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Bank Math

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

Bank Math

PrepBank · পাতা ১১৬ / ১৬১ · ১১,৫০১–১১,৬০০ / ১৬,১২৪

১১,৫০১.
In a box, there are 10 apples and 2/5th of the apples are rotten. If three apples are taken out from the box, what will be the probability that at least one apple is rotten.
  1. ক
    ক) 3/4
  2. খ
    খ) 5/6
  3. গ
    গ) 5/8
  4. ঘ
    ঘ) 8/13
ব্যাখ্যা

Let, rotten apples = 10 × 2/5 = 4, not rotten = 6

If 1 apple is rotten + 2 apples are not rotten 
= 4C1 × 6C2 = 60
If 2 apples are rotten + 1 apple is not rotten 
= 4C2 × 6C1 = 36
If 3 apples are rotten = 4C3 = 4

Total outcomes = 10C3 = 120
∴ Probability = (60 + 36 + 4) / 120 = 100/120 = 5/6

১১,৫০২.
Rana took a loan of Tk. 7500 with simple interest for as many years as the rate of interest. If he paid Tk. 1875 as interest at the end of the loan period, what was the rate of interest?
  1. ক
    4.5%
  2. খ
    5%
  3. গ
    5.5%
  4. ঘ
    7%
ব্যাখ্যা
Question: Rana took a loan of Tk. 7500 with simple interest for as many years as the rate of interest. If he paid Tk. 1875 as interest at the end of the loan period, what was the rate of interest?

Solution:
Let,
rate r = R%
and time n = R years

ATQ,
Prn = I
⇒ (7500 × R × R)/100 = 1875
⇒ 75R2 = 1875
⇒ R2 = 1875/75
⇒ R2 = 25
∴ R = 5
১১,৫০৩.
The contributions made by Rasel and Akash are in the ratio of 6 : 4. If 10% of the total profit is donated and Rasel gets 9000 as his share of profit, what is the total profit?
  1. ক
    15780.66
  2. খ
    16000.66
  3. গ
    16656.66
  4. ঘ
    16666.66
ব্যাখ্যা

Question: The contributions made by Rasel and Akash are in the ratio of 6 : 4. If 10% of the total profit is donated and Rasel gets 9000 as his share of profit, what is the total profit?

Solution:
Let,
the total profit after donated = p
The ratio of contribution by Rasel and Akash = 6 : 4
sum of ratios = 10

Rasel's share = (6/10) × p = 9000
⇒ p = (10 × 9000)/6
∴ p = 15000
so, total profit after donated = 15000

ATQ,
Rasel gets 9000 after 10% donated
Now,
90% = 15000
1% = 15000/90
And, 100% = (15000 × 100)/90 = 16,666.66

১১,৫০৪.
What are the two roots of the equation x2 - 7x + 12 = 0 ?
  1. ক
    ক) 1 & 7
  2. খ
    খ) 3 & 4
  3. গ
    গ) - 3 & 4
  4. ঘ
    ঘ) 2 & 3
ব্যাখ্যা
x2 - 7 x + 12 = 0 
⇒x2 - 7 x + 12 = 0
⇒ x2 - 3 x - 4x + 12 = 0
⇒ (x - 3)(x - 4) = 0
∴ x = 3, 4
১১,৫০৫.
A, B and C together can complete a piece of work in 12 days. All the three started working at it together and after 5 days A left. Then B and C together completed the work in 10 more days. A alone could complete the work in-
  1. ক
    36 days
  2. খ
    40 days
  3. গ
    42 days
  4. ঘ
    55 days
ব্যাখ্যা
Question: A, B and C together can complete a piece of work in 12 days. All the three started working at it together and after 5 days A left. Then B and C together completed the work in 10 more days. A alone could complete the work in-

Solution:
(A + B + C) do 1 work in 12 days.

So (A + B + C)'s 1 day work = 1/12 part
∴ (A + B + C)'s 5 day work = 5/12 part

Remaining work = 1 - (5/12) = 7/12 part

(B + C) takes 10 more days to complete 7/12 part.
∴ (B + C)'s 1 day work = 7/120 part

Now A's 1 day work = (A + B + C)'s 1 day work - (B + C)'s 1 day work
= 1/12 - 7/120
= (10 - 7)/120
= 3/120
= 1/40

∴ A does 1/40 work in in 1 day
∴ A alone could complete the work in 40 days
১১,৫০৬.
Find the missing number:
  1. ক
    ক) 20
  2. খ
    খ) 65
  3. গ
    গ) 91
  4. ঘ
    ঘ) 169
ব্যাখ্যা
Question: Find the missing number:


Solution: 
১ম চিত্রে,
√9 + √25
= 3 + 5
= 8

২য় চিত্রে, 
√16 + √49
= 4 + 7
= 11

৩য় চিত্রে, 
√121 + √81
= 11 + 9
= 20
১১,৫০৭.
A rectangular grassy plot 110 m by 65 m has a gravel path 2.5 m wide all round it on the inside. Find the cost of gravelling the path at 80 paisa per sq. metre.
  1. ক
    ক) Tk. 570
  2. খ
    খ) Tk. 620
  3. গ
    গ) Tk. 680
  4. ঘ
    ঘ) Tk. 750
ব্যাখ্যা

Area of the plot = (110 × 65)m2
= 7150m2
Area of the plot excluding the path = [(110 - 5) × (65 - 5)]m2
= 6300 m2

∴ Area of the path = (7150 - 6300) m2
= 850 m2

Cost of graveling the path = Tk. {850 × (80/100)}
= Tk. 680

১১,৫০৮.
What is the probability of rolling an even number on a standard six-sided die?
  1. ক
    1/6
  2. খ
    1/3
  3. গ
    1/2
  4. ঘ
    2/3
ব্যাখ্যা
Question: What is the probability of rolling an even number on a standard six-sided die?

Solution:
A standard six-sided die has the numbers,
1, 2, 3, 4, 5, 6
Even numbers between 1 and 6 are = 2, 4, 6
So, favorable outcomes = 3
And total possible outcomes = 6

∴ Probability = Favorable outcomes/Total outcomes ​
= 3/6 ​
= 1/​2
১১,৫০৯.
If n is an integer between 10 and 70, then any of the following could be n + 7 except-
  1. ক
    78
  2. খ
    70
  3. গ
    57
  4. ঘ
    46
ব্যাখ্যা
Question: If n is an integer between 10 and 70, then any of the following could be n + 7 except-

Solution: 
maximum value of n + 7 = 69 + 7 = 76 
78 > 76
So, the correct answer is A
১১,৫১০.
If two-third of three-fourth of a number is 34, find the 20% of that number?
  1. ক
    13
  2. খ
    13.6
  3. গ
    14
  4. ঘ
    14.6
ব্যাখ্যা
Question: If two-third of three-fourth of a number is 34, find the 20% of that number?

Solution:
Let, the number be x.

ATQ,
(2/3) × (3/4) × x = 34
⇒ (1/2) × x = 34
∴ x = 68

20% of 68 = (20 × 68)/100 = 13.6
১১,৫১১.
A fruit seller sells 20 oranges for Tk. 900 and suffers a loss equal to the cost price of 5 oranges. Find the cost price of one orange.
  1. ক
    Tk. 55
  2. খ
    Tk. 60
  3. গ
    Tk. 42
  4. ঘ
    Tk. 62
ব্যাখ্যা

Question: A fruit seller sells 20 oranges for Tk. 900 and suffers a loss equal to the cost price of 5 oranges. Find the cost price of one orange.

Solution:
Let,
cost price of 1 orange is = Tk. x
∴ cost price of 20 oranges is = Tk. 20x
∴ cost price of 5 oranges is = Tk. 5x

We know,
Cost price - Selling price = Loss
20x - 900 = 5x
⇒ 20x - 5x = 900
⇒ 15x = 900
⇒ x = 900/15
∴ x = 60

∴ Cost price of 1 orange is Tk. 60

১১,৫১২.
ln a square ABCD, diagonals AC and BD intersect at O. The angle bisector of ∠CAB meets BD and BC at F and G, respectively. OF : CG is equal to-
  1. ক
    1 : 2
  2. খ
    1 : 3
  3. গ
    2 : 3
  4. ঘ
    3 : 1
  5. ঙ
    2 : 1
ব্যাখ্যা
Question: ln a square ABCD, diagonals AC and BD intersect at O. The angle bisector of ∠CAB meets BD and BC at F and G, respectively. OF : CG is equal to-

Solution:

ABCD is a square
AC = √2AB
AO = OC = AC/2 = √2AB/2 = AB/√2
ΔAOF ∼ ΔABG

[By AA property]
AO/AB = OF/BG
(AB/√2)/AB = OF/BG
1/√2 = OF/BG
BG = √2OF . . . . . . (i)
AG is angle bisector of ΔABC
AB/AC = BG/GC =1/√2
[angle bisector theorem]
BG = (1/√2)GC . . . . . . (ii)

Compare (i) and (ii)
√2OF = (1/√2)GC
OF : CG = 1 : 2
১১,৫১৩.
By what percent the volume of a cube increases if the length of each edge was increased by 50%?
  1. ক
    ক) 50%
  2. খ
    খ) 125%
  3. গ
    গ) 237.5%
  4. ঘ
    ঘ) 273.5%
ব্যাখ্যা

Let original edge = a,
Then, original volume = a3

New edge = (150/100)a
= 3a/2
New volume = (3a/2)3
= 27a3/8

Increase in volume = (27a3)/8 - (a3)
= 19a3/8

∴ Increase% = {(19a3/8) × (1/a3) × 100}%
= 237.5%

১১,৫১৪.
If the height of a cone is doubled and its base diameter is trebled, then the ratio of the volume of the resultant cone to that of the original cone is? 
  1. ক
    ক) 9 : 1
  2. খ
    খ) 13 : 2
  3. গ
    গ) 4 : 11
  4. ঘ
    ঘ) 18 : 1
ব্যাখ্যা
Question: If the height of a cone is doubled and its base diameter is trebled, then the ratio of the volume of the resultant cone to that of the original cone is? 

Solution:  
Let the original radius and height of the cone be r and h respectively.
Then, new radius = 3r and new height = 2h 
∴ New volume/Original volume = ((1/3) × π × (3r)2 × 2h) : ((1/3) × π × r2 × h) = 18 : 1 
১১,৫১৫.
A cistern can be filled by two pipes A and B in 4 hours and 6 hours respectively. When full, the tank can be emptied by a third pipe C in 8 hours. If all the taps be turned on at the same time, the cistern will be full in?
  1. ক
    3 hrs. 26 min.
  2. খ
    3 hrs. 14 min.
  3. গ
    3 hrs. 08 min.
  4. ঘ
    3 hrs. 38 min.
ব্যাখ্যা
Question: A cistern can be filled by two pipes A and B in 4 hours and 6 hours respectively. When full, the tank can be emptied by a third pipe C in 8 hours. If all the taps be turned on at the same time, the cistern will be full in?

Solution:
Net filling in 1 hour = (1/4) + (1/6) - (1/8)
= (6 + 4 - 3)/24
= 7/24
∴ Time taken to fill the cistern = (24/7) hrs.
= 3 hrs. 26 min.
১১,৫১৬.
If (1/3)2y = 1/81, then find (0.3)y = ?
  1. ক
    0.09
  2. খ
    0.081
  3. গ
    1
  4. ঘ
    0.25
ব্যাখ্যা

Question: If (1/3)2y = 1/81, then find (0.3)y = ?

Solution:
দেওয়া আছে,
(1/3)2y = 1/81
⇒ (1/3)2y = (1/3)4
⇒ 2y = 4
⇒ y = 4/2
⇒ y = 2

এখন,
(0.3)y 
​= (0.3)2 
​= 0.09

∴ নির্ণেয় মান হলো 0.09

১১,৫১৭.
A parallelogram has a base of 30m and height is 10m long. Then its area is-
  1. ক
    200 m2
  2. খ
    250 m2
  3. গ
    300 m2
  4. ঘ
    320 m2
ব্যাখ্যা
Question: A parallelogram has a base of 30m and  height is 10m. Then its area is-

Solution: 
area = base × height
= 30 × 10
= 300 m2
১১,৫১৮.
A cement mixture is composed of 3 elements.By weight, ⅓ of the mixture is sand, 3/5 is water and the remaining 12 pounds of the mixture is gravel. What is the weight of the entire mixture in pounds?
  1. ক
    ক) 60
  2. খ
    খ) 80
  3. গ
    গ) 90
  4. ঘ
    ঘ) 180
  5. ঙ
    ঙ) None
ব্যাখ্যা
বালু এবং পানির মোট পরিমাণ (১/৩)+(৩/৫)
= (৫+৯)/১৫ = ১৪/১৫ অংশ
শেষে পাথরের পরিমাণ = ১-(১৪/১৫) = ১/১৫ অংশ
১/১৫ অংশ = ১২ পাউন্ড হলে
১ বা সম্পূর্ণ অংশ হবে = ১৫x১২ = ১৮০ পাউন্ড।
১১,৫১৯.
A football team is to be consisted out of 14 boys. In how many ways the team can be chosen so that the owner of the ball is always in the team?
  1. ক
    250
  2. খ
    272
  3. গ
    286
  4. ঘ
    300
ব্যাখ্যা
প্রশ্ন: A football team is to be consisted out of 14 boys. In how many ways the team can be chosen so that the owner of the ball is always in the team?

সমাধান:
14 জনের দল থেকে 1জন ঠিক রেখে বাকি 13জন থেকে (11 - 1) = 10 জনের টিম গঠন করা যাবে
= 13C10
=286
১১,৫২০.
70 grams is what percentage of 20 kilograms?
  1. ক
    0.5% 
  2. খ
    0.35% 
  3. গ
    0.15% 
  4. ঘ
    0.8% 
ব্যাখ্যা

Question: 70 grams is what percentage of 20 kilograms?

Solution:
We know,
20 kg = 20 × 1000 
= 20000 grams

∴ Percentage = (70 ÷ 20000) × 100
= 0.35%

∴ 70 grams is 0.35% of 20 kg.

১১,৫২১.
From the top of a lighthouse 60 m high above sea level, the angle of depression of a boat is 45°. How far is the boat from the foot of the lighthouse?
  1. ক
    40 m
  2. খ
    60 m
  3. গ
    30 m
  4. ঘ
    55 m
ব্যাখ্যা

Question: From the top of a lighthouse 60 m high above sea level, the angle of depression of a boat is 45°. How far is the boat from the foot of the lighthouse?

Solution:

Let the height of the lighthouse above sea be AC and it is given 60 m.
Angle of depression = 45°

Boat is at point B so the distance between the base of lighthouse A and Boat is AB.

 So, tan 45° = AC / AB
⇒ 1 = 60 / AB
⇒ AB = 60 m

∴ The boat is 60 m away from the foot of the lighthouse.

১১,৫২২.
If Px = Qy = Rz and Q/P = R/Q then 2z/(x + z) = ?
  1. ক
    x/y
  2. খ
    z/y
  3. গ
    y/x
  4. ঘ
    y/z
ব্যাখ্যা

[মূল প্রশ্নে Px = Qy = Rz এর পরিবর্তে Px = Qy = Rz হবে]

Question: If Px = Qy = Rz and Q/P = R/Q then 2z/(x + z) = ?

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

১১,৫২৩.
A two member committee comprising of one male and one female member is to be constitute out of five males and three females. Amongst the females. Ms. A refuses to be a member of the committee in which Mr. B is taken as the member. In how many different ways can the committee be constituted?
  1. ক
    11
  2. খ
    23
  3. গ
    14
  4. ঘ
    None
ব্যাখ্যা
Question: A two member committee comprising of one male and one female member is to be constitute out of five males and three females. Amongst the females. Ms. A refuses to be a member of the committee in which Mr. B is taken as the member. In how many different ways can the committee be constituted?

Solution:
Probability = 5C1 × 3C1 - 1
= 15 - 1
= 14
১১,৫২৪.
A can do a piece of work in 24 days. When he had worked for 4 days, B joined him. If the work was finished in 16 days, B can alone finish that work is -
  1. ক
    ক) 20 days
  2. খ
    খ) 30 days
  3. গ
    গ) 36 days
  4. ঘ
    ঘ) 32 days
ব্যাখ্যা
Question: A can do a piece of work in 24 days. When he had worked for 4 days, B joined him. If the work was finished in 16 days, B can alone finish that work is -

Solution:
A's 1 day's work = 1/24 part
A's 16 day's work = 16/24 part
= 2/3 part

Remaining work  = 1 - 2/3 = 1/3 part

1/3 part work done by B in 16 - 4 = 12 days
1 part work done by B in 12 × 3 = 36 days
১১,৫২৫.
A person invests in the stock market at 10% per annum, compounded annually, for 1 years. If the interest was compounded half-yearly, he would have received Tk. 75 more. Find the sum that invest in the stock market.
  1. ক
    ক) Tk. 20,000
  2. খ
    খ) Tk. 25,000
  3. গ
    গ) Tk. 30,000
  4. ঘ
    ঘ) Tk. 35,000
ব্যাখ্যা
When interest compounded annually, then person get 10% of the total sum
If interest compounded half-annually,
Interest rate = 5% and time t = 2

As we know, 5 + 5 + (5 × 5)/100 = 10.25%
If interest compounded half-annually, then person get 10.25% of the total sum

According to the question
⇒ 10.25% – 10% = 75
⇒ 0.25% = 75
⇒ 100% = 30,000

∴ The sum that invest in the stock market = Tk. 30000
১১,৫২৬.
(64)0.20/(4)0.10 = ?
  1. ক
    √2
  2. খ
    2
  3. গ
    0
  4. ঘ
    4
ব্যাখ্যা

Question: (64)0.20/(4)0.10 = ?

Solution: 
Given that, 
(64)0.20/(4)0.10
= (43)0.20/(4)0.10
= (4)0.60/(4)0.10
= (4)0.60 - 0.10
= (4)0.50
= (4)1/2
= √4
= 2

১১,৫২৭.
Soldier : Gun : : Blacksmith : ?
  1. ক
    ক) Wood
  2. খ
    খ) Sword
  3. গ
    গ) Iron
  4. ঘ
    ঘ) Hammer
ব্যাখ্যা
Soldier uses Gun and Blacksmith uses Hammer
১১,৫২৮.
The price of a variety of a commodity is Tk. 7 per kg and that of another is Tk. 12 per kg. Find the ratio in which two varieties should be mixed so that the price of the mixture is Tk. 10 per kg.
  1. ক
    3 : 4
  2. খ
    4 : 3
  3. গ
    2 : 3
  4. ঘ
    5 : 4
ব্যাখ্যা
Question: The price of a variety of a commodity is Tk. 7 per kg and that of another is Tk. 12 per kg. Find the ratio in which two varieties should be mixed so that the price of the mixture is Tk. 10 per kg.

Solution:
Let the mixed amount of the first and second commodity is A and B

∴ A : B = (12-10) : (10-7)
A : B = 2 : 3
১১,৫২৯.
There are n students in a school. If r% among the students are 12 years or younger, which of the following expressions represents the number of students who are older than 12?
  1. ক
    ক) n(1 - r)
  2. খ
    খ) 100 (1 - r)n
  3. গ
    গ) n(1 - r)/100
  4. ঘ
    ঘ) n(100 - r)/100
ব্যাখ্যা
Question: There are n students in a school. If r% among the students are 12 years or younger, which of the following expressions represents the number of students who are older than 12?

Solution: 
r% শিক্ষার্থীর বয়স ১২ বা তার কম হলে,

n - ( n এর r%) এর বয়স ১২ এর বেশি।
= n - (rn/100)
= (100n - rn)/100
= n(100 - r)/100
১১,৫৩০.
The value of a machine is Tk. 6,250. Its value decreases by 10% during the first year, 20% during second year and 30% during the third year. What will be the value of the machine after 3 years?
  1. ক
    ক) 2650
  2. খ
    খ) 3050
  3. গ
    গ) 3150
  4. ঘ
    ঘ) 3750
ব্যাখ্যা
প্রশ্ন: The value of a machine is Tk. 6,250. Its value decreases by 10% during the first year, 20% during second year and 30% during the third year. What will be the value of the machine after 3 years?

সমাধান: 
একটি মেশিনের মূল্য ৬২৫০ টাকা 

প্রথম বছরে এর মূল্য ১০% হ্রাস পায় 
প্রথম বছর পর এর মূল্য = ৬২৫০ - ৬২৫০ এর ১০% 
= ৬২৫০ - (৬২৫০ ×১/১০)
= ৬২৫০ - ৬২৫ টাকা 
= ৫৬২৫ টাকা 

দ্বিতীয় বছরে এর মূল্য ২০% হ্রাস পায় 
দ্বিতীয় বছর শেষে মূল্য = ৫৬২৫ - ৫৬২৫ এর ২০%
= ৫৬২৫ - ১১২৫ টাকা
= ৪৫০০ টাকা 

তৃতীয় বছরে, এর মূল্য ৩০% হ্রাস পায় 
∴ তৃতীয় বছর শেষে এর মূল্য = ৪৫০০ - ৪৫০০ এর ৩০%
= ৪৫০০ - ১৩৫০ টাকা 
= ৩১৫০ টাকা 
১১,৫৩১.
A batsman in his 17th innings makes a score of 85 and their by increasing his average by 3. What is his average after the 17th innings ?
  1. ক
    34
  2. খ
    36
  3. গ
    35
  4. ঘ
    37
  5. ঙ
    38
ব্যাখ্যা
16x + 85 = 17(x + 3)
x = 34 + 3 = 37
১১,৫৩২.
A person sold a book at a gain of 15%. Had he bought it for 25% less and sold it for Tk. 600 less, he would have made a profit of 32%. The cost price of the book was:
  1. ক
    ক) Tk. 2250
  2. খ
    খ) Tk. 2750
  3. গ
    গ) Tk. 2550
  4. ঘ
    ঘ) Tk. 3750
ব্যাখ্যা
Question: A person sold a book at a gain of 15%. Had he bought it for 25% less and sold it for Tk. 600 less, he would have made a profit of 32%. The cost price of the book was:

Solution:
Let the original cost price = Tk. x

Selling price = x + 15% of x
= 115x/100
= 23x/20

Cost price = x - 25% of x
= x - (25x/100)
= 75x/100
= 3x/4

Selling price = (3x/4) + 32% of (3x/4)
= (3x/4) + (32/100) × (3x/4)
= 99x/100

ATQ,
(23x/20) - (99x/100) = 600
⇒ (115x - 99x)/100 = 600
⇒ 16x = 600 × 100
⇒ x = (600 × 100)/16
∴ x = 3750
১১,৫৩৩.
A can do a piece of work in 10 days. While B alone can do it in 15 days. They work together for 5 days and the rest of the work is done by C in 2 days. If they get TK. 450 for the whole work, how much money C should get from the whole?
  1. ক
    ΤΚ. 150
  2. খ
    TK. 75
  3. গ
    TK. 225
  4. ঘ
    Tk. 200
  5. ঙ
    None of these
ব্যাখ্যা
Question: A can do a piece of work in 10 days. While B alone can do it in 15 days. They work together for 5 days and the rest of the work is done by C in 2 days. If they get TK. 450 for the whole work, how much money C should get from the whole?

Solution:
A's work of one day is 1/10
B's work of one day is 1/15

A and B's work of 5 days = 5 × (1/10 + 1/15) = 5 × {(3 + 2)/30}
= 5 × (1/6)
= 5/6

Remaining work = (1 - 5/6) = 1/6

The ratio of A, B, C's work = 5/10 : 5/15 : 1/6 = 1/2 : 1/3 : 1/6 = 3 : 2 : 1
∴ C should get = 450 × (1/6) = 75 Taka
১১,৫৩৪.
A pipe was used to fill a cistern in 10 hours but after working for 7 hours it stopped. another pipe that has the capacity to fill the tank in 20 hours was replaced to fill the rest of the tank. How much time will it take to fill the rest of the tank by the second pipe?
  1. ক
    4 hours
  2. খ
    6 hours
  3. গ
    8 hours
  4. ঘ
    9 hours
ব্যাখ্যা
Question: A pipe was used to fill a cistern in 10 hours but after working for 7 hours it stopped. another pipe that has the capacity to fill the tank in 20 hours was replaced to fill the rest of the tank. How much time will it take to fill the rest of the tank by the second pipe?

Solution: 
in 7 hours,
first pipe fill-up = 7/10
remaining = 3/10

∴ time to fill 3/10 of a tank by the second pipe is
= 3/10 × 20
= 6 hours
১১,৫৩৫.
An outlet pipe can empty a cistern in 3 hours. In what time will it empty 2/3 part of the cistern? 
  1. ক
    2 hrs
  2. খ
    3 hrs
  3. গ
    4 hrs
  4. ঘ
    5 hrs
ব্যাখ্যা
Question: An outlet pipe can empty a cistern in 3 hours. In what time will it empty 2/3 part of the cistern? 

Solution: 
সম্পূর্ণ অংশ খালি করতে সময় লাগে ৩ ঘণ্টা 
২/৩ অংশ খালি করতে সময় লাগে ৩ × ২/৩ ঘণ্টা 
= ২ ঘণ্টা 
১১,৫৩৬.
The 180 students in a group are to be seated in rows so that there is an equal number of students in each row. Each of the following could be the number of rows EXCEPT
  1. ক
    4
  2. খ
    20
  3. গ
    30
  4. ঘ
    40
  5. ঙ
    90
ব্যাখ্যা
Question: The 180 students in a group are to be seated in rows so that there is an equal number of students in each row. Each of the following could be the number of rows EXCEPT.

Solution:
Obviously the number of rows must be a factor of 180.
180/4=45
180/20=9
180/30=6
180/40=4.50
180/90 = 2

The only option which is not a factor of 180 is 40
১১,৫৩৭.
Solve the inequality: ∣x - 3∣ ≥ 4
  1. ক
    - 1 ≤ x ≤ 7
  2. খ
    - 7 ≤ x ≤ 1
  3. গ
    x ≤ - 7 or x ≥ 1
  4. ঘ
    x ≤ - 1 or x ≥ 7
  5. ঙ
    None of these
ব্যাখ্যা
Question: Solve the inequality: ∣x - 3∣ ≥ 4

Solution:
Consider two cases:
(i) x - 3 ≥ 4
Add 3 to both sides: x ≥ 7

(ii) x - 3 ≤ - 4
Add 3 to both sides:
x ≤ - 1

So, the solution to ∣x - 3∣ ≥ 4 is x ≤ - 1 or x ≥ 7
১১,৫৩৮.
If two distinct integers m and n are picked from {1, 2, 3, 4, .... 100} and multiplied, what is the probability that the resulting number has exactly 3 factors?
  1. ক
    2/(25 × 99)
  2. খ
    4/(25 × 99)
  3. গ
    8/(25 × 99)
  4. ঘ
    32/(25 × 99)
ব্যাখ্যা
Question: If two distinct integers m and n are picked from {1, 2, 3, 4, .... 100} and multiplied, what is the probability that the resulting number has exactly 3 factors?

Solution: 
যে কোনো ধনাত্মক পূর্ণসংখ্যার '1' এবং সংখ্যাটি নিজেই গুণনীয়ক হিসেবে থাকবে। এটি এটিকে ন্যূনতম 2টি ফ্যাক্টর তৈরি করে। যদি ধনাত্মক পূর্ণসংখ্যার আরও একটি গুণনীয়ক থাকে, তাহলে 1 এবং সংখ্যা ছাড়াও, সংখ্যাটির বর্গমূলটি শুধুমাত্র অন্য গুণনীয়ক হওয়া উচিত।
অতএব, যদি একটি ধনাত্মক পূর্ণসংখ্যার শুধুমাত্র 3টি গুণনীয়ক থাকে, তাহলে এটি একটি নিখুঁত বর্গ হওয়া উচিত এবং এটি একটি মৌলিক সংখ্যার বর্গ হওয়া উচিত।
এমন সংখ্যা আছে 4 টি 

১০০ থেকে ২ টি বাছাই করার উপায় = 100C2 = 100 × 99/2


সম্ভাব্যতা = 4/100 × 99/2
= 2/(25 × 99)
১১,৫৩৯.
A takes 5 days more than B to do a certain job and 9 days more than C. A and B together can do the job in the same time as C. How many days  A would take to do it?
  1. ক
    16 days
  2. খ
    15 days
  3. গ
    14 days
  4. ঘ
    12 days
  5. ঙ
    None
ব্যাখ্যা
Question: A takes 5 days more than B to do a certain job and 9 days more than C. A and B together can do the job in the same time as C. How many days  A would take to do it?

Solution:
Let
A takes x days to do the job alone.

Then,
B takes (x - 5) days
C takes (x - 9) days.

ATQ,
(1/x) + {1/(x - 5)} = 1/(x - 9)
⇒ {(x - 5) + x}/x(x - 5) = 1/(x - 9)
⇒ (2x - 5)/(x2 - 5x) = 1/(x - 9)
⇒ (2x - 5)(x - 9) = x2 - 5x
⇒ 2x2 - 18x - 5x + 45 = x2 - 5x
⇒ 2x2 - 23x + 45 - x2 + 5x = 0
⇒ x2 - 18x + 45 = 0
⇒ x2 - 15x - 3x + 45 = 0
⇒ x(x - 15) - 3(x - 15) = 0
⇒ (x - 15)(x - 3) = 0
∴ x = 15 or 3

But x ≠ 3 [because (3 - 5) = - 2 not exist]

∴ A takes 15 days to do the job alone.
১১,৫৪০.
How far apart are the centers of two circles with diameters of 16 cm and radii of 6 cm, when they touch each other externally? 
  1. ক
    14 cm
  2. খ
    16 cm
  3. গ
    22 cm
  4. ঘ
    10 cm
ব্যাখ্যা

Question: How far apart are the centers of two circles with diameters of 16 cm and radii of 6 cm, when they touch each other externally? 

Solution:
আমরা জানি,
দুইটি বৃত্ত পরস্পরকে বহিঃস্পর্শ করলে কেন্দ্রদ্বয়ের মধ্যবর্তী দূরত্ব বৃত্ত দুইটির ব্যাসার্ধের যোগফলের সমান।

এখানে,
১ম বৃত্তের ব্যাসার্ধ = 16/2 = 8 সে.মি.
২য় বৃত্তের ব্যাসার্ধ = ৬ সে.মি.

∴ কেন্দ্রদ্বয়ের মধ্যবর্তী দূরত্ব = (8 + 6) সে.মি.
= 14 সে.মি.

১১,৫৪১.
If the radius of a circle is reduced by 40%, its circumference is reduced by-
  1. ক
    60%
  2. খ
    40%
  3. গ
    35%
  4. ঘ
    30%
ব্যাখ্যা
Question:  If the radius of a circle is reduced by 40%, its circumference is reduced by-

Solution: 
If radius of a circle is r, circumference 2πr where 2π is constant 
So, if radius is changed, the circumference will change by the same amount.

The radius of a circle is reduced by 40%,then its circumference is reduced by 40%
১১,৫৪২.
If θ lies in the first quadrant and cos2θ - sin2θ = 1/2. then the value of tan22θ + sin23θ is-
  1. ক
    3
  2. খ
    4
  3. গ
    2
  4. ঘ
    None of the above
ব্যাখ্যা

Question: If θ lies in the first quadrant and cos2θ - sin2θ = 1/2. then the value of tan22θ + sin23θ is-

Solution:
cos2θ - sin2θ = 1/2
⇒ cos2θ = 1/2
⇒ cos2θ = cos60°
⇒ 2θ = 60°
⇒ θ = 30°

Now, tan22θ + sin23θ
= tan260° + sin290°
= 3 + 1
= 4

১১,৫৪৩.
A ladder is placed against a wall such that its foot is at a distance of 2.5 from the wall and its top reaches a window 6m above the ground. Find the length of the ladder.
  1. ক
    ক) 6.5m
  2. খ
    খ) 8.5m
  3. গ
    গ) 4.5m
  4. ঘ
    ঘ) 2.5m
ব্যাখ্যা


ধরি 
মইয়ের দৈর্ঘ্য x মিটার 
এখানে 
AC = x = মই 
AB = দেয়াল 

পিথাগোরাসের সূত্র অনুসারে 
x2 = 62 + 2.52
x2 = 36 + 6.25 
x2 = 42.25 
x = √42.25 
x = 6.5
১১,৫৪৪.
Find the volume of the cylinder having a radius of 5 units and a height of 8 units?
  1. ক
    314.57 Cubic units
  2. খ
    628.57 Cubic units
  3. গ
    125.71 Cubic units
  4. ঘ
    None of these
ব্যাখ্যা
Question: Find the volume of the cylinder having a radius of 5 units and a height of 8 units?

Solution:
We have, 
Radius,r = 5 units
Height,h = 8 units

Volume of the cylinder, V = πr2h cubic units.
V = (22/7) × 52 × 8
V = 22/7 × 25 × 8
V = 628.57 Cubic units.

Hence, the volume of the cylinder is 628.57 cubic units.
১১,৫৪৫.
৫৫০ এর ৯২.৫% সমান কত?
  1. ক
    ৫০৮.৭৫
  2. খ
    ৫০৬.৪৮
  3. গ
    ৫২১.৫৫
  4. ঘ
    ৫০২.৭৮
  5. ঙ
    ৫০৭
ব্যাখ্যা

প্রশ্ন: ৫৫০ এর ৯২.৫% সমান কত?

সমাধান:
৫৫০ এর ৯২.৫%
= ৫৫০ × (৯২.৫/১০০)
 = ৫৫০ × (৯২৫/১০০০)
= ৫০৮.৭৫

১১,৫৪৬.
A is west of B. B is north of C. D is south of A. So, where is D in relation to C?
  1. ক
    North
  2. খ
    South
  3. গ
    East-West
  4. ঘ
    West
ব্যাখ্যা
Question: A is west of B. B is north of C. D is south of A. So, where is D in relation to C?

Solution:

অতএব, D এর অবস্থান C এর পশ্চিমদিকে।
১১,৫৪৭.
A train running at the speed of 25m/s can cross a 200 metres platform in 11 seconds. What is the length of the train?
  1. ক
    ক) 50m
  2. খ
    খ) 60m
  3. গ
    গ) 75m
  4. ঘ
    ঘ) 80m
ব্যাখ্যা
Question: A train running at the speed of 25m/s can cross a 200 metres platform in 11 seconds. What is the length of the train?

Solution: 
Here, 
Speed, S = 25m/s
Time, T = 11s

Let, the distance is = D

We know that,
D = S × T
D = 25 × 11
= 275m 

The length of the train is = 275 - 200 = 75m
১১,৫৪৮.
If the radius of a circle is doubled, its area is increased by-
  1. ক
    ক) 200%
  2. খ
    খ) 100%
  3. গ
    গ) 300%
  4. ঘ
    ঘ) 50%
ব্যাখ্যা
Let
Radius of the circle = r
then Area of the circle = πr2
if we doubled the radius then radius = 2r
and Area of the circle = π(2r)2 = 4πr2 

Increased Area = 4πr2 - πr2=3πr2 
Increase % = (3πr2/πr2) ×100}% 
                  = 300%
১১,৫৪৯.
Angry. : Night : : ? : Day
  1. ক
    ক) Helpful
  2. খ
    খ) Pleased
  3. গ
    গ) Cruel
  4. ঘ
    ঘ) Loving
ব্যাখ্যা
First terms is the opposite of the IIIrd term as is the case with IInd and IVth terms
১১,৫৫০.
  1. ক
    0
  2. খ
    1
  3. গ
    2
  4. ঘ
    3
ব্যাখ্যা
Question:

Solution:
১১,৫৫১.
HCF and LCM of two numbers are 7 and 140 respectively. If the numbers are between 20 and 45, the sum of the numbers is ?
  1. ক
    48
  2. খ
    65
  3. গ
    54
  4. ঘ
    63
ব্যাখ্যা

Question: HCF and LCM of two numbers are 7 and 140 respectively. If the numbers are between 20 and 45, the sum of the numbers is ?

Solution:
Given that, 
HCF and LCM of two numbers are 7 and 140

Let,
The numbers are 7x and 7y
∴ LCM = 7xy

∴ 7xy = 140
⇒ xy = 20
⇒ Possible co-prime factors of xy = (1, 20), (4, 5)
⇒ Numbers are between 20 and 45
∴ Required number are = 4 × 7 = 28 and 5 × 7 = 35
∴ Sum of numbers are = 28 + 35 = 63

So the sum of the numbers is 63.

১১,৫৫২.
Find the value of,
  1. ক
    12
  2. খ
    16
  3. গ
    2
  4. ঘ
    4
ব্যাখ্যা
Question: Find the value of,


Solution:
১১,৫৫৩.
How many triangles are there?
  1. ক
    14
  2. খ
    12
  3. গ
    15
  4. ঘ
    16
ব্যাখ্যা
Question: How many triangles are there?



Solution:


The simplest triangles are AFB, FEB, EBC, DEC, DFE, and AFD so, 6 triangles.
The triangles are composed of two components: AEB, FBC, DFC, ADE, DBE, and ABD so, 6 triangles
The triangles composed of three components are ADC and ABC so, 2 trangles.
There is only one triangle DBC which is composed of four components.

Thus, there are 6 + 6 + 2 + 1 = 15 triangles in the figure.
১১,৫৫৪.
X and Y invest in business in the ratio 5 : 3. If 8% of the total profit goes to charity and X's share is Tk. 1380, the total profit is:
  1. ক
    Tk. 2400
  2. খ
    Tk. 2540
  3. গ
    Tk. 2620
  4. ঘ
    Tk. 2760
ব্যাখ্যা
Question: X and Y invest in business in the ratio 5 : 3. If 8% of the total profit goes to charity and X's share is Tk. 1380, the total profit is:

Solution:
Let the total profit be Tk. 100.
After paying to charity, X's share = (92 × 5/8) = 57.5
If X's share is Tk. 57.5, total profit = 100.
If X's share is Tk. 1380, total profit = (100/57.5 × 1380)
= 2400.
Therefore, the total profit is Tk. 2400.
১১,৫৫৫.
69% of the students of a certain class took Mathematics and 47% took Biology. If each student took Biology or Mathematics and 44 took both, what is the total number of students in the class?
  1. ক
    150
  2. খ
    250
  3. গ
    300
  4. ঘ
    275
ব্যাখ্যা
Question: 69% of the students of a certain class took Mathematics and 47% took Biology. If each' student took Biology or Mathematics and 44 took both, what is the total number of students in the class?

Solution:
দেওয়া আছে,
গণিত নিয়েছে = 69%
জীববিজ্ঞান নিয়েছে = 47%
উভয় বিষয় নিয়েছে = 44 জন 

ধরি,
উভয় বিষয় নিয়েছে = x জন

প্রশ্নমতে,
(69% - x) + x + (47% - x) = 100%
⇒ 116% - x = 100%
⇒ x = (116 - 100)%
⇒ x = 16%

আবার, 
শর্তমতে,
মোট শিক্ষার্থী × 16% = 44
⇒ মোট শিক্ষার্থী × (16/100) = 44
⇒ মোট শিক্ষার্থী = (44 × 100)/16
⇒ মোট শিক্ষার্থী = 275 

∴ মোট শিক্ষার্থীর সংখ্যা = 275 জন
১১,৫৫৬.
Two trains are running in opposite directions at the same speed. The length of each train is 120 meter. If they cross each other in 12 seconds, the speed of each train (in km/hr) is -
  1. ক
    ক) 42 km/hr
  2. খ
    খ) 36 km/hr
  3. গ
    গ) 28 km/hr
  4. ঘ
    ঘ) 20 km/hr
ব্যাখ্যা

Distance covered = (120 + 120)
= 240 meter.
Time = 12 seconds.
Relative speed = 240/12
= 20 m/s.
= 20 × (18/5) km/hr.
= 72 km/hr.
The relative speed, in this case, is the sum of the speeds of the trains and each train has the same speed,
speed as each train = 72/2 km/hr.
= 36 km/hr.

১১,৫৫৭.
If the angles of a triangle are in the ratio 1 : 2 : 6, what is the measure of the middle angle (in degrees)?
  1. ক
    75°
  2. খ
    20°
  3. গ
    80°
  4. ঘ
    60°
  5. ঙ
    None of these
ব্যাখ্যা
Question: If the angles of a triangle are in the ratio 1 : 2 : 6, what is the measure of the middle angle (in degrees)?

Solution:
Given that,
The angles of a triangle are in the ratio 1 : 2 : 6
Let,
x, 2x, 6x

We know that,
Sum of angles in a triangle = 180°

Now
x + 2x + 6x = 180°
⇒ 9x = 180°
⇒ x = 180°/9 = 20°
∴ x = 20°

∴ middle angle = 2x = 2 × 20 = 40°
১১,৫৫৮.
Pabel and Khalil started a business in the ratio of 2 : 3. After 2 years Pabel left the business but Khalil continued. After 3 years he had a profit of Tk 26000. What was the profit of Pabel?
  1. ক
    Tk. 8000
  2. খ
    Tk. 15600
  3. গ
    Tk.18000
  4. ঘ
    No Profit
ব্যাখ্যা
Question: Pabel and Khalil started a business in the ratio of 2 : 3. After 2 years Pabel left the business but Khalil continued. After 3 years he had a profit of Tk 26000. What was the profit of Pabel?

Solution:
Let,
the initial capital of Pabel and Khalil be Tk. 2x and Tk. 3x respectively.

Then, ratio of profits = (2x × 24) : (3x × 36)
= (2x × 2) : (3x × 3)
= 4x : 9x
= 4 : 9

∴ Profit of Pabel = 26000 × (4/13)
= 2000 × 4
= Tk. 8000
১১,৫৫৯.
A small company employs 3 men and 5 women. If a team of 4 employees is to be randomly selected to organize the company retreat, what is the probability that the team will have exactly 2 women?
  1. ক
    1/7
  2. খ
    3/7
  3. গ
    5/7
  4. ঘ
    6/7
ব্যাখ্যা
Question:  A small company employs 3 men and 5 women. If a team of 4 employees is to be randomly selected to organize the company retreat, what is the probability that the team will have exactly 2 women?

Solution:
the probability that the team will have exactly 2 women is = (5C2 × 3C2)/8C4
= 30/70
= 3/7
১১,৫৬০.
A shopkeeper sells a badminton racket, whose marked price is Tk.800, at a discount of 15% and gives a shuttle cock costing Tk.20 free with each racket. Even then he makes a profit of 25%. His cost price per racket is-
  1. ক
    488 Tk
  2. খ
    528 Tk
  3. গ
    552Tk
  4. ঘ
    None of the above
ব্যাখ্যা
Question: A shopkeeper sells a badminton racket, whose marked price is Tk.800, at a discount of 15% and gives a shuttle cock costing Tk.20 free with each racket. Even then he makes a profit of 25%. His cost price per racket is-

Solution:
Marked price = 800 tk
After 15% discount selling price = (100 - 15)% of 800
= 85% of 800
= (85/100) × 800
= 680
cost of shuttle cock = 20 Tk

So, actual selling price = (680 - 20) tk
= 660 tk

let, cost price per racket is = x tk
at 25% profit, selling price = 1.25x tk

 Now,
1.25x = 660
⇒ x = 660/1.25
= 528 Tk

His cost price per racket is 528 Tk.
১১,৫৬১.
The speed of a boat in still water is 25 kmph. If it can travel 10 km upstream in 1 hr, what time would it take to travel the same distance downstream?
  1. ক
    ক) 21 minute
  2. খ
    খ) 22 minute
  3. গ
    গ) 30 minute
  4. ঘ
    ঘ) 35 minute
  5. ঙ
    ঙ) 15 minute
ব্যাখ্যা

Speed of boat in still water = 25 km/hr
Speed upstream = 10 km/hr
Speed of the stream = (25 − 10) = 15 km/hr
Speed downstream = (25 + 15) = 40 km/hr

Time taken to travel 10 km downstream = 10/40 hour
= (10 × 60)/40
= 15 minute

১১,৫৬২.
There is an equilateral triangle with a square inscribed inside it. One of the sides of the square lies on a side of the equilateral triangle. What is the ratio of the area of the square to that of the equilateral triangle?
  1. ক
    6 : (5 + 7√3)
  2. খ
    24 : (24 + 7√3)
  3. গ
    12 : (12 + 7√3)
  4. ঘ
    18 : (12 + 7√3)
ব্যাখ্যা
Question: There is an equilateral triangle with a square inscribed inside it. One of the sides of the square lies on a side of the equilateral triangle. What is the ratio of the area of the square to that of the equilateral triangle?

Solution: 


let, side of equilateral triangle is a 
area =  (√3/4)a2 

let side of square r
BD = CE = r/tan60 = r/√3
BC = a =  r +  r/√3 +  r/√3 = (2r + √3r)/√3
= r (2 + √3)/√3

area of triangle = (√3/4) {r (2 + √3)/√3}2
= r2(√3/4) (7 + 4√3)/3
= r2 (7√3 + 12)/12

area of square = r2 

ratio = r2 : r2 (7√3 + 12)/12
= 12 : (12 + 7√3)
১১,৫৬৩.
What is the sum of the first 17 terms of an arithmetic progression if the first term is - 20 and last term is 28?
  1. ক
    68
  2. খ
    136
  3. গ
    34
  4. ঘ
    72
ব্যাখ্যা

Question: What is the sum of the first 17 terms of an arithmetic progression if the first term is - 20 and last term is 28?

Solution: 
First term of AP = a = - 20
and last term = l = 28
Number of terms = n = 17

∴ Sum of the first 17 terms, Sn = n/2(a + l) 
= 17/2(- 20 + 28)
= 17 × 4
= 68

১১,৫৬৪.
If p is an even integer and q is an odd integer, which of the following must be an odd integer?
  1. ক
    ক) p/q
  2. খ
    খ) 2p + q
  3. গ
    গ) pq
  4. ঘ
    ঘ) 2 (p + q)
  5. ঙ
    ঙ) 3p/q
ব্যাখ্যা
Question: If p is an even integer and q is an odd integer, which of the following must be an odd integer?

Solution: 
2 × even integer = even integer
So, 2p is an even integer

even integer + odd integer = odd integer

So,
2p + q must be an odd integer.
১১,৫৬৫.
A fisher man can push 2 km against the stream in 20 min. What's more, return in 15 min. What is the rate of the current?
  1. ক
    ক) 1 km/hr
  2. খ
    খ) 2 km/hr
  3. গ
    গ) 3 km/hr
  4. ঘ
    ঘ) None of these
ব্যাখ্যা

Speed upstream = (2/20)×60 km/hr
= 6 km/hr
Speed downstream = (2/15)×60 km /hr
= 8km/hr
Speed of the current
= 1/2(8-6) km /hr
= 1 km/hr

১১,৫৬৬.
Due to increase the rate of simple interest from 5% to 7%, the income of Rahim was increased by Tk. 1200 in 3 years. How much was his principal?
  1. ক
    Tk. 20000
  2. খ
    Tk. 15000
  3. গ
    Tk. 12000
  4. ঘ
    Tk. 11600
ব্যাখ্যা
Question: Due to increase the rate of simple interest from 5% to 7%, the income of Rahim was increased by Tk. 1200 in 3 years. How much was his principal?

Solution:
Let,
His principal = P

ATQ,
{P × (7/100) × 3} - {P × (5/100) × 3} = 1200
⇒ (21P/100) - (15P/100) = 1200
⇒ (21P - 15P)/100 = 1200
⇒ 6P/100 = 1200
⇒ 6P = 1200 × 100
⇒ P = 120000/6
∴ P = 20000
১১,৫৬৭.
What is the original price of a T-shirt, if the sale price after 16% discount Is 264?
  1. ক
    ক) 300
  2. খ
    খ) 214
  3. গ
    গ) 320
  4. ঘ
    ঘ) 314
ব্যাখ্যা

84% = 264
So, 100% = (264 × 100) / 84
= 314.29 ≈ 314

১১,৫৬৮.
Two buses start from a bus terminal with a speed of 60 km/h at interval of 15 minutes. What is the speed of a man coming from the opposite direction towards the bus terminal if he meets the buses at interval of 12 minutes?
  1. ক
    15 kmph
  2. খ
    9 kmph
  3. গ
    16 kmph
  4. ঘ
    20 kmph
ব্যাখ্যা

Question: Two buses start from a bus terminal with a speed of 60 km/h at interval of 15 minutes. What is the speed of a man coming from the opposite direction towards the bus terminal if he meets the buses at interval of 12 minutes?

Solution:
Let Speed of the man is x kmph.
Distance covered in 15 minutes at 60 kmph = distance covered in 12 minutes at (60 + x) kmph.
60 × (15/60) = (12/60)(60 + x)}
900 = 720 + 12x
12x = 180
x = 180/12
x = 15

So the speed of the man coming from the opposite direction is 15kmph 

১১,৫৬৯.
  1. ক
    0
  2. খ
    - 1
  3. গ
    √3
  4. ঘ
    1
  5. ঙ
    √(2/3)
ব্যাখ্যা

Question: 


Solution: 

১১,৫৭০.
If sec θ = 5/4, then what is the value of sinθ?
  1. ক
    3/5
  2. খ
    8/3
  3. গ
    3/4
  4. ঘ
    4/5
ব্যাখ্যা

Question: If sec θ = 5/4, then what is the value of sinθ?

Solution:
এখানে,
secθ = 5/4 = অতিভুজ/ভূমি
∴ অতিভুজ = 5, ভূমি = 4

পিথাগোরাসের উপপাদ্য অনুসারে, লম্ব নির্ণয় করি,
লম্ব = √(অতিভুজ2 - ভূমি2)
= √(52 - 42)
= √(25 - 16)
= √9
= 3

এখন,
sinθ = লম্ব/অতিভুজ 
= 3/5

সুতরাং, sinθ = 3/5।

১১,৫৭১.
A can do a job in 30 days. B alone can do the same job in 20 days. If A starts the work and joined by B after 10 days, in how many days the job will be done?
  1. ক
    15 days
  2. খ
    16 days
  3. গ
    17 days
  4. ঘ
    18 days
ব্যাখ্যা
Question: A can do a job in 30 days. B alone can do the same job in 20 days. If A starts the work and joined by B after 10 days, in how many days the job will be done?

Solution:
A's one day work = 1/30
A's ten-day work = (1/30) × 10 = 1/3

So the remaining work would be = 1 - 1/3 = 2/3

B's one day work = 1/20
A and B's one day work = 1/30 + 1/20 = (2 + 3)/60 = 5/60 = 1/12

1/12 of the job will be done by them in one day.
So, the remaining job 2/3 will be done in = (2/3) × 12 = 8 days

Therefore, the total number of days required to do the job would be = 10 + 8 = 18 days
১১,৫৭২.
The quadratic equation whose one rational root is 3 + √2 is-
  1. ক
    x2 - 7x + 6 = 0
  2. খ
    x2 + 7x + 6 = 0
  3. গ
    x2 - 7x + 5 = 0
  4. ঘ
    x2 - 6x + 7 = 0
  5. ঙ
    x2 - 8x + 7 = 0
ব্যাখ্যা

Question: The quadratic equation whose one rational root is 3 + √2 is-

Solution:
one root is 3 + √2
∴ other root is 3 - √2

Sum of roots = 3 + √2 + 3 - √2 = 6
Product of roots = (3 + √2)(3 - √2) = (3)2- (√2)2
= 9 - 2 = 7

∴ Required quadratic equation is x2- 6x + 7 = 0

১১,৫৭৩.
If the side of a square is increased by 20%, by what percent will the area be increased?
  1. ক
    21%
  2. খ
    42%
  3. গ
    44%
  4. ঘ
    20%
ব্যাখ্যা

Question: If the side of a square is increased by 20%, by what percent will the area be increased?

Solution:
Let the original side length = 10 units.

∴ Area = 10 × 10 = 100 square units

Again, 
After a 20% increase, the new side length = 10 + (20% of 10)
= 10 + 2 = 12 units

∴ New area = 12 × 12 = 144 square units

∴ Increase in area = (144 - 100) square units
= 44 square units

∴ Percentage increase in area = (44/100) × 100%
= 44%

So the area will increase by 44%

১১,৫৭৪.
A train starts from city Y at 2 PM and travels towards city X at 75 km/hr. Another train starts from city X at 1 PM and travels towards Y at 60 km/hr. If the distance between theses two cities is 330 km then at what time will they meet ? 
  1. ক
    ক) 6 : 00 PM 
  2. খ
    খ) 5 : 30 PM 
  3. গ
    গ) 5 : 00 PM 
  4. ঘ
    ঘ) 4 : 00 PM 
ব্যাখ্যা
দ্বিতীয় ট্রেনটি 1 ঘণ্টা পূর্বে ছেড়েছে ফলে ২য় ট্রেনটি 60 km এগিয়ে থাকবে। 

ট্রেন দুটির মধ্যবর্তী  দূরত্ব হবে = (330 - 60) km= 270km 
ট্রেন দুটি  মিলিত হতে সময় লাগবে = 270/(75 + 60) = 2 ঘণ্টা 

সময় লাগবে =  2 PM  + 2 = 4PM
১১,৫৭৫.
A book costs Tk. 750. It is sold after giving three successive discounts of 20%, 15%, and 10%. Find the final price of the book after all discounts.
  1. ক
    Tk. 580
  2. খ
    Tk. 530
  3. গ
    Tk. 489
  4. ঘ
    Tk. 459
ব্যাখ্যা
Question: A book costs Tk. 750. It is sold after giving three successive discounts of 20%, 15%, and 10%. Find the final price of the book after all discounts.

Solution:
20% discount on 750 = 750 × 20% = Tk. 150

∴ Price after 20% discount = (750 - 150) = Tk. 600

Again,
15% discount on 600 = 600 × 15% = Tk. 90

∴ Price after 15% discount on Tk. 600 = (600 - 90) = Tk. 510

And,
10% discount on 510 = 510 × 10% = Tk. 51

∴ Price after 10% discount on Tk. 510 = (510 - 51) = Tk. 459

∴ the final price of the book after all discounts = Tk. 459
১১,৫৭৬.
Rafsan invested 1/3 of his capital at 7%, 1/4 at 8% and the remainder at 10%. If his annual income is 561 taka, the capital is - 
  1. ক
    Tk. 5400
  2. খ
    Tk. 6600
  3. গ
    Tk. 7000
  4. ঘ
    Tk. 8200
ব্যাখ্যা
Question: Rafsan invested 1/3 of his capital at 7%, 1/4 at 8% and the remainder at 10%. If his annual income is 561 taka, the capital is - 

Solution: 
ধরি,
আসল x টাকা
আমরা জানি, I = Pnr

561 = {(x × 7)/(100 × 3)} + {(x × 8)/(100 × 4)} + {(5x × 10)/(12 × 100)} 
⇒ 561 = (7x/300) + (x/50) + (x/24)
⇒ 561 = 102x/1200
⇒ 561 = 51x/600
∴x = 6600 taka
১১,৫৭৭.
The average weight of A, B and C is 45kg. If the average weight of A and B is 40 kg and that of B and C is 43 kg, then the weight of B is-
  1. ক
    ক) 17 kg
  2. খ
    খ) 31 kg
  3. গ
    গ) 20 kg
  4. ঘ
    ঘ) None of these
ব্যাখ্যা
Question: The average weight of A, B and C is 45kg. If the average weight of A and B is 40 kg and that of B and C is 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.
১১,৫৭৮.
The average of the first nine prime numbers is-
  1. ক
    ক) 12.12
  2. খ
    খ) 11.11
  3. গ
    গ) 13.22
  4. ঘ
    ঘ) 14.05
ব্যাখ্যা
First 9 prime numbers.= 2, 3, 5, 7, 11, 13, 17, 19, 23
Average = Sum of all numbers / Total numbers.
              = (2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23)/9
              = 100/9
              = 11.11
Therefore the avg of first 9 prime numbers is 11.11
১১,৫৭৯.
The traffic lights at three different road crossings change after 12 seconds, 18 seconds, and 24 seconds respectively. If they all change simultaneously at 08 : 20 : 00 AM, then at what time will they again change simultaneously ?
  1. ক
    09: 32: 00 AM
  2. খ
    08: 40 : 20 AM
  3. গ
    08: 21: 12 AM
  4. ঘ
    09 : 17: 21 AM
ব্যাখ্যা
Question: The traffic lights at three different road crossings change after 12 seconds, 18 seconds, and 24 seconds respectively. If they all change simultaneously at 08 : 20 : 00 AM, then at what time will they again change simultaneously ?

Solution:
12 = 2 × 2 × 3
18 = 2 × 3 × 3
24 = 2 × 2 × 2× 3
∴ LCM of (12, 18, 24) is = 72

They will change simultaneously after every = 72 seconds
= 72/60 
= 1 minute 12 seconds

They change 1st at = 08 : 20 : 00 AM
So, again they change at = (08 : 20 : 00 + 1 minute 12 seconds) AM
= 08 : 21 : 12 AM
১১,৫৮০.
How many three-lettered words can be formed from letters A, B, C, D, E, G, H if repeats are not allowed?
  1. ক
    343
  2. খ
    35
  3. গ
    210
  4. ঘ
    18
ব্যাখ্যা
Question: How many three-lettered words can be formed from letters A, B, C, D, E, G, H if repeats are not allowed?

Solution:
For first position of word we can select 7 letters
For the 2nd position of the word we can select (7 - 1) = 6 letters
For the 3rd position of the word we can select (7 - 2) = 5 letters

∴ number of three-lettered words = 7 × 6 × 5 = 210 
১১,৫৮১.
A train of 250 m long is moving at 54 km/h. The time taken by the train to cross a bridge of 350 m long is-
  1. ক
    36 seconds
  2. খ
    40 seconds
  3. গ
    30 seconds
  4. ঘ
    45 seconds
ব্যাখ্যা

Question: A train of 250 m long is moving at 54 km/h. The time taken by the train to cross a bridge of 350 m long is-

Solution:
মোট অতিক্রান্ত দূরত্ব = ট্রেনের দৈর্ঘ্য + সেতুর দৈর্ঘ্য
= (250 + 350) মিটার
= 600 মিটার

ট্রেনের গতিবেগ = 54 কিমি/ঘন্টা
= (54 × 1000) মিটার/3600 সেকেন্ড
= 15 মিটার/সেকেন্ড

সময় = দূরত্ব ÷ গতিবেগ
= 600 মিটার ÷ 15 মিটার/সেকেন্ড
= 40 সেকেন্ড

সুতরাং, সেতুটি অতিক্রম করতে ট্রেনটির 40 সেকেন্ড সময় লাগবে।

১১,৫৮২.
A circle and a rectangle have the same perimeter. The sides of the rectangle are 18 cm and 26 cm. what is the area of the circle?
  1. ক
    1250 sq. cm
  2. খ
    616 sq. cm
  3. গ
    154 sq. cm
  4. ঘ
    88 sq. cm
ব্যাখ্যা
Question: A circle and a rectangle have the same perimeter. The sides of the rectangle are 18 cm and 26 cm. what is the area of the circle?

Solution:
Perimeter of the rectangle =2(18 + 26)
= 88 cm

∴ Circumference of circle = 88 cm
⇒ 2πr = 88
⇒ r = 88/2π
⇒ r = (88 × 7)/(2 × 22)
∴ r = 14 cm

∴ Area of circle = πr2
= (22/7) × (14)2
= 616 sq. cm
১১,৫৮৩.
In one hour, a boat goes 14 km/hr along the stream and 8 km/hr against the stream. The speed of the boat in still water (in km/hr) is:
  1. ক
    ক) 8 km/hr
  2. খ
    খ) 11 km/hr
  3. গ
    গ) 12 km/hr
  4. ঘ
    ঘ) 10 km/hr
ব্যাখ্যা

Let speed of the boat in still water = a and speed of the stream = b
Then
a+b = 14
a-b = 8
Adding these two equations, we get 2a = 22
=> a = 11
ie, speed of boat in still water = 11 km/hr

১১,৫৮৪.
A pipe can fill up an empty tank in 14 minutes. Another pipe flows out 12 liters of water per minute. If the two pipes are opened together and the empty tank is filled up in 98 minutes, how much water does the tank contain?
  1. ক
    ক) 178 liter
  2. খ
    খ) 184 liter
  3. গ
    গ) 192 liter
  4. ঘ
    ঘ) 196 liter
  5. ঙ
    ঙ) 216 liter
ব্যাখ্যা
Question: A pipe can fill up an empty tank in 14 minutes. Another pipe flows out 12 liters of water per minute. If the two pipes are opened together and the empty tank is filled up in 98 minutes, how much water does the tank contain?

Solution:
মনে করি, ট্যাংকটি খালি হয় x মিনিটে।
প্রশ্নমতে,
(1/14) - (1/x) = 1/98
⇒ (1/14) - (1/98) = 1/x
⇒ 6/98 = 1/x
⇒ 3/49 = 1/x
⇒ x = 49/3

অপর নল দ্বারা ট্যাংকটি 49/3 মিনিটে পুরো খালি হয়।

∴ ট্যাংকটির ধারণক্ষমতা = (49/3) × 12 = 196 লিটার
১১,৫৮৫.
If w is 10% less than x, and y is 30% less than z, than wy is what percent less than xz ?
  1. ক
    ক) 10%
  2. খ
    খ) 15%
  3. গ
    গ) 24%
  4. ঘ
    ঘ) 37%
ব্যাখ্যা
ধরি, x = 100
w = 100 - 100 এর 10% = 100 - 10 = 90
আবার,
z = 100 
y = 100 - 100 এর 30% = 100 - 30 = 70 

wy = 90 × 70 = 630 
xz = 100 × 100 = 10000

wy , xz এর চেয়ে কম = 10000 - 6300 = 3700
wy , xz এর চেয়ে শতকরা কম ={(3700/10000) × 100}% = 37%
১১,৫৮৬.
The average of first 101 natural numbers is:
  1. ক
    51
  2. খ
    5151
  3. গ
    5252
  4. ঘ
    52
ব্যাখ্যা
The average of first 101 natural numbers
= (101 + 1)/2
= 51
১১,৫৮৭.
Two men start together to walk a certain distance, one at 4 km/hr and another at 3 km/hr. The former arrives half an hour before the latter. Find the distance -
  1. ক
    ক) 8 km
  2. খ
    খ) 7 km
  3. গ
    গ) 6 km
  4. ঘ
    ঘ) 9 km
ব্যাখ্যা

If the required distance be x km, then
(x/3)−(x/4)= 1/2
⇒ (4x−3x)/12= 1/2
⇒ x/12= 1/2
⇒ x= 6 km

১১,৫৮৮.
Find 
  1. ক
    1
  2. খ
    1/4
  3. গ
    1/2
  4. ঘ
    2
ব্যাখ্যা

Question: find 

Solution: 

১১,৫৮৯.
In how many different ways can the letters of the word FORMULATE be arranged? 
  1. ক
    ক) 384580
  2. খ
    খ) 367580
  3. গ
    গ) 362880
  4. ঘ
    ঘ) 378580
ব্যাখ্যা
The given word is FORMULATE
The given word contains 9 letters, all different.
∴ Required number of ways =9P9
                                              =9!
                                              =(9 × 8 × 7× 6 × 5 × 4 × 3 × 2 × 1)
                                              =362880
১১,৫৯০.
A Petrol tank now is 1/2 full. After you remove 8 gallons petrol from the 1/2 full tank the tank is then 1/10 full. What is the capacity, in gallons, of the tank? 
  1. ক
    40
  2. খ
    20
  3. গ
    31/2
  4. ঘ
    30
  5. ঙ
    None of these
ব্যাখ্যা
Question: A Petrol tank now is 1/2 full. After you remove 8 gallons petrol from the 1/2 full tank the tank is then 1/10 full. What is the capacity, in gallons, of the tank?

Solution:
A Petrol tank now is 1/2 full
Here,
(1/2 - 1/10)
= (5 - 1)/10
= 4/10
= 2/5

So,
Capacity of 2/5 of the tank is 8 gallons
∴ Capacity of full tank is (8 × 5)/2 = 20 gallons
১১,৫৯১.
If |2x - 3| < 1 and p < 3x - 2 < q, then find the values of p, q.
  1. ক
    p = - 1 and q = 2
  2. খ
    p = 1 and q = 2
  3. গ
    p = 1 and q = 4
  4. ঘ
    p = - 2 and q = 3
ব্যাখ্যা
Question: If |2x - 3| < 1 and p < 3x - 2 < q, then find the values of p, q.

Solution:
Given that,
|2x - 3| < 1
⇒ - 1 < 2x - 3 < 1
⇒ - 1 + 3 < 2x - 3 + 3 < 1 + 3
⇒ 2 < 2x < 4
⇒ 1 < x < 2
⇒ 3 < 3x < 6
⇒ 3 - 2 < 3x - 2 < 6 - 2
∴ 1 < 3x - 2 < 4

∴ p = 1 and q = 4
১১,৫৯২.
When a number is divided by 13, the remainder is 11. When the same number is divided by 17, then the remainder is 9. What is the number?
  1. ক
    ক) 339
  2. খ
    খ) 349
  3. গ
    গ) 359
  4. ঘ
    ঘ) 369
ব্যাখ্যা

13p + 11 and x = 17q + 9
∵ 13p + 11 = 17q + 9
17q - 13p = 2
q = (2 + 13p)/17
∵ The least value of p for which q = (2 + 13p)/17 is a whole number p = 26
x = (13 x 26 + 11)
= (338 + 11)
= 349

১১,৫৯৩.
Solve: 2y2 = 13y + 45
  1. ক
    9
  2. খ
    9/2
  3. গ
    - 5
  4. ঘ
    - 5/9
ব্যাখ্যা
Question: Solve: 2y2 = 13y + 45

Solution:
2y2 = 13y + 45
⇒ 2y2 - 13y - 45 = 0
⇒ 2y2 - 18y + 5y - 45 = 0
⇒ 2y(y - 9) + 5(y - 9) = 0
⇒ (y - 9)(2y + 5) =0
∴ y - 9 = 0  or, 2y + 5 = 0
∴ y = 9        ∴ y = - 5/2
১১,৫৯৪.
313 + 313 + 313 =?
  1. ক
    339
  2. খ
    314
  3. গ
    913
  4. ঘ
    340
ব্যাখ্যা
Question: 313 + 313 + 313 =?

Solution:
313 + 313 + 313
= 3 × 313
= 31 + 13
= 314
১১,৫৯৫.
The percentage of metal extracted from a mine of lead ore is 50%. Now the percentage of gold is 4/5% of the metal and the rest is lead. If the mass of ore extracted from the mine is 6000kg, the mass of the lead is- 
  1. ক
    48 kg
  2. খ
    1976 kg
  3. গ
    2998 kg
  4. ঘ
    2976 kg
ব্যাখ্যা
Question: The percentage of metal extracted from a mine of lead ore is 50%. Now the percentage of gold is 4/5% of the metal and the rest is lead. If the mass of ore extracted from the mine is 6000kg, the mass of the lead is- 

Solution:
Given mass of lead ore = 6000 kg 
Mass of metal is 50% of 6000 = 3000 kg 
Mass of gold is 4/5% of 3000 = (4 × 3000)/(5 × 100) = 24 kg 

Mass of lead = Mass of metal - Mass of gold
⇒ Mass of lead = 3000 - 24 = 2976 kg
১১,৫৯৬.
The sum of the ages of A, B, and C is 120 years. Ten years ago, the ratio of their ages was 2 : 3 : 4. What is the present age of A?
  1. ক
    ক) 30 years
  2. খ
    খ) 40 years
  3. গ
    গ) 50 years
  4. ঘ
    ঘ) 60 years
ব্যাখ্যা

Question: The sum of the ages of A, B, and C is 120 years. Ten years ago, the ratio of their ages was 2 : 3 : 4. What is the present age of A?

Solution:
Let, A, B, and C's ages 10 years ago be 2x, 3x, and 4x years respectively

ATQ,
(2x + 10) + (3x + 10) + (4x + 10) = 120
⇒ 9x  + 30 = 120
⇒ 9x = 90
⇒ x = 10

So, the age of A = 2 × 10 + 10 = 30

১১,৫৯৭.
A father is 30 years older than his son. In 10 years, the father will be twice the son’s age. Find the present age of father.
  1. ক
    40
  2. খ
    45
  3. গ
    50
  4. ঘ
    55
ব্যাখ্যা

Question: A father is 30 years older than his son. In 10 years, the father will be twice the son’s age. Find the present age of father.

Solution:
Let the present age of the son be x years.
Then the present age of the father = x + 30 years.

In 10 years,
Son’s age = x + 10
Father’s age = x + 30 + 10 = x + 40

According to the problem, father’s age will be twice the son’s age,
x + 40 = 2(x + 10)
⇒ x + 40 = 2x + 20
⇒ 2x - x = 40 - 20
⇒ x = 20

∴ Present ages of father = x + 30 = 20 + 30 = 50

১১,৫৯৮.
If the average of three consecutive even numbers is 34, find the smallest of these numbers.
  1. ক
    30
  2. খ
    32
  3. গ
    34
  4. ঘ
    28
ব্যাখ্যা
Question: If the average of three consecutive even numbers is 34, find the smallest of these numbers.

Solution:
Let the first number is x, then the next two even numbers would be x + 2, x + 4

As per question;
(x + x + 2 + x + 4)/3 = 34
⇒ (3x + 6)/3 = 34
⇒ 3x + 6 = 102
⇒ 3x = 96
∴ x = 32

Smallest number would be = 32
১১,৫৯৯.
হাসান একটি পণ্য বিক্রি করে ২০% লাভ করে। যদি সে উক্ত পণ্যটি ১০% কম দামে কিনত এবং ৪০% লাভে বিক্রি করত, তাহলে সে ২৪ টাকা বেশি পেত। তাঁর প্রকৃত বিক্রয়মূল্য কত ছিল?
  1. ক
    ৪২০ টাকা
  2. খ
    ৪৪০ টাকা
  3. গ
    ৪৮০ টাকা
  4. ঘ
    ৫০০ টাকা
  5. ঙ
    কোনটিই নয়
ব্যাখ্যা

প্রশ্ন: হাসান একটি পণ্য বিক্রি করে ২০% লাভ করে। যদি সে উক্ত পণ্যটি ১০% কম দামে কিনত এবং ৪০% লাভে বিক্রি করত, তাহলে সে ২৪ টাকা বেশি পেত। তাঁর প্রকৃত বিক্রয়মূল্য কত ছিল?

সমাধান:
ধরি, প্রকৃত ক্রয়মূল্য = x টাকা
∴ ২০% লাভে বিক্রয়মূল্য = x + x এর ২০%
= x + ২০x/১০০
= ১২০x/১০০
= ১২x/১০
= ৬x/৫

নতুন ক্রয়মূল্য = x − x এর ১০%
= x − ১০x/১০০
= ৯০x/১০০
= ৯x/১০

৪০% লাভে নতুন বিক্রয়মূল্য = (৯x/১০) + (৯x/১০) এর ৪০%
= (৯x/১০) + (৩৬x/১০০)
= (৯০x + ৩৬x)/১০০
= ১২৬x/১০০
= ৬৩x/৫০

প্রশ্নমতে,
নতুন বিক্রয়মূল্য − পূর্ব বিক্রয়মূল্য = ২৪
⇒ ৬৩x/৫০ − ৬x/৫ = ২৪
⇒ (৬৩x - ৬০x)/৫০ = ২৪
⇒ ৩x/৫০ = ২৪
⇒ ৩x = ২৪ × ৫০
⇒ x = (২৪ × ৫০)/৩
∴ x = ৪০০

∴ প্রকৃত বিক্রয়মূল্য = ৬x/৫ = (৬×৪০০)/৫
= ৪৮০ টাকা

১১,৬০০.
The present age of Kajol and Rohit are in the ratio of 7 : 8 respectively. After 6 years, the respective ratio between the age of Kajol and Rohit will be 9 : 10. What is the age of Rohit after 5 years?
  1. ক
    ক) 22 years
  2. খ
    খ) 24 years
  3. গ
    গ) 29 years
  4. ঘ
    ঘ) 34 years
ব্যাখ্যা
Let the present age of Kajol and Rohit is 7x and 8x respectively.
After 6 years, the age of Kajol = 7x + 6
After 6 years, the age of Rohit = 8x + 6
The ratio of age of Kajol and Rohit after 6 years is 9 : 10.
⇒ (7x + 6)/(8x + 6) = 9/10
⇒ 70x + 60 = 72x + 54
⇒ 2x = 6
⇒ x = 3

The age of Rohit after 5 years = 8 × 3 + 5 = 24 + 5 = 29 years

∴The age of Rohit after 5 years is 29 years.