Greens theorem and also evaluate the integral J (6y + x)dx + (y + 2 x )dy ^{05}

C

where C : The circle (x - 2)^{2} + (y - 3)^{2} = 4.

Trace the curve r2 = a2 cos 23. 04

OR

Use Greens theorem to evaluate the integral j(y2 dx + x2 dy) where 05

C

C : The triangle bounded by x = 0, x + y = 1, y = 0.

^{ 2 }

Find the flux of F = yz j + z k outward through the surface S cut from the 05

22

cylinder y + z = 1, z > 0, by the planes x = 0 and x = 1.

Use Stokes theorem to evaluate J F. dr if 04

C

F = (x + y) i + (2x - z) j + (y + z) k and C is the boundary of the triangle (2,0,0), (0,3,0) and (0,0,6).

3

GUJARAT TECHNOLOGICAL UNIVERSITY

B.E. Sem-I Remedial examination March 2009

Seat No. Enrolment No.

Subject code: 110008 Subject Name: MATHS - I Date: 18 / 03 /2009 Time: 10:30am To 1:30pm

Instructions: Total Marks: 70

1. Attempt all questions.

2. Make suitable assumptions wherever necessary.

3. Figures to the right indicate full marks.

(a) Do as directed (Each of one mark) 08

i. State Lagranges Mean value theorem. What does it geometrically mean?.

ii. Define critical point. Does local extremum exist at x = 0 to the function y = |X, however it is not differentiable at x = 0 ?.

^{} n +1

iii. For which values of p does the series is convergent.

n=1 n

x^{n}

iv.

n + 2

Find the radius of convergence for the series

n=l

5 l

Can we solve the integral I --dx directly?. Give the

v.

reason.

vi. Find the directional derivative of the function

f (x, y) = ax + by; a, b are constants, at the point (0,0) which makes an angle of 30 with positive x-axis.

2l-sin 0

vii. Evaluate the integral | | r^{2} cos 0drd0

0 0

viii. Find the constants a, b, c so that

F = (x + 2y + az)i + (bx - 3 y - z) j + (4x + cy + 2 z)k is irrotational

(b) Attempt the following

02

02

i. If |x 1 < 1o , prove that +1| < 0.331

, , , , i x^{2} x sin x

ii. It can be shown that the inequalities 1--<-< 1

6 2 - 2 cos x

hold for all values of x close to zero. What, if anything, does

x sin x

lim- ?

this tell you about x 2 - 2 cos x

iii. Prove that f (x) = x- [x] xe R is discontinuous at all ^{02}

integral points.

Attempt the following questions

3

i. Evaluate J(x 2)dx by using an appropriate area formula.

2

ii. State First Fundamental Theorem of Calculus. Find the value of c by using MVT for integral, for the function

f( x) = sin x, x e

(a)

02

0,P

2

iii. Expand sin(p + x) in powers of x. Hence find the value of ^{03} sin 44 ^{0}

Attempt the following questions

(b)

1 tan^{1} x tan^{1} y 1 _{TT} , ,

that-- <-<--. Hence deduce that

1 + x x y 1 + y

p 3 _{1} 4 p 1

I--< tan <+.

4 25 3 4 6

ii. Can the Rolle s Theorem for f(x) = |x|, xe [1,1] applied?

iii. Define stationary point. Suppose that a manufacturing firm 02 produces x number of items. The profit function of the firm is

_{x}^{3}

given by P( x) = + 729x 2500. Find the number of items that the firm should produce to attain maximum profit.

Attempt the following questions

i. State Rolle s Theorem. Show that this theorem cannot be applied 02 for f(x) = [*1 xe [0,2] however f'(x) = 0 for all xe (1.2)

(b)

ii. Verify Cauchys Mean Value theorem for and

x

02

-^{}r, "x e [a, b], a > 0. x

iii. What is the necessary condition for the function to have a local extremum?. A soldier placed at a point (3, 4) wants to shoot the fighter plane of an enemy which is flying along the curve

03

y = x^{2} + 4 when it is nearest to him. Find such the distance.

Q.3

(a)

Attempt the following questions

log_{10} (x +1) = x^{log1} ^{6}.

^{10} 1 + 6x

sin^{2} x

^{J} 4x( x 1)

4

3

11 r cos3x

iii. Check the convergence of I dx.

* /->

0 x'^{2}

(b) Test the convergence of the following series

1 2 3 02

i. --1---1---+ ... .

2! 3! 4!

n^{p}

r7 _{r}. 02

n=1 v n +1 + v n

ii.

n^{3} + 2

02

iii.

n=1 2^{n} + 2

iv. ne ^{n}

n=1

02

OR

Q.3

(a) Attempt the following questions

02

i.

State Cauchys Mean Value Theorem. Verify it for f (x) = log x, g(x) = , xe [l, e] , and find the value of c.

x

3x + 5 02 ii. Check the convergence of J 4-dx.

2 _{x}^{2} _{02} iii. Find the area between the curve y^{2} =--- and its ^{02}

asymptote.

1 - x^{2}

(b) Check the convergence of the following series 08

. i n1. xn

n=1 6^{n} n=11 n+ 2)

1 - 2x+ 3x^{2} - 4x^{3} + ..., 0 < x < 1

^{(-1) }n _{x}n+1 ^{}2n -1

i.

iii.

iv.

Q.4

(a) Attempt the following questions

f 1/. V \

x^{4} + y^{7} 4

1 V x^{/5} + y^{/5}

du du 1

prove that x--+ y = . 02

dx dy 20

If u = sin ^{1}

i.

_{N} , du du du

ii. If u = f(x- y, y- z, z- x), prove that--1---1--= 0. 03

dx dy dz

iii. Find the extreme values of x^{3} + 3-15x^{2} -15y^{2} + 72x. 03

(b) Attempt the following questions

i. Find the area common to the cardioids r = a(1 - cos 6) and 04 r = a(1 + cos 6).

ii. Evaluate JJ (x^{2}+r )dA , by changing the variables, where R

R

is the region lying in the first quadrant and bounded by the hyperbolas x^{2} - y^{2} = 1, x^{2} - y^{2} = 9, xy = 2, and xy = 4.

OR

02

03

ii.

iii.

03

If u = log(^{3} + y^{3} + z^{3} - 3xyz), prove that

' A+A+A']^{2} _{u} = _{-} 9

_{v} dx dy dz J (x + y + z)^{2} If u = f( x^{2} + 2yz, y^{2} + 2 zx), prove that

^{(}y^{2} - zx)^{uu}+^{(x}*^{-} y^{z)duu}+^{(z2 -} = o.

ax dy dz

The temperature at any point (x, y, z) in space is T = 400xyz'^{2}. Find the highest temperature on the surface of the unit sphere x^{2} + y^{2} + z^{2} = 1 by the method of Lagranges multipliers.

(a) Attempt the following questions

i.

Attempt the following questions

(b)

i. Find the volume generated by the revolution of the loop of the 04 curve y^{2} (a + x) = x^{2} (3 a - x) about the x-axis.

ii. Evaluate J Jy^{2} x^{2} + y^{2} dydx, by changing into polar

02

0 0 coordinates.

(a) Attempt the following questions

i. Evaluate JJ xydA , where R is the region bounded by x-axis,

02

03

02

02

02

R

ordinate

x = 2a and the curve x^{2} = 4 ay.

ii. Evaluate JJJVx^{2} + y^{2} dV , where D is the solid bounded by

D

the surfaces x^{2} + y^{2} = z^{2} , z = 0, z = 1.

(b) Attempt the following questions

i. Find the directional derivative of the divergence of

F(x, y, z) = xyi + xy^{2} j + z^{2} k at the point (2,1,2) in the direction of the outer normal to the sphere x^{2} + y^{2} + z^{2} = 9.

ii. Prove that r^{n}r is irrotational.

(c) Attempt the following questions

i. Find the work done when a force

F = (x^{2} - y^{2} + x)i - (2xy + y) j moves a particle in the xy-plane from (0, 0) to (1, 1) along the parabola x^{2} = y ?.

ii. Use divergence theorem to evaluate

JJ (x^{3} dydz + x^{2} ydzdx + x^{2} zdzdx) , where S is the closed

S

surface consisting of the cylinder x^{2} + y^{2} = a^{2} and the circular discs z=0 and z=b.

(a) Attempt the following questions

_ y

no

-dA by changing the order of integration. 02

0 x ^{y}

W1-x^{2} V1-x^{2} - y^{2}

ii. Evaluate | | |xyzdzdydx. 03

00

0

(b) Attempt the following questions

i. The temperature at any point in space is given by T = xy + yz 02 + zx. Determine the derivative of T in the direction of the

vector 3i - 4k at the point (1, 1, 1).

ii. Show that F = 2xyzi + (x^{2} z + 2y)j + x^{2}yk is irrotational and find a scalar function f such that F = gradf.

02

(c) Attempt the following questions

i.

02

22 x^{2} + y^{2}

Find I F.dr where F = ~^{yi}^{xj} and C is the circle

j V^{2} -i- T/

C

x^{2} + y^{2} = 1 traversed counterclockwise.

ii. Evaluate the surface integral curlF.ds by using Stokes 03

5

theorem, where 5 is the part of the surface of the parabobloid z = 1 - x^{2} - y^{2}, for which z > 0 and F = yi + zj + xk.

5

GUJARAT TECHNOLOGICAL UNIVERSITY

B.E. Sem-I Remedial examination March 2009

Seat No. Enrolment No.

Subject code: 110008 Subject Name: MATHS - I Date: 18 / 03 /2009 Time: 10:30am To 1:30pm

Instructions: Total Marks: 70

1. Attempt all questions.

2. Make suitable assumptions wherever necessary.

3. Figures to the right indicate full marks.

(a) Do as directed (Each of one mark) 08

i. State Lagranges Mean value theorem. What does it geometrically mean?.

ii. Define critical point. Does local extremum exist at x = 0 to the function y = |X, however it is not differentiable at x = 0 ?.

^{} n +1

iii. For which values of p does the series is convergent.

n=1 n

x^{n}

iv.

n + 2

Find the radius of convergence for the series

n=l

5 l

Can we solve the integral I --dx directly?. Give the

v.

reason.

vi. Find the directional derivative of the function

f (x, y) = ax + by; a, b are constants, at the point (0,0) which makes an angle of 30 with positive x-axis.

2l-sin 0

vii. Evaluate the integral | | r^{2} cos 0drd0

0 0

viii. Find the constants a, b, c so that

F = (x + 2y + az)i + (bx - 3 y - z) j + (4x + cy + 2 z)k is irrotational

(b) Attempt the following

02

02

i. If |x 1 < 1o , prove that +1| < 0.331

, , , , i x^{2} x sin x

ii. It can be shown that the inequalities 1--<-< 1

6 2 - 2 cos x

hold for all values of x close to zero. What, if anything, does

x sin x

lim- ?

this tell you about x 2 - 2 cos x

iii. Prove that f (x) = x- [x] xe R is discontinuous at all ^{02}

integral points.

Attempt the following questions

3

i. Evaluate J(x 2)dx by using an appropriate area formula.

2

ii. State First Fundamental Theorem of Calculus. Find the value of c by using MVT for integral, for the function

f( x) = sin x, x e

(a)

02

0,P

2

iii. Expand sin(p + x) in powers of x. Hence find the value of ^{03} sin 44 ^{0}

Attempt the following questions

(b)

1 tan^{1} x tan^{1} y 1 _{TT} , ,

that-- <-<--. Hence deduce that

1 + x x y 1 + y

p 3 _{1} 4 p 1

I--< tan <+.

4 25 3 4 6

ii. Can the Rolle s Theorem for f(x) = |x|, xe [1,1] applied?

iii. Define stationary point. Suppose that a manufacturing firm 02 produces x number of items. The profit function of the firm is

_{x}^{3}

given by P( x) = + 729x 2500. Find the number of items that the firm should produce to attain maximum profit.

Attempt the following questions

i. State Rolle s Theorem. Show that this theorem cannot be applied 02 for f(x) = [*1 xe [0,2] however f'(x) = 0 for all xe (1.2)

(b)

ii. Verify Cauchys Mean Value theorem for and

x

02

-^{}r, "x e [a, b], a > 0. x

iii. What is the necessary condition for the function to have a local extremum?. A soldier placed at a point (3, 4) wants to shoot the fighter plane of an enemy which is flying along the curve

03

y = x^{2} + 4 when it is nearest to him. Find such the distance.

Q.3

(a)

Attempt the following questions

log_{10} (x +1) = x^{log1} ^{6}.

^{10} 1 + 6x

sin^{2} x

^{J} 4x( x 1)

4

3

11 r cos3x

iii. Check the convergence of I dx.

* /->

0 x'^{2}

(b) Test the convergence of the following series

1 2 3 02

i. --1---1---+ ... .

2! 3! 4!

n^{p}

r7 _{r}. 02

n=1 v n +1 + v n

ii.

n^{3} + 2

02

iii.

n=1 2^{n} + 2

iv. ne ^{n}

n=1

02

OR

Q.3

(a) Attempt the following questions

02

i.

State Cauchys Mean Value Theorem. Verify it for f (x) = log x, g(x) = , xe [l, e] , and find the value of c.

x

3x + 5 02 ii. Check the convergence of J 4-dx.

2 _{x}^{2} _{02} iii. Find the area between the curve y^{2} =--- and its ^{02}

asymptote.

1 - x^{2}

(b) Check the convergence of the following series 08

. i n1. xn

n=1 6^{n} n=11 n+ 2)

1 - 2x+ 3x^{2} - 4x^{3} + ..., 0 < x < 1

^{(-1) }n _{x}n+1 ^{}2n -1

i.

iii.

iv.

Q.4

(a) Attempt the following questions

f 1/. V \

x^{4} + y^{7} 4

1 V x^{/5} + y^{/5}

du du 1

prove that x--+ y = . 02

dx dy 20

If u = sin ^{1}

i.

_{N} , du du du

ii. If u = f(x- y, y- z, z- x), prove that--1---1--= 0. 03

dx dy dz

iii. Find the extreme values of x^{3} + 3-15x^{2} -15y^{2} + 72x. 03

(b) Attempt the following questions

i. Find the area common to the cardioids r = a(1 - cos 6) and 04 r = a(1 + cos 6).

ii. Evaluate JJ (x^{2}+r )dA , by changing the variables, where R

R

is the region lying in the first quadrant and bounded by the hyperbolas x^{2} - y^{2} = 1, x^{2} - y^{2} = 9, xy = 2, and xy = 4.

OR

02

03

ii.

iii.

03

If u = log(^{3} + y^{3} + z^{3} - 3xyz), prove that

' A+A+A']^{2} _{u} = _{-} 9

_{v} dx dy dz J (x + y + z)^{2} If u = f( x^{2} + 2yz, y^{2} + 2 zx), prove that

^{(}y^{2} - zx)^{uu}+^{(x}*^{-} y^{z)duu}+^{(z2 -} = o.

ax dy dz

The temperature at any point (x, y, z) in space is T = 400xyz'^{2}. Find the highest temperature on the surface of the unit sphere x^{2} + y^{2} + z^{2} = 1 by the method of Lagranges multipliers.

(a) Attempt the following questions

i.

Attempt the following questions

(b)

i. Find the volume generated by the revolution of the loop of the 04 curve y^{2} (a + x) = x^{2} (3 a - x) about the x-axis.

ii. Evaluate J Jy^{2} x^{2} + y^{2} dydx, by changing into polar

02

0 0 coordinates.

(a) Attempt the following questions

i. Evaluate JJ xydA , where R is the region bounded by x-axis,

02

03

02

02

02

R

ordinate

x = 2a and the curve x^{2} = 4 ay.

ii. Evaluate JJJVx^{2} + y^{2} dV , where D is the solid bounded by

D

the surfaces x^{2} + y^{2} = z^{2} , z = 0, z = 1.

(b) Attempt the following questions

i. Find the directional derivative of the divergence of

F(x, y, z) = xyi + xy^{2} j + z^{2} k at the point (2,1,2) in the direction of the outer normal to the sphere x^{2} + y^{2} + z^{2} = 9.

ii. Prove that r^{n}r is irrotational.

(c) Attempt the following questions

i. Find the work done when a force

F = (x^{2} - y^{2} + x)i - (2xy + y) j moves a particle in the xy-plane from (0, 0) to (1, 1) along the parabola x^{2} = y ?.

ii. Use divergence theorem to evaluate

JJ (x^{3} dydz + x^{2} ydzdx + x^{2} zdzdx) , where S is the closed

S

surface consisting of the cylinder x^{2} + y^{2} = a^{2} and the circular discs z=0 and z=b.

(a) Attempt the following questions

_ y

no

-dA by changing the order of integration. 02

0 x ^{y}

W1-x^{2} V1-x^{2} - y^{2}

ii. Evaluate | | |xyzdzdydx. 03

00

0

(b) Attempt the following questions

i. The temperature at any point in space is given by T = xy + yz 02 + zx. Determine the derivative of T in the direction of the

vector 3i - 4k at the point (1, 1, 1).

ii. Show that F = 2xyzi + (x^{2} z + 2y)j + x^{2}yk is irrotational and find a scalar function f such that F = gradf.

02

(c) Attempt the following questions

i.

02

22 x^{2} + y^{2}

Find I F.dr where F = ~^{yi}^{xj} and C is the circle

j V^{2} -i- T/

C

x^{2} + y^{2} = 1 traversed counterclockwise.

ii. Evaluate the surface integral curlF.ds by using Stokes 03

5

theorem, where 5 is the part of the surface of the parabobloid z = 1 - x^{2} - y^{2}, for which z > 0 and F = yi + zj + xk.

5

GUJARAT TECHNOLOGICAL UNIVERSITY

B.E. Sem-I Remedial examination March 2009

Seat No. Enrolment No.

Subject code: 110008 Subject Name: MATHS - I Date: 18 / 03 /2009 Time: 10:30am To 1:30pm

Instructions: Total Marks: 70

1. Attempt all questions.

2. Make suitable assumptions wherever necessary.

3. Figures to the right indicate full marks.

(a) Do as directed (Each of one mark) 08

i. State Lagranges Mean value theorem. What does it geometrically mean?.

ii. Define critical point. Does local extremum exist at x = 0 to the function y = |X, however it is not differentiable at x = 0 ?.

^{} n +1

iii. For which values of p does the series is convergent.

n=1 n

x^{n}

iv.

n + 2

Find the radius of convergence for the series

n=l

5 l

Can we solve the integral I --dx directly?. Give the

v.

reason.

vi. Find the directional derivative of the function

f (x, y) = ax + by; a, b are constants, at the point (0,0) which makes an angle of 30 with positive x-axis.

2l-sin 0

vii. Evaluate the integral | | r^{2} cos 0drd0

0 0

viii. Find the constants a, b, c so that

F = (x + 2y + az)i + (bx - 3 y - z) j + (4x + cy + 2 z)k is irrotational

(b) Attempt the following

02

02

i. If |x 1 < 1o , prove that +1| < 0.331

, , , , i x^{2} x sin x

ii. It can be shown that the inequalities 1--<-< 1

6 2 - 2 cos x

hold for all values of x close to zero. What, if anything, does

x sin x

lim- ?

this tell you about x 2 - 2 cos x

iii. Prove that f (x) = x- [x] xe R is discontinuous at all ^{02}

integral points.

Attempt the following questions

3

i. Evaluate J(x 2)dx by using an appropriate area formula.

2

ii. State First Fundamental Theorem of Calculus. Find the value of c by using MVT for integral, for the function

f( x) = sin x, x e

(a)

02

0,P

2

iii. Expand sin(p + x) in powers of x. Hence find the value of ^{03} sin 44 ^{0}

Attempt the following questions

(b)

1 tan^{1} x tan^{1} y 1 _{TT} , ,

that-- <-<--. Hence deduce that

1 + x x y 1 + y

p 3 _{1} 4 p 1

I--< tan <+.

4 25 3 4 6

ii. Can the Rolle s Theorem for f(x) = |x|, xe [1,1] applied?

iii. Define stationary point. Suppose that a manufacturing firm 02 produces x number of items. The profit function of the firm is

_{x}^{3}

given by P( x) = + 729x 2500. Find the number of items that the firm should produce to attain maximum profit.

Attempt the following questions

i. State Rolle s Theorem. Show that this theorem cannot be applied 02 for f(x) = [*1 xe [0,2] however f'(x) = 0 for all xe (1.2)

(b)

ii. Verify Cauchys Mean Value theorem for and

x

02

-^{}r, "x e [a, b], a > 0. x

iii. What is the necessary condition for the function to have a local extremum?. A soldier placed at a point (3, 4) wants to shoot the fighter plane of an enemy which is flying along the curve

03

y = x^{2} + 4 when it is nearest to him. Find such the distance.

Q.3

(a)

Attempt the following questions

log_{10} (x +1) = x^{log1} ^{6}.

^{10} 1 + 6x

sin^{2} x

^{J} 4x( x 1)

4

3

11 r cos3x

iii. Check the convergence of I dx.

* /->

0 x'^{2}

(b) Test the convergence of the following series

1 2 3 02

i. --1---1---+ ... .

2! 3! 4!

n^{p}

r7 _{r}. 02

n=1 v n +1 + v n

ii.

n^{3} + 2

02

iii.

n=1 2^{n} + 2

iv. ne ^{n}

n=1

02

OR

Q.3

(a) Attempt the following questions

02

i.

State Cauchys Mean Value Theorem. Verify it for f (x) = log x, g(x) = , xe [l, e] , and find the value of c.

x

3x + 5 02 ii. Check the convergence of J 4-dx.

2 _{x}^{2} _{02} iii. Find the area between the curve y^{2} =--- and its ^{02}

asymptote.

1 - x^{2}

(b) Check the convergence of the following series 08

. i n1. xn

n=1 6^{n} n=11 n+ 2)

1 - 2x+ 3x^{2} - 4x^{3} + ..., 0 < x < 1

^{(-1) }n _{x}n+1 ^{}2n -1

i.

iii.

iv.

Q.4

(a) Attempt the following questions

f 1/. V \

x^{4} + y^{7} 4

1 V x^{/5} + y^{/5}

du du 1

prove that x--+ y = . 02

dx dy 20

If u = sin ^{1}

i.

_{N} , du du du

ii. If u = f(x- y, y- z, z- x), prove that--1---1--= 0. 03

dx dy dz

iii. Find the extreme values of x^{3} + 3-15x^{2} -15y^{2} + 72x. 03

(b) Attempt the following questions

i. Find the area common to the cardioids r = a(1 - cos 6) and 04 r = a(1 + cos 6).

ii. Evaluate JJ (x^{2}+r )dA , by changing the variables, where R

R

is the region lying in the first quadrant and bounded by the hyperbolas x^{2} - y^{2} = 1, x^{2} - y^{2} = 9, xy = 2, and xy = 4.

OR

02

03

ii.

iii.

03

If u = log(^{3} + y^{3} + z^{3} - 3xyz), prove that

' A+A+A']^{2} _{u} = _{-} 9

_{v} dx dy dz J (x + y + z)^{2} If u = f( x^{2} + 2yz, y^{2} + 2 zx), prove that

^{(}y^{2} - zx)^{uu}+^{(x}*^{-} y^{z)duu}+^{(z2 -} = o.

ax dy dz

The temperature at any point (x, y, z) in space is T = 400xyz'^{2}. Find the highest temperature on the surface of the unit sphere x^{2} + y^{2} + z^{2} = 1 by the method of Lagranges multipliers.

(a) Attempt the following questions

i.

Attempt the following questions

(b)

i. Find the volume generated by the revolution of the loop of the 04 curve y^{2} (a + x) = x^{2} (3 a - x) about the x-axis.

ii. Evaluate J Jy^{2} x^{2} + y^{2} dydx, by changing into polar

02

0 0 coordinates.

(a) Attempt the following questions

i. Evaluate JJ xydA , where R is the region bounded by x-axis,

02

03

02

02

02

R

ordinate

x = 2a and the curve x^{2} = 4 ay.

ii. Evaluate JJJVx^{2} + y^{2} dV , where D is the solid bounded by

D

the surfaces x^{2} + y^{2} = z^{2} , z = 0, z = 1.

(b) Attempt the following questions

i. Find the directional derivative of the divergence of

F(x, y, z) = xyi + xy^{2} j + z^{2} k at the point (2,1,2) in the direction of the outer normal to the sphere x^{2} + y^{2} + z^{2} = 9.

ii. Prove that r^{n}r is irrotational.

(c) Attempt the following questions

i. Find the work done when a force

F = (x^{2} - y^{2} + x)i - (2xy + y) j moves a particle in the xy-plane from (0, 0) to (1, 1) along the parabola x^{2} = y ?.

ii. Use divergence theorem to evaluate

JJ (x^{3} dydz + x^{2} ydzdx + x^{2} zdzdx) , where S is the closed

S

surface consisting of the cylinder x^{2} + y^{2} = a^{2} and the circular discs z=0 and z=b.

(a) Attempt the following questions

_ y

no

-dA by changing the order of integration. 02

0 x ^{y}

W1-x^{2} V1-x^{2} - y^{2}

ii. Evaluate | | |xyzdzdydx. 03

00

0

(b) Attempt the following questions

i. The temperature at any point in space is given by T = xy + yz 02 + zx. Determine the derivative of T in the direction of the

vector 3i - 4k at the point (1, 1, 1).

ii. Show that F = 2xyzi + (x^{2} z + 2y)j + x^{2}yk is irrotational and find a scalar function f such that F = gradf.

02

(c) Attempt the following questions

i.

02

22 x^{2} + y^{2}

Find I F.dr where F = ~^{yi}^{xj} and C is the circle

j V^{2} -i- T/

C

x^{2} + y^{2} = 1 traversed counterclockwise.

ii. Evaluate the surface integral curlF.ds by using Stokes 03

5

theorem, where 5 is the part of the surface of the parabobloid z = 1 - x^{2} - y^{2}, for which z > 0 and F = yi + zj + xk.

5

Attachment: |

Earning: Approval pending. |