C++ FILE FOR BOARDS CLASS 12

Mock Phase Test

ADVANCED PATTERN

SECTION A (Single Option Correct)

1.

Domain of definition of f (x) =

cos ec−1(2x) + π

is

3

⎛

−1

⎤

⎛

−∞,

1

⎤

⎛

−1 ⎤

⎡ 1

⎞

(A) ⎜

−∞,

⎥

(B) R

(C) ⎜

⎥

(D) ⎜

−∞,

⎥

∪ ⎢

,∞⎟

2

⎝

3

⎦

⎝

⎦

⎝

3 ⎦

⎣2

⎠

2.

Suppose F(x) = sin (3x) + 5,

⎡

π

,

π ⎤

. If g(x) is the function whose graph is reflection of graph

x ∈ ⎢

⎥

⎣

6

2 ⎦

of F(x) with respect to line y = x, then g(x) equals

1

−

⎡

π

π ⎤

1

−

x ∈[4,6]

(A) g(x) =

sin 1(x − 5),

x ∈ ⎢

,

⎥

(B) g(x) =

sin 1(x − 5),

3

⎣

6

2 ⎦

3

	1		−		−		⎡	π		π ⎤
(C) g(x) =			sin 1(x) − 5, x ∈[4,6]	(D)	g(x) = sin 1(3x)	− 5,	x ∈ ⎢		,	⎥
	3						⎣	6		2 ⎦

3.If f(x) = {x}2 + {x} + 1 where {.} represent fractional part of x and g(x) = sin−1x.

Then range of

⎛

1

⎞

is

g⎜

⎟

⎝ f(x) ⎠

⎡

−

1

1

π ⎤

⎛

−

1

1

π ⎞

⎛

−

1

1

π ⎤

⎡

−

π

π ⎤

(A) ⎢sin

,

⎥

(B) ⎜ sin

,

⎟

(C) ⎜ sin

,

⎥

(D)

⎢

,

⎥

⎣

3

2 ⎦

⎝

3

2 ⎠

⎝

3

2 ⎦

⎣

2

2 ⎦

4.F(x) = (x − 1) | x2 − 3x + 2 | +(x + 1) | x2 + 3x + 2 | +(x + 3) | x2 − x −12 | is non-differentiable at

(A) x = 1, −1, −3 (B) 2, −2, 4 (C) x = 1, −1, −3, −2, 2, 4 (D) always differentiable

⎛

1

+ cos

1

⎞x

5.

lim ⎜ sin

⎟ is equal to

x

x

x→∞ ⎝

⎠

(A) 1

(B) 0

(C) e

(D)

1

e

sinax − n(ex cos x)

6.

If lim

=

1

(given) then ‘a’ and ‘b’ are respectively equal to

x sinbx

2

x→0

(A) 1, 1

(B) 1, 2

(C) 2, 1

(D) 2, 2

7.Let g(x) = ∫−x10 tf′(t)dt for x ≥ −10 where f is an increasing function then

(A)g(x) is an increasing function of x

(B)g(x) is an decreasing function of x

(C)g(x) is an increasing for x > 0 and decreasing for −10 < x < 0

(D)none of these

8.

Let f(x) = |sin3x| and g(x) = sin3x, both being defined for x in the interval

⎛

−

π

,

π ⎞

⎜

⎟

⎝

2

2 ⎠

′

′

(B)

′

′

′

′

(D)

′

′

(A) f (x) = g (x)

f (x) = −g (x)

(C) f (x) =| g (x) |

g (x) =| f (x) |

9.

lim

1

+

2

+ ....

n − 1

is equal to

n→∞

n

n

(A)

1

(B)

1

(C)

2

2

3

3

10.

If f(x) =

⎛ a + x ⎞a+b+2x

then f′(0) equals

⎜

⎟

⎝ b + x ⎠

b2 − a2

⎛ a ⎞a+b−1

⎛

a

+

b2

− a2

⎞ ⎛ a ⎞a+b

(A)

⎜

⎟

(B) ⎜

2log

⎟ ⎜

⎟

b2

b

ab

⎝ b ⎠

⎝

⎠ ⎝ b ⎠

(D) 0

(C) 2log	⎛ a ⎞			+	b2	− a2
	⎜		⎟
	⎝ b ⎠					ab

(D)2 log ⎛ a ⎞

⎜⎟

⎝ b ⎠

SECTION B (Multiple Option Correct)

11.Which of the following is /are true

(A) lim tan x = ∞

(B)

lim

tan x = ∞

(C) lim tan x = ∞ (D)

lim tan x does not exist

x→π / 2−

x→π / 2+

x→π / 2

x→π / 2

⎧| x − 3 |

x ≥ 1

12.

The function f(x) =

⎪

is

⎨

2

3x

13

⎪

x

−

+

x < 1

2

⎩ 4

4

(A) continuous at x = 1

(B) differentiable at x = 1

(C) continuous at x = 3

(D) differentiable at x = 3

13.Let h(x) = min{x, x2}, for every and real number of x. Then

(A)h is continuous

(B)h is differentiable for all x

(C)h’(x) = 1 for all x > 1

(D)h is not differentiable at two values of x.

14.If y = |x – 2| - | x + 1|, then

(A) for x < -2 , y = 3	(B) for x > 3 , y = 3
(C) for 0 ≤ x ≤ 1 , y = -2x + 1	(D) for 1 ≤ x ≤ 2 , y = -2x + 1

15.Let f be a function defined from set X to X such that f(f(x)) = x for all x∈ X, then

(A) f is one to one but need not be onto

(B) f is onto but need not be one to one

(C) f is both one to one and onto

(D) nothing can be said

SECTION C

16.

Which of the following represent the graph of function f(x) =

| 1− x | −1| −1

(A)

(B)

x

x

(C)

(D)

x

x

.

17.If g(x) = min [{x}, {−x}], where {.} represent fractional part of x, and graph of f(x) is given, then the range of function f(x) + g(x) is

f(x)

1

0

1/2

1

2

3

⎡

3 ⎤

⎡ 1

⎤

⎡

3 ⎤

(A)

⎢0,

⎥

(B)

⎢

,1⎥

(C) ⎢1,

⎥

(D) [0, 2]

⎣

2⎦

⎣ 2

⎦

⎣

2 ⎦

18.Number of solutions of x2 + sin2 x = 1 is

	(A) 4	(B) 0	(C) 2	(D) 3
19.	If the graph of function y = f(x) is given then f(x) may be define as			f(x)

(A)	⎧ | x |			| x |≥ 1
	⎨
	⎩ n | x | 0 <| x |< 1
	⎧| x |			| x |≥ 1
(C) ⎨⎪ 1				0 <| x |< 1
	⎪
	⎩| x |

20. If F(x) has the graph given as follow Which of follow is correct?

(B) ⎧⎨| x | ⎩ x2

⎧ 1

⎪

(D) ⎪⎨| x | ⎪ 2⎪⎩ x2

| x |≥ 1

0 <| x |< 1

| x |≥ 1

0 <| x |< 1

1

−1 1 x

(A)Local maximam at x = x1, x5Local minimum at x = x2, x3, x4

(B)Local max at x = x2, x3 Local min at x = x4

(C)Local max at x = x1, x2, x3 Local min at x = x2 x4, x5

(D)Local max at x = x1, x3 Local min at x = x4

x1 x2 x3 x4 x5

SECTION D

Pessage 1:

A function is called one-one if each element of domain has a distinct image in co-domain or for any two or more than two elements of domain, function doesn’t have same value. Otherwise function will be many-one. Function is called onto if co-domain = Range otherwise into. Function which is both one-one and onto, is called bijective. Inverse is defined only for bijective function.

21.Which of the following function is one-one for ∀ x∈R.

(A) f(x) = x2 + x

(B) f(x) = x |x|

(C) f(x) = sin

πx

2

22.

Let f: R → Y. f(x) =

x2

, then set

Y for which f(x) is onto

x2 + 2

⎛

1

1 ⎞

⎡1

⎞

(A) [0, 1)

(B)

⎜

,

⎟

(C) ⎢

,1⎟

(D)

⎝

3

2 ⎠

⎣ 3

⎠

(D) f(x) =[2x]

⎡ 1 ⎞ ⎢ ,1⎟⎣ 2 ⎠

23. Let f: X → Y if f(x) = 2x2 −1 is bijective then possible set of X and Y are

.

	⎛ 1		⎞	(B) X = (0,∞), Y = (0, ∞)
	(A) X = (0,∞), Y = ⎜		,∞ ⎟
		2
	⎝		⎠

⎛

1

⎞

(C) X = (−∞,0) Y = ⎜

−∞, −

⎟

(D) X = (−∞,0) Y = (0,∞)

2

⎝

⎠

24.

Let f: (−∞,1] → (−∞,1] such that f(x) = 2x − x2 then f−1(x) is

(A) −1 −

(B) −1 +

(C) 1 +

(D) 1 −

1 − x

1 − x

1 − x

1 − x

25.

If f: [0, ∞) →[0, ∞) and f(x) =

x

, then f(x) is

1 + x

(A) one-one & onto

(B) one-one & into

(C) many one & onto

(D) many one & into

Pessage 2:

Let f(x) is a cube polynomial which has local maximum at x = −1, if f(2) = 18, f(1) = −1 and f′(x) has local minima at x = 0, then

26.The cube polynomial f(x) is

	(A)	1		(x3	+ 45x − 54)	(B) (x3 − x − 1)
		8
	(C) x3 + x2 + 9x − 12					(D)	1	(19x3	− 57x + 34)
							4

27.f(x) is increasing for

⎡	1	⎞	⎡			⎤	(C) x ∈ R	(D) x ∈[1,∞)
(A) x ∈ ⎢		,∞⎟	(B) x ∈ ⎣−1,2	5		⎦
⎣	3	⎠

28.f(x) has local minimum at

(A) x = 0

(B) x = 1

(C) x = 2

(D) x = −

2

Answers:
1.	D	2.	B
3.	C	4.	B
5.	C	6.	A
7.	C	8.	C
9.	C	10.	B
11.	A, D	12.	A, B, C
13.	A, D	14.	A, C, D
15.	C	16	A
17	A	18.	C
19.	C	20.	D
21.	B	22.	C
23.	A	24.	D
25.	B	26.	D
27.	D	28.	B

.