MATHEMATICS PREVIOUS YEAR QUESTIONS

MATHEMATICS

PREVIOUS YEAR QUESTIONS

CREDITS

CHARAN VEGI

II YEAR, MCE

ANMOL K. MATHUR

II YEAR, ECE

AND

CHANGE ACCHA HAI TEAM

UNIT I

Infinite series: Tests for convergence of series (Comparison, Ratio, Root, Integral, Raebe's, logarithmic), Alternating series, Absolute convergence, Conditional convergence.

1. State and prove integral test for convergence of the infinite series.

(2015) 2. What is absolute convergence of an infinite series? Show that absolute convergence

implies convergence. What about the converse? Justify.

(2014) 3. Discuss the convergence of series

   

x

1 ₂

x

2 ₃

x

3 4 ! 3 3 ! 2 2 ! 1 1 -(2015, 2010) 4. Discuss the convergence and absolute convergence of the following series;

 

1 ... ... 3 2 1 3 2        n x x x x n n (2011) 5. State test to check for the convergence of an alternating series. Examine the

convergence/absolute convergence of the series





(1 2 3) ... 4 1 2 1 3 1 2 1 3 3 3       (2010) 6. Test the convergence of the series

  

3

2

1

3 3 3 3 sin 2 sin sinx x x -(2015)

7. Test the convergence of the series











_          2 3 ) 2 )( 1 ( 2 1 ) 1 ( 1 1 x n n n m m m x n n m m x n m _{……, m, n>0, x>0.}

8. State Cauchy's integral test. Hence or, otherwise discuss the convergence of



_np

1 .

(2011)

UNIT II

Differential & Integral Calculus of single variable: Taylor's & MaClaurin's expansion, Radius of curvature, Tracing of some standard curves, Applications of definite integral to Area, Arc length, Surface area and volume (in cartesian, parametric and polar co-ordinates).

1. Compute the value or cos 320_{upto four decimal places using Taylor series.} (2015) 2. Calculate the approximate value of by choosing a suitable function and writing its

Taylor's series, correct upto 4th_{decimal place.}

(2011) 3. Expand _ea_sin1x

upto the term containing x4_.

(2010) 4. Find the area common to two cardioids r = a(1 + cosϴ) and r = a(1 - cosϴ).

(2015) 5. Find the area included between the cardiod r = a(1 + cosϴ) and the circle r=a.

(2011) 6. A) Find the length of the curve 3

2 2      x y from x = 0 to x = 2.

B) Show that the radius of curvature at any point of the. Cardioid r = a(1 - cosϴ) varies as .

(2014, 2010) 7. Find the surface area of the solid generated by revolving an arc of the Cycloid x=a(ϴ

-sinϴ) ,

y =a(1- cosϴ ), 0 2 about the x- axis.

8. Find the surface area of the solid generated by revolving one arch of the cycloid x = a(ϴ + sinϴ) ,

y = a(1 — cosϴ) about the tangent at the vertex.

(2015)

9. The area lying inside the cardiod r= 2a(1 -cosϴ) and outside the parabola r(1+ cosϴ)= 2a is revolved about the initial line ϴ=0. Find the volume of the solid generated.

(2010)

10. Expand tan-1_{x up to the fifth power of x and using this evaluate the value ofπ.} (2014) 11. Expand f(xy) = tan-1_{xy in powers of (x -1) and (y-1) using Taylor's series expansion upto}

second degree terms and then compute f(1.1, 0.8).

(2010) 12. For the surface z = f(x, y), what is the geometrical interpretation for the partial

derivative of z with respect to x. Explain this by considering a surface of your choice. (2014)

13. Show that the curvature at the point (3a/2, 3a/2) on the folium x3_{+ y}3_{=3xy is} -a 3 2 8 _. (2011) UNIT III

Calculus of several variables: Partial differentiation, Euler's theorem, Total differential, Taylor's theorem, Maxima-Minima, Lagrange's method of multipliers, Application in estimation of error and approximation.

1. If           y x y x ec u _{1/3 1/3} 2 / 1 2 / 1 1/2 1

cos , prove that

                 12 tan 12 13 12 tan 2 2 2 2₂ 2 2 2 2 u u y u y y x u xy x u

x

(2015)

2. If y x y x u  

_sin1 _{,then prove that}

u u u y u y y x u xy x u x 2 2 2_{2 3} 2 2 2 cos 4 2 cos sin 2           (2010) 3. If z is a function of x and y, where x = and y = ,

show that y z y x z x v z u z            (2015) 4. Using Lagrange's multiplier method to find the volume of the greatest rectangular

parallelepiped that can be inscribed in the ellipsoid ₂2  ₂2  ₂2 1

c z b y a x (2015)

5. Give an example of a homogeneous function of order n in x and y. Verify that its partial derivative with respect to x is homogeneous function or Order n — 1.

(2014) 6. If             y x y x

u _sin 1 _{, then find the value Of}

(2014)

7. If u log(x3_{+ y}3_{+ z}3_{-3xyz), show that}





2 2 9 . z y x u z y x                  

8. Locate the stationary point of f (x, y)= x4_{+ y}4_{- 2x}2_{+ 4xy — 2y}2_{and determine their} nature.

9. At a distance of 50 meter from the foot of a tower, the elevation of its top is 300_{. the} possible errors in the measuring of distance and the elevation are 2 cm and 0.05 degree respectively. Find the approximate error in the calculated height.

(2011) 10. The diameter and altitude of a can in the shape of a right circular cylinder, with top and

bottom both closed, are measured as 4 cm and 6 cm respectively. The possible error in each measurement is at the most 0.1 cm. Find approximately the maximum possible error in the values computed for volume and surface.

(2014)

11. A balloon is in the form of a right circular cylinder of radius 1.5. meter and length 4 meter and is surmounted by hemispherical ends. If the radius is increased by 0.01 meter and length by 0.05 meter, find the percentage change in the volume of balloon.

(2010)

12. Using Lagrange's multipliers method find the smallest and the largest distances between the points P and Q such that P lies on the plane x+y+z= 2a and Q lies on the sphere x2₊ Y2_{+ z}2_{= a}2_{where a is any constant.}

(2010) UNIT IV

Multiple Integrals: Double integral (Cartesian and polar co-ordinates), Change of order of integration, Triple integrals (Cartesian, cylindrical and spherical co-ordinates), Beta and Gamma functions, Applications of multiple integration in area and volume.

1. Evaluate by changing the order of integration.

(2015) 2. Change the order of integration and hence evaluate the integral

 

2 



 0 0 / 20 2 2 a x a a x a xdydx xdydx

3. Using suitable multiple integral to find the volume bounded by the xy-plane, the cylinder = 1 and the plane x + y + z = 3.

(2011, 2015)

4. Prove the followings.

(i)







     

            2 / 0 3 4 4 3 4 1 2 1 sec tan  _    d (ii) . ) ( ) ,1 ( n m m n m n m       (2015)

5. Plot the area tying inside the circle and outside the circle . Using double integration evaluate it. Can you evaluate it using single integration? Just explain how?

6. Define gamma and beta functions. What is the relation between these two? Using the evaluate the integral

    d



/2 0 5 7 _cos sin (2014, 2010) 7. Find the volume of the sphere With centre at (2, 2, 2) and diameter equal to 4 units using

(I) Cylindrical coordinates, (Il) Spherical polar coordinates.

(2014)

8. Find the volume of the ellipsoid x2_/a2_{+ y}2_/b2_{+ z}2_/c2_{= 1 using} (i) Cylindrical co-ordinates.

(ii) Spherical polar co-ordinates.

(2010) 8. Evaluate

 

  1 0 2 2 2 2 x x x y

xdydx _{by changing in the polar co-ordinates}

(2011) 9. Show that



         1 0 1 . 1 log dy y n n (2011) 10. Find the volume bounded by cylinder x2_+y2_{= 4 hyperboloid x}2_{+ Y}2_{- z}2₌₁

(2011)

11. Using double integration calculate the area which is inside the cardioid r = 2(I+cosØ) and outside the circle r =2.

(2010)

Vector Differential Calculus: Continuity and differentiability of vector functions, Scalar and Vector point function, Gradient, Directional Derivative, Divergence, Curl and their

applications.

1. A vector field is given by _{F 2xyz}3^_ix2_z3 ^_j3x2_yz2_k^ _{. Show that it is irrotational, and} hence find its scalar potential.

(2015)

2. Define gradient of scalar field. What is its geometrical interpretation? Find the directional derivative of ф at the point (1, 1, 1) in the direction of the line

z y x _     2 3 2 1 (2014) 3. Define gradient of a scalar field. What does its direction and magnitude represent? Find

the angle between the two surfaces x2_+y2_+z2_{=9 and z = x}2_{+ y}2_{- 3 at (2,-1,2).}

(2010) 4. A) If and are constant vectors, then find *



a*(b*r)



, where is a position vector of

the point P(x, y, z).

B) Show that the vector * is solenoidal.

(2014)

5. (i) If u=x+y+z, v =x2_+y2_+z2_{w=yz+zx+xy. prove that grad u, grad v and grad w are coplanar.} (ii) Find whether the vector field F cosh(xy)(^i^j)is conservative. If it is so, find the potential function.

(2011)

6. Prove that div(rn_{)= (n + 3)r}n_{. Hence show that /r}3_{is solenoidal. Is}3_{irrotatinal also?} Verify your answer. Here is the position vector of a point P(x,y,z)and r = ||

(2010)

UNIT VI

Vector Integral Calculus: Line integral, Surface integral and Volume integral, Applications to work done by the force, Applications of Green's, Stoke's and Gauss divergence theorems.

1. Evaluate



s

s

d

r

.

_{where is the position vector, andSis the surface}

(2015) 2. Verify Stokes' Theorem for the function _{F x}2^_ixy^_{j integrated round the square in}

the plane z = 0 and bounded by the lines x = 0, Y = 0, and x = a, y = a.

(2015) 3. Evaluate









 

 





c dy y x dx xy

x2 2 2 _{, where c is the square formed by the line}

1 , 1    x y (2014) 4. Evaluate



_SF.n^ds, where ^ 2 ^ ^ 2

i

xy

j

y

k

z

F







_andSis the portion of the surface of the cylinder + , 0z4, included in the first octant.

(2014) 5. Evaluate SF nds

^

.



, where F axi^by^jczk^ andSis the surface of the sphere . (2014) 6. State the Divergence theorem. Verify it for F 4xz^iy^jyzk^taken over the cube

bounded by the planes x =0= y= z and x=1=y=z.

(2011) 7. Verify Green's theorem in the plane for



c[(3x 8y )dx(4y6xy)dy]

2

2 _{where C is the}

boundary of the region bounded by x=0, y =0 and x+y= 1.

(2011) 8. If ^ ^ 2 ^ 4xzi y j yzk F    , evaluate



s dS N

F. ^ , where S is the surface of the cube bounded by x=0, x =1, y=0, y = 1, :z=0, z=1.