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Bank Math

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

Bank Math

PrepBank · পাতা ৪৫ / ১৬১ · ৪,৪০১–৪,৫০০ / ১৬,১২৪

৪,৪০১.
How many numbers between 1 to 100 which are divisible by 3 or 5?
  1. ক
    ক) 53
  2. খ
    খ) 47
  3. গ
    গ) 49
  4. ঘ
    ঘ) 51
সঠিক উত্তর: খ
খ) 47
উত্তর
সঠিক উত্তর: খ
খ) 47
খ
ব্যাখ্যা
Question: How many numbers between 1 to 100 which are divisible by 3 or 5?

Solution:
The numbers which are divisible by 5 are: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100.
Total = 20

The numbers which are divisible by 3 are: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99
Total = 33

The numbers which are divisible by both 3 and 5 are: 15, 30, 45, 60, 75, 90
Total = 6

∴ Total number of numbers between 1 to 100 which are divisible by 3 or 5 = 20 + 33 - 6
= 53 - 6
= 47
৪,৪০২.
Rimon saves Tk. 3395 from his salary. He needs to pay this money as a milk bill, electricity bill and mobile phone bill in the ratio 42 : 32 : 23. Find the money to be paid for each bill.
  1. ক
    ক) Tk. 1245, Tk. 1150 and Tk 1000
  2. খ
    খ) Tk. 1470, Tk. 1120 and Tk. 805
  3. গ
    গ) Tk. 1550, Tk. 1235 and Tk. 610
  4. ঘ
    ঘ) Tk. 1764, Tk. 1022 and Tk. 529
সঠিক উত্তর: খ
খ) Tk. 1470, Tk. 1120 and Tk. 805
উত্তর
সঠিক উত্তর: খ
খ) Tk. 1470, Tk. 1120 and Tk. 805
খ
ব্যাখ্যা

Common factor helps in finding actual values easily
So, take 'A' as a common factor.
∴ 3 numbers will now be 42A, 32A and 23A
∴ 42A + 32A + 23 A = 3395
∴ 97A = 3395
∴ A = 35
3 parts of 3395 are
42A = 42 x 35 = 1470;
⇒ 32A = 32 x 35 = 1120
⇒ 23A = 23 x 35 = 805
These are the amounts to be paid.

৪,৪০৩.
Which phrase is the most similar in meaning to "pass the buck"?
  1. ক
    Face the consequences
  2. খ
    Accept responsibility
  3. গ
    Offer assistance
  4. ঘ
    Passing judgment
  5. ঙ
    None of the above
সঠিক উত্তর: ঙ
None of the above
উত্তর
সঠিক উত্তর: ঙ
None of the above
ঙ
ব্যাখ্যা
• The most similar in meaning to "pass the buck" - Shift the blame.
- অপশনে  "pass the buck" এর সামঞ্জস্যপূর্ণ অর্থ না থাকায়  "None of the above" উত্তর হবে।

• Pass the buck
English Meaning: to blame someone or make them responsible for a problem that others should deal with.
Bangla Meaning: অন্যায়ভাবে দোষারোপ করা / এক জনের দোষ অন্যের ঘাড়ে চাপানো।

Ex. Sentence: She's always trying to pass the buck and I'm sick of it!
Bangla Meaning: সে সময়ই অন্যায়ভাবে দোষারোপ করার চেষ্টা করে এবং আমি এটা নিয়ে বিরক্ত।

Source: Live MCQ Lecture.
৪,৪০৪.
The complementary angle of supplementary angle of 130°-
  1. ক
    50°
  2. খ
    30°
  3. গ
    40°
  4. ঘ
    60°
  5. ঙ
    70°
সঠিক উত্তর: গ
40°
উত্তর
সঠিক উত্তর: গ
40°
গ
ব্যাখ্যা
Question: The complementary angle of supplementary angle of 130°-

Solution:
For supplementary angle: The sum of two angles is 180°.
For complementary angle: The sum of two angles is 90°.

The supplement angle of 130° = 180° - 130° = 50°
The complement angle of 50° = 90° - 50° = 40°

∴ The complement angle of the supplement angle of 130° is 40°
৪,৪০৫.
The L.C.M of three different numbers is 120. Which of the following can't be their H.C.F? 
  1. ক
    8
  2. খ
    12
  3. গ
    24
  4. ঘ
    35
সঠিক উত্তর: ঘ
35
উত্তর
সঠিক উত্তর: ঘ
35
ঘ
ব্যাখ্যা
Question: The L.C.M of three different numbers is 120. Which of the following can't be their H.C.F? 

Solution: 
যেহেতু গ.সা.গু , ল.সা.গু র একটি উৎপাদক , তাই তিনটি সংখ্যার ল.সা.গু ১২০ হলে তাদের গ সা গু ৩৫ হবে না। 
কারণ, ১২০, ৩৫ দ্বারা নিঃশেষে বিভাজ্য নয়।  
৪,৪০৬.
A bag contains 8 red marbles, 12 yellow marbles, and 4 purple marbles. What is the probability of drawing a yellow marble?
  1. ক
    1/2
  2. খ
    1/3
  3. গ
    2/3
  4. ঘ
    1/4
সঠিক উত্তর: ক
1/2
উত্তর
সঠিক উত্তর: ক
1/2
ক
ব্যাখ্যা

Question: A bag contains 8 red marbles, 12 yellow marbles, and 4 purple marbles. What is the probability of drawing a yellow marble?

Solution:
Total number of marbles = 8 + 12 + 4
= 24

And number of yellow marbles (favorable outcomes) = 12

Probability of drawing a yellow marble = (Number of yellow marbles)/(Total number of marbles)
= 12/24
= 1/2

So the probability is 1/2.

৪,৪০৭.
What value of x satisfies to equation x - 2 = 2 - x?
  1. ক
    ক) 2
  2. খ
    খ) 1
  3. গ
    গ) 0
  4. ঘ
    ঘ) - 2
সঠিক উত্তর: ক
ক) 2
উত্তর
সঠিক উত্তর: ক
ক) 2
ক
ব্যাখ্যা
Question: What value of x satisfies to equation x - 2= 2 - x?

Solution: 
x - 2 = 2 - x
⇒ x + x = 2 + 2
⇒  2x = 4
∴ x = 2
৪,৪০৮.
What number should come next in the series:
5, 8, 14, 26, 50, .......?
  1. ক
    88
  2. খ
    92
  3. গ
    98
  4. ঘ
    102
  5. ঙ
    110
সঠিক উত্তর: গ
98
উত্তর
সঠিক উত্তর: গ
98
গ
ব্যাখ্যা

Question: What number should come next in the series:
5, 8, 14, 26, 50, .......?

Solution:
দেওয়া আছে, সিরিজটি হলো: 5, 8, 14, 26, 50, ...

পার্থক্যগুলোর মধ্যে একটি প্যাটার্ন রয়েছে। প্রতিটি পার্থক্য আগের পার্থক্যের 2 গুণ।

5 থেকে 8 পর্যন্ত পার্থক্য: 3
8 থেকে 14 পর্যন্ত পার্থক্য: 6 (3 × 2)
14 থেকে 26 পর্যন্ত পার্থক্য: 12 (6 × 2)
26 থেকে 50 পর্যন্ত পার্থক্য: 24 (12 × 2)
সুতরাং, পরবর্তী পার্থক্যটি হবে: 24 × 2 = 48

পরবর্তী সংখ্যাটি হবে শেষ সংখ্যা এবং এই পার্থক্যের যোগফল: 50 + 48 = 98

অতএব, পরবর্তী সংখ্যাটি হলো 98

৪,৪০৯.
The sum of the squares of three numbers is 1197 and the ratio of the first and the second as also of the second and the third is 3 : 2 . The third number is-
  1. ক
    10
  2. খ
    12
  3. গ
    15
  4. ঘ
    9
সঠিক উত্তর: খ
12
উত্তর
সঠিক উত্তর: খ
12
খ
ব্যাখ্যা
Question: The sum of the squares of three numbers is 1197 and the ratio of the first and the second as also of the second and the third is 3 : 2 . The third number is-

Solution:
Given,
First : Second = 3 : 2 = 9 : 6
Second : Third = 3 : 2 = 6 : 4

∴ First : Second : Third = 9 : 6 : 4

Let,
the number be 9x, 6x and 4x

ATQ,
(9x)2 + (6x)2 + (4x)2 = 1197
⇒ 81x2 + 36x2 + 16x2 = 1197
⇒ 133x2 = 1197
⇒ x2 = 9
∴ x = 3

So the third number = 4 × 3 = 12
৪,৪১০.
The difference between a two-digit number and the number obtained by interchanging the positions of its digits is 36. What is the difference between the two digits of that number?
  1. ক
    6
  2. খ
    4
  3. গ
    10
  4. ঘ
    8
সঠিক উত্তর: খ
4
উত্তর
সঠিক উত্তর: খ
4
খ
ব্যাখ্যা
Question: The difference between a two-digit number and the number obtained by interchanging the positions of its digits is 36. What is the difference between the two digits of that number?

Solution:
Let the ten's digit be x and the unit's digit be y.
So, the number = 10x + y

After interchanging the positions of the number's digits, the number will be = 10y + x

Then, (10x + y) - (10y + x) = 36
⇒ 9(x - y) = 36
⇒ x - y = 4
৪,৪১১.
The difference between the selling price and cost price of an article is Tk. 210. If the profit percent is 25, then the selling price of the article is =?
  1. ক
    Tk. 950
  2. খ
    Tk. 1050
  3. গ
    Tk. 2000
  4. ঘ
    Tk. 2250
সঠিক উত্তর: খ
Tk. 1050
উত্তর
সঠিক উত্তর: খ
Tk. 1050
খ
ব্যাখ্যা
Question: The difference between the selling price and cost price of an article is Tk. 210. If the profit percent is 25, then the selling price of the article is =?

Solution: 
Let, selling price of the article is x taka
cost price x - 210

ATQ, 
210/(x - 210) = 25/100
⇒ 210/(x - 210) = 1/4
⇒ x - 210 = 840 
∴ x = 1050 taka 
৪,৪১২.
P can build a wall in 24 days and Q can do it in 20 days. With help of R, they completed the work in 6 days. Find in how many days R alone can do the work.
  1. ক
    29/4 days
  2. খ
    40/3 days
  3. গ
    44/5 days
  4. ঘ
    56/3 days
সঠিক উত্তর: খ
40/3 days
উত্তর
সঠিক উত্তর: খ
40/3 days
খ
ব্যাখ্যা
Question: P can build a wall in 24 days and Q can do it in 20 days. With help of R, they completed the work in 6 days. Find in how many days R alone can do the work.

Solution:
P can do 1/24 part per day
Q can do 1/20 part per day
R can do per day = (1/6) - {(1/24) + (1/20)}
= (1/6) - (11/120)
= (20/120) - (11/120)
= 3/40
Time taken by R alone = 40/3 days
৪,৪১৩.
Rina and Shila entered into a partnership with their capitals in the ratio 5 : 6 At the end of 8 months, Rina withdraws her capital. If they receive the profit in the ratio of 5 : 9, find how long Shila's capital was used?
  1. ক
    ক) 12 months
  2. খ
    খ) 14 months
  3. গ
    গ) 16 months
  4. ঘ
    ঘ) 18 months
সঠিক উত্তর: ক
ক) 12 months
উত্তর
সঠিক উত্তর: ক
ক) 12 months
ক
ব্যাখ্যা

Question: Rina and Shila entered into a partnership with their capitals in the ratio 5 : 6 At the end of 8 months, Rina withdraws her capital. If they receive the profit in the ratio of 5 : 9, find how long Shila's capital was used?

Solution: 
Shila মূলধন x সময়ের জন্য খেটেছিলো 

প্রশ্নমতে,
(5 × 8)/(6 × x) = 5/9
30x = 360
x = 360/30
x = 12

৪,৪১৪.
A contractor undertook to complete a road project in 40 days. However, on the first day, 20 workers were absent. The remaining workers finished the project in 60 days. How many workers were originally supposed to work on the project?
  1. ক
    36
  2. খ
    50
  3. গ
    60
  4. ঘ
    80
সঠিক উত্তর: গ
60
উত্তর
সঠিক উত্তর: গ
60
গ
ব্যাখ্যা

Question: A contractor undertook to complete a road project in 40 days. However, on the first day, 20 workers were absent. The remaining workers finished the project in 60 days. How many workers were originally supposed to work on the project?

Solution:
ধরি, শুরুতে শ্রমিকের সংখ্যা ছিল = x জন
মোট কাজ = x × 40 

যেহেতু 20 জন শ্রমিক অনুপস্থিত ছিল, তাই বর্তমান শ্রমিক সংখ্যা = (x - 20) জন।
তারা কাজটি শেষ করে 60 দিনে।
∴ বর্তমান মোট কাজ = (x - 20) × 60

শর্তমতে,
40x = 60(x - 20)
⇒ 40x = 60x - 1200
⇒ 60x - 40x = 1200
⇒ 20x = 1200
⇒ x = 1200/20
⇒ x = 60

∴ শুরুতে 60 জন শ্রমিক নিয়োগ করা হয়েছিল।

৪,৪১৫.
If the length of each side of an equilateral triangle is increased by 2 meters, the area is found to be increased by 3 + √3 square meters. The length of each side of the triangle is:
  1. ক
    3 meters
  2. খ
    3√2 meters
  3. গ
    √3 meters
  4. ঘ
    5√3 meters
  5. ঙ
    None
সঠিক উত্তর: গ
√3 meters
উত্তর
সঠিক উত্তর: গ
√3 meters
গ
ব্যাখ্যা
Question: If the length of each side of an equilateral triangle is increased by 2 meters, the area is found to be increased by 3 + √3 square meters. The length of each side of the triangle is:

Solution:
Let,
the length of each side of the equilateral triangle = a meters
∴ Its area = √3/4 × a2 sq. meter

The area of the triangle when the length of each side increases by 2 meters = √3/4 × (a + 2)2 sq. metre

ATQ,
৪,৪১৬.
A sold a watch to B at a gain of 20% and B sold it to C at loss of 10%. If C bought the watch for TK.216 at what price did A purchase it?
  1. ক
    Tk. 180 
  2. খ
    Tk. 200 
  3. গ
    Tk. 210 
  4. ঘ
    Tk. 220 
সঠিক উত্তর: খ
Tk. 200 
উত্তর
সঠিক উত্তর: খ
Tk. 200 
খ
ব্যাখ্যা
Question: A sold a watch to B at a gain of 20% and B sold it to C at loss of 10%. If C bought the watch for TK.216 at what price did A purchase it?

Solution:
A purchase at x taka
selling price of A = purchasing price of B = x + 20% of x = 1.2x 
buying price of C = 1.2x - 10% of 1.2x = 1.2x - 0.12x = 1.08x 

1.08x = 216 
⇒ x = 216/1.08 = Tk. 200 
৪,৪১৭.
At the end of a banquet 20 people shake hands with each other. How many handshakes will there be in total?
  1. ক
    ক) 171
  2. খ
    খ) 190
  3. গ
    গ) 160
  4. ঘ
    ঘ) 180
সঠিক উত্তর: খ
খ) 190
উত্তর
সঠিক উত্তর: খ
খ) 190
খ
ব্যাখ্যা
Question: At the end of a banquet 20 people shake hands with each other. How many handshakes will there be in total?

Solution:
Two people can make a one handshake.
Number of handshakes = 20C2 = 190
৪,৪১৮.
From a container, full of pure milk, 20% is replaced by water and this process is repeated two times. At the end of second operation, the quantity of pure milk reduces to-
  1. ক
    38%
  2. খ
    45%
  3. গ
    55%
  4. ঘ
    64%
সঠিক উত্তর: ঘ
64%
উত্তর
সঠিক উত্তর: ঘ
64%
ঘ
ব্যাখ্যা
Question: From a container, full of pure milk, 20% is replaced by water and this process is repeated two times. At the end of second operation, the quantity of pure milk reduces to-

Solution: 
ধরি, শুরুতে দুধের পরিমাণ ১০০ লিটার 

২০% পানি দ্বারা প্রতিস্থাপন করলে, বিশুদ্ধ দুধের পরিমাণ = ১০০ - ২০ লিটার 
= ৮০ লিটার 

দ্বিতীয়বার পুনরায় ২০% দ্বারা পানি প্রতিস্থাপন করলে, বিশুদ্ধ দুধের পরিমাণ = ৮০ - ৮০ এর ২০%
= ৮০ - ১৬
= ৬৪ লিটার

অতএব, ৬৪% বিশুদ্ধ দুধ থাকে।
৪,৪১৯.
A map has a scale of 1 cm to 3 km, what length on actual ground does a 3 cm length on the map represent?
  1. ক
    ক) 9 km
  2. খ
    খ) 1 km
  3. গ
    গ) 6 km
  4. ঘ
    ঘ) 6 cm
  5. ঙ
    ঙ) 1 cm
সঠিক উত্তর: ক
ক) 9 km
উত্তর
সঠিক উত্তর: ক
ক) 9 km
ক
ব্যাখ্যা
Question: A map has a scale of 1 cm to 3 km, what length on actual ground does a 3 cm length on the map represent?

Solution: 
Given that,
1 cm represents 3 km 
∴ 3 cm represent (3 × 3) km 
= 9 km
৪,৪২০.
Determine sin(A + B) for A = π/2 and B = π/4.
  1. ক
    1/√3
  2. খ
    1/4
  3. গ
    1/√2
  4. ঘ
    1/2
সঠিক উত্তর: গ
1/√2
উত্তর
সঠিক উত্তর: গ
1/√2
গ
ব্যাখ্যা
Question: Determine sin(A + B) for A = π/2 and B = π/4.

Solution:
A = π/2
B = π/4

sin(A + B) = sin(π/2 + π/4)
= sin(90° + 45°)
= cos 45°
= 1/√2
৪,৪২১.
If 2x = 3y = 10, then 12xy =?
  1. ক
    1200
  2. খ
    200
  3. গ
    120
  4. ঘ
    40
সঠিক উত্তর: খ
200
উত্তর
সঠিক উত্তর: খ
200
খ
ব্যাখ্যা
Question: If 2x = 3y = 10, then 12xy =?

Solution:
2x = 3y = 10

12xy 
= 2x × 6y
= 10 × 2 × 3y
= 20 × 10
= 200
৪,৪২২.
Simi is walking at a speed of 10 kmph. After every km she takes rests for 5 minutes. The taken to cover a distance of 5 kilometers by Simi is -
  1. ক
    ক) 55 minutes
  2. খ
    খ) 35 minutes
  3. গ
    গ) 50 minutes
  4. ঘ
    ঘ) 75 minutes
সঠিক উত্তর: গ
গ) 50 minutes
উত্তর
সঠিক উত্তর: গ
গ) 50 minutes
গ
ব্যাখ্যা
Question: Simi is walking at a speed of 10 kmph. After every km she takes rests for 5 minutes. The taken to cover a distance of 5 kilometers by Simi is -

Solution:
Time taken to 5 km at 10 kmph = 5/10 hr = 1/2 hr = 30 minutes
In 5 km she will rest 4 times
Time taken to rest = 4 × 5 = 20 minutes

So, total time taken = 30 + 20 = 50 minutes
৪,৪২৩.
A group of 8 people has their average weight increased by 2.5 kg after replacing a 65 kg person with someone new. What is the possible weight of the replacement?
  1. ক
    70 Kg
  2. খ
    78 Kg
  3. গ
    82 Kg
  4. ঘ
    85 Kg
সঠিক উত্তর: ঘ
85 Kg
উত্তর
সঠিক উত্তর: ঘ
85 Kg
ঘ
ব্যাখ্যা
Question: A group of 8 people has their average weight increased by 2.5 kg after replacing a 65 kg person with someone new. What is the possible weight of the replacement?

Solution:
ধরি,
 8 জন ব্যাক্তির গড় ওজন = x কেজি
তাহলে, মোট ওজন = 8x
65 কেজি ওজনের ব্যক্তি চলে যাওয়ার পর মোট ওজন= (8x - 65) কেজি

আবার,
মনে করি, নতুন ব্যাক্তির ওজন= y kg

তাহলে নতুন ব্যক্তি প্রবেশ করার পর নতুন মোট ওজন হবে = (8x - 65 + y) কেজি
এবং নতুন গড় = x + 2.5

প্রশ্নমতে,
(8x - 65 + y)/8 = x + 2.5
⇒ 8x - 65 + y = 8x + 20
⇒ 8x + y - 8x = 20 + 65
⇒ y = 85

সুতরাং, নতুন ব্যাক্তির ওজন =  85 কেজি।
৪,৪২৪.
The average temperature for the first 4 days of a week is 38.2°C and that of the last 4 days is 39.3°C. If the average temperature for the whole week is 38.6°C, then the temperature on the fourth day is-
  1. ক
    36.5°C
  2. খ
    38.8°C
  3. গ
    39.8°C
  4. ঘ
    39.3°C
সঠিক উত্তর: গ
39.8°C
উত্তর
সঠিক উত্তর: গ
39.8°C
গ
ব্যাখ্যা
Question: The average temperature for the first 4 days of a week is 38.2°C and that of the last 4 days is 39.3°C. If the average temperature for the whole week is 38.6°C, then the temperature on the fourth day is-

Solution:
Temperature on the fourth day
= [(38.2 × 4 + 39.3 × 4) - (38.6 × 7)]° C
= 39.8° C
৪,৪২৫.
The length of the tangent drawn from a point P to a circle of radius 6 cm is 8 cm. What is the distance of P from the center of the circle?
  1. ক
    8 cm
  2. খ
    10 cm
  3. গ
    12 cm
  4. ঘ
    14 cm
সঠিক উত্তর: খ
10 cm
উত্তর
সঠিক উত্তর: খ
10 cm
খ
ব্যাখ্যা
Question: The length of the tangent drawn from a point P to a circle of radius 6 cm is 8 cm. What is the distance of P from the center of the circle?

Solution:

O be the center of the circle.
P be the point from which the tangent is drawn.
PT be the tangent from P to the circle, where T is the point of tangency.

The radius OT = 6 cm.
The length of the tangent PT = 8 cm.
Since OT is the radius and PT is a tangent, OT ⊥ PT, forming a right triangle △OPT.
OP2 = OT2 + PT2
⇒ OP2 = 62 + 82
⇒ OP2 = 36 + 64
⇒ OP2 = 100
⇒ OP = 10 cm
৪,৪২৬.
A sum of money at simple interest becomes Tk. 3000 in 2 years and Tk. 3540 in 5 years. Find the approximate rate of interest.
  1. ক
    4.75%
  2. খ
    4.5%
  3. গ
    6%
  4. ঘ
    6.8%
সঠিক উত্তর: ঘ
6.8%
উত্তর
সঠিক উত্তর: ঘ
6.8%
ঘ
ব্যাখ্যা
Question: A sum of money at simple interest becomes Tk. 3000 in 2 years and Tk. 3540 in 5 years. Find the approximate rate of interest.

Solution:
After five years the amount is 3540
After two years the amount is 3000
∴ Interest for three years 540
∴ Interest for 1 years 540/3
∴ Interest for 2 years (540 × 2)/3
= 360

So, we have:
P = 3000 - 360 = 2640
Interest, I = 360
Number of years, n = 2 years

Rate of interest, r = I/(Pn)
= (360× 100)/(2640 × 2)
= 6.8
৪,৪২৭.
If 10 men or 18 women can do a work in 50 days then how many days would it take 25 men and 15 women to do twice the work?
  1. ক
    25 days
  2. খ
    30 days
  3. গ
    32 days
  4. ঘ
    40 days
সঠিক উত্তর: খ
30 days
উত্তর
সঠিক উত্তর: খ
30 days
খ
ব্যাখ্যা
Question: If 10 men or 18 women can do a work in 50 days then how many days would it take 25 men and 15 women to do twice the work?

Solution:
Here,
10 men = 18 women
∴ 1 men = 18/10 women
∴ 25 men = (18 × 25)/10 women
= 45 women

∴ 25 men and 15 women = (45 + 15) = 60 women

ATQ,
18 women can do the work in 50 days
∴ 1 women can do the work in = (50 × 18) days
∴ 60 women can do the work in (50 × 18)/60 days
= 15 days

Hence to do twice the work time need = (15 × 2) = 30 days
৪,৪২৮.
If 
  1. ক
    8
  2. খ
    12
  3. গ
    16
  4. ঘ
    14
সঠিক উত্তর: ক
8
উত্তর
সঠিক উত্তর: ক
8
ক
ব্যাখ্যা

Question: If 

Solution:

৪,৪২৯.
The average of 8 numbers is 8. If 4 is subtracted from each of 6 of the numbers, what is the new average?
  1. ক
    3.5
  2. খ
    5
  3. গ
    4
  4. ঘ
    6.5
সঠিক উত্তর: খ
5
উত্তর
সঠিক উত্তর: খ
5
খ
ব্যাখ্যা

Question: The average of 8 numbers is 8. If 4 is subtracted from each of 6 of the numbers, what is the new average?

Solution:
Given that, 
The average of 8 numbers = 8
∴ Sum of the 8 numbers = (8 × 8) = 64

If 4 is subtracted from each of 6 of these numbers, then the new sum becomes,
= 64 - (6 × 4)
= 64 - 24
= 40

Therefore, the new average of the 8 numbers will be,
= 40/8
= 5

So the new average is 5.

৪,৪৩০.
If A = {1, 4, 9, 16, 25}, the number of proper subsets of A is
  1. ক
    15
  2. খ
    16
  3. গ
    31
  4. ঘ
    32
সঠিক উত্তর: গ
31
উত্তর
সঠিক উত্তর: গ
31
গ
ব্যাখ্যা
Question: If A = {1, 4, 9, 16, 25}, the number of proper subsets of A is

Solution:
দেওয়া আছে,
A = {1, 4, 9, 16, 25}

সেটের উপাদান সংখ্যা = 5

∴ প্রকৃত উপসেট সংখ্যা = 2n - 1
= 25 - 1
= 32 - 1
= 31
৪,৪৩১.
During a certain season, a team won 80 percent of its first 100 games and 50 percent of its remaining games. If the team won 70 percent of its games for the entire season, what was the total number of games that the team played?
  1. ক
    ক) 180
  2. খ
    খ) 170
  3. গ
    গ) 156
  4. ঘ
    ঘ) 150
সঠিক উত্তর: ঘ
ঘ) 150
উত্তর
সঠিক উত্তর: ঘ
ঘ) 150
ঘ
ব্যাখ্যা

Let the number of the remaining games be x
then,
0.8×100 + 0.5×x = 0.7×(100+x)
⇒ 80 - 70 = 0.7x - 0.5x = 0.2x
⇒ x = 50
∴ total number of games thus equal to = 100+x
= 100+50
= 150

৪,৪৩২.
Peter is twice as good as workman as Tom. When they work together they can finish a task in 16 days. If Tom works alone, in how many days he will complete the task?
  1. ক
    46 days
  2. খ
    48 days
  3. গ
    50 days
  4. ঘ
    52 days
সঠিক উত্তর: খ
48 days
উত্তর
সঠিক উত্তর: খ
48 days
খ
ব্যাখ্যা
Question: Peter is twice as good as workman as Tom. When they work together they can finish a task in 16 days. If Tom works alone, in how many days he will complete the task?

Solution:
(Peter + Tom)'s one-day work = 1/16
As per the question, Peter can finish twice as much work as finished by Tom in a given duration of time.

Therefore, 2/3 of their one day's work will be completed by Peter and 1/3 of their one day work will be completed by Tom.

So, Tom's one day work will be = (1/16) × (1/3) = 1/48
So, Tom will take 48 days to complete the task.
৪,৪৩৩.
The angle of depression of a point situated at a distance of 70 m from the base of a tower is 60°. The height of the tower is -
  1. ক
    65√3 meter
  2. খ
    55√3 meter
  3. গ
    60√3 meter
  4. ঘ
    70√3 meter
সঠিক উত্তর: ঘ
70√3 meter
উত্তর
সঠিক উত্তর: ঘ
70√3 meter
ঘ
ব্যাখ্যা
Question: The angle of depression of a point situated at a distance of 70 m from the base of a tower is 60°. The height of the tower is -

Solution:

Let,
Height of the tower AB = h meter.
Now, ∠DAC = ∠ACB = 60° and BC = 70 meter.

In ΔABC 
tan60° = AB/BC
⇒ √3 = h/70
∴ h = 70√3

Height of the tower 70√3 meter.
৪,৪৩৪.
The average of a positive natural number and its cube is 13 times the number. The number is - 
  1. ক
    6
  2. খ
    5
  3. গ
    8
  4. ঘ
    10
সঠিক উত্তর: খ
5
উত্তর
সঠিক উত্তর: খ
5
খ
ব্যাখ্যা

Question: The average of a positive natural number and its cube is 13 times the number. The number is -

Solution:
Let the number be a positive natural number = x (x ≠ 0).

ATQ,
The average of the number and its cube is 13 times the number.
⇒ (x + x3)/2 = 13x
⇒ x + x3 = 26x
⇒ x3 + x - 26x = 0
⇒ x3 - 25x = 0
⇒ x(x2 - 25) = 0
⇒ x(x - 5)(x + 5) = 0

So, x = 0 or x = 5 or x = - 5
Therefore, the positive natural number is 5.

৪,৪৩৫.
By interchanging the digits of a two digit number we get a number which is four times the original number minus 24. If the unit’s digit of the original number exceeds its ten’s digit by 7, then original number is?
  1. ক
    22
  2. খ
    29
  3. গ
    31
  4. ঘ
    36
সঠিক উত্তর: খ
29
উত্তর
সঠিক উত্তর: খ
29
খ
ব্যাখ্যা
Question: By interchanging the digits of a two digit number we get a number which is four times the original number minus 24. If the unit’s digit of the original number exceeds its ten’s digit by 7, then original number is?

Solution:
Let, the two–digit number be = (10x + y) where x < y.
Number obtained on reversing the digits = (10y + x)

ATQ,
10y + x = 4 (10x + y) - 24
⇒ 40x + 4y - 10y - x = 24
⇒ 39x - 6y = 24
⇒ 13x - 2y = 8 ....(i)

Again, y - x = 7
y = x + 7 ....(ii)
⇒ 13x - 2 (x + 7) = 8
⇒ 13x - 2x - 14 = 8
⇒ 11x = 14 + 8 = 22
⇒ x = 22/11
∴ x = 2

From equation (ii),
y - 2 = 7
⇒ y = 2 + 7
∴ y = 9

So, Number = 10x + y = (10 × 2) + 9 = 29
৪,৪৩৬.
For which of the following values of n is (100 + n)/n NOT an integer?
  1. ক
    1
  2. খ
    2
  3. গ
    3
  4. ঘ
    4
  5. ঙ
    5
সঠিক উত্তর: গ
3
উত্তর
সঠিক উত্তর: গ
3
গ
ব্যাখ্যা
Question: For which of the following values of n is (100 + n)/n NOT an integer?

Solution:
(100 + n)/n = 100/n + n/n = 100/n + 1

Since 1 is already an integer, we can see that 100/n + 1 will be an integer whenever 100/n is an integer.
In other words, 100/n + 1 is an integer whenever 100 is divisible by n.

Conversely, 100/n + 1 is NOT an integer whenever 100 is NOT divisible by n.
Since 100 is NOT divisible by 3, the correct answer is C
৪,৪৩৭.
The angle of elevation of a ladder leaning against a wall is 60° and the foot of the ladder is 4.5 m away from the wall. The length of the ladder is:
  1. ক
    4.5 m
  2. খ
    8 m
  3. গ
    9 m
  4. ঘ
    11 m
সঠিক উত্তর: গ
9 m
উত্তর
সঠিক উত্তর: গ
9 m
গ
ব্যাখ্যা
Question: The angle of elevation of a ladder leaning against a wall is 60° and the foot of the ladder is 4.5 m away from the wall. The length of the ladder is:

Solution:
Let,
AB be the wall and BC be the ladder.
Then, ∠ACB = 60° and AC = 4.5 m.

Here, 
AC/BC = cos60°
⇒ AC/BC = 1/2 
⇒ BC = 2 × AC 
⇒ BC = 2 × 4.5
∴ BC = 9 

∴ The length of the ladder is 9 m.


৪,৪৩৮.
A dealer buys an article marked at Tk. 25000 with 20% and 5% off. He spends Tk. 1000 for its repairs and sells it for Tk. 25000. What is his profit or loss percent?
  1. ক
    ক) loss of 20%
  2. খ
    খ) loss of 25%
  3. গ
    গ) profit of 25%
  4. ঘ
    ঘ) profit of 15%
সঠিক উত্তর: গ
গ) profit of 25%
উত্তর
সঠিক উত্তর: গ
গ) profit of 25%
গ
ব্যাখ্যা
Question: A dealer buys an article marked at Tk. 25000 with 20% and 5% off. He spends Tk. 1000 for its repairs and sells it for Tk. 25000. What is his proft or loss percent?

Solution:
Given,
Marked Price = 25000.

After first discount it become,
= 25000 - 20% of 25000
= Tk. 20000.

After second discount, it becomes
= 20000 - 5% of 20000
= Tk. 19000.

So, Selling price = Tk. 19000.

Cost price for the man who bought it, as he spends tk. 1000 on repair.

∴ Cost price = (19000 + 1000) = Tk. 20000

∴ Profit = 25000 - 20000 = Tk. 5000.
% Profit = (5000 × 100)/20000
= 25%
৪,৪৩৯.
The sum of three consecutive even integers is 156. If the product of the smallest and largest integers is 2700, what is the middle integer?
  1. ক
    50
  2. খ
    48
  3. গ
    54
  4. ঘ
    65
  5. ঙ
    52
সঠিক উত্তর: ঙ
52
উত্তর
সঠিক উত্তর: ঙ
52
ঙ
ব্যাখ্যা
Question: The sum of three consecutive even integers is 156. If the product of the smallest and largest integers is 2700, what is the middle integer?

Solution:
Let the three consecutive even integers are, n - 2, n and n + 2, where n is the middle integer and even.

And given
sum of the integers is 156

ATQ
(n - 2) + n + (n + 2) = 156
⇒ 3n = 156
⇒ n = 156/3
∴ n = 52

So, the integers are 52 - 2 = 50, 52 and 52 + 2 = 54

Now, we verify the 2nd condition, The product of the smallest (50) and largest (54) integers.
50 × 54 = 2700

This satisfies the product condition exactly.
∴ The middle integer is n = 52.
 
৪,৪৪০.
The value of √0.000529 is
  1. ক
    ক) 2.03
  2. খ
    খ) 0.0023
  3. গ
    গ) 0.23
  4. ঘ
    ঘ) 0.023
সঠিক উত্তর: ঘ
ঘ) 0.023
উত্তর
সঠিক উত্তর: ঘ
ঘ) 0.023
ঘ
ব্যাখ্যা
√0.000529 =√{529/1000000} 
                  = √{232/106}
                  = 23/103
                  = 23/1000
                   = 0.023
৪,৪৪১.
The average marks of a student in 5 subjects is 75. Later, marks in one subject was increased by 12 and marks in another subject was decreased by 17. Find the corrected average of marks.
  1. ক
    ক) 74
  2. খ
    খ) 72
  3. গ
    গ) 71
  4. ঘ
    ঘ) 69
সঠিক উত্তর: ক
ক) 74
উত্তর
সঠিক উত্তর: ক
ক) 74
ক
ব্যাখ্যা
Sum of all numbers = 75 × 5 = 375
Later, marks in one subject was increased by 12 and marks in another subject was decreased by 17.
Corrected marks = 375 + 12 – 17
                       = 370
Corrected Average = Correct sum / number of subjects
                           = 370/5
                           = 74
∴ Corrected average is 74
৪,৪৪২.
Find the initial sum that grows to Tk. 1500 in 5 years and Tk. 1620 in 7 years at simple interest.
  1. ক
    1100 Tk.
  2. খ
    1285 Tk.
  3. গ
    1200 Tk.
  4. ঘ
    1160 Tk.
সঠিক উত্তর: গ
1200 Tk.
উত্তর
সঠিক উত্তর: গ
1200 Tk.
গ
ব্যাখ্যা

Question: Find the initial sum that grows to Tk. 1500 in 5 years and Tk. 1620 in 7 years at simple interest.

Solution:
in two years the interest becomes = 1620 - 1500 = 120Tk.
in 5 years, interest = (120/2)/5 =300 Tk.

so , the initial sum is = 1500 - 300 = 1200 Tk.

৪,৪৪৩.
A boat makes a return journey from point A to point B and back in 5 hours 36 minutes. One way it travels with the stream and on the return it travels against the stream. If the speed of the stream increases by 2 km/hr, the return journey takes 9 hours 20 minutes. What is the speed of the boat in still water? (The distance between A and B is 16 km.)
  1. ক
    10 km/hr
  2. খ
    9 km/hr
  3. গ
    3 km/hr
  4. ঘ
    5 km/hr
  5. ঙ
    7 km/hr
সঠিক উত্তর: ঙ
7 km/hr
উত্তর
সঠিক উত্তর: ঙ
7 km/hr
ঙ
ব্যাখ্যা
Question: A boat makes a return journey from point A to point B and back in 5 hours 36 minutes. One way it travels with the stream and on the return it travels against the stream. If the speed of the stream increases by 2 km/hr, the return journey takes 9 hours 20 minutes. What is the speed of the boat in still water? (The distance between A and B is 16 km.)

Solution:
Let x be speed of u/s
and y be the speed of d/s.

∴ (16/x) + (16/y) = (28/5)
and 16/(y+2) + 16/(x-2) = 28/3

Solving these 2 equations,
we get x = 4km/hr
and y = 10km/hr

∴ speed of boat in still water = (4 + 10)/2 = 7 km/hr.
৪,৪৪৪.
If θ = 45°, what is the value of sec2θ - tan2θ?
  1. ক
    - √2
  2. খ
    - 1
  3. গ
    √2
  4. ঘ
    1
সঠিক উত্তর: ঘ
1
উত্তর
সঠিক উত্তর: ঘ
1
ঘ
ব্যাখ্যা
Question: If θ = 45°, what is the value of sec2θ - tan2θ?

Solution:
θ = 45°

given,
sec2θ - tan2θ
= (√2)2 - (1)2
= 2 - 1
= 1
৪,৪৪৫.
The minimum value of 2sin2θ + 3cos2θ is ?
  1. ক
    0
  2. খ
    2
  3. গ
    3
  4. ঘ
    1
সঠিক উত্তর: খ
2
উত্তর
সঠিক উত্তর: খ
2
খ
ব্যাখ্যা

Question: The minimum value of 2sin2θ + 3cos2θ is ?

Solution:
Let, x = 2sin2θ + 3cos2θ
⇒ x = 2sin2θ + 2cos2θ + cos2θ
⇒ x = 2(sin2θ + cos2θ) + cos2θ
⇒ x = 2 + cos2θ     ;[since sin2θ + cos2θ = 1]

Therefore x will be the minimum when cosθ = 0. i.e. minimum value of x will 2

৪,৪৪৬.
If 4 workers can assemble a car in 6 hours, how long would it take 15 workers to assemble the same car?
  1. ক
    102 minutes
  2. খ
    80 minutes
  3. গ
    96 minutes
  4. ঘ
    72 minutes
সঠিক উত্তর: গ
96 minutes
উত্তর
সঠিক উত্তর: গ
96 minutes
গ
ব্যাখ্যা
Question: If 4 workers can assemble a car in 6 hours, how long would it take 15 workers to assemble the same car?

Solution:
Given,
4 workers can assemble a car in 6 hours
∴ 1 workers can assemble a car in (6 × 4) hours
∴ 15 workers can assemble a car in (24/15) hours
= (24 × 60)/15 minutes
= 96 minutes
৪,৪৪৭.
If logx(27/64) = - 3, then x = ?
  1. ক
    1/3
  2. খ
    4/3
  3. গ
    3/4
  4. ঘ
    2/3
সঠিক উত্তর: খ
4/3
উত্তর
সঠিক উত্তর: খ
4/3
খ
ব্যাখ্যা
Question: If logx (27/64) = - 3, then x = ?

Solution:
logx (27/64) = - 3
⇒ x- 3 = 27/64
⇒ x- 3 = (3/4)3
⇒ x- 3 = 1/(4/3)3
⇒ x- 3 = (4/3)- 3
⇒ x = 4/3
৪,৪৪৮.
If C = {a, b, 1, 2} and D = {3, 4, 6, 8} then C union D is-
  1. ক
    {a, b, 1, 2, 3, 4, 6, 8}
  2. খ
    {a, b}
  3. গ
    {1, 2, 3, 4, 6, 8}
  4. ঘ
    {a, 1, 2, 3, 4, 6}
সঠিক উত্তর: ক
{a, b, 1, 2, 3, 4, 6, 8}
উত্তর
সঠিক উত্তর: ক
{a, b, 1, 2, 3, 4, 6, 8}
ক
ব্যাখ্যা
Question: If C = {a, b, 1, 2} and D = {3, 4, 6, 8} then C union D is-

Solution:
C union D = C ∪ D = {a, b, 1, 2} ∪ {3, 4, 6, 8}
= {a, b, 1, 2, 3, 4, 6, 8}
৪,৪৪৯.
Nila and Nipa entered into a partnership with their capitals in the ratio 5 : 6 At the end of 8 months, Nila withdraws her capital. If they receive the profit in the ratio of 5 : 9, find how long Nipa's capital was used?
  1. ক
    ক) 8 months
  2. খ
    খ) 12 months
  3. গ
    গ) 10 months
  4. ঘ
    ঘ) 14 months
সঠিক উত্তর: খ
খ) 12 months
উত্তর
সঠিক উত্তর: খ
খ) 12 months
খ
ব্যাখ্যা
Question: Nila and Nipa entered into a partnership with their capitals in the ratio 5 : 6 At the end of 8 months, Nila withdraws her capital. If they receive the profit in the ratio of 5 : 9, find how long Nipa's capital was used?

Solution: 
Nipa মূলধন x সময়ের জন্য খেটেছিলো 

প্রশ্নমতে,
(5 × 8)/(6 × x) = 5/9
30x = 360
x = 360/30
x = 12
৪,৪৫০.
Rafiq bought 120 storybooks at Tk. 250 each and sold them at a profit of 15%. Find the total profit he made.
  1. ক
    4500 Tk.
  2. খ
    500 Tk.
  3. গ
    5500 Tk.
  4. ঘ
    45000 Tk.
সঠিক উত্তর: ক
4500 Tk.
উত্তর
সঠিক উত্তর: ক
4500 Tk.
ক
ব্যাখ্যা

Question: Rafiq bought 120 storybooks at Tk. 250 each and sold them at a profit of 15%. Find the total profit he made.

Solution:
Cost Price of 120 storybooks = 120 × Taka 250
= Taka 3,0000

Profit Percentage = 15%
∴ Profit Amount = Profit Percentage × Cost Price 
= 15% × Taka 3,0000
= Taka 4500

∴ Total Profit = Profit Amount = 4500

৪,৪৫১.
Parimal has two grandchildren, Jasmine, aged 2, and Holly, aged 4. Parimal divides Tk. 30 between them in the ratio of their ages. How much does Jasmine get?
  1. ক
    Tk. 10
  2. খ
    Tk. 12
  3. গ
    Tk. 15
  4. ঘ
    Tk. 18
  5. ঙ
    Tk. 20
সঠিক উত্তর: ক
Tk. 10
উত্তর
সঠিক উত্তর: ক
Tk. 10
ক
ব্যাখ্যা
Question: Parimal has two grandchildren, Jasmine, aged 2, and Holly, aged 4. Parimal divides Tk. 30 between them in the ratio of their ages. How much does Jasmine get?

Solution:
Tk. 30 is the whole amount.
Parimal divides Tk. 30 in the ratio 2 : 4.

The total number of shares is 2 + 4 = 6.
Each share is worth Tk. 30 ÷ 6 = Tk. 5.
∴ Jasmine gets 2 shares, 2 × 5 = Tk. 10
৪,৪৫২.
A runner crosses a square field diagonally with a side length of 200√2 m in just 20 seconds. How much time will it take to complete the full perimeter of the field?
  1. ক
    40√2 seconds
  2. খ
    30√2 seconds
  3. গ
    40 seconds
  4. ঘ
    20√2 seconds
সঠিক উত্তর: ক
40√2 seconds
উত্তর
সঠিক উত্তর: ক
40√2 seconds
ক
ব্যাখ্যা
Question: A runner crosses a square field diagonally with a side length of 200√2 m in just 20 seconds. How much time will it take to complete the full perimeter of the field?

Solution: 
Diagonal of the square field is = √2 × 200√2 = 400 m

speed = 400 / 20 = 20 mps

perimeter = 4 × 200√2 m

time = 800√2 / 20 = 40√2 seconds
৪,৪৫৩.
A boat travelled upstream and then returned, stopping 4 km short of the place from which it started. Speed of the stream is 2 km/hr. If total time taken by the boat is 6 hour and its speed in still water is 4 km/hr, distance travelled upstream is :
  1. ক
    ক) 12 km
  2. খ
    খ) 8 km
  3. গ
    গ) 6 km
  4. ঘ
    ঘ) 9 km
  5. ঙ
    ঙ) 10 km
সঠিক উত্তর: ঙ
ঙ) 10 km
উত্তর
সঠিক উত্তর: ঙ
ঙ) 10 km
ঙ
ব্যাখ্যা

Let distance travelled upstream be x km
x/(4 − 2) +( x − 4)/( 4 + 2) = 6
⇒ x/2 + (x − 4)/6 = 6
⇒ 3 x + x − 4 = 36
⇒ 4 x = 40
⇒ x = 10

৪,৪৫৪.
A and B are two alloys of gold and copper prepared by mixing metals in the ratios 7 : 2 and 7 : 11 respectively. If equal quantities of the alloys are melted to form a third alloy C, then the ratio of gold and copper in alloy C will be -
  1. ক
    7 : 8
  2. খ
    7 : 9
  3. গ
    7 : 5
  4. ঘ
    6 : 5
সঠিক উত্তর: গ
7 : 5
উত্তর
সঠিক উত্তর: গ
7 : 5
গ
ব্যাখ্যা
Question: A and B are two alloys of gold and copper prepared by mixing metals in the ratios 7:2 and 7:11 respectively. If equal quantities of the alloys are melted to form a third alloy C, then the ratio of gold and copper in alloy C will be -

Solution: A and B are two alloys of gold and copper prepared by mixing metals in the ratios 7:2 and 7:11 respectively. If equal quantities of the alloys are melted to form a third alloy C

 Gold in C


Copper in C 


∴ Gold: Copper = 7/6:5/6 = 7:5
৪,৪৫৫.
If log8x + log8(1/6) = 1/3, then the value of x is -
  1. ক
    ক) 8
  2. খ
    খ) 6
  3. গ
    গ) 16
  4. ঘ
    ঘ) 12
সঠিক উত্তর: ঘ
ঘ) 12
উত্তর
সঠিক উত্তর: ঘ
ঘ) 12
ঘ
ব্যাখ্যা
Question: If log8x + log8(1/6) = 1/3, then the value of x is -

Solution:
Given that
log8x + log8(1/6) = 1/3
⇒ log8{x × (1/6)} = 1/3
⇒ log8(x/6) = 1/3
⇒ 81/3 = x/6
⇒ (23)1/3 = x/6
⇒ 2 = x/6
x = 12
৪,৪৫৬.
A train crosses a platform 90 m long in 50 seconds at a speed of 36 km/hr. What is the time taken by the train to cross an electric pole?
  1. ক
    ক) 30 sec
  2. খ
    খ) 41 sec 
  3. গ
    গ) 45 sec 
  4. ঘ
    ঘ) 53 sec 
সঠিক উত্তর: খ
খ) 41 sec 
উত্তর
সঠিক উত্তর: খ
খ) 41 sec 
খ
ব্যাখ্যা
Question: A train crosses a platform 90 m long in 50 seconds at a speed of 36 km/hr. What is the time taken by the train to cross an electric pole?

Solution:
Speed of the train = (36 × 1000)/(60 × 60) = 10 m/sec
So, in 50 sec it goes = (10 × 50) = 500 m

Since the platform is 90 m so, only train = (500 - 90) m = 410 m

∴ Time taken by the train to cross an electric pole = (410/10) sec = 41 sec
৪,৪৫৭.
The value of log5 (125 × 625)/25 is equal to-
  1. ক
    5
  2. খ
    25
  3. গ
    1/5
  4. ঘ
    725
সঠিক উত্তর: ক
5
উত্তর
সঠিক উত্তর: ক
5
ক
ব্যাখ্যা
Question: The value of log5 (125 × 625)/25 is equal to-

Solution:
log5 {(125 ⋅ 625)/25}
= log5 {(53 ⋅ 54)/52}
= log5 53 + 4 - 2
= log5 55
= 5 log5 5
= 5 ⋅ 1
= 5
৪,৪৫৮.
In a garden, there are 10 rows and 12 columns of mango trees. The distance between two trees is 2 meters and a distance of one meter is left from all sides of the boundary of the garden. What is the length of the garden?
  1. ক
    ক) 20 m
  2. খ
    খ) 22 m
  3. গ
    গ) 24 m
  4. ঘ
    ঘ) 26 m
  5. ঙ
    ঙ) None
সঠিক উত্তর: গ
গ) 24 m
উত্তর
সঠিক উত্তর: গ
গ) 24 m
গ
ব্যাখ্যা
Each row contains 12 plants.
There are 11 gapes between the two corner trees (11 x 2) metres and 1 metre on each side is left.
Therefore Length = (22 + 2) m = 24 m.
৪,৪৫৯.
If the sum of an infinite geometric series is 20 and the common ratio r = 1/2, what is the first term?
  1. ক
    19
  2. খ
    13
  3. গ
    15
  4. ঘ
    10
সঠিক উত্তর: ঘ
10
উত্তর
সঠিক উত্তর: ঘ
10
ঘ
ব্যাখ্যা
Question: If the sum of an infinite geometric series is 20 and the common ratio r = 1/2, what is the first term?

Solution:
Here,
r = 1/2 
a = ?

We know that,
S∞ = a/(1 - r)
⇒ 20 = a/(1 - 1/2)
⇒ 20 = a/(1/2)
⇒ 20 = 2a
∴ a = 10
৪,৪৬০.
The volume V of a right circular cylinder is V =πr2h where r is the radius of the base and h is the height of the cylinder. If the volume of a right circular cylinder is 64π and its height is 4, what is the circumference of its base?
  1. ক
    2π
  2. খ
    3π
  3. গ
    8π
  4. ঘ
    5π
সঠিক উত্তর: গ
8π
উত্তর
সঠিক উত্তর: গ
8π
গ
ব্যাখ্যা
Question: The volume V of a right circular cylinder is V = πr2h where r is the radius of the base and h is the height of the cylinder. If the volume of a right circular cylinder is 64π and its height is 4, what is the circumference of its base?
(একটি সোজা বৃত্তাকার সিলিন্ডারের ভলিউম V হল V = πr²h, যেখানে r হল ভিত্তির ব্যাসার্ধ এবং h হল সিলিন্ডারের উচ্চতা। যদি সোজা বৃত্তাকার সিলিন্ডারের ভলিউম 64π হয় এবং এর উচ্চতা 4 হয়, তবে তার ভিত্তির পরিধি কত হবে?)

Solution: 
একটি সিলিন্ডারের উচ্চতা h একক ও ব্যাসার্ধ r একক হলে,
উক্ত সিলিন্ডারের আয়তন = πr2h ঘন একক
 
প্রশ্নমতে,
   πr2 × h = 64π
 বা, πr2  × 4 = 64π
 বা, r2 = 16
 বা, r = 4
 
সুতরাং বৃত্তের পরিধি =2πr = 2π × 4 = 8π
৪,৪৬১.
A can finish a work in 18 days and B can do the same work in 15 days. B worked for 10 days and left the job. In how many days, A alone can finish the remaining work?
  1. ক
    ক) 4
  2. খ
    খ) 5.5
  3. গ
    গ) 6
  4. ঘ
    ঘ) 8
সঠিক উত্তর: গ
গ) 6
উত্তর
সঠিক উত্তর: গ
গ) 6
গ
ব্যাখ্যা

B's 10day's work = 1/15 × 10 = 2/3
Remaining work =
(1−2)/3 = 1/3
Now,
1/18 work is done by A in 1 day
∴ 1/3 work is done by A in 18 × 1/3 = 6 days

৪,৪৬২.
If cos(x - 15°) = √3/2, what is the value of tan(x + 15°)?
  1. ক
    √3
  2. খ
    1
  3. গ
    1/√3
  4. ঘ
    0
সঠিক উত্তর: ক
√3
উত্তর
সঠিক উত্তর: ক
√3
ক
ব্যাখ্যা

Question: If cos(x - 15°) = √3/2, what is the value of tan(x + 15°)?
​
​Solution: 
​cos(x - 15°) = √3/2
​⇒ cos(x - 15°) = cos 30°
​​⇒ x - 15° = 30°
​​⇒ x = 30° + 15°
​∴ x = 45°
​
​Now,
tan(x + 15°)
​= ​tan(45° + 15°)
​= tan 60°
​= √3

৪,৪৬৩.
a, b, c, d and e are five consecutive integers in increasing order of size. Which one of the following expression is not odd?
  1. ক
    ক) a + b + c
  2. খ
    খ) ab + c
  3. গ
    গ) ac + e
  4. ঘ
    ঘ) ac + d
সঠিক উত্তর: গ
গ) ac + e
উত্তর
সঠিক উত্তর: গ
গ) ac + e
গ
ব্যাখ্যা
ধরি,
a = 1; b = 2; c = 3; d = 4; e = 5
অপশন ক a + b + c  = 1 + 2 + 3 = 6
অপশন খ  ab + c = 1 × 2 + 3 = 5
অপশন গ ac + e = 1 × 3 + 5 = 8
অপশন ঘ ac + d = 1 × 3 + 4 =7

আবার, 
ধরি, 
a = 2; b = 3; c = 4; d = 5; e = 6
অপশন ক a + b + c  = 2 + 3 + 4 = 9
অপশন খ  ab + c = 2 × 3 + 4 = 10
অপশন গ ac + e = 2 × 4 + 6 = 14
অপশন ঘ ac + d = 2 × 4 + 5 = 13

অপশন গ সকল ক্ষেত্রে জোড় 
সঠিক উত্তর গ
৪,৪৬৪.
What percentage of numbers from 1 to 50 has 1 or 9 in the unit digits?
  1. ক
    ক) 10%
  2. খ
    খ) 20%
  3. গ
    গ) 15%
  4. ঘ
    ঘ) 25%
সঠিক উত্তর: খ
খ) 20%
উত্তর
সঠিক উত্তর: খ
খ) 20%
খ
ব্যাখ্যা
Question: What percentage of numbers from 1 to 50 has 1 or 9 in the unit digits?

Solution:
Numbers from 1 to 50 has 1 or 9 in the unit digits
= 1, 9, 11, 19, 21, 29, 31, 39, 41, 49
= 10 Numbers

Amount in percentage
= 10/50 × 100%
= 20%
৪,৪৬৫.
If 15 is the 6th number in a series of 6 consecutive odd numbers what is 4th number in the series?
  1. ক
    ক) 7
  2. খ
    খ) 9
  3. গ
    গ) 11
  4. ঘ
    ঘ) 13
সঠিক উত্তর: গ
গ) 11
উত্তর
সঠিক উত্তর: গ
গ) 11
গ
ব্যাখ্যা
If 15 is the 6th number, then the series is 5, 7, 9, 11,13 and 15
So 4th number is 11
৪,৪৬৬.
If A = {1, 2, 3, 4, 5}, then the number of proper subsets of A is-
  1. ক
    30
  2. খ
    31
  3. গ
    32
  4. ঘ
    22
সঠিক উত্তর: খ
31
উত্তর
সঠিক উত্তর: খ
31
খ
ব্যাখ্যা
প্রশ্ন: If A = {1, 2, 3, 4, 5}, then the number of proper subsets of A is-

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

∴ A সেটের প্রকৃত উপসেট এর  সংখ্যা (৩২ - ১) টি = ৩১টি
৪,৪৬৭.
A train without stopping travels 60 km/h and with stoppage 40 km/h. What is the time taken for stoppage on a route 300 km?
  1. ক
    ক) 10 hours
  2. খ
    খ) 20 hours
  3. গ
    গ) 5 hours
  4. ঘ
    ঘ) 2.5 hours
  5. ঙ
    ঙ) 25 hours
সঠিক উত্তর: ঘ
ঘ) 2.5 hours
উত্তর
সঠিক উত্তর: ঘ
ঘ) 2.5 hours
ঘ
ব্যাখ্যা

Since, the train travels at 60 km/h, it's speed per minute is 1 km per minute. Hence, if it's speed with stoppage is 40 km/h, it will travel 40 minutes per hour i.e. train stops 20 min per hour.
Time taken to travel 300 km with stoppage,
= 300/40 = 7.5 hours
Time taken for stoppage as it stops for 20 min per hour
= (7 × 20 + 10)
= 140 + 10
= 150 minutes.
= 150/60 hours
= 2.5 hours.

৪,৪৬৮.
A 180 m long train crosses a milestone in 18 seconds and a train of same length coming from the opposite direction in 12 seconds. The speed of the other train is -
  1. ক
    ক) 56 km/hr
  2. খ
    খ) 62 km/hr
  3. গ
    গ) 64 km/hr
  4. ঘ
    ঘ) 72 km/hr
সঠিক উত্তর: ঘ
ঘ) 72 km/hr
উত্তর
সঠিক উত্তর: ঘ
ঘ) 72 km/hr
ঘ
ব্যাখ্যা
Speed of first train = 180/18 m/sec
                               = 10 m/sec

Let the speed of second train be x m/sec
Relative speed = (10 + x) m/sec

Here
360/(10 + x) = 12
12(10 + x) = 360
120 + 12x = 360
12x = 360 - 120 
12x = 240 
x = 20 m/sec
x = 20 × (18/5) km/hr
x = 72 km/hr
৪,৪৬৯.
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 hours
  2. খ
    3 hours
  3. গ
    4 hours
  4. ঘ
    5 hours
সঠিক উত্তর: ক
2 hours
উত্তর
সঠিক উত্তর: ক
2 hours
ক
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 
৪,৪৭০.
Working alone, R can complete a certain kind of job in 9 hours. R and S, working together at their respective rates, can complete one of these jobs in 6 hours. In how many hours can S, working alone, complete one of these jobs?
  1. ক
    18
  2. খ
    12
  3. গ
    9
  4. ঘ
    6
সঠিক উত্তর: ক
18
উত্তর
সঠিক উত্তর: ক
18
ক
ব্যাখ্যা
Question: Working alone, R can complete a certain kind of job in 9 hours. R and S, working together at their respective rates, can complete one of these jobs in 6 hours. In how many hours can S, working alone, complete one of these jobs?

Solution:
working together R and S can complete the job in 6 hours
which means
1/R + 1/S =1/6
R can complete the job in 9 hours
∴ 1/9 + 1/S = 1/6
⇒ 1/S = 1/6 - 1/9
⇒ 1/S = 1/18
∴ S working alone can complete the job in 18 hours
৪,৪৭১.
A train 900 meters long is running at a speed 78 kmph. If it crosses a tunnel in 1 min, then the length of the tunnel is-
  1. ক
    200 m
  2. খ
    300 m
  3. গ
    400 m
  4. ঘ
    500 m
সঠিক উত্তর: গ
400 m
উত্তর
সঠিক উত্তর: গ
400 m
গ
ব্যাখ্যা
Question: A train 900 meters long is running at a speed 78 kmph. If it crosses a tunnel in 1 min, then the length of the tunnel is-

Solution:
৩৬০০ সেকেন্ডে অতিক্রম করে ৭৮০০০ মি.
১ সেকেন্ডে অতিক্রম করে ৭৮০০০/৩৬০০ মি.
৬০ সেকেন্ডে অতিক্রম করে (৭৮০০০ × ৬০)/৩৬০০ মি.
= ১৩০০ মি.

টানেলের দৈর্ঘ্য = ১৩০০ - ৯০০ = ৪০০ মি.
৪,৪৭২.
Two pipes, Pipe X and Pipe Y, can fill a tank in 15 minutes and 30 minutes, respectively. If both pipes are opened together, how long will it take to fill the tank?
  1. ক
    16 minutes
  2. খ
    5 minutes
  3. গ
    18 minutes
  4. ঘ
    12 minutes
  5. ঙ
    10 minutes
সঠিক উত্তর: ঙ
10 minutes
উত্তর
সঠিক উত্তর: ঙ
10 minutes
ঙ
ব্যাখ্যা
Question: Two pipes, Pipe X and Pipe Y, can fill a tank in 15 minutes and 30 minutes, respectively. If both pipes are opened together, how long will it take to fill the tank?

Solution:
pipe X fill a tank in 15 minutes so, it fills in one minute (1/15)
pipe Y fill a tank in 30 minutes so, it fills in one minute (1/30)

∴ Both pipes fill in one minute = (1/15) + (1/30) = (2 + 1)/30 = 3/30 = 1/10

so, it will take 1/(1/10) or 10 minutes to fill the tank.
৪,৪৭৩.
A man can row three-quarters of a kilometer against the stream in 45/4 minutes and down the stream in 15/2  minutes. The speed (in km/hr) of the man in still water is:
  1. ক
    ক) 8 km/hr
  2. খ
    খ) 7 km/hr
  3. গ
    গ) 6 km/hr
  4. ঘ
    ঘ) 5 km/hr
সঠিক উত্তর: ঘ
ঘ) 5 km/hr
উত্তর
সঠিক উত্তর: ঘ
ঘ) 5 km/hr
ঘ
ব্যাখ্যা
Question: A man can row three-quarters of a kilometer against the stream in 45/4 minutes and down the stream in 15/2 minutes. The speed (in km/hr) of the man in still water is:

Solution: 
three - quarters of a kilometer = 750 meters
45/4 minutes = (45/4) × 60 = 675 sec
15/2 minutes = (15/2) × 60 = 450sec


Rate upstream = 750/ 675 m/sec
                         = 10/9 
Rate downstream =750/450 m/sec
 =5/3 m/sec


∴Rate in still water= (1/2){(10/9) + (5/3)} m/sec
= (1/2){(10 + 15)/9}
= (1/2)(25/9)
= 25/18
= (25/18)(18/5) km/hr.
= 5 km/hr
৪,৪৭৪.
While going to office, Salman travels at a speed of 30 kmph and on his way back, he travels at a speed of 45 kmph. What is his average speed of the whole journey?
  1. ক
    45 kmph
  2. খ
    36 kmph
  3. গ
    32 kmph
  4. ঘ
    42 kmph
সঠিক উত্তর: খ
36 kmph
উত্তর
সঠিক উত্তর: খ
36 kmph
খ
ব্যাখ্যা
Question: While going to office, Salman travels at a speed of 30 kmph and on his way back, he travels at a speed of 45 kmph. What is his average speed of the whole journey?

Solution:
When distance travelled is same, then average speed = (2ab)/(a + b); (where a and b are two different speeds)

∴ The Average Speed = (2 × 45 × 30)/(45 + 30)
= 2700/75 kmph
= 36 kmph
৪,৪৭৫.
What will be the next word of the series? ZAS, YBR, XCQ, ........
  1. ক
    DRW
  2. খ
    WES
  3. গ
    WDP
  4. ঘ
    HDP
সঠিক উত্তর: গ
WDP
উত্তর
সঠিক উত্তর: গ
WDP
গ
ব্যাখ্যা
Question: What will be the next word of the series?
ZAS, YBR, XCQ, ........

Solution:
ZAS, YBR, XCQ,
from the series
the first letter of each word is in the opposite direction of X, Y, and Z. So, the letter before X is W

next letter is in the sequence of A, B, C. so the next letter will be D
third letter of the series is also in the opposite direction Q, R, and S . So, the letter before Q is P
৪,৪৭৬.
Atul sells a bicycle to Bijoy at a profit of 20%. Bijoy sells it to Jalal at a profit of 25%. If Jalal pays Tk. 225 for it, the cost price of the bicycle for Atul is =?
  1. ক
    Tk. 150
  2. খ
    Tk. 130
  3. গ
    Tk. 125
  4. ঘ
    Tk. 120
সঠিক উত্তর: ক
Tk. 150
উত্তর
সঠিক উত্তর: ক
Tk. 150
ক
ব্যাখ্যা
Question: Atul sells a bicycle to Bijoy at a profit of 20%. Bijoy sells it to Jalal at a profit of 25%. If Jalal pays Tk. 225 for it, the cost price of the bicycle for Atul is =?

Solution: 
cost price of the bicycle for Bijoy = 225/1.25
= (225 × 100)/125
= 180 taka 

cost price of the bicycle for Atul = 180/1.2
= (180 × 10)/12
= 150 taka 
৪,৪৭৭.
What is the difference between simple and compound interest at 10% per annum on a sum of Tk. 6000 at the end of 2 years?
  1. ক
    Tk. 120
  2. খ
    Tk. 90
  3. গ
    Tk. 60
  4. ঘ
    Tk. 75
সঠিক উত্তর: গ
Tk. 60
উত্তর
সঠিক উত্তর: গ
Tk. 60
গ
ব্যাখ্যা

Question: What is the difference between simple and compound interest at 10% per annum on a sum of Tk. 6000 at the end of 2 years?

Solution:
Principal (P) = Tk. 6000
Rate (r) = 10% per annum
Time (n) = 2 years

Simple Interest (SI):
SI = (P × R × T)/100
= (6000 × 10 × 2)/100
= 120000/100
= Tk. 1200

Compound Interest (CI):
Amount (A) = P × (1 + r/100)n
= 6000 × (1 + 10/100)2
= 6000 × (1.1)2
= 6000 × 1.21
= Tk. 7260

∴ CI = A - P = 7260 - 6000
= Tk. 1260

∴ Difference between CI and SI = 1260 - 1200
= Tk. 60

৪,৪৭৮.
Nabil's monthly income is reduced by 15%. By what percentage his monthly income must be increased so that it is again equal to his original monthly income?
  1. ক
    16.6%
  2. খ
    15.6%
  3. গ
    14.6%
  4. ঘ
    17.6%
সঠিক উত্তর: ঘ
17.6%
উত্তর
সঠিক উত্তর: ঘ
17.6%
ঘ
ব্যাখ্যা
Question: Nabil's monthly income is reduced by 15%. By what percentage his monthly income must be increased so that it is again equal to his original monthly income?

Solution:
Let,
Nabil's monthly income is Tk. 100.
After reduced 15%,
Nabil's monthly income after reduced (100 - 15) = 85

Nabil's monthly income must be increased Tk. 15  so that it is again equal to his original monthly income.

∴ Percentage his monthly income must be increased = (15/85) × 100%
= 17.6%
৪,৪৭৯.
If 2x + 3y = 11 and xy = 5, then find the value of 8x3 + 27y3 =?
  1. ক
    471
  2. খ
    321
  3. গ
    288
  4. ঘ
    341
সঠিক উত্তর: ঘ
341
উত্তর
সঠিক উত্তর: ঘ
341
ঘ
ব্যাখ্যা
Question: If 2x + 3y = 11 and xy = 5, then find the value of 8x3 + 27y3 = ?

Solution:
Given that,
2x + 3y = 11 and xy = 5

Now,
8x3 + 27y3
= (2x)3 + (3y)3
= (2x + 3y)3 - 3 × 2x × 3y (2x + 3y)  ;[a3 + b3 = (a + b)3 - 3ab(a + b)]
= (2x + 3y)3 - 18xy(2x + 3y)
= 113 - 18 × 5 × 11
= 1331 - 990
= 341
∴ The required answer is 341.
৪,৪৮০.
Two pipes A and B can fill a cistern in 36 and 54 minutes respectively. Both pipes are opened together, after how many minutes should B be turned off, so that the cistern be fill in 24 minutes?
  1. ক
    21 minutes
  2. খ
    16 minutes
  3. গ
    18 minutes
  4. ঘ
    24 minutes
সঠিক উত্তর: গ
18 minutes
উত্তর
সঠিক উত্তর: গ
18 minutes
গ
ব্যাখ্যা
Question: Two pipes A and B can fill a cistern in 36 and 54 minutes respectively. Both pipes are opened together, after how many minutes should B be turned off, so that the cistern be fill in 24 minutes?

Solution:
A can fill the cistern in 36 minutes
So in 1 min A can fill the cistern = 1/36 th part
In 24 min, A can fill the cistern = 24/36 = 2/3 rd

∴ Remaining part = 1 - (2/3) = 1/3 rd

As B can fill full cistern in 54 minutes
So it will fill 1/3 rd part in = (1/3) × 54) = 18 minutes.
৪,৪৮১.
Working alone, two pipes A and B require 9 hours and 6.25 hours more respectively to fill a pool than if they were working together. Find the total time taken to fill the pool if both were working together.
  1. ক
    6 hours
  2. খ
    6.5 hours
  3. গ
    7 hours
  4. ঘ
    7.5 hours
  5. ঙ
    None of these
সঠিক উত্তর: ঘ
7.5 hours
উত্তর
সঠিক উত্তর: ঘ
7.5 hours
ঘ
ব্যাখ্যা
Question: Working alone, two pipes A and B require 9 hours and 6.25 hours more respectively to fill a pool than if they were working together. Find the total time taken to fill the pool if both were working together.

Solution:
Let the time taken if both were working together be 'n' hours.
Time taken by A = n + 9
Time taken by B = n + 6.25  

In such kind of problems, we apply the formula :
n2 = a × b, where 'a' and 'b' are the extra time taken if both work individually than if both work together.
Therefore,
n2 = 9 × 6.25
⇒ n = 3 × 2.5 = 7.5  

Thus, working together, pipes A and B require 7.5 hours
৪,৪৮২.
A question paper consists of three sections 4,5 and 6 questions respectively. Attempting one question from each section is compulsory but a candidate need not attempt all the questions. In how many ways can a candidate attempt the questions?
  1. ক
    43439
  2. খ
    13137
  3. গ
    29295
  4. ঘ
    None
সঠিক উত্তর: গ
29295
উত্তর
সঠিক উত্তর: গ
29295
গ
ব্যাখ্যা
Question: A question paper consists of three sections 4,5 and 6 questions respectively. Attempting one question from each section is compulsory but a candidate need not attempt all the questions. In how many ways can a candidate attempt the questions?

Solution:
At least 1 question from each section is compulsory, so from the 1st section the candidate can attempt 1 or 2 or 3 or 4 questions.
In each section each question can be dealt with in 2 ways, i.e. either he attempts it or leaves it.
So far 4 question there are 2 × 2 × 2 × 2 ways to attempt.
As he has to attempt at least 1 question, the total number of ways in which he can attempt questions from 1st section is 24 - 1
Similarly for the 2nd section there are 25 - 1 ways in which he can attempt and for the 3rd section there are 26 - 1 ways.
The ways in which the attempts one or more questions in any section is independent of the number of ways in which he attempts one or more questions from the other sections.
Thus, total number of ways in which he can attempt questions in that paper:
= (24 - 1)(25 - 1)(26 - 1)
= 15 × 31 × 63
= 29295
৪,৪৮৩.
If you subtract - 1 from + 1, What will be the result?
  1. ক
    ক) -2
  2. খ
    খ) 0
  3. গ
    গ) 2
  4. ঘ
    ঘ) 1
সঠিক উত্তর: গ
গ) 2
উত্তর
সঠিক উত্তর: গ
গ) 2
গ
ব্যাখ্যা
Question: If you subtract - 1 from + 1, What will be the result?

Solution:
(+ 1) - (- 1)
= (+ 1 + 1)
= 2
৪,৪৮৪.
In a business Titu invested twice as Rasel for a year. The total amount is 21000. After one year Titu received a total of 17000 Taka. What is Rasel's profit?
  1. ক
    ক) 8500 Tk
  2. খ
    খ) 1500 Tk
  3. গ
    গ) 1850 Tk
  4. ঘ
    ঘ) 1000 tk
সঠিক উত্তর: খ
খ) 1500 Tk
উত্তর
সঠিক উত্তর: খ
খ) 1500 Tk
খ
ব্যাখ্যা
Question: In a business Titu invested twice as Rasel for a year. The total amount is 21000. After one year Titu received a total of 17000 Taka. What is Rasel's profit?

Solution:
ধরি, রাসেলের বিনিয়োগ = ক টাকা
টিটুর বিনিয়োগ = ২ক টাকা
প্রশ্নমতে,
৩ক = ২১০০০
ক = ৭০০০ টাকা
∴ টিটুর বিনিয়োগ = ১৪০০০ টাকা
টিটুর লাভ = ১৭০০০ - ১৪০০০ = ৩০০০ টাকা

১৪০০০ টাকায় লাভ হয় ৩০০০ টাকা
৭০০০ টাকায় লাভ হয় = ৩০০০/২ = ১৫০০ টাকা

∴ রাসেলের লাভ হবে ১৫০০ টাকা।
৪,৪৮৫.
Two trains are running in opposite directions with the same speed. If the length of each train is 180 meters and they cross each other in 9 seconds, then the speed of each train (in km/hr) is
  1. ক
    48km/hr
  2. খ
    72km/hr
  3. গ
    36km/hr
  4. ঘ
    60km/hr
  5. ঙ
    None of these
সঠিক উত্তর: খ
72km/hr
উত্তর
সঠিক উত্তর: খ
72km/hr
খ
ব্যাখ্যা
Question: Two trains are running in opposite directions with the same speed. If the length of each train is 180 meters and they cross each other in 9 seconds, then the speed of each train (in km/hr) is

Solution:
Let the speed of each train be x m/sec.
Then, relative speed of the two trains = 2xm/sec

Now
⇒ 2x = (180 + 180)/9
⇒ 2x = 360/9
⇒ 2x = 40
∴ x = 20

∴ Speed of each train = 20m /sec = 20(18/5) = 72km/hr
৪,৪৮৬.
A train is moving at a speed of 132 km/hr. If the length of the train is 110 meters, how long it will take to cross a railway platform 165 meter long?
  1. ক
    ক) 4.5 seconds
  2. খ
    খ) 5.5 seconds
  3. গ
    গ) 6.5 seconds
  4. ঘ
    ঘ) 7.5 seconds
সঠিক উত্তর: ঘ
ঘ) 7.5 seconds
উত্তর
সঠিক উত্তর: ঘ
ঘ) 7.5 seconds
ঘ
ব্যাখ্যা
Speed = 132 km/hr = 132 × 5/18 m/s = 110/3 m/s
time = distance/speed = (110 + 165)/(110/3) = 7.5 seconds
৪,৪৮৭.
Two trains are running at 40 km/hr and 20 km/hr respectively in the same direction. If the faster train completely passes a man sitting in the slower train in 5 seconds, the length of the faster train is -
  1. ক
    19
  2. খ
    13(2/9)
  3. গ
    27(7/9)
  4. ঘ
    33
সঠিক উত্তর: গ
27(7/9)
উত্তর
সঠিক উত্তর: গ
27(7/9)
গ
ব্যাখ্যা

Relative speed = 40 - 20
= 20 km/hr
= 20 × (5/18)
= 50/9 m/s
Time = 5 s
Distance = (50/9) × 5
= 250/9
= 27(7/9).

৪,৪৮৮.
Copper is 9 times as heavy as water, and tin is 3 times as heavy as water. In what ratio should these be mixed to get an alloy 6 times as heavy as water? 
  1. ক
    1 : 3
  2. খ
    1 : 2
  3. গ
    1 : 1
  4. ঘ
    2 : 1
  5. ঙ
    None
সঠিক উত্তর: গ
1 : 1
উত্তর
সঠিক উত্তর: গ
1 : 1
গ
ব্যাখ্যা

Question: Copper is 9 times as heavy as water, and tin is 3 times as heavy as water. In what ratio should these be mixed to get an alloy 6 times as heavy as water?

Solution:
Let copper be 9x times as heavy and tin 3y times as heavy as water.

9x + 3y = 6(x + y)
⇒ 9x + 3y = 6x + 6y
⇒ 9x − 6x = 6y − 3y
⇒ 3x = 3y
⇒ x/y = 1/1

∴ Copper and tin should be mixed in the ratio 1 : 1.

৪,৪৮৯.
The two diagonals of a rhombus are respectively 10 and 20 m. If the area of a square is same as that of rhombus, find the perimeter of the square. 
  1. ক
    ক) 20 m
  2. খ
    খ) 30 m
  3. গ
    গ) 40 m
  4. ঘ
    ঘ) 50 m
সঠিক উত্তর: গ
গ) 40 m
উত্তর
সঠিক উত্তর: গ
গ) 40 m
গ
ব্যাখ্যা
Question: The two diagonals of a rhombus are respectively 10 and 20 m. If the area of a square is same as that of rhombus, find the perimeter of the square. 

Solution:
area of rhombus = (1/2) × 10 × 20
= 100 m2

area of square = 100 m2
side of square = √100 m
= 10 m

∴ perimeter of the square = 4 × 10 m
= 40 m
৪,৪৯০.
If x = 3 + 2√2, find the value of .
  1. ক
    2
  2. খ
    4
  3. গ
    2√2
  4. ঘ
    √2
  5. ঙ
    3√2
সঠিক উত্তর: ক
2
উত্তর
সঠিক উত্তর: ক
2
ক
ব্যাখ্যা

Question: If x = 3 + 2√2, find the value of .

Solution:
Given,

৪,৪৯১.
One woman and one man can build a wall together in two hours, but the woman would need the help of two girls in order to complete the same job in the same amount of time. If one man and one girl worked together, it would take them four hours to build the wall. Assuming that rates for men, women and girls remain constant, how many hours would it take one woman, one man, and one girl, working together, to build the wall?
  1. ক
    10/7
  2. খ
    1
  3. গ
    5/7
  4. ঘ
    12/7
  5. ঙ
    None
সঠিক উত্তর: ঘ
12/7
উত্তর
সঠিক উত্তর: ঘ
12/7
ঘ
ব্যাখ্যা
Question: One woman and one man can build a wall together in two hours, but the woman would need the help of two girls in order to complete the same job in the same amount of time. If one man and one girl worked together, it would take them four hours to build the wall. Assuming that rates for men, women and girls remain constant, how many hours would it take one woman, one man, and one girl, working together, to build the wall?

Solution:
One woman and one man can build a wall together in 2 hrs.
so, 1w + 1w = 1/2 [walls per hour]

The woman would need the help of 2 girls to complete the same job in the same amount of time.
so, 1w + 2g = 1/2 [walls per hour]
From (I) and (II), we deduce that: 1m = 2g

"If one man and one girl worked together, it would take them 4 hours to build the wall."
This gives the equation: 1m + 1g = 1/4 [walls per hour]
⇒ 2g + g = 1/4
⇒ 3g = 1/4
⇒ g = 1/12

we now know:
1m = 2g = 1/6
1w = 4g = 1/3

Now, 1 woman, 1 man, and 1 girl working together to build the wall
৪,৪৯২.
In a vessel, milk and water are in a ratio of 3 : 5. In the second vessel, the milk and water are in the ratio of 3 : 2. In what ratio should these two mixtures be mixed to form a new mixture in which the milk and water are in the ratio 2 : 3?
  1. ক
    1 : 8
  2. খ
    6 : 1
  3. গ
    7 : 1
  4. ঘ
    8 : 1
সঠিক উত্তর: ঘ
8 : 1
উত্তর
সঠিক উত্তর: ঘ
8 : 1
ঘ
ব্যাখ্যা
Question: In a vessel, milk and water are in a ratio of 3 : 5. In the second vessel, the milk and water are in the ratio of 3 : 2. In what ratio should these two mixtures be mixed to form a new mixture in which the milk and water are in the ratio 2 : 3?

Solution:
The ratio of milk and water in the first vessel = 3 : 5
The ratio of milk and water in the second vessel = 3 : 2
The ratio of milk and water in the final mixture = 2 : 3

Let
The mixture from the two-vessel be mixed in the ratio x : y.

Quantity of milk taken from the first vessel = 3x/8
Quantity of water taken from the first vessel = 5x/8

Quantity of milk taken from the second vessel = 3y/5
Quantity of water taken from the second vessel = 2y/5

Then,
{(3x/8) + (3y/5}/{(5x/8) + (2y/5)} = 2/3
⇒ {(15x + 24y)/40}/{(25x + 16y)/40} = 2/3
⇒ 3(15x + 24y) = 2(25x + 16y)
⇒ 45x + 72y = 50x + 32y
⇒ 50x - 45x = 72y - 32y
⇒ 5x = 40y
⇒ x/y = 8/1

∴ The first and second mixure are mixed in the ratio 8 : 1
৪,৪৯৩.
If θ is a positive acute angle and 4cos2θ - 1 = 0, then the value of tan(θ - 30°) is equal to?
  1. ক
    1/√3
  2. খ
    √3
  3. গ
    1
  4. ঘ
    0
সঠিক উত্তর: ক
1/√3
উত্তর
সঠিক উত্তর: ক
1/√3
ক
ব্যাখ্যা
Question: If θ is a positive acute angle and 4cos2θ - 1 = 0, then the value of tan(θ - 30°) is equal to?

Solution:
Given,
4cos2θ - 1 = 0
⇒ 4cos2θ = 1
⇒ cos2θ = 1/4
⇒ cosθ = 1/2
⇒ cosθ = cos60°
∴ θ = 60°

Now, 
tan(θ - 30°) = tan(60° - 30°)
= tan 30°
= 1/√3
৪,৪৯৪.
If a rectangle's length and width are both doubled, by what percent is the rectangle's area increased?
  1. ক
    ক) 50%
  2. খ
    খ) 100%
  3. গ
    গ) 200%
  4. ঘ
    ঘ) 300%
সঠিক উত্তর: ঘ
ঘ) 300%
উত্তর
সঠিক উত্তর: ঘ
ঘ) 300%
ঘ
ব্যাখ্যা
Question: If a rectangle's length and width are both doubled, by what percent is the rectangle's area increased?

Solution:
ধরি,
আয়তক্ষেত্রের দৈর্ঘ্য ক একক
আয়তক্ষেত্রের প্রস্থ খ একক
∴ আয়তক্ষেত্রের ক্ষেত্রফল কখ বর্গএকক 


দৈর্ঘ্য ও প্রস্থ দ্বিগুণ হলে ক্ষেত্রফল হবে = ২ক × ২খ বর্গএকক 
= ৪কখ বর্গএকক 

ক্ষেত্রফল বৃদ্ধি পায় = (৪কখ - কখ) বর্গএকক 
= ৩কখ 

কখ বর্গএকক এ বৃদ্ধি পায় ৩কখ বর্গএকক 
∴ ১ বর্গএকক এ বৃদ্ধি পায় (৩কখ)/(কখ) = ৩ বর্গএকক 
∴ ১০০ বর্গএকক এ বৃদ্ধি পায় ৩ × ১০০ বর্গএকক 
= ৩০০ বর্গএকক 

∴ আয়তক্ষেত্রের ক্ষেত্রফল ৩০০% বৃদ্ধি পাবে।

৪,৪৯৫.
A merchant marks up his goods by 25% above the cost price. He then offers a discount of 10% on the marked price. What is the overall percentage profit?
  1. ক
    10%
  2. খ
    12.5%
  3. গ
    15%
  4. ঘ
    20%
সঠিক উত্তর: খ
12.5%
উত্তর
সঠিক উত্তর: খ
12.5%
খ
ব্যাখ্যা

Question: A merchant marks up his goods by 25% above the cost price. He then offers a discount of 10% on the marked price. What is the overall percentage profit?

Solution:
Let,
The cost price (CP) be Tk. 100

Marked Price = 25% more than the cost price
= 100 + 25
= Tk. 125

Discount = 10% of 125
= (10/100) × 125
= Tk. 12.5

Selling Price (SP) = 125 - 12.5 = Tk. 112.5

∴ Profit = SP - CP = 112.5 - 100 = Tk. 12.5

∴ Overall percentage profit = (profit/cost price) × 100%
= (12.5/100) × 100%
= 12.5%

৪,৪৯৬.
Solution set of the inequality: x - 5 > 4x + 7 is
  1. ক
    (- ∞, - 4)
  2. খ
    [- ∞, - 4)
  3. গ
    (- ∞, - 4]
  4. ঘ
    [- ∞, - 4]
সঠিক উত্তর: ক
(- ∞, - 4)
উত্তর
সঠিক উত্তর: ক
(- ∞, - 4)
ক
ব্যাখ্যা
Question: Solution set of the inequality: x - 5 > 4x + 7 Is

Solution:
x - 5 > 4x + 7
⇒ - 5 > 4x - x + 7
⇒ - 5 > 3x + 7
⇒ - 5 - 7 > 3x
⇒ - 12 > 3x
⇒ - 12/3 > 3x/3
⇒ - 4 > x
⇒ x < - 4

∴ নির্ণেয় সমাধান সেট: (- ∞, - 4)
৪,৪৯৭.
In a group of 60 people 27 people like Coca Cola and 42 people like Pepsi Cola. Each person likes at least one of the two drinks. How many of these people like both Coca Cola and Pepsi Cola?
  1. ক
    ক) 7
  2. খ
    খ) 9
  3. গ
    গ) 11
  4. ঘ
    ঘ) 13
সঠিক উত্তর: খ
খ) 9
উত্তর
সঠিক উত্তর: খ
খ) 9
খ
ব্যাখ্যা

সবাই দুইটির যেকোনো একটি পছন্দ করে, 
∴ T = n(c) + n(p) - n(c∩p) 
⇒ 60 = 27 + 42 - n(c∩p) 
⇒ n(c∩p) = 69 - 60
⇒ n(c∩p) = 9

৪,৪৯৮.
0, 2, 6, 8, 16, 30, 54,? 
  1. ক
    88
  2. খ
    95
  3. গ
    100
  4. ঘ
    122
সঠিক উত্তর: গ
100
উত্তর
সঠিক উত্তর: গ
100
গ
ব্যাখ্যা
Question: 0, 2, 6, 8, 16, 30, 54,?  

Solution: 
0 + 2 + 6 = 8
2 + 6 + 8 = 16
6 + 8 + 16 = 30
8 + 16 + 30 = 54

16 + 30 + 54 = 100
৪,৪৯৯.
What mathematical sign should be inserted between 2 and 3 so that the sum of the two digits becomes more than 2, but less than 3?
  1. ক
    ÷
  2. খ
    √
  3. গ
    ×
  4. ঘ
    .
সঠিক উত্তর: ঘ
.
উত্তর
সঠিক উত্তর: ঘ
.
ঘ
ব্যাখ্যা
Question: What mathematical sign should be inserted between 2 and 3 so that the sum of the two digits becomes more than 2, but less than 3?

Solution:
এখানে, 2 ও 3 এর মধ্যে দশমিক বসালে সংখ্যাটি হয় 2.3, যা 2 থেকে বড় এবং 3 থেকে ছোট।
৪,৫০০.
A and B take part in 100 m race. A runs at 5 kmph. A gives B a start of 8 m and still beats him by 8 seconds. The speed of B is-
  1. ক
    5.15 kmph
  2. খ
    4.14 kmph
  3. গ
    4.25 kmph
  4. ঘ
    4.4 kmph
সঠিক উত্তর: খ
4.14 kmph
উত্তর
সঠিক উত্তর: খ
4.14 kmph
খ
ব্যাখ্যা
Question: A and B take part in 100 m race. A runs at 5 kmph. A gives B a start of 8 m and still beats him by 8 seconds. The speed of B is-

Solution:
A's speed = 5 × (5/18)m/sec = 25/18 m/sec

Time taken by A to cover 100 m = 100 × (18/25) sec = 72 sec

∴ Time taken by B to cover 92 m = (72 + 8) = 80 sec

∴ B's speed = (92/80) × (18/5) kmph = 4.14 kmph