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মোট প্রশ্ন১৬,১২৪এই পাতা১০০প্রতি পাতা১০০
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Bank Math

PrepBank · পাতা ৫৫ / ১৬১ · ৫,৪০১–৫,৫০০ / ১৬,১২৪

৫,৪০১.
The average of Amit's five quiz scores is 88. What score does Amit need to get on a sixth quiz to raise his average for all six quizzes to 90?
  1. ক
    98
  2. খ
    100
  3. গ
    94
  4. ঘ
    99
ব্যাখ্যা
Question: The average of Amit's five quiz scores is 88. What score does Amit need to get on a sixth quiz to raise his average for all six quizzes to 90?

Solution:
Sum of 5 scores = 88 × 5 = 440
Sum of 6 scores = 90 × 6 = 540
Sixth quiz score = 540 - 440 = 100
৫,৪০২.
A can finish a job in 18 days, B in 12 days, and C in 6 days. B and C begin the work together but have to stop after working for 2 days. How many days will A alone take to complete the remaining work? 
  1. ক
    9 days
  2. খ
    25 days
  3. গ
    15 days
  4. ঘ
    3 days
ব্যাখ্যা

Question: A can finish a job in 18 days, B in 12 days, and C in 6 days. B and C begin the work together but have to stop after working for 2 days. How many days will A alone take to complete the remaining work?

Solution:
Work done by B and C in 1 day:
B's 1-day work = 1/12,
C's 1-day work = 1/6
B + C in 1 day = 1/12 + 1/6
= (1 + 2)/12
= 3/12
= 1/4

Work done by B and C in 2 days = 2 × 1/4 = 1/2

Remaining work = 1 - 1/2 = 1/2

A's 1-day work = 1/18

∴ Time for A to finish remaining work = (1/2) ÷ (1/18) 
= 1/2 × 18
= 9 days

৫,৪০৩.
A ladder is leaning against a wall. It makes a 60° angle with the wall. If the distance between foot of ladder and wall is 5.5 meters, find the length of the ladder.
  1. ক
    11 m
  2. খ
    10.5 m
  3. গ
    10 m
  4. ঘ
    9.5 m
ব্যাখ্যা

Question: A ladder is leaning against a wall. It makes a 60° angle with the wall. If the distance between foot of ladder and wall is 5.5 meters, find the length of the ladder.

Solution:

Let BC be the wall and AC be the ladder.
∠BAC = 60° and AB = 5.5 meter
In ΔABC,
cos60° = AB/AC
⇒ 1/2 = 5.5/AC
⇒ AC = 5.5 × 2
∴ AC = 11

৫,৪০৪.
 √0.01 + √0.81 +√1.21+√0.0009 = ?
  1. ক
    2.13
  2. খ
    2.03
  3. গ
    3.03
  4. ঘ
    3.01
  5. ঙ
    2.06
ব্যাখ্যা
The sum of 
√0.01 + √0.81 +√1.21+√0.0009
= 0.1 + 0.9 + 1.1 + 0.03
= 2.13
৫,৪০৫.
In how many ways a President, VP and Water-boy can be selected from a group of 10 people.
  1. ক
    ক) 7C3
  2. খ
    খ) 10C3
  3. গ
    গ) 7P3
  4. ঘ
    ঘ) 10P3
ব্যাখ্যা
We are selecting three different posts here, so order matters.
Thus, total ways of selecting a President, VP and Water-boy from a group of 10 people would be 10P3
------------------------------------------------------------------------------------
Alternative way:

৫,৪০৬.
Merun invested Tk. 333000 in 5(½) % stocks at 110. If brokerage is TK. 1, what is his annual income from his investment?
  1. ক
    ক) 16300
  2. খ
    খ) 17300
  3. গ
    গ) 16500
  4. ঘ
    ঘ) 18547
  5. ঙ
    ঙ) 17098
ব্যাখ্যা

Investment = Tk. 333000
Since face value is not given, we can take it as Tk. 100
and dividend per share = Tk.11/2
Market Value = 110 + 1 = 111
Number of shares purchased = 333000/111 = 3000
Total income = 3000 × 11/2 = Tk.16500

৫,৪০৭.
The distance from the point P to two vertices A and B of an equilateral triangle are ।PA।= 2 and ।PB।= 3. What is the greatest possible value of ।PC। = ?
  1. ক
    ক) 5
  2. খ
    খ) 9
  3. গ
    গ) 6
  4. ঘ
    ঘ) 12
ব্যাখ্যা


ধরি,
 । PA।= a =  2, ।PB।= b = 3 এবং । PC। = x 

অর্ধ পরিসীমা = S = (a + b + x)/2 
                            = (2 + 3 + x)/2 
                             =(5 + x)/2 
ΔABC = (1/2)[(√3/4)(a2 + b2 + x2) + 3√{s(s - a)(s - b)(s - c)}]
           = (1/2)[(√3/4)(22 + 32 + x2) + 3√{(5 + x)/2 ((5 + x)/2  - 2)((5 + x)/2  - 3)((5 + x)/2  - x)}]
          = (1/2)[(√3/4)(13 + x2) + 3√{((5 + x)/2)((x + 1)/2)((x - 1)/2)((5 - x)/2)}]
           = (1/2)[(√3/4)(13 + x2) + 3√{((5 + x)/2)((5 - x)/2) × ((x + 1)/2)((x - 1)/2)}]
           = (1/2)[(√3/4)(13 + x2) + 3√{((25 - x2)/4)((x2 - 1)/4)}]
            = (1/2)[(√3/4)(13 + x2) + 3√{(25 - x2)(x2 - 1)/16}]
             = (1/2)[(√3/4)(13 + x2) + (3/4)√{(25 - x2)(x2 - 1)

ΔABC এর মান পাওয়া যাবে যখন (25 - x2)(x2 - 1) ≥ 0

25 - x2 ≥ 0 এবং x2 - 1 ≥ 0
25 - x2 ≥ 0 সত্য হবে যদি যখন 
x = ±1, ±2, ±3, ±4, ±5
- 5 < x < 5 
x এর সর্বোচ্চ মান হবে 5
। PC। = x = 5
৫,৪০৮.
If Lalon loses 8 pounds, he will weigh twice as much as his sister. Together they now weigh 278 pounds. What is Lalon’s present weight, in pounds?
  1. ক
    ক) 135
  2. খ
    খ) 139
  3. গ
    গ) 147
  4. ঘ
    ঘ) 188
ব্যাখ্যা

Let, L = Lalon’s current weight, in pounds
S = Sister’s current weight, in pounds

We are told that “If Lalon loses 8 pounds, he will weigh twice as much as his sister.'' We put this into an equation:
L – 8 = 2S
∴ L = 2S + 8...... (i)

Next, we are told that “Together they now weigh 278 pounds.” We can also put this into an equation.
L + S = 278........ (ii)
To solve this equation, we can substitute 2S + 8 from Equation (i) for the variable L in Equation 2:
2S + 8 + S = 278
3S = 270
S = 90

From equation (ii) we can find,
L = 278 - 90 = 188

৫,৪০৯.
A toaster machine is priced at Tk. 25,000 and sold with two successive discounts of 15% and 10%. What is its final selling price? 
  1. ক
    Tk. 19,125
  2. খ
    Tk. 15,125
  3. গ
    Tk. 18,125
  4. ঘ
    Tk. 19,000
ব্যাখ্যা

 Question: A toaster machine is priced at Tk. 25,000 and sold with two successive discounts of 15% and 10%. What is its final selling price?

Solution: 
Given,
Marked Price = Tk. 25000
Price after 15% discount
= 25000 - (15% of 25000)
= 25000 - 3750
= Tk. 21,250

Price after 10% discount
= 21250 - (10% of 21250)
= 21250 - 2125
= Tk. 19,125

∴ Final Selling Price = Tk. 19,125

৫,৪১০.
The average of four consecutive even numbers is 27. Find the largest of these numbers.
  1. ক
    ক) 28
  2. খ
    খ) 30
  3. গ
    গ) 32
  4. ঘ
    ঘ) 40
ব্যাখ্যা

Consider the consecutive even numbers as : x, (x + 2), (x + 4) and (x+ 6)
Average = Sum of Quantities/Number of Quantities
{x + (x + 2) + (x + 4) + (x + 6)}/4 = 27
⇒ (4x + 12)/4 = 27
⇒ x + 3 = 27
⇒ x = 27 - 3
⇒ x = 24.

Therefore,
Largest number = (x + 6) = (24 + 6) = 30
Smallest number = 24.
Hence, the answer is 30.

৫,৪১১.
The angle between the minute hand and the hour hand of a clock when the time is 4:20, is:
  1. ক
    ক) 0º
  2. খ
    খ) 10º
  3. গ
    গ) 5º
  4. ঘ
    ঘ) 20º
ব্যাখ্যা

Angle traced by hour hand in 13/ 3 hrs
=( 360/12 × 13/3)∘ =130∘
Angle traced by min. hand in 20 min
=( 360/60 ×20)∘=120∘
∴Required angle
=(130−120)∘ =10∘

৫,৪১২.
If the salary of an employee is reduced by 10 percent for his late attendance and then increased by 10 percent on a pardon, how much does he lose?
  1. ক
    ক) 1%
  2. খ
    খ) (1/2)%
  3. গ
    গ) 99%
  4. ঘ
    ঘ) 90%
ব্যাখ্যা
At 10% decrease,
salary = 100 - 10 = 90

And a 10% increase,
the salary = 90 + 90×10/100 = 99

∴ He is at = 100 - 99 = 1% loss
৫,৪১৩.
A man buys tk. 50 shares in a company which pays 10% dividend. If the man gets 12.5% on his investment, at what price did he buy the shares?
  1. ক
    ক) 35.50
  2. খ
    খ) 40
  3. গ
    গ) 48
  4. ঘ
    ঘ) 52
ব্যাখ্যা

Dividend on 1 share
= ( 10/ 100 ×50)
= Tk. 5
Tk. 12.50 is an income on an investment of tk. 100
Tk. 5 is an income on an investment of :
= Tk. (100× 2/25 ×5)
= Tk. 40.
∴ Cost of 1 share = Tk. 40

৫,৪১৪.
What is the area of the triangle BCD?
  1. ক
    52
  2. খ
    48
  3. গ
    54
  4. ঘ
    42
  5. ঙ
    46
ব্যাখ্যা
Question: What is the area of the triangle BCD?

Solution:
ΔBAC থেকে পিথাগোরাসের সূত্রানুযায়ী,
BC2 = AB2 - AC2 [BCA এক সমকোণ বলে]
⇒ BC2 = 102 - 62
⇒ BC2 = 64
∴ BC = 8

আবার, ΔDEB তে DEB এক সমকোণ বলে,
BE2 = BD2 - DE2
⇒ BE2 = 132 - 52
⇒ BE2 = 144
∴ BE = 12

DEBC এর ক্ষেত্রফল = (1/2) × (BC + ED) × BE [ট্রাপিজিয়াম DEBC তে BC এবং DE সমান্তরাল এবং তাদের মধ্যে লম্ব দূরত্ব EB]
= (1/2) × (8 + 5) × 12 = 78
ΔDEB এর ক্ষেত্রফল = (1/2) × 12 × 5 = 30
∴ ΔBCD এর ক্ষেত্রফল = 78 - 30 = 48
৫,৪১৫.
A rectangular block 6 cm by 12 cm by 15 cm is cut up into an exact number of equal cubes. Find the least possible number of cubes.
  1. ক
    ক) 30
  2. খ
    খ) 40
  3. গ
    গ) 50
  4. ঘ
    ঘ) 60
ব্যাখ্যা
Volume of the block = (6 × 12 × 15) cm3
= 1080 cm3

Side of the largest cube
= H.C.F of 6 cm, 12 cm, 15 cm
= 3 cm.

Volume of this cube = (3 × 3 × 3) cm3
= 27 cm3

Number of cubes = 1080/27
= 40.
৫,৪১৬.
A shirt with a list price of Tk. 150 is sold for Tk. 105 after two successive discounts. If the second discount is 12.5%, what was the rate of the first discount? 
  1. ক
    20%
  2. খ
    30%
  3. গ
    25%
  4. ঘ
    33%
ব্যাখ্যা

Question: A shirt with a list price of Tk. 150 is sold for Tk. 105 after two successive discounts. If the second discount is 12.5%, what was the rate of the first discount?

Solution:
Let the first discount be x%.

∴ After the first discount,
∴ the price = (100 − x)% of 150 
= (100 − x)/100 × 150

After the second discount of 12.5%,
the selling price = 87.5% of the first discounted price
= 87.5/100 × (100 − x)/100 × 150

Given selling price = 105,
ATQ,
87.5/100 × (100 − x)/100 × 150 = 105
⇒ 100 − x = (105 × 100 × 100) / (87.5 × 150)
⇒ 100 − x = 80

∴ x = 100 − 80
= 20

∴ The first discount is 20%.

৫,৪১৭.
A person's present age is two-fifth of the age of his mother. After 8 years, he will be one-half of the age of his mother. How old is the mother at present?
  1. ক
    ক) 36 years
  2. খ
    খ) 38 years
  3. গ
    গ) 40 years
  4. ঘ
    ঘ) 42 years
ব্যাখ্যা
Let mother's present age be x years
Therefore, person's present age = 2x/5 years
2x/5 + 8 = (x + 8)/2
or, x = 40 years
৫,৪১৮.
If p and n are integers such that p > n > 0 and p2 - n2 = 12, which of the following value of p - n?
  1. ক
    ক) - 1
  2. খ
    খ) 2
  3. গ
    গ) 8
  4. ঘ
    ঘ) 18
ব্যাখ্যা
p > n > 0 এবং p2 - n2 = 12
অপশন চেক 
ধরি 
p = 4 , n = 2
p2 - n2 = 12
42 - 22 = 12
p - n = 4 - 2 = 2
৫,৪১৯.
If 100 square marbles of equal size were required to pave a corridor of dimension 6m x 24m then the length of each marble is -
  1. ক
    120 cm
  2. খ
    144 cm
  3. গ
    250cm
  4. ঘ
    100 cm
ব্যাখ্যা

Dimension of the corridor = 6m x 24 m
Area of the corridor = 6 x 24 m2.
It is given that 100 square marbles are needed to cover the corridor of area 6 x 24 m2.
Area of each marble = 6 x 24 / 100 m2= 144 / 100 m2= 1.44 m2
Since the marbles are in square shape, the length of each marble = sqrt(1.44) m = 1.2 m
Hence the answer is 1.2m = 1.2 x 100 cm = 120 cm.

৫,৪২০.
The difference between the length and the breadth of a blackboard is 8cm. If the breadth is decreased by 4cm and the length increased by 7cm, the area remains the same. Find the dimensions of the blackboard?
  1. ক
    ক) 30, 22
  2. খ
    খ) 28, 20
  3. গ
    গ) 34, 26
  4. ঘ
    ঘ) 56, 48
ব্যাখ্যা
Question: The difference between the length and the breadth of a blackboard is 8cm. If the breadth is decreased by 4cm and the length increased by 7cm, the area remains the same. Find the dimensions of the blackboard?

Solution: 
blackboard এর প্রস্থ = x সে.মি.
blackboard এর  দৈর্ঘ্য = x + 8 সে.মি.
blackboard এর ক্ষেত্রফল = x(x + 8)  বর্গ সে.মি.
= x2 + 8x বর্গ সে.মি.

প্রশ্নমতে 
(x - 4)(x + 8 + 7) = x2 + 8x
(x - 4)(x + 15) = x2 + 8x
x2 + 15x - 4x - 60 = x2 + 8x
x2 - x2 + 11x - 8x  = 60
3x = 60
x = 20

blackboard এর প্রস্থ = 20 সে.মি.
blackboard এর দৈর্ঘ্য = 20 + 8 = 28 সে.মি.
৫,৪২১.
The sum of first 17 terms of the series 5, 9, 13, 17, ............ is
  1. ক
    529
  2. খ
    462
  3. গ
    629
  4. ঘ
    523
ব্যাখ্যা
Question: The sum of first 17 terms of the series 5, 9, 13, 17, ............ is

Solution:
৯ - ৫ = ৪
১৩ - ৯ = ৪
∴ সাধারণ অন্তর, d = ৪ 
প্রথম পদ, a = ৫
পদের সংখ্যা, n = ১৭

প্রথম ১৭ পদের সমষ্টি = (n/2){2a + (n - 1)d}
= (১৭/২){২ × ৫ + (১৭ -১) × ৪}
= (১৭/২) (১০ + ৬৪)
= (১৭/২) × ৭৪
= ১৭ × ৩৭ 
= ৬২৯
৫,৪২২.
A pendulum strikes 5 times in 3 seconds and another pendulum strikes 7 times in 4 seconds. If both pendulums start striking at the same time, how many clear strikes can be listened to in a minute?
  1. ক
    195
  2. খ
    199
  3. গ
    200
  4. ঘ
    205
ব্যাখ্যা

First pendulum strikes once in 3/5 seconds.
Second pendulum strikes once in 4/7 seconds
L.C.M of 3/5 and 4/7
= (L.C.M of 3 and 4)/(H.C.F of 5 and 7)
= 12.
So, they strike together after every 12 seconds.
Thus,
they strike together {(60/12) + 1}
= 6 times in 1 minute.
∴ Total number of clear strikes heard
= [{60/(3/5)} + {60/(4/7)}] - 6
= {60 × (5/3) + 60 × (7/4)} - 6
= (100 + 105) - 6
= 199.

৫,৪২৩.
60% of a number added to 60 gives the result as the number itself, then the number is:
  1. ক
    ক) 125
  2. খ
    খ) 150
  3. গ
    গ) 175
  4. ঘ
    ঘ) 200
ব্যাখ্যা
Question: 60% of a number added to 60 gives the result as the number itself, then the number is:

Solution:
Let x be the number which is added to 60.
Now,
60% of x = 0.6x

ATQ,
⇒ 60 + 0.6x = x
⇒ x - 0.6x = 60
⇒ 0.4x = 60
⇒ x = 60/0.4
∴ x = 150
৫,৪২৪.
Three-fifth of the square of a certain number is 126.15. What is the numbers?
  1. ক
    ক) 13.5
  2. খ
    খ) 14.5
  3. গ
    গ) 75.5
  4. ঘ
    ঘ) 145
ব্যাখ্যা

Let the number be x
Then,⇔3/5 x2 = 126.15
⇔x2=(126.15× 5/3)
⇔x2=210.25
⇔x= √ 210.25
⇔x=14.5

৫,৪২৫.
A hall is 15m long and 12m broad. If the sum of the areas of the floor and the ceiling is equal to of the areas of four walls, the volume of the hall is:
  1. ক
    ক) 720 m3
  2. খ
    খ) 1,200 m3
  3. গ
    গ) 900 m3
  4. ঘ
    ঘ) 1,800 m3
ব্যাখ্যা
ধরি 
হল ঘরের উচ্চতা = h
দেওয়া আছে, 
         ঘরের দৈর্ঘ্য ১৫ মি.
          ঘরের প্রস্থ ১২ মি. 

প্রশ্নমতে, 
 ২ ( ১৫ × ১২ ) = ২ × ( ১৫ + ১২ ) × h 
বা, ২ × ২৭ × h = ২ ( ১৫ × ১২ )
বা, h  × ২৭ = ১৫ × ১২
বা, h = (১৫ × ১২)/২৭
      h = ২০/৩
আমরা জানি,
আয়তন= (১৫ × ১২×২০)/৩ 
             = ১২০০ ঘন মিটার
৫,৪২৬.
What is the sum of first five prime numbers?
  1. ক
    21
  2. খ
    23
  3. গ
    28
  4. ঘ
    32
ব্যাখ্যা
Question: What is the sum of first five prime numbers?

Solution:
First five prime numbers are 2, 3, 5, 7, 11

sum of first five prime numbers = 2 + 3 + 5 + 7 + 11 = 28
৫,৪২৭.
The ratio of three numbers is 3 : 4 : 5 and the sum of their squares is 1250. The sum of the numbers is-
  1. ক
    90
  2. খ
    65
  3. গ
    60
  4. ঘ
    30
ব্যাখ্যা
Question: The ratio of three numbers is 3 : 4 : 5 and the sum of their squares is 1250. The sum of the numbers is-

Solution:
Let, the number be 3x, 4x, 5x

According to the question,
(3x)2 + (4x)2 + (5x)2 = 1250
⇒ 9x2 + 16x2 + 25x2 = 1250
⇒ 50x2 = 1250
⇒ x2 = 1250/50
⇒ x2 = 25
∴ x = 5

∴ The sum of the numbers = 3x + 4x + 5x
= 12x
= 12 × 5
= 60
৫,৪২৮.
A cube has a total surface area of 294 square meters. What is the length of its diagonal?
  1. ক
    7√3 m
  2. খ
    7 m
  3. গ
    5√3 m
  4. ঘ
    6√2 m
ব্যাখ্যা

Question: A cube has a total surface area of 294 square meters. What is the length of its diagonal?

Solution:
আমরা জানি, একটি ঘনকের মোট পৃষ্ঠের ক্ষেত্রফল = 6a2
প্রশ্নমতে, 6a2 = 294
⇒ a2 = 294/6
⇒ a2 = 49
⇒ a = √49
⇒ a = 7 মিটার।

আমরা জানি, 
একটি ঘনকের কর্ণের দৈর্ঘ্য = a√3
এখানে, a = 7
সুতরাং, কর্ণের দৈর্ঘ্য = 7√3 মিটার।

সুতরাং, ঘনকটির কর্ণের দৈর্ঘ্য হলো 7√3 মিটার।

৫,৪২৯.
40% boys in a school. If 30% of the boys and 20% of the girls went to the debate club, what percentage of the students went to the debate club?
  1. ক
    21%
  2. খ
    27%
  3. গ
    25%
  4. ঘ
    24%
  5. ঙ
    20%
ব্যাখ্যা
ধরি, মোট ছাত্র-ছাত্রী ১০০ জন।
৪০% ছাত্র হলে ছাত্রী ৬০%।
ছাত্র অংশ নেয় ৪০×৩০%= ১২ জন।
ছাত্রী অংশ নেয় ৬০×২০%= ১২ জন।
∴ শতকরা মোট অংশগ্রহণকারী ছাত্র-ছাত্রী = ২৪%।
৫,৪৩০.
If the average of 5 consecutive number is 12, what is the sum of the least and the greatest of the integers?
  1. ক
    ক) 10
  2. খ
    খ) 24
  3. গ
    গ) 12
  4. ঘ
    ঘ) 14
ব্যাখ্যা
প্রশ্ন: If the average of 5 consecutive number is 12, what is the sum of the least and the greatest of the integers?

সমাধান: 
ধরি, পাঁচটি ক্রমিক সংখ্যা x, x + 1, x + 2, x + 3, x + 4

পাঁচটি ক্রমিক সংখ্যার গড়  ১২
পাঁচটি সংখ্যার সমষ্টি = ১২ × ৫ 
= ৬০ 

প্রশ্নমতে,
x + x + 1 + x + 2 + x + 3 + x + 4 = 60
⇒ 5x + 10 = 60 
⇒ 5x = 50 
∴ x = 10 

সবচেয়ে ছোট সংখ্যাটি ১০
সবচেয়ে বড় সংখ্যাটি ১০ + ৪ = ১৪

∴সবচেয়ে বড় সংখ্যা ও সবচেয়ে ছোট সংখ্যার সমষ্টি = ১০ + ১৪
= ২৪ 
৫,৪৩১.
Two dice are tossed. The probability that the total score is a prime number is:
  1. ক
    ক) 5/18
  2. খ
    খ) 1/4
  3. গ
    গ) 1/2
  4. ঘ
    ঘ) 5/12
ব্যাখ্যা
Question: Two dice are tossed. The probability that the total score is a prime number is:

Solution: 
Clearly, n(S) = (6 × 6) = 36.
Let E = Event that the sum is a prime number.

Then E = {(1, 1), (1, 2), (1, 4), (1, 6), (2, 1), (2, 3), (2, 5), (3, 2), (3, 4), (4, 1), (4, 3), (5, 2), (5, 6), (6, 1), (6, 5) }
  n(E) = 15.

P(E) = n(E)/n(S) = 15/36 = 5/12
৫,৪৩২.
If Px = Qy = Rz and Q/P = R/Q then [2z/(x + z)]3 = ?
  1. ক
    y3/x3
  2. খ
    y3/z2
  3. গ
    z/y
  4. ঘ
     x/y
ব্যাখ্যা

Question: If Px = Qy = Rz and Q/P = R/Q then [2z/(x + z)]3 = ?
(Officer Cash 22 এর অনুরূপ) )

Solution:
ধরি,
Px = Qy = Rz = k
এখন,
Px = k
∴ P = k(1/x)

অনুরুপভাবে,
Qy = k
∴ Q = k(1/y)
এবং
Rz = k
∴ R = k(1/z)

আবার,
⇒ Q/p = R/Q
⇒ Q2 = PR
⇒ {k(1/y)}2 = k(1/x) × k(1/z)
⇒ k(2/y) = k(z + x)/xz
⇒ 2/y = (z + x)/xz
⇒ 2xz = y(z + x)
∴ 2z/(x + z) = y/x

∴ [2z/(x + z)]3 = y3/x3

৫,৪৩৩.
The capital stock of a company is Tk. 300000 and is divided into 3000 shares. If the company declares a total dividend of Tk. 45000, how much will Rina receive for her 40 shares?
  1. ক
    Tk. 800
  2. খ
    Tk. 650
  3. গ
    Tk. 720
  4. ঘ
    Tk. 600
ব্যাখ্যা
Question: The capital stock of a company is Tk. 300000 and is divided into 3000 shares. If the company declares a total dividend of Tk. 45000, how much will Rina receive for her 40 shares?

Solution:
3000 shares income Tk. 45000
1 share incomes Tk. 45000/3000
40 shares income Tk. (45000 × 40)/3000
= Tk. 600

∴ Rina will receive Tk. 600 as her share of the dividend.
৫,৪৩৪.
Which of the number below is not equivalent to 4%?
  1. ক
    ক) 1/25
  2. খ
    খ) 4/100
  3. গ
    গ) 0.40
  4. ঘ
    ঘ) 0.04
ব্যাখ্যা

আমরা অপশন গুলো বিবেচনা করি -
A. 1/25 = 1/25 × 100% = 4%
B. 4/100 = 1/25 = 4%
C. .04 = 4/100 = 40/100 × 100% = 40%
D. 0.04 = 4/100 = 1/25 = 4%
সুতরাং, অপশন গ) 40% সঠিক উত্তর।

৫,৪৩৫.
Two pipes A and B can fill a cistern in 24 and 36 minutes respectively. Both pipes are opened together, after how many minutes should B be turned off, so that the cistern be fill in 16 minutes?
  1. ক
    12 minutes.
  2. খ
    14 minutes.
  3. গ
    18 minutes.
  4. ঘ
    20 minutes.
ব্যাখ্যা
Question: Two pipes A and B can fill a cistern in 24 and 36 minutes respectively. Both pipes are opened together, after how many minutes should B be turned off, so that the cistern be fill in 16 minutes?

Solution:
A can fill the cistern in 24 minutes
So in 1 min A can fill the cistern = 1/24 part
In 16 min, A can fill the cistern = 16/24 part
= 2/3 part

Remaining part = 1 - 2/3 = 1/3 rd

As B can fill full cistern in 36 minutes
So it will fill 1/3 rd part in = (1/3 × 36) minutes.
= 12 minutes.
৫,৪৩৬.
Lopa invested a part of Tk. 12000 in 12% stock at Tk. 120 and remainder in 15% stock at Tk. 125. If his total dividend per annum is Tk. 1360, how much does he invest in 12% stock at Tk. 120?
  1. ক
    ক) Tk. 4000
  2. খ
    খ) Tk. 4500
  3. গ
    গ) Tk. 5500
  4. ঘ
    ঘ) Tk. 6000
ব্যাখ্যা

Let investment in 12% stock be Tk. x.
Then,
investment in 15% stock = Tk. (12000 - x)

∴ (12/120) × x + (15/125) × (12000 - x) = 1360
⇒ x/10 + 3/25(12000 - x) = 1360
⇒ 5x + 72000 - 6x = (1360 × 50)
⇒ x = 4000

Hence, Investment in 12% stock is Tk. 4000

৫,৪৩৭.
A number whose fifth part increased by 4 is equal to its fourth part reduced by 10. Find the number.
  1. ক
    ক) 260
  2. খ
    খ) 270
  3. গ
    গ) 280
  4. ঘ
    ঘ) 290
ব্যাখ্যা
Let
the number be x
According to problems condition 
(x/5) + 4 = (x/4) - 10
(x/5) - (x/4) = - 10 - 4
(4x - 5x)/20 = - 14
-x/20 = - 14 
x = 280
৫,৪৩৮.
A sum of money at simple interest amounts to Tk 960 in 5 years and to Tk 1000 in 6 years. The sum is-
  1. ক
    720 tk
  2. খ
    760 tk
  3. গ
    780 tk
  4. ঘ
    740 tk
ব্যাখ্যা
Question: A sum of money at simple interest amounts to Tk 960 in 5 years and to Tk 1000 in 6 years. The sum is-

Solution:
Simple interest for 1 years = 1000 - 960 tk
= 40 tk

∴ Simple interest for 5 years = 40 × 5
= 200 tk

∴ Sum = Amount after 5 years - SI for 5 years
= 960 - 200 tk
= 760 tk
৫,৪৩৯.
4 years ago, the ratio of 1/2 of A’s age that time and four times of B’s age at the time were 5:12. Eight years hence, 1/2 of A’s age at that time will be less than B’s age at that time by 2 years. What is B’s present age?
  1. ক
    ক) 10 years
  2. খ
    খ) 14 years
  3. গ
    গ) 12 years
  4. ঘ
    ঘ) 5 years
ব্যাখ্যা

Let the present age of A be a years and that of B be b years
Then, 4 years ago,
A's age = (a - 4)
B's age = (b - 4)
Now, according to the given information in question,
{(a - b)/2}/4(b - 4) = 5/12 or (a - 4)/2(4b - 16) = 5/12 or (a - 4)/(4b - 16) = 5/6
By cross multiplying we get
or, 6a - 24 = 20b - 80
or, 6a - 20b = -56
or, 10b - 3a = 28
After 8 years, 
(a + 8)2 + 2 = b + 8
or, a/2 + 4 + 2 = b + 8
or, b - a/2 = -2
or, 2b - a = -4 .......(i)
a = 2b + 4 ......(ii)
Putting the value of a in equation (i), we get
10b - 3(2b + 4) = 28
or, 4b = 40
∴ b = 10
Hence, the present age of B is 10 years.

৫,৪৪০.
Rahman is a boatman. He can row a boat at the speed of 5 km/hr upstream and 15 km/hr downstream. Find the speed of the stream.
  1. ক
    5 km/hr
  2. খ
    8 km/hr
  3. গ
    6 km/hr
  4. ঘ
    4 km/hr
ব্যাখ্যা

Question: Rahman is a boatman. He can row a boat at the speed of 5 km/hr upstream and 15 km/hr downstream. Find the speed of the stream.

Solution:
Let’s denote:
B as Speed of the boat in still water (km/h)
S as Speed of the stream (km/h)

Speed of the boat upstream is the speed of the boat in still water minus the speed of the stream:
B - S = 5 km/hr -------- (1)

Speed of the boat downstream is the speed of the boat in still water plus the speed of the stream:
B + S = 15 km/hr -------- (2)

(1) + (2)
B - S = 5
B + S = 15
2B = 20
∴ B = 10 km/hr

Putting the value of B in (2)
∴ S = (15 - 10) km/hr = 5 km/hr

∴ The speed of the stream is 5 km/hr

৫,৪৪১.
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
    36
  2. খ
    45
  3. গ
    54
  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:
 

Let,
Number of people who can speak both languages = x persons
∴ Number of people who speak only French = (55 - x) persons
∴ Number of people who speak only Spanish = (40 - x) persons

Given that,
Number of people who speak none of the languages = 20 persons

According to the question,
Only French + Both + Only Spanish = Total students - Those who speak none
⇒ (55 - x) + x + (40 - x) = 100 - 20 
⇒ 95 - x = 80
⇒ x = 95 - 80
∴ x = 15

∴ Only French = (55 - 15) = 40 persons
∴ Only Spanish = (40 - 15) = 25 persons

∴ Number of people who speak only one language (French or Spanish) = (40 + 25) = 65 persons

৫,৪৪২.
If tanθ = 3/4, then cosθ = ?
  1. ক
    5/3
  2. খ
    4/3
  3. গ
    4/5
  4. ঘ
    3/5
ব্যাখ্যা

Question: If tanθ = 3/4, then cosθ = ?

Solution:
এখানে,
tanθ = 3/4 = লম্ব/ভূমি

∴ লম্ব = 3, ভূমি = 4
∴ অতিভুজ = √(32+ 42)
= √25 = 5

∴ cosθ = ভূমি/অতিভুজ
= 4/5

৫,৪৪৩.
Mr. Tamal purchased stocks for Tk.1,500 and sold 2/3 of it after its value doubled. He sold the remaining stock at 5 times of its purchase price. What is the total profit?
  1. ক
    ক) Tk. 2,000
  2. খ
    খ) Tk. 2,500
  3. গ
    গ) Tk. 3,000
  4. ঘ
    ঘ) Tk. 6,000
ব্যাখ্যা
Question: Mr. Tamal purchased stocks for Tk.1,500 and sold 2nd/3 of it after its value doubled. He sold the remaining stock at 5 times of its purchase price. What v. as the total profit?

Solution: 
স্টকের মূল্য = ১৫০০ টাকা
দ্বিগুণ বাড়ার পর , এর মূল্য = ১৫০০ × ২ 
= ৩০০০ টাকা 

৩০০০ এর ২/৩ অংশ 
= ২০০০ টাকা 
বাকি আছে = ১ - ২/৩
= ১/৩ অংশ 

বৃদ্ধি পাবার আগে ১/৩ অংশের দাম = ১৫০০ ×১/৩ টাকা
= ৫০০ টাকা 

১/৩ অংশের নতুন দাম = ৫০০ × ৫
= ২৫০০ টাকা 

মোট বিক্রয়মূল্য = ২০০০ + ২৫০০ টাকা 
= ৪৫০০ টাকা 

∴ মোট লাভ = ৪৫০০ - ১৫০০ টাকা 
= ৩০০০ টাকা 
৫,৪৪৪.
If the sum of two numbers is 26 and their H. C. F and L. C. M are 1 and 120 respectively, the sum of the reciprocals of the two numbers is- 
  1. ক
    13/60
  2. খ
    13/62
  3. গ
    11/60
  4. ঘ
    17/60
ব্যাখ্যা

Question: If the sum of two numbers is 26 and their H. C. F and L. C. M are 1 and 120 respectively, the sum of the reciprocals of the two numbers is-

Solution:
Let the two numbers are x and y then
x + y = 26
and
xy = H. C. F × L. C. M = 1 × 120 = 120

Sum of their reciprocals = (1/x) + (1/y)
= (x + y)/xy
= 26/120
= 13/60

৫,৪৪৫.
The L.C.M. of two numbers is 48. The numbers are in the ratio 2 : 3. Then the sum of the numbers is:
  1. ক
    64
  2. খ
    32
  3. গ
    40
  4. ঘ
    28
ব্যাখ্যা
Question: The L.C.M. of two numbers is 48. The numbers are in the ratio 2 : 3. Then the sum of the numbers is: 

Solution: 
Let the numbers be 2x and 3x
Then, their L.C.M. = 6x
 
ATQ,
6x = 48
or x = 8
∴ The numbers are 16 and 24
Hence, required sum = (16 + 24) = 40
৫,৪৪৬.
A sum of money amounts to Tk. 15000 in 4 years at 25% simple interest per annum. Find the sum.
  1. ক
    Tk. 9000
  2. খ
    Tk. 6300
  3. গ
    Tk. 10000
  4. ঘ
    Tk. 7500
ব্যাখ্যা

Question: A sum of money amounts to Tk. 15000 in 4 years at 25% simple interest per annum. Find the sum. 

Solution:
Given, 
A = 15000 
T = 4
R = 25%

We know,
SI = P × R × T
= P × (25/100) × 4
= P

A = P + SI 
⇒ A = P + P
⇒ A = 2P
⇒ 15000 = 2P
⇒ P = 15000/2
∴ P = 7500

৫,৪৪৭.
A bank offers 5% compound interest calculated on half-yearly basis. A customer deposits Tk. 1600 each on 1st January and 1st July of a year. At the end of the year, the amount he would have gained by way of interest is:
  1. ক
    ক) Tk. 119
  2. খ
    খ) Tk. 121
  3. গ
    গ) Tk. 120
  4. ঘ
    ঘ) Tk. 122
ব্যাখ্যা
প্রশ্ন: A bank offers 5% compound interest calculated on half-yearly basis. A customer deposits Tk. 1600 each on 1st January and 1st July of a year. At the end of the year, the amount he would have gained by way of interest is:

সমাধান:
১ম ১৬০০ টাকার ক্ষেত্রে চক্রবৃদ্ধি মূলধন = 1600 × {1 + 5/(2 × 100)}2  
= 1600 × (41/40)  × (41/40) 
= 1681

২য় ১৬০০ টাকার ক্ষেত্রে চক্রবৃদ্ধি মূলধন = 1600 × {1 + 5/(2 × 100)}  
=  1600 × (41/40) 
= 1640

মোট চক্রবৃদ্ধি মূলধন = 1681 + 1640 
= 3321

মোট মুনাফা = 3321 - (1600 + 1600)
= 3321 - 3200
= 121
৫,৪৪৮.
Find the value of a,
√(2a) = 64
  1. ক
    - 2
  2. খ
    8
  3. গ
    12
  4. ঘ
    16
ব্যাখ্যা
Question: Find the value of a,
√(2a) = 64

Solution:
Given,
√(2a) = 64
⇒ (2a)1/2 = 26
⇒ 2(a/2) = 26
⇒ a/2 = 6
⇒ a = 6 × 2
⇒ a = 12
৫,৪৪৯.
A petrol tank is half full. If 10 gallons of petrol are removed, the tank becomes one-tenth full. What is the total capacity of the tank in gallons?
  1. ক
    25 gallons
  2. খ
    20 gallons
  3. গ
    30 gallons
  4. ঘ
    40 gallons
ব্যাখ্যা

Question: A petrol tank is half full. If 10 gallons of petrol are removed, the tank becomes one-tenth full. What is the total capacity of the tank in gallons?

Solution:
Let,
The capacity of the tank in gallons is x gallons.

According to question,
⇒ (x/2) - 10 = x/10
⇒ (x - 20)/2 = x/10
⇒ 10(x - 20) = 2x
⇒ 10x - 200 = 2x
⇒ 10x - 2x = 200
⇒  8x = 200
∴ x = 200/8 = 25 gallons

৫,৪৫০.
One year ago, Muna was four times as old as her daughter. Six years hence, Muna's age will exceed her daughter's age by 9 years. The ratio of the present ages of Muna and her gaughter is-
  1. ক
    10 : 3
  2. খ
    11 : 4
  3. গ
    12 : 5
  4. ঘ
    13 : 4
ব্যাখ্যা
Solution: One year ago, Muna was four times as old as her daughter. Six years hence, Muna's age will exceed her daughter's age by 9 years. The ratio of the present ages of Muna and her gaughter is-

Solution:
Let,
daughter's age one years ago be = x years
Then, Muna's age one years ago be = 4x years

ATQ,
(4x + 1) + 6 = (x + 1 + 6) + 9
⇒ (4x + 7) = (x + 16)
⇒ 4x - x = 16 - 7
⇒ 3x = 9
∴ x = 3

Ratio of Muna and her daughter age now = (4x + 1)/(x + 1)
= (4 · 3 + 1)/(3 + 1) = 13/4
= 13 : 4
৫,৪৫১.
If dividing Q(x) = 4x3 - 3x2 + bx - 5 by (x + 1) results the remainder 8 then find the value of b.
  1. ক
    - 20
  2. খ
    18
  3. গ
    - 16
  4. ঘ
    10
ব্যাখ্যা
Question: If dividing Q(x) = 4x3 - 3x2 + bx - 5 by (x + 1) results the remainder 8 then find the value of b.

Solution:
Dividing Q(x) by x + 1 we will get the remainder
∴  Q(- 1) = 4(- 1)3 - 3(- 1)2 + b(- 1) - 5
= - 4 - 3 - b - 5
= - 12 - b

ATQ,
- 12 - b = 8
⇒ b = - 12 - 8
∴ b = - 20
৫,৪৫২.
Think of a number and then double the number. Add 6 and then multiply the number by 10. Now divide the number by 20, then subtract the number you first thought of. What is the result?
  1. ক
    ক) 5
  2. খ
    খ) 4
  3. গ
    গ) 3
  4. ঘ
    ঘ) 2
ব্যাখ্যা
ধরি,
সংখ্যাটি 10x

প্রশ্নমতে,
10x × 2 = 20x
আবার,
পরবর্তী সংখ্যা = 20x + 6 

এখন 
{10(20x + 6)/20} - 10x
= {(20x + 6)/2} - 10x
= (20x + 6 - 20x)/2
= 6/2
= 3
৫,৪৫৩.
  1. ক
    1/64
  2. খ
    64
  3. গ
    256
  4. ঘ
    4
ব্যাখ্যা
Question:


Solution:
৫,৪৫৪.
If A = {1, 3, 5, 7, 9} and B = {2, 4, 6, 8}, what is A ∩ B?
  1. ক
    {1, 3, 5, 7, 9}
  2. খ
    {2, 4, 6, 8}
  3. গ
    {1, 2, 3, 4, 5, 6, 7, 8, 9}
  4. ঘ
    { }
ব্যাখ্যা
Question: If A = {1, 3, 5, 7, 9} and B = {2, 4, 6, 8}, what is A ∩ B?

Solution:
Given,
A = {1, 3, 5, 7, 9}
and B = {2, 4, 6, 8}

A ∩ B = {1, 3, 5, 7, 9} ∩ {2, 4, 6, 8}
= { }
৫,৪৫৫.
The average monthly income of P and Q is Tk. 4000 that of Q and R is Tk. 3250 and that P and R is Tk. 3500. what is P's monthly income? 
  1. ক
    ক) Tk. 3550
  2. খ
    খ) Tk.3250
  3. গ
    গ) Tk. 4550
  4. ঘ
    ঘ) Tk. 4250
ব্যাখ্যা
Average monthly income of P and Q = Tk. 4000
Average monthly income of Q and R = Tk. 3250
Average monthly income of P and R = Tk. 3500
Total income of P + Q
= 2 × 4000
= Tk. 8000.....(i)

Total income of Q + R
= 2 × 3250
= Tk. 6500 .....(ii)
Total income of R + P
= 2 × 3500
= Tk. 7000.....(iii)
On adding equation (i) (ii) and (iii), we have
2 (P + Q + R) = 8000 + 6500 + 7000
⇒ P + Q + R = 21500/2
⇒ P + Q + R = Tk. 10750.....(iv)

By equation (iv) - (ii)
P's monthly income
= Tk.(10750 - 6500)
= Tk. 4250
৫,৪৫৬.
A bag contains 6 black and 8 white balls. One ball is drawn at random. What is the probability that the ball drawn is white?
  1. ক
    3/4
  2. খ
    4/7
  3. গ
    1/8
  4. ঘ
    3/7
  5. ঙ
    None of these
ব্যাখ্যা
Question: A bag contains 6 black and 8 white balls. One ball is drawn at random. What is the probability that the ball drawn is white?

Solution:
Let number of balls = (6 + 8) = 14.
Number of white balls = 8.

∴ P (drawing a white ball) = 8/14 = 4/7
৫,৪৫৭.
The average weight of A, B and C is 45 kg. If the average weight of A and B be 40 kg and that of B and C be 43 kg, then the weight of B is-
  1. ক
    17 kg
  2. খ
    20 kg
  3. গ
    31 kg
  4. ঘ
    37 kg
ব্যাখ্যা
Question: The average weight of A, B and C is 45 kg. If the average weight of A and B be 40 kg and that of B and C be 43 kg, then the weight of B is-

Solution:
Let A, B, C represent their respective weights.

Then, we have:
A + B + C =(45 × 3) = 135..............(i)
A + B = (40 × 2) = 80.................(ii)
B + C=(43 × 2) = 86.................(iii)

Adding (ii) and (iii),
we get: A + 2B + C =80 + 86
A + 2B + C =166 .....(iv)

Subtracting (i) from (iv),
we get:
A + 2B + C - (A + B + C) = 166 - 135 
B = 31

∴ B's weight =31 kg.
৫,৪৫৮.
In what ratio must a grocer mix two varieties of tea worth Tk. 60 a kg and Tk. 65 a kg so that by selling the mixture at Tk. 68.20 a kg he may gain 10%?
  1. ক
    3 : 2
  2. খ
    3 : 4
  3. গ
    3 : 5
  4. ঘ
    4 : 5
ব্যাখ্যা
Question: In what ratio must a grocer mix two varieties of tea worth Tk. 60 a kg and Tk. 65 a kg so that by selling the mixture at Tk. 68.20 a kg he may gain 10%?

Solution:
Quantity of Tk. 60 tea is x kg.
Quantity of Tk. 65 tea is y kg

S.P. of 1 kg of the mixture = Tk. 68.20,
Gain = 10%.
C.P of 1 kg of the mixture = Tk. (100/110 × 68.20) = Tk. 62

ATQ,
60x + 65y = (x + y)62
⇒ 60x + 65y = 62x + 62y
⇒ 62x - 60x = 65y - 62y
⇒ 2x = 3y
∴ x/y = 3/2
৫,৪৫৯.
The average temperature for the first 4 days of a week is 40.2°C and that of the last 4 days is 41.3°C. If the average temperature for the whole week is 40.6°C, then the temperature on the fourth day is-
  1. ক
    38.5°C
  2. খ
    40.8°C
  3. গ
    41.8°C
  4. ঘ
    41.3°C
ব্যাখ্যা
Question: The average temperature for the first 4 days of a week is 40.2°C and that of the last 4 days is 41.3°C. If the average temperature for the whole week is 40.6°C, then the temperature on the fourth day is-
 
Solution:
Temperature on the fourth day
= [(40.2 × 4 + 41.3 × 4) - (40.6 × 7)]° C
= 41.8° C
৫,৪৬০.
Average mark in a class test of 40 students is 40. Average mark of all the 25 boys is 46. Then the average mark obtained by the girls is
  1. ক
    30
  2. খ
    32
  3. গ
    35
  4. ঘ
    36
ব্যাখ্যা
Question: Average mark in a class test of 40 students is 40. Average mark of all the 25 boys is 46. Then the average mark obtained by the girls is

Solution: 
Given,
Average mark of 40 students = 40
Total mark of 40 students = (40 × 40)
= 1600 

Average mark of all the 25 boys = 46
Total mark of all the 25 boys = (46 × 25)
= 1150

∴ Total marks of all the 15 girls is (1600 - 1150) = 450

So the average mark of 15 girls = 450/15 = 30
৫,৪৬১.
Robin is 28th from left end of a row of 50 students and Siam is 27th from right end of the row. How many students are sitting between them?
  1. ক
    5
  2. খ
    4
  3. গ
    3
  4. ঘ
    2
  5. ঙ
    1
ব্যাখ্যা

Question: Robin is 28th from left end of a row of 50 students and Siam is 27th from right end of the row. How many students are sitting between them?

Solution:



Siam's position from left = 50 - 27 + 1 = 24

So,
Robin's position from left = 28
Siam's position from left = 24

∴ Number of students are sitting between them = (28 - 24) - 1 
= 4 - 1
= 3

৫,৪৬২.
An amount is invested in a saving account for two years. It increases by Tk. 630 in two years after annual compounding at the rate of 10% per year. What is the amount, in Tk, invested initially? 
  1. ক
    ক) Tk. 2000
  2. খ
    খ) Tk. 2500
  3. গ
    গ) Tk. 3000
  4. ঘ
    ঘ) None of these
ব্যাখ্যা
Question: An amount is invested in a saving account for two years. It increases by Tk. 630 in two years after annual compounding at the rate of 10% per year. What is the amount, in Tk, invested initially? 

Solution:
Let the principal be x
We know that,
C = P{1 + (r/100)}n

ATQ,
x + 630 = x{1 + (10/100)}2
⇒ x + 630 = x{11/10}2
⇒ x + 630 = 121x/100
⇒ 100x + 63000 = 121x
⇒ 21x = 63000
∴ x = 3000

∴ The amount invested initially is Tk. 3000
৫,৪৬৩.
If the square of the sum of two numbers is equal to 4 times of their product, Then the ratio of these numbers is: 
  1. ক
    ক) 2 : 1
  2. খ
    খ) 1 : 3
  3. গ
    গ) 1 : 1
  4. ঘ
    ঘ) 1 : 2
ব্যাখ্যা
(x + y)2 = 4xy
=> (x + y)2 - 4xy = 0
=> (x - y)2 = 0
=> x = y
=> x/y = 1
∴ x : y = 1 : 1
৫,৪৬৪.
25, 36, 49, 81, 121, 169, 225
  1. ক
    ক) 36
  2. খ
    খ) 49
  3. গ
    গ) 121
  4. ঘ
    ঘ) 169
ব্যাখ্যা
The numbers are squares of odd natural numbers, starting from 5 up to 15. So, 36 is wrong.
৫,৪৬৫.
If x = 5 + √3 and y = 5 - √3, find the value of (x2 + y2)2.
  1. ক
    2136
  2. খ
    313
  3. গ
    3136
  4. ঘ
    3236
ব্যাখ্যা

Question: If x = 5 + √3 and y = 5 - √3, find the value of (x2 + y2)2

Solution:
We are given:
x = 5 + √3, y = 5 - √3

Now,
x + y
= (5 + √3) + (5 - √3)
= 10

And,
⇒ xy
= (5 + √3)(5 - √3)
= 52 - (√3)2
= 25 - 3
= 22

We know,
x2 + y2 = (x + y)2 - 2xy
⇒ x2 + y2 = (10)2 - 2(22) [Substitute the values]
⇒ x2 + y2 = 100 - 44
⇒ x2 + y2 = 56

∴ (x2 + y2)2 = 562 = 3136

৫,৪৬৬.
What is the average of the following numbers: 35.5, 32.5, 34.0, 35.5 and 34.5?
  1. ক
    ক) 33.0
  2. খ
    খ) 33.3
  3. গ
    গ) 34.0
  4. ঘ
    ঘ) 34.4
ব্যাখ্যা
Question: What is the average of the following numbers: 35.5, 32.5, 34.0, 35.5 and 34.5?

Solution:
Average = (35.5 + 32.5 + 34.0 +35.5 + 34.5) / 5
= 172/5
= 34.4
৫,৪৬৭.
The dimensions of an open box are 50 cm, 40 cm and 23 cm. Its thickness is 3 cm. If a cubic cm of metal used in the box weighs 0.5 gms, Find the weight of the box.
  1. ক
    ক) 8.04 kg
  2. খ
    খ) 5.06 kg
  3. গ
    গ) 4.03 kg
  4. ঘ
    ঘ) 9.44 kg
ব্যাখ্যা

External dimensions,
l = 50 cm,
b = 40 cm,
h = 23 cm

Internal dimension,
l' = 50 - (2 × 3) = 44 cm
b' = 40 - (2 × 3) = 34 cm
h' = 23 - 3 = 20 cm

The volume of the metal used in the box = External Volume - Internal Volume
= [( 50 × 40 × 23) - (44 × 34 × 20)] cm3
= 16080 cm3.

∴ Weight of the metal =
(16080 × 0.5)/1000 kg
= 8.04 kg

৫,৪৬৮.
When a plot is sold for Tk. 18,700, the owner loses 15%. At what price must that plot be sold in order to gain 15%?
  1. ক
    Tk. 21,000
  2. খ
    Tk. 22,500
  3. গ
    Tk. 25,300
  4. ঘ
    Tk. 25,800
ব্যাখ্যা
Question: When a plot is sold for Tk. 18,700, the owner loses 15%. At what price must that plot be sold in order to gain 15%?

Solution:
15% ক্ষতিতে
ক্রয়মূল্য 100 টাকা হলে বিক্রয়মূল্য (100 - 15) টাকা বা 85 টাকা

বিক্রয়মূল্য 85 টাকা হলে ক্রয়মূল্য 100 টাকা
বিক্রয়মূল্য 1 টাকা হলে ক্রয়মূল্য 100/85 টাকা
বিক্রয়মূল্য 18700 টাকা হলে ক্রয়মূল্য (100 × 18700)/85 টাকা
= 22000 টাকা

15% লাভে
ক্রয়মূল্য 100 টাকা হলে বিক্রয়মূল্য (100 + 15) টাকা বা 115 টাকা

ক্রয়মূল্য 100 টাকা হলে বিক্রয়মূল্য 115 টাকা
ক্রয়মূল্য 100 টাকা হলে বিক্রয়মূল্য 115/100 টাকা
ক্রয়মূল্য 22000 টাকা হলে বিক্রয়মূল্য (115 × 22000)/100 টাকা
= 25300 টাকা
৫,৪৬৯.
Which of the following is not a leap year?
  1. ক
    2000
  2. খ
    800
  3. গ
    700
  4. ঘ
    1200
ব্যাখ্যা
Question: Which of the following is not a leap year?

Solution: 
The century divisible by 400 is a leap year. [2000, 800 and 1200 is divisible by 400]
∴ The year 700 is not a leap year. [700 is not divisible by 400]

[Rules for leap years:
• Must be divisible by 4,
• If it's a century year (divisible by 100), it must also be divisible by 400.]
৫,৪৭০.
The mean of x, x + 4, x + 8, x + 12 is 20. Find x.
  1. ক
    12
  2. খ
    14
  3. গ
    16
  4. ঘ
    18
ব্যাখ্যা

Question: The mean of x, x + 4, x + 8, x + 12 is 20. Find x.

Solution:
Given,
Total numbers = 4
Mean = 20

Sum of numbers: 
x+ (x + 4) + (x + 8) + (x + 12) 
= 4x + 24

ATQ,
(4x + 24)/4 = 20
⇒ 4x + 24 = 80
⇒ 4x = 56
∴ x = 14

৫,৪৭১.
How many prime number are in between 45 to 72?
  1. ক
    ক) 5
  2. খ
    খ) 6
  3. গ
    গ) 7
  4. ঘ
    ঘ) 8
ব্যাখ্যা
45 থেকে 72 পর্যন্ত মৌলিক সংখ্যা 47, 53, 59, 61, 67, 71

45 থেকে 72 পর্যন্ত মৌলিক সংখ্যা = ৬টি
৫,৪৭২.
The monthly incomes of two workers are in the ratio 3 : 4 and their monthly expenditures are in the ratio 5 : 7. If each saves TK. 100 per month, find their monthly incomes.
  1. ক
    400 and 600
  2. খ
    650 and 950
  3. গ
    600 and 800
  4. ঘ
    300 and 600
ব্যাখ্যা
Question: The monthly incomes of two workers are in the ratio 3 : 4 and their monthly expenditures are in the ratio 5 : 7. If each saves TK. 100 per month, find their monthly incomes.

Solution:
Let,
Incomes be 3x and 4x
Expenditures be 5y and 7y

Savings Equations,
3x - 5y = 100 .......(1)
4x - 7y = 100 ........(2)
Now,
(1) × 4 ⇒ 12x - 20y = 400
(2) × 3 ⇒ 12x - 21y = 300

Now subtract ⇒ (12x - 20y) - (12x - 21y) = 400 - 300
⇒ 12x - 20y - 12x + 21y = 100
⇒ y = 100

From (1),
3x - 5(100) = 100
⇒ 3x = 600
∴ x = 200

∴ The monthly incomes are,
First worker = 3x = (3 × 200) = 600
Second worker = 4x = (4 × 200) = 800
৫,৪৭৩.
Find the least number that leaves a remainder of 5 when divided by 6, 9, 15, and 20.
  1. ক
    170
  2. খ
    159
  3. গ
    185
  4. ঘ
    190
ব্যাখ্যা

Question: Find the least number that leaves a remainder of 5 when divided by 6, 9, 15, and 20.

Solution:
We have to find the least number,
therefore, we find out the LCM of 6, 9, 15, and 20.

6 = 3 × 2
9 = 3 × 3
15 = 3 × 5
20 = 2 × 2 × 5

∴ LCM = 2 × 2 × 3 × 3 × 5
= 180

This is the least number which is exactly divisible by 6, 9, 15, and 20.
So, required number leaves remainder of 5 is = 180 + 5 = 185

৫,৪৭৪.
Find the compound interest at the rate of 10% per annum for four years on the principal which in four years at the rate of 4% per annum gives Tk. 1600 as simple interest.
  1. ক
    ক) Tk. 4641
  2. খ
    খ) Tk. 4732
  3. গ
    গ) Tk. 4321
  4. ঘ
    ঘ) Tk. 4899
ব্যাখ্যা

∴ P = I/nr = 1600/(4/100 x 4) = 10000
So, I = P(1 + r)n - P
= 10000(1 + 10/100)4 - 10000
= 14641 - 10000
= 4641

৫,৪৭৫.
In an exam, there are 3 multiple choice questions, and each question has 4 choices. Only one answer per question is correct. How many ways can a student fail to get all answers correct?
  1. ক
    63
  2. খ
    64
  3. গ
    65
  4. ঘ
    66
  5. ঙ
    None of the above
ব্যাখ্যা

Question: In an exam, there are 3 multiple choice questions, and each question has 4 choices. Only one answer per question is correct. How many ways can a student fail to get all answers correct?

Solution:
Each question has 4 options, so the total number of ways to answer all 3 questions is = 43
= 4 × 4 × 4
= 64

Number of ways, getting correct answers = 13 = 1

∴ Number of ways of not getting all answers correct = 64 - 1 = 63

৫,৪৭৬.
Eight points are situated on a plane, 4 of them in a straight line, the other 4 elsewhere. How many triangles can be formed by joining 3 points at a time?
  1. ক
    42​ ways
  2. খ
    38​ ways
  3. গ
    56​ ways
  4. ঘ
    72​ ways
  5. ঙ
    52​ ways
ব্যাখ্যা

Question: Eight points are situated on a plane, 4 of them in a straight line, the other 4 elsewhere. How many triangles can be formed by joining 3 points at a time?

Solution: 
Total combinations of 3 points from 8 = 8C3
= 56 ways

Given,
there are 4 collinear points

From the 4 collinear points, no triangle can be formed using any 3 of them (since they lie on the same line).
Total combinations of 3 points from the 4 collinear points = 4C3 = 4 ways

So the valid triangles = 56 − 4 = 52​ ways

৫,৪৭৭.
Which of the following is equivalent to (2x + 4)/(2x2 + 8x + 8) for all values of x for which both expressions are defined?
  1. ক
    1/(2x2 + 6)
  2. খ
    2/(x + 6)
  3. গ
    1/(x + 4)
  4. ঘ
    1/(x + 2)
ব্যাখ্যা
Question: Which of the following is equivalent to (2x + 4)/(2x2 + 8x + 8) for all values of x for which both expressions are defined?

Solution:
(2x + 4)/(2x2 + 8x + 8)
= {2(x + 2)}/{2(x2 + 4x + 4)}
= (x + 2)/(x + 2)2
= 1/(x + 2)
৫,৪৭৮.
If n = (33)43 + (43)33, what is the units digit of n?
  1. ক
    ক) 0
  2. খ
    খ) 2
  3. গ
    গ) 4
  4. ঘ
    ঘ) 6
ব্যাখ্যা

First of all, the units digit of (33)43 is the same as that of 343 and the units digit of (43)33 is the same as that of 333. So, we need to find the units digit of 343 + 333.

Next, the units digit of 3 in positive integer power repeats in blocks of four {3, 9, 7, 1}:
31 = 3 (the units digit is 3)
32 = 9 (the units digit is 9)
33 = 27 (the units digit is 7)
34 = 81 (the units digit is 1)
35 = 243 (the units digit is 3 again!)
...

Thus:
The units digit of 343 is the same as the units digit of 33, so 7 (43 divided by the cyclicity of 4 gives the remainder of 3).
The units digit of 333 is the same as the units digit of 31, so 3 (33 divided by the cyclicity of 4 gives the remainder of 1).

Therefore, the units digit of (33)43 + (43)33 is 7 + 3 = 0.

৫,৪৭৯.
The price of a model M camera is $209 and the price of a special lens is $69. When the camera and lens are purchased together, the price is $239. The amount saved by purchasing the camera and lens together is approximately what percent of the total price of the camera and lens when purchased separately?
  1. ক
    14%
  2. খ
    16%
  3. গ
    33%
  4. ঘ
    86%
ব্যাখ্যা
Question: The price of a model M camera is $209 and the price of a special lens is $69. When the camera and lens are purchased together, the price is $239. The amount saved by purchasing the camera and lens together is approximately what percent of the total price of the camera and lens when purchased separately?

Solution:
Price of the camera and lens when purchased separately = 209 + 69 = 278
Total price when purchased together = 239

Discount in $ = 278 - 239 = 39

Discount % = (39/278) × 100% = 14.02% ≅ 14%

Hence the correct answer is A
৫,৪৮০.
A sum of money amounts to Tk. 5200 in 5 years and to Tk. 5680 in 7 years at simple interest. The rate of interest per annum is?
  1. ক
    4%
  2. খ
    5%
  3. গ
    6%
  4. ঘ
    8%
ব্যাখ্যা
Question: A sum of money amounts to Tk. 5200 in 5 years and to Tk. 5680 in 7 years at simple interest. The rate of interest per annum is?

Solution:
Simple interest for 2 years = (5680 - 5200) Tk.
= Tk. 480
∴ Simple Interest for 2 years =  Tk. 480
∴ Simple Interest for 5 years =  Tk. (480 × 5/2)
= Tk. 1200
∴ Principal = 5200 - 1200 = Tk. 4000.

We know, 
I = Pnr
⇒ r = I/Pn
⇒ r = (1200 × 100)/(4000 × 5)
∴ r = 6%
৫,৪৮১.
A trader sells his goods at a discount of 20%. He still makes a profit of 25%. If he sells the goods at the marked price only, his profit will be:
  1. ক
    ক) 56. 25%
  2. খ
    খ) 25.56 %
  3. গ
    গ) 50.25%
  4. ঘ
    ঘ) 54.25%v
ব্যাখ্যা

ধরি,
দ্রব্যটির ক্রয়মূল্য 100 টাকা
25% লাভে দ্রব্যটির বিক্রয়মূল্য = 100 + 25 = 125 টাকা
অর্থাৎ দ্রব্যটি দামের 80% হলো = 125 টাকা
∴ দ্রব্যটি দামের 100% হলো = (125 × 100)/80
= 156.25 টাকা
∴ শতকরা লাভ হতো = 156.25 - 100 = 56.25%

৫,৪৮২.
If x and y are positive integers such that 3x + y = 94 and 2x - y = 16, what is the value of x2 - y2?
  1. ক
    12
  2. খ
    16
  3. গ
    24
  4. ঘ
    32
ব্যাখ্যা

Question: If x and y are positive integers such that 3x + y = 94 and 2x - y = 16, what is the value of x2 - y2?

Solution: 
Given that, 
3x + y = 94
⇒ 3x + y = (32)4
⇒ 3x + y = 38
∴ x + y = 8 ........(1)

And,
2x - y = 16
⇒ 2x - y = 24
∴ x - y = 4 ....... (2)

Now (1) + (2) than we get,
⇒ (x + y) + (x - y) = 8 + 4
⇒ 2x = 12
⇒ x = 12/2
∴ x = 6

From (1),
6 + y = 8
⇒ y = 8 - 6
∴ y = 2

∴ x2 - y2 = (6)2 - (2)2
= 36 - 4
= 32

৫,৪৮৩.
Find the number which replaces the question mark.
  1. ক
    ক) 6
  2. খ
    খ) 4
  3. গ
    গ) 8
  4. ঘ
    ঘ) 10
ব্যাখ্যা
Question: Find the number which replaces the question mark.

Solution:

∴ Correct answer 6.
৫,৪৮৪.
A shopkeeper buys a mobile phone for Tk. 8,000 and sells it to a retailer at a profit of 15%. The retailer then sells it to a customer at a profit of 10%. How much does the customer pay for the mobile phone?
  1. ক
    Tk. 10,120
  2. খ
    Tk. 9,900
  3. গ
    Tk. 10,000
  4. ঘ
    Tk. 10,240
  5. ঙ
    Tk. 9,840
ব্যাখ্যা

Question: A shopkeeper buys a mobile phone for Tk. 8,000 and sells it to a retailer at a profit of 15%. The retailer then sells it to a customer at a profit of 10%. How much does the customer pay for the mobile phone?

Solution:
দোকানদারের 15% লাভে বিক্রয়মূল্য = 8000 + 8000 এর 15%
= 8000 + (8000 × 15 / 100)
= 8000 + 1200
= Tk. 9200

দোকানদারের বিক্রয়মূল্য = খুচরা বিক্রেতার ক্রয়মূল্য

খুচরা বিক্রেতার 10% লাভে বিক্রয়মূল্য = 9200 + 9200 এর 10%
= 9200 + (9200 × 10 / 100)
= 9200 + 920
= Tk. 10120

সুতরাং, খুচরা বিক্রেতার বিক্রয়মূল্য = ক্রেতার ক্রয়মূল্য = Tk. 10,120

৫,৪৮৫.
A person pays Tk. 8000 as an amount on the sum of Tk. 6000 that he had borrowed for 3 years. What will be the rate of interest?
  1. ক
    11.11%
  2. খ
    6.5%
  3. গ
    10%
  4. ঘ
    6%
ব্যাখ্যা

Question: A person pays Tk. 8000 as an amount on the sum of Tk. 6000 that he had borrowed for 3 years. What will be the rate of interest?

Solution:
Amount, A = Tk. 8000
Principal, P= Tk. 6000
Time, T = 3 years
Interest Rate, R =?

Amount = Principal  + Simple Interest
SI = A – P
= 8000 – 6000
= Tk. 2000

SI = (P × R ×T)/100
⇒ R = (SI × 100)/(P × T)
= (2000 × 100)/(6000 × 3) 
= 11.11 %
∴ The rate of interest is 11.11 %.

৫,৪৮৬.
A boatman goes 4 km against the current of the stream in 1 hour and goes 1 km along the current in 10 minutes. How long will it take to go 7 km in stationary water?
  1. ক
    1 h 12 min
  2. খ
    1 h 24 min
  3. গ
    2 h 12 min
  4. ঘ
    2 h 24 min
  5. ঙ
    None of this
ব্যাখ্যা

Question: A boatman goes 4 km against the current of the stream in 1 hour and goes 1 km along the current in 10 minutes. How long will it take to go 7 km in stationary water?

Solution: 
Speed along current = 60/10 km/h
= 6 km/h
Speed against current  = 4 km/h

∴ Speed in still water = (Speed against current + Speed along current)/2
= (4 + 6)/2
= 10/2
= 5

∴ Required time = 7/5 h
= [(7/5) × 60] min
= 84 min
= 60 min + 24 min 
= 1 h 24 min 

৫,৪৮৭.
At the beginning of a class period, half of the students in a class go to the library. Later in the period, half of the remaining students go to the computer lab. If there are 8 students remaining in the class, how many students were originally in the class?
  1. ক
    ক) 12
  2. খ
    খ) 16
  3. গ
    গ) 24
  4. ঘ
    ঘ) 32
ব্যাখ্যা

Let the initial number of students = x
Goes to library = x/2
So, remaining students are = x - x/2 = x/2
Then, goes to computer lab = (x/2)/2 = x/4
ATQ, x/2 - x/4 = x/4 = 8
∴ x = 32
The initial number of students were 32

৫,৪৮৮.
Of the three numbers, the first is twice the second and the second is twice the third. The average of the reciprocal of the numbers is (7/72). The numbers are:
  1. ক
    16, 8, 32
  2. খ
    26, 8, 12
  3. গ
    16, 8, 4
  4. ঘ
    16, 8, 40
  5. ঙ
    24, 12, 6
ব্যাখ্যা

Question: Of the three numbers, the first is twice the second and the second is twice the third. The average of the reciprocal of the numbers is (7/72). The numbers are:

Solution:
Let the number be x.
Then, second number = 2x.
First number = 4x.

∴ (1/x) + (1/2x) + (1/4x) = (7/72) × 3
⇒ (4 + 2 + 1)/4x = (7/24)
⇒ 7/4x = 7/24
⇒ 4x = 24
⇒ x = 24/4
∴ x = 6

Therefore, the numbers are (4 × 6) or 24,  (2 × 6) or 12 and 6.

৫,৪৮৯.
The sum of squares of 3 consecutive integers is less than 97. What is the greatest possible value of the smallest one?
  1. ক
    3
  2. খ
    4
  3. গ
    5
  4. ঘ
    7
  5. ঙ
    6
ব্যাখ্যা
Let's assume two sets of integers are {4, 5, 6} and {5, 6, 7}
42 + 52 + 62 = 16 + 25 + 36 = 77
52 + 62 + 72 = 25 + 36 + 49 = 110 
As there sum should be less than 97, so the least number of these 3 consecutive integers is 4.
৫,৪৯০.
Working 5 hours a day, Samiya can complete a work in 8 days and working 6 hours a day, Fahima can complete the same work in 10 days. Working 6 hours a day, they can jointly complete the work in:
  1. ক
    8 days
  2. খ
    6 days
  3. গ
    3 days
  4. ঘ
    4 days
  5. ঙ
    None
ব্যাখ্যা

Question: Working 5 hours a day, Samiya can complete a work in 8 days and working 6 hours a day, Fahima can complete the same work in 10 days. Working 6 hours a day, they can jointly complete the work in:

Solution: 
Working 5 hours a day, Samiya can complete a work in 8 days = 8 × 5 = 40 hours
Working 6 hours a day, Fahima can complete a work in 10 days = 6 × 10 = 60 hours

(Samiya and Fahima)'s 1 hour's work = (1/40) + (1/60)
= (3 + 2)/120
= 5/120
= 1/24

They can jointly complete the work in 24 hours
Working 6 hours a day, they can jointly complete the work in = 24/6 = 4 days

৫,৪৯১.
Each side of a rectangular field diminished by 40%. By how much percent is the area of the field diminished?
  1. ক
    53.25%
  2. খ
    60% 
  3. গ
    64% 
  4. ঘ
    67.9% 
ব্যাখ্যা
Question: Each side of a rectangular field diminished by 40%. By how much percent is the area of the field diminished?

Solution: 
Let, length is 80 m and breadth is 40 m 
Area = 80 × 40 = 3200 m2 

diminished by 40%, 
New length = 80 - 80 × .4 
= 48 m 
New breadth = 40 - 40 × .4
= 24 m 
New area = 48 × 24
= 1152 m2 

percent is the area of the field diminished = {(3200 - 1152)/3200} × 100%
= 64% 
৫,৪৯২.
A’s current age is three times B’s current age. Eight years ago, A’s age was five times B’s age at that time. Find the difference between their present ages.
  1. ক
    28 years
  2. খ
    36 years
  3. গ
    32 years
  4. ঘ
    24 years
ব্যাখ্যা
Question: A’s current age is three times B’s current age. Eight years ago, A’s age was five times B’s age at that time. Find the difference between their present ages.

Solution:
Let,
B's current age = x 
A's current age = 3x

ATQ,
3x - 8 = 5(x - 8)
⇒3x - 8 = 5x - 40
⇒ 5x - 3x= 40 - 8
⇒ 2x = 32
∴ x = 16

∴ B's current age = 16 years
∴ A's current age = (3 × 16) = 48 years

So the difference between their present ages = (48 - 16) = 32 years
৫,৪৯৩.
A lamp post 18 meters tall broke in such a way that the broken part makes a 30-degree angle with the ground. At what height did the lamp post break?
  1. ক
    5 meters
  2. খ
    6 meters 
  3. গ
    4 meters 
  4. ঘ
    8 meters 
ব্যাখ্যা

Question: A lamp post 18 meters tall broke in such a way that the broken part makes a 30-degree angle with the ground. At what height did the lamp post break?

Solution:


Let,
height from ground (A) to broken part (C) = h
rest = 18 - h
The broken part makes a 30-degree angle with the ground at B.
It creates a triangle ABC where,
BC = 18 - h
AC = h

Now,
sin 30° = AC/BC
⇒ 1/2 = h/(18 - h)
⇒ 2h = 18 - h
⇒ 3h = 18
∴ h = 6

∴ the lamp post broke at 6 m height from the ground.

৫,৪৯৪.
The average age of a family of 5 members is 20 years. If the age of the youngest member be 10 years then what was the average age of the family at the time of the birth of the youngest member?
  1. ক
    13.5 
  2. খ
    14
  3. গ
    15
  4. ঘ
    None of these
ব্যাখ্যা

Question: The average age of a family of 5 members is 20 years. If the age of the youngest member be 10 years then what was the average age of the family at the time of the birth of the youngest member?

Solution: 
At present the total age of the family = 5 × 20 =100
The total age of the family at the time of the birth of the youngest member,
= 100 - 10 - (10 × 4)
= 50

Therefore, average age of the family at the time of birth of the youngest member,
= 50/5
= 10

৫,৪৯৫.
Jeff takes 20 minutes to jog around the race course one time, and 25 minutes to jog around a second time. What is his average speed in miles per hour for the whole jog if the course is 3 miles long?
  1. ক
    ক) 6
  2. খ
    খ) 8
  3. গ
    গ) 10
  4. ঘ
    ঘ) 12
ব্যাখ্যা
Question: Jeff takes 20 minutes to jog around the race course one time, and 25 minutes to jog around a second time. What is his average speed in miles per hour for the whole jog if the course is 3 miles long?

Solution: 
নির্ণেয় গড় গতিবেগ = (3 + 3)/{(20/60) + (25/60)}
                                = 6/{(20 + 25)/60}
                                = 6/(45/60)
                                = 6/(3/4)
                                = 6 × (4/3)
                                = 8
৫,৪৯৬.
Find the equation of the line with an x-intercept of - 6 and a y-intercept of 3.
  1. ক
    x + 2y - 6 = 0
  2. খ
    x - 2y + 6 = 0
  3. গ
    2x - y - 6 = 0
  4. ঘ
    2x + y - 6 = 0
ব্যাখ্যা

Question: Find the equation of the line with an x-intercept of - 6 and a y-intercept of 3.
​
​Solution:
​যখন একটি রেখা x-অক্ষকে 'a' বিন্দুতে এবং y-অক্ষকে 'b' বিন্দুতে ছেদ করে, 
​তখন এর সমীকরণ হয়: (x/a) + (y/b) = 1

এখানে, a = - 6 এবং b = 3
মানগুলো সমীকরণে বসিয়ে পাই,
x/(- 6) + y/3 = 1
⇒ (- x + 2y)/6 = 1
⇒ - x + 2y = 6
⇒ x - 2y + 6 = 0

৫,৪৯৭.
In an election between two candidates, the winner got 65% of the total votes cast and won the election by a majority of 2748 votes. What is the total number of votes cast if no vote is declared invalid?
  1. ক
    ক) 8580
  2. খ
    খ) 8720
  3. গ
    গ) 9000
  4. ঘ
    ঘ) 9160
ব্যাখ্যা

Winner gets 65% 0f valid votes and loser gets 35% of votes.
Difference between this two = 2748
(65-35)% = 2748
30% = 2748
Total number of voters, 100%
= (2748×100)/30
= 9160

৫,৪৯৮.
How long will it take for two pipes to fill a tank together when they can fill it alone in 14 hours and 21 hours respectively?
  1. ক
    8.4 hours
  2. খ
    8.6 hours
  3. গ
    8.8 hours
  4. ঘ
    8 hours
ব্যাখ্যা
Question: How long will it take for two pipes to fill a tank together when they can fill it alone in 14 hours and 21 hours respectively?

Solution:
together in one hour they can fill = 1/14 + 1/21 = 5/42

so, the total time to fill the tank is = 42/5 hours =  8.4 hours
৫,৪৯৯.
In a mixture of 45 liters, the ratio of milk and water is 4 : 1. How much water must be added to make the mixture ratio 3 : 2?
  1. ক
    15 liters
  2. খ
    20 liters
  3. গ
    25 liters
  4. ঘ
    10 liters
ব্যাখ্যা
Question: In a mixture of 45 liters, the ratio of milk and water is 4 : 1. How much water must be added to make the mixture ratio 3 : 2?

Solution:
Quantity of milk in 45 litres of mixture 45 × (4/5) litres = 36 litres
∴ Quantity of water in the mixture = 45 - 36 litres = 9 litres

Let x litres of water be added to the mixture.
Then,
36/(9 + x) = 3/2
⇒ 72 = 27 + 3x
⇒ 3x = 45
∴ x = 15
৫,৫০০.
A contract is to be completed in 46 days and 117 men were set to work, each working 8 hours a day. After 33 days (4/7) th of work is completed. How many additional men may be employed so that the work may be completed in time, each man now working 9 hours a day?
  1. ক
    51 men
  2. খ
    68 men
  3. গ
    81 men
  4. ঘ
    98 men
ব্যাখ্যা
Question: A contract is to be completed in 46 days and 117 men were set to work, each working 8 hours a day. After 33 days (4/7) th of work is completed. How many additional men may be employed so that the work may be completed in time, each man now working 9 hours a day?

Solution:
Remaining Work = (1 - 4/7) = 3/7
Remaining Time = (46 - 33) = 13 days

33 days working 8 hrs a day =  33 × 8 hrs = 264 hrs
13 days working 9 hrs a day = 13 × 9 hrs = 117 hrs

In 264 hrs to complete (4/7) the of work it requires 117 men
∴ In 117 hrs to complete (3/7) the of work it requires =

= 198 men

∴ Additional men required = 198 - 117 = 81 person.