BE-102
Model Test Paper -I
Engineering Mathematics-I
Time: 3 Hours
MM: 70
Note: 1. Attempt all questions. Each question carries equal marks.
UNIT-I
1.
(a) Expand Y =
by help of maclaurin’s theorem
and prove that.
= 1 + sin
+ …………
(b)Find the maxima or minima value of x3y2(1-x-y).
or
(a) If U = f(r) where r2 =x2+y2 then prove that
+
= f ''(r) + f '(r)
(b)Prove that the radious of curvature at any point of the curve
x = 3a cost – acos3t
y = 3a sint – a sin3t
UNIT-II
Q.2 (a) Evaluate
dx as of limit of summation ?
(b) Evaluate
by 0
x
1, 0
where R is the region determined
y
x2 ,0
or
z
x+y .
(a) Prove that
=
(b) Change the order of integration
dxdy, and hence evaluate it.
UNIT-III
Q.3
+4y = x2 +cos2x
(a) solve
(b) x2
-3x
+y =
or
(a)Solve
+ 2y =
,
-
=
(b) (D2 + 1)y = secx
UNIT-IV
Q.4
(a) Reduce the following matrices to normal form and hence find its rank:
(b) Verify Cayley-Hamilton theorem for the matrix A and find its inverse: A=
Or
(a) Investigate the values of
x+y+z=6
x+2y+3z=10
x+2y+ z=
in the simultaneous equations:
(b) Find the characteristics roots and the characteristics vectors of
the matrix:
UNIT-V
Q.5
(a) In a Boolean algebra,prove that:
(i)Complement of each element in B is unique.
(ii)(a')' = a for any, a
B
(b) Draw the simplified circuit of the function:
f(x,y,z) = x.z+y.(y+z).(x+y.z)
or
(a) Write the function:
F(x,y,z) = x.y' +x.z+(x.y)
Into conjunctive normal forms in three variables x,y and z.
(b) If a graph has exactly two vertices of odd degree, there must be a path between these
two vertices.
BE-102
Model Test Paper -II
Engineering Mathematics-I
Time -3 Hrs
Max Marks-70
Note-Attempt any five question. All question carry equal marks.
Unit I
1. (a) If u =loge(x3+y3+z3-3xyz) show that (
+
+
)=
(b) Using Maclaurin’s series to prove that loge (1+ex) = log2 +
+
-
----
Or
(a) The straight line
touch the curve (
n
+(
n
= 2 for all values of n. find the point of
contact.
(b) Find the maximum and minimum value of the function f(x) = x2-4x+5 in the interval 1≤ x ≤ 4
Unit II
2. (a) prove that
B(m,n)
(b) Evaluate
Or
(a) Evaluate
(b) Evaluate ∫∫R x2y2dxdy where R is the region bounded by x = 0,y = 0 and x2+y2 = 1, x≥0 , y≥0
Unit III
3. (a) solve
(b) Solve
+ x =cost given that x = 2 and y = 0 when t = 0
Or
(a) solve by method of variation of parameters
(b)
solve x2
Unit IV
4. (a) verify Cayley –Hamilton theorem of the matrix A =
and hence find A-1
(b) Investigate the value of λ and µ in the simultaneous equation
X+y+z= 6
X+2y+3z=10
X+2y+λz= µ
Or
(a) Find the Eigen values and Eigen vector of the matrix A=
(b) Test for consistency and solve 5x+3y+7z=4 , 3x+20y+2z=9 ,7x+2y+10z=5
Unit V
5 (a) Prove that the number of vertices of odd degree in a graph is always even.
(b) For every element of Boolean algebra B, Prove that (i) (a·b)’=a’+b’ (ii) a+a·b =a
Or
(a) Draw a simplified circuit of the function F(x,y,z)=x·y’·z + (z+y)·x’
(b) Define the following terms (i) Sub graph (ii) Degree of a vertex (iii) Tree (iv) Spanning Tree