Engineering Mathematics-1
Question
S. No
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UNIT-I THEORY OF MATRICES
1 3i 4 7 1 3i
1
Find the eigen values of the matrix
2 3
If A is Hermitian matrix Prove that it is skew- Hermitian matrix State Cayley- Hamilton Theorem
4
Prove that
1 2
1 i 1 i 1 i 1 i is a unitary matrix.
1 2 3 5 Find the value of k such that rank of 2 k 7 is 2 . 3 6 10 3 4i 2 6 Find the eigen values of the matrix 2 3 4i 7
1|P ag e
Question
S. No
Find A
9
1 1 Find the Skew- symmetric part of the matrix 3
1 1 1
2 1 2
10 If 2, 3, 4 are the eigen values of A then find the eigen values of adj A UNIT-II DIFFERENTIAL CALCULUS METHODS
4
5
6 7 8 9
Define Rolle’s Mean value theorem. Verify Lagrange’s Mean Value theorem for f(x) = log x in [1, e] Verify Lagrange’s Mean Value theorem for function f(x) = cos x in [0, π/2]. Verify Cauchy’s Mean Value theorem for f(x) =x2, g(x) =x3 in[1, 2]. Find first and second order partial derivatives of ax2+2hxy+by2 and
verify
2 f 2 f . xy xy
When two functions u,v of independent variables x,y are functional dependent If x = u(1-v), y = uv prove that JJ”=1
x y 3axy 3
Find the maximum and minimum values of If
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3 1 i 2 i 4 2i
Define modal matrix.
2 3
Course Outcome
if A =
8
1
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u e x siny ,v e x cos y then find .
3
(u , v) ( x , y )
10 Verify Rolle’s Mean value theorem for f(x)=(x+2)3(x-3)4 in [-2,3] Analyze UNIT-III IMPROPER INTEGRALS, MULTIPLE INTEGRALS AND ITS APPLICATIONS 1 Prove that (m, n) (n, m) . Analyze a
( (a x)
m1
x n1dx (a b)mn1 (m, n), m 0, n 0
2
Prove that
3
Compute (11 / 2), (1 / 2), (7 / 2) .
4
Write the value of
5
Evaluate
5
7
Analyze
7
Apply
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Remember
7
Evaluate
8
Evaluate
8
Evaluate
8
Evaluate
9
0
2
(1) .
x
7
ydydx Evaluate rdrd . Evaluate xy (x y )dxdy .
8
Evaluate
6
0 0
a sin
0
0
3 1
0 0
2|P ag e
e log y
ex
logzdxdydz . 1 1
2
Question
S. No 9
find the value of
1
2
3
1 2 3
dxdydz .
10 Write the spherical polar coordinates
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Remember Remember Apply Apply Apply Apply Apply Remember Apply
12 12 12 13 12 12 12 12 14
UNIT-1V DIFFERENTIAL EQUATIONS AND APPLICATIONS 1 2 3 4 5 6 7 8
3x
2
Solve (x+1)dy/dx –y=e (x+1) Write the working rule to find orthogonal trajectory in Cartesian form. Form the D.E.by eliminate c in 2 y=1+c√1-x Solve (x+y+1) dy/dx =1 Prove that the system of parabolas y2 = 4a (x+a) is self orthogonal. Find the O.T. of the family of curves State Newtons law of cooling A bacterial culture, growing exponentially, increases from 200 to 500 grams in the period from 6 a.m to 9 a.m. . How many grams will be present at noon.
9
Solve -3 + 2y =0 10 Define S.H.M. and give its D.E UNIT-V LAPLACE TRANSFORMS AND ITS APPLICATIONS 1 2 3 4 5 6 7 8 9 10
Find Laplace transform of State and prove first shifting theorem. Define change of scale property of Find If is periodic function with period T then find Find Find Find ) Define inverse L.T. of f(s) Use L.T to solve D.E. when t=0
1. Group - B (Long Answer Questions) S. No
1
2
Question UNIT-I THEORY OF MATRICES Show that only real number for which the system x+2y+3z= x , 3x+y+2z= y , 2x+3y+z= z has non-zero solution is 6 and solve them when =6 Express the matrix matrix and skew- Hermitian matrix.
3|P ag e
as the sum of Hermitian
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S. No 3
Question Given that
show that
is unitary
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1
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1
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3
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2
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4
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4
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1
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Remember
1
matrix.
4
5 6
7 8
Find Inverse by elementary row operations of
Find whether the following equations are consistent if so solve them Solve the equations using partial pivoting Guass Elimination method of
1 1 1 2 1 and find A4 Diagonalize the matrix A= 0 4 4 3 Prove The Eigen Values of Real symmetric matrix are Real.
Reduce the Quadratic form form. Reduce the quadratic form 10 canonical form by orthogonal reduction. 9
11
12
to the
Find rank by reducing to Normal form of matrix
Solve the following System of equations 4x+2y+z+3w=0, 6x+3y+4z+7w=0, 2x+y+w=0
Find 13
-4xy-4yz to the Canonical
and
such that x 2 y z 1 , x 2y z ,
x 2 y z 1 has (i) no solution (ii) Unique solution (iii) Many solutions
14
15
1 1 Find the eigen values and eigen vectors of 1 1 0 0 State Cayley-Hamilton theorem and verify the matrix A =
4|P ag e
0 0 0
S. No
Question
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1 1 1 0 1 2 1 0 1
16
17
2 2 3 2 1 Diagonalize the matrix A 1 2 2 1 1 1 2 1 2 by an orthogonal Diagonalize the matrix A 1 1 2 1 transformation. Reduce to sum of squares, the quadratic form
18
x12 2 x22 7 x32 4 x1 x2 8x1 x3 and find the rank, index and signature.
19
Prove that the eigen values of a Skew-Hermitian matrix are purely imaginary or zeros. If A is any square matrix then prove that i) A + A
20
are Hermitian iii) A - A
ii) AA , A A
are skew – Hermitian
UNIT-II DIFFERENTIAL CALCULUS METHODS 1
if f ( x) =
x , g ( x) =
1 prove that ‘c’ of the CMVT is the geometric x
mean of a and b for any a>0,b>0 2 3 4
x 2 y 2 z 2 given x y z 3a prove using mean value theorem sin u sin v u v find the minimum value of
u
yz xz xy (u, v, w) ,v ,w find x y z ( x, y, z )
5 Apply
5
Analyze
5
Apply
5
Apply
5
Apply
5
if using mean value theorem prove that the function 5 6
f ( x) x 2 4 x 7 is increases when x 2 decreasing when x 2 Calculate approximately
by using LMVT 2
7 8
find the region in which f ( x) =1-4 x - x is increasing and the region in which it is decreasing using mean value theorem find three positive numbers whose sum is 100 and whose product is maximum
5|P ag e
S. No
Question
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Find the volume of the greatest rectangular parallelepiped that can be 9
10 11 12 13
14
x2 y 2 z 2 inscribed in the ellipsoid 2 2 2 1 a b c Using mean value theorem for 0 a b prove that a b b 1 6 1 1 log 1 and hence Show that log 6 5 5 b a a 2 2 4 4 find the maxima and minima of f ( x) 2( x y ) x y Using mean value theorem prove that tanx>x in 0 x 2 1 3 1 cos 1 using LMVT Prove that 3 5 3 5 3 8 If
Also Show that =1
Prove that u xy yz zx, v x y z , w x y z are functionally dependent and find the relation between them If throughout an interval [a,b] prove using mean value theorem f(x) is a constant in that interval. Divide 24 into three points such that the continued product of the first, square of the second and cube of the third is maximum 2
15 16 17 18
using rolle’s theorem show that zero between 0 and 1
19
If
20
If
2
2
g ( x) 8x3 6 x 2 2 x 1 has a
x y z u, y z uv, z uvw show that ( x, y, z ) u 2v (u, v, w) x u(1 v), y uv prove that JJ 1 1
UNIT-III IMPROPER INTEGRATION, MULTIPLE INTEGRATION AND APPLICATIONS
x2 y 2 x2 y 2 dxdy over 2 2 2 the annular region between the circles x y a and By transforming into polar coordinates Evaluate
1
Evaluate
8
Evaluate
9
Evaluate
9
Apply
9
x 2 y 2 b2 with b a Evaluate 2
( x y z)dzdydx where R is the region bounded by the R
planes x 0, x 1, y 0, y 1, z 0, z 1 1
1 z 1 y z
3
0
0
xyzdxdydz
0
Evaluate Find the volume of the tetrahedron bounded by the plane 4
x y z 1 and the coordinate planes by triple integration a b c
6|P ag e
S. No
Question a (1 cos )
5
0
Evaluate
ellipse
0
Evaluate 8 9 10
a sin
2
a2 r 2 2
7
0
rdzdrd
8
Apply
8
Evaluate
9
Analyze
7
Analyze
7
Evaluate
7
Evaluate
8
0
ProveThat
where p>0, q>0
Show that (1 / 2)
Evaluate a
11
Evaluate
xydxdy taken over the positive quadrant of the
x2 y 2 1 a 2 b2
Course Outcome
0
Find the value of 6
r 2 cos drd
Blooms Taxonomy Level
Evaluate the double integral
a2 y2
0
( x 2 y 2 )dydx
0
12
Show that
Analyze
9
13
Show that
Analyze
8
Evaluate
9
Evaluate
9
Evaluate
7
Evaluate
7
r2 0 (r 2 a 2 )2 drd
2
14 Evaluate
0
1 2 x
15
By changing the order of integration, evaluate
xydxdy
0 0
By changing the order of integration, evaluate a
16
a2 x2
0
a 2 x 2 y 2 dydx
0
sin
17 Express
7|P ag e
0
p
cos q d in terms of beta function.
Blooms Taxonomy Level
Course Outcome
Evaluate
9
Evaluate
9
Evaluate
8
Analyze
10
Apply
10
Solve
a 2 ( xdy ydx) x2 y 2
3
Solve
2 xydy ( x2 y 2 1)dx 0
Apply
11
4
Find the orthogonal trajectories of the family of curves
Apply
10
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10
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11
Analyze
10
S. No
Question
Evaluate 18
x dxdy over the region bounded by hyperbola 2
xy 4, y 0, x 1, x 4 x
x log y
0
0
log 2
19
Evaluate
0
e x y z dxdydz
Evaluate by changing the order of integration 20
1
a
a
0
ax
y2
UNIT-IV DIFFERENTIAL EQUATIONS AND APPLICATIONS A bacterial culture, growing exponentially, in increases from 200 to 500 grams in the period from 6 am to 9 am. How many grams will be present at noon?
xdx ydy
2
( x 1)
5 6
dydx
y 4 a2 x2
x2 y 2 a2
dy y e3 x ( x 1)2 dx
Solve the D.E Obtain the orthogonal trajectories of the family of curves
r (1 cos ) 2a
7
A particle is executing S.H.M, with amplitude 5 meters time 4 sec.find the time required by the Particle in passing between points which are at distances 4 &2 meters from the centre of force and are on the same side of it.
8
Solve
( D2 3D 2) y 2cos(2 x 3) 2e x x 2
Apply
10
9
Solve
D2 ( D2 4) y 96 x2 sin 2 x k
Apply
10
10
By using method of variation of parameters solve
Apply
10
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10
Analyze
10
11
x Solve
( D2 1) y cos ecx
dy y x3 y 6 dx
If the air is maintained at 25 c and the temperature of the body cools
12
from 100 c to 80 c in 10 minutes, find the temperature of the body
after 20 minutes and when the temperature will be 40 c
8|P ag e
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( D4 2D2 1) y x2 cos2 x
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10
1 hours? 2
Analyze
10
Evaluate
12
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12
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13
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12
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13
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12
S. No
Question
13
Solve ( D 1) y sin x sin 2 x e x
14
x (1 e y )dx e y (1 )dy 0 y Solve
2
x
x
2
x
A copper ball is heated to a temperature of 80 c and time t=0, then it is
placed in water which is maintained at 30 c . If at t=3minutes,the 15
temperature of the ball is reduced to 50 c find the time at which the
temperature of the ball is 40 c 16
Solve ( D 6D 11D 6) y e 3
2
x3 sec2 y
2 x
e3 x
dy 3x 2 tan y cos x dx
17
Solve
18
A body weighing 10kgs is hung from a spring pull of 20kgs will stretch the spring to 10 cms. The body is pulled down to 20 cms below the static equilibrium position and then released. Find the displacement of the body from its equilibrium position at time‘t’ seconds, the maximum Velocity and the period of oscillation
19
20
Solve the D.E The number N of bacteria in a culture grew at a rate proportional to N. The value of N was initially 100 and increased to 332 in one hour. What was the value of N after 1
UNIT-V LAPLACE TRANSFORMS AND ITS APPLICATIONS
1
Using L.T Evaluate
2
Find L
3 4
5
6
et e2t 0 t dt
s 2 ( s 1)( s 9)( s 25) if0
2
2
cos 4t sin 2t t
Find L
2 cos(t), 3 Find L g (t ) where g(t)= 0, 2S 2 6S 5 Find the .L.T of 3 S 6S 2 11S 6
9|P ag e
2 3 2 if t< 3
if t>
S. No
Question
7
Solve the D.E using L.T, y
8
Solve the D.E using L.T,
9 10 11
12 13 14
15
16
18
3 y ' 2 y 4t e3t , y(o) y ' (0) 1
d2y dy 2 2 y 5sin t , y(0) y ' (0) 0 2 dt dt 2 s 4 Find the .L.T of log( 2 ) s 9 Use Laplace transforms, solve
( D2 1) x t cos 2t , givenx(0) x' (0) 0 2S 2 6S 5 S 3 6S 2 11S 6 e2 s 1 Find L 2 s 4s 5 2 ' Using L.T solve ( D 2D 3) y sin x, y(0) y (0) 0 Find the Inverse L.T of
2t
Find the Laplace transform of te sin 3t Find the L.T of periodic function f (t ) with period T where
T 4ET T -E, if 0 t 2 f(t)= 3E- 4ET , if T t T T 2 2S 3 -1 Find L 3 2 S 6S 11S 6 Solve the D.E using L.T
17
''
d 2x dx 2 x e2t with 2 dt dt
dx x(0) 2, 1 at t=0 dt Find L t sin 3t cos 2t
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14
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13
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12
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14
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12
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14
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12
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Program Outcome
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1
Using L.T solve 19 20
( D3 D2 4D 4) y 68e x sin 2 x, y 1, Dy 19, D2 y 37 at x 0 3t Find L e sinh 3t using change of scale property
3. Group - III (Analytical Questions) S. No
Questions UNIT-I THEORY OF MATRICES
1
i j
If A is n rowed matrix A aij where aij , . denotes greatest integer then find the value of det A
10 | P a g e
S. No
Questions
Blooms Taxonomy Level Apply
Program Outcome 1
2
If a=diagonal(1 -1 2)and b=diagonal (2 3 -1)then find 3a+4b
3
If a=
Apply
1
4
1 0 k 3 Is invertible. Find the value of k for which matrix A= 0 1 k 0 1
Apply
1
Remember
1
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2
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2
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1
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1
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1
Remember
5
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5
Remember
5
Remember
5
Apply Remember
6 6
Remember
6
Remember Apply
6 6
Apply
5
5 6
7
i 0 4n then find A 0 i
What is rank of 4x5 matrix If 1,2,3 are eigen value of A then find eigen value of Adj A.
1 0 0 & 1 are two orthogonal vectors of 3x3 matrix then find third If 1 0 vector
8
Ik 0
If A is nxn matrix , rank is k and normal form is of null matrix below side of
9 10
1 2 3 4 5 6 7 8 9 10
1
0 then find order 0
Ik
3 0 0 3 If A= 5 4 0 then express A in terms of A. 3 6 1 Find the rank of quadratic form whose eigen values are 0 ,0 ,6 UNIT-II DIFFERENTIAL CALCULUS METHODS When the Jacobian Transformation is used? Find the functional relationship between u=x + y +z, v=x y +y z + z x, w = x2+y2+z2 What are critical points? Write the relationship between u 3 2
xy , v tan1 x tan1 y . 1 xy
Find the stationary values of x y (1-x-y). What are saddle points? What is condition for f( x , y)to have maximum and minimum values at (a, b)? What is the demerit of Lagrange’s method of undetermined multipliers? If f(x, y) = x y+(x-y) then find stationary points.
u
If u=x y then x
.
UNIT-III IMPROPER INTEGRALS, MULTIPLE INTEGRALS AND ITS APPLICATIONS Write the relationship between beta and gamma functions. Remember
11 | P a g e
7
Blooms Taxonomy Level
Program Outcome
s in cos d using -function
Remember
7
(p+1,q) + (p,q+1).
Remember
7
Apply
7
Analyze
8
Remember Remember Remember
8 9 8
Analyze
8
Analyze
8
Apply
10
Analyze
10
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10
S. No
Questions
2
What is the value of
2
4
2
0
3 4
What is the value of
(m, m)
Find
An equivalent iterated integral with order of integration reversed for 5
1 ex
dydx is 0 1
6 7 8
How to find the area of bounded region. How to find the volume of closed surface. What is difference between proper and improper integrals
9
Convert
10
a
a2 x2
0 1
What is the area of
( x 2 y 2 )dydx to polar co-ordinates.
r drd over the region included between the 3
circles r sin , r 4 sin .
UNIT-IV DIFFERENTIAL EQUATIONS 1
1
6 d 2 y dy Find the order and degree of y dy 2 dx
2
A spherical rain drop evaporates at a rate proportional to its surface area at any instant t. The differential equation giving the rate of change of the radius (r) of the rain drop is.
3
If x
4 5 6
When the differential equation is said to be homogeneous? Mention two applications of higher order differential equations. What is general solution of higher order differential equations?
Remember Remember Remember
10 10 10
7
what is orthogonal form of the function f (x , y ,
dx dr ), f (r , , ) dy d
Remember
11
8
Remember
10
Remember
10
Analyze
10
Remember
12
Remember
12
3 4
what is general solution of linear differential equation when the Bernoull’s differential equation becomes linear differential equation Give the complementary function for (D2+6D+9)y=0 UNIT-V LAPLACE TRANSFORMS Give example where the Laplace Transforms technique is used What are the conditions that the functions has to satisfy to Apply laplace transform Where the convolution theorem is in laplace transforms What are periodic functions
Remember Remember
12 13
5
Find L
Apply
12
9 10
1 2
4
dy x 2 y 2, y(1) 1 then y(2) is dx
1
12 | P a g e
1 s a
S. No 6
Questions What is Laplace Transform of unit impulse function
L f ' (t)
7
If f(0)=0 find
8
Find the value of
9
When L1
10
If y’’+ y=sin 3t with y=y’=0 then find
L 2t
1 is possible n s
13 | P a g e
Lf (t)
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12
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12
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14
