Functions Test

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Functions Test - 3

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Weekly Quiz Competition

Question 1
3 / -1

Find the domain of the definition of the function y = lxl

A
0 ≤ x

B
— ∞< x < +∞

C
x < + ∞

D
0 ≤ x < + ∞

Solution
Question 2
3 / -1

If f(x) is an even function, then the graph y = f(x) will be symmetrical about

A
x-axis

B
y-axis

C
Both the axes

D
None of these

Solution

y — axis by definition.
Question 3
3 / -1

The function f(x) = x² + 4x + 4 is

A
Neither odd nor even

B
even

C
odd

D
periodic

Solution

Given: f(x) = x² + 4x + 4
Replace x by -x,
⇒ f(-x) = (-x)² + 4(-x) + 4
= x² - 4x + 4 (∵ (-x)² = x²)
⇒ f(-x) ≠ ± f(x)
Hence function is neither odd nor even.
Question 4
3 / -1

Find the minimum value of the function fix) = log₂ (x² - 2x + 5).

A
-4

B
2

C
4

D
-2

Solution

The minimum value of the function would occur at the minimum value of (x² - 2x + 5) as this quadratic function has imaginary roots.
Thus, minimum value of the argument of the log is 4. So minimum value of the function is log₂ 4 = 2.
Question 5
3 / -1

If f(x) is a polynomial satisfying f(x) f(1/x) = f(x) + f(1/x) and f(3) = 28, then f(4) = ?

A
63

B
65

C
17

D
None of these

Solution
Question 6
3 / -1

Which of the following is an even function?

A
x^–8

B
x³

C
x^–33

D
x⁷³

Solution

x^–8 is even since f(x) = f(–x) in this case.
Question 7
3 / -1

For what value of x, x² + 10x + 11 will give the minimum value?

A
5

B
+10

C
–5

D
–10

Solution

dy/dx = 2x + 10 = 0 fi x = –5.
Question 8
3 / -1

Find the maximum value of the function 1/(x² – 3x + 2).

A
11/4

B
1/4

C
0

D
None of these

Solution

Since the denominator x² – 3x + 2 has real roots, the maximum value would be infinity.
Question 9
3 / -1

Read the instructions below and solve.
f(x) = f(x – 2) – f(x – 1), x is a natural number
f(1) = 0, f(2) = 1
The value of f(8) is

A
0

B
13

C
–5

D
–9

Solution

f(1) = 0, f(2) = 1,
f(3) = f(1) – f(2) = –1
f(4) = f(2) – f(3) = 2
f(5) = f(3) – f(4) = –3
f(6) = f(4) – f(5) = 5
f(7) = f(5) – f(6) = –8
f(8) = f(6) – f(7) = 13
f(9) = f(7) – f(8) = –21
13
Question 10
3 / -1

Read the instructions below and solve.
f(x) = f(x – 2) – f(x – 1), x is a natural number
f(1) = 0, f(2) = 1
What will be the domain of the definition of the function f(x) = 8–^xC_5–x for positive values of x?

A
{1, 2, 3}

B
{1, 2, 3, 4}

C
{1, 2, 3, 4, 5}

D
{1, 2, 3, 4, 5, 6, 7, 8}

Solution

f(1) = 0, f(2) = 1,
f(3) = f(1) – f(2) = –1
f(4) = f(2) – f(3) = 2
f(5) = f(3) – f(4) = –3
f(6) = f(4) – f(5) = 5
f(7) = f(5) – f(6) = –8
f(8) = f(6) – f(7) = 13
f(9) = f(7) – f(8) = –21
For any ⁿC_r, n should be positive and r ≥ 0.
Thus, for positive x, 5 – x ≥ 0
fi x = 1, 2, 3, 4, 5.
Question 11
3 / -1

Ajesh saves Rs 50,000 every year and deposits the money in a bank at compound interest of 10%(compunded annually).What would be his total saving at the end of the 5th year?

A
Rs 3,05,255

B
Rs 1,05,255

C
Rs 8,05,255

D
Rs 4,05,255

Solution

At the end of the 1st year, he will get Rs 50000, it will give him interest for 4 years compounded annually
Hence at the end of 5 years, this amount will become 50000(1.1)⁴
Similarly, the amount deposited in the 2nd year will give interest for 3 years. Hence it will become 50000(1.1)³
Similarly, we can calculate for the remaining years.
The total saving at the end of the 5th year would be a GP, given by
Net saving = 50000(1.1)⁴+ 50000(1.1)³ ..... 50000
Thus net saving = = Rs 3,05,255
Question 12
3 / -1

Define the following functions:
(a) (a M b) = a – b (b) (a D b) = a + b
(c) (a H b) = (ab) (d) (a P b) = a/b
Q.
Which of the following functions will represent a² – b²?

A
(a M b) H (a D b)

B
(a H b) M (a P b)

C
(a D b)/(a M b)

D
None of these

Solution

Option a = (a – b) (a + b) = a² – b²
Question 13
3 / -1

Define the following functions:
(a) (a M b) = a – b (b) (a D b) = a + b
(c) (a H b) = (ab) (d) (a P b) = a/b
Q.
What is the value of (3M4H2D4P8M2)?

A
6.5

B
6

C
–6.5

D
None of these

Solution

3 – 4 × 2 + 4/8 – 2 = 3 – 8 + 0.5 – 2 = – 6.5
(using BODMAS rule)
Question 14
3 / -1

Define the following functions:
(a) (a M b) = a – b (b) (a D b) = a + b
(c) (a H b) = (ab) (d) (a P b) = a/b
Q.
Which of the four functions defined has the minimum value?

A
(a M b)

B
(a D b)

C
(a H b)

D
Cannot be determined

Solution

The minimum would depend on the values of a and b. Thus, cannot be determined.
Question 15
3 / -1

If 0 < a <1 and 0 < b < 1 and if a < b, which of the following expressions will have the highestvalue?

A
(a M b)

B
(a D b)

C
(a P b)

D
Cannot be determined

Solution

Again (a + b) or a/b can both be greater than each other depending on the values we take for a and b.
E.g. for a = 0.9 and b = 0.91, a + b > a/b.
For a = 0.1 and b = 0.11, a + b < a/b

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Functions Test - 3

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Functions Test - 3

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SHARING IS CARING

Question 1

0 ≤ x

— ∞< x < +∞

x < + ∞

0 ≤ x < + ∞

Solution

Question 2

x-axis

y-axis

Both the axes

None of these

Solution

Question 3

Neither odd nor even

even

odd

periodic

Solution

Question 4

-4

2

4

-2

Solution

Question 5

63

65

17

None of these

Solution

Question 6

x–8

x3

x–33

x73

Solution

Question 7

5

+10

–5

–10

Solution

Question 8

11/4

1/4

0

None of these

Solution

Question 9

0

13

–5

–9

Solution

Question 10

{1, 2, 3}

{1, 2, 3, 4}

{1, 2, 3, 4, 5}

{1, 2, 3, 4, 5, 6, 7, 8}

Solution

Question 11

Rs 3,05,255

Rs 1,05,255

Rs 8,05,255

Rs 4,05,255

Solution

Question 12

(a M b) H (a D b)

(a H b) M (a P b)

(a D b)/(a M b)

None of these

Solution

x^–8

x³

x^–33

x⁷³