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Algebra

মোট প্রশ্ন১,৩৮০এই পাতা১০০প্রতি পাতা১০০
প্রশ্ন দেখুনঅনুশীলন
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উত্তরিতবর্তমানপুনরায় দেখুনঅসম্পূর্ণ

Algebra

PrepBank · পাতা ১ / ১৪ · ১–১০০ / ১,৩৮০

১.
The equation 12x2 + 4kx + 3 = 0 has real and equal roots, if-
  1. ক
    k = ± 2
  2. খ
    k = ± 3
  3. গ
    k = ± 4
  4. ঘ
    k = ± 6
ব্যাখ্যা
Question: The equation 12x2 + 4kx + 3 = 0 has real and equal roots, if-

Solution:
Here a = 12, b = 4k, c = 3
Since the given equation has real and equal roots
∴ b2 - 4ac = 0
⇒ (4k)2 - 4 × 12 × 3 = 0
⇒ 16k2 - 144 = 0
⇒ 16k2 = 144
⇒ k2 = 9
⇒ k = ± 3
২.
A vegetable cart sells a potato for $0.24 and a tomato for $0.76. Fred bought 12 vegetables in total, he only bought potatoes and tomatoes. If Fred paid $6.52 total, how many potatoes did he buy?
  1. ক
    ক) 4
  2. খ
    খ) 5
  3. গ
    গ) 8
  4. ঘ
    ঘ) 9
ব্যাখ্যা

Let, Fred bought x potatoes and (12 - x) tomatoes

ATQ, 
0.24x + 0.76(12 - x) = 6.52
Or, 0.24x + 9.12 - 0.76x = 6.52
Or, 0.52x = 2.6
Or, x = 5

৩.
If {1/|2q - 7|} > 1/5, then what is the value of q?
  1. ক
    - 1 < q < 6
  2. খ
    - 1 < q < 3
  3. গ
    1 < q < - 4
  4. ঘ
    1 < q < 6
ব্যাখ্যা

Question: If {1/|2q - 7|} > 1/5, then what is the value of q?

Solution:
Given that, 
{1/|2q - 7|} > 1/5
⇒ |2q - 7| < 5
⇒ - 5 < 2q - 7 < 5
⇒ - 5 + 7 < 2q - 7 + 7 < 5 + 7
⇒ 2 < 2q < 12
∴ 1 < q < 6

৪.
If a3 - 8b3 = - 2 and a = - 1, then b =?
  1. ক
    - 1/2
  2. খ
    2
  3. গ
    - 2/3
  4. ঘ
    1/2
ব্যাখ্যা
Question: If a3 - 8b3 = - 2 and a = - 1, then b =?

Solution:
a = - 1

a3 - 8b3 = - 2
⇒ (- 1)3 - 8b3 = - 2
⇒ - 1 - 8b3 = - 2
⇒ - 8b3 = - 2 + 1
⇒ - 8b3 = - 1
⇒ 8b3 = 1
⇒ b3 = 1/8
⇒ b3 = (1/2)3
∴ b = 1/2
৫.
If a2 - 8 = 2√15, Than what is the value of a. 
  1. ক
    √3 + √2
  2. খ
    √15
  3. গ
    √3 - √2
  4. ঘ
    √5 + √3
ব্যাখ্যা

Question: If a2 - 8 = 2√15, Than what is the value of a. 

Solution: 
Given that, 
a2 - 8 = 2√15
⇒ a2 = 8 + 2√15
⇒ a2 = 5 + 2 × √5 × √3 + 3
⇒ a2 = (√5)2 + + 2 × √5 × √3 + (√3)2
⇒ a2 = (√5 + √3)2 ; [(a + b)2 = a2 + 2ab + b2]
∴ a = √5 + √3

৬.
The range of f(x) = 1/(x + 1) is:
  1. ক
    R\{0}
  2. খ
    x > -1
  3. গ
    x < -1
  4. ঘ
    R\{- 1}
ব্যাখ্যা
Question: The range of f(x) = 1/(x + 1) is:

Solution:
দেওয়া আছে,
f(x) = 1/(x + 1)
⇒ y = 1/(x + 1)
⇒1/y = x + 1
⇒ x = (1/y) - 1
⇒ x = (1 - y)/y

∴ f-1(x) = y = (1 - x)/x
x এর মান 0 ব্যতীত যেকোনো বাস্তব সংখ্যা হবে। কারণ x এর মান 0 হলে ফাংশনটি অসঙ্গায়িত হবে।

অতএব, নির্ণেয় রেঞ্জ: R\{0}
৭.
The factors of the polynomial a3 - 6a2 + 12a - 9
  1. ক
    (a - 1) (a2 - 3a + 1)
  2. খ
    (a + 3) (a2 + 3a + 3)
  3. গ
    (a - 3) (a2 - 3a + 2)
  4. ঘ
    (a - 3) (a2 - 3a + 3)
ব্যাখ্যা
Question: The factors of the polynomial a3 - 6a2 + 12a - 9 are:

Solution:
a3 - 6a2 + 12a - 9
= a3 - 3 . a2 . 2 + 3 . a . 22 - 23 - 1
= (a - 2)3 - 13
= (a - 2 - 1) {(a - 2)2 + (a - 2) . 1 + 12}
= (a - 3) (a2 - 4a + 4 + a - 2 + 1)
= (a - 3) (a2 - 3a + 3)
৮.
If - 3 is 6 more than x, what is the value of x/3? 
  1. ক
    - 9
  2. খ
    - 6
  3. গ
    - 3
  4. ঘ
    1
ব্যাখ্যা
Question: If - 3 is 6 more than x, what is the value of x/3? 

Solution: 
x + 6 = - 3 
⇒ x = - 3 - 6
⇒ x = - 9 
⇒ x/3 = - 9/3
= - 3 
৯.
Solve |2x - 3| ≤ 1
  1. ক
    ক) - 1 ≤ x ≤ - 2
  2. খ
    খ) - 1 < x < 12
  3. গ
    গ) 1 ≤ x ≤ 2
  4. ঘ
    ঘ) 1 ≤ x ≤ - 2
ব্যাখ্যা
Question: Solve |2x - 3| ≤ 1

Solution:
|2x - 3| ≤ 1
বা, - 1 ≤ 2x - 3 ≤ 1
বা, - 1 + 3 ≤ 2x - 3 + 3 ≤ 1 + 3
বা, 2 ≤ 2x ≤ 4
বা, 2/2 ≤ 2x/2 ≤ 4/2
∴ 1 ≤ x ≤ 2
১০.
If x = 1 + √2 and y = 1 - √2, find the value of x2 + y2.
  1. ক
    ক) 6
  2. খ
    খ) 8
  3. গ
    গ) 10
  4. ঘ
    ঘ) 12
ব্যাখ্যা

Question: If x = 1 + √2 and y = 1 - √2, find the value of x2 + y2.

Solution: 
Given that,
x = 1 + √2,
y = 1 - √2

∴ x + y = 1 + √2 + 1 - √2
= 2

And,
xy = (1 + √2)(1 - √2)
= 12 - (√2)2
= 1 - 2
= - 1 

Now,
x2 + y2 = (x + y)2 - 2xy
= (2)2 - 2(- 1)
= 4 + 2
= 6

১১.
If x2 - 2x + 1 = 0 then the value of (x4 + 2x2 + 1)/x2 is -
  1. ক
    ক) 1
  2. খ
    খ) 2
  3. গ
    গ) 3
  4. ঘ
    ঘ) 4
ব্যাখ্যা
Question: If x2 - 2x + 1 = 0 then the value of (x4 + 2x2 + 1)/x2 is -

Solution:
Given, 
x2 - 2x + 1 = 0
⇒ (x - 1)2 = 0
⇒ x - 1 = 0
∴ x = 1

(x4 + 2x2 + 1)/x2 = (14 + 2 . 12 + 1)/12
= 4/1
= 4
১২.
Find the equation of the line with x-intercept = 6 and y-intercept = 5.
  1. ক
    6x + 5y - 30 = 0
  2. খ
    6x + 5y + 30 = 0
  3. গ
    5x - 6y - 30 = 0
  4. ঘ
    5x + 6y - 30 = 0
ব্যাখ্যা

Question: Find the equation of the line with x-intercept = 6 and y-intercept = 5.

Solution:
Given,
x-intercept = 6, So, the line passes through (6, 0).
y-intercept = 5, So, the line passes through (0, 5).

We know, The intercept form of a line is:
(x/a) + (y/b) = 1, where a = x-intercept, b = y-intercept.
⇒ (x/6) + (y/5) = 1
⇒ (5x + 6y)/30 = 1
⇒ 5x + 6y = 30
⇒ 5x + 6y - 30 = 0

∴ The equation of the line is 5x + 6y - 30 = 0

১৩.
If 5n + 4 = 11, what is the value of 10n - 2?
  1. ক
    68
  2. খ
    14
  3. গ
    12
  4. ঘ
    7
  5. ঙ
    - 1
ব্যাখ্যা
Question: If 5n + 4 = 11, what is the value of 10n - 2?

Solution:
5n + 4 = 11
⇒ 5n = 7
⇒ 5n × 2 = 7 × 2
∴ 10n = 14

∴ 10n - 2 = 14 - 2 = 12
১৪.
When x = (y + 3)2 , which of the following matches (- 2y - 6)2?
  1. ক
    4x
  2. খ
    2x
  3. গ
    3x
  4. ঘ
    - 2x
  5. ঙ
    - 4x
ব্যাখ্যা

Question: When x = (y + 3)2 , which of the following matches (- 2y - 6)2?

Solution:
Here,
x = (y + 3)2

∴ (- 2y - 6)2 ={- 2 (y + 3)}2
= 4 × (y + 3)2
= 4x

১৫.
If a - b = 3 and ab = 2, then the value of a3 - b3 - 3ab will be?
  1. ক
    20
  2. খ
    27
  3. গ
    29
  4. ঘ
    39
ব্যাখ্যা

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

১৬.
If px2 + 24x + 16 is a perfect square number, then p = ?
  1. ক
    6
  2. খ
    3
  3. গ
    4
  4. ঘ
    9
ব্যাখ্যা

Question: If px2 + 24x + 16 is a perfect square number, then p = ?

Solution:
দেওয়া আছে, রাশিটি একটি পূর্ণবর্গ সংখ্যা।
px2 + 24x + 16
= (3x)2 + 2(3x)(4) + (4)2

একটি রাশি পূর্ণবর্গ হওয়ার জন্য, এটি (a2 + 2ab + b2) অথবা (a2 - 2ab + b2) আকারের হতে হবে। এখানে,
a = 3x এবং b = 4

যেহেতু প্রথম পদটি a2 এর সমান, 
∴ px2 = a2
⇒ px2 = (3x)2
⇒ px2 = 9x2
⇒ p = 9

অতএব, p এর মান হলো 9।

১৭.
If 5xy + 28x - 16 = 0, and y = - 4 then what is the value of 2x + y ?
  1. ক
    0
  2. খ
    - 2
  3. গ
    2
  4. ঘ
    - 4
ব্যাখ্যা
Question: If 5xy + 28x - 16 = 0, and y = - 4 then what is the value of 2x + y ?

Solution: 
5xy + 28x - 16 = 0
or, - 20x + 28x - 16 = 0
or, 8x = 16
x = 2

2x + y = 4 - 4 = 0
১৮.
When the integer n is divided by 17, the quotient is x and the remainder is 5. When n is divided by 23, the quotient is y and the remainder is 14. Which of the following is true?
  1. ক
    23x + 17y = 19
  2. খ
    17x - 23y = 9
  3. গ
    17x + 23y = 19
  4. ঘ
    14x + 5y = 6
  5. ঙ
    5x - 14y = - 6
ব্যাখ্যা
Question: When the integer n is divided by 17, the quotient is x and the remainder is 5. When n is divided by 23, the quotient is y and the remainder is 14. Which of the following is true?

Solution:
From the problem it follows:
n = 17x + 5
n = 23y + 14

So,
17x + 5 = 23y + 14
⇒ 17x - 23y = 9
১৯.
What is the minimum value of 4x2 + 16x - 17?
  1. ক
    0
  2. খ
    - 29
  3. গ
    - 30
  4. ঘ
    - 33
  5. ঙ
    None
ব্যাখ্যা
Question: What is the minimum value of 4x2 + 16x - 17?

Solution:
Method 1:
Given function: 4x2 + 16x - 17
= 4(x2 + 4x) - 17
= 4(x2 + 4x + 4 - 4) - 17
= 4(x2 + 4x + 4) - 16 - 17
= 4(x + 2)2 - 33

Therefore,
4x2 + 16x - 17 = 4(x + 2)2 - 33
Since (x + 2)2 ≥ 0 for all real x 4(x + 2)2 ≥ 0
Therefore minimum value of 4(x + 2)2 - 33 is -33

Method 2:
The minimum of a quadratic function occurs at x = - b/2a
If a is positive, the minimum value of the function is f(- b/2a)
In the given function 4x2 + 16x - 17, a = 4, b = 16 and c = -17

So, f(- b/2a) or, f(-2) = 4(-2)2 + 16(-2) - 17
= 16 - 32 - 17
= - 33
২০.
The daily rate for a hotel room having 4 beds is Tk. 390 for one person and X taka for each additional person. If 3 people take the room for one day and each pays Tk. 210 for the room, then what is the value of X?
  1. ক
    ক) 60
  2. খ
    খ) 80
  3. গ
    গ) 240
  4. ঘ
    ঘ) 120
ব্যাখ্যা
Question: The daily rate for a hotel room having 4 beds is Tk. 390 for one person and X taka for each additional person. If 3 people take the room for one day and each pays Tk. 210 for the room, then what is the value of X?

Solution: 
৩ জনের প্রত্যেকে ২১০ টাকা দেয়। 
মোট = ২১০ × ৩ টাকা 
= ৬৩০ টাকা 

প্রশ্নমতে, 
৩৯০ + ২X = ৬৩০ 
⇒ ২X = ৬৩০ - ৩৯০ 
⇒ ২X = ২৪০
∴ X = ১২০ টাকা 
২১.
If n(U) = 100, n(A) = 40, n(B) = 35, and n(A ∩ B) = 15, then what is n(A ∪ B)′?
  1. ক
    56
  2. খ
    40
  3. গ
    45
  4. ঘ
    60
  5. ঙ
    30
ব্যাখ্যা
Question: If n(U) = 100, n(A) = 40, n(B) = 35, and n(A ∩ B) = 15, then what is n(A ∪ B)′?

Solution:
Given that,
n(U) = 100
n(A) = 40
n(B) = 35
and n(A ∩ B) = 15

We know,
n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
= 40 + 35 - 15
= 60
∴ n(A ∪ B) = 60

Now,
n(A ∪ B)′ = n(U) - n(A ∪ B)
= 100 - 60 = 40

∴ n(A ∪ B)′ = 40
২২.
If a + b + c = 13 and a2 + b2 + c2 = 69, then what is the value of ab + bc + ca?
  1. ক
    - 81
  2. খ
    50
  3. গ
    100
  4. ঘ
    - 80
ব্যাখ্যা

Question: If a + b + c = 13 and a2 + b2 + c2 = 69, then what is the value of ab + bc + ca?

Solution: 
We know,
(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
⇒ 2(ab + bc + ca) = (a + b + c)² - ( a² + b² + c²)
⇒ 2(ab + bc + ca) = 13² - 69 [given, a + b + c = 13 and a² + b² + c² = 69]
⇒ 2(ab + bc + ca) = 169 - 69 = 100
⇒ (ab + bc + ca) = 100/2
∴ (ab + bc + ca) = 50

২৩.
Find the smallest number by which 5808 should be multiplied so that the product becomes a perfect square.
  1. ক
    ক) 3
  2. খ
    খ) 7
  3. গ
    গ) 11
  4. ঘ
    ঘ) 2
ব্যাখ্যা

5808 = (4 × 4) × (11 × 11) × 3 = 42 × 112 × 3 
∴ 5808 should be multiplied by another 3 to make it a full square. 

২৪.
If b is one-fourth of a, then what is the value of  
  1. ক
    1/4
  2. খ
    1/3
  3. গ
    1
  4. ঘ
    2/3
  5. ঙ
    5/2
ব্যাখ্যা

Question: If b is one-fourth of a, then what is the value of  

Solution:
Given, b = 1/4 of a = a/4

Now,

২৫.
Solution set of the inequality,
2x + 3 > 5x - 6 is-
  1. ক
    (∞, 6)
  2. খ
    (- ∞, 3)
  3. গ
    [- ∞, - 3)
  4. ঘ
    (- ∞, - 3]
ব্যাখ্যা
Question: Solution set of the inequality,
2x + 3 > 5x - 6 is-

Solution:
2x + 3 > 5x - 6
⇒ 3 > 5x - 2x - 6
⇒ 3 + 6 > 3x
⇒ 9 > 3x
⇒ 3 > x
⇒ x < 3

∴ নির্ণেয় সমাধান সেট: (- ∞, 3)
২৬.
If p, q, r  are non-zero numbers and p + q = r, which of the following is equal to 1? 
  1. ক
    ক) (p - q)/r
  2. খ
    খ) (p - r)/q
  3. গ
    গ) (q - r)/p
  4. ঘ
    ঘ) (r - q)/p
ব্যাখ্যা
Question: If p, q, r  are non-zero numbers and p + q = r, which of the following is equal to 1? 

Solution: 
p + q = r
⇒ p = r - q

(r - q)/p
= p/p
= 1
২৭.
The average salary of all the workers in a workshop is Tk. 6000. The average salary of 7 technicians is Tk. 12000 and the average salary of the rest is Tk. 5000. How many workers are there?
  1. ক
    40
  2. খ
    45
  3. গ
    49
  4. ঘ
    55
ব্যাখ্যা
Question: The average salary of all the workers in a workshop is Tk. 6000. The average salary of 7 technicians is Tk. 12000 and the average salary of the rest is Tk. 5000. How many workers are there?

Solution:
let, there are x number of workers
 The average salary of all the workers in a workshop is Rs. 6000
∴ total salary = 6000 × x = 6000x

The average salary of 7 technicians is Tk. 12000
∴ salary of 7 technicians is = (12000 × 7) = 84000 tk

the average salary of the rest is Tk.  5000
∴ salary of the rest = 5000 × (x - 7) tk

 5000 × (x - 7) +  84000 = 6000x
⇒ 5000x - 35000 + 84000 = 6000x

⇒ 1000x = 49000
∴ x = 49

so, there are 49 workers.
২৮.
98.98 ÷ 11.03 + 7.014 × 15.99 = ?
  1. ক
    ক) 132
  2. খ
    খ) 144
  3. গ
    গ) 12
  4. ঘ
    ঘ) 121
ব্যাখ্যা

98.98 ÷ 11.03 + 7.014 × 15.99
= 121.128 ≈ 121 

Another approach:
98.98 ÷ 11.03 + 7.014 × 15.99
Let consider the equation as = 99 ÷ 11 + 7 × 16 = 9 + 112 = 121

২৯.
(2√27 - √75 + √12) is equal to -
  1. ক
    ক) 4√3
  2. খ
    খ) √3
  3. গ
    গ) 2√3
  4. ঘ
    ঘ) 3√3
ব্যাখ্যা

(2√27 – √75 + √12)
= 2√(32.3) - √(3 × 52) + √(3 × 22)
= 6√3 – 5√3 + 2√3
= 3√3

৩০.
If x2 + y2 = 50 and xy = 21, what is the value of (x - y)2?
  1. ক
    8
  2. খ
    12
  3. গ
    16
  4. ঘ
    25
ব্যাখ্যা

Question: If x2 + y2 = 50 and xy = 21, what is the value of (x - y)2?
 
Solution:
We are given:
x2 + y2 = 50
xy = 21

Use the identity:
(x - y)2 = x2 + y2 - 2xy

Substitute the values:
⇒ (x - y)2 = x2 + y2 - 2xy
⇒ (x - y)2 = 50 - 2 × 21
⇒ (x - y)2 = 50 - 42
∴ (x - y)2 = 8

৩১.
If a2 - 5a - 1 = 0; what is the value of a2 + (1/a2)?
  1. ক
    23
  2. খ
    25
  3. গ
    27
  4. ঘ
    29
ব্যাখ্যা
Question: If a2 - 5a - 1 = 0; what is the value of a2 + (1/a2)?

Solution: 
a2 - 5a - 1 = 0
⇒ a2 - 1 = 5a
⇒a - 1/a = 5
⇒ (a - 1/a)2 = (5)2
⇒ a2 + 1/a2 - 2 = 25
∴ a2 + 1/a2 = 27
৩২.
If x2 - √7x + 1 = 0, then the value of x2 + x- 2 = ?
  1. ক
    3
  2. খ
    5
  3. গ
    8
  4. ঘ
    3√2
ব্যাখ্যা

Question: If x2 - √7x + 1 = 0, then the value of x2 + x- 2 = ?

Solution:
দেয়া আছে,
x2 - √7x + 1 = 0
⇒ x2 + 1 = √7x
⇒ x + (1/x) = √7 [উভয়পক্ষকে x দ্বারা ভাগ করে]

প্রদত্ত রাশি = x2 + x- 2
= x2 + (1/x2)
= (x + 1/x)2 - 2 . x . (1/x)
= (√7)2 - 2
= 7 - 2
= 5

∴ নির্ণেয় মান হলো 5

৩৩.
  1. ক
    √5 + √3
  2. খ
    √5 + √2
  3. গ
    √5 - √3
  4. ঘ
    √5 - √2
ব্যাখ্যা
৩৪.
If x + 1 > 1 - 2x then -
  1. ক
    ক) x > 1/2
  2. খ
    খ) x > 0
  3. গ
    গ) x > 3
  4. ঘ
    ঘ) x < 0
ব্যাখ্যা
Question: If x + 1 > 1 - 2x then -

Solution:
x + 1 > 1 - 2x
⇒ x + 2x > 1 - 1
⇒ 3x > 0
∴ x > 0
৩৫.
If a + b = √11 and a - b = √5, what is the value of 8ab(a2 + b2)?
  1. ক
    96
  2. খ
    100
  3. গ
    56
  4. ঘ
    84
ব্যাখ্যা

Question: If a + b = √11 and a - b = √5, what is the value of 8ab(a2 + b2)?

Solution:
দেওয়া আছে, a + b = √11 এবং a - b = √5

আমরা জানি,
4ab = (a + b)2 - (a - b)2
2(a2 + b2) = (a + b)2 + (a - b)2

এখন,
8ab(a2 + b2) = (4ab) × 2(a2 + b2)
= [(a + b)2 - (a - b)2] × [(a + b)2 + (a - b)2]
= [(√11)2 - (√5)2] × [(√11)2 + (√5)2]
= (11 - 5) × (11 + 5)
= 6 × 16
= 96

৩৬.
The equation of the given curve is :
  1. ক
    ক) y = -x
  2. খ
    খ) y = 1/x
  3. গ
    গ) y = x
  4. ঘ
    ঘ) y = -1
ব্যাখ্যা

According to the figure x and y are in negative relation but as they are in (+ +) coordination, so they have built a positive correlation. Which is satisfied only by y = 1/x equation. And equation a, b and d are equation of straightline. 

৩৭.
If a * b = 2a - 3b + ab, then 3 * 5 + 5 * 3 is equal to?
  1. ক
    18
  2. খ
    20
  3. গ
    22
  4. ঘ
    24
ব্যাখ্যা
3 ∗ 5 + 5 ∗ 3
⇒ 3 ∗ 5 = 2 × 3 − 3 × 5 + 3 × 5 = 6 − 15 + 15 = 6
⇒ 5 ∗ 3 = 2 × 5 − 3 × 3 + 3 × 5 = 10 − 9 + 15 = 16
∴ 3 ∗ 5 + 5 ∗ 3 ⇒ 6 + 16 = 22
৩৮.
If P = {1, 4, 9, 16, 25, 36, 49, 64}, the number of proper subsets of P is
  1. ক
    215
  2. খ
    272
  3. গ
    265
  4. ঘ
    255
ব্যাখ্যা

Question: If P = {1, 4, 9, 16, 25, 36, 49, 64}, the number of proper subsets of P is
(Janata RC 2022 অনুযায়ী)

Solution:
দেওয়া আছে,
P = {1, 4, 9, 16, 25, 36, 49, 64}

সেটের উপাদান সংখ্যা = 8

∴ প্রকৃত উপসেট সংখ্যা = 2n - 1
= 28 - 1
= 256 - 1
= 255

৩৯.
If (x/y) + (y/x) = √7, what is the value of (x4/y4) + (y4/x4)?
  1. ক
    22
  2. খ
    23
  3. গ
    25
  4. ঘ
    27
ব্যাখ্যা

Question: If x/y + y/x = √7, what is the value of (x4/y4) + (y4/x4)?

Solution: 
Given that, 
x/y + y/x = √7

Now, 
(x4/y4) + (y4/x4)
= (x2/y2)2 + (y2/x2)2
= (x2/y2 + y2/x2)2 - 2 ; [a2 + b2 = (a + b)2 - 2ab]
= {(x/y)2 + (y/x)2}2 - 2
= {(x/y + y/x)2 - 2}2 - 2 ; [a2 + b2 = (a + b)2 - 2ab]
= {(√7)2 - 2}2 - 2
= (7 - 2)2 - 2
= 25 - 2
= 23

৪০.
  1. ক
    ক) 16
  2. খ
    খ) 12
  3. গ
    গ) 8
  4. ঘ
    ঘ) 4
ব্যাখ্যা

(4/5) × (√x/5) × (16/25) = 256/625
√x (64/625) = 256/625
√x × 64 = 256
√x = 256/64
√x = 4
(√x )2 = 42
x = 16
৪১.
If x + y = a, x2 + y2 = b2 and x3 + y3 = c3 then find the value of a3 + 2c3 = ?
  1. ক
    2ab2
  2. খ
    a3b2
  3. গ
    ab
  4. ঘ
    3ab2
ব্যাখ্যা

Question: If x + y = a, x2 + y2 = b2 and x3 + y3 = c3 then find the value of a3 + 2c3 = ? 

Solution: 
Given that, 
x + y = a .........(1)
x2 + y2 = b2 .........(2)
And x3 + y3 = c3

Now, 
a3 + 2c3
= (x + y)3 + 2(x3 + y3)
= x3 + 3x2y + 3xy2 + y3 + 2x3 + 2y3   ; [(a + b)3 = a3 + 3a2b + 3ab2 + b3]
= 3x3 + 3y3 + 3xy(x + y)
= 3(x3 + y3) + 3xy(x + y)
= 3{(x + y)(x2 - xy + y2)} + 3xy(x + y)  ; [a3 + b3 = (a + b)(a2 - ab + b2)]
= 3(x + y)(x2 - xy + y2 + xy)
= 3(x + y)(x2 + y2)
= 3ab2 ; [From 1 and 2]

৪২.
If m = 1 + √7​ and n = 1 - √7​, find the value of m3 + n3 = ?
  1. ক
    44
  2. খ
    - 28
  3. গ
    36
  4. ঘ
    - 6
  5. ঙ
    38
ব্যাখ্যা
Question: If m = 1 + √7​ and n = 1 - √7​, find the value of m3 + n3 = ?

Solution:
Given that,
m = 1 + √7
n = 1 - √7

Now,
m + n = 1 + √7 + 1 - √7 = 2
And, mn = (1 + √7)(1 - √7) = 12 - (√7)2 = 1 - 7 = - 6

We know that,
m3 + n3 = (m + n)3 - 3mn(m + n)
= 23 - 3 × (- 6) × 2
= 8 + 36 = 44
∴ m3 + n3 = 44
৪৩.
20 + 8 × 0.5/(20 - ?) = 12. Find the value in place of (?).
  1. ক
    ক) 2
  2. খ
    খ) 8
  3. গ
    গ) 18
  4. ঘ
    ঘ) 27
ব্যাখ্যা

Let the missing number be x.
Given,
20 + 8 × 0.5/(20 - x) = 12
⇒ (20 + 4)/(20 - x) = 12
⇒ 24/(20 - x) = 12
⇒ 20 - x = 24/12
⇒ 20 - x = 2
⇒ x = 20 - 2
⇒ x = 18.

৪৪.
The value of - 5 - (- 5) is how much greater than the value of (- 5 - 5)?
  1. ক
    0
  2. খ
    5
  3. গ
    8
  4. ঘ
    10
ব্যাখ্যা
Question: The value of - 5 - (- 5) is how much greater than the value of (- 5 - 5)?

Solution:
- 5 -(- 5)
= - 5 + 5
= 0

And,
- 5 - 5
= - 10 

Now,
0 - (- 10)
= 10

∴ The value of - 5 - (- 5) is 10 greater than the value of - 10 -(- 5)
৪৫.
If A = {a, b, c, d, e, f, g} and B = {d, e, f, g}, then A - B = ?
  1. ক
    {a, b, c}
  2. খ
    {a, b, c, d}
  3. গ
    {d, e, f}
  4. ঘ
    {d, e, f, g}
ব্যাখ্যা

Question: If A = {a, b, c, d, e, f, g} and B = {d, e, f, g}, then A - B = ?

Solution:
A - B = {a, b, c, d, e, f, g} - {d, e, f, g}
= {a, b, c}

৪৬.
If a > b > 1, then which of the following is true?
  1. ক
    ক) a² > b2
  2. খ
    খ) a2 < ab
  3. গ
    গ) b + a > 2a
  4. ঘ
    ঘ) a - b < 0
ব্যাখ্যা
Question: If a > b > 1, then which of the following is true?

Solution: 
let, a = 3, b = 2.
a2 = 9, b2 = 4; a2 > b2 is true
a2 = 9, ab = 6; a2 < ab is false
b + a = 5, 2a = 6; b + a > 2a is false
a - b = 1; a - b < 0 is false
৪৭.
What should be the value of "Q" so that the expression (25 - 30x + Qx2) becomes a perfect square?
  1. ক
    5
  2. খ
    9
  3. গ
    4
  4. ঘ
    16
ব্যাখ্যা

Question: What should be the value of "Q" so that the expression (25 - 30x + Qx2) becomes a perfect square?

Solution:
(25 - 30x + Qx2)
= (5)2 - 2 × 5 × 3x + (3x)2 + Qx2 - (3x)2
= (5 - 3x)2 + Qx2 - 9x2

∴ The expression becomes a perfect square if,
Qx2 - 9x2 = 0
⇒ Qx2 = 9x2
∴ Q = 9

৪৮.
If y = √8 + √7, then what is the value of y3 + (1/y3)?
  1. ক
    34√2
  2. খ
    64√3
  3. গ
    81√2
  4. ঘ
    116√2
ব্যাখ্যা

Question: If y = √8 + √7, then what is the value of y3 + (1/y3)?

Solution:
দেওয়া আছে,
y = √8 + √7
⇒ 1/y = 1/(√8 + √7)
⇒ 1/y = (√8 - √7)/(√8 + √7)(√8 - √7)
⇒ 1/y = (√8 - √7)/{(√8)2 - (√7)2}
⇒ 1/y = (√8 - √7)/(8 - 7)
∴ 1/y = √8 - √7

এখন, y + 1/y = (√8 + √7) + (√8 - √7)
= 2√8 = 2 × 2√2 = 4√2

এখন,
y3 + (1/y3)
= (y + 1/y)3 - 3(y)(1/y)(y + 1/y)
= (y + 1/y)3 - 3(y + 1/y)
= (4√2)3 - 3(4√2)
= (43)(√2)3 - 12√2
= 64(2√2) - 12√2
= 128√2 - 12√2
= 116√2

সুতরাং, নির্ণেয় মান হলো 116√2

৪৯.
If x + 1/x = 5 , then x3 + 1/x3 = ?
  1. ক
    90
  2. খ
    105
  3. গ
    110
  4. ঘ
    140
ব্যাখ্যা
Question: If x + 1/x = 5 , then x3 + 1/x3 = ?

Solution:
x3 + 1/x3
=(x + 1/x)3 - 3. x. (1/x)(x + 1/x)
= 53 - (3 × 5)
= 125 - 15
= 110
৫০.
Solve the inequality (x + 4)2.(x - 3) < 0
  1. ক
    (3, ∞)
  2. খ
    (- 4, 3)
  3. গ
    (- ∞, - 4) ∪ (- 4, 3)
  4. ঘ
    (- ∞, 3)
  5. ঙ
    None
ব্যাখ্যা
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)
৫১.
  1. ক
    ক) 0
  2. খ
    খ) 1
  3. গ
    গ) 2
  4. ঘ
    ঘ) 3
ব্যাখ্যা
= x/(x - 1) + 1/(x +1) - 2x/(x2 - 1)
= {x(x + 1) + 1(x - 1) - 2x}/(x2 - 1)
= (x2 + x + x - 1 - 2x)/(x2 - 1)
=(x2 - 1)/(x2 - 1)
= 1
৫২.
In a group of 100 people, 45 play football, 30 play cricket, and 20 play both. How many play neither football nor cricket?
  1. ক
    35
  2. খ
    45
  3. গ
    50
  4. ঘ
    55
ব্যাখ্যা
Question: In a group of 100 people, 45 play football, 30 play cricket, and 20 play both. How many play neither football nor cricket?

Solution:
উভয় খেলা খেলে ২০ জন
শুধুমাত্র ফুটবল খেলে ৪৫ - ২০ = ২৫ জন
শুধুমাত্র ক্রিকেট খেলে ৩০ - ২০ = ১০ জন

∴ কমপক্ষে একটি খেলা খেলে = ২০ + ২৫ + ১০ = ৫৫ জন

∴ ফুটবল বা ক্রিকেট কোনটিই খেলে না = ১০০ - ৫৫ = ৪৫ জন
৫৩.
If 2 m, 60 cm cloths is required for one shirt, then the cloth required for 7 shirts is?
  1. ক
    ক) 14 m 80 cm
  2. খ
    খ) 18 m 20 cm
  3. গ
    গ) 15 m 20 cm
  4. ঘ
    ঘ) 16 m 80 cm
  5. ঙ
    ঙ) 13 m 60 cm
ব্যাখ্যা

Cloth is required for 1 shirt
= 2 m, 60 cm or 260 cm
Cloth is required for 7 shirt
= 260 × 7
= 1820 cm or 18 m 20 cm

৫৪.
What is the solution of  - 19 < 3x + 2 ≤ 17?
  1. ক
    (- 7, 5)
  2. খ
    [- 7, 5]
  3. গ
    (- 7, 5]
  4. ঘ
    (- 7, 7]
ব্যাখ্যা
Question: What is the solution of  - 19 < 3x + 2 ≤ 17?

Solution: 
- 19 < 3x + 2 ≤ 17
- 19 - 2 < 3x + 2 - 2 ≤ 17 - 2
- 21 < 3x ≤ 15
- 7 < x ≤ 5

∴ x ∈ (- 7, 5]
৫৫.
If (4P - 4)/(3P - 3) = (4/P) where P ≠ 1, what is the value of P2 - 4P + 3 = ?
  1. ক
    0
  2. খ
    1
  3. গ
    2
  4. ঘ
    3
ব্যাখ্যা

Question: If (4P - 4)/(3P - 3) = (4/P) where P ≠ 1, what is the value of P2 - 4P + 3 = ?

Solution:
Given that,
(4P - 4)/(3P - 3) = 4/P
or, P(4P - 4) = 4(3P - 3)
or, 4P2- 4P = 12P - 12
or, 4P2- 4P - 12P + 12 = 0
or, 4P2- 16P + 12 = 0
or, 4(P2 - 4P + 3) = 0
or, P2 - 4P + 3 = 0/4
∴ P2 - 4P + 3 = 0

৫৬.
Which of the following is equivalent to a - b ≥ a + b
  1. ক
    ক) a ≤ b
  2. খ
    খ) a ≤ 0
  3. গ
    গ) b ≤ a
  4. ঘ
    ঘ) b ≤ 0
ব্যাখ্যা

a - b ≥ a + b
⇒ a - b - b ≥ a
⇒ -2b ≥ 0
⇒ 2b ≤ 0 [-1 দ্বারা গুণ করে] 
∴ b ≤ 0

৫৭.
Solve the inequality |x - 2| < 5
  1. ক
    3 < x < 7
  2. খ
    3 < x < - 7
  3. গ
    - 3 < x < 7
  4. ঘ
    - 3 < x < - 7
ব্যাখ্যা

Question: Solve the inequality |x - 2| < 5

Solution:
|x - 2| < 5
⇒ - 5 < x - 2 < 5
⇒ - 5 + 2 < x - 2 + 2 < 5 + 2
⇒ - 3 < x < 7

৫৮.
P = {x ∈ N : 2 < x ≤ 6} and Q = {x ∈ N : x is an even number and x ≤ 8}. Find the value of P ∩ Q.
  1. ক
    {3, 5}
  2. খ
    {4, 6}
  3. গ
    {3, 4, 5, 6}
  4. ঘ
    {2, 8}
ব্যাখ্যা

Question: P = {x ∈ N : 2 < x ≤ 6} and Q = {x ∈ N : x is an even number and x ≤ 8}. Find the value of P ∩ Q.

Solution:
Given that, 
P = {x ∈ N : 2 < x ≤ 6}
∴ P = {3, 4, 5, 6}

Q = {x ∈ N : x is even and x ≤ 8}
∴ Q = {2, 4, 6, 8}

Now,
P ∩ Q = {3, 4, 5, 6} ∩ {2, 4, 6, 8}
= {4, 6}

৫৯.
(x + y)2 = ?
  1. ক
    x2 + y2 - 2xy
  2. খ
    x2 - y2 + 2xy
  3. গ
    (x + y)2 - 4xy
  4. ঘ
    (x - y)2 + 4xy
ব্যাখ্যা
Question: (x + y)2 = ?

Solution: 
(x + y)2 = x2 + y2 + 2xy
(x + y)2 = (x - y)2 + 4xy
৬০.
Find an equation for the line with x-intercept = 2, Y - intercept = - 1
  1. ক
    2x - y = 1
  2. খ
    x - 2y = 2
  3. গ
    y = -1
  4. ঘ
    x = 2
ব্যাখ্যা

Question: Find an equation for the line with x-intercept = 2, Y - intercept = - 1

Solution:
দেওয়া আছে, 
x-অক্ষের ছেদবিন্দু 2   ∴ বিন্দু (2, 0)
y-অক্ষের ছেদবিন্দু - 1  ∴ বিন্দু (0, - 1)

এখন, (2, 0) এবং (0, - 1) বিন্দুর ঢাল, 
m = (y2 - y1)/(x2 - x1)
⇒ m = (- 1 - 0)/(0 - 2) = 1/2
∴ m = 1/2

এখন, রেখার সমীকরণ
y = mx + b, [যেখানে, ঢাল m এবং y-অক্ষের ছেদবিন্দু, b ].
⇒ y = (1/2)x - 1  ; [m = 1/2, y-অক্ষের ছেদবিন্দু, b = - 1]
⇒ 2y = x - 2
∴ x - 2y = 2

৬১.
Which of the following describes all values of x for which 1 - x2 ≥ 0?
  1. ক
    x ≤ - 2
  2. খ
    x ≤ - 2 or x ≥ 1
  3. গ
    - 1 ≤ x ≤ 1
  4. ঘ
    x ≥ 1
ব্যাখ্যা
Question: Which of the following describes all values of x for which 1 - x2 ≥ 0?

Solution: 
1 - x2 ≥ 0
⇒ 1 ≥  x2
⇒ x2 ≤ 1
⇒ √(x2) ≤ √1
⇒ |x| ≤ 1

We must consider that x can be either positive or negative because the variable x is inside the absolute value sign. Therefore, we’ll need to solve the inequality twice.

When x is positive:
|x| ≤ 1
= x ≤ 1

When x is negative:
|x| ≤ 1
= - x ≤ 1
= x ≥ - 1

We combine the two resulting inequalities to get:
-1 ≤ x ≤ 1
৬২.
If y + (3/y) = 5, what is the value of y3 + (27/y3)?
  1. ক
    70
  2. খ
    80
  3. গ
    100
  4. ঘ
    90
ব্যাখ্যা

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

৬৩.
If (a/b) + (b/a) = 3, then (a/b)2 + (b/a)2 =?
  1. ক
    9
  2. খ
    6
  3. গ
    3
  4. ঘ
    7
ব্যাখ্যা
প্রশ্ন: If (a/b) + (b/a) = 3, then (a/b)2 + (b/a)2 =? 

সমাধান: 
দেয়া আছে,
(a/b) + (b/a) = 3 

(a/b)2 + (b/a)2 = {(a/b) + (b/a)}2 - 2(a/b). (b/a) 
= 32 - 2
= 9 - 2
= 7
৬৪.
If ‍a2 + b2 = 63 and ab = 9 then = ?
  1. ক
    ক) 1/2
  2. খ
    খ) 2
  3. গ
    গ) 1
  4. ঘ
    ঘ) 1/3
ব্যাখ্যা
Question: If ‍a2 + b2 = 63 and ab = 9 then = ?

Solution:
দেওয়া আছে,
a2 + b2 = 63
ab = 9

এখন,
(1/a) + (1/b)
= (b + a)/ab
= √(a + b)2/ab
= √(a2 + b2 + 2ab)/ab
= √{63 + (2 × 9)}/9
= √81/9
= 9/9
= 1
৬৫.
What is the unit digit in the product 84 × 59 × 13 × 77?
  1. ক
    ক) 6
  2. খ
    খ) 9
  3. গ
    গ) 8
  4. ঘ
    ঘ) 7
ব্যাখ্যা
প্রশ্ন : What is the unit digit in the product 84 × 59 × 13 × 77?
সমাধান: 84 × 59 × 13 × 77 = 4960956
Without the use of calculator, to count the unit digit = 4 × 9 × 3 × 7 = 756

So, 6 is the unit digit
৬৬.
Find the value of the expression a2 - 2ab + b2 for a = 1, b = 1.
  1. ক
    1
  2. খ
    - 1
  3. গ
    2
  4. ঘ
    0
ব্যাখ্যা
Question: Find the value of the expression a2 - 2ab + b2 for a = 1, b = 1.

Solution:
a2 - 2ab + b2
= (a - b)2
= (1 - 1)2
= (0)2
= 0
৬৭.
If p + 3 + 1/p = 0 then what is the value of p2 + 1/p2 = ?
  1. ক
    11
  2. খ
    7
  3. গ
    8
  4. ঘ
    9
ব্যাখ্যা
Question: If p + 3 + 1/p = 0 then what is the value of p2 + 1/p2 = ?

Solution:
Given,
p + 3 + 1/p = 0
⇒ p + 1/p = - 3
⇒ (p + 1/p)2 = (- 3)2 [
⇒ p2 + 2. p. 1/p + 1/p2 = 9
⇒ p2 + 1/p2 = 9 - 2
∴ p2 + 1/p2 = 7
৬৮.
A farm rears geese and dogs. The headcount in the farm is 84 and the leg count is 282. How many geese are there?
  1. ক
    27
  2. খ
    30
  3. গ
    54
  4. ঘ
    57
ব্যাখ্যা

Let geese be denoted by 'G' and Dogs by 'D'
Geese have 2 legs; Dogs have 4 legs.

Total Heads = G + D = 84 ------------------------- (1)
Total Legs = 2G + 4D = 282 --------------------- (2)

Divide equation 2 by 2, we get,
G + 2D = 141 -------------------------------------- (3)
Equation 3 - Equation 2
G + 2D - G - D = 141 - 84
∴ D = 57

So, Geese = 84 - 57 = 27

৬৯.
If x2 + y2 = 40 and xy = 12, what is the value of (x - y)2?
  1. ক
    12
  2. খ
    16
  3. গ
    10
  4. ঘ
    15
ব্যাখ্যা

Question: If x2 + y2 = 40 and xy = 12, what is the value of (x - y)2?
 
Solution:
We are given:
x2 + y2 = 40
xy = 12

Use the identity:
(x - y)2 = x2 + y2 - 2xy

Substitute the values:
⇒ (x - y)2 = x2 + y2 - 2xy
⇒ (x - y)2 = 40 - 2 × 12
⇒ (x - y)2 = 40 - 24
∴ (x - y)2 = 16

৭০.
If two jeans and three shirts cost Tk. 4000 and three jeans and two shirts cost Tk. 3500, then how much does a jeans-
  1. ক
    400 taka
  2. খ
    500 taka
  3. গ
    800 taka
  4. ঘ
    1000 taka
ব্যাখ্যা
Question: If two jeans and three shirts cost Tk. 4000 and three jeans and two shirts cost Tk. 3500, then how much does a jeans

Solution: 
মনেকরি 
১টি  jeans এর দাম = x টাকা 
১টি shirt এর দাম = y টাকা 

এখানে 
2x + 3y = 4000.......................(1)
3x + 2y = 3500.........................(2)

(1) × 2 - (2) × 3 ⇒
4x + 6y - 9x - 6y = 8000 - 10500
- 5x = - 2500
x = 500

১টি  jeans এর দাম = 500 টাকা  
৭১.
Solve the inequality |1 - 2x| < 7
  1. ক
    3 < x < 2
  2. খ
    - 3 < x < 4
  3. গ
    4 < x < - 3
  4. ঘ
    - 3 < x < 3
ব্যাখ্যা

Question: Solve the inequality |1 - 2x| < 7

Solution:
Given that, 
|1 - 2x| < 7
⇒ - 7 < 1 - 2x < 7
⇒ - 7 - 1 < 1 - 1 - 2x < 7 - 1
⇒ - 8 < - 2x < 6
⇒ - 4 < - x < 3    (dividing by - 2 and reversing the inequality signs)
⇒ 4 > x > - 3
∴ - 3 < x < 4

৭২.
A natural number when increased by 12, equals 160 times its reciprocal. The number is-
  1. ক
    20
  2. খ
    15
  3. গ
    12
  4. ঘ
    8
ব্যাখ্যা
Question: A natural number when increased by 12, equals 160 times its reciprocal. The number is-

Solution: 
Let the number be x. Then,
x + 12 = 160 × (1/x)
⇒ x2 + 12x - 160 = 0
⇒ x2 + 20x - 8x - 160 = 0
⇒ x(x + 20) - 8(x + 20) = 0
⇒ (x + 20)(x - 8) = 0
⇒ x= - 20, 8

 Therefore, the required number is 8.
৭৩.
{(0.6)4 - (0.5)4}/{(0.6)2 + (0.5)2} is equal to-
  1. ক
    ক) 0.11
  2. খ
    খ) 0.011
  3. গ
    গ) 1.11
  4. ঘ
    ঘ) 0.001
ব্যাখ্যা
{(0.6)4 - (0.5)4}/{(0.6)2 + (0.5)2}
= [{(0.6)2}2 - {(0.5)2}2]/{(0.6)2 + (0.5)2}
= {(0.6)2 + (0.5)2}{(0.6)2 - (0.5)2}/{(0.6)2 + (0.5)2}
= {(0.6)2 - (0.5)2}
= (0.6 + 0.5)(0.6 - 0.5)
= 1.1 × 0.1
= 0.11
৭৪.
What are the solutions to the equation x2 - 2x - 2 = 0
  1. ক
    ক) (1 + √2), (1 - √2)
  2. খ
    খ) (1 + √3), (1 - √3)  
  3. গ
    গ) (1 + √5), (1 - √5)
  4. ঘ
    ঘ) (1 + √7), (1 - √7)
ব্যাখ্যা
Question: What are the solutions to the equation x2 - 2x - 2 = 0

Solution: 
Given that 
x2 - 2x - 2 = 0.........(1)
Comparing ax2 + bx + c = 0 with (1) get, a = 1, b = - 2 and c = - 2

We know
x = {(- b) ± √(b2 - 4ac)}/2a
   = [{- (- 2)} ± √{(- 2)2 - 4.1(- 2)]/2.1
    = (2 ± √12)/2
    =(2 ± 2√3)/2
    = 1 ± √3
    = (1 + √3), (1 - √3)
৭৫.
  1. ক
    9
  2. খ
    5/4
  3. গ
    10/9
  4. ঘ
    7/8
ব্যাখ্যা
Question: 

Solution:
৭৬.
If (2a - 2b) = 2 and (3a + 2b) = 8 then find the value of (3a + 4).
  1. ক
    5
  2. খ
    6
  3. গ
    8
  4. ঘ
    10
ব্যাখ্যা
Question: If (2a - 2b) = 2 and (3a + 2b) = 8 then find the value of (3a + 4).

Solution:
2a - 2b = 2 ...........(1)
3a + 2b = 8 .............(2)

From (1) we get,
2a - 2b = 2
⇒ a - b = 1
⇒ a = b + 1 .......(3)

From (2) we get,
3(b + 1) + 2b = 8
⇒ 3b + 3 + 2b = 8
 ⇒ 5b = 5 
∴ b = 1
From (3) we get,
a = 1 + 1
∴ a = 2

Now,
3a + 4
= (3 × 2) + 4
= 6 + 4
= 10
৭৭.
    ব্যাখ্যা
    Question:
     
    Solution:
    ৭৮.
    If (1/x) + (1/y) = 1/3, then xy/(x + y) = ?
    1. ক
      3
    2. খ
      1
    3. গ
      1/3
    4. ঘ
      1/5
    ব্যাখ্যা
    Question: If (1/x) + (1/y) = 1/3, then xy/(x + y) = ?
     
    Solution:
    (1/x) + (1/y) = 1/3
    ⇒ (y + x)/xy = 1/3
    ∴ xy/(x + y) = 3
    ৭৯.
    In a class of 78 students, 41 are taking French, 22 are taking German and 9 are taking both courses. How many students are not enrolled in either course?
    1. ক
      6
    2. খ
      15
    3. গ
      24
    4. ঘ
      33
    ব্যাখ্যা
    Question: In a class of 78 students, 41 are taking French, 22 are taking German and 9 are taking both courses. How many students are not enrolled in either course?

    Solution:
    Total students = 78
    Students taking French n(F) = 41
    Students taking German n(G) = 22
    Students taking both French and German = 9

    We know,
    n(F ∪ G) = n(F) + n(G) - n(F ∩ G)
    n(F ∪ G) = 41 + 22 - 9 = 54

    ∴ Not enrolled = Total students - n(F ∪ G) = 78 - 54 = 24
    ৮০.
    Find the first term of an AP whose 8th and 12th terms are respectively 39 and 59.
    1. ক
      ক) 5
    2. খ
      খ) 6
    3. গ
      গ) 4
    4. ঘ
      ঘ) 3
    ব্যাখ্যা
    প্রশ্ন: Find the first term of an AP whose 8th and 12th terms are respectively 39 and 59.

    সমাধান: 
    Let,
    First term = a
    Common difference = d

    8th term = a + 7d = 39 ........... (1)
    12th term = a + 11d = 59 ........... (2)

    By (2) - (1) we  get,
     a + 11d - a - 7d = 59 - 39
    ⇒ 4d = 20
    ∴ d = 5

    Hence,
    a + 7 × 5 = 39
    ⇒ a = 39 - 35
    ∴ a = 4
    ৮১.
    What is the solution of the inequality ।5x - 3। < 4 ?
    1. ক
      ক) - 1/5 < x < 7/5
    2. খ
      খ) - 7/5 < x < 1/5
    3. গ
      গ) - 3/5 < x < 1/5
    4. ঘ
      ঘ) - 7/5 < x < 7/5
    ব্যাখ্যা
    Question: What is the solution of the inequality ।5x - 3। < 4 ?

    Solution: 
    ।5x - 3। < 4
    - 4 < 5x - 3 < 4
     - 4  + 3 < 5x - 3 + 3 < 4 + 3
    - 1 < 5x < 7
    - 1/5 < 5x/5 < 7/5
    - 1/5 < x < 7/5
    ৮২.
    1. ক
      ক) 14
    2. খ
      খ) 12
    3. গ
      গ) 144
    4. ঘ
      ঘ) 196
    ব্যাখ্যা
    প্রশ্ন:

    সমাধান: 
    ৮৩.
    Solve the inequality, (x/4) < (4x - 1)/15 
    1. ক
      x < - 2
    2. খ
      x > 4 
    3. গ
      x < - 4
    4. ঘ
      x < 2
    ব্যাখ্যা

    Question: Solve the inequality, (x/4) < (4x - 1)/15

    Solution: 
    Given that, 
    x/4 < (4x - 1)/15
    ⇒ 15x < 16x - 4
    ⇒ 15x - 16x < - 4
    ⇒ - x < - 4
    ⇒ x > 4 

    ৮৪.
    If a/b = 2/3 and b/c = 4/5 what is the value of (a + b)/(b + c)?
    1. ক
      ক) 4/9
    2. খ
      খ) 20/27
    3. গ
      গ) 5/9
    4. ঘ
      ঘ) 10/13
    ব্যাখ্যা
    Question: If a/b = 2/3 and b/c = 4/5 what is the value of (a + b)/(b + c)?

    Solution:
    a/b = 2/3
    b/c = 4/5
    ৮৫.
    If a = 1 + √2 and b = 1 - √2, find the value of a2 + b2.
    1. ক
      2
    2. খ
      4
    3. গ
      6
    4. ঘ
      8
    ব্যাখ্যা
    Question: If a = 1 + √2 and b = 1 - √2, find the value of a2 + b2.

    Solution: 
    Given that,
    a = 1 + √2,
    b = 1 - √2

    ∴ a + b = 1 + √2 + 1 - √2
    = 2

    And,
    ab = (1 + √2)(1 - √2)
    = 12 - (√2)2
    = 1 - 2
    = - 1 

    Now,
    a2 + b2 = (a + b)2 - 2ab
    = (2)2 - 2(- 1)
    = 4 + 2
    = 6
    ৮৬.
    Find the equation of the line with x-intercept = 4 and y-intercept = 3.
    1. ক
      4x - 3y - 12 = 0
    2. খ
      4x + 3y - 12 = 0
    3. গ
      3x + 4y - 12 = 0
    4. ঘ
      3x - 4y + 12 = 0 
    ব্যাখ্যা

    Question: Find the equation of the line with x-intercept = 4 and y-intercept = 3.

    Solution:
    Given, x-intercept = 4,
    So, the line passes through (4, 0).
    y-intercept = 3,
    So, the line passes through (0, 3).

    We know,
    The intercept form of a line is:
    (x/a) + (y/b) = 1, where a = x-intercept, b = y-intercept.
    ⇒ (x/4) + (y/3) = 1
    ⇒ (3x + 4y)/12 = 1
    ⇒ 3x + 4y = 12
    ⇒ 3x + 4y - 12 = 0

    ∴ The equation of the line is 3x + 4y - 12 = 0

    ৮৭.
    What is the 12th term of the sequence - 2, - 4, - 6, ...... , - 100?
    1. ক
      ক) - 28
    2. খ
      খ) - 26
    3. গ
      গ) - 24
    4. ঘ
      ঘ) - 20
    ব্যাখ্যা
    Question: What is the 12th term of the sequence - 2, - 4, - 6, ...... , - 100?

    Solution:
    Here,
    - 4 - (- 2) = - 4 + 2 = - 2
    - 6 - (- 4) = - 6 + 4 = - 2
    ∴ d = - 2
    a = - 2
    n = 12

    ∴ The 12th term of the sequence = a + (n - 1)d
    = - 2 + (12 - 1)(- 2)
    = - 2 + 11(- 2)
    = - 2 - 22
    = - 24 
    ৮৮.
    If (x + y) = 9 and (x - y) = 7, what is the value of xy? 
    1. ক
      ক) 7
    2. খ
      খ) 8
    3. গ
      গ) 9
    4. ঘ
      ঘ) 10
    ব্যাখ্যা
    দেয়া আছে,
    x + y = 9
    x - y = 7 

    আমরা জানি,
    4xy = (x + y )2 - (x - y)2
    4xy = 92 - 72 
    4xy = 81 - 49 
    4xy = 32 
    xy = 8
    ৮৯.
    1. ক
      0
    2. খ
      12
    3. গ
      112
    4. ঘ
      None of these
    ব্যাখ্যা
    Question:

    Solution:
    Here,
    Mean  = 20
    Number of observation = 10
    ৯০.
    If 4x - y = 6 and 2x + 3y = 10 find the value of 2x + 5y.
    1. ক
      14
    2. খ
      12
    3. গ
      10
    4. ঘ
      8
    ব্যাখ্যা
    Question: If 4x - y = 6 and 2x + 3y = 10 find the value of 2x + 5y.

    Solution:
    4x - y = 6 ------------ (1)
    2x + 3y = 10 ---------- (2)

    Multiplying equ (1) by 2
    8x - 2y = 12 --------- (3)

    (2) + (3) ⇒
    2x + 3y = 10
    8x - 2y = 12
    10x + y = 22
    y = 22 - 10x --------- (4)

    Putting the value of y in (2)
    2x + 3(22 - 10x) = 10
    ⇒ 2x + 66 - 30x = 10
    ⇒ - 28x = 10 - 66
    ⇒ - 28x = - 56
    ∴ x = 2

    Putting x = 2 in (4),
    y = 22 - 10 × 2
    ⇒ y = 22 - 20
    ∴ y = 2

    Now, 2x + 5y = 2 . 2 + 5 . 2 = 4 + 10 = 14
    ৯১.
    Which of the following fractions is greater than 3/4 and less than 5/6?
    1. ক
      ক) 2/3
    2. খ
      খ) 4/5
    3. গ
      গ) 9/10
    4. ঘ
      ঘ) 1/2
    ব্যাখ্যা

    3/4 = 0.75
    5/6 = 0.83
    2/3 = 0.667
    4/5 = 0.80
    9/10 = 0.9
    1/2 = 0.50

    So, the fraction 4/5 is greater than 3/4 and less than 5/6

    ৯২.
    If a= √3/2, then √(1 + a) + √(1 - a)= ?
    1. ক
      ক) √3
    2. খ
      খ) 2 - √3
    3. গ
      গ) 2 + √3
    4. ঘ
      ঘ) √3/2
    ব্যাখ্যা
    let 
    √(1 +a) + √(1 - a) = x 
    {√(1 +a) + √(1 - a)}2 = x2
    (1 +a)  + (1 - a) + 2 {√(1 +a) × √(1 - a)} =x2
    2 + 2 {√(1 +a) × √(1 - a)}  = x2 
    2 + 2 √(1 - a2) = x2
    2 + 2 √{1- (√3/2)2} = x2
    2 + 2 √{ 1- 3/4} = x2   
    2 + 2 √{( 4- 3)/4}=  x2
    2 + 2 × 1/2 = x2
    2 + 1 = x2 
    x2 = 3 
    ∴ x = √3
    ৯৩.
    What should come in place of both x in the equation x /√128 = √162/x.
    1. ক
      12
    2. খ
      14
    3. গ
      144
    4. ঘ
      196
    ব্যাখ্যা
    Question: What should come in place of both x in the equation x /√128 = √162/x.

    Solution:
     x /√128 = √162/x
    ⇒ x2 = √(128 × 162)
    ⇒ x2 = √(64 × 2 × 18 × 9)
    ⇒ x2 = √(82 × 62 × 32)
    ⇒ x2 = 8 × 6 × 3
    ⇒ x2 = 144
    ∴ x = 12
    ৯৪.
    If a2 + (2a/3) + 1/9 = 0, then (a - 2/3)2 = ?
    1. ক
      ক) 1
    2. খ
      খ) 1/3
    3. গ
      গ) 9
    4. ঘ
      ঘ) 27
    ব্যাখ্যা
    Question: If a2 + (2a/3) + 1/9 = 0, then (a - 2/3)2 = ?

    Solution:
    a2 + 2a/3 + 1/9 = 0
    ⇒ a2 + (1/3)2 + 2 × a × (1/3) = 0
    ⇒ (a + 1/3)2 = 0
    ⇒ a + 1/3 = 0
    ⇒ a = - 1/3

    Now,
    {(- 1/3) - (2/3)}2
    = {(- 1 - 2)/3}2
    = {(- 3)/3}2
    = (- 1)2
    = 1
    ৯৫.
    What is the 10th term of the geometric sequence: 4 + 8 +16 + ........?
    1. ক
      2048
    2. খ
      1080
    3. গ
      1640
    4. ঘ
      1264
    5. ঙ
      None
    ব্যাখ্যা
    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 = arn - 1
    ∴ ধারাটির 10 তম পদ = 4 × 210 - 1
    = 4 × 29
    = 4 × 512
    = 2048
    ৯৬.
    There are eight teachers and four administrators in a school. A task force of 5 people needs to be formed. How many different ways can a task force be formed that contains three teachers and two administrators?
    1. ক
      286
    2. খ
      242
    3. গ
      336
    4. ঘ
      384
    5. ঙ
      None
    ব্যাখ্যা
    Question: There are eight teachers and four administrators in a school. A task force of 5 people needs to be formed. How many different ways can a task force be formed that contains three teachers and two administrators?

    Solution:
    Have: 8 teachers, 4 administrators
    Want: 3 teachers AND 2 administrators

    Total ways = 8C3 × 4C2
    = 56 × 6
    = 336
    Therefore, there are 336 different ways to form a task force that contains three teachers and two administrators.
    ৯৭.
    In a proportion the product of 1st and 4th terms is 40 and that of 2nd and 3rd terms is 2.5x. Then the value of x is-
    1. ক
      ক) 16
    2. খ
      খ) 26
    3. গ
      গ) 56
    4. ঘ
      ঘ) 76
    5. ঙ
      ঙ) 96
    ব্যাখ্যা

    Product of 1st and 4th terms (extremes) = product of 2nd and 3rd terms (means)
    ⇒ 2.5x = 40
    ⇒ x = 40/2.5 = 16

    ৯৮.
    In a room of 36 people, 20 players play chase while 28 players play poker. How many players play both?
    1. ক
      ক) 45
    2. খ
      খ) 20
    3. গ
      গ) 12
    4. ঘ
      ঘ) 28
    ব্যাখ্যা

    We know,
    Total = n(A) + n(B) - both + none
    ⇒ 36 = 20 + 28 - both + 0
    ⇒ 36 = 48 - both
    ⇒ both = 48 - 36
    ⇒ both = 12

    ৯৯.
    Murad bought equal number of 20-paisa and 30-paisa stamps. If the total cost of the stamps was tk 10, what was the total number of stamps that Murad bought?
    1. ক
      25
    2. খ
      34
    3. গ
      40
    4. ঘ
      46
    5. ঙ
      None of these
    ব্যাখ্যা
    Question: Murad bought equal number of 20-paisa and 30-paisa stamps. If the total cost of the stamps was tk 10, what was the total number of stamps that Murad bought?

    Solution:
    Let's represent the number of 20-paisa stamps as 'x' and the number of 30-paisa stamps as 'y'.
    Since Murad bought an equal number of 20-paisa and 30-paisa stamps, we know that x = y.

    ATQ,
    20x + 30y = 1000 [10 tk = 1000 paisa]
    ⇒ 50x = 1000
    ⇒ x = 20

    ∴ Total stamps = x + y = 20 + 20 = 40
    Therefore, the total number of stamps Murad bought is 40.
    ১০০.
    In the series 3, 9, 15, ….. what will be the 21st term?
    1. ক
      117
    2. খ
      121
    3. গ
      123
    4. ঘ
      129
    ব্যাখ্যা
    Question: In the series 3, 9, 15, ….. what will be the 21st term?

    Solution:
    Here,
    9 - 3 = 6
    15 - 9 = 6
    So, the series is an A.P. in which a = 3 and d = 6.

    ∴ 21st term = a + (21 - 1) d
    = a + 20d
    = 3 + 20 × 6
    = 123