Bank Math Master

Question: If x ≥ 15 and y ≤ 9 , which of the following must be true?

Solution:
Here,
x ≥ 15 ..............(1)
and,
y ≤ 9
⇒ - y ≥ - 9 .............(2)

From (1) + (2) we get,
x - y ≥ 15 - 9
∴ x - y ≥ 6

Question: For a geometric sequence, the first term a = 7 and the common ratio r = 2 . What is the sum of the first 5 terms?

Solution:
Given,
a = 7
r = 2 > 1
n = 5

S = {a(rⁿ - 1)}/(r - 1)
= {7(2⁵ - 1)}/(2 - 1)
= 7 × 31
= 217

Question: In a box, there are 6 red, 8 blue and 10 green balls. One ball is picked up randomly. What is the probability that it is neither blue nor green?

Solution:
Number of total balls = (6 + 8 + 10) = 24

Since the ball is neithere Blue nor Green so the ball must be Red.

∴ Probability = 6/24 = 1/4

Question: if - 1 < x < 0 then which of the following is biggest?

Solution:
let,
x = - 1/2

a) x/2 = (- 1/2)/2
= - 1/4

b) x² = (- 1/2)²
= 1/4

c) 1/(x²) = 1/(- 1/2)²
= 1/(1/4)
= 4

d) x + 1 = (- 1/2) + 1
= (- 1 + 2)/2
= 1/2

Question: If A = {x ∈ N : 4x < 16} then, how many subsets does set "A" have?

Solution:
Given,
A = {x ∈ N : 4x < 16}
∴ A = {1, 2, 3}

We know,
the number of subsets = 2ⁿ
Here, n=3

the number of subsets A have = 2³ = 8

Question: Two cards are drawn together from a pack of 52 cards. The probability that one is a diamond and one is a heart, is?

Solution:
Two cards can be drawn together from a standard deck in = ⁵²C₂ = 1326 ways.

One diamond can be drawn in = ¹³C₁ ways
and a heart too can be drawn in = ¹³C₁ ways.

Therefore, the number of ways that a diamond and a heart can be drawn = ¹³C₁ × ¹³C₁
= 13 × 13
= 169

Therefore, the required probability = 169/1326
= 13/102

Question: If |x - 5| < 6 , then what is the solution-

Solution:
Given
|x - 5| < 6
⇒ - 6 < x - 5 < 6
⇒ - 6 + 5 < x - 5 + 5 < 6 + 5
⇒ - 1 < x < 11

Question: solve |x - 4| > 1

Solution:
Given,
|x - 4| >1

x - 4 > 1
⇒ x - 4 + 4 > 1 + 4
∴ x > 5

or
x - 4 < - 1
⇒ x - 4 + 4 < - 1 + 4
∴ x < 3

∴ x > 5 or x < 3

Question: If C = {a, b, 1, 2} and D = {3, 4, 6, 8} then C union D is-

Solution:
C union D = C ∪ D = {a, b, 1, 2} ∪ {3, 4, 6, 8}
= {a, b, 1, 2, 3, 4, 6, 8}

Question: Two cards are drawn from a pack of 52 cards. What is the probability that both are black cards?

Solution:
number of black cards = 26

probability both card black = ²⁶C₂/⁵²C₂
= 325/1326
= 25/102

Question: If x² − 8x +15 < 0 then solve the inequality-

Solution:
Given,
x² − 8x +15 < 0
⇒ x² - 3x - 5x + 15 < 0
⇒ x(x - 3) - 5(x - 3) < 0
⇒ (x - 3)(x - 5) < 0

The inequality will be true if x - 3 > 0 and x - 5 < 0 .
x - 3 > 0
or, x > 3

x - 5 < 0
or, x < 5
The inequality will be true if 3 < x < 5

∴ The solution of the inequality is 3 < x < 5

Question: A committee of 5 members is to be formed by selecting out of 6 men and 7 women. What is the probability that the committee has exactly 2 men and 3 women?

Solution:
Total member = 3 + 2 = 5

2 men can be selected out of 6 men = ⁶C₂ ways
3 women can be selected out of 7 women in = ⁷C₃ ways
Required number of ways = ⁶C₂ × ⁷C₃ = 15 × 35 = 525

The total number of ways to make committee with all members = ¹³C₅ = 1287

∴ The probability that the committee has exactly 2 men and 3 women = 525/1287
= 175/429

Question: If x² − 18x + 72 ≥ 0 then solve the inequality-

Solution:
Given,
x² − 18x + 72 ≥ 0
⇒ (x − 6)(x − 12) ≥ 0

The inequality will be true if x - 6 ≤ 0 or x - 12 ≥ 0 .
x - 6 ≤ 0
⇒ x ≤ 6

x - 12 ≥ 0
⇒ x ≥ 12
The inequality will be true if x ≤ 6 or x ≥ 12

∴ The solution of the inequality is x ≤ 6 or x ≥ 12

Question: If the sum of an infinite geometric series is 150 and the common ratio r = 1/2 what is the first term?

Solution:
Here,
r = 1/2
a = ?

We know that,
S_∞ = a/(1 - r)
⇒ 150 = a/(1 - 1/2)
⇒ 150 = a/(1/2)
⇒ 150 = 2a
∴ a = 75

Question: Which of the following describes all values of x for which 25 - x² ≥ 0

Solution:
25 - x² ≥ 0
⇒ - x² ≥ - 25
⇒ x² ≤ 25
⇒ x² ≤ 5²
∴ - 5 ≤ x ≤ 5 [x² ≤ a² ⇒ - a ≤ x ≤ a]

Question: Express C = {x : x positive integer and x² < 18} by using roster method.

Solution:
1, 2, 3, 4, 5,... are positive integers.
Here,
if x = 1 then x² = 1² = 1
if x = 2 then x² = 2² = 4
If x = 3 then x² = 3² = 9
if x = 4 then x² = 4² = 16
If x = 5 then x² = 5² = 25 ; which is greater than 18.

∴ as per conditions the acceptable positive integers are 1, 2, 3 and 4.
∴ the given set is C = {1, 2, 3, 4}