# I/I SUBJECT NAME – MATHEMATICS-I SUBJECT CODE – (MAT-101)

YEAR/SEMESTER – I/I SUBJECT NAME – MATHEMATICS-I SUBJECT CODE – (MAT-101)

SECTION-A (Very Short Answer Type Questions) UNIT-I

S.No. Question COURSE OUTCOMES

a) Find???? if?? = log 3?? CO5

b) Find ???? if ?? = ??????2?? CO5

c) Find first order partial derivative of ?? = log(??2 + ??2) CO5

d) Find the nth derivation of CO1

e) State Leibnitz Theorem for the nth differentiation of the product of two functions CO1

f) If ?? = ??????+???? ??(???? – ????), ??h???? ??h???? ?? ???? + ?? ???? = 2??????

???? ????

CO2

g) If u = f(x/y, y/z, z/x), then prove that x ?u +y ?u +z ?u = 0

?x ?y ?z CO2

h) Find the (n-1)th derivative of xn. CO4

i) If y=(sin-1x)2,prove that (1-x2)??2??2 – ?? ????????-2=0.

???? CO2

j) State Euler’s theorem for homogeneous functions of two variables CO1

UNIT-II

S.No. Question COURSE OUTCOMES

a) State Taylor’s Theorem for a function of two variables CO1

b) State Maclaurin’s Theorem for a function of two variables CO1

c) Expand the exsiny in powers of xand y as far as terms of the third degree. CO3

d) Expand the excosy in powers of xand y as far as terms of the third degree. CO3

e) Write the Jacobian for implicit function. CO1

f) If ?? = ??(1 + ??), ?? = ??(1 + ??) Find the value of ?? (?? ??)

??(?? ??) CO1

g) If x=rcos??,y=rsin?? find ??(??,??) and ??(??,??).

??(??,??) ??(??,??) CO2

h) If u=exsin y and v=excos y; evaluate:??(??,??).

??(??,??) CO2

i) Find the extream values of theorem ??3 + ??3 – 3?????? CO1

j) Write the Lagrange,s method of undetermined multipliers CO1

UNIT-III

S.No. Question COURSE OUTCOMES

a) Define Singular & Non Singular Matrices. CO1

b) Define Triangular matrix. CO1

c) ? 3 5+2i -3?

If H=?5-2i 7 4i??, show that H is a

?

?? -3 -4i 5 ??

Hermitian matrix. Verify that iH is a skew Hermitian matrix. CO1

d) ?-1 2 1?

If A=? 3 -1 2?? , find the values of ? for which the matrix equation ?

?? 0 1 ???

AX=O has (i) Unique solution (ii) More than one solution. CO1

e) ?2 Find the sum and product of the eigen values of the matrix A= ?3 ?

??1 1

4

0 – 1?

2??

2?? CO1

f) Define orthogonal matrix with examples CO1

g) Define Unitary Matrix with examples CO1

h) Find the eigen val

? 8 -6 2 ?

? ? ,

-6 7 -4

? ?

?? 2 -4 3 ?? ues of the following matrices, CO1

i) 1 3

Transform [2 4

3 8

transformation. 3

10] into a unit matrix by using elementary

4 CO4

j) State Cayley-Hamilton theorem CO1

UNIT-IV

S.No. Question COURSE OUTCOMES

a) If ??? =xî+yj+z??^ then show that gradr=???

?? CO4

b) If ??? =xî+yj+z??^ then show that grad 1= – ???3

?? ?? CO4

c) If f(x,y,z)= 3xz2y-y3z2, fine grad f at the (1, -2,-1) CO4

d) Find a unit vector normal to the surface x3+ y3+3xyz=3 at the point (1,2,-1). CO4

e) What is greatest rate of increase of u= xyz2 at the point (1,0,3). CO4

f) 1 r=(??2 + ??2 + ??2)2 , evaluate ??2(log ??). CO3

g) If ??? =xî+yj+z??^ then show that curl r=0 CO3

h) if ??????(?x,y,z) = xz3î-2x2yzj +2yz4??^ , find divergence CO4

I) If ??? =xî+yj+z??^ then show that ??????(??^) = 2

?? CO3

J) Write the statement of Stoke’s Theorem CO1

SECTION-B (Short Answer Type Questions)

UNIT-I

S.No. Question COURSE OUTCOMES

a) If y = , find yn CO5

b) ???? ?? = ??????-1??,

?????????? ??h???? (1 – ??2)????+2 – (2?? + 1)??????+1 – ??2???? = 0 CO4

c) If xxyyzz=c, show that at x=y=z, ?2z =-(xlogex)-1

?x?y CO4

d) If u=log x4 + y4 , show that, x?u+y?u =3 x + y ?x ?y CO4

e) If u=exyz, prove that ?3u = (1+3xyz+x2 y2 z2)exyz.

?x?y?z CO4

f) If y=xnlogx prove that: Yn+1

??

CO4

g) If u= (x1/4+ y1/4)(x1/5+ y1/5),apply Euler’s theorem to find the values of X?u/?x+y ?u/?y CO5

h) If u=cos(???? 2++??????2++??????2),prove that x???? ????+?? ????????+?? ???? ???? =0 ??

CO2

i) If u=tan- ),prove that x????+y????= sin2u

??-?? ???? ???? CO2

j) If u= sin- ),prove that x tan u

???? ???? 2 CO2

UNIT-II

S.No. Question COURSE OUTCOMES

a) Expand the eaxsinby in powers of x and y as far as terms of the third degree. CO3

b) Expand tan-1[x+h], by Taylor series CO3

c) Expand cosx in powers of(x – -?? ) upto 4 degree

4 CO2

d) If x + y+ z=u, y+ z=vu and z=uvw, find (x,y,z) . ?

?(u,v,w) CO4

e) Find the extreme values of function x3-3axy+y3 CO4

f) Verify the chain rule for jacobians if x=u,y=u tan v,z=w. CO3

g) Calculate the jacobian??(??,??,??) of the following_u= x+2y+z, v= x+2y+3z,

??(??,??,??)

w= 2x+3y+5z CO3

h) Divide a number say 120 so that the sum of their products taken two at a time shall be maximum CO4

i) Find the max. and min. distances of the point (3,4,12)from the sphere x2+y2+z2=1 CO3

j) Find the min. value of x2+y2+z2,given that ax+by+cz=p. CO5

UNIT-III

S.No. Question COURSE OUTCOMES

a) a) Show that the system of equations3x+4y+5z =a,4x+5y+6z 5x+6y+7z = cdoes not have a solution unless a+c=2b. = b CO4

b) a) Find the rank following matrix by reducing in Normal form:

?1 3 2 5 1 ? ?2 2 -1 6 3 ?

? ?

?1 1 2 3 -1?

? ?

?0 2 5 2 -3? CO4

c) 2 0 -1

Find the inverse of the matrix[5 1 0 ]by using elementary row

0 1 3

operation. CO3

d) Investigate for what values of a & b the following equations,

?? + ?? + ?? = 6,?? + 2?? + 3?? = 10 & ?? + 2?? + ???? = ??

Have (i) Unique Solution, (ii) Infinite Solutions, (iii) No Solution. CO3

e) Using elementary transformations ,find the inverse of the matrix:

0 1 2

A=[1 2 3]

3 1 1 CO4

f) Test the consistency of the following system of linear equations and hence find the solution if

2x – y + 3z =8,-x +2y +z =4,3x +y -4z =0 CO4

g) Test the consistency of the following system of linear equations and hence find the solution if exists : 7×1+2×2+3×3=16, 2×1+11×2+5×3=25, X1+3×2+4×3=13 CO4

h) Using elem

1

A=[1

2 enta

2

4

6 ry transformations ,find the rank of the matrix:

3

2].

5 CO3

i) For what values of a & b the following equations,

?? + ?? + ?? = 1,?? + 2?? + 4?? = ?? & ?? + 4?? + 10?? = ??2 Have a solution and solve them completely in each case.. CO3

j) ?3 -2 1??x? ? b ?

Determine the values of a, b for which the system ??5 -8 9????y??=?? 3 ??

??2 1 a????z?? ??-1??

(i) a unique solution (ii) no solution (iii) infinitely many solutions. CO4

UNIT-IV

S.No. Question COURSE OUTCOMES

a) Show that the vector v = 3y4z2 i^+4x3z2 j – 3x2y2 k^ is solenoidal CO5

b) Prove that (y2 –z2+3yz-2x)i^ +(3xz+2xy)j +(3xy-2xz+2z)??^ is both solenoidal and irrotational. CO5

c) Find grad Ø,whereØ = 3??2?? – ??3??2 ???? ?????????? (1 ,-2 ,-1) CO2

d) f(x, y ,z) = 2×2 +3y2 +z2 at the point P(2,1,3) in the direction of the vector ?a? =î -2??^ CO4

e) If u=x+y+z, v=x2+y2+z2 ,w= yz+zx+xy, prove that grad u, grad v and grad w are coplanar vectors. CO4

f) Find the directional derivative of the function f= x2-y2+2z2 at the point p(1,2,3) in the direction of the line PQ where Q is the point (5,0,4) CO5

g) Find the direction derivative of f =5x2y-5y2z+5z2xat the point p(1,1,1)

2

in the direction of the line ??-1 = ?? -3 = ??.

2 -2 1 CO3

h) If ??? =xî+yj+z??^ then show that 1)gradr=??? (2) grad 1= – ???3 (3)grad

?? ?? ?? rn= nrn-2??? ,where ???= (xi^ +yj+z??^). CO5

I) if ??????(?x,y,z) = xz3î-2x2yzj +2yz4??^ , find divergence and curl of ??????(?x,y,z) CO4

J) Find the divergence and curl of the vector field ???(x,y,z)= ????????(xy2î + yz2j +zx2??^ ) at the point ( 1,2,3). CO5

SECTION-C [Descriptive Answer Type Questions]

UNIT-I

S.No. Question COURSE OUTCOMES

a) If tany = (a+x)/(a-x), prove that (a2+x2)yn+2 + 2(n+1)xyn+1 +n(n+1)yn = 0 CO5

b) If u=f(r), wherer2=x2 +y2 , prove that ?2u + ??y2u2 =f’’(r) + 1r f’(r)

?x2 CO3

c) If y= a cos(logx)+b(sinx),prove that

X2y2+xy1+y=0 and x2yn+2+(2n+1)xyn+1+(n2+1)yn=0. CO5

d) If y =(sin-1 x)2 , prove that(1-x2)yn+2 -(2n+1)xyn+1 -n2yn = 0. Hence find the value of yn at x=0 CO4

e) If ?? = ?????? (?? ??????-1?? )Prove that (1 – ??2)??2 – ????1 + ??2?? = 0 CO5

f) Find ????

???? ?? 2 + 5 ?? + 6

?? CO3

g) If y=??1 +x?? , prove that

?1-x?

(1-x2)yn -?2(n-1)x+1?yn-1 -(n-1)(n-2)yn-2 =0 CO5

h) If u= sin- ),prove that x tan u

???? ???? 2 CO4

i) If 2??+2?? + ??2??+2?? +??2??+2??=1,prove that

??

(?u/?x)2+(?u/?y)2+(?u/?z)2=2(x?u/?x+y?u/?y+z?u/?z). CO5

j) If u=sin-1 },show that x????+?? ????+?? ????+3tan u=0.

?? +?? +?? ???? ???? ???? CO4

UNIT-II

S.No. Question COURSE OUTCOMES

a) Expand the series for loge(1+x)and then find the series for loge? ?, and hence determine the value of loge??11 ?? up to five

?1+ x ?

?1-x ? ? 9 ?

places of decimal CO5

b) Expand tan-1 y in the neighborhood of (1, 1) and inclusive of x

second-degree terms. Hence compute f(1.1,0.9) approximately. CO3

c) Test the function f(x,y)=x3y2(6-x-y)for maxima and minima for point not at the origin. CO5

d) Find the minimum distance from the point (1, 2, 0) to the cone

??2 = ??2 + ??2 CO4

e) If u= xyz , v=x2+y2+z2, w=x+y+z, find the jacobian ??(??,??,??).

??(??,??,??) CO5

f) If u,v,w are the roots of the equation (x-a)3 +(x-b)3 +(x-c)3

?(u,v,w)

=0,then find .

?(a,b,c) CO3

g) Ifx+y+z=u,y+z=uv,z=uvw, then show that =u2v. CO5

h) Show that u = y+z, v = x+2z2, w = x – 4yz – 2y2 are not independent. Find the relation between them CO4

i) Using Lagrange’s method, find the maximum & minimum distances from the origin to the curve 3×2+4xy+6y2=140 CO5

j) The sum of three positive numbers is constant .Prove that their product is max. when they are equal. CO4

UNIT-III

S.No. Question COURSE OUTCOMES

a) ?4 3 1 ?

Verify Cayley Hamilton theorem, for?2 1 -2??

?

??1 2 1 ?? CO3

b) follow

?2 ?2

??-1 Find the eigen values and corresponding eigen vectors of the ing matrices,

1 1 ?

3 4?

-1 -2?? CO5

c) ? 2 -1 1 ?

Using Hamilton theorem find the inverse of ?-1 2 -1??

?

?? 1 -1 2 ?? CO4

d) Solve with the help of matrices the simultaneous equations:

?? + ?? + ?? = 3, ?? + 2?? + 3?? = 4, ?? + 4?? + 9?? = 6 CO5

e) ?2 1

Find the characteristic equation of the matrix A= ?0 1

?

??1 1

compute A-1.Also find the matrix represented by A8-5 A5+ A4-5 A3+8 A2 -2A+I. 1?

0?? and hence

2??

A7 +7 A6 -3 CO4

f) ?2+i 3 -1+3i?

If A=?-5 i 4-2i ?? , verify that A?A is a Hermitian m

? is the conjugate transpose of A. atrix where A? CO3

g) 2

Verify Cayley –Hamilton theorem for the matrixA=[-1

1 Hence compute A-1. -1 1

2 -1].

-1 2 CO5

h) ?1 0 0?

ForA = ??1 0 1?? , verify Cayley-Hamilton theorem.

??0 1 0?? CO4

I) Find the eigen values an

6 -2 2

[-2 3 -1]

2 -1 3 d eigen vectors of the following matrix CO5

j) 1

Reduce the matrixA=[0

0

transformation. -1 2

2 -1]to diagonal form similarity

0 3 CO4

UNIT-IV

S.No. Question COURSE OUTCOMES

a) Show that ??[??¯.3??¯] = ????¯3 – 3(??¯??.5??¯).??¯

?? CO3

b) Find the divergence and curl of the vector

??¯ = (??????)?? + 3??2???? + (????2 – ??2??)??^ CO3

c) A fluid motion is given by ??¯ = (ysinz –sinx ) i^ +(x sin z + 2yz ) j + ( xy cosz +y2)??^is the motion irrotational ? If so , find the velocity potential . CO4

d) Find the scalar potential function f for ????? = y2 i^+2xyj-??????^. CO5

e) Use divergence th, to evaluate??s ??s?????^ ds where S is the surface of the sphere x2+y2+z2=9 CO4

f) Verify Gauss’ divergence th. for ???=(x3-yz)î+(y3-zx)j+(z3-xy) ??^ taken over the cube bounded by the planes x=0, x=1 ,y=o,y=1 and z=o,z=1. CO5

g) Using Green theorem, find the area of the region in the first quadrant bounded by the curves y=x, y= 1/x, y=x/4 CO5

h) verify greens theorem by evaluating ?c (x3-xy3)dx+(y2-2xy)dy where c is the square having the vertices at the points (o,o),(2,0),(2,2) and (o,2). CO4

i) Evaluate stocke’s theorem, ??¯ = ??2??^ + ??^ – (?? + ??)??^ and m the boundary of the triangle with vertices (0, 0, 0), (1, 0, 0) and (1, 1, 0). CO5

j) If ?? = ?? + ?? + ??, ?? = ??2 + ??2 + ??2, ?? = ???? + ???? + ????, Prove that grad u, grad v and grad w are coplanar vector. CO4