পেট্রোবাংলা প্রিলি ও লিখিত সমন্বিত প্রস্তুতি [Archived]

Question: Solve the inequality (7x + 5)/(3x - 1) < 5

Solution:

The critical points are x = 1/3, 5/4
Plot these points on the number line,

The given inequality is positive, the solution is x ∈ (- ∞, 1/3) ∪ (5/4, ∞).

Question: The value of 5^1/4 × (125)^0.25 is -

Solution:
5^1/4 × (125)^0.25
= 5^1/4 × (5³)^1/4
= 5^{(1/4 + 3/4)}
= 5^(4/4)
= 5

Question: If 4P² - 5P - 6 = 0, where P is a natural number, find the value of P.

Solution:
4P² - 5P - 6 = 0
⇒ 4P² - 8P + 3P - 6 = 0
⇒ 4P(P - 2) + 3(P - 2) = 0
⇒ (4P + 3)(P - 2) = 0
⇒ P = (-3 / 4) or P = 2

Question: Find the value of log₉59049

Solution:
log₉59049
= log₉9⁵
= 5 × log₉9
= 5 × 1
= 5

Question: If 2^{x - 1} + 2^{x + 1} = 320, then x is equal to-

Solution:

∴ x - 6 = 1
⇒ x = 7

Question: Solve |x + 7| < 11

Solution:
We have |x + 7| < 11
⇒ - 11 < x + 7 < 11
⇒ - 11 - 7 < x + 7 - 7 < 11 - 7
⇒ - 18 < x < 4

Question: If 21168 = 2^a × 3^b × 7^c, where a, b and c are natural numbers, then what is the value of (4a - 5b + c)?

Solution:
21168 = 2^a × 3^b × 7^c
Factors of 21168 = 2⁴ × 3³ × 7²
∴ a = 4
⇒ b = 3
⇒ c = 2

According to the question:
(4a - 5b + c)
= {(4 × 4) - (5 × 3) + (2)}
= 16 - 15 + 2
= 3

Question: If a and b are roots of P² - 4P - 21 = 0, where a > b, find the equation whose roots are a/b and b/a.

Solution:
Given the equation as P² - 4P - 21 = 0.
As a and b are its roots,
∴ a + b = - {(- 4)/1} = 4
ab = - 21

Given new roots are (a/b) and (b/a)
Hence sum of roots = (a/b) + (b/a) = (a² + b²)/ab
= {(a + b)² - 2ab}/ab
= (16 + 42)/(- 21)
= - 58/21

So new equation = P² - (- 58/21)P + 1 = 0
⇒ 21P² + 58P + 21 = 0.

Question: If x, y and z are the sides of a right angled triangle, where ‘z’ is the hypotenuse, then find the value of 1/(log_{x + z}y) + 1/(log_{x - z}y).

Solution:
Here x, y and z are the sides of a right angled triangle, so z² = x² + y².

ঋণাত্মক সংখ্যার লগারিদম হয় না বিধায়, সঠিক উত্তর: ঙ) None of these

Question: If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, what is A ∩ B?

Solution:
A ∩ B = {1, 2, 3, 4} ∩ {3, 4, 5, 6}
= {3, 4}

Question: If (3/5)^x = 81/625, then what is the value of x^x?

Solution:
(3/5)^x = 81/625
We know,
3⁴ = 81 and
5⁴ = 625
∴ (3/5)⁴ = 81/625

∴ On comparing both the equation,
we get x = 4
Now, x^x = 4⁴ = 256

Question: Solve the inequality (3x - 8)/(x + 7) > 8

Solution:


The solution is x ∈ (- 64/5, - 7)

Question: If (13² - 5²)^3/2 = 6³ × A, then the value of A is-

Solution:
(13² - 5²)^3/2 = 6³ × A

Solving the given expression,
⇒ (169 - 25)^3/2 = 6³ × A
⇒ 144^3/2 = 6³ × A
⇒ 12³ = 6³ × A
⇒ A = (12/6)³
⇒ A = 2³
∴ The value of A is 2³

Question: If one root of the equation P² - 5P - 36 = 0 is same as P² - 25P + Q = 0, then find the value of Q.

Solution:
1st equation is P² - 5P - 36 = 0
⇒ (P - 9)(P + 4) = 0
⇒ P = 9 or P = -4

If P = 9 then P² - 25P + Q = 0
⇒ (9)² - 25(9) + Q = 0
⇒ Q = 144

If P = - 4 then P² - 25P + Q = 0
⇒ (- 4)² - 25 (- 4) + Q = 0
⇒ Q = - 116.

∴ Hence answer is 2nd option.

Question: Find the value of log₂2 + log₂2² + log₂2³ + ........ + log₂2ⁿ.

Solution:
log₂2 + log₂2² + log₂2³ + ........ + log₂2ⁿ
= log₂2 + 2log₂2 + 3log₂2 + ........ + nlog₂2
= 1 + 2 + 3 + ........ + n
= n(n+1)/2

Question: In a survey of 100 people, 60 people like coffee, 50 people like tea, and 20 people like both coffee and tea. How many people like only coffee?

Solution:
Total number of people surveyed = 100
People who like coffee = 60
People who like tea = 50
People who like both coffee and tea = 20
People who like only coffee can be calculated as follows:

People who like only coffee = People who like coffee - People who like both coffee and tea
= 60 - 20
= 40

Therefore, the number of people who like only coffee is 40.

Question: 2³ × 3⁴ × 1080 ÷ 15 = 6^x, find the value of x.

Solution:
2³ × 3⁴ × 1080 ÷ 15 = 6^x
⇒ 2³ × 3⁴ × 72 = 6^x
⇒ 2³ × 3⁴ × (2 × 6²) = 6^x
⇒ 2⁴ × 34 × 6² = 6^x
⇒ (2 × 3)⁴ × 6² = 6^x [∵ x^m × y^m = (xy)^m]
⇒ 6⁴ × 6² = 6^x
⇒ 6^{(4 + 2)} = 6^x
⇒ x = 6

Question: Solve |5x + 5| - 8 ≤ 17

Solution:
We have
|5x + 5| - 8 ≤ 17
⇒ |5x + 5| ≤ 25
⇒ - 25 < 5x + 5 ≤ 25
⇒ - 30 < 5x ≤ 20
⇒ - 6 ≤ x ≤ 4

Question: If m and n are whole numbers such that, mⁿ = 121, then the value of (m - 1)^{n + 1} is-

Solution:
If mⁿ = 121
∴ mⁿ = 11²

∴ n = 2; m = 11;
then
(m -1) ^{n + 1} = (11 - 1)^{2 +1} = 10³ = 1000

Question: Find the larger of the two positive numbers, such that sum of the numbers is 18 and difference of their squares is 9 times the larger number.

Solution:
Let
the numbers be x, y where x > y
∴ x + y = 18
⇒ y = 18 - x

ATQ,
x² - y² = 9x
⇒ x² - (18 - x)² = 9x
⇒ x² - 324 + 36x - x² = 9x
⇒ 36x - 9x = 324
⇒ 27x = 324
⇒ x = 324/27
∴ x = 12

Question: If ‘x’ is an integer then solve (log₂x)² - log₂x⁴ - 32 = 0.

Solution:
We have (log₂x)² - log₂x⁴ - 32 = 0.
⇒ (log₂x)² - 4log₂x - 32 = 0 ......(1)
Let log₂x = y
(i) ⇒ y² - 4y - 32 = 0
⇒ y² - 8y + 4y - 32 = 0
⇒ y (y - 8) + 4(y - 8) = 0
⇒ (y - 8) (y + 4) = 0
⇒ y = 8, - 4
⇒ log₂x = 8 or log₂x = - 4
⇒ x = 2⁸ = 256 or x = 2^{- 4} = 1/16
Since ‘x’ is an integer so x = 256.

Question: Observe the following diagram and answer the question. Find the number of students who play any two of the three sports.

Solution:
The shaded part represents the students who play any two of the three sports which is shown below:

Hence, the students who play any two of the three sports are 6 + 5 = 11.

Question: Solve the inequality (x + 4)².(x - 3) < 0

Solution:
We have (x + 4)².(x - 3) < 0
The critical points are x = - 4, 3

The solution is (- ∞, - 4) ∪ (- 4, 3)

Question: If a² + b² = 45 and ab = 18, find (1/a) + (1/b).

Solution:
দেওয়া আছে,
ab = 18
a² + b² = 45
⇒ (a + b)² - 2ab = 45
⇒ (a + b)² - 36 = 45
⇒ (a + b)² = 81
∴ a + b = 9

এখন,
(1/a) + (1/b)
= (b + a)/ab
= 9/18
= 1/2

Question: If log₃{log₂(x² - 4x - 37)} = 1, where ‘x’ is a natural number, find the value of x.

Solution:
We have log₃{log₂ (x² - 4x - 37)} = 1
⇒ log₂(x² - 4x - 37) = 3
⇒ x² - 4x - 37 = 8
⇒ x² - 4x - 45 = 0
⇒ (x - 9) (x + 5) = 0
⇒ x = 9, - 5
Since x is a natural number, so x = 9.

Question: If A = {2, 4, 6, 8} and B = {1, 2, 3, 4}, what is A \ B?

Solution:
A \ B = {2, 4, 6, 8}\{1, 2, 3, 4}
= {6, 8}

Question: Solve the inequality {1/(x + 3)} ≤ {1/(2x + 5)}

Solution:

The critical points are x = - 3, - 5/2, - 2

The solution is (- ∞, - 3) ∪ (- 5/2, - 2)