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Algebra

মোট প্রশ্ন১,৩৮০এই পাতা১০০প্রতি পাতা১০০
প্রশ্ন দেখুনঅনুশীলন
ঘনত্ব
উত্তর
উত্তরিতবর্তমানপুনরায় দেখুনঅসম্পূর্ণ

Algebra

PrepBank · পাতা ৩ / ১৪ · ২০১–৩০০ / ১,৩৮০

২০১.
If a + b = 7 and ab = 12, then what is 1/a + 1/b?
  1. ক
    19
  2. খ
    5
  3. গ
    5/12
  4. ঘ
    7/12
ব্যাখ্যা
Question: If a + b = 7 and ab = 12, then what is 1/a + 1/b?

Solution:
1/a + 1/b
= (b + a)/ab
= (a + b)/(ab)
= 7/12
২০২.
Solve the inequality {1/(x + 3)} ≤ {1/(2x + 5)}
  1. ক
    (- 3, 2) ∪ (2, ∞)
  2. খ
    (- 5/2, 3)
  3. গ
    (- ∞, - 5/2) ∪ (3, 7)
  4. ঘ
    (- ∞, - 3) ∪ (- 5/2, - 2)
  5. ঙ
    None of these
ব্যাখ্যা
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)
২০৩.
If a + b = √13 and a - b = √5, what is the value of 8ab(a2 + b2)?
  1. ক
    96
  2. খ
    64
  3. গ
    104
  4. ঘ
    144
ব্যাখ্যা

Question: If a + b = √13 and a - b = √5, what is the value of 8ab(a2 + b2)?

Solution:
দেওয়া আছে,
a + b = √13
a - b = √5

আমরা জানি,
2(a2 + b2) = (a + b)2 + (a - b)2
4ab = (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]
= [(√13)2 - (√5)2][(√13)2 + (√5)2]
= (13 - 5)(13 + 5)
= (8)(18)
= 144

২০৪.
In a tourist group of 100 people, 55 speak French, 40 speak Spanish, and 20 speak none of the languages. How many of them speak just one language?
  1. ক
    15
  2. খ
    25
  3. গ
    40
  4. ঘ
    65
ব্যাখ্যা
Question: In a tourist group of 100 people, 55 speak French, 40 speak Spanish, and 20 speak none of the languages. How many of them speak just one language?

Solution:

ধরি,
উভয় ভাষায় কথা বলতে পারে = x জন
∴ শুধু ফ্রেঞ্চ বলতে পারে = (55 - x) জন
∴ শুধু স্প্যানিশ বলতে পারে = (40 - x) জন

দেওয়া আছে,
কোনো ভাষায় কথা বলে না = 20 জন

প্রশ্নমতে,
(55 - x) + x + (40 - x) = 100 - 20
95 - x = 80
x = 95 - 80
x = 15

শুধু ফ্রেঞ্চ বলতে পারে = (55 - 15) জন = 40 জন 
শুধু স্প্যানিশ বলতে পারে = (40 - 15) জন = 25 জন

∴ শুধুমাত্র একটি ভাষায় (ফ্রেঞ্চ বা স্প্যানিশ) কথা বলতে পারে = (40 + 25) জন = 65 জন 
২০৫.
If Nita is 15 ahead in rank of Mita who ranks 13th from the last, then how many students are there in the class if Nita ranks 5th in order of merit?
  1. ক
    34
  2. খ
    32
  3. গ
    33
  4. ঘ
    31
ব্যাখ্যা

Question: If Nita is 15 ahead in rank of Mita who ranks 13th from the last, then how many students are there in the class if Nita ranks 5th in order of merit?

Solution:
Let the total number of students = n

Now,
Mita’s position from the last = 13th
∴ Mita’s position from the front = n - 13 + 1 = n - 12

Again,
Nita’s position from the front = 5th.

And, Nita is 15 ranks ahead of Mita.
∴ Mita’s position = Nita’s position + 15.

According to the problem,
n - 12 = 5 + 15
⇒ n - 12 = 20
⇒ n = 20 + 12
∴ n = 32 

Therefore, the total number of students in the class = 32.

২০৬.
Solution set of the inequality, 4x - 7 ≤ 2x + 5 is-
  1. ক
    [- ∞, 4]
  2. খ
    (- ∞, - 6)
  3. গ
    (- ∞, 6]
  4. ঘ
    None of these
ব্যাখ্যা

Question: Solution set of the inequality, 4x - 7 ≤ 2x + 5 is-

Solution:
Given that,
4x - 7 ≤ 2x + 5
⇒ 4x - 2x ≤ 7 + 5
⇒ 2x ≤ 12
⇒ x ≤ 6
∴ x ≤ 6

∴ Solution set of the inequality is (- ∞, 6]

২০৭.
If a3 + b3 =  a2 - ab + b2 and a = 13/17, find the vale of b.
  1. ক
    17/13
  2. খ
    4/13
  3. গ
    1
  4. ঘ
    4/17 
ব্যাখ্যা
Question: If a3 + b3 =  a2 - ab + b2 and a = 13/17, find the vale of b. 

Solution:
Given, 
a3 + b3 =  a2 - ab + b2
⇒ (a + b)(a2 - ab + b2) =  (a2 - ab + b2)     [ a3 + b3 = (a + b)(a2 - ab + b2) ]
⇒ (a + b) = 1
⇒ b = 1 - a
⇒ b = 1 - (13/17)
⇒ b = 4/17
২০৮.
If b < 2 and 2x - 3b = 0, which of the following must be true?
  1. ক
    ক) x > - 3
  2. খ
    খ) x < 2
  3. গ
    গ) x = 3
  4. ঘ
    ঘ) x < 3
  5. ঙ
    ঙ) x > 3
ব্যাখ্যা
Question: If b < 2 and 2x - 3b = 0, which of the following must be true?

Solution: 
Given that,
2x - 3b = 0
⇒ 2x = 3b
⇒ 2x < (3×2) [ i.e b < 2]
∴ x < 3
২০৯.
If a3 - b3 = 56 and a - b = 2, then the value of a2 + b2 =?
  1. ক
    15
  2. খ
    20
  3. গ
    24
  4. ঘ
    26
ব্যাখ্যা
Question: If a3 - b3 = 56 and a - b = 2, then the value of a2 + b2 =? 

Solution: 
a3 - b3 = 56
⇒ (a - b) (a2 + ab + b2) = 56
⇒ a2 + ab + b2 = 56/2 = 28
⇒ (a - b)2 + 3ab = 28 
⇒ 3ab = 28 - 4 = 24 
⇒ ab = 24/3 = 8

a2 + ab + b2 = 28
⇒ a2 + b2 = 28 - 8 = 20
২১০.
The 2nd term of a geometric sequence is 27/8, and the 5th term is 1. What is the common ratio?
  1. ক
    2/3
  2. খ
    5/8
  3. গ
    1/2
  4. ঘ
    2/5
  5. ঙ
    - 3/8
ব্যাখ্যা

Question: The 2nd term of a geometric sequence is 27/8, and the 5th term is 1. What is the common ratio?

Solution:
আমরা জানি, একটি গুণোত্তর ধারার n-তম পদ, an = a.rn - 1

দেয়া আছে,
2য় পদ, a2 = 27/8
⇒ ar = 27/8 …...(1)

5ম পদ, a5 = 1
⇒ a.r4 = 1 …....(2)

এখন, সমীকরণ (2) ÷ সমীকরণ (1) ⇒
(ar4)/(ar) = 1/(27/8)
⇒ r3 = 8/27
⇒ r3 = (2/3)3
⇒ r = 2/3

∴ সাধারণ অনুপাত (common ratio) হলো 2/3

২১১.
If a ⊕ b = (a × b) + b then 7 ⊕ 6 equals to - 
  1. ক
    ক) 48
  2. খ
    খ) 42
  3. গ
    গ) 38
  4. ঘ
    ঘ) 32
ব্যাখ্যা
Question: If a  ⊕ b = (a × b) + b then 7 ⊕ 6 equals to - 

Solution: 
⇒7 ⊕ 6 =(7 × 6) + 6
             = 42 + 6
             = 48
২১২.
Let, m and n be natural numbers. If m2 + mn + m = 14 and n2 + nm + n = 28, then (2m + n) =?
  1. ক
    7
  2. খ
    8
  3. গ
    10
  4. ঘ
    12
ব্যাখ্যা
Question: Let, m and n be natural numbers. If m2 + mn + m = 14 and n2 + nm + n = 28, then (2m + n) =? 

Solution: 
n2 + nm + n + m2 + mn + m = 28 + 14 
⇒ m2 + 2mn + n2 + m + n = 42  
⇒ (m + n)2 + m + n  = 42
⇒ (m + n) (m + n + 1) = 42 

m + n = 6, m + n + 1 = 7 

Given, 
m2 + mn + m = 14 
⇒ m(m + n + 1) = 14
⇒ m × 7 = 14
⇒ m = 2

n = 6 - m = 6 - 2 = 4

∴ 2m + n  = 2 × 2 + 4 = 4 + 4 = 8
২১৩.
If 2x - 1 ≥ - 3, then
  1. ক
    x ≤ - 2
  2. খ
    x ≥ - 2
  3. গ
    x ≤ - 1
  4. ঘ
    x ≥ - 1
ব্যাখ্যা
Question: If 2x - 1 ≥ - 3, then

Solution:
2x - 1 ≥ - 3
2x - 1 + 1 ≥ - 3 + 1
2x ≥ - 2
2x/2 ≥ - 2/2
x ≥ - 1
 
২১৪.
In a graph there are two curves, y1 = 2x1 - 5 and y2 = - x2 + 10. y2 will be greater than y1 when -
  1. ক
    ক) x > 5
  2. খ
    খ) x < 5
  3. গ
    গ) - 1 < x
  4. ঘ
    ঘ) x < 9
ব্যাখ্যা

Given,
y1 = 2x - 5 and y2 = - x + 10
As, y2 > y1
Or, - x + 10 > 2x - 5
Or, 15 > 3x
Or, 3x < 15
Or, x<5

২১৫.
If 2x + 3y = 24 and y = 2x, then find y - x. 
  1. ক
    3
  2. খ
    4
  3. গ
    5
  4. ঘ
    6
ব্যাখ্যা

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

২১৬.
If (a/b) + (b/a) = 4, find the value of (a3/b3) + (b3/a3).
  1. ক
    52
  2. খ
    64
  3. গ
    76
  4. ঘ
    112
ব্যাখ্যা

Question: If (a/b) + (b/a) = 4, find the value of (a3/b3) + (b3/a3).

Solution:
দেওয়া আছে, (a/b) + (b/a) = 4

প্রদত্ত রাশি = (a3/b3) + (b3/a3)
= (a/b)3 + (b/a)3
= {(a/b) + (b/a)}3 - 3 (a/b)(b/a) . {(a/b) + (b/a)}
= 43 - (3 × 1 × 4)
= 64 - 12
= 52

∴ প্রদত্ত রাশির মান 52

২১৭.
Solution set of the inequality: 3y + 4 ≥ 2y - 5 is:
  1. ক
    [- 9,  ∞)
  2. খ
    (- 9,  ∞)
  3. গ
    (- 9,  ∞]
  4. ঘ
    (9, - ∞)
ব্যাখ্যা

Question: Solution set of the inequality: 3y + 4 ≥ 2y - 5 is-

Solution:
Given that,
3y + 4 ≥ 2y - 5
⇒ 3y + 4 - 2y ≥ 2y - 5 - 2y
⇒ y + 4 ≥ - 5
⇒ y ≥ - 5 - 4
⇒ y ≥ - 9

∴ Solution set of the inequality is  [- 9,  ∞)

২১৮.
If x = 3 + 22, then find the value of √x - 1/√x.
  1. ক
    5/24
  2. খ
    5
  3. গ
    24/5
  4. ঘ
    25/4
ব্যাখ্যা
Question: If x = 3 + 22, then find the value of √x - 1/√x.

Solution:
Given,
x = 3 + 22
⇒ x = 25
⇒ 52 = x
⇒ 5 = √x
∴ √x = 5

Now,
√x - 1/√x
= 5 - 1/5
= (25 - 1)/5
= 24/5
২১৯.
If b < 2 and 2x - 3b = 0, which of the following must be true?
  1. ক
    x > - 3
  2. খ
    x < 2
  3. গ
    x = 3
  4. ঘ
    x < 3
  5. ঙ
    x > 3
ব্যাখ্যা
Question: If b < 2 and 2x - 3b = 0, which of the following must be true?

Solution:
2x - 3b = 0
⇒ 3b = 2x
∴ b = (2x)/3

b < 2
∴ (2x)/3 < 2
⇒ 2x < 6
⇒ x < 3
২২০.
If set A = {5, 15, 20, 30} and B = {5, 15, 20} which of the following represents (A ∩ B) ?
  1. ক
    ক) {5, 15, 20, 30}
  2. খ
    খ) {20}
  3. গ
    গ) {5, 15, 20}
  4. ঘ
    ঘ) {5, 20, 30}
ব্যাখ্যা
Question: If set A = {5, 15, 20, 30} and B = {5, 15, 20} which of the following respresents (A ∩ B) ?

Solution:
দেওয়া আছে,
A = {5, 15, 20, 30}
B = {5, 15, 20}
এখন,
A ∩ B = {5, 15, 20, 30} ∩ {5, 15, 20}
= {5, 15, 20}
২২১.
What is the solution set of the inequality, - 2x + 11 ≥ 5?
  1. ক
    (- ∞, 3]
  2. খ
    (3, - ∞)
  3. গ
    (- ∞, 4]
  4. ঘ
    [3, ∞)
ব্যাখ্যা

Question: What is the solution set of the inequality, - 2x + 11 ≥ 5?

Solution:
Given that, 
- 2x + 11 ≥ 5
⇒ - 2x + 11 - 11 ≥ 5 - 11 ; [Subtract 11 from both sides]
⇒ - 2x ≥ - 6
∴ x ≤ 3  ; [Divide both sides by - 2]

Solution set: x ≤ 3
or in interval notation: (- ∞, 3]

২২২.
The 3rd term of a geometric sequence is 48, and the 6th term is 384. What is the common ratio is?
  1. ক
    2
  2. খ
    6
  3. গ
    3
  4. ঘ
    4
  5. ঙ
    5
ব্যাখ্যা
Question: The 3rd term of a geometric sequence is 48, and the 6th term is 384. What is the common ratio is?

Solution:
Given, 
The 3rd term of a geometric sequence, a3 = 48
The 6th term of the same sequence, a6 = 384

In a geometric sequence we know,
an = arn - 1

So,
a3 = ar3 - 1= ar2 .......(1)
And
a6 = ar6 - 1= ar5........ (2)

Now (2) ÷ (1), 
ar5/ar2 = 384/48
⇒ r3 = 8 = 23
∴ r =  2

So the common ratio is 2.
২২৩.
What is the sum of the reciprocals of the values of zeroes of the polynomial 6x2 + 3x2 - 5x + 1?
  1. ক
    5
  2. খ
    5/9
  3. গ
    9
  4. ঘ
    8
ব্যাখ্যা
Question: What is the sum of the reciprocals of the values of zeroes of the polynomial 6x2 + 3x2 - 5x + 1?

Solution: 
6x2 + 3x2 - 5x + 1
⇒ 9x2 - 5x + 1
Let α and β are two roots of the equations

As we know,
Sum of roots (α + β) = 5/9
Product of roots (αβ) = 1/9

ATQ,
1/α + 1/β = (α + β)/αβ
= (5/9)/(1/9)
= (5/9) × (9/1)
= 5
২২৪.
What is the distance of the origin from the line 12x - 5y + 26 = 0?
  1. ক
    ক) 2
  2. খ
    খ) 3
  3. গ
    গ) 4
  4. ঘ
    ঘ) 5
ব্যাখ্যা
প্রদত্ত রেখার সমীকরণ 12x - 5y + 26 = 0
মূলবিন্দুর স্থানাংক (0,0)
এখানে,
A = 12 ,B = - 5  C = 26

নির্ণেয় দূরুত্ব = । Ax + By + C /√(A2 + B2)।
                    = ।(12 × 0) - (5 × 0) + 26 / √{122 + (- 5)2}।
                    = । 26/√169।
                    =। 26/13।
                     = 2
২২৫.
Which of the following is not equivalent to 25x2 = y2 - 4?
  1. ক
    x2 = (y2 - 4)/25
  2. খ
    25x2 = (y + 2)(y - 2)
  3. গ
    75x2 = 3y2 - 12
  4. ঘ
    5x = y - 2
ব্যাখ্যা
Question: Which of the following is not equivalent to 25x2 = y2 - 4?
 
Solution:
25x2 = y2 - 4
= y2 - 22
= (y + 2) (y - 2)
 
25x2 = y2 - 4
⇒ 3 × 25x2 = 3(y2 - 4)
⇒ 75x2 = 3y2 - 12
 
25x2 = y2 - 4
⇒ x2 = (y2 - 4)/25 
 
So, last option is not equivalent to 25x2 = y2 - 4
২২৬.
Ι3a - 15Ι = 18. What is the product of all possible values of a?
  1. ক
    - 13
  2. খ
    - 11
  3. গ
    - 10
  4. ঘ
    - 9
  5. ঙ
    None
ব্যাখ্যা
Question: Ι3a - 15Ι = 18. What is the product of all possible values of a?

Solution:
Given, |3a - 15| = 18

Solve the absolute value equation for both cases.
3a - 15 = 18
⇒ 3a = 15 + 18 = 33
∴ a = 11

or, 3a - 15 = - 18
⇒ 3a = - 18 + 15
⇒ 3a = -3
∴ a = - 1

The product of all possible values of a is = 11 × (-1 ) = - 11
২২৭.
How many terms are there in the geometric progression,
3, 6, 12, 24, …, 768
  1. ক
    8
  2. খ
    10
  3. গ
    7
  4. ঘ
    9
ব্যাখ্যা
Question: How many terms are there in the geometric progression,
3, 6, 12, 24, …, 768

Solution:
First term, a = 3
Common ratio, r = 6/3 = 2

Last term or nth term of GP = arn - 1
⇒ 768 = 3 × (2n - 1)
⇒ 2n - 1 = 768/3
⇒ 2n - 1 = 256
⇒ 2n - 1 = 28
So, comparing the power,
Thus, n - 1 = 8
∴ n = 9

∴ Number of terms = 9
২২৮.
If x = y = 4z and xyz = 128 than x =?
  1. ক
    12
  2. খ
    8
  3. গ
    14
  4. ঘ
    9
ব্যাখ্যা
Question: If x = y = 4z and xyz = 128 than x =?

Solution:
Given that,
x = y = 4z
Now,
xyz = 128
⇒ x × x × (x/4) = 128
⇒ x3/4 = 128
⇒ x3 = 128 × 4
⇒ x3 = 512
⇒ x3 = 83
∴ x = 8
২২৯.
What is the sum of the squares of the digits from 1 to 9?
  1. ক
    ক) 260
  2. খ
    খ) 105
  3. গ
    গ) 285
  4. ঘ
    ঘ) 385
ব্যাখ্যা

12  + 22  + 32  + ........ + 92 
n সংখ্যক স্বাভাবিক সংখ্যার বর্গের সমষ্টি  = n(n + 1)(2n + 1)/6
= 9(9 + 1)(2×9 + 1)/6
= (9×10×19)/6
= 285

২৩০.
If x4 - x2 + 1 = 0, then x3 + 1/x3 =?
  1. ক
    3
  2. খ
    2
  3. গ
    1
  4. ঘ
    0
ব্যাখ্যা
Question: If x4 - x2 + 1 = 0, then x3 + 1/x3 =?

Solution:
x4 - x2 + 1 = 0
⇒ x4 + 1 = x2
⇒ x4/x2 + 1/x2 = x2/x2
⇒ x2  + 1/x2 = 1
⇒ (x + 1/x)2 - 2.x.1/x = 1
⇒ (x + 1/x)2 - 2 = 1
⇒ (x + 1/x)2 = 1 + 2
⇒ (x + 1/x)2 = 3
⇒ (x + 1/x) = √3

x3 + 1/x3 =(x + 1/x)3 - 3.x.1/x(x + 1/x)
= (√3)3 - 3√3
= 3√3  - 3√3
= 0
২৩১.
If a = b = 2c and abc = 256 then, b2 = ?
  1. ক
    64
  2. খ
    48
  3. গ
    36
  4. ঘ
    8
ব্যাখ্যা
Question: If a = b = 2c and abc = 256 then, b2 = ?

Solution:
Given,
a = b = 2c
∴ a = b 
and b = 2c
⇒ c = b/2

ATQ,
abc = 256
⇒ b × b × (b/2) = 256
⇒ b3/2 = 256
⇒ b3 = 256 × 2
⇒ b3 = 512
⇒ b3 = 83
∴ b = 8

∴ b2 = 82 = 64
২৩২.
Find the HCF of: x3 - 1, x2 - 1, x2 - 2x + 1
  1. ক
    x - 3
  2. খ
    0
  3. গ
    x
  4. ঘ
    x - 1
ব্যাখ্যা
Question: Find the HCF of: x3 - 1, x2 - 1, x2 - 2x + 1

Solution:
x3 - 1
= (x - 1)(x2 + 2x + 1)
= (x - 1)(x + 1)2
= (x - 1)(x + 1)(x + 1)

x2 - 1
= (x + 1)(x - 1)

x2 - 2x + 1
= (x - 1)2
= (x - 1)(x - 1)

∴ HCF = (x - 1)
২৩৩.
If f(x) = 3x2 + bx + 4 and f(- 1) = 0; Then b = ?
  1. ক
    ক) 3
  2. খ
    খ) 4
  3. গ
    গ) 6
  4. ঘ
    ঘ) 7
ব্যাখ্যা
Question:  If f(x) = 3x2 + bx + 4 and f(- 1) = 0; Then b = ?

Solution:
f(x) = 3x2 + bx + 4 
∴ f(- 1) = 3(- 1)2 + b(- 1) + 4
= 3 - b +4
= 7 - b 

A.T.Q,
7 - b = 0
∴ b = 7
২৩৪.
√{(16)(20) + (8)(32)} = ?
  1. ক
    ক) 4√20
  2. খ
    খ) 24
  3. গ
    গ) 25
  4. ঘ
    ঘ) 32
ব্যাখ্যা

√{(16)(20) + (8)(32)} 
= √{(320) + (256)} 
= √(576)
= 24

২৩৫.
x2 - (a + b)x + ab = 0; x = ?
  1. ক
    a, b
  2. খ
    a
  3. গ
    b
  4. ঘ
    ab
ব্যাখ্যা
Question: x2 - (a + b)x + ab = 0; x = ?

Solution:
Given that
x2 - (a + b)x + ab = 0
x2 - ax - bx + ab = 0
x(x - a) - b(x - a) = 0
(x - a)(x - b) = 0

হয় 
x - a = 0
x = a

অথবা
x - b = 0
x = b

নির্ণেয় সমধান x = a, b
২৩৬.
Find the first term of an arithmetic progression(A.P.) whose 8th and 12th terms are respectively 39 and 59.
  1. ক
    5
  2. খ
    4
  3. গ
    6
  4. ঘ
    3
ব্যাখ্যা

Question: Find the first term of an arithmetic progression(A.P.) whose 8th and 12th terms are respectively 39 and 59.

Solution:
We know,
Arithmetic progression, nth term is  = a + (n - 1)d,
where a is the first term and d is the common difference.
Given that, 
8th term = 39 and 12th term = 59

Now,
8th term = a + 7d = 39 ........... (i)
12th term = a + 11d = 59 ........... (ii)
(i) - (ii) ⇒ a + 7d - a - 11d = 39 - 59
⇒ 4d = 20
 ∴ d = 5
Hence, a + 7 × 5 = 39
Thus, a = 39 - 35 = 4

 so the first term of the A.P. is 4.

২৩৭.
Solve the following equation.
  1. ক
    1
  2. খ
    2
  3. গ
    3
  4. ঘ
    4
ব্যাখ্যা
Question: Solve the following equation.


Solution:
২৩৮.
M men agree to purchase a gift for Taka D. If three men drop out how much more will each have to contribute towards the purchase of the gift?
  1. ক
    ক) D/(M - 3)
  2. খ
    খ) MD/3
  3. গ
    গ) M/(D - 3)
  4. ঘ
    ঘ) 3D/(M2 - 3M)
  5. ঙ
    ঙ) None of these
ব্যাখ্যা
Question: M men agree to purchase a gift for Taka D. If three men drop out how much more will each have to contribute towards the purchase of the gift?

Solution:
M men agree to purchase a gift for Taka D. 
∴ 1 man purchase for the gift = D/M Taka 

If three men drop out then remaining = M - 3 men 
∴ 1 man has to purchase D/(M - 3) Taka 

Now, 
D/(M - 3) - D/M
⇒ {DM - D(M - 3)}/M(M - 3)
⇒ {DM - DM + 3D}/(M2 - 3M)
∴ 3D/(M2 - 3M)
২৩৯.
A factory produces 500 bottles of soda in 2 hours. How many bottles will it produce in 6 hours, working at the same rate?
  1. ক
    ক) 500 bottles
  2. খ
    খ) 1000 bottles
  3. গ
    গ) 1200 bottles
  4. ঘ
    ঘ) 1500 bottles
ব্যাখ্যা
Question: A factory produces 500 bottles of soda in 2 hours. How many bottles will it produce in 6 hours, working at the same rate?

Solution:
Production rate = Number of items produced / Time
Production rate = 500 bottles / 2 hours = 250 bottles per hour

Bottles produced in 6 hours = Production rate × Time
Bottles produced in 6 hours = 250 bottles/hour × 6 hours = 1500 bottles
২৪০.
If x - y = 15 and xy = 54, then what is the value of x + y?
  1. ক
    21
  2. খ
    23
  3. গ
    18
  4. ঘ
    27
ব্যাখ্যা

Question: If x - y = 15 and xy = 54, then what is the value of x + y?

Solution:
দেওয়া আছে,
x - y = 15
xy = 54

আমরা জানি,
(x + y)2 = (x - y)2 + 4xy
⇒ (x + y)2 = (15)2 + 4 × 54
⇒ (x + y)2 = 225 + 216
⇒ (x + y)2 = 441
⇒ x + y = √441
⇒ x + y = 21

সুতরাং, x + y এর মান হলো 21।

২৪১.
If xy = 60 and x2 + y2 = 169, then x + y =?
  1. ক
    15
  2. খ
    17
  3. গ
    19
  4. ঘ
    21
ব্যাখ্যা
Question: If xy = 60 and x2 + y2 = 169, then x + y =?

Solution:
xy = 60 ............(1)
x2 + y2 = 169 ...............(2)

(x + y)2 = x2 + y2 + 2xy
⇒ (x + y)2 = 169 + (2 × 60)
⇒ (x + y)2 = 289
∴ x + y = 17
২৪২.
Find the equation of the vertical line passing through the point (7, - 4).
  1. ক
    y = - 4
  2. খ
    x = 7
  3. গ
    y = 7
  4. ঘ
    x = - 4
ব্যাখ্যা

Question: Find the equation of the vertical line passing through the point (7, - 4).

Solution:
একটি উল্লম্ব রেখা (vertical line) হলো এমন একটি সরলরেখা যা Y-অক্ষের সমান্তরাল। এই ধরনের রেখার একটি বিশেষ বৈশিষ্ট্য হলো, এই রেখার উপর অবস্থিত প্রতিটি বিন্দুর x-স্থানাঙ্ক (x-coordinate) একই থাকে, কিন্তু y-স্থানাঙ্ক (y-coordinate) পরিবর্তিত হতে পারে।

উল্লম্ব রেখার (vertical line) সাধারণ সমীকরণ হলো: x = a
যেখানে a একটি ধ্রুবক সংখ্যা এবং রেখার প্রতিটি বিন্দুর x এর মান একই থাকে।

প্রশ্নে বলা হয়েছে রেখাটি (7, - 4) বিন্দুর মধ্য দিয়ে যায়।
যেহেতু, আমরা জানি, একটি উল্লম্ব রেখার প্রতিটি বিন্দুর x-স্থানাঙ্ক একই থাকে, এবং এই বিন্দুর x-স্থানাঙ্ক হলো 7, সুতরাং, রেখাটির সমীকরণ হবে:
x = 7

২৪৩.
If x + 4/x = 4, then find the value of x?
  1. ক
    5
  2. খ
    2
  3. গ
    6
  4. ঘ
    4
ব্যাখ্যা
x + 4/x = 4
বা, (x2 + 4)/x = 4
বা, x2 + 4 = 4x
বা, x2 - 4x + 4 = 0
বা, x2 - 2.2x.1 + 22 = 0
বা, (x - 2)2 = 0
বা, x = 2
২৪৪.
If 2x - 3y = 6, then 6y - 4x = ?
  1. ক
    ক) -12
  2. খ
    খ) -6
  3. গ
    গ) 6
  4. ঘ
    ঘ) 12
ব্যাখ্যা

Given, 2x - 3y = 6 .... (i)
Lets (i) × (-2),
So, - 4x + 6y = - 12
Or, 6y - 4x = - 12

২৪৫.
To fill a tank, 28 buckets of water are required. How many buckets of water will be required to fill the same tank if the capacity of the bucket is reduced to four-fifth of its present?
  1. ক
    35
  2. খ
    28
  3. গ
     25
  4. ঘ
    None of the above
ব্যাখ্যা

Question: To fill a tank, 28 buckets of water are required. How many buckets of water will be required to fill the same tank if the capacity of the bucket is reduced to four-fifth of its present?

Solution: 

ধরি,
বালতির ধারণক্ষমতা x
ট্যাঙ্কের ধারণক্ষমতা 28x

আবার,
বালতির নতুন ধারণক্ষমতা (4x/5) 

∴ প্রয়োজনীয় বালতির সংখ্যা = 28x/(4x/5) 
= 28x × (5/4x) 
= (7 × 5) টি
= 35 টি

২৪৬.
If x = 2 then the value of x3 + 27x2 + 243x + 631 is -
  1. ক
    ক) 1222
  2. খ
    খ) 1232
  3. গ
    গ) 1133
  4. ঘ
    ঘ) 1233
ব্যাখ্যা
Question: If x = 2 then the value of x3 + 27x2 + 243x + 631 is -

Solution:
Given,
x = 2
Now, 
 x3 + 27x2 + 243x + 631
= x3 + 3 . x2 . 9 + 3 . x . 92 + 93 - 98
= (x + 9)3 - 98
= (2 + 9)3 - 98
= 1331 - 98
= 1233
২৪৭.
If x + y = 7 and x - y =3, then x2 - y2 =
  1. ক
    ক) -4
  2. খ
    খ) 4
  3. গ
    গ) 10
  4. ঘ
    ঘ) 21
ব্যাখ্যা
x2 - y2
= (x + y)(x - y)
= 7 × 3
= 21
২৪৮.
The sum of two numbers is 21 and the sum of their squares is 333. The product of two numbers is -
  1. ক
    ক) 40
  2. খ
    খ) 49
  3. গ
    গ) 54
  4. ঘ
    ঘ) 58
ব্যাখ্যা

Let, the numbers be x and y
According to the question, x  +  y = 21 ......(i) and x2  +  y2  =  333 ......(ii) 
∴ (x + y)2 = x2 + y2 + 2xy
⇒ (21)2 = 333 + 2xy
⇒ 441 = 333 + 2xy
⇒ 2xy = 108
⇒ xy = 54

২৪৯.
Which of the following is equivalent to the pair of inequalities 2x - 5 > 3 and x + 4 ≤ 12?
  1. ক
    4 ≤ x < 16
  2. খ
    1 ≤ x < 4
  3. গ
    4 < x ≤ 8
  4. ঘ
    4 ≤ x < 8
ব্যাখ্যা

Question: Which of the following is equivalent to the pair of inequalities 2x - 5 > 3 and x + 4 ≤ 12?

Solution:
প্রথম অসমতাটি হলো:
⇒ 2x - 5 > 3
⇒ 2x > 3 + 5
⇒ 2x > 8
⇒ x > 4

দ্বিতীয় অসমতাটি হলো:
⇒ x + 4 ≤ 12
⇒ x ≤ 12 - 4
⇒ x ≤ 8

এখন প্রাপ্ত ফলাফল দুটিকে একত্রে সাজালে পাই:
4 < x এবং x ≤ 8
∴ 4 < x ≤ 8

২৫০.
If A = {a, b, r, t, s, y} and B = {e, d, f, r, s}, then B - A =
  1. ক
    {a, b, t, y}
  2. খ
    {r, s}
  3. গ
    {e, d, f}
  4. ঘ
    {a, b, d, e, f, t, y}
ব্যাখ্যা
Question: If A = {a, b, r, t, s, y} and B = {e, d, f, r, s}, then B - A =

Solution:
B - A = {e, d, f, r, s} - {a, b, r, t, s, y}
= {e, d, f}
২৫১.
x + y = 6 এবং 2x = 4 হলে, y এর মান কত?
  1. ক
    2
  2. খ
    4
  3. গ
    6
  4. ঘ
    8
ব্যাখ্যা
প্রশ্ন: x + y = 6 এবং 2x = 4 হলে, y এর মান কত?

সমাধান:
2x = 4
⇒ x = 4/2
∴ x = 2

এখন,
x + y = 6
⇒ y = 6 - 2
∴ y = 4
২৫২.
If L + M = 2N and N + O = 2L, then-
  1. ক
    L = M
  2. খ
    L + O = N + M
  3. গ
    M + O = N + L
  4. ঘ
    L = M + O
ব্যাখ্যা
Question: If L + M = 2N and N + O = 2L, then-

Solution: 
Let
L + M = 2N .............(1)
N + O = 2L .............(2)

(1) + (2)⇒
L + M = 2N
N + O = 2L
L + M + N + O = 2N + 2L
⇒ L + M + N + O - L - N = 2N + 2L - L - N
∴ M + O = N + L
২৫৩.
Simplify: z2 + 11z2 - 5z - 11z2 + 5z.
  1. ক
    z2
  2. খ
    - z2
  3. গ
    5z
  4. ঘ
    - 5z
ব্যাখ্যা
Question: Simplify: z2 + 11z2 - 5z - 11z2 + 5z.

Solution:
z2 + 11z2 - 5z - 11z2 + 5z
= z2
২৫৪.
In an examination, 41% of students failed in Economics, 35% of students failed in Geography and 39% of students failed in History, 5% of students failed in all the three subjects, 14% of students failed in Economics and Geography, 21% of students failed in Geography and History and 18% of students failed in History and Economics. Find the percentage of students who failed in only Economics.
  1. ক
    16%
  2. খ
    14%
  3. গ
    12%
  4. ঘ
    10%
ব্যাখ্যা
Question: In an examination, 41% of students failed in Economics, 35% of students failed in Geography and 39% of students failed in History, 5% of students failed in all the three subjects, 14% of students failed in Economics and Geography, 21% of students failed in Geography and History and 18% of students failed in History and Economics. Find the percentage of students who failed in only Economics.

Solution:

Now,
e = 5%
b + e = 14%
⇒ b = 9%

and, d + e = 18%
⇒ d = 13%
Therefore, Percentage of students who failed only in Economics = a = 41% - (b + e + d)
⇒ a = 41% - (9 + 5 + 13)%
⇒ a = 41% - 27%
⇒ a = 14%
Hence, 14% is the correct answer.
২৫৫.
If a + b + c = 15 and a2 + b2 + c2 = 77, what is the value of ab + bc + ca = ?
  1. ক
    82
  2. খ
    74
  3. গ
    66
  4. ঘ
    70
ব্যাখ্যা

Question: If a + b + c = 15 and a2 + b2 + c2 = 77, what is the value of ab + bc + ca = ?

Solution:
Given that, 
a + b + c = 15 and a2 + b2 + c2 = 77

We know,
(a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
⇒ 152 = 77 + 2(ab + bc + ca) ; [Substitute the given values]
⇒ 225 = 77 + 2(ab + bc + ca)
⇒ 2(ab + bc + ca) = 225 - 77
⇒ 2(ab + bc + ca) = 148
⇒ ab + bc + ca = 148/2
∴ ab + bc + ca = 74

২৫৬.
If A = {x : x is a natural number and a factor of 18} and B = {x : x is a natural number and less than 6}, then find A ∩ B. 
  1. ক
    { }
  2. খ
    {1, 2, 3}
  3. গ
    {1, 2, 3, 4, 5} 
  4. ঘ
    {1, 2, 3, 4, 5, 6, 9, 18} 
ব্যাখ্যা
Question: If A = {x : x is a natural number and a factor of 18} and B = {x : x is a natural number and less than 6}, then find A ∩ B. 

Solution: 
Here,
A = {1, 2, 3, 6, 9, 18} 
B = {1, 2, 3, 4, 5} 

Therefore,
A ∩ B = {1, 2, 3, 6, 9, 18} ∩ {1, 2, 3, 4, 5}
= {1, 2, 3}
২৫৭.
If f(x) = x/(x - 3), x ≠ 3, then what is the value of f -1(3)?
  1. ক
    4
  2. খ
    5
  3. গ
    6.5
  4. ঘ
    4.5
ব্যাখ্যা
Question: If f(x) = x/(x - 3), x ≠ 3, then what is the value of f -1(3)?

Solution:
let, f -1(3) = a
f(a) = 3

f(a) = a/(a - 3) 
So,
a/(a - 3) = 3
⇒ a = 3a - 9
⇒ 2a = 9
∴ a = 4.5

So, f -1(3) = 4.5
২৫৮.
Find the median of the given set of numbers 2, 6, 6, 8, 4, 2, 7, 9.
  1. ক
    6
  2. খ
    8
  3. গ
    4
  4. ঘ
    5
ব্যাখ্যা
Question: Find the median of the given set of numbers 2, 6, 6, 8, 4, 2, 7, 9.

Solution:
Arranging the given data in ascending order 2, 2, 4, 6, 6, 7, 8, 9
Total terms in the given data, (n) = 8 (It is even)

Median = {(8/2)th term + (8/2 + 1)th term}/2
= (4th term + 5th term)/2
= (6 + 6)/2
= 12/2
= 6
২৫৯.
Which letter in the alphabet is as far from G as T is from M?
  1. ক
    ক) N
  2. খ
    খ) O
  3. গ
    গ) P
  4. ঘ
    ঘ) M
ব্যাখ্যা

a b c d e f G h i j k l m N o p q r s t u v w x y z

২৬০.
If |2x + 5| < 3, then for what values of p and q will p < 3x - 2 < q hold?
  1. ক
    p = - 12,  q= - 3
  2. খ
    p = - 8, q = - 2
  3. গ
    p = - 10, q = - 4
  4. ঘ
    p = - 14, q = - 5
ব্যাখ্যা

Question: If |2x + 5| < 3, then for what values of p and q will p < 3x - 2 < q hold?

Solution:
Given that, 
|2x + 5| < 3
⇒ - 3 < 2x + 5 < 3
⇒ - 3 - 5 < 2x + 5 - 5 < 3 - 5
⇒ - 8 < 2x < -2
⇒ - 4 < x < -1    ; [dividing by 2]
⇒ - 12 < 3x < - 3  ; [Now multiply all parts by 3]
⇒ - 12 - 2 < 3x - 2 < - 3 - 2  ; [Subtract 2 from all parts]
⇒ - 14 < 3x - 2 < - 5

Now comparing with p < 3x - 2 < q, Then we get,
∴ p = - 14 and q = - 5

২৬১.
If 2x - 7 ≤ 11, then-
  1. ক
    x ≤ 3
  2. খ
    x ≥ - 3
  3. গ
    x ≤ 6
  4. ঘ
    x ≥ 9
  5. ঙ
    None of them
ব্যাখ্যা

Question: If 2x - 7 ≤ 11, then-

Solution:
Given, 
⇒ 2x - 7 ≤ 11
⇒ 2x - 7 + 7 ≤ 11 + 7
⇒ 2x ≤ 18
∴ x ≤ 9 

২৬২.
If a2 - 4a + 1 = 0, then the value of a3 + 1/a3 = ?
  1. ক
    64
  2. খ
    76
  3. গ
    52
  4. ঘ
    48
ব্যাখ্যা

Question: If a2 - 4a + 1 = 0, then the value of a3 + 1/a3 = ?

Solution: 
Given that, 
a2 - 4a + 1 = 0
⇒ a2 + 1 = 4a
⇒ (a2/a) + 1/a = 4a/a
∴ a + 1/a = 4

Now, 
a3 + 1/a3
= (a + 1/a)3 - 3 × a × (1/a) (a + 1/a)
= 43 - (3  × 4)
= 64 - 12
= 52

২৬৩.
Eight telephone poles are each 15 feet apart. What is the distance, in feet from the first to the last pole?
  1. ক
    ক) 60
  2. খ
    খ) 84
  3. গ
    গ) 105
  4. ঘ
    ঘ) 120
ব্যাখ্যা
Question: Eight telephone poles are each 15 feet apart. What is the distance, in feet from the first to the last pole?

Solution: 
Eight telephone poles are each 15 feet apart.

∴ the distance, in feet from the first to the last pole is = 15 × (8 - 1)
= 15 × 7
= 105  
২৬৪.
If 0 ≤ x ≤ 4 and y < 6, which of the following cannot be the value of xy?
  1. ক
    - 2
  2. খ
    0
  3. গ
    10
  4. ঘ
    24
ব্যাখ্যা

Question: If 0 ≤ x ≤ 4 and y < 6, which of the following cannot be the value of xy?

Solution:
0 ≤ x ≤ 4; x = 0, 1, 2, 3, 4 এবং y < 6

এখন,
x = 2 হলে,
a) xy = 2 × (- 1) = - 2   [y = - 1]
b) xy = 2 × 0 = 0          [y = 0]
c) xy = 2 × 5 = 10          [y = 5]

কিন্তু,
d) xy = 24 
= 1 × 24
= 2 × 12
= 3 × 8
= 4 × 6 

সুতরাং, 24 সম্ভব নয়। কারণ, প্রশ্নমতে y < 6.

২৬৫.
Find the midpoint of the line segment joining the points P1 = (4, -3) and P2 = (6, 1).
  1. ক
    (5, -1)
  2. খ
    (5, 1)
  3. গ
    (-1, 5)
  4. ঘ
    (0, 1)
ব্যাখ্যা

Question: Find the midpoint of the line segment joining the points P1 = (4, -3) and P2 = (6, 1).

Solution: 
দেওয়া আছে, 
P1 = (4, -3) এবং P2 = (6, 1)

আমরা জানি, 
দুটি বিন্দু (x1, y1) এবং (x2, y2)-এর জন্য মধ্যবিন্দু,
M = {(x1​ + x2)/2​​, (y1​ + y2​​)/2}

যেখানে, x1​ = 4, y1​ = - 3, x2 ​= 6, y2 ​= 1

এখন, 
Mx​ = (4 + 6)/2​ = 10​/2 = 5
এবং,
My ​= (- 3 + 1​)/2 = - 2​/2 = - 1 

∴ মধ্যবিন্দু = (5, - 1) 

২৬৬.
The cost of 21 pencils and 9 clippers is Tk. 819. The cost price of 7 pencils and 3 clippers is = ?
  1. ক
    ক) Tk. 204
  2. খ
    খ) Tk. 223
  3. গ
    গ) Tk. 409
  4. ঘ
    ঘ) Tk. 273
  5. ঙ
    ঙ) Tk. 208
ব্যাখ্যা

Cost of 21 pencils and 9 clippers = Tk. 819
Cost of 7 pencils and 3 clippers = 819/3 = Tk. 273

২৬৭.
In a group of 100 students, 72 students can play Football and 43 can play Cricket. How many can play both Football and Cricket?
  1. ক
    ক) 20
  2. খ
    খ) 13
  3. গ
    গ) 17
  4. ঘ
    ঘ) 15
ব্যাখ্যা
Question: In a group of 100 students, 72 students can play Football and 43 can play Cricket. How many can play both Football and Cricket?

Solution:
Let,
x people can play both Football and Cricket.
∴ People can only play football 72 - x.
People can only play cricket 43 - x 

Here,
72 - x + 43 - x + x = 100
⇒ 115 - x = 100
∴ x = 15 
 
২৬৮.
If in an infinite G.P. first term is equal to the twice of the sum of the remaining terms, then its common ratio is
  1. ক
    2
  2. খ
    1/2
  3. গ
    1/3
  4. ঘ
    2/3
ব্যাখ্যা
Question: If in an infinite G.P. first term is equal to the twice of the sum of the remaining terms, then its common ratio is

Solution: 
Let the G.P. = a + ar + ar2 + ....

ATQ, 
a = 2 ( ar + ar2 + ....)
⇒ 1/2 = r + r2 + ....
⇒ 1/2 = r/(1 - r)
⇒ 1 - r = 2r 
⇒ r = 1/3
২৬৯.
If m = 7 - 4√3, then what is the value of √m + (1/√m)?
  1. ক
    2
  2. খ
    3
  3. গ
    4
  4. ঘ
    5
  5. ঙ
    None
ব্যাখ্যা
Question: If m = 7 - 4√3, then what is the value of √m + (1/√m)?

Solution:
m = 7 - 4√3
⇒ m = 4 + 3 - 4√3
⇒ m = (2)2 + (√3)2 - 2 × 2√3
⇒ m = (2 - √3)2 [ a2 + b2 - 2ab = (a + b)2]
⇒ √m = 2 -√3

Now, 1/√m = 1/(2 - √3)
⇒ 1/√m = [1/(2 - √3) × {(2 + √3)/(2 + √3)}]
⇒ 1/√m = (4 - 3)/(2 + √3)
⇒ 1/√m = 1/(2 + √3)

∴ √m + 1/√m = 2 - √3 + 2 + √3 = 4.
২৭০.
If A = {1, 2, 3, 4, 5} then the number of proper subsets of A is
  1. ক
    120
  2. খ
    30
  3. গ
    31
  4. ঘ
    32
ব্যাখ্যা
প্রশ্ন: If A = {1, 2, 3, 4, 5} then the number of proper subsets of A is

সমাধান:
এখানে A সেটের উপাদান সংখ্যা ৫টি
A সেটের উপসেট সংখ্যা হবে ২৫ টি = ৩২টি 

∴ A সেটের প্রকৃত উপসেট এর  সংখ্যা =(৩২ - ১) টি
= ৩১টি
২৭১.
Find the greatest number-
  1. ক
    ক) 0.3
  2. খ
    খ) √0.3
  3. গ
    গ) 1/3
  4. ঘ
    ঘ) 2/5
ব্যাখ্যা
Question: Find the greatest number-

Solution: 
(0.3)2 = 0.09

(√0.3)2 = 0.3

(1/9)2 
= 1/81
= 0.012

(2/5)2 
= (0.4)2
= 0.16

∴ √0.3 সংখ্যাটি বৃহত্তম। 
২৭২.
If x - 1/x = 4; what is the value of x + 1/x?
  1. ক
    ক) 25
  2. খ
    খ) 3√5
  3. গ
    গ) 2√3
  4. ঘ
    ঘ) 2√5
ব্যাখ্যা
Question: If x - 1/x = 4; what is the value of x + 1/x?

Solution: 
(x - 1/x)2 = (4)2
x2 + 1/x2 - 2 = 16
x2 + 1/x2 = 18
x2 + 1/x2 + 2 = 18 + 2
(x + 1/x)2 = 20
x + 1/x = √20
x + 1/x = 2√5
২৭৩.
The value of (0.125 + 0.027)/(0.5 × 0.5 + 0.09 - 0.15) is- 
  1. ক
    0.2
  2. খ
    0.4
  3. গ
    0.5
  4. ঘ
    0.8
ব্যাখ্যা
Question: The value of (0.125 + 0.027)/(0.5 × 0.5 + 0.09 - 0.15) is- 

Solution: 
let,
0.5 = a
0.3 = b 

(0.125 + 0.027)/(0.5 × 0.5 + 0.09 - 0.15)
= {(0.5)3 + (0.3)3}/(0.5 × 0.5 + 0.3 × 0.3 - 0.5 × 0.3)
= (a3 + b3)/(a2 + b2 - ab)
= (a + b)(a2 + b2 - ab)/(a2 + b2 - ab)
= a + b
= 0.5 + 0.3
= 0.8
২৭৪.
In a class of 60 students, 30 take Bengali, 20 take Hindi, 15 takes no language. How many take both Bengali and Hindi? 
  1. ক
    ক) 5
  2. খ
    খ) 10
  3. গ
    গ) 15
  4. ঘ
    ঘ) 35
ব্যাখ্যা
Question: In a class of 60 students, 30 take Bengali, 20 take Hindi, 15 takes no language. How many take both Bengali and Hindi? 

Solution: 
Number of students who took one or both the languages = 60 - 15 = 45

Here,
n(B) = 30
n(H) = 20
n(B∪H) = 45

Now,
n(B∩H) = n(B) + n(H) - n(B∪H) 
= 30 + 20 - 45
= 5

∴ 5 students take both Bengali and Hindi.
২৭৫.
What is the value of (6x2 - 5y2)(6x2 + 5y2), if x = 1/√3 and y = 1/√5?
  1. ক
    2
  2. খ
    3
  3. গ
    4
  4. ঘ
    5
ব্যাখ্যা
Question: What is the value of (6x2 - 5y2)(6x2 + 5y2), if x = 1/√3 and y = 1/√5?

Solution:
(6x2 - 5y2)(6x2 + 5y2)
= (6x2)2 - (5y2)2
= 36x4 - 25y4
= 36 × (1/√3)4 - 25 × (1/√5)4
= 36 × (1/9) - 25 × (1/25)
= 4 - 1
= 3
২৭৬.
State the order of the matrix-
  1. ক
    2 × 3
  2. খ
    3
  3. গ
    3 × 2
  4. ঘ
    2
ব্যাখ্যা

Question: State the order of the matrix-

Solution:
ম্যাট্রিক্সের মাত্রা বা ক্রম(Order of Matrix): একটি ম্যাট্রিক্সের সারি ও কলামের সংখ্যা যথাক্রমে m ও n হলে, ঐ ম্যাট্রিক্সকে m × n ক্রমের বা আকারের ম্যাট্রিক্স বলা হয়।
অর্থাৎ ম্যাট্রিক্সের আকার বা মাত্রা বোঝাতে প্রথমে সারি এবং পরে কলাম উল্লেখ করা হয়।
প্রদত্ত ম্যাট্রিক্সটি একটি আয়তাকার ম্যাট্রিক্স কারণ এর সারি ও কলাম অসমান।
এখানে,
সারি m = 3 এবং কলাম n = 2

∴ প্রদত্ত ম্যাট্রিক্সটি একটি 3 × 2 আকারের ম্যাট্রিক্স।

২৭৭.
If 2x + y = 10 and x - y = 2, then x + y = ?
  1. ক
    8
  2. খ
    5
  3. গ
    6
  4. ঘ
    4
ব্যাখ্যা
Question: If 2x + y = 10 and x - y = 2, then x + y = ?

Solution:
Given,
2x + y = 10.............(1)
and x - y = 2.........(2)

from, (1) + (2) we get
2x + y + x - y = 10 + 2
⇒ 3x = 12
∴ x = 4

Now,
4 - y = 2
⇒ 4 - 2 = y
∴ y = 2

∴ x + y = 4 + 2 = 6
২৭৮.
একটি বাগানে ক টি গাছ এবং খ টি দোয়েল পাখি আছে। যদি প্রতিটি দোয়েল ১টি করে গাছে বসে তবে ১টি দোয়েল উড়তে থাকে। যদি প্রতি গাছে ২টি করে দোয়েল বসে তাহলে ১টি গাছে দোয়েল বসেনা বা খালি থাকে। গাছ ও দোয়েলের সংখ্যা যথাক্রমে-
  1. ক
    ৩ এবং ৪
  2. খ
    ৪ এবং ৫
  3. গ
    ৫ এবং ৬
  4. ঘ
    ৬ এবং ৭
  5. ঙ
    কোনটিই নয়
ব্যাখ্যা
প্রশ্ন: একটি বাগানে ক টি গাছ এবং খ টি দোয়েল পাখি আছে। যদি প্রতিটি দোয়েল ১টি করে গাছে বসে তবে ১টি দোয়েল উড়তে থাকে। যদি প্রতি গাছে ২টি করে দোয়েল বসে তাহলে ১টি গাছে দোয়েল বসেনা বা খালি থাকে। গাছ ও দোয়েলের সংখ্যা যথাক্রমে-

সমাধান:
প্রশ্নমতে,
ক + ১ = খ
এবং, ২(ক - ১) = খ
⇒ ২ক - ২ = ক + ১
∴ ক = ৩
এবং  খ = ৩ + ১ = ৪
২৭৯.
  1. ক
    10
  2. খ
    8
  3. গ
    6
  4. ঘ
    4
ব্যাখ্যা
Question:

Solution:
২৮০.
The sum of three consecutive multiples of 4 is 300. What is the largest number?
  1. ক
    96
  2. খ
    100
  3. গ
    104
  4. ঘ
    108
ব্যাখ্যা

Question: The sum of three consecutive multiples of 4 is 300. What is the largest number?

Solution: 
Let,
First multiple 4x
Second multiple 4(x + 1) = 4x + 4
Third multiple 4(x + 2) = 4x + 8

ATQ,
4x + 4x + 4 + 4x + 8 = 300
⇒ 12x + 12 = 300
⇒ 12x = 288
⇒ x = 288/12
∴ x = 24

So, largest number = 4x + 8
= (4 × 24) + 8
= 104

২৮১.
If x + 1/x = 7, find the value of 3x/(x2 - 6x + 1)
  1. ক
    1
  2. খ
    3
  3. গ
    0
  4. ঘ
    - 2
ব্যাখ্যা
Question: If x + 1/x = 7, find the value of 3x/(x2 - 6x + 1)

Solution:
Given,
x + 1/x = 7
⇒ (x2 + 1)/x = 7
∴ x2 + 1 = 7x

Now,
3x/(x2 - 6x + 1)
= 3x/(x2 + 1 - 6x)
= 3x/(7x - 6x)
= 3x/x
= 3
২৮২.
Solve the equation x2 + 4x - 5 = 0.
  1. ক
    5, 1
  2. খ
    5, - 1
  3. গ
    - 5, 1
  4. ঘ
    - 5, - 1
ব্যাখ্যা
Question: Solve the equation x2 + 4x - 5 = 0.

Solution:
x2 + 4x - 5 = 0
⇒ x2 - 1x + 5x - 5 = 0
⇒ x(x - 1) + 5(x - 1) = 0
⇒ (x - 1)(x + 5) = 0
Hence, (x - 1) = 0, and (x + 5) =0
⇒ x - 1 = 0
∴ x = 1

similarly, x + 5 = 0
∴ x = - 5.

Therefore,
x = - 5 & x = 1
২৮৩.
Let U = {1,2,3,4,5,6,7,8}, A = {2,3,6}, and B = {1,4,5}. Find Ac∪Bc.
  1. ক
    {1,2,3,4,5,6,7,8}
  2. খ
    { }
  3. গ
    {2,3,6}
  4. ঘ
    {1,4,5,7,8}
  5. ঙ
    {7,8}
ব্যাখ্যা

Question: Let U = {1,2,3,4,5,6,7,8}, A = {2,3,6}, and B = {1,4,5}. Find Ac ∪ Bc.

Solution:
Complement of A:
A = {2,3,6}
U = {1,2,3,4,5,6,7,8}
Ac = U - A = {1,4,5,7,8}

Complement of B:
B = {1,4,5}
Bc = U - B = {2,3,6,7,8}

Union of Complements:
Ac ∪ Bc = {1,4,5,7,8} ∪ {2,3,6,7,8} = {1,2,3,4,5,6,7,8} = U

∴Ac ∪ Bc = {1,2,3,4,5,6,7,8}

২৮৪.
If 2x2 - 5x + 3 = 0 and 4x2 - (k + 1)x + 6 = 0, Then find the values of k for each of the following quadratic equations, so that they have a common root?
  1. ক
    9
  2. খ
    5
  3. গ
    - 3
  4. ঘ
    8
  5. ঙ
    12
ব্যাখ্যা
Question: If 2x2 - 5x + 3 = 0 and 4x2 - (k + 1)x + 6 = 0, Then find the values of k for each of the following quadratic equations, so that they have a common root?

Solution:
দেওয়া আছে,
দ্বিঘাত সমীকরণ দুইটি,
2x2 - 5x + 3 = 0 .....(1)
এবং 4x2 - (k + 1)x + 6 = 0 ........(2)

যদি দুটি দ্বিঘাত সমীকরণের উভয় উৎপাদক সাধারণ হয়, তাহলে তাদের সহগগুলি সমানুপাতিক হবে। অর্থাৎ
a1/a2 = b1/b2 = c1/c2 .......(3)

এখন,
1 নং সমীকরণ হতে সহগগুলির তুলনা করে পাই,
a1 = 2 , b1 = - 5, c1 = 3
এবং 2 নং সমীকরণ হতে সহগগুলির তুলনা করে পাই,
a2 = 4, b2 = - (k + 1), c2 = 6

এখন,
3 নং সমীকরণ হতে সমানুপাতিকতার শর্ত ব্যবহার করে,
2/4 = - 5/- (k + 1) = 3/6

প্রথম দুইটি অনুপাত হতে পাই,
⇒ 2/4 = - 5/- (k + 1)
⇒ 2k + 2 = 20
⇒ 2k = 18
∴ k = 9

আবার, শেষ দুইটি অনুপাত হতে পাই,
⇒ - 5/- (k + 1) = 3/6
⇒ 3k + 3 = 30
⇒ 3k = 27
∴ k = 9

∴ উভয় সমীকরণে একটি সাধারণ মূল 9 থাকবে।

২৮৫.
What should be the value of "P" so that the expression (9 − 24x + Px2) becomes a perfect square?
  1. ক
    4
  2. খ
    6
  3. গ
    8
  4. ঘ
    16
ব্যাখ্যা

Question: What should be the value of "P" so that the expression (9 − 24x + Px2) becomes a perfect square?

Solution:
(9 − 24x + Px2)
= (3)2 − 2 × 3 × 4x + (4x)2 + Px2 - (4x)2
= (3 - 4x)2 + Px2 - 16x2

∴ the expression becomes a perfect square if,
Px2 - 16x2 = 0
⇒ Px2 = 16x2
∴ P = 16

২৮৬.
If (x - y)2 = 12 and xy = 1, then what is the value of (x2 + y2)?
  1. ক
    ক) 12
  2. খ
    খ) 14
  3. গ
    গ) 17
  4. ঘ
    ঘ) 18
ব্যাখ্যা
Question: If (x - y)2 = 12 and xy = 1, then what is the value of (x2 + y2)?

Solution: 
x2 + y2
= (x - y)2 + 2xy 
= 12 + 2 × 1
= 12 + 2
= 14
২৮৭.
If (a - 18)2 + (b - 12)2 + (c - 6)2 = 0 then, What is the value of (a + b + c)1/2 = ?
  1. ক
    9
  2. খ
    6
  3. গ
    12
  4. ঘ
    14
ব্যাখ্যা

Question: If (a - 18)2 + (b - 12)2 + (c - 6)2 = 0 then, What is the value of (a + b + c)1/2 = ?
​
​Solution:
​দেওয়া আছে,
​(a - 18)2 + (b - 12)2 + (c - 6)2 = 0
​
​আমরা জানি, যেকোনো বর্গের যোগফল শূন্য হলে প্রতিটি বর্গই শূন্য হবে।
​সুতরাং,
​​(a - 18)2 = 0
⇒ ​a - 18 = 0
∴ ​a = 18
​
​আবার,
​(b - 12)2 = 0
⇒ ​​b - 12 = 0
∴ ​​b = 12
​
​এবং
​(c - 6)2 = 0
⇒ ​​c - 6 = 0
∴ ​c = 6
​
​প্রদত্ত রাশি,
​ (a + b + c)1/2
​= (18 + 12 + 6)1/2
​= (36)1/2
​= 6

২৮৮.
If x/2 = y/3 = z/4 = (2x - 3y - + 5z)/k, then the value of k is -
  1. ক
    ক) 12
  2. খ
    খ) 15
  3. গ
    গ) 16
  4. ঘ
    ঘ) 18
ব্যাখ্যা

Let,
x/2 = y/3 = z/4 = m
Then, x = 2m, y = 3m, z = 4m
∴ x/2 = (2x - 3y + 5z)/k = 2m/2
⇒ (2 × 2m - 3 × 3m + 5 × 4m)/k = m
⇒ k = 4 - 9 + 20
= 15.

২৮৯.
If x8 - 1442x4 + 1 = 0, then a possible value of x - (1/x) is:
  1. ক
    4
  2. খ
    6
  3. গ
    8
  4. ঘ
    None of the above
ব্যাখ্যা
Question: If x8 - 1442x4 + 1 = 0, then a possible value of x - (1/x) is:

Solution:
Given,
x8 - 1442x4 + 1 = 0
⇒ (x8/x4) - (1442x4/x4) + (1/x4) = 0
⇒ x4 - 1442 + (1/x4) = 0
⇒ x4 + (1/x4) = 1442
⇒ x4 + (1/x4) + 2 = 1444
⇒ {x2 + (1/x2)}2 = (38)2
⇒ {x2 + (1/x2)} = 38
⇒ {x2 + (1/x2)} - 2 = 36
⇒ {x - (1/x)}2 = 36
∴ x - (1/x) = 6
২৯০.
If x = 2y = 4z and xyz = 64, find the value of x.
  1. ক
    6
  2. খ
    12
  3. গ
    8
  4. ঘ
    11
ব্যাখ্যা

Question: If x = 2y = 4z and xyz = 64, find the value of x.

Solution:
Given,
x = 2y = 4z
So, y = x / 2 and z = x / 4

Now,
xyz = 64
⇒ x × (x/2) × (x/4) = 64
⇒ x3/8 = 64
⇒ x3 = 64 × 8
⇒ x3 = 512
⇒ x = ∛512
∴ x = 8

২৯১.
The value of (0.125 + 0.027)/(0.5 × 0.5 + 0.09 - 0.15) is- 
  1. ক
    ক) 0.02
  2. খ
    খ) 0.2
  3. গ
    গ) 0.08
  4. ঘ
    ঘ) 0.8
ব্যাখ্যা
let 
0.5 = a
0.3 = b 

(0.125 + 0.027)/(0.5 × 0.5 + 0.09 - 0.15)
= {(0.5)3 + (0.3)3}/(0.5 × 0.5 + 0.3 × 0.3 - 0.5 × 0.3)
= (a3 + b3)/(a2 + b2 - ab)
= (a + b)(a2 + b2 - ab)/(a2 + b2 - ab)
= a + b
= 0.5 + 0.3
= 0.8
২৯২.
  1. ক
    1
  2. খ
    1/2
  3. গ
    - 3/2
  4. ঘ
    - 1
ব্যাখ্যা
Question:


Solution:
২৯৩.
What is the sum of the first 20 terms of the series 6, 11, 16, 21, ...?
  1. ক
    1010
  2. খ
    1190
  3. গ
    1260
  4. ঘ
    1070
ব্যাখ্যা

Question: What is the sum of the first 20 terms of the series 6, 11, 16, 21, ...?

Solution:
এটি একটি সমান্তর ধারা (arithmetic series)।
এখানে,
​প্রথম পদ, a = 6
সাধারণ অন্তর, d = 11 - 6 = 5
পদের সংখ্যা, n = 20

সমান্তর ধারার n-সংখ্যক পদের সমষ্টি সূত্র:
Sn = (n/2) [2a + (n - 1)d]

∴ প্রথম 20টি পদের সমষ্টি:
S20 = (20/2) [2 × 6 + (20 - 1) × 5]
= 10 [12 + 19 × 5]
= 10 [12 + 95]
= 10 × 107
= 1070

∴ প্রথম 20টি পদের সমষ্টি = 1070

২৯৪.
If (2p + 1/p) = 4, the value of (p3 + 1/8p3) is -
  1. ক
    ক) 3
  2. খ
    খ) 4
  3. গ
    গ) 5
  4. ঘ
    ঘ) 5/2
ব্যাখ্যা
Question: If (2p + 1/p) = 4, the value of (p3 + 1/8p3) is -

Solution:
Given,
(2p + 1/p) = 4
Or, (1/2) (2p + 1/p) = 4 × 1/2
Or, p + 1/2p = 2

Now, (p3 + 1/8p3) = (p)3 + (1/2p)3
= (p + 1/2p)3 - 3 . p . 1/2p . (p + 1/2p)
= (2)3 - 3 . (1/2) . 2
= 8 - 3
= 5
২৯৫.
If 9a2 - 9a - 4 is divided by 3a - 4, the result is:
  1. ক
    3a + 1
  2. খ
    2a + 1
  3. গ
    3a - 2
  4. ঘ
    3a - 1
ব্যাখ্যা
Question: If 9a2 - 9a - 4 is divided by 3a - 4, the result is:

Solution:
9a2 - 9a - 4
= 9a2 - 12a + 3a - 4
= 3a(3a - 4) + 1(3a - 4)
= (3a - 4)(3a + 1)

Now,
(9a2 - 9a - 4)/(3a - 4) = {(3a - 4)(3a + 1)}/(3a + 1) = (3a + 1)
২৯৬.
If the first and sixth term of a geometric series are respectively 1/2 and 1/64, then the common ratio is____
  1. ক
    ক) 1/4
  2. খ
    খ) 1/2
  3. গ
    গ) 1
  4. ঘ
    ঘ) 2
ব্যাখ্যা

Given, ar0 = 1/2
Or, a = 1/2
And, ar5 = 1/64
Or, r5 = 1/32
Or, r5 = (1/2)5
∴ Ratio, r = 1/2

২৯৭.
If, a - (1/a) = 2, then the value of a3 - (1/a3) is -
  1. ক
    15
  2. খ
    2
  3. গ
    14
  4. ঘ
    11
ব্যাখ্যা

Given,
a - (1/a) = 2
On cubing both of the sides,
a3 - (1/a3) - 3 × a × (1/a) (a - 1/a) = 23
⇒ a3 - (1/a3) - 3(a - 1/a) = 8
⇒ a3 - (1/a3) - 3 × 2 = 8
⇒ a3 - (1/a3) = 8 + 6
∴ a3 - (1/a3) = 14.

২৯৮.
Complete the following series: 9, 11, 15, 23, 39, ?
  1. ক
    ক) 64
  2. খ
    খ) 42
  3. গ
    গ) 56
  4. ঘ
    ঘ) 71
  5. ঙ
    ঙ) 60
ব্যাখ্যা
9+2 = 11
11+4 = 15
15+8 = 23
23+16 = 39
39+32 = 71
⇒ ? = 71
২৯৯.
If (11z - 1)2 = 441, then which one of the following could equal z?
  1. ক
    2
  2. খ
    5
  3. গ
    9
  4. ঘ
    15
ব্যাখ্যা
Question: If (11z - 1)2 = 441, then which one of the following could equal z?

Solution:
(11z - 1)2 = 441
⇒ 11z - 1 = √441
⇒ 11z - 1 = 21
⇒ 11z = 22
∴ z = 2
৩০০.
If a + b + c = 2s, then [(s - a)2 + (s - b)2 + (s - c)2 + s2] =?
  1. ক
    (s2 - a2 - b2 - c2)
  2. খ
    (s2 + a2 + b2 + c2)
  3. গ
    (a2 + b2 + c2)
  4. ঘ
    (4s2 - a2 - b2 - c2)
ব্যাখ্যা
Question: If a + b + c = 2s, then [(s - a)2 + (s - b)2 + (s - c)2 + s2] =?

Solution:
[(s - a)2 + (s - b)2 + (s - c)2 + s2]
= (s2 + a2 - 2as) + (s2 + b2 - 2sb) + (s2 + c2 - 2sc) + s2
= 4s2 + (a2 + b2 + c2) - 2s(a + b + c)
= 4s2 + a2 + b2 + c2 - 4s2
= a2 + b2 + c2