Let's break this down step-by-step:* There are 2 boxes containing coins* Box I contains 5 gold coins * Box II contains 3 gold coins and 2 silver coins* The probability of choosing Box I is 1/2* The probability of choosing a gold coin from Box I is 5/5 = 100%* The probability of choosing Box II is 1/2 * The probability of choosing a gold coin from Box II is 3/5 = 60%* Using the formula for total probability:P(gold coin) = P(Box I) × P(gold coin from Box I) + P(Box II) × P(gold coin from Box II

~~,

65/1

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into three

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each.

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'

~3:r

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SECTION
:

A

";4;!'ii1Y~&

c/fi~1~f(:!

?

Evaluate:

cos 75°

:'"

,.

-2

.

-2

If a matrix has 5 elements, write all possible orders it can have.

P.T.G.

~

~

-.c~'-

1:fT";f~

.

dx

b)3

~
f

:

(ax

+ b)3

dx

Evaluate:

In

7.

+

(ax

Evaluate:

f

6.

~

:

In

1:fT";f
~

Write

of the

(1, 0, 0) "(f~ (0, 1, 1) ~

Write

the

points

(1,

~

I

vector

1

- j

i

1

on the vector

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7

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~~'1:&r'qjT~

2

z-6
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. y+4
=

x-5

-

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=

.

-

1.

y

0

equatIon
~

=

~

=

~
I

65/1

7

1

i + j.

~~~I

+j

f

.
vector

1

3

3

0, 0) and

1

i

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'qjT~

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10

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me

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the

a

8.

2

'0"

~~,!",",'

B
~~

a+b,

{

each.

function

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Find

7.

+
JR'

if

:

as

defined

is

5}

4,

3,

2,

1,

{O,

set

the

on

*

operation

binary

A

OR

=

I

lOx

ii'

=

3iq;

fog

=

gof

that

marks

4

i?Ii

f(x)

JfR

as

Jffflq;

4

carry

22

11

'(fq:;

R

~

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such

R

R

f:

~

R

Let

:

g

it

11

defined

mr

JfR

11.

numbers

be

Question

to

I

i

I

SECTION

a+b<6

6

and

element

'a'

~

~

'a'.

I

t

~

of

JR'

=

each

inverse

W~~

the

~

~

b

operation

being

:

t

~

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~

6

*
<

b

+

~
a

~~

~

~

~
b,

5}
a

{

+

4,

3,

2,

1,

{O,

~~

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=

gof

this

a,

7

-

+

6

lOx

=
~

+

a

for

with

R,

~

if

identity

"[T'{T

the

invertible

~

R
R

~

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g

is

is

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set

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the

f(x)

that

of

~

Show

-

a + b

6,

a*b=

12.

Prove

that:

[

,Ji-,;-x

-

.J:f+x

-1

+.J~

.Jl-=-x

]

x

1

7t

=

4

-

"2

-1

cos

'a'

~

~

-~:S:
x1
:S:
1

'

x,

~~~:

tan

q;r

~

6

~

~

b

+

"(fm

a

I

t

t

~

a

-

6

d~~~

6,
~

q;r

-

b
~

+
~

~

~

t

cYj~~un(.j

~

~

~

a

a:*b=

65/1

!

~

c'

5

1

:S:

x

~

-

x,

"2

1

-1

:S:

1

7t

cos

]

-

.Ji-=x

4

-

.JI-,;-x +,Ji -=-x

.JI-,;-x

=

[

-1

tan

P.T.G.

- 2

2x - 3
2x - 9
2x - 27

:

~

~

3x - 16 = 0

x- 8

~

3x - 4

x- 4

~

x

~

RJor1I(.1I~(1

~

m

qjT

""J]um

~

~1fTJl~1

3x - 64

x

14. Find the relationship

between 'a' and 'b' so that the function 'f' defined

by:
{ ax + 1,

if

x:S 3

f(x) =

is continuous at x = 3.
if

x > 3

~,

"qft~
3

>
I

x

~

~

~

+
~

3,

~

3

x:S

~

1,

ax
qjT

~

~

'b'

~

'a'

m

bx

~,

{log (x e)}

=

f(x)

~

'f',

~

~

{

+

dx

.

2

log x

-

-

a

ow

s

Y

,

-

x

dy -

t

th

h

e

-

x

Y

If

-

OR

bx + 3,

cosO)

7t]
2

+ cose)

0.03~~F~,
I

~

~

F

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~

~~~~~9~~~~,
~

65/1

6

m~~

3"{?AaJ

cm,
I

~

~

~

~

.q:

its surface area.

2.

~

0.03

of

error

an

with

cm

9

]

~

[

0,

e,

-

e

error in calculating

sin

4

-

=

y

fCfi

~

(2

as

measured

is

sphere

a

of

radius

the

If
~

then find the approximate

.

[0

.

-

t"

,

f

m

.

Ion

.

.

OR

+

unc

- e .

4 sin e
(2

(xe)}

mcreasmg

=

an

Prove t h at y

IS

.

{log

.

2

dx

15

X

log

=

~

~

I~

~...~~..'
G~II~~

~
(11

~
~,

y

-

x
e

-

Y
X

'-11~

-n&

3"{?AaJ

-{

(1 + x2) d2

show

that

d
a) -L
dx

-

+ (2x

~

m

~,

d2y
-

x2)

(1 +

y)

log

(;

tan

=

-

+ (2x

a)

dx2
17. Evaluate:
n/2

f

0

dx

:

~

f

n/2

x + sin x dx
1+cosx
C

0

x dy

- y dx = .J;i-~7dx

Solve the following
dx
(y + 3x 2) -

""

,.

:

~

~

-qj)

~1.i1~,UI

,

~~;

"'-': "."
'.

dx

equation:

R-~7

=

differe~tial

dx

~

RI'--1I(1~d

19.

y

following .
-

the
dy

Solve

x

18.

i

~

dy
- =

x + sin x dx
1 + cosx

0
1=fr;:r

0

==

,

x

~

dx

fCfi

(; log y ),

= tan

16. If x

differential

-

equation:

=x

'

.

:

~

~1.i1~,UI

~

RI'--1I(1I(9d ~

-qj)

dy

=x

(y + 3x 2dx )

dy
20. Using vectors, find the area of the triangle with vertices A(1, 1, 2),

5)

3,

B(2,

2),

1,

A(l,

m

~

7

---~

P.T.D.

~

-

65/1

~-

~

~

~

-q)1

~

~
I

'q}""{,
~

5)

m

5,

-q)1
C(l,

~

~

B(2, 3, 5) and C(l, 5, 5).

J"'--

b
~n'l

~"l

--~

:

~11~~

dx
~

~J"t;;

J

n/3

:

--=~~+

the

and

2k)

+

4]

+

(3i

A

+

~

1
2k)

+

':'
4J

':'

+

(31

A

+

1
2k)

+

':'

In

coins.

I

two

~

~
containing

~
each

~

~

III

and

~~

II

~

I,

5

=

and takes

out a coin. If the coin is of gold, what

box

a

chooses

person

A

coin.

silver

one

and

gold

one

is

there

III,

box

box I, both coins are gold coins, in box II, both are silver coins and in
at random

is the

10

~

~
~

~

~

"G1";1)
~
~

~

~

-q:;r

I~,
m
~
fu"iiiit

fiij

~

~

~

I

~
t,

t

III~,

~
}fll(.f~r:ff

~

~

~
~

~

'i;fTm-q:;rt?

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~

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t,

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m

~

~I<!~~J

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~

~

~

~

~
~

III

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~

~

~

I

"G1";1)

II

I,
~

t

n~,

~
~

I

t

~

~,

~

-q:;r

~
R~I(.'1ctl

~

m

~

probability that the other coin in the box is also of gold?

65/1

~

6

O.

-

=

=

x

~

~

x-~

~

I

3

+

x

I

=

-

(21

=

r

~

~

1

~
k)

boxes

+

1

10)

-

j

1

5,

identical

(i

.

1

-

1,

(-

three

r

~

~
~
Given

28.

to x

= 5.

j + k)
1

Y

= (2i - ]

J"

-

1

~

. (i

=- 6

x

(- 1, - 5, - 10), from the point of

point

2k)

of the

+

distance

r

~

'(fP-1T
~

~

above x-axis and between

intersection of the line "1
plane

+ 31 and evaluate the area under the

':'

the

= Ix

y

~

q:jf

I

3

+

Find

-q:;r

x

I

'(f'qj

0

Y

=
=

x

27.

31

x +

I

of

ml:fi

=

curve y

dx

--

)(:;":

Sketch the graph

~

26.

7

-5

.J-c:x

~

J

I

~

~

1:Jr;r

3:r~

nl6

",

~

rr

Determine

desktQP

40,000

a
computers

Rs.

-

the number

of

demand

and

computers

25,000
monthly

Rs.

personal

cost

of
total

will

types

will not exceed 250 units.

the

that

that

two

sell
estimates

model

to

plans

portable

a

He

and

merchant
respectively.

model

A

-

of units

of each type

of computers which the merchant should stock to get maximum profit
if he does not want to invest more than Rs. 70 lakhs and his profit on
the desktop model is Rs. 4,500 and on the portable

model is Rs. 5,000.

'

~

~

,

~

I

~

"(f4T

.

~

~

~

4,500~.

~W?J1R
~

~

m

~

~
~

~

m1ft

-;rf:r

'"!iA1m

~

~

~

~

~~
~

~m~

~

~

~

~
o;f1i't

~
0

5

2

~

~,

mG-'fq

~

1:{lfuCfi

~

m

~
70
5,000~.~,

40,000~.

-q;~

~

~
-q;~
31f~

fC:ti

25,000~.

"5fqjT{

"GT

~

~~

~. ~ "(f4T i-~ifq

~

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~~

"(f4T

~ml:fi~~~1

~

r

~~

~

~

Wm'{T

~

W?J1R

~
~

~

~

~

~

Make an L.P.P. and solve it graphically.

65/1

.

rll.li..iiiiii...iii_..ii~.;;;;;;;;;;;;;

11

~~~~~

'.

t,-~

,

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