Babu Banarasi Das Northern India Institute of Technology, Lko
DEPARTMENT OF MATHEMATICS
B.TECH FIRST SEMESTER
ODD SEMESTER (2013 -14)
MATHS – I (NAS -103)
QUESTION BANK
UNIT – I
Q 1 If
=
) , Prove that
(
+ ……………+
Q 2 If x = sin(
+( − 1)! . Also show that
=
) then evaluate the value (1 −
)
with usual symbols .
Q 3 If y = (sin
) prove that (
n even.
Q 4 If
=
ℎ
ℎ
) = 0 ,for n odd (
=
=
Q 6 If u = f(r) and x = rcos ,y = sin ,,prove that
+
.
= 1, find
+3
)
( ) + tan
+
=
Q 8 find the first six terms of the espansion of the function
in the neighborhood of the point (0,0).
Q 9 Expand
Q 11 Trace the curve
+
)
( )
′′ (
)+
′(
log (1 + ) ina Taylor’s series
in the powers of (X-1) and (y-1) upto the third degree terms.
(2 − ) =
Q 12 Trace the Folium of Descarte
(cissoid)
+
Q 13 Trace th lamniscate of Bernoulli
=3
=
=
)
π
cosy near the point, (1, ) by Taylor’s theorem
Q 10 Expand function
−(
+1+
) = 2. 2 . 4 . 6 ……( − 2) ,n ≠2for
= −(
,
Q 5 Verify Euler’s theorem for the function u =sin
Q 7 If u = x log xy,where
− (2 + 1 )
= !
a
2 a
1
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UNIT - II
Q 1 If
,
,
.,
=
is 4.
=
,
. Show that the jacobian of
=
Q 2 If u = x+y+z ,uv =y+z, uvw = z evaluate
( , , )
,
,
to
ℎ
.
( . . )
Q. 3 If x = r cosθ , y = r sin θ prove that JJ/ = 1, where J = J(x, y)
Q 4 If u=
,v =
,
=
relation between them.
(
)
show that u.v,w,are not independent and find the
Q 5 If u,v, w are the roots of equation ( − ) + ( − ) + ( − ) = 0
, ,
find jacobian of
to , , .
ℎ
Q 6 The power ‘P’ required to propel a steamer of length ‘l’ at a speed ‘u’ is given by P =
where
is constant if u is increased by 3% and l is decreased by 1% find the
corresponding increase in P.
Q 7 In estimating the cost of a pile of bricks measured as 6’×50’× 4′,the tape is stretched 1%
beyond the standard length .If counts is 12 bricks to
,and bricks cost Rs 100 per 1000 ,find
the approximate error in the cost.
Q 8 Find [(3.82) + 2(2.1) ] .
Q 9 Show that the minimum value of f(x,y) =xy +
Q 10 Test the function f(x,Y) = (
not on the circle
+
=1
+
(
)
)
( + ) is 3
for the maxima and minima for the points
Q 11 Find the dimension of rectangular box of maximum capacity whose surface aria is
given when (a) box is open at the top (b) box is closed
Q 12 use the method of Lagrange’s multipliers to find the extreme value of f(x,y,z) =
2x+3y+z subject to
+
= 5 and x+z = 1
UNIT - III
0
1 Find by elementary transformation row transformation the inverse of the matrix 1
3
2
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1
2
1
2
3
1
2. Use E-Transformation to find the triangular form and hence find rank of matrix A
−7
8
−4
=
2
3
−3
−2
2
0
3. Use the elementary transformation to reduce the matrix A into Normal Form and hence
find the rank of A
1
1

2

1
2 3 1
3 3 2
4 3 3

1 1 1
4. Find the non singular matrices P and Q so that PAQ is a normal form, where A =
 2 1 − 3 − 6
3 − 3 1
2 

1 1
1
2 
5: Test the consistency of the following system of linear equation and hence find the solution
4 −
= 12, − + 5 − 2 = 0, −2 + 4 = −8
6: Determine for what values of & the following equations
x+2y+3z=10, x+2y+ z=
+
+ = 6,
have (i) no solution (ii) unique solution (iii) infinite solution.
7. Determine b such that the system of homogeneous equation 2x + y + 2z = 0, x + y +3z = 0,
4x + 3y + bz = 0 has (i) trivial sol (ii) non trivial sol, find the non trivial sol using matrix
method.
1
8: Show that row vectors of the matrix −1
0
2
3
−2
9: Verify Cayley hamiltion theorem for the matrix
−2
0 are linearly independent
1
=
1
2
2
and hence find
−1
10: Find all the eigen values and eigen vectors of the matrices
(i)
=
−2
2
−1
2 −3
1 −6
−2 0
11: A square matrix A is defend by
resulting Diagonal matrix D of A.
(ii)
=
8
= −6
2
−1
1
−1
2
2
−1
−6 2
7 −4
−4 3
−2
1 . Find the modal matrix P and the
0
3
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12: Express the matrix
1+
2
−1 +
=
2
2+
−4
skew Hermitian matrix .
13: Show that the matrix
+
+
=
5−5
4+2
7
− +
−
as the some of Hermitian matrix and
is a unitary matrix if
+
+
+
=1
14: Let V be the set of all pairs (x,y) of real numbers and let F the fieald of real numbers
define as ( x, y,) + ((
15: Let V =
{( , , ):
+
,
)
(
+
) and C(x ,y) = ( Cx, y)
, +
Show that W is not a subspace of , where (i) W ={ ( , , ):
≥ 1}
+
16: Is the vectors ( 3, -1, 0,-1) in the subspace of
≥ 0} (ii) W=
Spanned by the vectors (2,-1,3,2) , (-
1,1,1,-3) & (1.1,9,-5)
17: Determine whether (1,1,1,1) , ( 1,2,3,2), ( 2,5,6,4) , (2,6,8,5) form a basses of
. If not find
the dimension of the subspaces they span.
18: Which of the following function T:
(1+
,
(ii) T( ,
19: Let T:
)
)=(
) (iii) T( ,
,
→
)
(
are linear Transformation (i) T(
→
,
,
)=
)
be the Linear Transformation defined by T(x,y,z,t) = ( x-y+z+t, 2x-
2y+3y+3z+4t, 3x_3y+4z+5t)
UNIT IV
1 x2
1. Evaluate
∫ ∫e
y
x
dydx
0 0
2. Evaluate ∬
quadrants.
3. Evaluate ∬
4cos
y x = 0 ,y =
ℎ
etween the circle r = 2cos and r =
ℎ
4. By changing the order of integration evaluate ∫ ∫
5. By changing the order of integration of ∫ ∫
∫
, x+y = 2 in first
=
6. By changing into the polar coordinates evaluate ∫ ∫
4
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(
)(
)
dxdy.
dxdy.show that
(
(
+
dxdy
7. Evaluate
∫∫ (x + y ) dxdy where R is a parallelogram in xy plane with vertices (1,0),
2
R
(3,1), (2, 2), (0, 1) using the transformation u = x +y and v = x -2y
8. Using the transformation x + y = u, y =uv show that
∫∫ [xy(1 − x − y )]
1/ 2
dxdy =
2π
,
105
integration being taken over the area of the triangle bounded by the lines x = 0, y =0
and x + y = 1.
9. By double integration find the whole area of the curve
10. Find the area of the curve
11. Evaluate the ∭(
+
+
.
=
+
+
cos2 .
=
13. Evaluate Γ (−1 / 2)
∞
∫e
−h 2 x 2
(2 − ).
over the first octant of the sphere
)
12. Find the volume cut from the paraboloid 4z =
14. Evaluate
=
+
by plane z = 4 .
dx
0
15. Prove that β (m +1, n ) =
(
∞
16. Prove that
17. Assuming
m
β (m, n )
m+n
)
x 4 1+ x 5
1
∫0 (1+ x )15 dx = 5005
! − 1 !=
<1.
,0 <
18. For a beta function show that ( , )= ( + 1 , ) ( , + 1).
19. Prove that ∬
=
≤ℎ
20. Evaluate the ∭( +
+ )
ℎ
where D is domain x≥ 0,
≥0
+
extending over all positive and zero values of
x,y ,z subject to the condition x+y+z < 1 .
21. Find the mass of the an octant of the ellipsoid ( ) + ( ) + ( ) = 1 the density at
any point being
=
.
UNIT V
Q 1 at any point of the curve x = 3cost y = 3sint ,z =4t find (i) tangent vector (ii) unit
tangent vector .
Q 2 Find the directional derivative of x
= at sint ,
+y
at the point (2,-1,1 ) along to the curve x
y = a cost , z = at at t = S
5
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Q 3 Find the directional derivative of ∅ =
−
direction of the line PQ where Q is the point (5,0,4).
at the point (1,2, 3)in the
+2
Q 4 Find the directional derivative of ∅( , , ) =
direction of 2 ̂ − ̂ − 2 . Find the greatest rate of increase of ∅
Q 5 Determine the constant a and b such that the curl of vector
(
+
Q 6 If
Q 7 If
Q 8 If
−4
=
(
̂
) ̂ − (3
̂
( , , )=
+
at the point (1,-2, 1)in the
+4
) is zero.
̅ = (2
+3
) ̂+
find the value of div .
show that
̂+
from the (1,2,3) to (3,5,7).
̂+
Q 9 Evaluate ∬ (
̂+
̂+
× ( ∇ × ̅ ) = ∇ ( a. ̅ ) – ( . ∇ ) ̅
is the force field . Find the work done by
).
along the line
where S is surface of the sphere .
Q 10 Using green’s theorem,find the area of the region in the first quadrant bounded by the
curves y =x y = , y = .
Q 11 Use the Stoke’s theorem to evaluate ∫ ( + 2 )
+( − )
+( − )
where C is
the boundary of triangle with vetices (2,0,0) ,(0,3,0),and (0,0,6) oriented in the anticlockwise
direction
Q 12 Use the Divergence Theorem to evaluate ∬ (
+
portion of the plane x+2y +3z =6 which lies in the first octant.
6
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+
) where S is the