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Bank Math

মোট প্রশ্ন১৬,১২৪এই পাতা১০০প্রতি পাতা১০০
প্রশ্ন দেখুনঅনুশীলন
ঘনত্ব
উত্তর
উত্তরিতবর্তমানপুনরায় দেখুনঅসম্পূর্ণ

Bank Math

PrepBank · পাতা ৭৫ / ১৬১ · ৭,৪০১–৭,৫০০ / ১৬,১২৪

৭,৪০১.
What kind of number is 1?
  1. ক
    ক) Prime number
  2. খ
    খ) Composite Number
  3. গ
    গ) Prime & Composite Number
  4. ঘ
    ঘ) None
ব্যাখ্যা
১ এর চেয়ে বড় যে সকল সংখ্যাকে শুধু ১ এবং ঐ সংখ্যা ছাড়া আর কোনো সংখ্যা দ্বারা ভাগ করা যায় না, তাদেরকে মৌলিক সংখ্যা বলে। 
অর্থাৎ মৌলিক সংখ্যার উৎপাদক হবে দুইটি: ১ এবং শুধুমাত্র সেই সংখ্যাটি।
 
অর্থাৎ, ১ মৌলিক বা যৌগিক কোনটিই নয়।
৭,৪০২.
The four walls and ceiling of a room of length 25 ft., breadth 12 ft. and height 10 ft. are to be painted. Painter A can Paint 200 sqr.ft in 5 days, painter B can paint 250 sqr.ft in 2 days. If A & B work together, in how many days do they finish the job?
  1. ক
    ক) 4(9/11)
  2. খ
    খ) 5(8/13)
  3. গ
    গ) 5(11/12)
  4. ঘ
    ঘ) 6(10/33)
  5. ঙ
    ঙ) 7(6/11)
ব্যাখ্যা

Total area to be painted = 25×12 +2(10×12 + 10×25) = 1040 sqr.ft

A paints = 200/5 = 40 sqr.ft per day
B paints = 250/2 = 125 sqr.ft per day

A + B = 40 + 125 = 165 sqr.ft
Number of days = 1040/165 = 6(10/33)

৭,৪০৩.
If x ≠ 0 and x = √(4xy - 4y2), then in terms of y, x = ?
  1. ক
    2y
  2. খ
    y
  3. গ
    y/2
  4. ঘ
    - 2y
ব্যাখ্যা
Question: If x ≠ 0 and x = √(4xy - 4y2), then in terms of y, x = ?

Solution:
x = √(4xy - 4y2)
⇒ x2 = 4xy - 4y2
⇒ x2 - 4xy + 4y2 = 0
⇒ (x - 2y)2 = 0
∴ x = 2y
৭,৪০৪.
The salaries of A, B and C are in the ratio 1 : 3 : 4. If the salaries are increased by 5%, 10% and 10% respectively, then the increased salaries will be in the ratio - 
  1. ক
    11 : 23 : 34
  2. খ
    21 : 66 : 88
  3. গ
    22 : 36 : 28
  4. ঘ
    12 : 61 : 18
ব্যাখ্যা

Question: The salaries of A, B and C are in the ratio 1 : 3 : 4. If the salaries are increased by 5%, 10% and 10% respectively, then the increased salaries will be in the ratio -

Solution:
Given that
Salary has  in 1 : 3 : 4 ratio
Let
A's Salary = Tk. 100
B's Salary = Tk. 300
C's Salary = Tk. 400

Now,
5% increase in A's Salary,
A's new Salary = (100 + 5% of 100) = Tk. 105

B's Salary increases by 10%, Then,
B's new Salary = (300 + 10% of 300) = Tk. 330

C's Salary increases by 10%,
C's new Salary = (400 + 10% of 400) = Tk. 440

Then, ratio of increased Salary,
A : B : C = 105 : 330 : 440 = 21 : 66 : 88

৭,৪০৫.
A, B, C enter into a partnership investing Tk. 40,000, Tk. 50,000 and Tk. 60,000 respectively. Total profit of A, B, C is Tk. 45,000. So B's profit is-
  1. ক
    Tk. 10,000
  2. খ
    Tk. 25,000
  3. গ
    Tk. 15,000
  4. ঘ
    Tk. 20,000
ব্যাখ্যা

Question: A, B, C enter into a partnership investing Tk. 40,000, Tk. 50,000 and Tk. 60,000 respectively. Total profit of A, B, C is Tk. 45,000. So B's profit is-

Solution:
A : B : C = 40000 : 50000 : 60000
A : B : C = 4 : 5 : 6

∴ Sum of the ratio = 4 + 5 + 6 = 15

∴ B's profit = 45000 × (5/15)
= 45000 × (1/3)
= 15,000

∴ B's profit is Tk. 15,000

৭,৪০৬.
The earnings of Rahim is 12000 Tk every month and Anish is 191520 per year. If the monthly expenses of every person are around 9960 Tk. Find the ratio of the savings.
  1. ক
    12 : 30
  2. খ
    17 : 18
  3. গ
    8 : 25
  4. ঘ
    17 : 50
ব্যাখ্যা
Question: The earnings of Rahim is 12000 Tk every month and Anish is 191520 per year. If the monthly expenses of every person are around 9960 Tk. Find the ratio of the savings.

Solution: 
Savings of Rohan per month = Tk. (12000-9960) = Tk. 2040

Yearly income of Anish = Tk. 191520
Hence, the monthly income of Anish = Tk. 191520/12 = Tk. 15960.
So, the savings of Anish per month = Tk. (15960 – 9960) = Tk. 6000

Thus, the ratio of savings of Rohan and Anish is 2040 : 6000 = 17 : 50.
৭,৪০৭.
Average of 80 numbers are 42. When 5 more numbers are included, the average of 85 numbers become 45. Find the average of 5 numbers.
  1. ক
    ক) 93
  2. খ
    খ) 112
  3. গ
    গ) 115
  4. ঘ
    ঘ) 119
ব্যাখ্যা
Sum of 80 numbers
= 80 × 42 = 3360
Now, sum of 85 numbers
= 85 × 45 = 3825
Hence, sum of 5 numbers
= 3825 - 3360 = 465
Average of five numbers
= 465/5
 = 93
৭,৪০৮.
The values of k for equation 3x2 - 6x + k = 0 to have real roots is —
  1. ক
    k < 3
  2. খ
    k ≤ 4
  3. গ
    k ≥ 3
  4. ঘ
    k ≤ 3
ব্যাখ্যা

Question: The values of k for equation 3x2 - 6x + k = 0 to have real roots is —

Solution:
এখানে, 3x2 - 5x + k = 0 সমীকরণকে ax2 + bx + c = 0 সমীকরণের সাথে তুলনা করলে করি।

আমরা জানি,
বাস্তবমূলের ক্ষেত্রে,
b2 - 4ac ≥ 0
⇒ (- 6)2 - 4 × 3 × k ≥ 0
⇒ 36 - 12k ≥ 0
⇒ 12k ≤ 36
⇒ k ≤ 3

৭,৪০৯.
The present ages of three cousins are in the ratio of 4 : 5 : 6. Five years ago, their total age was 60 years. In three years, what will be the age of the youngest cousin?
  1. ক
    19 years
  2. খ
    23 years
  3. গ
    25 years
  4. ঘ
    28 years
ব্যাখ্যা

Question: The present ages of three cousins are in the ratio of 4 : 5 : 6. Five years ago, their total age was 60 years. In three years, what will be the age of the youngest cousin?

Solution: Present age ratio of three cousins is 4 : 5 : 6
Let, their age is 4X, 5X and 6X respectively.

ATQ,
4X - 5 + 5X - 5 + 6X - 5 = 60
Or, 15X - 15 = 60
Or, 15X = 75
Or, X = 5

∴ The present age of the youngest cousin is = 4X = 4 × 5 = 20 years.
In three years, his age will be = (20 + 3) = 23 years.

৭,৪১০.
After travelling 50km, a train is meeting with an accident and travels at (3/4)th of the usual Speed and reaches 45 min late. Had the accident happened 10km further on it would have reached 35 min late. Find the usual Speed?
  1. ক
    ক) 20 km/hr
  2. খ
    খ) 25 km/hr
  3. গ
    গ) 30 km/hr
  4. ঘ
    ঘ) 32 km/hr
ব্যাখ্যা

Here there are 2 cases
Case 1: accident happens at 50 km
Case 2: accident happens at 60 km
Difference between two cases is only for the 10 kms between 50 and 60.
Time difference of 10 minutes is only due to these 10 kms.
In case 1, 10 kms between 50 and 60 is covered at (3/4)th Speed.
In case 2, 10 kms between 50 and 60 is covered at usual Speed.
So the usual Time 't' taken to cover 10 kms, can be found out as below.
(4/3)t – t = 10 mins [s = vt]
⇒ t = 30 mins, d = 10 kms
so usual
Speed = Distance/Time
= 10/30min
= 10/(1/2) km/hr.
= 20 km/hr.

৭,৪১১.
A hollow iron pipe is 21 cm long and its external diameter is 8 cm. If the thickness of the pipe is 1 cm and iron weighs 8 g/cm3, then the weight of the pipe is:
  1. ক
    36 kg
  2. খ
    3.696 g
  3. গ
    3.696 kg
  4. ঘ
    6.96 g
ব্যাখ্যা
Question: A hollow iron pipe is 21 cm long and its external diameter is 8 cm. If the thickness of the pipe is 1 cm and iron weighs 8 g/cm3, then the weight of the pipe is:

Solution: 
Volume of iron = π (42 - 32) 21
=  π (16 - 9) 21
= π 7 × 21
= 462 cm3

weight of the pipe is = 462 × 8 g
= 3696 g
= 3.696 kg
৭,৪১২.
It was Monday on Jan 1, 2018. What was the day of the week on Jan 1, 2022?
  1. ক
    Monday
  2. খ
    Friday
  3. গ
    Saturday
  4. ঘ
    Sunday
ব্যাখ্যা
Question: It was Monday on Jan 1, 2018. What was the day of the week on Jan 1, 2022?

Solution: 
We know that,
The first day and the last day of a year are same day of the week, except leap year.
In a leap year, the week day of the last day of the year is next to the week day of the first day of the year.

Now,
It was Monday on Jan 1, 2018 so, it was Monday on Dec 31, 2018.
It was Tuesday on Jan 1, 2019 so, it was Tuesday on Dec 31, 2019.
It was Wednesday on Jan 1, 2020 (which was leap year) so, it was Thursday on Dec 31, 2020.
It was Friday on Jan 1, 2021 so, it was Friday on Dec 31, 2021.
It was Saturday on Jan 1, 2022.
৭,৪১৩.
The difference between simple and compound interests compounded annually on a certain sum of money for 2 years at 4% per annum is 10 taka. The sum is:
  1. ক
    ক) 250 taka
  2. খ
    খ) 6050 taka
  3. গ
    গ) 6150 taka
  4. ঘ
    ঘ) 6250 taka
ব্যাখ্যা
Question: The difference between simple and compound interests compounded annually on a certain sum of money for 2 years at 4% per annum is 10 taka. The sum is:

Solution: 
ধরি,
আসল x টাকা 

সরল সুদ = x × 2 × 4/100
= 2x/25
= 0.08 x

চক্রবৃদ্ধি সুদ = x (1 + 0.04)2 - x
= x × 1.04 × 1.04 - x
= 1.0816 x - x
= 0.0816 x 

প্রশ্নমতে,
0.0816 x - 0.08 x = 10
⇒ 0.0016 x = 10
⇒ x = 10/0.0016
∴ x = 6250
৭,৪১৪.
If (x2 - x + 2)/2 = 4 then x could be equal to -
  1. ক
    3
  2. খ
    2
  3. গ
    4
  4. ঘ
    - 3
ব্যাখ্যা
Question: If (x2 - x + 2)/2 = 4 then x could be equal to -

Solution:
(x2 - x + 2)/2 = 4
⇒ x2 - x + 2 = 8
⇒ x2 - x + 2 - 8 = 0
⇒ x2 - x - 6 = 0
⇒ x2 - 3x + 2x - 6 = 0
⇒ x(x - 3) + 2(x - 3) = 0
⇒ (x - 3) (x + 2) = 0

Either,
x - 3 = 0
∴ x = 3

Or,
x + 2 = 0
∴ x = - 2

∴ x = (3, - 2)
৭,৪১৫.
If a person walks at 18 km/h instead of 12 km/h, he would have walked 24 km more. The actual distance travelled by him is-
  1. ক
    48 km
  2. খ
    50 km
  3. গ
    52 km
  4. ঘ
    53 km
ব্যাখ্যা
Question: If a person walks at 18 km/h instead of 12 km/h, he would have walked 24 km more. The actual distance travelled by him is-

Solution:
Let
the actual distance travelled be x km.

Then,
x/12 = (x + 24)/18
⇒ 18x = 12x + 288
⇒ 6x = 288
∴ x = 48 km
৭,৪১৬.
The average age of husband, wife and their child 3 years ago was 27 years and that of wife and the child 5 years ago was 20 years. The present age of the husband is:
  1. ক
    35 years
  2. খ
    40 years
  3. গ
    45 years
  4. ঘ
    50 years
ব্যাখ্যা
Question: The average age of husband, wife and their child 3 years ago was 27 years and that of wife and the child 5 years ago was 20 years. The present age of the husband is:

Solution:
Sum of the present ages of husband, wife and child =(27 × 3 + 3 × 3) years
= (81 + 9) years
= 90 years

Sum of the present ages of wife and child =(20 × 2 + 5 × 2) years 
= (40 + 10) years 
= 50 years 

∴ Husband's present age =(90 - 50) years = 40 years.
৭,৪১৭.
Mohan lent some amount of money at 8% simple interest and an equal amount of money at 10% simple interest each for two years. If his total interest was TK. 720, what amount was lent in each case ?
  1. ক
    TK. 1700
  2. খ
    TK. 1800
  3. গ
    TK. 1900
  4. ঘ
    TK. 2000
ব্যাখ্যা

Question: Mohan lent some amount of money at 8% simple interest and an equal amount of money at 10% simple interest each for two years. If his total interest was TK. 720, what amount was lent in each case ?

Solution:
Let the amount invested = TK. P
According to the questions,
⇔ (P × 8 × 2)/100 + (P × 10 × 2)/100 = 720
⇔ (16P + 20P)/100 = 720
⇔ 36P = 72000
⇔ P = 2000

৭,৪১৮.
The average of 20 numbers is 35. If two numbers, 40 and 50 are discarded, then the average of the remaining numbers is nearly -
  1. ক
    ক) 32.89
  2. খ
    খ) 33.89
  3. গ
    গ) 34.59
  4. ঘ
    ঘ) 33.1
ব্যাখ্যা
Question: The average of 20 numbers is 35. If two numbers, 40 and 50 are discarded, then the average of the remaining numbers is nearly -

Solution:
Sum of 20 numbers = (20 × 35) = 700

after discarding 2 numbers of 40 and 50 the new average is = (700 - 40 - 50)/18
= 33.89

৭,৪১৯.
A rectangle is 14 cm long and 10 cm wide. If the length is reduced by x cm and its width is increased also by x cm so as to make it a square, then its area changes by:
  1. ক
    4
  2. খ
    144
  3. গ
    12
  4. ঘ
    2
ব্যাখ্যা
Question: A rectangle is 14 cm long and 10 cm wide. If the length is reduced by x cm and its width is increased also by x cm so as to make it a square, then its area changes by:

Solution: 
ATQ,
14 - x = 10 + x
⇒ x + x = 14 - 10
⇒ 2x = 4
∴ x = 2

Area of rectangle = 14 × 10 cm2
= 140 cm2

Area of square = 122
= 144 cm2

∴ Area changes by = 144 - 140 cm2 
= 4 cm2
৭,৪২০.
If x = a + (1/a) and y = a - (1/a) then x4 + y4 - 2x2y2 = ?
  1. ক
    4
  2. খ
    8
  3. গ
    16
  4. ঘ
    22
ব্যাখ্যা
Question: If x = a + (1/a) and y = a - (1/a) then x4 + y4 - 2x2y2 = ?

Solution:
Given,
x = a + 1/a
y = a - 1/a

x + y = a + (1/a) + a - (1/a) = 2a
x - y = a + (1/a) - a + (1/a) = 2/a

Now,
x2 + y2- 2. x2. y2 = (x2)2 + (y2)2 - 2. x2 .y2
= (x2 - y2)2
= {(x + y)(x - y)}2
= {(2a)(2/a)}2
= 42
= 16
৭,৪২১.
What is the sum of the first nine terms of the given sequence?
5, 7, 12, 19, ..
  1. ক
    ক) 645
  2. খ
    খ) 548
  3. গ
    গ) 642
  4. ঘ
    ঘ) 523
ব্যাখ্যা
প্রশ্ন: What is the sum of the first nine terms of the given sequence?
         5, 7, 12, 19, ...
সমাধান:
প্রদত্ত অনুক্রমটি একটি ফিবোনাক্কি অনুক্রম। 
অতএব অনুক্রমটি হবে  5, 7, 12, 19, 31, 50, 81,131, 212, 343, 555, 898

প্ৰথম নয়টি পদের সমষ্টি, 
F(9) = F(11) - F(2) = 555 - 7 = 548
৭,৪২২.
Of the four numbers, the first is twice the second, the second is one - third of the third and the third is 5 times the fourth. The average of the numbers is 24.75. The lowest of these numbers is-
  1. ক
    ক) 6
  2. খ
    খ) 9
  3. গ
    গ) 15
  4. ঘ
    ঘ) 18
ব্যাখ্যা
Question: Of the four numbers, the first is twice the second, the second is one - third of the third and the third is 5 times the fourth. The average of the numbers is 24.75. The lowest of these numbers is-

Solution: 
The average of the numbers is 24.75.
The sum of these numbers = (24.75 × 4)
= 99

Let the fourth number be x.
Then,
third number = 5x,
second number = 5x/3 and
first number = 10x/3
x + 5x + 5x/3 + 10x/3 = (24.75 × 4)
or, 11x = 99
or, x = 9
So, the numbers are 9, 45, 15 and 30.
∴ Lowest number = 9.
৭,৪২৩.
31, 29, 24, 22, 17, ... What number should come next?
  1. ক
    19
  2. খ
    15
  3. গ
    14
  4. ঘ
    13
ব্যাখ্যা
Question: 31, 29, 24, 22, 17, ... What number should come next?

Solution: 
31 - 29 = 2
29 - 24 = 5 
24 - 22 = 2
22 - 17 = 5 
17 - 15 = 2 

next number is 15 
৭,৪২৪.
Titu, Pintu and Mithu can complete a piece of work in 18, 6 and 12 days respectively. Working together, they will complete the same work in:
  1. ক
    ক) 36/5 days 
  2. খ
    খ) 36/7 days 
  3. গ
    গ) 36/13 days 
  4. ঘ
    ঘ) 36/11 days
ব্যাখ্যা
(Titu + Pintu + Mithu)'s 1 day's work =(1/18) + (1/6) + (1/12)
                                                           =(2 + 6 + 3)/36
                                                           = 11/36

All the three together will complete the job in 36/11 days
৭,৪২৫.
In a group of five men, no two men have the same age. The oldest man is 50 years old, and the youngest is 30 years old. If X is the average age of the men in the group, which of the following best indicates all and only possible values of X? (All ages are in whole numbers)
  1. ক
    30 < X < 50
  2. খ
    31 < X < 49
  3. গ
    32 < X < 48
  4. ঘ
    35 < X < 45
ব্যাখ্যা

Question: In a group of five men, no two men have the same age. The oldest man is 50 years old, and the youngest is 30 years old. If X is the average age of the men in the group, which of the following best indicates all and only possible values of X? (All ages are in whole numbers)

Solution: 
Here, 
Oldest man = 50 years
Youngest man = 30 years
Average age = X

Since no two people are the same age, the minimum average = (30 + 31 + 32 + 33 + 50)/5
= 176/5
= 35.2 
and the maximum average = (30 + 47 + 48 + 49 + 50)/5
= 224/5
= 44.8

That is, the value of X is greater than 35 and less than 45.
i.e. 35 < X < 45 

৭,৪২৬.
A is thrice as good a workman as B and so takes 40 days less than B for doing a job. The time in which they can do the job together is-
  1. ক
    ক) 12 days 
  2. খ
    খ) 15 days 
  3. গ
    গ) 18 days 
  4. ঘ
    ঘ) 20 days 
ব্যাখ্যা
Question: A is thrice as good a workman as B and so takes 40 days less than B for doing a job. The time in which they can do the job together is- 

Solution:
Let, A alone takes x days and B alone takes 3x days to complete the work.

According to the question,
3x - x = 40
⇒ 2x = 40
⇒ x = 20

So, A takes 20 days and B takes ( 3 × 20) = 60 days 

Now, ( A+B )'s 1 days work = (1/20) + (1/60)
= 4/60
= 1/15 

∴ A & B together can do the work in 15 days.
৭,৪২৭.
Two numbers are in the ratio 2 : 5. If 16 is added to both the numbers, their ratio becomes 1 : 2. The numbers are
  1. ক
    ক) 16, 40
  2. খ
    খ) 20, 50
  3. গ
    গ) 28, 70
  4. ঘ
    ঘ) 32, 80
ব্যাখ্যা

Let the numbers be 2x, 5x
ATQ,
(2x + 16) / (5x + 16) = 1/2
Or, 4x + 32 = 5x + 16
Or, x = 16
∴ The numbers: 2x = 2.16 = 32
5x = 5.16 = 80

৭,৪২৮.
A boat covers 143 km upstream in 13 hours and the same distance downstream in 11 hours. What is the speed (in km/h) of the boat in still (without stream) water?
  1. ক
    8 km/h
  2. খ
    10 km/h
  3. গ
    15 km/h
  4. ঘ
    12 km/h
ব্যাখ্যা

Question: A boat covers 143 km upstream in 13 hours and the same distance downstream in 11 hours. What is the speed (in km/h) of the boat in still (without stream) water?

Solution:
Let, the speed of the boat in still water = x km/h
and the speed of the stream = y km/h

According to the question,
143/(x - y) = 13
⇒ 13(x - y) = 143
⇒ x - y = 11 ............(1)

Again,
143/(x + y) = 11
⇒ 11(x + y) = 143
⇒ x + y = 13 ............(2)

Adding (1) and (2):
x - y + x + y = 11 + 13
2x = 24
⇒ x = 12 km/h

৭,৪২৯.
If a train of length 300 meters takes 81 seconds to cover a 600 meters wall then which of the following will be its speed in km/hr?
  1. ক
    38
  2. খ
    42
  3. গ
    35
  4. ঘ
    40
ব্যাখ্যা

We know,
Time taken by a train of length l metres to pass a stationary object of length b metres is
the time taken by the train to cover (l + b) metres.
Given that,
Time taken by a train of length 300 meters to pass a wall of length 600 meters is
the time taken by the train to covers (300+600) meters.
That is, 81 seconds.
We know that,
distance/time = speed
Therefore, required speed = (300+600)/81 m/sec
= 900/81 m/sec
Converting the above to km/hr
= (900/81) x (18/5) km/hr [m/s to km/hr conversion: a m/s = a x 18/5 km/hr.]
= 40 km/hr
Hence the answer is 40 km/hr.

৭,৪৩০.
If One-third of one-fourth of a number is 15, then three-tenth of that number is:
  1. ক
    54
  2. খ
    45
  3. গ
    36
  4. ঘ
    35
  5. ঙ
    None
ব্যাখ্যা
Question: If One-third of one-fourth of a number is 15, then three-tenth of that number is:

Solution:
Let,
the number is 'x'
then ,
(1/3) × (1/4) × x = 15
⇒ x/12 = 15
⇒ x = 180

Now,
(3/10) × x = (3/10) × 180 = 18 × 3 = 54.
∴ three-tenths of that number is 54.
৭,৪৩১.
The diagonal of a rectangular field is 15 metres and the difference between its length its length and width is 3 metres. The area of the rectangular field is-
  1. ক
    21m2
  2. খ
    9m2
  3. গ
    12m2
  4. ঘ
    108m2
ব্যাখ্যা

Question: The diagonal of a rectangular field is 15 metres and the difference between its length its length and width is 3 metres. The area of the rectangular field is-

Solution:
Let l and b be the length and breadth of the rectangle respectively.
Then,
⇒ √(l2 + b2) = 15
⇒ (l2 + b2) = (15)2
⇒ l2 + b2 = 225

And,
∴  l - b = 3
⇒ (l - b)2 = 9
⇒ l2 + b2 - 2lb = 9
⇒ 225 - 2lb = 9
⇒ 2lb = 216
∴ lb = 108

Hence, area of the field = lb = 108m2

৭,৪৩২.
The angle measure of base angles of an isosceles triangle are represented by x and the vertex angle is 3x+10. Find the measure of base angle.
  1. ক
    ক) 112°
  2. খ
    খ) 42.5°
  3. গ
    গ) 34°
  4. ঘ
    ঘ) 16°
ব্যাখ্যা
ATQ, x + x + 3x + 10 = 180°
Or, 5x = 170°
Or, x = 34°
৭,৪৩৩.
Zisan completed the school project in 20 days. How many days will Ali take to complete the same work if he is 25% more efficient than Zisan?
  1. ক
    12 days
  2. খ
    14 days
  3. গ
    16 days
  4. ঘ
    18 days
ব্যাখ্যা
  প্রশ্ন: Zisan completed the school project in 20 days. How many days will Ali take to complete the same work if he is 25% more efficient than Zisan?

সমাধান:
আলী জিসানের চেয়ে ২৫% দক্ষ।

অর্থাৎ,
জিসান ১২৫ দিন সময় নিলে আলী নিবে ১০০ দিন 
∴ জিসান ১ দিন সময় নিলে আলী নিবে ১০০/১২৫ দিন
∴ জিসান ২০ দিন সময় নিলে আলী নিবে (১০০ × ২০)/১২৫ দিন
= ১৬ দিন 
৭,৪৩৪.
If the ratio of radius of two spheres is 4 : 7, the ratio of their volume is-
  1. ক
    4 : 7
  2. খ
    64 : 343
  3. গ
    49 : 16
  4. ঘ
    16 : 49
  5. ঙ
    None of these
ব্যাখ্যা
Question: If the ratio of radius of two spheres is 4 : 7, the ratio of their volume is-

Solution:
Ratio of radii of 2 spheres is 4 : 7.
∴ratio of their volume = 43 : 73 = 64 : 343
৭,৪৩৫.
If log8x = 2/3, then the value of x is-
  1. ক
    ক) 4
  2. খ
    খ) 4/3
  3. গ
    গ) 3/4
  4. ঘ
    ঘ) - 4
ব্যাখ্যা
log8x = 2/3
x = 8(2/3)
x = (23)(2/3)
x = 22
x = 4
৭,৪৩৬.
A man walks a at a rate of 10 mph. After every ten miles, he rests for 6 minutes. How much time does he take to walk 50 miles?
  1. ক
    300
  2. খ
    318
  3. গ
    322
  4. ঘ
    324
ব্যাখ্যা

Question: A man walks a at a rate of 10 mph. After every ten miles, he rests for 6 minutes. How much time does he take to walk 50 miles?

Solution:
The man needs time = 50/10 = 5 hours
= (5 × 60)
= 300 minutes

He will also rest 4 times after 10, 20, 30 and 40 miles
Total resting time = 4 × 6 = 24 minutes.

Total time = 300 + 24 = 324 minutes

৭,৪৩৭.
Karim started a business with Tk. 2100 and is joined afterward by Tanim with Tk. 3600. After how many months did Tanim join if the profits at the end of the year are divided equally?
  1. ক
    4 months
  2. খ
    6 months
  3. গ
    5 months
  4. ঘ
    8 months
ব্যাখ্যা
Question: Karim started a business with Tk. 2100 and is joined afterward by Tanim with Tk. 3600. After how many months did Tanim join if the profits at the end of the year are divided equally?

Solution:
Suppose,
Tanim joined after x months.

Then,
2100 × 12 = 3600 × (12 - x)
⇒ 252 = 432 - 36x
⇒ 36x = 180
∴ x = 5

∴ Tanim joined after 5 months.
৭,৪৩৮.
Kalam bought two varieties of rice, costing Tk. 50/kg and Tk. 60/kg each, and mixed them in some ratio. Then he sold the mixture at Tk. 70/kg. Making a profit of 20%. What was the ratio of the mixture?
  1. ক
    ক) 1 : 10
  2. খ
    খ) 1 : 5
  3. গ
    গ) 2 : 7
  4. ঘ
    ঘ) 3 : 8
  5. ঙ
    ঙ) None
ব্যাখ্যা
ধরি, কালাম 50 টাকায় চাল কেনে x পরিমাণ এবং 60 টাকায় কেনে y পরিমাণ চাল।
অর্থাৎ, তার মোট খরচ হয় (50x + 60y) টাকা।
যেহেতু সে 20% লাভে বিক্রয় করে চালের মিশ্রণ সেহেতু,
প্রশ্নমতে, 120/100(50x+60y) = 70(x+y)
বা, 60x + 72y = 70x + 70y
বা, 10x = 2y
বা, x:y = 1:5
৭,৪৩৯.
Find the simple interest on BDT 8,000 at 6% per annum for 9 months.
  1. ক
    BDT 360
  2. খ
    BDT 480
  3. গ
    BDT 540
  4. ঘ
    BDT 720
ব্যাখ্যা

Question: Find the simple interest on BDT 8,000 at 6% per annum for 9 months.

Solution: 
Given, P = 8000 Taka
n = 9 months = 0.75 years
r = 6% 

Simple Interest, I = Pnr
= 8000 × 0.75 × 6/100
= 80 × 4.5
= 360 Taka

৭,৪৪০.
A tradesman makes his marks his goods 30% above the C.P If he allows a discount of (25/4)%, then his gain percent is-
  1. ক
    ক) 20.875%
  2. খ
    খ) 18.875%
  3. গ
    গ) 21.875%
  4. ঘ
    ঘ) 16.875%
ব্যাখ্যা
Let C.P be 100
then the marked price Tk. 130 

Sell price = {100 - (25/4)} % of 130 
                = {(400 - 25)/4}% of 130 
                = (375/400) × 130 
                 = 121.875

Gain% = (121.875 - 100)%
           = 21.875%
৭,৪৪১.
In triangle ABC, AB = AC and ∠C = 30°. Find the measure of ∠A.
  1. ক
    180°
  2. খ
    90°
  3. গ
    120°
  4. ঘ
    60°
ব্যাখ্যা
Question: In triangle ABC, AB = AC and ∠C = 30°. Find the measure of ∠A.

Solution:

AB = AC
∴  ∠B  = ∠C = 30°

∴ ∠A = 180° - ∠B - ∠C
= 180° - 30° - 30° 
= 180° - 60°
= 120°
৭,৪৪২.
Average of ten positive numbers is x. If each number increases by 10%, then x-
  1. ক
    remains unchanged
  2. খ
    may increase or decrease
  3. গ
    will increase
  4. ঘ
    is increased by 10%
ব্যাখ্যা

Question: Average of ten positive numbers is x. If each number increases by 10%, then x-

Solution:
Let the ten positive numbers be n1, n2,..., n10
Then, n1 + n2 + n3 + ........... + n10 = 10x ............(1)

According to the question,
each number is increased by 10%
10% = 10/100 = 0.1
Suppose the average of the ten numbers = X1
Therefore, X1 =( 1.1n1 + 1.1n2 + 1.1n3 + ........... + 1.1n10)/10
⇒ X1 = 1.1(n1 + n2 + n3 + ........... + n10)/10
⇒ X1 = 1.1X [We get from equation (1)]
⇒ X1/ X = 1.1
⇒ X1/X = (1.1 x 100)%
⇒ X1/X =  110%

Total percentage gains = (110 - 100)% = 10%

∴ The new average increases by 10%.

৭,৪৪৩.
A man bought some eggs of which 10% are rotten. He gives 80% of the remainder to his neighbors. Now he is left out with 36 eggs. Then he ate two eggs. How many eggs did he buy?
  1. ক
    180
  2. খ
    195
  3. গ
    200
  4. ঘ
    206
ব্যাখ্যা
Question: A man bought some eggs of which 10% are rotten. He gives 80% of the remainder to his neighbors. Now he is left out with 36 eggs. Then he ate two eggs. How many eggs did he buy?

Solution:
let, the man bought  x eggs
10% are rotten

so eggs remained = x - x × 10%
= x - x/10
= 9x/10

80% of 9x/10
= (9x/10) × (80/100)
= 18x/25

he is left with = (9x/10) - (18x/25)
= (45x - 36x)/50
= 9x/50

So, (9x/50) = 36
⇒ 9x = 36 × 50
⇒ x = (36 × 50)/9
= 200

∴ He bought 200 eggs
৭,৪৪৪.
What is the absolute value (modulus) of x if - 8 < x < 2?
  1. ক
    |x + 3| < 5
  2. খ
    |x + 2| < 5
  3. গ
    |x + 2| > 5
  4. ঘ
    |x + 3| ≤ 5
ব্যাখ্যা

Question: What is the absolute value (modulus) of x if - 8 < x < 2?

Solution:
Given that, 
- 8 < x < 2
⇒ - 8 + 3 < x + 3 < 2 + 3
⇒ - 5 < x + 3 < 5
∴ |x + 3| < 5

৭,৪৪৫.
The compound interest on Tk. 3000 for 2 years at 10% is-
  1. ক
    ক) Tk. 630
  2. খ
    খ) Tk. 600
  3. গ
    গ) Tk. 3600
  4. ঘ
    ঘ) Tk. 3630
ব্যাখ্যা
Question: The compound interest on Tk. 3000 for 2 years at 10% is-

Solution:
Givent that,
P = Tk. 3000
n = 2 years
r = 10%

Required amount
= P(1 + r)n 
= [3000 × {1 + (10/100)2 }] 
= [3000 × (11/10) × (11/10) ]
= 3630

Compound Interest = 3630 - 3000 = 630
৭,৪৪৬.
The square root of (7 + 3√5) (7 - 3√5) is
  1. ক
    √3
  2. খ
    4
  3. গ
    √5
  4. ঘ
    7
  5. ঙ
    2
ব্যাখ্যা
Question: The square root of (7 + 3√5) (7 - 3√5) is

Solution:
√{(7 + 3√5) (7 - 3√5)}
= √{(7)2 - (3√5)2}
= √(49 - 45)
= √4
= 2
৭,৪৪৭.
Find the least multiple of 23. which when divided by 18, 21 and 24 leaves remainders 7, 10 and 13 respectively.
  1. ক
    3002
  2. খ
    3013
  3. গ
    3024
  4. ঘ
    3036
ব্যাখ্যা
Question: Find the least multiple of 23. which when divided by 18, 21 and 24 leaves remainders 7, 10 and 13 respectively.

Solution:
Here, (18 - 7) = 11, (21 - 10) = 11 and (24 - 13) = 11
L.C.M of 18, 21 and 24 is 504.

Let, the required number be 504x - 11.
Least value of x for which (504x - 11) is divisible by 23 is x = 6

Required number = (504 × 6) - 11 = 3024 - 11 = 3013.
৭,৪৪৮.
(33 - 56 ) : Read the following questions carefully and choose the right answer.
33. In an office, 44% of the workers prefer coffee and 72% prefer tea. If each of them prefers coffee or tea and 40 like both, the total number of workers in the office is
  1. ক
    ক) 200
  2. খ
    খ) 250
  3. গ
    গ) 240
  4. ঘ
    ঘ) 210
ব্যাখ্যা
Question: In an office, 44% of the workers prefer coffee and 72% prefer tea. If each of them prefers coffee or tea and 40 like both, the total number of workers in the office is-

Solution: 
Out of 100%, 44% drink coffee and 72% drink tea,

Using the formula,
n(A ∪ B) = n(A) + n(B) - n(A∩ B)
⇒ 100 = 72 + 44 - n(A∩ B)
⇒ 100 = 116 - n(A ∩ B)
⇒ n(A ∩ B) = 16

According to the question,
⇒ 16% = 40
⇒ 1%= 40/16
⇒ 100% = 100 × 40/16
              = 25 × 10
              = 250 
৭,৪৪৯.
A shopkeeper offers two successive discounts of 10% and 20% on an item marked at Tk. 1,200. What is the final selling price?
  1. ক
    880 Tk
  2. খ
    900 Tk
  3. গ
    864 Tk
  4. ঘ
    960 Tk
ব্যাখ্যা

Question: A shopkeeper offers two successive discounts of 10% and 20% on an item marked at Tk. 1,200. What is the final selling price?

Solution: 
New Price after First Discount : 
1200 - 10% of 1200
= 1200 - 120
= 1080 Taka

Final Selling Price after Second Discount:
1080 - 20% of 1080
= 1080 - 216
= 864

So, Final Selling Price = Tk. 864

৭,৪৫০.
The ratio of the present ages of Tina and Bina is 9 : 10 respectively. Twenty years ago, the ratio of their ages was 4 : 5 respectively. What is the present age of Bina?
  1. ক
    ক) 30 years
  2. খ
    খ) 35 years
  3. গ
    গ) 40 years
  4. ঘ
    ঘ) 42 years
ব্যাখ্যা
Ratio of the present ages of Tina and Bina = 9 : 10
Let's consider the present ages of Tina and Bina as 9x and 10x

According to the question 
(9x - 20) : (10x - 20) = 4 : 5
(9x - 20) / (10x - 20) = 4/5
5(9x - 20) = 4(10x - 20)
45x - 100 = 40x - 80
45x - 40x = 100 - 80
5x = 20
x = 4 

the present age of Bina = 40 years
৭,৪৫১.
α and β are the roots of 5x2 + 3x + 1 = 0 then, the value of (1/α) + (1/β) is-
  1. ক
    2
  2. খ
    - 1
  3. গ
    5
  4. ঘ
    - 3
ব্যাখ্যা
Question: α and β are the roots of 5x2 + 3x + 1 = 0 then, the value of (1/α) + (1/β) is-

Solution:
Here,
5x2 + 3x + 1 = 0
where, a = 5, b = 6 and c = 1

∴ α + β = - (b/a) = - 3/5
and αβ = c/a = 1/5

∴ (1/α) + (1/β)
= (β + α)/αβ
= {- (3/5)}/(1/5)
= - 3
৭,৪৫২.
What should be the value of "P" so that the expression (9 − 12x + Px2) becomes a perfect square?
  1. ক
    5
  2. খ
    4
  3. গ
    6
  4. ঘ
    8
ব্যাখ্যা

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

Solution:
(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

৭,৪৫৩.
A and B working together can finish a work in 12 days, B and C working together can finish the work in 16 days. If A works for 5 days, B works for 7 days, and C completes the remaining work in 11 days, C alone can complete the work in how many days?
  1. ক
    22 days
  2. খ
    24 days
  3. গ
    26 days
  4. ঘ
    28 days
ব্যাখ্যা
Question: A and B working together can finish a work in 12 days, B and C working together can finish the work in 16 days. If A works for 5 days, B works for 7 days, and C completes the remaining work in 11 days, C alone can complete the work in how many days?

Solution:
ATQ,
A + B = 12 days
B + C = 16 days

Note: Assume the total work = LCM of the given days
Take the LCM of days = LCM of (12 and 16) = 48
Let the total work = 48

Note: One day work = (total work/ days)
Now,
(A + B)'s one day work = 48/12 = 4 unit
(B + C)'s one day work = 48/16 = 3 unit

As per the question:
A works for 5 days
B works for 7 days or (5 + 2) days, that means B works 5 days with A and remaining 2 days with C.
C works 13 days or (2 + 11) days, that means C works 2 days with B and remaining 11 days alone.

That means total work done by (A + B) in 5 days
A + B = 5 days × 4 unit = 20 units
And, total work done by (B + C) = 2 days × 3 unit = 6 units
So, A + B + C finish the 26 units of work.

Remaining work = 48 - 26 = 22 unit work
And C completes the remaining work in 11 days.
i.e., C's one day's works = 22/11 = 2 units.

C alone can finish total work in [total work/ C's one day work] = [48/2] = 24 days.
৭,৪৫৪.
The perimeter of one face of a cube is 20 cm. Its volume must be-
  1. ক
    100 cm3
  2. খ
    115 cm3
  3. গ
    125 cm3
  4. ঘ
    150 cm3
ব্যাখ্যা
Question: The perimeter of one face of a cube is 20 cm. Its volume must be-

Solution: 
perimeter of one face is 20 cm

let, length of one side is a cm
perimeter = 4a cm

⇒ 4a = 20
⇒ a = 20/4
= 5 cm

volume = a3
= 53
= 125 cm3
৭,৪৫৫.
যদি কোনও পণ্যের দাম ৫০% বৃদ্ধি পায়, তবে তার ব্যয় একই রাখতে এর ব্যবহারের পরিমাণ কত ভগ্নাংশ কমাতে হবে?
  1. ক
    ১/২
  2. খ
    ১/৩
  3. গ
    ১/৪
  4. ঘ
    ২/৩
  5. ঙ
    কোনটি নয়
ব্যাখ্যা
প্রশ্ন: যদি কোনও পণ্যের দাম ৫০% বৃদ্ধি পায়, তবে তার ব্যয় একই রাখতে এর ব্যবহারের পরিমাণ কত ভগ্নাংশে কমাতে হবে?

সমাধান:
দেওয়া আছে,
'ক' কেজি পণ্যের মূল্য ১০০ টাকা।

৫০% বৃদ্ধি পেলে,
∴  ক কেজির মূল্য = ১০০ + ১০০ × ০.৫ = ১৫০ টাকা

১০০ টাকায় নতুন পরিমাণ পণ্য = (১০০ × ক)/১৫০ = ২ক/৩ কেজি

ভোগের পরিমাণ যে ভগ্নাংশ কমাতে হবে = {ক - (২ক/৩)}/ক
= (ক/৩)/ক
= ১/৩
৭,৪৫৬.
A train 300 m long is running at a speed of 90 km/hr. If it passes through a tunnel in 40 seconds, then the length of the tunnel is-
  1. ক
    600 meters
  2. খ
    700 meters
  3. গ
    900 meters
  4. ঘ
    1000 meters
ব্যাখ্যা

Question: A train 300 m long is running at a speed of 90 km/hr. If it passes through a tunnel in 40 seconds, then the length of the tunnel is-

Solution:
ট্রেনের গতিবেগ = 90 কিমি/ঘন্টা
= (90 × 5/18) মিটার/সেকেন্ড
= 25 মিটার/সেকেন্ড

একটি সুড়ঙ্গ অতিক্রম করার সময় ট্রেনটি তার নিজের দৈর্ঘ্যের সাথে সুড়ঙ্গের দৈর্ঘ্যও অতিক্রম করে।
সুতরাং, মোট অতিক্রান্ত দূরত্ব = ট্রেনের গতিবেগ × সময়
= 25 মিটার/সেকেন্ড × 40 সেকেন্ড
= 1000 মিটার

ধরা যাক, সুড়ঙ্গটির দৈর্ঘ্য = L মিটার
মোট অতিক্রান্ত দূরত্ব = ট্রেনের দৈর্ঘ্য + সুড়ঙ্গের দৈর্ঘ্য
⇒ 1000 = 300 + L
⇒ L = 1000 - 300
⇒ L = 700 মিটার

সুতরাং, সুড়ঙ্গটির দৈর্ঘ্য হলো 700 মিটার।

৭,৪৫৭.
If x + y = 3, x2 + y2 = 5, then x3 + y3 =?
  1. ক
    9
  2. খ
    11
  3. গ
    13
  4. ঘ
    7
ব্যাখ্যা
Question: If x + y = 3, x2 + y2 = 5, then x3 + y3 =?

Solution:
Given that,
x + y = 3
x2 + y2 = 5

(x + y)2 = x2 + 2xy + y2
⇒ 2xy = (x + y)2 - (x2 + y2)
⇒ 2xy = 32 - 5
⇒ 2xy = 9 - 5
⇒ 2xy = 4
∴ xy = 2

Now,
x3 + y3 = (x + y)3 - 3xy(x + y)
= 33 - 3 × 2 × 3
= 27 - 18
= 9
৭,৪৫৮.
A monkey climbing up a pole ascends 6 meters and slips 3 meters in alternative minutes. If the pole is 60 meters high, how long will it take the monkey to reach the top?
  1. ক
    ক) 31 miles
  2. খ
    খ) 33 miles
  3. গ
    গ) 35 miles
  4. ঘ
    ঘ) 37 miles
ব্যাখ্যা

Net height ascended in 2 min = (6 - 3) m = 3 m.
Net height ascended in 36 min = (3/2 × 36) = 54m.
In the 37th min,
the monkey ascends 6m and reaches the top.
Hence,
Total time taken = 37 minutes.

৭,৪৫৯.
A wheel rotates 10 times per minute and moves 20 m during each rotation. How many meters does the wheel move in 1 hour?
  1. ক
    ক) 10000
  2. খ
    খ) 20000
  3. গ
    গ) 18000
  4. ঘ
    ঘ) 12000
ব্যাখ্যা

Number of rotation in one hour = 10 × 60 = 600
So, Distance moved = (600 × 20) = 12000 m

৭,৪৬০.
What is the least common multiple (LCM) of 18 and 24?
  1. ক
    72
  2. খ
    48
  3. গ
    108
  4. ঘ
    36
ব্যাখ্যা
Question: What is the least common multiple (LCM) of 18 and 24? 

Solution: 
18 = 2 × 3 × 3 
24 = 23 × 3 

the least common multiple (LCM) of 18 and 24 = 23 × 3 × 3 
= 72
৭,৪৬১.
A merchant buys an article for Tk. 27 and sells it at a profit of 10 percent of the selling price. The selling price of the article is -
  1. ক
    Tk. 30
  2. খ
    Tk. 35
  3. গ
    Tk. 40
  4. ঘ
    Tk. 32
  5. ঙ
    None
ব্যাখ্যা
Question: A merchant buys an article for Tk. 27 and sells it at a profit of 10 percent of the selling price. The selling price of the article is -

Solution:
Let, the selling price of the article be Tk. p

Cost price = Tk. 27
Profit = 10% of p

Again,
Profit = Selling price - Cost price
⇒ 10% of p = p - 27
⇒ p - 10% of p = 27
⇒ 90% of p = 27
⇒ 9p/10 = 27
⇒ p = 27 × (10/9)
⇒ p = 30

Hence, The selling price of the article is Tk. 30
 
৭,৪৬২.
Eight years back, Adil's age was 1/8th of Zaher's age. Ten years from now, Zaher's age will be double of Adil's age. How many years old is Adil now?
  1. ক
    ক) 23
  2. খ
    খ) 21
  3. গ
    গ) 17
  4. ঘ
    ঘ) 11
ব্যাখ্যা
প্রশ্ন: Eight years back, Adil's age was 1/8th of Zaher's age. Ten years from now, Zaher's age will be double of Adil's age. How many years old is Adil now?

সমাধান: 
ধরি,
জহিরের বর্তমান বয়স ক বছর
৮ বছর পূর্বে জহিরের বয়স ক - ৮ বছর 
∴ ৮ বছর পূর্বে আদিলের বয়স (ক - ৮)/৮ বছর 

বর্তমানে,
আদিলের বয়স = (ক - ৮)/৮ + ৮ বছর 
= (ক - ৮ + ৬৪)/৮ বছর 
= (ক + ৫৬)/৮ বছর 

১০ বছর পর জহিরের বয়স হবে ক + ১০ বছর 
১০ বছর পর আদিলের বয়স হবে (ক + ৫৬)/৮ + ১০ বছর 
= (ক + ৫৬ + ৮০)/৮ বছর 
= (ক + ১৩৬)/৮ বছর 

প্রশ্নমতে, 
ক + ১০ = ২ ×{(ক + ১৩৬)/৮}
বা, ক + ১০ = (ক + ১৩৬)/৪  
বা, ৪(ক + ১০) = ক + ১৩৬
বা, ৪ক + ৪০  = ক + ১৩৬
বা, ৪ক - ক = ১৩৬ - ৪০
বা, ৩ক = ৯৬
বা, ক = ৯৬/৩
∴ ক = ৩২

∴ আদিলের বর্তমান বয়স = ( ৩২ + ৫৬)/৮ বছর 
= ৮৮/৮ বছর 
= ১১ বছর 
৭,৪৬৩.
Didar is younger to Rohan by 9 years. If their ages are in the respective ratio of 4 : 5, how old is Didar?
  1. ক
    36 years
  2. খ
    23 years
  3. গ
    29 years
  4. ঘ
    Cannot be determined
  5. ঙ
    None of these
ব্যাখ্যা
Question: Didar is younger to Rohan by 9 years. If their ages are in the respective ratio of 4 : 5, how old is Didar?

Solution:
Let Rohan's age be x years.
Then,
Didar's age = (x - 9) years.

(x - 9)/x = 4/5
⇒ 5x - 45 = 4x
∴ x = 45
Hence, Didar's age = (x - 9) = 45 - 9 = 36 years.
৭,৪৬৪.
The length of a tangent from point A at a distance of 5 cm from the centre of the circle is 4 cm. Find the radius of the circle.
  1. ক
    3 cm
  2. খ
    4 cm
  3. গ
    5 cm
  4. ঘ
    6 cm
ব্যাখ্যা
Question: The length of a tangent from point A at a distance of 5 cm from the centre of the circle is 4 cm. Find the radius of the circle.

Solution: 
Let O be the centre of the circle and AB is the tangent.


OB ⊥ AB then ∆OAB is a right-angled triangle.

OA2 = AB2 + OB2 (by Pythagoras Theorem)
⇒ 52 = 42 + OB2
⇒ OB2 = 52 – 42
⇒ OB2 = 25 – 16
⇒ OB2 = 9
⇒ OB = √9 = 3

Hence, the radius of the circle = 3 cm.
৭,৪৬৫.
The plural of 'Phenomenon'.
  1. ক
    Phenomenon
  2. খ
    Phenomenones
  3. গ
    Phenomena
  4. ঘ
    Phenomeni
ব্যাখ্যা
• The plural of 'Phenomenon' is - গ) Phenomena.

• Phenomenon: [singular]
- English meaning: an observable fact or event; a rare or significant fact or event.
- Bangla meaning: (১) ইন্দ্রিয়গোচর বস্তু বা বিষয়। (২) বিস্ময়কর ব্যক্তি/বিষয়/ঘটনা।

- Phenomenon এর plural হলো Phenomena এবং Phenomenons.

• Ex. Sentences:
- Gravity is a natural phenomenon.
- Glaciers are interesting natural phenomena.

Source:
1. Oxford Dictionary.
2. Merriam-Webster Dictionary.
৭,৪৬৬.
A mother was 27 years old when her daughter was born. Currently, the mother's age is 9 years more than three times her daughter's age. How old will the daughter be in 6 years?
  1. ক
    12
  2. খ
    15
  3. গ
    21
  4. ঘ
    18
ব্যাখ্যা

Question: A mother was 27 years old when her daughter was born. Currently, the mother's age is 9 years more than three times her daughter's age. How old will the daughter be in 6 years?

Solution:
Let the daughter's current age be x years.
∴ Mother's current age = 3x + 9 years.

According to the question,
3x + 9 - x = 27
⇒ 2x = 27 - 9
⇒ 2x = 18
∴ x  = 9

∴ Daughter's current age 9 years.
∴ Daughter's age after 6 years = 9 + 6 = 15 years.

৭,৪৬৭.
3 pumps working 8 hours a day, can empty a tank in 2 days. How many hours a day must 4 pumps work to empty the tank in 1 day?
  1. ক
    10 hours
  2. খ
    12 hours
  3. গ
    14 hours
  4. ঘ
    16 hours
ব্যাখ্যা
Question: 3 pumps working 8 hours a day, can empty a tank in 2 days. How many hours a day must 4 pumps work to empty the tank in 1 day?

Solution: 
3 pumps need 2 × 8 hours = 16 hours
1 pump needs  16 × 3  hours
∴ 4 pumps need (16 × 3)/4 hours
= 12 hours
৭,৪৬৮.
A man buy 10 articles for Tk. 8 and sells them at the rate of Tk. 1.25 per article. His gain percent is:
  1. ক
    ক) (215/4)%
  2. খ
    খ) (225/4)%
  3. গ
    গ) (225/2)%
  4. ঘ
    ঘ) (325/4)%
ব্যাখ্যা
Cost price of 10 articles (C.P.) = Tk. 8
Selling price of 10 articles (S.P.) = 1.25×100
                                                   = Tk.12.50
                           Profit = S.P - C.P.
                                     =12.50 - 8
                                     =Tk. 4.50.

∴ Gain % ={(4.50×100)/8}%
                = (450/8)%
                 = (225/4)%
৭,৪৬৯.
12 workers can dig a 36-meter long trench working 8 hours per day. How many extra workers are required to dig a 45-meter long trench working 5 hours per day?
  1. ক
    6 workers
  2. খ
    0 workers
  3. গ
    10 workers
  4. ঘ
    12 workers
ব্যাখ্যা

Question: 12 workers can dig a 36-meter long trench working 8 hours per day. How many extra workers are required to dig a 45-meter long trench working 5 hours per day?

Solution:
8 ঘণ্টা কাজ করে 36 মিটার পরিখা খনন করে 12 জন শ্রমিক।
1 ঘণ্টা কাজ করে 1 মিটার পরিখা খনন করতে প্রয়োজন = (12 × 8)/36 জন শ্রমিক
5 ঘণ্টা কাজ করে 45 মিটার পরিখা খনন করতে প্রয়োজন = (12 × 8 × 45)/(36 × 5) জন শ্রমিক
= (12 × 8 × 45)/180 জন শ্রমিক
= 4320/180 জন শ্রমিক
= 24 জন শ্রমিক

∴ অতিরিক্ত শ্রমিকের সংখ্যা = 24 - 12 = 12 জন

৭,৪৭০.
A boy was asked to multiply a number by 25 but by mistake he multiplied by 35 and the answer was 120 more than the correct answer. What was the number?
  1. ক
    7
  2. খ
    8
  3. গ
    9
  4. ঘ
    10
  5. ঙ
    12
ব্যাখ্যা
Question: A boy was asked to multiply a number by 25 but by mistake he multiplied by 35 and the answer was 120 more than the correct answer. What was the number?

Solution:
Let,
The number be x

ATQ,
35x - 120 = 25x
⇒ 35x - 25x = 120
⇒ 10x = 120
∴ x = 12
৭,৪৭১.
Two vessels of equal capacity contain juice and water in the ratio of 7 : 2 and 11 : 7 respectively. The mixture of both vessels is mixed and transferred into a bigger vessel. What is the ratio of juice and water in the new mixture?
  1. ক
    18 : 9
  2. খ
    21 : 6
  3. গ
    22 : 7
  4. ঘ
    25 : 11
ব্যাখ্যা
Question: Two vessels of equal capacity contain juice and water in the ratio of 7 : 2 and 11 : 7 respectively. The mixture of both vessels is mixed and transferred into a bigger vessel. What is the ratio of juice and water in the new mixture?

Solution:
The ratio of juice and water in the first vessel = 7 : 2  ................(1)
Total capacity of first vessel = 7 + 2 = 9 units

The ratio of juice and water in the second vessel = 11 : 7 ...............(2)
Total capacity of second vessel = 11 + 7 = 18 units

We will have to equal the total capacity of both vessels, so multiply by 2 in equation (1).
The ratio of juice and water in the first vessel = 14 : 4  ................(3)

∴ Ratio of juice and water in bigger vessel = (14 + 11) : (4 + 7) = 25 : 11
৭,৪৭২.
If principal M becomes N in 2 years when interest R% is compounded half-yearly. And if the same principal M becomes N in 2 years when interest S% is compound annually, then which of the following is true?
  1. ক
    R > S
  2. খ
    R = S
  3. গ
    R < S
  4. ঘ
    None
ব্যাখ্যা

Question: If principal M becomes N in 2 years when interest R% is compounded half-yearly. And if the same principal M becomes N in 2 years when interest S% is compound annually, then which of the following is true?

Solution:
ধরি, আসল = M এবং সবৃদ্ধি মূলধন = N
সময় n = 2 বছর

অর্ধবার্ষিক চক্রবৃদ্ধির ক্ষেত্রে (মুনাফার হার R%):
N = M(1 + (R/2)/100)2 × 2
= M(1 + R/200)4

বার্ষিক চক্রবৃদ্ধির ক্ষেত্রে (মুনাফার হার S%):
N = M(1 + S/100)2

যেহেতু উভয় ক্ষেত্রে আসল (M), সময় (2 বছর) এবং সবৃদ্ধি মূলধন (N) একই,
সেহেতু যে পদ্ধতিতে বছরে বেশিবার মুনাফা গণনা করা হয় (Half-yearly), সেখানে কাঙ্ক্ষিত মুনাফা পেতে তুলনামূলক কম সুদের হার প্রয়োজন।
∴ R < S

৭,৪৭৩.
One-fourth of the sum of prime numbers, greater than 4 but less than 16, is =?
  1. ক
    ক) 36
  2. খ
    খ) 24
  3. গ
    গ) 3
  4. ঘ
    ঘ) 9
ব্যাখ্যা
Question: One-fourth of the sum of prime numbers, greater than 4 but less than 16, is =?

Solution: 
prime numbers, greater than 4 but less than 16 is = 5,  7, 11, 13

One-fourth of the sum = (5 + 7 + 11 + 13)/4
= 36/4
= 9
৭,৪৭৪.
If (a - 1) is an odd number, what are the two other odd numbers nearest to it? 
  1. ক
    a, a - 2
  2. খ
    a - 3, a + 1
  3. গ
    a, a - 1
  4. ঘ
    a - 3, a + 5
ব্যাখ্যা
Question: If (a - 1) is an odd number, what are the two other odd numbers nearest to it? 

Solution: 
a - 1 is an odd number 
previous odd number = a - 1 - 2 = a - 3
next odd number = a - 1 + 2 = a + 1
৭,৪৭৫.
A steamer goes downstream from one part to another in 4 hours. It covers the same distance upstream in 5 hours. If the speed of stream is 2 km/hr, the distance between the two ports is:
  1. ক
    60 km
  2. খ
    72 km
  3. গ
    80 km
  4. ঘ
    96 km
ব্যাখ্যা
Question: A steamer goes downstream from one part to another in 4 hours. It covers the same distance upstream in 5 hours. If the speed of stream is 2 km/hr, the distance between the two ports is:

Solution:
Let, the distance between the two parts = 'x' km
and the speed of steamer in still water = 'y' km/hr

ATQ,
x/(y + 2) = 4
⇒ x = 4y + 8 …. (1)

And, x/(y - 2) = 5
⇒ x = 5y - 10 ….. (2)

From (1) and (2)⇒
4y + 8 = 5y - 10
⇒ y = 18

∴ From (1)
x = 4 × 18 + 8
⇒ x = 80 km.
৭,৪৭৬.
An agent sells goods worth Tk. 18,000. If his commission rate was 12.5​%, what was the amount of his commission?
  1. ক
    1125
  2. খ
    1875
  3. গ
    2250
  4. ঘ
    2500
ব্যাখ্যা

Question: An agent sells goods worth Tk. 18,000. If his commission rate was 12.5​%, what was the amount of his commission?

Solution:

Commission = 12.5% of 18000
= (125/10) × (1/100) × 18000
= (125/1000) × 18000
= 2250

৭,৪৭৭.
The one-sixth of the complementary angle to 60° is
  1. ক
    5°
  2. খ
    10°
  3. গ
    15°
  4. ঘ
    20°
ব্যাখ্যা
প্রশ্ন: The one-sixth of the complementary angle to 60° is

সমাধান:
60° এর পূরক কোণ = 90° - 60° = 30°
30° এর 1/6 = 5°
৭,৪৭৮.
If 5x + 7y = 41 and 7x + 5y = 43, what is the average of x and y?
  1. ক
    2.8
  2. খ
    3
  3. গ
    3.5
  4. ঘ
    4.6
ব্যাখ্যা

Question: If 5x + 7y = 41 and 7x + 5y = 43, what is the average of x and y?

Solution:
দেয়া আছে,
5x + 7y = 41 ...........(1)
7x + 5y = 43 ...........(2)

(1) ও (2) যোগ করে পাই, 
5x + 7y + 7x + 5y = 41 + 43
⇒ 12x + 12y = 84
⇒ 12(x + y) = 84
⇒ x + y = 84/12
∴ x + y = 7

∴ গড় = (x + y) /2
= 7/2
= 3.5

৭,৪৭৯.
Digit 1 is occurring 134 times on writing all of the page numbers of a book. What will be the number of pages in the book?
  1. ক
    192
  2. খ
    193
  3. গ
    194
  4. ঘ
    195
  5. ঙ
    196
ব্যাখ্যা
Question: Digit 1 is occurring 134 times on writing all of the page numbers of a book. What will be the number of pages in the book?

Solution:
From 1 - 99, the digit 1 occurs 20 times,
from 100 - 199, the digit 1 occurs 120 times.
So, from 1 to 199, the digit 1 occurs 20 + 120 = 140 times
According to question 1 is occurring only 134 times, which means we need to remove 194, 195 196, 197, 198, and 199.
So, the required number of pages will be 193.
৭,৪৮০.
A large tanker can be filled by two pipes A and B in 60 and 40 minutes respectively. How many minutes will it take to fill the tanker from empty state if B is used for half the time and A and B fill it together for the other half ?
  1. ক
    ক) 20 min
  2. খ
    খ) 30 min
  3. গ
    গ) 40 min
  4. ঘ
    ঘ) 50 min
ব্যাখ্যা
Question: A large tanker can be filled by two pipes A and B in 60 and 40 minutes respectively. How many minutes will it take to fill the tanker from empty state if B is used for half the time and A and B fill it together for the other half ?

Solution:
A ৬০ মিনিটে পূর্ণ করে ১ অংশ 
১ মিনিটে পূর্ণ করে ১/৬০ অংশ 

B ৪০ মিনিটে পূর্ণ করে ১ অংশ 
১ মিনিটে পূর্ণ করে ১/৪০ অংশ 

A,B একসাথে ১ মিনিটে পূর্ণ করে ১/৬০ + ১/৪০ 
= ৫/১২০ 
= ১/২৪ অংশ 

ধরি, সময় লাগে x মিনিট 

B ৪০ মিনিটে পূর্ণ করে ১ অংশ 
১ মিনিটে পূর্ণ করে ১/৪০ অংশ
x/২ মিনিটে পূর্ণ করে x/৮০ অংশ 

A,B একসাথে ১ মিনিটে পূর্ণ করে ১/২৪ অংশ 
x/২ মিনিটে পূর্ণ করে x/৪৮ অংশ 

x/৮০ + x/৪৮ = ১
⇒ ৩x + ৫x / ২৪০ = ১
⇒ ৮x = ২৪০ 
∴ x = ৩০ মিনিট 
৭,৪৮১.
A student is required to solve 6 out of the 10 questions in a test. The questions are divided into two sections of 5 questions each. In how many ways can the student select the questions to solve if not more than 4 questions can be chosen from either section?
  1. ক
    100 ways
  2. খ
    1800 ways
  3. গ
    200 ways
  4. ঘ
    290 ways
ব্যাখ্যা
Question: A student is required to solve 6 out of the 10 questions in a test. The questions are divided into two sections of 5 questions each. In how many ways can the student select the questions to solve if not more than 4 questions can be chosen from either section?

Solution:
Possibility 1: This can be done in 5C4 × 5C2
= 5 × 10
= 50 ways

Possibility 2: This can be done in 5C3 × 5C3
= 10 × 10
= 100 ways

Possibility 3: This can be done in 5C2 × 5C4
= 10 × 5
= 50 ways

Total number of ways = 50 + 100 + 50 = 200 ways
৭,৪৮২.
A thief is noticed by a policeman from a distance of 200 m. The thief starts running and the policeman chases him. The thief and the policeman run at the rate of 10 km and 11 km per hour respectively. What is the distance between them after 6 minutes?
  1. ক
    220 m
  2. খ
    195 m
  3. গ
    180 m
  4. ঘ
    100 m
ব্যাখ্যা

Question: A thief is noticed by a policeman from a distance of 200 m. The thief starts running and the policeman chases him. The thief and the policeman run at the rate of 10 km and 11 km per hour respectively. What is the distance between them after 6 minutes?

Solution:
Relative speed of the thief and policeman  =  (11 - 10) km/hr = 1 km/hr 
Distance covered in 6 minutes  = (1/60) × 6 km   = 1/10 km = 100 m
Therefore, Distance between the thief and policeman = (200 - 100) m = 100 m.

৭,৪৮৩.
The total area of 64 small squares of a chessboard is 400 cm2. There is a 3 cm wide border around the chessboard. What is the length of the side of the chessboard?
  1. ক
    20 cm
  2. খ
    23 cm
  3. গ
    26 cm
  4. ঘ
    28 cm
ব্যাখ্যা
Question: The total area of 64 small squares of a chessboard is 400 cm2. There is a 3 cm wide border around the chessboard. What is the length of the side of the chessboard?

Solution: 
Length of 64 small squares = √400 
= 20 cm

The length of the side of the chess board is = 20 + 3 + 3
= 26 cm
৭,৪৮৪.
42x+2 × 8x-2 = 512,  x=?
  1. ক
    11/7
  2. খ
    9/7
  3. গ
    8
  4. ঘ
    8/7
ব্যাখ্যা
Question: 42x + 2 × 8x - 2 = 512,  x=?

Solution:
42x + 2 × 8x - 2 = 512
⇒ 22(2x + 2) × 23(x - 2) = 29
⇒ 24x + 4 × 23x - 6 = 29
⇒ 24x + 4 + 3x - 6 = 29
⇒ 4x + 4 + 3x - 6 = 9
⇒ 7x - 2 = 9
⇒ 7x = 11
∴ x = 11/7
৭,৪৮৫.
A boat takes 12 hours to cover a certain distance upstream and 8 hours to cover the same distance downstream. What is the ratio between the speed of the boat and the speed of the water current, respectively?
  1. ক
    7 : 1 
  2. খ
    4 : 1 
  3. গ
    5 : 1 
  4. ঘ
    3 : 1 
ব্যাখ্যা

Question: A boat takes 12 hours to cover a certain distance upstream and 8 hours to cover the same distance downstream. What is the ratio between the speed of the boat and the speed of the water current, respectively?

Solution:
Let,
Upstream speed = x kmph
Downstream speed = y kmph

Since distance is same,
x × 12 = y × 8
⇒ 12x = 8y
⇒ y = 3x/2

Now,
Speed of boat = (y + x)/2
Speed of current = (y - x)/2

∴ Required ratio
= (y + x) : (y - x)
= [(3x/2) + x] : [(3x/2) - x]
= (5x/2) : (x/2)
= 5x : x
= 5 : 1

∴ The ratio of the speed of the boat to the speed of the water current is 5 : 1.

৭,৪৮৬.
At present, the ratio between the ages of Haseena and Anushka is 4 : 3. After 6 years, Haseena's age will be 26 years. What is the present age of Anushka?
  1. ক
    15 years
  2. খ
    14 years
  3. গ
    13 years
  4. ঘ
    12 years
ব্যাখ্যা
Question: At present, the ratio between the ages of Haseena and Anushka is 4 : 3. After 6 years, Haseena's age will be 26 years. What is the present age of Anushka?

Solution:
After 6 years, Haseena's age will be 26 years
Therefore, the Present age of Haseena = 26 - 6 = 20
Let the present age of Anushka = x
Then,
20/x = 4/3
⇒ x = (20 × 3)/4
∴ x = 15 Years
৭,৪৮৭.
In how many ways can a committee of 4 people be formed from 7 men and 6 women such that the number of women is greater than the number of men?
  1. ক
    105 ways.
  2. খ
    120 ways.
  3. গ
    135 ways.
  4. ঘ
    155 ways.
  5. ঙ
    None
ব্যাখ্যা
Question: In how many ways can a committee of 4 people be formed from 7 men and 6 women such that the number of women is greater than the number of men?

Solution:
Given,
the number of women is greater than the number of men
We have to select women > men

If we want women > men, so there are 2 possible combinations:
1st combination:
3 women out of 6 women, 1 man out of 7 men = 6C3 × 7C1 = 20 × 7 = 140

2nd combination:
4 women out of 6 women = 6C4 = 15
                           
∴ Total number of valid committees = 140 + 15 = 155 ways.
৭,৪৮৮.
Today is Monday. After 61 days, it will be:
  1. ক
    Monday
  2. খ
    Saturday
  3. গ
    Sunday
  4. ঘ
    Friday
ব্যাখ্যা
Question: Today is Monday. After 61 days, it will be:

Solution:
Each day of the week is repeated after 7 days.
So, after 63 days, it will be Monday.
∴ After 61 days, it will be Saturday.
৭,৪৮৯.
A man buys a pen for 10% less than its value and sells it for 10% more than its value. His gain or loss percentage is-
  1. ক
    No profit, no loss
  2. খ
    20% profit
  3. গ
    Less than 20% profit
  4. ঘ
    More than 20% profit
  5. ঙ
    None of the above
ব্যাখ্যা
Question: A man buys a pen for 10% less than its value and sells it for 10% more than its value. His gain or loss percentage is-

Solution:
Let, the value of pen is x taka

Buying price = x - 0.1x
= 0.9x taka

Selling price = x + 0.1x
= 1.1x taka

Profit = 1.1x - 0.9x = 0.2x taka

Profit percentage = (0.2x/0.9x) × 100%
= (200/9)%
= 22.22%
৭,৪৯০.
Rabi borrowed Tk 20,000 at 10% compound interest to invest in business. What was his actual profit or loss after 2 years if he earned 12% per year on the borrowed amount? 
  1. ক
    Tk 600 profit
  2. খ
    Tk 800 profit
  3. গ
    Tk 800 loss
  4. ঘ
    Tk 4800 profit
ব্যাখ্যা
Question: Rabi borrowed Tk 20,000 at 10% compound interest to invest in business. What was his actual profit or loss after 2 years if he earned 12% per year on the borrowed amount? 

Solution: 
If he earned 12% on the borrowed Tk 20,000 after 2 years,
His profit  = Tk 20000 × 12 × (2/100)
= Tk 4800 

∴ He got in total = Tk (20000 + 4800 )
= Tk 24800

At 10% compound interest on Tk 20000, he had to pay = Tk 20000{(1 + 10)/100}2
= Tk 20000(11/10)2
= Tk 20000 × (11/10) × (11/10)
= Tk 200 × 121
= Tk 24200

∴ His actual profit = Tk (24800 - 24200)
= Tk 600
৭,৪৯১.
Count the number of triangles and squares in the following figure:
  1. ক
    44 triangles 8 squares
  2. খ
    46 triangles 8 squares
  3. গ
    44 triangles 10 squares
  4. ঘ
    44 triangles 8 squares.
ব্যাখ্যা
Question: Count the number of triangles and squares in the following figure:

Solution:

ত্রিভুজ গণনা:
১টি ফাঁকা স্থান নিয়ে ত্রিভুজ আছে = AEI, EOI, OHI, HAI, EBJ, BFJ, FOJ, OEJ, HOL, OGL, GDL, DHL, OFK, FCK, CGK এবং GOK =16 টি
২টি ফাঁকা স্থান নিয়ে ত্রিভুজ আছে = HAE, AEO, EOH, OHA, OEB, EBF, BFO, FOE, DHO, HOG, OGD, GDH, GOF, OFC, FCG এবং CGO = 16 টি
৪টি ফাঁকা স্থান নিয়ে ত্রিভুজ আছে = HEF, EFG, FGH, GHE, ABO, BGO, CDO এবং DAO = 8 টি 
৮টি ফাঁকা স্থান নিয়ে ত্রিভুজ আছে = DAB, ABC, BCD এবং CDA = 4 টি
মোট ত্রিভুজ আছে = 16 + 16 + 8 + 4 = 44 টি

বর্গক্ষেত্র গণনা:
২টি ফাঁকা স্থান নিয়ে বর্গ আছে = HIOL, IEJO, JFKO এবং KGLO = 4টি 
৪টি ফাঁকা স্থান নিয়ে বর্গ আছে = AEOH, EBFO, OFGC এবং HOGD = 4 টি 
৮ টি ফাঁকা স্থান নিয়ে বর্গ আছে = EFGH = 1 টি 
১৬ টি ফাঁকা স্থান নিয়ে বর্গ আছে = ABCD = 1 টি 

মোট বর্গ আছে = 4 + 4 + 1 + 1 = 10 টি 
-----------------------------------------------------------------

ত্রিভুজ গণনা:
১টি ফাঁকা স্থান নিয়ে ত্রিভুজ আছে = 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16 = 16 টি
২টি ফাঁকা স্থান নিয়ে ত্রিভুজ আছে = 12, 43,14, 32, 56, 87, 85, 76, 1314, 1615, 1316, 1415, 910, 1211, 912, 1011 = 16 টি
৪টি ফাঁকা স্থান নিয়ে ত্রিভুজ আছে = 1285, 67910, 11121415, 431316, 3287, 87912, 9121413, 231314 = 8 টি
৮টি ফাঁকা স্থান নিয়ে ত্রিভুজ আছে = 1582341316, 6791011121415, 128567910, 43131614151211 = 4  টি
মোট ত্রিভুজ আছে = 16 + 16 + 8 + 4 = 44 টি

বর্গক্ষেত্র গণনা:
২টি ফাঁকা স্থান নিয়ে বর্গ আছে = 313, 79, 28, 1412  = 4টি 
৪টি ফাঁকা স্থান নিয়ে বর্গ আছে = 1234, 5678, 9101112, 13141516,  = 4 টি 
৮ টি ফাঁকা স্থান নিয়ে বর্গ আছে = 32879121413= 1 টি 
১৬ টি ফাঁকা স্থান নিয়ে বর্গ আছে = 1 টি 

মোট বর্গ আছে = 4 + 4 + 1 + 1 = 10 টি
৭,৪৯২.
How long will it take to row 20 km upstream if one can row 10 km in 10 minutes in still water and the same distance in 8 minutes with the stream?
  1. ক
    ক) 13.33 min
  2. খ
    খ) 24 min
  3. গ
    গ) 25.55 min
  4. ঘ
    ঘ) 26.67 min
ব্যাখ্যা
Question: How long will it take to row 20 km upstream if one can row 10 km in 10 minutes in still water and the same distance in 8 minutes with the stream?

Solution:
Let,
x be the speed of man in still water 
y be the speed of stream.
∴ Speed of man x = 10/10 km/min
= 1 km/min
= 60 km/hr

In downstream in 8 min he can row 10 km
∴ In downstream in 60 min he can row (10 × 60)/8 km
= 75 km

∴ x + y = 75
⇒ y = 75 - 60
∴ y = 15
∴ Speed of stream = 15 km/hr.

Hence upstream speed = 60 – 15 = 45 km/hr.

Time taken to cover 45 km = 60 min
∴ Time taken to cover 20 km = (60 × 20)/45 min  
= 26.67 min
৭,৪৯৩.
A school has only four classes that contain 10, 20, 30 and 40 students respectively. The pass percentage of these classes are 20%, 30%, 60% and 100% respectively. Find the pass % of the entire school.
  1. ক
    72%
  2. খ
    78%
  3. গ
    66%
  4. ঘ
    62%
ব্যাখ্যা

Question: A school has only four classes that contain 10, 20, 30 and 40 students respectively. The pass percentage of these classes are 20%, 30%, 60% and 100% respectively. Find the pass % of the entire school.

Solution:
Here,
10 of 20% = 2
20 of 30% = 6
30 of 60% = 18
40 of 100% = 40

The number of pass candidates are 2 + 6 + 18 + 40 = 66 out of total 100.

Hence, Pass percentage = 66%

৭,৪৯৪.
In a meeting, every person shakes hands with every other person exactly once. If the total number of handshakes was 45, how many people were in the meeting?
  1. ক
    10
  2. খ
    11
  3. গ
    12
  4. ঘ
    15
  5. ঙ
    None
ব্যাখ্যা
Question: In a meeting, every person shakes hands with every other person exactly once. If the total number of handshakes was 45, how many people were in the meeting?

Solution: 
Let,
the number of people be n.

ATQ,
number of total handshakes,
n(n - 1)/2 = 45
⇒ n(n - 1) = 45 × 2 
⇒ n2 - n = 90 
⇒ n2 - n - 90 = 0
⇒ n2 - 10n + 9n - 90 = 0
⇒ n(n - 10) + 9(n - 10) = 0
⇒ (n - 10)(n + 9) = 0
∴ n = 10, - 9

So the number of people be 10
৭,৪৯৫.
What is the average of the first six multiples of 4?
  1. ক
    16
  2. খ
    18
  3. গ
    14
  4. ঘ
    24
  5. ঙ
    12
ব্যাখ্যা
Question: What is the average of the first six multiples of 4?
 
Solution:
First six multiples of 4 is 4, 8, 12, 16, 20, 24
Average = (4 + 8 + 12 + 16 + 20 + 24)/6
= 84/6
= 14
৭,৪৯৬.
If 2/3 of A = 75% of B = 0.6 of C, then A : B : C is- 
  1. ক
    27 : 24 : 40
  2. খ
    9 : 8 : 10 
  3. গ
    9 : 8 : 4
  4. ঘ
    None of these
ব্যাখ্যা
Question: If 2/3 of A = 75% of B = 0.6 of C, then A : B : C is- 

Solution: 
2/3 of A=75%B
⇒ 2A/3 = 75B/100 = 3B/4
⇒ A/B = 9/8
⇒ A : B = 9 : 8 

75% of B = 0.6 of C
⇒ 3B/4 = 3C/5
⇒ B/C = 12/15 = 4/5
 ⇒ B : C = 4 : 5 = 8 : 10 

A : B : C = 9 : 8 : 10 
৭,৪৯৭.
At 7 : 15, how many degrees are included between hands of hour and minute of the clock?
  1. ক
    62.5°
  2. খ
    95°
  3. গ
    115°
  4. ঘ
    127.5°
ব্যাখ্যা
Question: At 7 : 15, how many degrees are included between hands of hour and minute of the clock?

Solution:
Value of angle = {(11 × 15) - (60 × 7)}/2
= (165 - 420)/2
= 255/2
= 127.5°
৭,৪৯৮.
An angle which is less than 360° and larger than 180° is classified as:
  1. ক
    acute angle
  2. খ
    reflex angle
  3. গ
    obtuse angle
  4. ঘ
    adjacent angle
ব্যাখ্যা
Question: An angle which is less than 360&deg; and larger than 180&deg; is classified as:

Solution: 
- ৯০° অপেক্ষা অপেক্ষা ছোট কোণকে সূক্ষ্মকোণ বলে।
- ৯০° অপেক্ষা বড় কিন্তু ১৮০° অপেক্ষা ছোট কোণকে স্থূলকোণ বলে।
- ১৮০° অপেক্ষা বড় কিন্তু ৩৬০° অপেক্ষা ছোট কোণকে প্রবৃদ্ধ কোণ বলে।
- একটি সরলরেখার উপর আরেকটি সরলরেখা লম্বভাবে দন্ডায়মান হলে যে দুইটি সন্নিহিত কোণ উৎপন্ন হয় এবং তাদের মান সমান হলে (৯০°) তাদের প্রত্যেককেটিকে সমকোণ বলে।
৭,৪৯৯.
A person has deposited Tk. 13200 in a bank which pays 14% interest. He withdraws the money and invests in Tk. 100 stock at Tk. 110 which pays a dividend of 15%. How much does he gain or lose?
  1. ক
    ক) Loses Tk. 48
  2. খ
    খ) Gains Tk. 48
  3. গ
    গ) Loses Tk. 132
  4. ঘ
    ঘ) Gains Tk. 132
ব্যাখ্যা

Income from bank = 14% of Tk. 13200 = Tk. 1848
Number of shares purchased
= Tk. (13200/110)
= Tk. 120
Income from stock
= (15% of Tk. 100) × 120
= Tk. (15 × 20)
= Tk. 1800
∴ Loss = Tk. (1848 - 1800)
= Tk. 48

৭,৫০০.
What is the slope of a line perpendicular to the line whose equation is 7x + 3y = 12?
  1. ক
    - 7/3
  2. খ
    3/4
  3. গ
    3/7
  4. ঘ
    - 4/3
ব্যাখ্যা
Question: What is the slope of a line perpendicular to the line whose equation is 7x + 3y = 12?

Solution:
সরল রেখার সাধারণ সমীকরণ,
y = mx + c ......(1) (এখানেm = ঢাল)

যদি কোনো রেখার ঢাল হয় m, তবে তার লম্ব (perpendicular) রেখার ঢাল হবে,
m' = - (1/m)

এখন,
7x + 3y = 12
3y = - 7x + 12
y = - (7/3)x + 4
(1) নং এর সাথে তুলনা করে পাই,
m = - (7/3)

∴ লম্ব (perpendicular) রেখার ঢাল হবে, m' = - {1/- (7/3)} = 3/7