ব্যাংক ডেইলি কুইজ [লং কোর্সের অংশ]

Question: If 2x + 3y = 24 and y = 2x, then find y - x.

Solution:
দেওয়া আছে, 2x + 3y = 24 এবং y = 2x
⇒ 2x + 3(2x) = 24
⇒ 2x + 6x = 24
⇒ 8x = 24
⇒ x = 3

∴ y = 2×3 = 6
তাহলে, y - x = 6 - 3 = 3

Question: If a - b = 3 and ab = 2, then the value of a³ - b³ - 3ab will be?
 
Solution:
We know,
a³ - b³ = (a - b)(a² + ab + b²)

So,
a³ - b³ - 3ab = (a - b)(a² + ab + b²) - 3ab
= 3 × (a² + b² + ab) - 3ab
= 3(a² + b²) + 3ab - 3ab
= 3(a² + b²)

Also, a² + b² = (a - b)²+ 2ab = 3² + 2×2 = 9 + 4 = 13

Therefore,
∴ a³- b³ - 3ab = 3 × 13 = 39

Question: If a - b = 2 and a² + b² = 20, then the value of a + b will be?

Solution:
We know, (a - b)² = a² - 2ab + b²
⇒ (a - b)² = a² + b² - 2ab ----------------(1)

Given,
a - b = 2
⇒ (a - b)² = 4
Also given, a² + b² = 20

Substitute the values in equation (1): 
20 - 2ab = 4
⇒ 2ab = 16
⇒ ab = 8

Now, (a + b)² = a² + b² + 2ab
= 20 + 16
= 36
∴ a + b = ±6

Question: If p + q + r = 10 and p² + q² + r² = 38, then what is the value of pq + qr + rp?

Solution:
We know the identity:
(p + q + r)² = p² + q² + r² + 2(pq + qr + rp)
⇒ 10² = 38 + 2(pq + qr + rp)
⇒ 100 = 38 + 2(pq + qr + rp)
⇒ 2(pq + qr + rp) = 100 - 38 = 62
⇒ pq + qr + rp = 62/2
∴ pq + qr + rp = 31

Question: If y + (3/y) = 5, what is the value of y³ + (27/y³)?

Solution:
দেওয়া আছে y + 3/y = 5

∴ y³ + 27/y³ = (y + 3/y)³ - 3 × (y + 3/y) × 3
= 5³ - 9 × 5
= 125 - 45
= 80

Question: If x² - 3x + 1 = 0, and x > 1, then what is the value of x - 1/x?

Solution:
Given, x² - 3x + 1 = 0
⇒ (x²/x) - (3x/x) + (1/x) = 0 [উভয়পক্ষকে x দ্বারা ভাগ করে]
⇒ x - 3 + 1/x = 0
⇒ x + 1/x = 3

এখন, (x - 1/x)² = (x + 1/x)² - 4 . x . 1/x
⇒ (x - 1/x)² = (3)² - 4
⇒ (x - 1/x)² = 9 - 4
⇒ (x - 1/x)² = 5
∴ x - 1/x = ±√5

যেহেতু x > 1 দেওয়া আছে, তাই x - 1/x > 0
সুতরাং, x - 1/x = √5

Question: What should be the value of "P" so that, the expression (9 - 12x + Px²) becomes a perfect square?

Solution:
Given Expression,
(9 - 12x + Px²)
= (3)² - 2 × 3 × 2x + (2x)² + Px² - (2x)²
= (3 - 2x)² + Px² - 4x²

∴ the expression becomes a perfect square if,
Px² - 4x² = 0
⇒ Px² = 4x²
∴ P = 4

Question: In a class of 50 students, 18 students like Biology, 20 students like Chemistry, and 22 students like Physics. It is found that 4 students like both Biology and Chemistry, 5 students like both Biology and Physics, and 6 students like both Chemistry and Physics. If 3 students like none of these subjects, find the number of students who like all three subjects.

Solution:
Let the number of students who like all three subjects = x.
Students liking at least one subject = Total students - Students liking none
= 50 - 3
= 47

According to question,
⇒ Biology + Chemistry + Physics - (Bio & Chem + Bio & Phys + Chem & Phys) + x = 47
⇒ 18 + 20 + 22 - (4 + 5 + 6) + x = 47
⇒ 60 - 15 + x = 47
⇒ 45 + x = 47
∴ x = 2

Question: If log₁₀4 + log₁₀(3x + 30)= log₁₀(2x + 8) + 1, then what is the value of x?

Solution:
Given equation:
⇒ log₁₀4 + log₁₀(3x + 30) = log₁₀(2x + 8) + 1
⇒ log₁₀4 + log₁₀(3x + 30) = log₁₀(2x + 8) + log₁₀10  [যেহেতু, log₁₀10 = 1]
⇒ log₁₀[4(3x + 30)] = log₁₀[10(2x + 8)]  [যেহেতু, logA + logB = log(AB)]
⇒ 4(3x + 30) = 10(2x + 8)
⇒ 12x + 120 = 20x + 80
⇒ 20x − 12x = 120 − 80
⇒ 8x = 40
⇒ x = 5

Question: If x = 3t, y = 1/2 × (t + 1), then the value of t for which x = 2y is?

Solution:
Here given,
x = 3t -------(1)
and, y = 1/2 × (t + 1) ----(2)

Now, when:
x = 2y
⇒ x = 2 × 1/2 × (t + 1) [from equation 2]
⇒ x = t + 1 -------(3)
∴ 3t = t + 1 [From equation (1)]
⇒ 2t = 1
⇒ t = 1/2

Question: Find the determinant of the matrix:
 

Solution:
2 × 2 ম্যাট্রিক্সের জন্য: 

⇒ det(A) = ad - bc

Given matrix,

∴ det(A) = 1 × 4 - 2 × 3 = 4 - 6 = - 2

Question:

Solution:

Question: Three times a whole number is equal to four less than the square of the number. Find the number.

Solution:
Let the number be x.
Then, 3x = x² - 4
⇒ x² - 3x - 4 = 0
⇒ x² - 4x + x - 4 = 0
⇒ x (x - 4) + 1 (x - 4) = 0
⇒ (x - 4) (x + 1) = 0
⇒ x = 4, - 1 
Solving, x = 4 is the only whole number solution.