Bank Math Master

Question: If a + (1/a) = √3 then what is the value of {a³ + (1/a³) + a + (1/a)}²?

Solution:
Now,
a³ + (1/a³) + a + (1/a)
= a³ + (1/a³) + a + (1/a)
= {a + (1/a)}³ - 3 · a · (1/a){a + (1/a)} + {a + (1/a)}
= (√3)³ - 3√3 + √3
= 3√3 - 3√3 + √3
= √3

∴ {a³ + (1/a³) + a + (1/a)}² = (√3)² = 3

Question: Which number will complete the series
3, 10, 31, 94, ____, 850

Solution:
Here,
3
(3 × 3) + 1 = 10
(10 × 3) + 1 = 31
(31 × 3) + 1 = 94
(94 × 3) + 1 = 283
(283 × 3) + 1 = 850

Question: If p = 3 + √2 then, Find the value of p².

Solution:
Given,
p = 3 + √2
⇒ p² = (3 + √2)²
= 3² + 2 · 3 · √2 + (√2)²
= 9 + 6√2 + 2
= 11 + 6√2

Question: If England is written as 1234526 and France is written as 785291, then Greece is written as-

Solution:
England is written as 1234526
E = 1
n = 2
g = 3
l = 4
a = 5
n = 2
d = 6

France is written as 785291
F = 7
r = 8
a = 5
n = 2
c = 9
e = 1

So, GREECE will be = 381191

Question: If 9a² - 9a - 4 is divided by 3a - 4, the result is:

Solution:
9a² - 9a - 4
= 9a² - 12a + 3a - 4
= 3a(3a - 4) + 1(3a - 4)
= (3a - 4)(3a + 1)

Now,
(9a² - 9a - 4)/(3a - 4) = {(3a - 4)(3a + 1)}/(3a + 1) = (3a + 1)

সঠিক উত্তর: REACHET
অপশনে সঠিক উত্তর না থাকায় প্রশ্নটি বাতিল করা হলো। 
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Question: In a certain language, the word "MANAGER" is coded as "RENAGAM." How is "TEACHER" coded in that code?

Solution: 
MANAGER ⇔ RENAGAM
M ⇒ A ⇒ N ⇒ A ⇒ G ⇒ E ⇒ R
1 ⇒ 2 ⇒ 3 ⇒ 4 ⇒ 5 ⇒ 6 ⇒ 7
R ⇒ E ⇒ N ⇒ A ⇒ G ⇒ A ⇒ M

Here, 
The letters in the 1st and 7th positions are interchanged.
The letters in the 2nd and 6th positions are interchanged.
The letters remain the same.
Now, let's apply this pattern to the word TEACHER:

T ⇒ E ⇒ A ⇒ C ⇒ H ⇒ E ⇒ R
1 ⇒ 2 ⇒ 3 ⇒ 4 ⇒ 5 ⇒ 6 ⇒ 7
R ⇒ E ⇒ A ⇒ C ⇒ H ⇒ E ⇒ T

So, "TEACHER" would be coded as "REACHAET."

Question: If 9x² - px + 16 is a square number, then p/2 = ?

Solution:
9x² - px + 16
= (3a)² - 2 · 3a · 4 + 4²
= (3a)² - 24a + 4²

Here, p= 24
∴ p/2 = 24/2 = 12

Question: If A and B are prime numbers, which of the following cannot be the sum of A and B?

Solution:
Let's consider the prime numbers less than 35:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31

Now, let's examine each option:
28 = 5 + 23
14 = 3 + 11
22 = 3 + 19

We can see that 35 cannot be written as the sum of two prime numbers. Therefore, 35 is the answer that cannot be the sum of two prime numbers A and B.

Question: If x = (√3 + 1)/(√3 - 1) and y =(√3 - 1)/ (√3 + 1), then the value of x² + y² is?

Solution:

Question: If CAT = 48 and DOG = 52 then SUN will be equal to-

Solution:

CAT = (3 × 2) + (1 × 2) + (20 × 2) = 48
DOG = (4 × 2) + (15 × 2) + (7 × 2) = 52
So, SUN = (19 × 2) + (21 × 2) + (14 × 2) = 108

Question: If a = 0.202, then what is the value of 

Solution:

Question: Himel is bigger than Rajib. Tofail is bigger than Rakib. Tanveer is not as big as Tofail but is bigger than Rajib. Rakib is not as big as Rajib. Which is the smallest?

Solution:
Hime is bigger than Rajib. ⇒ Himel > Rajib
Tofail is bigger than Rakib. ⇒ Tofail > Rakib
Tanveer is not as big as Tofail but is bigger than Rajib. ⇒ Tofail > Tanveer > Rajib
Rakib is not as big as Rajib. ⇒ Rajib > Rakib

∴ Rakib is the smallest.

Question: If a + (1/a) = 4, then the value of a³ + (1/a³) is:

Solution:
a³ + (1/a³)
= {a + (1/a)}³ - 3 · a · (1/a){a + (1/a)}
= 4³ - 3 · 4
= 64 - 12
= 52

Question: P, Q, R, S, T and U were sitting around a circular table, facing the centre. They were sitting at equal distances from one another. T and R were sitting exactly next to each other. P was at the immediate right of U. Q was at the immediate left of T. R is third to the left of S. Who is sitting to the immediate left of Q?

Solution:

- Q was at the immediate left of T.
- T and R were sitting exactly next to each other.
- R is third to the left of S.
- P was at the immediate right of U.
- S is sitting to the immediate left of Q.

Hence, the correct answer is S.

Solution: If ab + bc + ca = 31 and a² + b² + c² = 19 then, a + b + c = ?

Solution:
Given,
ab + bc + ca = 31
a² + b² + c² = 19

Now,
(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
⇒ (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
⇒ (a + b + c)² = 19 + (2 × 31)
⇒ (a + b + c)² = 19 + 62
⇒ (a + b + c)² = 81
⇒ a + b + c = √81
∴ a + b + c = 9

Question: Five friends are standing in a line for a photo. Alice is to Bob's immediate right. Charlie is between David and Eve. Bob is to the left of David. Who is in the middle of the line?

Solution:
Alice is to Bob's immediate right: Bob ⇔ Alice
Charlie is between David and Eve: David ⇔ Charlie ⇔ Eve
Bob is to the left of David: Bob ⇔ David

Combining these pieces of information:
Bob ⇔ Alice ⇔ David ⇔ Charlie ⇔ Eve

The person standing in the middle of the line is David.

Question: The square root of  (7 + 3√5)(7 - 3√5) is:

Solution:

Question: If HAPPY is coded as 'JCRRA', then SMILE is coded as-

Solution:
Let's first analyze the coding pattern for the word HAPPY:
H - J (Difference 2)
A - C (Difference 2)
P - R (Difference 2)
P - R (Difference 2)
Y - A (Difference 2)
We can see that each letter is replaced by the letter that is two positions ahead in the alphabet.

Now, applying the same rule to the word SMILE:
S - U (Difference 2)
M - O (Difference 2)
I - K (Difference 2)
L - N (Difference 2)
E - G (Difference 2)
Therefore, if HAPPY is coded as JCRRA, then SMILE will be coded as UOKNG.

Solution: If a + b = m, a² + b² = n and a³ + b³ = p³ then, m³ + 2p³ = ?

Solution:
m³ + 2p³
= (a + b)³ + 2(a³ + b³)
= a³ + 3a²b + 3ab² + b³ + 2a³ + 2b³
= 3a³ + 3a²b + 3ab² + 3b³
= 3(a³ + a²b + ab² + b³)
= 3{a²(a + b) + b²(a + b)}
= 3(a + b)(a²+b²)
= 3mn

Question: 31, 29, 24, 22, 17, . . . What number should come next?

Solution:
In this series at first subtracts 2, then 5.

31 - 2 = 29
29 - 5 = 24
24 - 2 = 22
22 - 5 = 17
17 - 2 = 15