®
HIGHER MATH
HIGHER MATH
CLASS - 3
CALCULUS AND VECTORS
DIRECTIONS for questions 1 to 3: Choose the correct alternative. 1.
f(x) = x + 5 g(x) = 2x + 4 Find f[g(x)] – g[f(x)] 1] 4x – 5
2.
3] –5
f(x) = ax 2 + bx + 2 and f(1) = 3. f(3) = 1] –3, –2
3.
2] 4x + 23
2] -2, 3
4] Cannot be determined
–7. Find the value of a and b 3] -3, 2
4] 2, 3
Find x if f(x) = 1 + x – x 2 and f(x + 1) = f(x + 2) 1] 0
2] 1
3] – 1
4] 2
DIRECTIONS for questions 4 and 5: Refer to the function defined below and answer the questions that follow. f(x) = (x – 1) (x – 2) (x + 3), = (5 – x)
4.
otherwise
g(x)= x 3 – 1, x > 0 1 = when x < 0 x Find the values of x for which f(x) = 0 1] 3, 5
5.
–2 < x < 4
2] 1, 2, –3, 5
3] –1, –2, 3, –5
4] 1, 2, 5
Which of the following is/are true? (i) If x = 1, f(x) = g(x) (ii) f(6) = g(–1) f(0)
(iii) g(0) = –6 1] (i) and (iii)
2] (ii) and (iii)
3] (i), (ii) and (iii) 4] (i) and (ii)
1
IMS-34-UG-AL-HM-Class-3
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HIGHER MATH
DIRECTIONS for questions 6 to 10: Refer to the data below and answer the questions that follows. A function, f(x), is defined as follows: f(x) = x + x 2 + x 3 if x > 0 = g(x) g(x) 6.
= 2x + 30
Find f(2) + f(–2) + f (– 1] 69
7.
if x < 0 1 2
)
2] 71
3] 0
Which of the following is true? 1] f(1) = g(–15) 3] f(0) = g(1)
8.
3] g(2) < f(2)
2] f(2) & g(2)
3] f(3) & g(3)
2] 228
4]
150 7
f(x) = x 2 + 1, g(x) = x 2 – 1, h(x) = x. Which of the following is true?
3] h(x) =
2] g(x) =
(f(x) g(x))2 2
4] h(x) =
h(x) f(x) x2
f(x) g(x) 2
What is the relationship between f(x) and g(x)?
1] g(x) = f(x) 3] g(x) = –f(x) 13.
4] f(4) & g(4)
3] 230
1] g(x) = f(x) × (h(x)) 2
12.
4] g(2) = f(2)
f(x, y) = 4x2 – 3xy + 3y2 + 6 and g(x) = x 2 – 3x + 20. Find g(f(1,–1)). 1] 16
11.
2] g(2) > f(2)
Which of the following given pairs is the closest in terms of value? 1] f(1) & g(1)
10.
2] f(–2305) = g(-2305) 4] None of these
Which of the following is true? 1] g(1) = f(1)
9.
4] –60
2] g(x) = f(–x) 4] g(x) = –f(–x)
If f(x) = ax 2 – bx + 7 and f(2) = 5 and f(4) = 11 find the value of a + b. 1] 1
IMS-34-UG-AL-HM-Class-3
2] 3
3] 2
2
4] 4
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HIGHER MATH
14.
Which of the following is true for the given graph? 1] f(x) =
f(x) 2
2] f(x) = 3f(–x)
x 3] f(x) = f 2
15.
Find f –1(5) for f(x) = x 3 – 3. 1]
16.
1 5
2] 2
3] f(x) =
px px
4] None of these
3
2] f(x) = (x x2 1)
(x2 4)(x 1) x
4] f(x) = sin 2x + cos 2x
If f(x) = –3x 3 +
1] Only I
19.
3
px px
I. f(x) = f(–x)
18.
3]
Which of the following is an odd function? 1] f(x) =
17.
4] None of these
What is
ò tan
5 x2
+ 5x 2 –
3 x3
1 II. f = f(x) x
2] Only II 4
which of the following is/are true?
3] Both I and II
4] Neither I nor II
x dx = ?
1]
1 tan 3 x - tan x + x 3
2]
1 tan 3 x + tan x - x 3
3]
1 tan 3 x - tan x - x 3
4]
1 tan 3 x + tan x + x 3
ò 1]
cot x sin x 2
dx = ?
sin x
2]
-2 sin x
3] –2 sin x
3
4] None of these
IMS-34-UG-AL-HM-Class-3
®
HIGHER MATH
20.
1
ò sin x + cos x 1] 3]
21.
dx = ?
pö æx log tan ç + ÷ 8ø 2 è2
2]
1 pö æ log tan ç x + ÷ 2 4ø è
pö æx log tan ç + ÷ 2 4ø 2 è
4]
1 pö æx log log ç + ÷ 2 2 8ø è
1
1
4x ; 4x + 2 What is the value of f(x) + f(1 – x)?
If f(x) =
1] 0 22.
2] 1
(4 x - 1) (2 x - 1) . ® 0 x x 2] (log 2) 2 3] 4 (log 2)2
x
2] ab
3] ab sin q
2] 1
4] ab cos q
3] 2
4] 7
The vector ‘a’ lies in the plane of vectors ‘b’ and ‘c’, then which of the following is correct? 1] a . (b × c) = 0
26.
4] Indeterminate
If a = 2i + 3j + 6k, b = 3i – 6j + 2k, c = 6i + 2j – 3k. Find a × (b × c). 1] 0
25.
4] 4x – 2
What will be the value of a × b, if the vectors are collinear? 1] 0
24.
4x 4x - 2
Find the value of lim 1] 2(log 2)2
23.
3]
The derivative of e (5 x 1] (5x – 10) e (5 x
2
3] (5x + 10) e (5 x
2
2] a . b × c = 1 2
- x + 3
- x + 3
3] a . b × c = –1 4] a . b × c = 0
) with respect to x is:
)
2] (10x – 1) e (5 x
)
4] (10x + 1) e (5 x
- x + 3
2
2
- x + 3
)
- x + 3
)
cos x dx, where c is any constant. 1 + cos x 1 + cos x 1 - cos x 1] x – + c 2] x – + c cos x sin x 3] x + tanx + c 4] x – cotx + secx + c
27.
Evaluate ò
28.
Evaluate
1/
1]
ò
0
2
x dx. 1 - x2
2 3
IMS-34-UG-AL-HM-Class-3
2] 1 –
1 2
3] log 2
4
4] 2 2