IBA ফ্যাকাল্টি ভিত্তিক প্রস্তুতি

Question: If m and n are even numbers, which of the following is always even?

Solution:
Let m = 2 and n = 4 (both are even numbers)
a) m + n + 1 = 2 + 4 + 1 = 7 ......... Odd
b) mn + 3 = (2 × 4) + 3 = 8 + 3 = 11 ......... Odd
c) 2m + n = (2 × 2) + 4 = 4 + 4 = 8 ......... Even
d) m² + n + 1 = (2)² + 4 + 1 = 4 + 4 + 1 = 9 ......... Odd

Answer: c) 2m + n is always even

Question: Let N be the smallest positive integer that is divisible by both 20 and 30. How many distinct prime factors does N have?

Solution:
এখানে, N হলো 20 এবং 30 দ্বারা বিভাজ্য ক্ষুদ্রতম সংখ্যা।
সুতরাং, N হবে 20 এবং 30 এর ল.সা.গু।

এখন, 20 = 2 × 2 × 5 = 2² × 5¹
এবং 30 = 2 × 3 × 5 = 2¹ × 3¹ × 5¹

LCM(20, 30) = 2² × 3¹ × 5¹ = 60
অতএব, N = 60

60 এর মৌলিক উৎপাদক = 2² × 3 × 5

স্বতন্ত্র মৌলিক উৎপাদকগুলি হলো 2, 3 এবং 5।

∴ N এর স্বতন্ত্র মৌলিক উৎপাদকের সংখ্যা হলো 3টি।

Question: If x > 2 and x < 5, then which of the following expressions is positive?
I. (x - 2)(x - 5)
II. (2 - x)(x - 5)
III. (2 - x)(5 - x)

Solution:
Given,
x > 2 and x < 5

For expression I: (x - 2)(x - 5)
Since x > 2, (x - 2) will be positive.
Since x < 5, (x - 5) will be negative.
(x - 2)(x - 5) = positive × negative = negative

For expression II: (2 - x)(x - 5)
Since x > 2, (2 - x) will be negative.
Since x < 5, (x - 5) will be negative.
(2 - x)(x - 5) = negative × negative = positive

For expression III: (2 - x)(5 - x)
Since x > 2, (2 - x) will be negative.
Since x < 5, (5 - x) will be positive.
(2 - x)(5 - x) = negative × positive = negative

∴ Only expression II is positive.

Question: The sum of three consecutive odd integers is 48 more than the last of the numbers. What is the middle number?

Solution:
Let the first odd number be x
∴ 2nd odd number is x + 2
∴ 3rd odd number is x + 4

According to the question,
Sum of the odd numbers = (x + 4) + 48

The equation,
x + (x + 2) + (x + 4) = (x + 4) + 48
⇒ 3x + 6 = x + 4 + 48
⇒ 3x + 6 = x + 52
⇒ 3x - x = 52 - 6
⇒ 2x = 46
⇒ x = 46/2
⇒ x = 23

First number is 23
The second number is 25
The third number is 27

∴ The middle number is 25

Question: Which one of the following is a rational number?

Solution:
ক) √3 × √5 = √15 ........ irrational
খ) √11 × √2 = √22 ........ irrational
গ) √3 × √27 = √81 = 9 ........ rational
ঘ) √6 × √16 = √96 ........ irrational
ঙ) √7 × √13 = √91 ........ irrational

Answer: গ) √3 × √27 = 9 is a rational number

Question: The difference between two numbers is 840. When the larger number is divided by the smaller, the quotient is 6 and the remainder is 10. What is the smaller number?

Solution:
Given that,
The difference of two numbers = 840
Quotient when the larger number is divided by the smaller number = 6
Remainder when the larger number is divided by the smaller number = 10

Now, Let the smaller number be x.
Larger number = 6x + 10

ATQ,
⇒ 6x + 10 - x = 840
⇒ 5x + 10 = 840
⇒ 5x = 840 - 10
⇒ 5x = 830
⇒ x = 830/5
⇒ x = 166

So the smaller number is 166.

Question: Find the H.C.F. of p(x) = x³ - 4x and q(x) = x² + 3x - 10

Solution:
Given that, p(x) = x³ - 4x and q(x) = x² + 3x - 10

Now,
The factors of p(x) = x³ - 4x
⇒ x(x² - 4)
⇒ x(x + 2)(x - 2)

And,
The factors of q(x) = x² + 3x - 10
⇒ x² + 5x - 2x - 10
⇒ x(x + 5) - 2(x + 5)
⇒ (x - 2)(x + 5)

∴ The required H.C.F. is (x - 2)

Question: A merchant has three different types of milk: 324 litres, 351 litres, and 459 litres. Find the minimum number of casks of equal size that can store the milk without mixing.

Solution:
The size of each cask should be the greatest possible that divides all three quantities, i.e., H.C.F of 324, 351, and 459.

H.C.F(324, 351, 459) = 27 litres

Total milk = 324 + 351 + 459 = 1134 litres

Number of casks required = 1134 ÷ 27 = 42

Question: A gardener planted trees in rows and columns such that the number of rows is five more than the number of columns. If the total number of rows and columns is 105, find the number of trees.

Solution:
Let the number of columns = x.
Then, number of rows = x + 5

According to the question: x + (x + 5) = 105
⇒ 2x + 5 = 105
⇒ 2x = 100
⇒ x = 50

Number of rows = x + 5 = 55

Total number of trees = rows × columns = 55 × 50 = 2750

Question: The traffic lights at three different road crossings change after every 24 sec, 36 sec, and 72 sec respectively. If they all change simultaneously at 8 : 20 : 00 hrs, then they will again change simultaneously at:

Solution:

Time interval for simultaneous change = L.C.M. of 24, 36, 72 = 72 sec

72 seconds = 1 min 12 sec

Next simultaneous change = 8 : 20 : 00 + 1 min 12 sec = 8 : 21 : 12 hrs

Question: If 7543P is divisible by 9, what is the value of P?

 Solution:
একটি সংখ্যা 9 দ্বারা বিভাজ্য হবে যদি সংখ্যাটির অঙ্কগুলোর সমষ্টি 9 দ্বারা বিভাজ্য হয়।

7 + 5 + 4 + 3 = 19 ; এর সাথে P যোগ করলে (19 + P) হবে, যা 9 দ্বারা বিভাজ্য হতে হবে।

19 + P = 27 (যা 9 দ্বারা বিভাজ্য নিকটতম সংখ্যা)
∴ P = 27 - 19 = 8
∴ P = 8

Question: A factory manufactures products in batches of 12, 18, and 30 units. What is the minimum number of units the factory needs to produce so that each batch can be formed exactly?

Solution:
To find the minimum number of units the factory needs to produce so that each batch size (12, 18, and 30) can be formed exactly, we need to find the least common multiple (LCM) of these batch sizes.

The prime factorization of each batch size is:
12 = 2 × 2 × 3 = 2² × 3¹

18 = 2 × 3 × 3 = 2¹ × 3²

30 = 2 × 3 × 5 = 2¹ × 3¹ × 5¹

Now,
The highest power of 2 is 2²
The highest power of 3 is 3²
The highest power of 5 is 5¹

So, the LCM of 12, 18, and 30 is:
= 2² × 3² × 5¹
= 4 × 9 × 5
= 180

∴ The minimum number of units the factory needs to produce is 180.

Question: A and B are two positive integers such that AB = 72. Which of the following cannot be the value of A + B?

Solution:
Factor pairs of 72:
(1, 72) → A + B = 73
(2, 36) → A + B = 38
(3, 24) → A + B = 27
(4, 18) → A + B = 22
(6, 12) → A + B = 18
(8, 9) → A + B = 17

So, possible values of A + B are: 73, 38, 27, 22, 18, 17.

Among the options, 20 is not possible.

Question: What is the least number which when doubled is exactly divisible by 8, 12, 16, and 20?

Solution:
Let the number be x.
Doubled the number is 2x.

8 = 2 × 2 × 2
12 = 2 × 2 × 3
16 = 2 × 2 × 2 × 2
20 = 2 × 2 × 5

∴ LCM = 2 × 2 × 2 × 2 × 3 × 5 = 240
∴ x = 240/2 = 120

Question: The square root of (3 + 2√5)(3 - 2√5) is:

Solution:
√{(3 + 2√5)(3 - 2√5)}
= √{3² - (2√5)²}
= √{9 - (4 × 5)}
= √{9 - 20}
= √(- 11)
= √{11(- 1)}
= √11 × √(- 1)
= i√11 [যেখানে i² = - 1]

Question: What is the H.C.F. of 4/9, 8/12, and 16/18?

Solution:
We know,
H.C.F. of fractions = (H.C.F. of numerators)/(L.C.M. of denominators)

H.C.F. of numerators:
H.C.F.(4, 8, 16) = 4

L.C.M. of denominators:
9 = 3²
12 = 2² × 3
18 = 2 × 3²
∴ L.C.M. = 2² × 3² = 4 × 9 = 36

∴ Required H.C.F. = 4/36 = 1/9

Question: A student multiplied 765 by a certain number and got 448835 as their answer. If in the answer both 8s are wrong, but the other digits are correct, then what will be the correct answer?

Solution:
The correct answer must be divisible by 765.

By checking possible corrections:

446435 ÷ 765 → remainder ≠ 0
445935 ÷ 765 → remainder ≠ 0
444635 ÷ 765 → remainder ≠ 0
442935 ÷ 765 → remainder = 0 → quotient = 579

Hence, the correct answer = 442935

Question: What is the least number which when divided by 3, 5, 6, 8, 10, and 12 leaves a remainder 2 in each case, but when divided by 22 leaves no remainder?

Solution:
The number, when divided by 3, 5, 6, 8, 10, and 12 leaves a remainder of 2. This means the number is 2 more than a multiple of their L.C.M.
LCM of 3, 5, 6, 8, 10, 12 = 120

The required number, N, must be in the form:
N = 120K + 2, where K is a positive integer.

N divisible by 22 ⇒ (120K + 2) ÷ 22 has remainder 0

120K + 2 = 22 × m ⇒ Check for smallest K

Try K = 2 ⇒ 120 × 2 + 2 = 242, divisible by 22

Hence, required number = 242

Question: The least number by which 150 must be multiplied to make it a perfect square is-

Solution:
একটি সংখ্যা পূর্ণবর্গ সংখ্যা হতে হলে তার মৌলিক গুণনীয়কগুলোকে অবশ্যই জোড় সংখ্যায় (even power) থাকতে হবে।

150 = 2 × 3 × 5 × 5 = 2¹ × 3¹ × 5²

জোড়া গঠন করে পাই, 2 × 3 × (5 × 5)
এখানে জোড়া বিহীন সংখ্যা 2 এবং 3

∴ 150 কে (2 × 3) = 6 দ্বারা গুণ করলে এটি পূর্ণবর্গ সংখ্যা হবে।

Question: The ratio of two numbers is 2 : 5 and their H.C.F is 6. Their L.C.M is -

Solution:
ধরি, সংখ্যা দুটি হলো 2x এবং 5x
∴ গসাগু (H.C.F) = x = 6

∴ সংখ্যা দুটি হলো: 2 × 6 = 12 এবং 5 × 6 = 30

∴ সংখ্যাদ্বয়ের গুণফল = 12 × 30 = 360
এবং H.C.F = 6

আমরা জানি,
L.C.M = (Product of two numbers)/H.C.F
= 360/6
= 60

∴ সংখ্যা দুটির লসাগু (L.C.M) = 60