Bank Math Master

Question: If x + (1/x) = 3, then the value of (3x² - 4x + 3)/(x² - x + 1) is?
 
Solution: 
Given that, 
x + (1/x) = 3
⇒ (x² + 1)/x = 3
∴ x² + 1 = 3x

Now, 
(3x² - 4x + 3)/(x² - x + 1)
= (3x² + 3 - 4x)/(x² + 1 - x)
= {3(x² + 1) - 4x}/(x² + 1 - x)
= {(3 × 3x) - 4x}/(3x - x) [মান বসিয়ে]
= (9x - 4x)/2x
= 5x/2x
= 5/2

Question: 2, 8, 14, 20, 26, 32, 38, ... choose which pair of numbers comes next?

Solution: 
This is an arithmetic sequence where each term increases by 6.
Check the differences, 
8 - 2 = 6
14 - 8 = 6
20 - 14 = 6
26 - 20 = 6
32 - 26 = 6
38 - 32 = 6
So the pattern is very clear that add 6 each time.

Next two numbers after 38 is-
38 + 6 = 44, 44 + 6 = 50

Therefore, the next pair is 44, 50.

Question: Two fair dice are thrown together. What is the probability that the product of the two numbers that appear is 20?

Solution:
Total number of possible outcomes when throwing two dice = 6 × 6 = 36
Favorable outcomes where the product is 20.
The possible pairs (first die, second die) are, (4, 5) and (5, 4)
∴ favorable outcomes = 2
∴ Probability = Number of favorable outcomes/Total outcomes
= 2/36
= 1/18

∴ The probability is 1/18.

Question: Solve the inequality |1 - 2x| < 7

Solution:
Given that, 
|1 - 2x| < 7
⇒ - 7 < 1 - 2x < 7
⇒ - 7 - 1 < 1 - 1 - 2x < 7 - 1
⇒ - 8 < - 2x < 6
⇒ - 4 < - x < 3    (dividing by - 2 and reversing the inequality signs)
⇒ 4 > x > - 3
∴ - 3 < x < 4

Question: If a² - 4a + 1 = 0, then the value of a³ + 1/a³ = ?

Solution: 
Given that, 
a² - 4a + 1 = 0
⇒ a² + 1 = 4a
⇒ (a²/a) + 1/a = 4a/a
∴ a + 1/a = 4

Now, 
a³ + 1/a³
= (a + 1/a)³ - 3 × a × (1/a) (a + 1/a)
= 4³ - (3  × 4)
= 64 - 12
= 52

Question: If LATER = 13579 and CHAIR = 20349, then CHEAT = ?

Solution:
Given that,
L    A   T   E    R
↓    ↓   ↓    ↓    ↓
1    3   5   7    9

and
C   H   A   I    R
↓    ↓   ↓    ↓    ↓
2   0    3   4    9

So
C   H   E    A   T
↓    ↓   ↓    ↓    ↓
2   0    7   3    5

Question: P = {x ∈ N : x³ < 216}. Then, how many elements are there in set P?

Solution:
Here, N = the set of natural numbers
= {1, 2, 3, 4, 5, 6, 7, 8, ……}

Given that, 
P = {x ∈ N : x³ < 216}

Now check each value.
When x = 1, 1³ = 1 < 216
When x = 2, 2³ = 8 < 216
When x = 3, 3³ = 27 < 216
When x = 4, 4³ = 64 < 216
When x = 5, 5³ = 125 < 216
When x = 6, 6³ = 216 < 216 ; false (not true)

Therefore, the set P contains only the values that satisfy the condition.
P = {1, 2, 3, 4, 5}
∴ The number of elements in set P = 5

Question: A bag contains 8 red marbles, 12 yellow marbles, and 4 purple marbles. What is the probability of drawing a yellow marble?

Solution:
Total number of marbles = 8 + 12 + 4
= 24

And number of yellow marbles (favorable outcomes) = 12

Probability of drawing a yellow marble = (Number of yellow marbles)/(Total number of marbles)
= 12/24
= 1/2

So the probability is 1/2.

Question: If |2x + 5| < 3, then for what values of p and q will p < 3x - 2 < q hold?

Solution:
Given that, 
|2x + 5| < 3
⇒ - 3 < 2x + 5 < 3
⇒ - 3 - 5 < 2x + 5 - 5 < 3 - 5
⇒ - 8 < 2x < -2
⇒ - 4 < x < -1    ; [dividing by 2]
⇒ - 12 < 3x < - 3  ; [Now multiply all parts by 3]
⇒ - 12 - 2 < 3x - 2 < - 3 - 2  ; [Subtract 2 from all parts]
⇒ - 14 < 3x - 2 < - 5

Now comparing with p < 3x - 2 < q, Then we get,
∴ p = - 14 and q = - 5

Question: If (x + 7)² = 81, which of the following can be the value of (x - 5)? 

Solution: 
Given that,
(x + 7)² = 81
⇒ x + 7 = ± √81
x + 7 = ± 9
So there are two possible solutions.

Case 1: (Positive value) 
x + 7 = 9
⇒ x = 9 - 7
⇒ x = 2

Case 2: (Negative value)
⇒ x + 7 = - 9
⇒ x = - 9 - 7
∴ x = - 16

So x = 2, - 16

Now, x - 5 = 2 - 5 = - 3 ; [x = 2]

Question: Find out the wrong term: 8, 14, 26, 48, 98, 194, 386 

Solution:
Each term = (previous term × 2) - 2
Let’s check term by term.
1st term = 8
2nd = 8 × 2 - 2 = 16 - 2 = 14
3rd = 14 × 2 - 2 = 28 - 2 = 26
4th = 26 × 2 - 2 = 52 - 2 = 50 ; but the given term is 48 (wrong here)
5th = 50 × 2 - 2 = 100 - 2 = 98
6th = 98 × 2 - 2 = 196 - 2 = 194
7th = 194 × 2 - 2 = 388 - 2 = 386

The first mistake occurs at the 4th term. 26 should become 50, but it is written as 48.
So 48 is the wrong term. It should be replaced by 50.
Correct series should be 8, 14, 26, 50, 98, 194, 386

Question: If A = {p, q, r, s, t}, then how many proper subsets does A have?

Solution:
Given that, 
A = {p, q, r, s, t}
The number of elements in set A is 5.

We know that,
Number of proper subsets = 2ⁿ - 1  ; [where n = number of elements in the set]
∴ Number of proper subsets of A = 2⁵ - 1
= 32 - 1
= 31

Question: A fair die is rolled once. What is the probability of getting an even number?

Solution:
A standard fair die has 6 faces.
{1, 2, 3, 4, 5, 6}

∴ Total possible outcomes = 6

And even numbers on a die, {2, 4, 6}
∴ Number of favorable outcomes = 3

∴ Probability of getting an even number = (Number of favorable outcomes)/(Total number of possible outcomes)
= 3/6
= 1/2

So the probability is 1/2.

Question: Express the following inequality using absolute value notation: 1 < x < 9

Solution:
1 < x < 9
∴ The midpoint = (1 + 9)/2
= 10/2
= 5
Now subtract the midpoint from all parts. then we get,
1 - 5 < x - 5 < 9 - 5
⇒ - 4 < x - 5 < 4
∴ |x - 5| < 4

Question: CMM, EOO, GQQ, _____, KUU

Solution:
প্রথম অক্ষর,
C → E → G → ? → K ; [+ 2 করে বাড়ছে]
C + 2 = E, E + 2 = G, G + 2 = I, I + 2 = K)
∴ ফাঁকা জায়গায় প্রথম অক্ষর = I

দ্বিতীয় অক্ষর,
M → O → Q → ? → U  ; [+ 2 করে বাড়ছে]
M + 2 = O, O + 2 = Q, Q + 2 = S, S + 2 = U
∴ ফাঁকা জায়গায় দ্বিতীয় অক্ষর = S

তৃতীয় অক্ষর, দ্বিতীয় অক্ষরের মত
∴ ফাঁকা জায়গায় তৃতীয় অক্ষর = S

সুতরাং ফাঁকা জায়গায় আসবে- ISS

Question: P = {x ∈ N : 2 < x ≤ 6} and Q = {x ∈ N : x is an even number and x ≤ 8}. Find the value of P ∩ Q.

Solution:
Given that, 
P = {x ∈ N : 2 < x ≤ 6}
∴ P = {3, 4, 5, 6}

Q = {x ∈ N : x is even and x ≤ 8}
∴ Q = {2, 4, 6, 8}

Now,
P ∩ Q = {3, 4, 5, 6} ∩ {2, 4, 6, 8}
= {4, 6}

Question: Numbers from 10 to 20 are written on cards. One card is drawn at random. What is the probability that the number is prime or divisible by 3?

Solution:
The numbers are, 
10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20
∴ Total numbers = 11

∴ Prime number are, 11, 13, 17, 19 = 4

And, List the numbers divisible by 3 = 3 numbers (12, 15, 18)

∴ Total favorable outcomes = Primes + Divisible by 3 - Overlap
= 4 + 3 - 0 = 7

∴ Probability = Favorable outcomes/Total outcomes
= 7/11

So the probability is 7/11.

Question: What is the solution set of the inequality, - 2x + 11 ≥ 5?

Solution:
Given that, 
- 2x + 11 ≥ 5
⇒ - 2x + 11 - 11 ≥ 5 - 11 ; [Subtract 11 from both sides]
⇒ - 2x ≥ - 6
∴ x ≤ 3  ; [Divide both sides by - 2]

Solution set: x ≤ 3
or in interval notation: (- ∞, 3]