Mathematics Test 230

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Mathematics Test 230

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Weekly Quiz Competition

Question 1
4 / -1

If |z + 1 – i| = 1 and arg then minimum value of |z – ω| is

A
√2

B
√2 – 1

C
√2 + 1

D
2√2

Solution

Concept:

|z – z₀| represents the distance between z and z₀.

|z – z₀| = r represents a circle of radius r, centered at z₀.

Calculation:

Given, |z + 1 – i| = 1

⇒ |z – (– 1 + i)| = 1, which represents a circle of radius 1 centered at (– 1, 1)

∴ Minimum value of |z – ω|

= Minimum distance between z and ω

= OA

= (OC + CB) – AB

= (√2 + 1) – 2

= √2 – 1

∴ The minimum value of |z – ω| is √2 – 1.

The correct answer is Option 2.
Question 2
4 / -1

If (1 + x + x²)ⁿ = a₀ + a₁ x + ... + a₂ x² + ... + a_2n x²ⁿ, then the value of a₀ + a₃ + a₆ + ... is

A
1

B
2ⁿ

C
2^n–1

D
3^n–1

Solution

Concept:

The roots of equation x³ = 1 are given by the cube roots of unity.

The correct answer is Option 4.
Question 3
4 / -1

Let f : [0, 1) → [0, ∞) be a function such that

A

B

C

D
can’t be defined

Solution

Concept:

The set of all possible input values is known as the Domain.
Question 4
4 / -1

If the equation x² + 2 (k + 1) x + 9k – 5 = 0 has only negative roots, then

A
k ≤ 0

B
k ≥ 0

C
k ≥ 6

D
k ≤ 6

Solution
Question 5
4 / -1

then the number of ways of selecting two numbers from the set {1, 2, 3, ....., 12} whose sum is divisible by 3 is

A
mn

B
m

C
n

D
m + n

Solution

Now, any natural number is either of the form 3k or 3k - 1 or 3k +1.

Sum of two numbers will be divisible by 3 iff either both are of the form 3k or one is of the form 3k - 1 and other is of the form 3k + 1.

This can be done in ⁴C₂ + ⁴C₁ x ⁴C₁ = 6 + 16 = 22 = 10 + 12 = m + n

∴ The number of ways of selecting two numbers from the set {1, 2, 3, ....., 12} whose sum is divisible by 3 is m + n.

The correct answer is Option 4.
Question 6
4 / -1

If n is a natural number ≥ 2, then roots of equation zⁿ= (1 + z)ⁿ lie on

A
straight line parallel to x-axis

B
Re(z) = –1/2

C
Im(z) = 1

D
none of these

Solution
Question 7
4 / -1

The number of integral values of x satisfying the equation 2^x (4 – x) = 2x + 4

A
0

B
1

C
2

D
3

Solution

∴ The number of integral values of x satisfying the equation 2^x (4 – x) = 2x + 4 is 3.

The correct answer is Option 4.
Question 8
4 / -1

Let f : [– 10, 10] → R, where f(x) = sin x + [x²/a] and [.] denotes the greatest integer function be an odd function. Then set of values of parameter ‘a’ is/are

A
(–10, 10) – {0}

B
(0, 10)

C
[100, ∞)

D
(100, ∞)

Solution

Calculation:

Given, f(x) is an odd function

⇒ f(x) = − f(− x)

⇒ sin x + [x²/a] = − (− sin x + [x²/a])

⇒ [x²/a] = 0

Now, x ∈ [ − 10, 10]

⇒ 0 ≤ x²/a < 1

⇒ a > 100 ∀ x ∈ [ − 10, 10]

∴ Then set of values of parameter ‘a’ is (100, ∞).

The correct answer is Option 4.
Question 9
4 / -1

Consider the sequence 1, 2, 2, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8, .... Then 1025th term will be

A
2⁹

B
2¹¹

C
2¹⁰

D
2¹²

Solution

Calculation:

Let m be number of times a number n is repeated in the given sequence(n)

Given sequence 1, 2, 2, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8, ⋯

By observing the sequence, 2⁰ = 1 is the first term.

2¹ = 2 (repeated 2 times)

2² = 4 (repeated 4 times)

2³ = 8 (repeated 8 times)

Next term will be 2⁴ = 16. It will be repeated 16 times.

Since we double the number each time we know the pattern will go: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024...

The number 1024 will start on term 1024, therefore the 1025^th term will be 1024 = 2¹⁰.

∴ The 1025th term is 2¹⁰.

The correct answer is Option 3.
Question 10
4 / -1

The set of values of a, such that the equation has two distinct real roots is given by

A
[0, 1]

B
(–∞, 0)

C
[0, ∞)

D

Solution