P T Engineering
Resume:
Total No. of Questions : *]
[Total No. of Pages : 4
[5868]-101
F.E. (Semester- I & II)
ENGINEERING MATHEMATICS - I
(2019 Pattern) (107001)
Time : 2 Hours] [Max. Marks : 70
Instructions to the candidates:
1) Q. 1 is compulsory.
2) Attempt Q2 or Q3, Q4 or Q5,Q6 or Q7, Q8 or Q9.
3) Neat diagrams must be drawn wherever necessary. 4) Figures to the right indicate full marks.
5) Use of electronic pocket calculator is allowed. 6) Assume suitable data, if necessary.
P6485
Q1) Write the correct option for the following multiple choice questions. a) If eigen value of a square matrix A is zero then. [1] i) A is non-singular ii) A is orthogonal
iii) A is singular iv) None of these
b) If u = yx then
u
x
is equal to [1]
i) 0 ii) xyx–1
iii) yx log y iv) None of these
c) The orthogonal transformation x = py transforms the quadratic form 222
Qx =1232 +332 x +x - x x 3 to the canonical form 222 Qyyy ¢ =12 +2 +3 .
The rank of quadratic from is [2]
i) 2 ii) 3
iii) 1 iv) 0
d)
22
1
u sec x 2 y
xy
= - éù êú ëû +
. Find the value of
x uu y
x y
+
[2]
i) –tan u ii) –cot u
iii) tan u iv) cot u
SEAT No. :
P.T.O.
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[5868]-101 2
e) If u = x2–y2 and v = 2xy then the value of,
,
uv
x y
is [2]
i) 4(x2 + y2) ii) – 4 (x2 + y2)
iii) 4(x2–y2) iv) 0
f) A system of linear equations Ax = B, where B is a null (zero) matrix is [2] i) Always consistent
ii) Consistent only if A = 0
iii) Consistent only if A 0
iv) In consistent if (A) < No. of variables
Q2) a) If 3
zy =tan + ax + y -ax 2 find value of
22
2
22
zz a
x y
-
. [5]
b) If
33
u tan 1 x y
x y
=- ç ç ç èøthen æö - + prove that
222 22 2
x uuu 22 21 xy y 4sin u sin2 u
xx y y
++ = -
[5]
c) If uf =( xy 22 -;yz 22 -; zx 22 - ) find value of 111 uuu
x xy yz z
++
[5]
OR
Q3) a) If uaxby =+;vbxay =-find value of
yvxu
uxyv
x uvy
çççç èø æö ççç èø æö èø æö èø ç æö ç [5] b) If ux =sin - 122 ( + y ) then find value of
222
22
xxyy uuu 22 2
x xy y
++
[5]
c) If uf = ( r, s ) where rxy =22 +; S = x 22 - y then show that yxxy uuu 4
x yr
+=
. [5]
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[5868]-101 3
Q4) a) If x = uv and
y uv
uv
+
=
-
, find,
uv
x y
. [5]
b) Examine for functional dependence,tan 11 tan
1
uvx xy
xy
- --
==-
+
y
and if depedent find the relation between them. [5] c) Discuss maxima and minima of fx (,y ) =x 22 +y +6 x + 12 [5] OR
Q5) a) Prove JJ' = 1 for x = u cosv, y = u sinv. [5] b) In calculating the volume of a right circular cone, errors of 2% and 1% are made in measuring the height and radius of base respectively find the error in the calculated volume. [5]
c) Find maximum value of ux = 234 y z such that 234 x +yz += a by Langrange's method. [5]
Q6) a) Investigate for what values of & the equations x+y+z = 6, x+2y+3z =10, x+2y+ z = have i) No solution ii) Infinitely many solutions. [5]
b) Examine for linear dependence and independence the vectros (1,1,3),
(1,2,4), (1,0,2). If dependent, find the relation between them. [5] c) Verify whether matrix
cos 0 sin
010
sin 0 cos
A
éù
êú
= êú
êú ëû -
is orthogonal or not. [5]
OR
Q7) a) Solve the system of equations x+y+2z = 0, x+2y+3z=0, x+3y+ z 5] b) Examine following vectors for linear dependence and independence
(1,–1,1), (2,1,1), (3,0,2). If dependent, find the relation between them.[5] c) Determine the currents in the network given in the figure . [5] CEGP013091
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Q8) a) Find the eigen values of the matrix
12
54
A
éù -
= êú
êú ëû -
. [5]
Find eigen vector corresponding to the highest eigen value. b) Verify cayley-Hamilton theorem for
12
34
A
éù
= êú êú ëû
. Hence find A–1 if it exists.
[5]
c) Find the modal matrix p which diagonalises
53
35
A
éù
= êú
êú ëû
. [5]
OR
Q9) a) Find the eigen values of
123
032
00 2
A
éù
êú
= êú
êú ëû -
. [5]
Find eigen vector corresponding to the highest eigen value. b) Verify cayley-Hamilton theorem for
100
101
010
A
éù
êú
= êú
êú ëû
[5]
c) Reduce the quadratic form 222
Qx =12 +22 xx +32 + x x 33 - 2 x x 11 + 2 x x 2 to canonical form by congruent transformations. [5]
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P.T.O.
Total No. of Questions—8] [Total No. of Printed Pages—4+1 Seat
No. [5667]-1001
F.E. (I Semester) EXAMINATION, 2019
ENGINEERING MATHEMATICS—I
(Phase–II)
(2019 PATTERN)
Time : 2 Hours Maximum Marks : 70
N.B. :— (i) Attempt Q. No. 1 or Q. No. 2, Q. No. 3 or Q. No. 4, Q. No. 5 or Q. No. 6, Q. No. 7 or Q. No. 8.
(ii) Use of electronic pocket calculator is allowed.
(iii) Assume suitable data, if necessary.
(iv) Neat diagrams must be drawn wherever necessary.
(v) Figures to the right indicate full marks.
1. (a) If z = tan (y + ax) + (y – ax)3/2, find the value of 2 2
2
2 2
–
z z
a
x y
· [6]
(b) If T = sin 2 2
2 2
,
xy
x y
x y
by using Euler’s theorem
find
T T
x y
x y
· [6]
(c) If u = x2 – y2, v = 2xy and z = f(u, v), then show that
– 2 2 2
z z z
x y u v
x y u
· [6]
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[5667]-1001 2
Or
2. (a) If x = u tan v, y = u sec v, prove that : [6] y y x x
u v u v
x x y y
·
(b) If u = sin–1,
x y
x y
by using Euler’s theorem.
prove that : [6]
2 2 2
2 2 3
2 2
1
2 (tan – tan ).
4
u u u
x xy y u u
x x y y
(c) If x =
cos
,
u
y =
sin
u
and z = f(x, y), then show that : [6]
– z z z z
u y x y x
u x y
·
3. (a) If u = x + y + z, v = x2 + y2 + z2, w = xy + yz + zx find u v w
x y z
· [6]
(b) Examine whether u =
–
,
1
x y
xy
v = tan–1 x – tan–1 y are
functionally dependent, if so find the relation between them. [5]
(c) Find the extreme values of x2 + y2 +
2 2
x y
· [6]
Or
4. (a) If u = x + y2, v = y + z2, w = z + x2, using Jacobian find
x
u
· [6]
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[5667]-1001 3 P.T.O.
(b) A power dissipated in a resistor is given by P = 2
R
· If errors
of 3% and 2% are found in and R respectively, find the percentage error in P. [5]
(c) Using Lagrange’s method find extreme value of xyz if x + y + z = a. [6]
5. (a) Examine for consistency of the system of linear equations and solve if consistent : [6]
x1 + x2 + x3 = 0
–2x1 + 5x2 + 2x3 = 1
8x1 + x2 + 4x3 = –1
(b) Examine for linear dependence or independence the vectors
(1, 1, 1, 3), (1, 2, 3, 4), (2, 3, 4, 7). Find the relation between them if dependent. [6]
(c) Determine the values of a, b, c when A is orthogonal where : [5]
0 2
A –
–
b c
a b c
a b c
·
Or
6. (a) Investigate for what values of a and b, the system of equations 2x – y + 3z = 2, x + y + 2z = 2, 5x – y + az = b have :
(1) No solution
(2) A unique solution
(3) An infinite number of solutions. [6]
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[5667]-1001 4
(b) Examine for linear dependence or independence the vectors x1 = (2, 3, 4, –2), x2 = (1, 1, 2, –1), x3 =
–1 1
, – 1, – 1,
2 2
·
Find the relation between them if dependent. [6]
(c) Determine the currents in the network given in figure below : [5]
7. (a) Find the eigen values and the corresponding eigen vectors for the following matrix : [6]
4 0 1
A –2 1 0
–2 0 1
·
(b) Verify Cayley-Hemilton theorem for A =
1 –1 0
2 3 –2
–2 0 1
and
use it to find A–1. [6]
(c) Find a matrix P that diagonalizes the matrix
A =
1 6 1
1 2 0
0 0 3
· [6]
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[5667]-1001 5 P.T.O.
Or
8. (a) Find the eigen values and the corresponding eigen vectors for the following matrix : [6]
5 0 1
A 0 –2 0
1 0 5
·
(b) Verify Cayley-Hamilton theorem for A =
0 1 0
0 0 1
1 –3 3
and use
it to find A–1. [6]
(c) Reduce the following quadratic form to the sum of the squares form : [6]
Q = 2x2 + 9y2 + 6z2 + 8xy + 8yz + 6xz.
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