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Evaluate
i) (1 | x 3|) dx
4
0
2
∫ + −
ii) ∫
−
+
3
3
[f(x) g(x)]dx , where f and g are odd functions.
iii) cos x dx ∫
iv) ∫sin θsin 2θsin 3θdθ
v) ∫ +
−
dx
1 x
x tan (x )
6
2 1 3
i) (1 | x 3|) dx
4
0
2
∫ + −
ii) ∫
−
+
3
3
[f(x) g(x)]dx , where f and g are odd functions.
iii) cos x dx ∫
iv) ∫sin θsin 2θsin 3θdθ
v) ∫ +
−
dx
1 x
x tan (x )
6
2 1 3
6. a) Find dx
dy , when
i) t x 3cost 2cos t, y 3sin t 2sin 3 3 = − = −
ii) [ \{1} 2 ( ) (sin ) , ] 0, 1 sin π = + ∈ − y nx x x x x l . (3 × 2 = 6)
b) If
⎪⎩
⎪
⎨
⎧
=
≠ = −
0 ; x 0
; x 0
x
1
x tan f(x) 1
, show that f is continuous but not differentiable at
x = 0 .
dy , when
i) t x 3cost 2cos t, y 3sin t 2sin 3 3 = − = −
ii) [ \{1} 2 ( ) (sin ) , ] 0, 1 sin π = + ∈ − y nx x x x x l . (3 × 2 = 6)
b) If
⎪⎩
⎪
⎨
⎧
=
≠ = −
0 ; x 0
; x 0
x
1
x tan f(x) 1
, show that f is continuous but not differentiable at
x = 0 .
5. a) An efficiency study of the workers at a factory shows that an average worker who
comes to work at 8:00 am will have produced t Q(t) t 9t 12 3 2 = − + + units t hours
later. At what time during the morning is the worker performing most efficiently. (5)
b) If x y tan−1 = , obtain an equation showing the relationship between n 2 n 1 y , y + + and
n y .
comes to work at 8:00 am will have produced t Q(t) t 9t 12 3 2 = − + + units t hours
later. At what time during the morning is the worker performing most efficiently. (5)
b) If x y tan−1 = , obtain an equation showing the relationship between n 2 n 1 y , y + + and
n y .
4. Sketch the graph of the function f defined by 4 3 f(x) = x + 8x , clearly giving all the
properties used in it.
properties used in it.
3. a) Obtain the largest possible domain and range of the function f , defined by
x 2
x 1 f(x)
+
+ = . Further, check whether or not limf(x) x→a exists for a = −1, 2 . (5)
b) Find dx
df , where ⎥
⎦
⎤
x 2
x 1 f(x)
+
+ = . Further, check whether or not limf(x) x→a exists for a = −1, 2 . (5)
b) Find dx
df , where ⎥
⎦
⎤
1. Which of the following statements are true or false? Give reasons for your answers. (2 × 5 =10)
i) The function f : R → R , given by f(x) n | x 1 x | 2 = l + + is neither even nor odd.
ii) ∫ = −
0
x
2 2 sin(t )dt sin x
dx
d .
iii) The area enclosed by the x-axis and the curve y = cosx over the interval ⎥
⎦
⎤ ⎢
⎣
⎡ π π − 2
3
,
2
is 0 .
iv) If f and g are functions over R such that f + g is continuous, then f must be
continuous.
v) x − y + 2 = 0 is a tangent to the curve 3 2 (x + y) = (x − y + 2) at (−1, 1).
i) The function f : R → R , given by f(x) n | x 1 x | 2 = l + + is neither even nor odd.
ii) ∫ = −
0
x
2 2 sin(t )dt sin x
dx
d .
iii) The area enclosed by the x-axis and the curve y = cosx over the interval ⎥
⎦
⎤ ⎢
⎣
⎡ π π − 2
3
,
2
is 0 .
iv) If f and g are functions over R such that f + g is continuous, then f must be
continuous.
v) x − y + 2 = 0 is a tangent to the curve 3 2 (x + y) = (x − y + 2) at (−1, 1).
A manager wants to appoint 4 sales-persons to 4 different cities. If the expected profit when different persons are appointed to different cities is as given in the following table, find the assignment that will maximize the profit.
Sales- persons
I II III IV
Cities A 9 7 8 8
B 7 3 5 14
C 5 4 12 10
D 8 3 4 13Q
Sales- persons
I II III IV
Cities A 9 7 8 8
B 7 3 5 14
C 5 4 12 10
D 8 3 4 13Q
Consider a small post office with a single staff member operating a single postal
counter. Suppose that the probability pk, that there are k customers in the post
office, is given by pk = p0p^k k = 0,1,2,... where 0 < p < 1.
(a) Show that p0 = 1 - p.
(b) Determine the probability that a newly arriving customer has to wait to be
served.
counter. Suppose that the probability pk, that there are k customers in the post
office, is given by pk = p0p^k k = 0,1,2,... where 0 < p < 1.
(a) Show that p0 = 1 - p.
(b) Determine the probability that a newly arriving customer has to wait to be
served.
Verify that the Pfaffian differential equation yz dx + (x^2y- zx) dy + (x^2z-xy) dz =0 is integrable and hence find its integral.
The differential equation of a damped vibrating system under the action of an external periodic force is: d^2x/dt^2 +2 m0 dx/dt +n^2x = a cospt Show that, if n>m0>0 the complementary function of the differential equation represents vibrations which are soon damped out. Find the particular integral in terms of periodic functions.
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#340153 on Dec 2023
#340153 on Dec 2023