IBA ফ্যাকাল্টি ভিত্তিক প্রস্তুতি

Question: In a class 3/4th of the students do not know either English or Spanish. But 1/6th of the students know English. How much students know both English and Spanish if students who know Spanish are 1/8th of total students in the class?

Solution:
Let's say total number of students = x

Let's say number of students who know both languages = y
According to set theory:
Total students who know at least one language = Students who know English + Students who know Spanish - Students who know both languages
⇒ x - (3/4)x = (1/6)x + (1/8)x - y
⇒ (1/4)x = (1/6 + 1/8)x - y
⇒ (1/4)x = (24/144 + 18/144)x - y
⇒ (36/144)x = (42/144)x - y
⇒ y = (42/144)x - (36/144)x
⇒ y = (6/144)x
∴ y = (1/24)x

Therefore, 1/24th of the total students know both English and Spanish.

Question: The average of five numbers is 7. If three new numbers would be added, then the new average comes out to be 8.5. What is the average of those three new numbers?

Solution:
Total of five numbers = 7 × 5 = 35
Total of 8 numbers = 8 × 8.5 = 68
Total of three new number = 68 - 35 = 33

∴ Average of those three new numbers = 33/3 = 11

Question: Alfi got married 8 years ago. Today his age is 7/5 times his age at the time of marriage. Her daughter's age is 1/7 times her age. Her daughter's age is:

Solution:
Let the age of Anita at the time of marriage be x
Now, hos age = x + 8

ATQ,
x + 8 = 7x/5
⇒ 5(x + 8) = 7x
⇒ 5x + 40 = 7x
⇒ 7x - 5x = 40
⇒ 2x = 40
∴ x = 20
Alfi's current age = 20 + 8 = 28 years
Her daughter's age = (1/7) × 28 = 4 years

Question: What is the 10th term of the geometric sequence: 4 + 8 +16 + ........?

Solution:
দেওয়া আছে,
প্রদত্ত ধারাটির প্রথম পদ, a = 4,
সাধারণ অনুপাত, r = 8/4 = 2

আমরা জানি,
গুণোত্তর ধারার n তম পদ = ar^{n - 1}
∴ ধারাটির 10 তম পদ = 4 × 2^{10 - 1}
= 4 × 2⁹
= 4 × 512
= 2048

Question: Jubin's front lawn is 1/3 the size of his back lawn. If John mows 1/2 of his front lawn and 2/3 of his back lawn, what fraction of his lawn is left unmowed?

Solution:
Let say back lawn = 6
Therefore front lawn = (1/3) × 6 =2
Total = 6 + 2 = 8
Now moved lawn= {(1/2) × 2} + {(2/3) × 6} = 5
Therefore unmoved lawn = total - moved = 8 - 5 =3
∴ Fraction = unmoved/total = 3/8

Question: A batsman in his 12th innings makes a score of 120, and thereby increases his average by 5. The average score after 12th innings is-

Solution:
Let, average runs after 12 innings = x 
Average runs after 11 innings = x - 5

ATQ,
12x = (x - 5) × 11 + 120
⇒ 12x = 11x - 55 + 120
⇒ 12x - 11x = 65
∴ x = 65

Question: The present ages of A and B are 42 and 36 years, respectively. After K years, the ratio of ages of B to A will be 15 : 17.What is value of K?

Solution:
ATQ after k years,
(36 + k)/(42 + k) = 15/17
⇒ 612 + 17k = 630 + 15k
⇒ 17k - 15k = 630 - 612
⇒ 2k = 18
∴ k = 9

প্রশ্ন: 11² + 12² + 13² + ........ + 20² = ?

সমাধান:
11² + 12² + 13² + ........ + 20²
= (1² + 2² + 3² + .... + 20²) - (1² + 2² + 3² + .... + 10²)
= [{20(20 + 1)(2 × 20 + 1}/6] - [{10(10 + 1)(2 × 10 + 1)}/6]
= {(20 × 21 × 41)/6} - {(10 × 11 × 21)/6}
= 2870 - 385
= 2485

Question: The value of is-

Solution:

Question: The ratio between the present ages of A and B is 3 : 5 respectively. If the difference between B's present age and A's age after 4 years is 2, what is the total of A's and B's present ages?

Solution:
Let the present ages of A and B be 3x years and 5x years respectively.
5x - (3x + 4) = 2
⇒ 2x = 2 + 4
⇒ 2x = 6
∴ x = 3

Therefore,
Required sum = 3x + 5x
= 8x
= 8 × 3
= 24 years

Question: If the 5th term of an arithmetic sequence is 45, what is the sum of the first 15 terms?
লাইভ পরীক্ষার সময় প্রশ্নটি ভুল ছিলো । প্রশ্নটি হওয়ার কথা 'If the 8th term of an arithmetic sequence is 45, what is the sum of the first 15 terms?

Solution:
ধরি,
সমান্তর ধারার ১ম পদ, a
সাধারণ অন্তর, d
সমান্তর ধারার 8ম পদ 45 হলে,
a + 7d = 45

∴ প্রথম 15টি পদের সমষ্টি = (15/2){2a + (15 - 1)d}
= (15/2){2a + 14d}
= (15/2) × 2(a + 7d)
= 15 × (a + 7d)
= 15 × 45
= 675

লাইভ পরীক্ষার সময় প্রশ্নে ভুল থাকায় প্রশ্নটি বাতিল করা হলো।

Question: The average monthly income of Abir and Sabbir is Tk. 8000. The average monthly income of Sabbir and Rana is Tk. 9500 and the average monthly income of Abir and Rana is Tk. 9000. What is the monthly income of Abir?

Solution:
Abir + Sabbir (total income) = (8000 × 2) = 16000 .......... (1)
Sabbir + Rana (total income) = (9500× 2) = 19000 ........ (2)
Abir + Rana (total income) = (9000 × 2) = 18000 ......... (3)

(1) + (2) + (3) ⇒
2(Abir + Sabbir+ Rana) = 16000 + 19000 + 18000
⇒ 2(Abir + Sabbir + Rana) = 53000
⇒ Abir + Sabbir + Rana = 26500 ....... (4)

(4) - (2) ⇒
Abir = 26500 - 19000  = 7500
∴ Abir's monthly income Tk. 7500.

Question: What is the missing number in the sequence:
32, 48, 72, ..., 162, 243.

Solution:
Each number is being multiplied by 3/2 to get the next number.
32 × (3/2) = 48
48 × (3/2) = 72
72 × (3/2) = 108
108 × (3/2) = 162
162 × (3/2) = 243

Question: If 0 < x < 1, which of the following expressions is greatest?

Solution:
Since x is > 0 and x < 1, it has to be a decimal/fraction.
Take a value that is a perfect square, as some options have a root value. Let's take x = 0.25

A) 1/√x = 1/√0.25 = 1/0.5 = 2
B) √x = √0.25 = 0.5
C) (1/π)x = 0.25/(22/7) = 0.08
D) x³ = (0.25)3 = 0.015
E) x⁴ = (0.25)4 = 0.004

So, If 0 < x < 1, then the expressions 1/√x is greatest.

Question: A teacher accidentally entered a student's marks as 82 instead of 67. Due to this error, the class average increased by 0.3 marks. Find the number of students in the class.

Solution:
Let the number of students in the class be x.
Total increase in marks = x × 0.3
According to the question
⇒ x × 0.3 = (82 - 67)
⇒ x × 0.3 = 15
⇒ x = 15/0.3
⇒ x = 50

Question: A father said to his son, 'I was as old as you are at the present at the time of your birth'. If the father's age is 52 years now, the son's age five years back was-

Solution:
let, at present son is x years old
so, at time of his birth the father was x years old

∴ at present the age of father is = x + x year
= 2x year

2x = 52
⇒ x = 52/2
= 26 year

∴ the son's age five years back was = 26 - 5
= 21 year

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

Solution:
This is a simple alternating subtraction series, which subtracts 2, then 5.
31 - 2 = 29,
29 - 5 = 24,
24 - 2 = 22,
22 - 5 = 17,
17 - 2 = 15.

Question: What decimal of an hour is a second (approximate)?

Solution:
1 hour = 60 minute
and 1 minute = 60 second

Thus, required decimal = 1/(60 × 60)
= 1/3600
≈ 0.00028