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TEST PAPER-05

CLASS - XI MATHEMATICS (Limits and Derivative)

[ANSWERS]

Ans1 2

2 0

1lim 2

2

h

h

e

h→

− ×

1 2 2= × =

Ans2 ( )( )

6

20

1 1lim

1 1x

x

x→

+ −

+ −

6 6

6

2 221 1

5

1

0, 1

11 1

lim lim111

6 1 63

2 1 2

y y

let x y

x y

yy y

yyy

→ →

+ =→ →

−− −=

−−−

×= = =×

Ans3 7 7

lim 7x a

x a

x a→

+ =+

7 7

7

6

6

7

27

2

7

7

a a

a a

a

a

a

a

+= =+

= =

= =

= =

Ans4 ( )3 5 3d

x xdx

− +

3 2

4 3

4 3

5 3

5 3 3 2

15 6

dx x

dx

x x

x x

− −

− −

= +

= × − + × −−= −

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Ans5 ( )2

2

1 cos 2lim

2x

x

xπ π→

+−

( )2 20 0

2

20

2

20

2

02

2

2

2

, 02

1 cos 1 coslim lim

2sin2lim

44

sin1 2lim2

2

sin1 2lim2

21 1

12 2

y y

y

y

y

let x y

x y

x y

y y

y y

y

y

y

y

y

y

πππ

π→ →

→

→

→

− == −

→ →

+ − −=

=×

= ×

=

= × =

Ans6 ( )( )

( )2 3 1

2 5

x xy

x

+ −=

+

( ) ( )( )

( ) ( )( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]

( )( ) ( ) ( )

( )( )[ ]

( )

2

2

2

2

2

2

2 3 1

2 5

2 5 2 3 1 2 3 1 2 5

2 5

2 5 2 3 1 3 1 2 2 3 1 2 0

2 5

2 5 2 3 3 1 1 2 3 6 2

2 5

2 5 3 6 3 1 6 12 2 4

2 5

x xdy d

dx dx x

d dx x x x x x

dx dxx

d dx x x x x x x

dx dx

x

x x x x x x

x

x x x x x x

x

+ −=

+

+ + − − + − +=

+

+ + − + − + − + − + =+

+ + × + − × − + − − =+

+ + + − − − + +=

+

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212 30 10x x x+ +=225 6 10x x+ − −

( )

( )

2

2

2

4

2 5

6 30 29

2 5

x

x x

x

++

+ +=+

Ans7 ( )5

. . . limx

L H S f x−→

( )

( )

( )

( ) ( )( )( )

0

0

5

0 0

5 5

5

5

5

5, 0

lim 5

lim | 5 | 5

0

. . . lim

5

5, 0

lim 5 lim | 5 | 5

0

. . . . . .

lim lim

lim e

lim 0

h

h

x

h h

x x

x

x

x h

x h

f h

h

R H S f x

put x h

x h

f h h

R H S R H S

f x f x

f x xist

f x

+

− +

→

→

→

→ →

→ →

→

→

= −→ →

−

− −

=

= +→ →

+ = + −

==

∴ =

∴

∴ =

Ans8 ( ) ( )2 3, 0

3 1 , 0

x xgiven f x

x x

+ ≤ = + >

( )( ) ( )

( )( )

( ) ( )

00

00

0

0

11

0

. . . lim lim 2 3 3

. . . lim lim3 1 3

. . . . . .

lim

lim 3

1

. . .

lim lim3 1

xx

xx

x

x

xx

for x

L H S f x x

R H S f x x

L H S R H S

f x exist

f x

for x

L H S

f x x

−

+

−

→→

→→

→

→

→→

== + =

= + =

=∴

∴ =

=

∴ = +

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( )

( ) ( )( )

( )( )

11

1

1

3 1 1 6

. . .

lim lim3 1

3 1 1 6

. . . . . .

lim

lim 6

xx

x

x

R H S

f x x

L H S R H S

f x exist

f x

+ →→

→

→

= + =

= +

= + ==

∴

=

Ans9 ( ) seclet f x x=

( ) ( )

( ) ( ) ( )

( )

( )

( )( )

( )

( )

1

0

0

0

0

0

0

sec

lim

sec seclim

1 1cos cos

lim

cos coslim

cos cos

22sin sin

2 2lim

cos cos .

22sin sin

2 2lim sin

cos cos

h

h

h

h

h

h

f x h x h

f x h f xf x

hx h x

h

s h x

hx x h

x h xh

x h h

x h x h

x h h

x h xh

→

→

→

→

→

→

+ = +

+ −=

+ −=

−+

=

− +=

+

+ − − =+

+ = −

+( )

( )

( )

( )

0

00

sin

22sin limsin

2 2limlim cos cos2

22 0

sinsin2

1cos 0 cos cos cos

tan sec

h

hh

x hh

h x h x

xx

x x x x

x x

θ θ

→

→→

= −

+

= ×+

+ = × =

+=

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Ans10 ( ) 4 5sin

3 7 cos

x xf x

x x

+=+

( )( ) ( ) ( ) ( )

( )( )( ) ( )( )

( )

12

2

3 7 cos 4 5sin 4 5sin 3 7 cos

3 7cos

3 7cos 4 5cos 4 5sin 3 7sin

3 7 cos

12

d dx x x x x x x x

dx dxf xx x

x x x x x x

x x

x

+ + − + × +=

+

+ + − + −=

+

=215 cos 28cos 35cos 12x x x x x+ + + −

( )

( )

( )

2

2

2 2

2

2

28sin 15sin 35sin

3 7 cos

15 cos 35 sin cos 28 cos 43sin

3 7 cos

15 cos 35 28cos 45sin

3 7cos

x x x

x x

x x x x x x

x x

x x x x

x x

+ + ++

+ + + + + =+

+ + +=+

Ans11 ( ) ( )2 2

0

sin sinlimh

a h a x a a

h→

+ + −

( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( )

2 2 2

0

2 2 2

0

2 2

0

2

0 0 0

2

2 sin sinlim

sin 2 sin sin sinlim

sin sin 2 sin sinlim

22cos sin

2 2lim lim 2 sin lim sin

22

2 0cos 1 2 s

2

h

h

h

h h h

a ah h a h a a

h

a a h ah a h h a h a a

h

a a h a ah a h h a h

ha h h

aa a h h a h

h

aa a

→

→

→

→ → →

+ + + −=

+ + + + + −=

+ − + + + + =

+ = + + + +

+ = × + [ ]

2

in 0 0 sin

cos 2 sin

a a

a a a a

+ + ×

= +

Ans12 (i) 4 2

cosd a b

xdx x x − +

( ) ( )

4 2

5 3

cos

4 2 sin

d d dax bx x

dx dx dx

a x b x x

− −

− −

= − +

= − − − −

5 3

4 2sin

a bx

x x

−= + −

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(ii) ( ) ( )cos tand

x x x xdx

+ −

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )2

2 2

2

cos tan tan cos

cos 1 sec tan 1 sin

sec cos cos sec sin tan sin

2 sec cos

d dx x x x x x x x

dx dx

x x x x x x

x x x x x x x x x tacx x x

x x x x

= + − + − +

= + − + − −

= − + − + − − +

= − + cos x−2

sin tan tan sin

2 sec sin tan tan sin

x x x x x

x x x x x x x x

− − += − − − +