Bank Math

PrepBank · পাতা ৬১ / ১৬১ · ৬,০০১–৬,১০০ / ১৬,১২৪

৬,০০১.

When 35 - [30 - {35 - (15 - *)}] = 60, then * is equal to-

ক
- 19
খ
35
গ
20
ঘ
- 29

ব্যাখ্যা

Question: When 35 - [30 - {35 - (15 - *)}] = 60, then * is equal to-

Solution:
35 - [30 - {35 - (15 - *)}] = 60
⇒ 35 - [30 - {35 -15 +*}] = 60
⇒ 35 - [30 - {20 + *}] = 60
⇒ 35 - [30 - 20 - *] = 60
⇒ 35 - [10 - *] = 60
⇒ 35 - 10 + * = 60
⇒ 25 + * = 60
∴ * = 60 - 25 = 35

৬,০০২.

If 8 people make 48 chairs in 12 days by working 6 hours a day, then how long will it take 12 people working 8 hours a day to make 192 chairs?

ক
6 days
খ
12 days
গ
24 days
ঘ
30 days

ব্যাখ্যা

Question: If 8 people make 48 chairs in 12 days by working 6 hours a day, then how long will it take 12 people working 8 hours a day to make 192 chairs?

Solution:
48 টি চেয়ার তৈরি করতে 8 জন লোক প্রতিদিন 6 ঘণ্টা কাজ করে = 12 দিনে
∴ 1 টি চেয়ার তৈরি করতে 1 জন লোক প্রতিদিন 1 ঘণ্টা কাজ করে = (8 × 6 × 12)/48 দিনে
∴ 192 টি চেয়ার তৈরি করতে 12 জন লোক প্রতিদিন 8 ঘণ্টা কাজ করে = (576 × 192)/(48 × 8 × 12)
= 24 দিনে

৬,০০৩.

When 4 is added to 1/2 of a number, the result is 14. What is the number?

ক
ক) 10
খ
খ) 15
গ
গ) 20
ঘ
ঘ) 25

ব্যাখ্যা

Question: When 4 is added to 1/2 of a number, the result is 14. What is the number?

Solution:
let the number be x

(x/2) + 4 = 14
⇒ x/2 = 10
∴ x = 20

৬,০০৪.

Raju, Bappa and Samir divide Tk. 1,500 among them in such a way that if Tk. 50, Tk. 40 and Tk. 60 are removed from the sums that Raju, Bappa and Samir received respectively, then the shares will be in the ratio of 2 : 3 : 4. How much did Raju receive?

ক
Tk. 300
খ
Tk. 490
গ
Tk. 660
ঘ
Tk. 350

ব্যাখ্যা

Question: Raju, Bappa and Samir divide Tk. 1,500 among them in such a way that if Tk. 50, Tk. 40 and Tk. 60 are removed from the sums that Raju, Bappa and Samir received respectively, then the shares will be in the ratio of 2 : 3 : 4. How much did Raju receive?

সমাধান:
ধরি,
টাকা সরিয়ে নেওয়ার পর রাজু, বাপ্পা এবং সমীরের প্রাপ্ত অংশের অনুপাত যথাক্রমে 2x, 3x এবং 4x।

তাহলে, টাকা সরিয়ে নেওয়ার আগে তাদের প্রাপ্ত অংশ ছিল:
রাজু = 2x + 50
বাপ্পা = 3x + 40
সামীর = 4x + 60

তাদের প্রাপ্ত মোট টাকা হলো Tk. 1,500।
প্রশ্নমতে,
⇒ (2x + 50) + (3x + 40) + (4x + 60) = 1500
⇒ (2x + 3x + 4x) + (50 + 40 + 60) = 1500
⇒ 9x + 150 = 1500
⇒ 9x = 1500 - 150
⇒ 9x = 1350
⇒ x = 1350/9
∴ x = 150

রাজুর প্রাপ্ত টাকা = 2x + 50
= 2(150) + 50
= 300 + 50
= 350
সুতরাং, রাজু Tk. 350 পেয়েছিল।

৬,০০৫.

Pipe B is two times efficient as pipe C. Pipe A and B together can fill an empty tank in 8 4/7 hours. Pipe A and C together can fill the same tank in 12 hours. In how many hours required filling by pipe B alone?

ক
ক) 15 hours
খ
খ) 12 hours
গ
গ) 20 hours
ঘ
ঘ) 30 hours

ব্যাখ্যা

Let, B alone filled the pipe by x hours.

Efficiency ratio of B and C =2 : 1
Time ratio of B and C = 1 : 2

Given,
(1/A + 1/B)- (1/A + 1/C) =7/60 - 1/12
⇒ 1/B - 1/C = 2/60 = 1/30
⇒ 1/x - 1/2x=1/30
⇒ 1/2x=1/30
⇒ 1/x=1/15
⇒ x = 15

B alone filled the pipe by 15 hours.

৬,০০৬.

A can do a work in 15 days and B in 20 days. If they work on it together for 4 days, then the fraction of the work that is left is -

ক
ক) 8/15
খ
খ) 7/15
গ
গ) 3/19
ঘ
ঘ) 7/25

ব্যাখ্যা

Question: A can do a work in 15 days and B in 20 days. If they work on it together for 4 days, then the fraction of the work that is left is -

Solution:
একদিনে,
A করে 1/15 অংশ
B করে 1/20 অংশ

মোট করে = 1/15 + 1/20
= 7/60

চারদিনে মোট কাজ করে = 4 × 7/60 = 28/60 = 7/15

∴ অবশিষ্ট কাজ = 1 - 7/15 = 8/15

৬,০০৭.

A geometric series has its first term as 1 divided by square root of 2, and its common ratio is √2. Which term in the sequence is 8√2?

ক
9^th term
খ
8^th term
গ
6^th term
ঘ
11^th term

ব্যাখ্যা

Question: A geometric series has its first term as 1 divided by square root of 2, and its common ratio is √2. Which term in the sequence is 8√2?

Solution:
প্রথম পদ, a = 1/√2
সাধারণ অন্তর, r = √2

ধরি, n তম পদ = ar^{n - 1} = 8√2
⇒ (1/√2) (√2)^{n - 1} = 8√2
⇒ (√2)^{n - 1} = 16
⇒ (√2)^{n - 1} = (√2)⁸
⇒ n - 1 = 8
∴ n = 9

অতএব, ধারাটির 9 তম পদ 8√2 হবে।

৬,০০৮.

A box contains 400 marbles, of which 40% are blue. You pick some marbles of which 25% are blue. Of the remaining marbles, 50% are blue marbles. How many marbles did you pick?

ক
130
খ
140
গ
150
ঘ
160
ঙ
None

ব্যাখ্যা

Question: A box contains 400 marbles, of which 40% are blue. You pick some marbles of which 25% are blue. Of the remaining marbles, 50% are blue marbles. How many marbles did you pick?

Solution:
Given that,
Total marbles = 400
Blue marbles initially = 40% of 400 = (40/100) × 400 = 160
Non-blue marbles = 400 - 160 = 240

Let the number picked be x.
∴ Blue picked = 25% of x = 0.25x
and non-blue picked = 0.75x

And,
Remaining marbles = 400 - x
∴ Remaining blue marbles = 160 - 0.25x

ATQ, Of the remaining marbles, 50% are blue then we get,
160 - 0.25x = 0.5(400 - x)
⇒ 160 - 0.25x = 200 - 0.5x
⇒ 0.5x - 0.25x = 200 - 160
⇒ 0.25x = 40
⇒ x = 40/0.25
∴ x = 160

So the number of marbles you picked is 160.

৬,০০৯.

ক
4/3
খ
5/2
গ
6/5
ঘ
9/4
ঙ
2/3

৬,০১০.

One tap (A) fills a reservoir four times as fast as another tap (B). If both taps running together can fill the reservoir in 20 minutes, then how long will the slower tap (B) alone take to fill the reservoir?

ক
40 minutes
খ
1 hour
গ
1 hour 20 minutes
ঘ
1 hour 40 minutes

ব্যাখ্যা

Question: One tap (A) fills a reservoir four times as fast as another tap (B). If both taps running together can fill the reservoir in 20 minutes, then how long will the slower tap (B) alone take to fill the reservoir?

সমাধান:
ধরি,
ধীরগতির নল B একা চৌবাচ্চাটি পূর্ণ করতে সময় নেয় x মিনিট।
তাহলে, দ্রুতগতির নল A একা চৌবাচ্চাটি পূর্ণ করতে সময় নেবে x/4 মিনিট।

প্রশ্নমতে, তারা একত্রে 20 মিনিটে পূর্ণ করে। অর্থাৎ,
1/x + 1/(x/4) = 1/20
⇒ 1/x + 4/x = 1/20
⇒ (1 + 4)/x = 1/20
⇒ 5/x = 1/20
⇒ x = 5 × 20
⇒ x = 100 মিনিট
∴ x = 1 ঘণ্টা 40 মিনিট [ 60 মিনিট = 1 ঘণ্টা]

∴ ধীরগতির নলটি (B) একা চৌবাচ্চাটি পূর্ণ করতে 1 ঘন্টা 40 মিনিট সময় নেবে।

৬,০১১.

A man sold 18 cots for Tk. 14,700, gaining thereby the cost price of 3 cots. The cost price of a cot is-

ক
ক) Tk. 600
খ
খ) Tk. 700
গ
গ) Tk. 800
ঘ
ঘ) Tk. 900

ব্যাখ্যা

Here
(S.P of 18 cots) - (C.P of 18 cots) = C.P of 3cots
S.P of 18 cots = C.P of 21 cots

S.P of 18 cots = Tk. 14,700

C.P of 21 cots = Tk. 14,700
C.P of 1 cot = Tk. (14,700/21)
= Tk. 700

৬,০১২.

The average of 5 consecutive numbers is n. What will be the average if the next two numbers are included?

ক
n + 2
খ
n - 1
গ
n - 2
ঘ
n + 1
ঙ
None of these

ব্যাখ্যা

Question: The average of 5 consecutive numbers is n. What will be the average if the next two numbers are included?

Solution:
The average of 5 consecutive terms is n, implies that the 3rd term is n. Now as the next 2 terms are included implies that the new average for 7 terms would be the 4th term. So, the 4th term would be n + 1.

Example:
(1 + 2 + 3 + 4 + 5)/5
= 15/5
= 3

(1 + 2 + 3 + 4 + 5 + 6 + 7)/7
= 28/7
= 4

৬,০১৩.

If the difference between the circumference and diameter of a circle is 120 cm, then the diameter of the circle is -

ক
42.50 cm
খ
56 cm
গ
48.27 cm
ঘ
64 cm

ব্যাখ্যা

Question: If the difference between the circumference and diameter of a circle is 120 cm, then the diameter of the circle is -

Solution:
ধরি,
বৃত্তের ব্যাসার্ধ = r
বৃত্তের ব্যাস = 2r
বৃত্তের পরিধি = 2πr

প্রশ্নমতে,
2πr - 2r = 120
⇒ 2r(π - 1) = 120
⇒ r = (120/2){(22/7) - 1}
⇒ r = 60/(22 - 7)/7
⇒ r = (60 × 7)/15
∴ r = 28

∴ বৃত্তের ব্যাস = 2r = 2 × 28 = 56 সে.মি.

৬,০১৪.

A committee of 5 members is to be formed by selecting out of 7 men and 6 women. In how many different ways the committee can be formed if it should have at least 3 men?

ক
756
খ
735
গ
645
ঘ
1287

ব্যাখ্যা

Question: A committee of 5 members is to be formed by selecting out of 7 men and 6 women. In how many different ways the committee can be formed if it should have at least 3 men?

Solution:
Men Women Ways
-------------------------------------------------------
3 2 ⁷C₃ × ⁶C₂ = 35 × 15 = 525
4 1 ⁷C₄ × ⁶C₁ = 35 × 6 = 210
5 0 ⁷C₅ × ⁶C₀ = 21 × 1 = 21

∴ Total number of ways = 525 + 210 + 21 = 756

৬,০১৫.

In a simultaneous throw of a pair of dice, what is the probability of getting a total more than 8?

ক
5/18
খ
12/18
গ
1/4
ঘ
7/18

ব্যাখ্যা

Question: In a simultaneous throw of a pair of dice, what is the probability of getting a total more than 8?

Solution:
The total number of possible outcomes for the pair of dice is the product of the outcomes for each die is-
= 6 × 6 = 36

And,
The favorable outcomes are the combinations where the sum of the two dice is more than 8. These sums can be 9, 10, 11, or 12. The combinations for each sum are-
Sum of 9: (3, 6), (4, 5), (5, 4), (6, 3) - 4 outcomes
Sum of 10: (4, 6), (5, 5), (6, 4) - 3 outcomes
Sum of 11: (5, 6), (6, 5) - 2 outcomes
Sum of 12: (6, 6) - 1 outcome

∴ The total number of favorable outcomes is the sum of these outcomes = 4 + 3 + 2 + 1 = 10

∴ P(sum > 8) = Favorable Outcomes/Total Outcomes = 10/36 = 5/18

৬,০১৬.

Find the domain of f(x) = 1/(x + 3)

ক
- 3
খ
3
গ
x ≠ - 3
ঘ
R - {- 3}

ব্যাখ্যা

Question: Find the domain of f(x) = 1/x + 3

Solution:
দেওয়া আছে,
f(x) = 1/x + 3

আমরা জানি,
একটি ভগ্নাংশের হর (denominator) শূন্য হতে পারবে না।
অর্থাৎ,
⇒ x + 3 ≠ 0
∴ x ≠ - 3

সুতরাং, f(x) এর ডোমেইন = R - {- 3}

৬,০১৭.

The speed of a car increases by 2 kms after every one hour. If the distance travelling in the first one hour was 35 kms. what was the total distance travelled in 12 hours?

ক
456 kms
খ
482 kms
গ
552 kms
ঘ
556 kms
ঙ
None of these

ব্যাখ্যা

Question: The speed of a car increases by 2 kms after every one hour. If the distance travelling in the first one hour was 35 kms. what was the total distance travelled in 12 hours?

Solution:
Total distance travelled in 12 hours = (35 + 37 + 39 +..... upto 12 terms)
This is an A.P with first term, a = 35,
number of terms, n = 12
d = 2.

Required distance = (12/2){2 × 35 + (12 - 1) × 2}
= 6 (70 + 22)
= 552 kms.

৬,০১৮.

A train leaves Dhaka at 4.10 P.M. and reaches Cumilla at 7.25 PM. The average speed of the train is 40 km/hr. What is the distance from Dhaka to Cumilla?

ক
115 km
খ
120 km
গ
125 km
ঘ
130 km
ঙ
124 km

ব্যাখ্যা

Time taken = 3 hrs 15 min = 3(1/4) hrs = 13/4 hrs
∴ Required distance :
=(40×13/4) km=130 km

৬,০১৯.

Which number logically follows the sequence? 4 6 9 6 14 6 ........

ক
ক) 6
খ
খ) 17
গ
গ) 19
ঘ
ঘ) 21

ব্যাখ্যা

4 6 9 6 14 6 .......
এখানে, ২য়, ৪র্থ, ৬ষ্ঠ, ৮ম ......... ইত্যাদি অবস্থানে নির্দিষ্ট সংখ্যা 6 বিদ্যমান
অবশিষ্ঠ সংখ্যাগুলোর ধারাঃ 4 9 14 19
পার্থক্যঃ 5 5 5
সুতরাং নির্ণেয় সংখ্যাটি হচ্ছে 14 + 5 = 19.

৬,০২০.

In a shop, shirts are usually sold at 40% above the cost price. During a sale, the shopkeeper offers a discount of 10% off the usual selling price. If he manages to sell 72 shirts for tk 13,608, then his cost price per shirt, (in tk) is

ক
135 tk
খ
150 tk
গ
145 tk
ঘ
155 tk

ব্যাখ্যা

Question: In a shop, shirts are usually sold at 40% above the cost price. During a sale, the shopkeeper offers a discount of 10% off the usual selling price. If he manages to sell 72 shirts for tk 13,608, then his cost price per shirt, (in tk) is

Solution:
Let the CP of each shirt be tk 100, then SP = 140.

∴ New SP = (140 × 90)/100
= 126 tk

∴ When S.P. is tk 126, C.P. = tk 100
∴ When S.P. is tk 13608/72, then C.P. = (100/126) × (13608/72)
= 150 tk

৬,০২১.

Twice the age of X is thrice the age of Y. 8 years back, the difference between the ages of X and Y was 18 years. What is the present age of X?

ক
50 years
খ
53 years
গ
54 years
ঘ
55 years

ব্যাখ্যা

Question: Twice the age of X is thrice the age of Y. 8 years back, the difference between the ages of X and Y was 18 years. What is the present age of X?

Solution:
2 times the age of X = 3 times the age of Y
Difference between X and Y 8 years back = 18 years

2X = 3Y
⇒ X : Y = 3 : 2

Let the present age of X and Y be 3R and 2R respectively
⇒ 3R - 2R = 18
⇒ R = 18

∴ 3R = 3 × 18 = 54
∴ The present age of X is 54 years

৬,০২২.

ক
ক) 2
খ
খ) 3
গ
গ) 5
ঘ
ঘ) 7

ব্যাখ্যা

Question:

Solution:
{³√(2⁴)⁶} - 1
= (2⁴)^6/3 - 1
= (2⁴)² - 1
= (2⁴ + 1) (2⁴ - 1)
= 17 × 15
= 17 × 3 × 5

So, the smallest prime factor is 3

৬,০২৩.

A sum of money on compound interest amounts to Tk. 13200 in 3 years and Tk. 12000 in 2 years. The rate of interest per annum is?

ক
15%
খ
12%
গ
10%
ঘ
8.5%

ব্যাখ্যা

Question: A sum of money on compound interest amounts to Tk. 13200 in 3 years and Tk. 12000 in 2 years. The rate of interest per annum is?

Solution:
Let,
Sum = Tk. P
Rate = r% = r/100

ATQ,
P(1 + r/100)² = 12000............(1)
P(1 + r/100)³ = 13200.........(2)

Dividing equation (2) by (1):
1 + r/100 = 13200/12000
⇒ (100 + r)/100 = 11/10
⇒ 100 +r = (11 × 100)/10
⇒ 100 +r = 110
⇒ r = 110 - 100
∴ r = 10

৬,০২৪.

The students in three batches in school is in the ratio of 2 : 3 : 5. If 20 students in each batch are increased than the ratio changes to 4 : 5 : 7. The total number of students in the three before the increase was-

ক
150
খ
100
গ
90
ঘ
10

ব্যাখ্যা

Question: The students in three batches in school is in the ratio of 2 : 3 : 5. If 20 students in each batch are increased than the ratio changes to 4 : 5 : 7. The total number of students in the three before the increase was-

Solution:
Let, students in the three before the increase were 2x, 3x, 5x

After increase, 2x + 20, 3x + 20, 5x + 20

(2x + 20)/ (3x + 20) = 4/5
⇒ 10x + 100 = 12x + 80
⇒ 2x = 20
⇒ x = 10

The total number of students in the three before the increase was = (2x + 5x + 3x)
= 10x
= 10 × 10
= 100

৬,০২৫.

The sum of the squares of three numbers is 532 and the ratio of the first and the second as also of the second and the third is 3 : 2. The third number is -

ক
8
খ
12
গ
28
ঘ
25
ঙ
40

ব্যাখ্যা

Question: The sum of the squares of three numbers is 532 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 that,
First : Second = 3 : 2
Second : third = 3 : 2
= {3 × (2/3)} : {2 × (2/3)}
= 2 : (4/3)

∴ Ratio between the numbers = 3 : 2 : (4/3)
= 9 : 6 : 4
Let the numbers be 9x, 6x and 4x
Then,
⇒ (9x)² + (6x)² + (4x)² = 532
⇒ 81x² + 36x² + 16x² = 532
⇒ 133x² = 532
⇒ x² = 4
⇒ x = 2

So, third number = 4x = 4 × 2 = 8

৬,০২৬.

What is the difference between the simple interest on a principal of Tk. 2500 being calculated at 5% per annum for 3 years and 6% per annum for 4 years?

ক
Tk. 200
খ
Tk. 175
গ
Tk. 320
ঘ
Tk. 225
ঙ
None of these

ব্যাখ্যা

Question: What is the difference between the simple interest on a principal of Tk. 2500 being calculated at 5% per annum for 3 years and 6% per annum for 4 years?

Solution:
We know that,
SI₁ = Prn/100
= (2500 × 5 × 3)/100 ;[P = 2500, r = 5%, n = 3]
= 375

And,
SI₂ = Prn/100
= (2500 × 6 × 4)/100 ;[P = 2500, r = 6%, n = 4]
= 600

∴ Difference = SI₂ - SI₁
= 600 - 375
= 225

৬,০২৭.

How many years will it take for an investment of Tk.10000 to earn Tk. 1200 in simple interest rate of 6%?

ক
2.5
খ
4
গ
3
ঘ
2

ব্যাখ্যা

Question: How many years will it take for an investment of Tk.10000 to earn Tk. 1200 in simple interest rate of 6%?
(Officer Cash 2022 অনুযায়ী)

Solution:
Given that,
Principal, P = 10000
Simple Interest, SI = 1200
Rate of interest, r = 6%
Time, n = ?

We know,
n = I/Pr
= 1200/(10000 × 6%)
= (1200 × 100)/(10000 × 6)
= 2

So, it will take 2 years for the investment to earn Tk. 1200 at 6% simple interest.

৬,০২৮.

A mother is twice as old as her son. If 20 years ago, the age of the mother was 10 times the age of the son, what is the present age of the mother?

ক
38 years
খ
40 years
গ
43 years
ঘ
45 years

ব্যাখ্যা

Let the age of son = X years

∴Age of mother would be =2X

As per question 20 years ago;
10 (X -20) = 2X - 20
⇒ 10X - 200 = 2X - 20
⇒ 10X - 2X= - 20 + 200
⇒ 8X = 180
⇒ X= 180/8
= 22.5 years

∴Age of mother = 22.5 × 2 = 45 years.

৬,০২৯.

What is the greatest number that divides 84, 144 or 18 without any remainder?

ক
6
খ
12
গ
18
ঘ
24

ব্যাখ্যা

Question: What is the greatest number that divides 84, 144 or 18 without any remainder?

Solution:
HCF of the given numbers will be the greatest number which can divide 48, 84 and 144
18 = 2 × 3 × 3
84 = 2 × 2 × 3 × 7
144 = 2 × 2 × 2 × 2 × 3 × 3
∴ HCF = 2 × 3 = 6
Hence 6 is the greatest number which divides 18, 84 and 144 without leaving any remainder

৬,০৩০.

What is the profit of Tk. 650 in 6 years at the simple rate of profit Tk. 7 percent per annum?

ক
ক) 223 tk
খ
খ) 250 tk
গ
গ) 273 tk
ঘ
ঘ) 280 tk

ব্যাখ্যা

Question: What is the profit of Tk. 650 in 6 years at the simple rate of profit Tk. 7 percent per annum?

Solution:
profit = Pnr
= 650 × 6 × 7/100
= 273 tk.

৬,০৩১.

A and B are two alloys in which the ratios of gold and copper are 5 : 3 and 5 : 11 respectively. If these equal amounts of two alloys are melted and made into alloy C. What will be the ratio of gold and copper in alloy C?

ক
3 : 5
খ
5 : 11
গ
7 : 15
ঘ
15 : 17

ব্যাখ্যা

Question: A and B are two alloys in which the ratios of gold and copper are 5 : 3 and 5 : 11 respectively. If these equal amounts of two alloys are melted and made into alloy C. What will be the ratio of gold and copper in alloy C?

Solution:
Ratio of Gold and Copper in Alloy A = 5 : 3
Ratio of Gold and Copper in Alloy B = 5 : 11

Amount of Gold in Alloy A = 5/8
Amount of Gold In Alloy B = 5/16

Amount of Copper in A = 3/8
Amount of Copper in B = 11/16

∴ Amount of Gold In C = Amount of gold in A + Amount of gold in B
= (5/8) + (5/16)
= (10 + 5)/16
= 15/16

∴ Amount of Copper in C = Amount of Copper in A + Amount of Copper in B
= (3/8) + (11/16)
= (6 + 11)/16
= 17/16

∴ Ratio of Gold and Copper in C = (15/16) : (17/16)
= 15 : 17

৬,০৩২.

The product of two co-prime numbers is 820. Then their LCM is =?

ক
410
খ
1640
গ
820
ঘ
1230

ব্যাখ্যা

Question: The product of two co-prime numbers is 820. Then their LCM is =?

Solution:
If hcf of two or more numbers is 1, then two or more numbers are co-prime numbers.
HCF of co-prime number is always 1

∴ Product of number = LCM × HCF
⇒ LCM × 1 = 820
⇒ LCM = 820

৬,০৩৩.

If x - 1/x = √5, then x + 1/x =?

ক
√3
খ
3√3
গ
1/3
ঘ
3

ব্যাখ্যা

Question: If x - 1/x = √5, then x + 1/x =?

Solution:
Given,
x - 1/x = √5

We know,
(x + 1/x)² = (x - 1/x)² + 4x.1/x
⇒ (x + 1/x)² = ( √5)² + 4
⇒ (x + 1/x)² = 5 + 4
⇒ (x + 1/x)² = 9
⇒ (x + 1/x) = √9
∴ (x + 1/x) = 3

৬,০৩৪.

If x > 7 and y > - 3 then which of the following is true?

ক
ক) - x > 2y
খ
খ) xy < - 21
গ
গ) xy > - 21
ঘ
ঘ) x > - 2y

ব্যাখ্যা

Question: If x > 7 and y > - 3 then which of the following is true?

Solution:
Given,
x > 7 and y > - 3
Or, xy > 7 × (- 3)
Or, xy > - 21

৬,০৩৫.

If A bought an article at Tk. 200 and sold it to B at 20% profit. Again B sold the article at 10% profit to C. Find the amount paid by C.

ক
Tk. 250
খ
Tk. 256
গ
Tk. 264
ঘ
Tk. 268

ব্যাখ্যা

Question: If A bought an article at Tk. 200 and sold it to B at 20% profit. Again B sold the article at 10% profit to C. Find the amount paid by C.

Solution:
Price paid by B = 200 + (200/100 × 20) = 200 + 40 = 240

∴ Price paid by C = 240 + (240/100 × 10) = 240 + 24 = 264

৬,০৩৬.

Find a positive number which when increased by 11 is equal to 60 times the reciprocal of the number.

ক
3
খ
4
গ
6
ঘ
9

ব্যাখ্যা

Question: Find a positive number which when increased by 11 is equal to 60 times the reciprocal of the number.

Solution:
Let the number be x.
A positive number (x) increased by 11 is equal to 60 times the reciprocal of the number (1/x)

x + 11 = 60/x
⇒ x² + 11x - 60 = 0
⇒ x² + 15x - 4x - 60 = 0
⇒ x(x + 15) - 4(x + 15) = 0
⇒ (x + 15)(x - 4) = 0
∴x = 4

The positive number (x) = 4

৬,০৩৭.

A, B and C enter into a partnership with the capital in the ratio 7/2 : 4/3 : 6/5. After 4 months A increases his share of capital by 50%. If at the end of the year the total profit earned is Tk. 2430, find the share of each in the profit.

ক
ক) 1680, 580, 450
খ
খ) 1575, 450, 405,
গ
গ) 1460, 415, 380
ঘ
ঘ) 1245, 380, 240

ব্যাখ্যা

Ratio of capitals = 7/2 : 4/3 : 6/5 = (7/2 × 30) : (4/3 × 30) : (6/5 × 30)
= 105 : 40 : 36.
Let the initial capitals of A, B and C be Tk. 105x, 40x, 36x respectively.
Then, ratio of profit = [105x × 4 + (150% of 105x) × 8] : (40x × 12 ) : (36x × 12)
1680 : 480 : 432
= 35 : 10 : 9.
∴ A's share = Tk. (2430 × 35/54) = Tk. 1575; B's share = Tk (2430 × 10/54) = Tk. 450
C's share = Tk. (2430 × 9/54) = Tk. 405.

৬,০৩৮.

A runs twice as fast as B and B runs thrice as fast as C. The distance covered by C in 72 minutes, will be covered by A in :

ক
8 min
খ
12 min
গ
14 min
ঘ
16 min

ব্যাখ্যা

Question: A runs twice as fast as B and B runs thrice as fast as C. The distance covered by C in 72 minutes, will be covered by A in :

Solution:
Let C's speed = x km/hr
Then, B's speed = 3x km/hr
And A's speed = 6x km/hr
∴ Ratio of speeds of A, B, C
= 6x : 3x : x
= 6 : 3 : 1
Ratio of times taken
= 1/6 : 1/3 : 1
= 1 : 2 : 6

If C takes 6 minutes, then A takes 1 minute
If C takes 72 minutes, then A takes (1/6) × 72 min = 12 min

৬,০৩৯.

A container is 2/5 full. After adding 12 liters, it becomes 4/5 full. What is the total capacity of the container?

ক
25 liters
খ
60 liters
গ
30 liters
ঘ
40 liters
ঙ
None of these

ব্যাখ্যা

Question: A container is 2/5 full. After adding 12 liters, it becomes 4/5 full. What is the total capacity of the container?

Solution:
Given that,
The container is initially 2/5 full.
After adding 12 liters, it is 4/5 full.

That means,
(4/5) - (2/5) = 2/5
So, 2/5 of the container is equal to 12 liters.

Let total capacity be x liters,
⇒ 2x/5 = 12
⇒ x = (12 × 5)/2
⇒ x = 30

So the total capacity of the container is 30 liters.

৬,০৪০.

Three friends P, Q, and R share the cost of renting a storage unit. P stores goods weighing 300 kg for 4 months, Q stores 200 kg for 6 months, and R stores 400 kg for 3 months. If the total rent is Tk 840, how much should R pay as their share of rent?

ক
Tk. 280
খ
Tk. 190
গ
Tk. 300
ঘ
Tk. 220

ব্যাখ্যা

Question: Three friends P, Q, and R share the cost of renting a storage unit. P stores goods weighing 300 kg for 4 months, Q stores 200 kg for 6 months, and R stores 400 kg for 3 months. If the total rent is Tk 840, how much should R pay as their share of rent?

Solution:
Here,
P : Q : R = (300 × 4) : (200 × 6) : (400 × 3)
= 1200 : 1200 : 1200
= 1 : 1 : 1

∴ R's rent = {840 × (1/3)}
= Tk 280

৬,০৪১.

A two-digit number has 5 in its ten's digit. The sum of its digits is one-sixth of the number itself. What is the number?

ক
54
খ
56
গ
60
ঘ
58

ব্যাখ্যা

Question: A two-digit number has 5 in its ten's digit. The sum of its digits is one-sixth of the number itself. What is the number?

Solution:
ধরি,
একক স্থানীয় অঙ্ক = x
দশক স্থানীয় অঙ্ক = 5

∴ সংখ্যাটি = 50 + x

প্রশ্নমতে,
5 + x = (50 + x)/6
⇒ 6(5 + x) = 50 + x
⇒ 30 + 6x = 50 + x
⇒ 6x - x = 50 - 30
⇒ 5x = 20
∴ x = 4

∴ সংখ্যাটি = 50 + 4 = 54

৬,০৪২.

To gain 25% on selling sample of milk at the cost price of pure milk, the quantity of water to be mixed with 50 kg. of pure milk is-

ক
12.5 kg
খ
10 kg
গ
8.5 kg
ঘ
9 kg

ব্যাখ্যা

Question: To gain 25% on selling sample of milk at the cost price of pure milk, the quantity of water to be mixed with 50 kg. of pure milk is:

Solution:
Let the quantity of water mixed be x kg
Let CP of 1 kg of pure milk = Tk 1

Hence,
% gain = x × (100/50)
⇒ 25 = 100x/50
⇒ 100x = 1250
∴ x = 12.5

৬,০৪৩.

A product is bought and sold at the profit of 15% the ratio of selling and buying cost is -

ক
ক) 17:20
খ
খ) 23:20
গ
গ) 20:23
ঘ
ঘ) 20:17

ব্যাখ্যা

Let, the cost price be = 100 and selling price = 115
Their ratio = sp : cp = 115 : 100 = 23 : 20

৬,০৪৪.

3 pumps, working 8 hours a day, can empty a tank in 2 days, How many hours a day must 4 pumps work to empty the tank in 1 day?

ক
9 hours
খ
10 hours
গ
11 hours
ঘ
12 hours

ব্যাখ্যা

Question: 3 pumps, working 8 hours a day, can empty a tank in 2 days, How many hours a day must 4 pumps work to empty the tank in 1 day?

Solution:
3 pumps, working 8 hours a day, can empty a tank in 2 days
Formula used: M1 × T1 = M2 × T2
Where M1 and M2 is men and T1 and T2 is time

Calculation:
Let H hours be the number of hours required Applying the above formula
⇒ 3 × 8 × 2 = 4 × 1 × H
⇒ H = 48/4
⇒ H = 12 hours

∴ 4 pump need to work 12 hours to complete the work in 1 day.

৬,০৪৫.

Find the average of the first 20 natural numbers.

ক
ক) 10.5
খ
খ) 11
গ
গ) 11.5
ঘ
ঘ) 12

ব্যাখ্যা

Question: Find the average of the first 20 natural numbers.

Solution:
প্রথম ২০ টি স্বাভাবিক সংখ্যার সমষ্টি = ২০(২০ + ১)/২
= ২১০

∴ গড় = ২১০/২০ = ১০.৫

৬,০৪৬.

When two dice are rolled, what is the probability that the sum of the numbers appeared on them is 11?

ক
1/15
খ
2/13
গ
1/6
ঘ
1/18

ব্যাখ্যা

Question: When two dice are rolled, what is the probability that the sum of the numbers appeared on them is 11?

Solution:
n(S) = 6²
= 36
n(E) = {(5, 6), (6, 5)} = 2

∴ p(E) = n(E)/n(S)
= 2/36
= 1/18

৬,০৪৭.

The clock showed 10 : 30 in the mirror. What is the actual time?

ক
2 : 30
খ
1 : 30
গ
3 : 30
ঘ
4 : 30

ব্যাখ্যা

Question: The clock showed 10 : 30 in the mirror. What is the actual time?

Solution:
আমরা জানি,
প্রকৃতপক্ষে সময়
= 11 : 60 - আয়নায় দেখা সময়
= 11 : 60 - 10 : 30
= 1 : 30

৬,০৪৮.

Parrot : Cage : : Man : ?

ক
ক) Home
খ
খ) Motor Car
গ
গ) Prison
ঘ
ঘ) Forest

ব্যাখ্যা

Parrot is kept into Cage. Similarly Man is kept into Prison after trial

৬,০৪৯.

In a sugar-water solution, the ratio of water to sugar is 8 : 3. If you add 2 kgs of sugar, the ratio becomes 2 : 1. What is the amount of sugar in the original solution in kg?

ক
ক) 3
খ
খ) 4.5
গ
গ) 6
ঘ
ঘ) 8

ব্যাখ্যা

Let
amount of water be 8x.
So, the amount of sugar is 3x.
According to question,
8x/(3x + 2) = 2/1
Solving this equation, we get, x = 2
Therefore, the amount of sugar in the original solution = 3 × 2 = 6 kg.

৬,০৫০.

The diagonal of a rectangle is √41 cm and its area is 20 sq. cm. The perimeter of the rectangle must be:

ক
36 cm
খ
22 cm
গ
9 cm
ঘ
18 cm

ব্যাখ্যা

Question: The diagonal of a rectangle is √41 cm and its area is 20 sq. cm. The perimeter of the rectangle must be:

Solution:
Let the length and breadth of the rectangle be x cm and y cm.
We are given two conditions are,
Diagonal = √41 cm
By Pythagoras theorem:
√(x² + y²) = √41
⇒ x² + y² = 41 ........ (1)
And Area, xy = 20 cm² .......(2)

We know,
(x + y)² = x² + y² + 2xy
= 41 + 2 × 20
= 41 + 40
= 81
⇒ x + y = √81
∴ x + y = 9

∴ Perimeter = 2(x + y) = 2 × 9 = 18 cm

So the perimeter of the rectangle is 18 cm.

৬,০৫১.

If a² + b² = 45 and ab = 18, find (1/a) + (1/b).

ক
1/3
খ
1/2
গ
2/3
ঘ
1/4
ঙ
None of these

ব্যাখ্যা

Question: If a² + b² = 45 and ab = 18, find (1/a) + (1/b).

Solution:
দেওয়া আছে,
ab = 18
a² + b² = 45
⇒ (a + b)² - 2ab = 45
⇒ (a + b)² - 36 = 45
⇒ (a + b)² = 81
∴ a + b = 9

এখন,
(1/a) + (1/b)
= (b + a)/ab
= 9/18
= 1/2

৬,০৫২.

What is the least perfect square that is a multiple of 7, 11 and 12?

ক
421231
খ
242131
গ
223121
ঘ
213444

ব্যাখ্যা

Question: What is the least perfect square that is a multiple of 7, 11 and 12?

Solution:
Let us assume the least perfect square be X
⇒ 7 = 7 × 1
⇒ 11 = 11 × 1
⇒ 12 = 2 × 2 × 3

The LCM of (7, 11, 12) = 2× 2 × 3 × 11 × 7

⇒ The least perfect square = 2² × 3² × 11² × 7² = 213444
∴ The required result will be 213444.

৬,০৫৩.

The average of the first four multiples of 5 is:

ক
10
খ
12.5
গ
15
ঘ
17.5

ব্যাখ্যা

The first four multiples of 5 are 5, 10, 15 and 20.
Required average
= total sum of multiple of 5 / 4
= (5 + 10 + 15 + 20)/4
= 50/4
= 12.5

৬,০৫৪.

How many 3 digit integers are multiple of 5?

ক
ক) 178
খ
খ) 179
গ
গ) 180
ঘ
ঘ) 181
ঙ
ঙ) None

ব্যাখ্যা

Between 100 and 200 (and including 100) there are 21 numbers evenly divisible by 5.
201 to 900 only net 20 numbers per century range that qualify. So, now there are 20×7 = 140 numbers evenly divisible by 5.

In last 100 (901 to 999) we only have 19 (the 20th one is 1,000 which is not a 3 digit number).

So, in total, we have 21 + 140 + 19 = 180

৬,০৫৫.

When six fair coins are tossed simultaneously, in how many of the outcomes will at most three of the coins turn up as heads?

ক
42
খ
32
গ
64
ঘ
40

ব্যাখ্যা

Question: When six fair coins are tossed simultaneously, in how many of the outcomes will at most three of the coins turn up as heads?

Solution:
outcomes in which 0 coins turn heads are,
⁶C₀ = 1 outcome.
outcomes in which 1 coin turn head are,
⁶C₁ = 6 outcomes.
outcomes in which 2 coins turn heads are,
⁶C₂ = 15 outcomes.
outcomes in which 3 coins turn heads are,
⁶C₃ = 20 outcomes.
Therefore, the total number of outcomes
= 1 + 6 + 15 + 20
= 42 outcomes.

৬,০৫৬.

Which value of y will satisfy the given inequality, 2(y - 3) ≥ 3y - 4 ?

ক
y ≤ - 2
খ
y ≥ 2
গ
x < y
ঘ
x ≤ y

ব্যাখ্যা

Question: Which value of y will satisfy the given inequality, 2(y - 3) ≥ 3y - 4 ?

Solution:
Given,
2(y - 3) ≥ 3y - 4
⇒ 2y - 6 ≥ 3y - 4
⇒ 2y - 3y ≥ - 4 + 6
⇒ - y ≥ 2
⇒ y ≤ - 2

৬,০৫৭.

If 7^b= 343, then the value of 7^{(b - 2)}is:

ক
7
খ
14
গ
21
ঘ
49

ব্যাখ্যা

Question: If 7^b= 343, then the value of 7^{(b - 2)} is:

Solution:
Given that,
7^b = 343
⇒ 7^b = 7³
∴ b = 3

Now,
7^{(b - 2)}
= 7^{(3 - 2)}
= 7¹
= 7

৬,০৫৮.

A 320 metre long train crosses a platform twice its length in 48 seconds. What is the speed of the train in km/hr?

ক
55 km/hr
খ
64 km/hr
গ
84 km/hr
ঘ
72km/hr

ব্যাখ্যা

Question: A 320 metre long train crosses a platform twice its length in 48 seconds. What is the speed of the train in km/hr?

Solution:
দেওয়া আছে,
ট্রেনটির দৈর্ঘ্য = 320 মিটার
প্ল্যাটফর্মটির দৈর্ঘ্য = 2 × 320 = 640 মিটার

অতিক্রান্ত মোট দূরত্ব = (ট্রেনের দৈর্ঘ্য + প্ল্যাটফর্মের দৈর্ঘ্য)
= (320 + 640) মিটার
= 960 মিটার

সময় লেগেছে = 48 সেকেন্ড

∴ ট্রেনটির গতিবেগ = দূরত্ব / সময়
= 960 / 48 মিটার/সেকেন্ড
= 20 মিটার/সেকেন্ড

এখন,
1 মিটার/সেকেন্ড = (1/1000)/(1/3600) কিমি/ঘন্টা
= 3.6 কিমি/ঘন্টা

1 মিটার/সেকেন্ড = 3.6 কিলোমিটার/ঘন্টা
∴ 20 মিটার/সেকেন্ড = 20 × 3.6 কিলোমিটার/ঘন্টা
= 72 কিলোমিটার/ঘন্টা

সুতরাং, ট্রেনটির গতিবেগ 72 কিমি/ঘন্টা।

৬,০৫৯.

A man sells one camera A for Tk. 7500 at a gain of 20% and another camera B for Tk. 8550 at a loss of 5%. Find his total loss or gain%.

ক
ক) 2.7%
খ
খ) 5.2%
গ
গ) 4..2%
ঘ
ঘ) 5.1%

ব্যাখ্যা

Here,
we just know the selling price and the gain and loss incurred, on two cameras.
Therefore,
first calculate the cost price of both the cameras, because gain or loss is calculated on the cost price of the material.
When a shopkeeper earns profit, Cost Price = 100/(100 + Gain%) × S.P.
When shopkeeper incurs a loss, Cost Price = 100/(100 – Loss%) × S.P.

C.P. of camera A = 100/(100 +20) × 7500
= 100/120 × 7500
= Tk. 6250
C.P of camera B = 100/(100 – 5) × 8550
= 100/95 × 8550
= Tk. 9000

Total C.P. = Cost of camera A + Cost of camera B
Total C.P. = 6250 + 9000 = Tk. 15250
Total S.P. = 7500 + 8550 = Tk. 16050
Selling Price > Cost Price, hence man gains during this transaction.

Gain = S.P. – C.P.
= 16050 – 15250
= Tk. 800
Gain% = Gain/C.P. × 100
Gain% = 800/15250 × 100
= 5.24%

৬,০৬০.

A man's regular pay is Tk. 30 per hour up to 40 hours, Overtime is twice the regular payment. If he was paid Tk. 1,680, how many hours overtime did he work?

ক
ক) 7
খ
খ) 16
গ
গ) 9
ঘ
ঘ) 8

ব্যাখ্যা

ধরি,
Overtime করেছিল x ঘণ্টা

প্রশ্নমতে,
(30 × 40) + (30 × 2 × x ) = 1,680
⇒ 1,200 + 60x = 1,680
⇒ 60x = 1,680 - 1,200
⇒ 60x = 480
⇒ x = 480/60
x = 8

অতএব
সে ৪ ঘন্টা Overtime করেছিল।

৬,০৬১.

The area of a triangle with sides 3 cm, 5 cm, 6 cm is -

ক
ক) 28 cm²
খ
খ) 2√14 cm²
গ
গ) 3√14 cm²
ঘ
ঘ) √14 cm²

ব্যাখ্যা

Semi perimeter, s = (3 + 5 + 6)/2
= 7 cm
∴ Area = √{s(s - a)(s - b)(s - c)}
= √{7 (7 - 3) (7 - 5) (7 - 6)} Sq cm
= √(7 × 4 × 2 × 1)Sq cm
= 2√14 Sq cm

৬,০৬২.

Find HCF of 84, 126, and 210.

ক
24
খ
38
গ
40
ঘ
42

ব্যাখ্যা

Question: Find HCF of 84, 126, and 210.

Solution:
Factorize each number into prime factors:
84 = 2² × 3 × 7
126 = 2 × 3² × 7
210 = 2 × 3 × 5 × 7

HCF is the product of common prime factors with smallest powers:
Common primes: 2, 3, 7
Smallest powers: 2¹, 3¹, 7¹
HCF = 2 × 3 × 7 = 42

৬,০৬৩.

The angle of elevation of the top of a building from a point on the ground, which is 30 m away from the foot of the building, is 30°. The height of the building is:

ক
ক) 10 m
খ
খ) 30/√3 m
গ
গ) √3/10 m
ঘ
ঘ) 30 m
ঙ
ঙ) 10√5 m

ব্যাখ্যা

Say x is the height of the building.
a is a point 30 m away from the foot of the building.
Here, height is the perpendicular and distance between point a and foot of building is the base.
The angle of elevation formed is 30°.

Hence,
tan 30° = perpendicular/base = x/30
1/√3 = x/30
x = 30/√3

৬,০৬৪.

An amount of Tk. 735 was divided between A, B and C. If each of them had received Tk. 25 less, their shares would have been in the ratio of 1 : 3 : 2. The money received by C was :

ক
ক) 195
খ
খ) 200
গ
গ) 225
ঘ
ঘ) 245

ব্যাখ্যা

ধরি,
A. পায় (x - 25) টাকা,
B পায় (3x - 25) টাকা
C পায় (2x - 25) টাকা।
প্রশ্নমতে,
(x - 25) + (3x - 25) + (2x - 25) = 735
6x - 75 = 735
6x = 735 + 75
6x = 810
x = 135

C পায় = 2x - 25 টাকা
= (2 × 135 - 25) টাকা
= 245 টাকা

৬,০৬৫.

A boat takes 8 hours to travel 32 km upstream (against the current). If it were traveling downstream (with the current), it would take only 4 hours to cover the same distance. What is the speed of the current?

ক
3 km/hr
খ
2.5 km/hr
গ
2 km/hr
ঘ
1 km/hr
ঙ
none of these

ব্যাখ্যা

Question: A boat takes 8 hours to travel 32 km upstream (against the current). If it were traveling downstream (with the current), it would take only 4 hours to cover the same distance. What is the speed of the current?

Solution:
দেওয়া আছে,
স্রোতের প্রতিকূলে 32 কিমি যেতে সময় লাগে 8 ঘণ্টা।
∴ প্রতিকূলে নৌকার গতিবেগ = 32/8 = 4 কিমি/ঘণ্টা।

আবার, স্রোতের অনুকূলে 32 কিমি যেতে সময় লাগে 4 ঘণ্টা।
∴ অনুকূলে নৌকার গতিবেগ = 32/4 = 8 কিমি/ঘণ্টা।

আমরা জানি,
স্রোতের গতিবেগ = (অনুকূলে গতিবেগ - প্রতিকূলে গতিবেগ) / 2
= (8 - 4)/2 কিমি/ঘণ্টা
= 4/2 কিমি/ঘণ্টা
= 2 কিমি/ঘণ্টা

সুতরাং, স্রোতের গতিবেগ 2 কিমি/ঘণ্টা।

৬,০৬৬.

If sec(3x - 40°) = cosec(50° - x), then the value of x is?

ক
10°
খ
20°
গ
30°
ঘ
40°

ব্যাখ্যা

Question: If sec(3x - 40°) = cosec(50° - x), then the value of x is?

Solution:
sec(3x - 40°) = cosec(50° - x)
⇒ sec(3x - 40°) = cosec{90° - (40° + x)}
⇒ sec(3x - 40°) = sec(40° + x)
⇒ 3x - 40° = 40° + x
⇒ 2x = 80°

∴ x = 40°

৬,০৬৭.

A salesman sells two shirts at BDT 594 each. On one he gains 10% and on the other, he loses 10%. Find his gain or loss percent on the whole.

ক
ক) 1% loss
খ
খ) 1% profit
গ
গ) 99% loss
ঘ
ঘ) 99% profit

ব্যাখ্যা

At 10% profit, The cost price of the first shirt
= 100/(100 + 10) × 594
= Tk. 540

At 10% loss, The cost price of the second shirt
= 100/(100 - 10) × 594
= Tk. 660

Now, the total cost price of two shirts
= 540 + 660
= Tk. 1200

and the total sell price of two shirts
= 594 × 2
= Tk. 1188

Loss = 1200 - 1188
= Tk. 12

Percentage of loss = (12/1200 × 100)%
= 1%
--------------------------------------------------------
Shortcut:
Loss percentage = (common gain or loss/10)% = (10/10)% = 1%
[ When a salesman sells two shirts at same price each and on one there is gain is equal to on other there is loss in percentage, then there shall be always loss percent on the whole and loss percentage will be (common gain or loss/10)% ]

৬,০৬৮.

What is the difference between the 6 digit largest and smallest numbers?

ক
ক) 888889
খ
খ) 899999
গ
গ) 999888
ঘ
ঘ) 988888

ব্যাখ্যা

6 অঙ্কের বৃহত্তম সংখ্যা = 999999
6 অঙ্কের ক্ষুদ্রতম সংখ্যা = 100000
পার্থক্য = 999999 - 100000
= 899,999

৬,০৬৯.

To finish a work, A sets aside half additional time than B. In the event that together they take 18 days to finish the work, what amount of time might B take to do it?

ক
ক) 30 days
খ
খ) 35 days
গ
গ) 40 days
ঘ
ঘ) 45 days

ব্যাখ্যা

Suppose B takes x days. Then,
A takes 150x/100 days, i.e. 3x/2 days
∴1/x+ 2/3x= 1/18 ⇒5/3x= 1/18 ⇒3x = 90 ⇒x= 30
Hence B takes 30 days.

৬,০৭০.

If LATER = 13579 and CHAIR = 20349, then CHEAT = ?

ক
20735
খ
20375
গ
20753
ঘ
20345

ব্যাখ্যা

Question: If LATER = 13579 and CHAIR = 20349, then CHEAT = ?

Solution:
Given that,
L A T E R
↓ ↓ ↓ ↓ ↓
1 3 5 7 9

and
C H A I R
↓ ↓ ↓ ↓ ↓
2 0 3 4 9

So
C H E A T
↓ ↓ ↓ ↓ ↓
2 0 7 3 5

৬,০৭১.

An office opens at 9 a.m. and closes at 5.30 p.m. with a lunch break of 17 minutes. What is the ratio of lunch breaks to the total period in the office?

ক
1 : 30
খ
1 : 17
গ
3 : 10
ঘ
5 : 27

ব্যাখ্যা

Question: An office opens at 9 a.m. and closes at 5.30 p.m. with a lunch break of 17 minutes. What is the ratio of lunch breaks to the total period in the office?

Solution:
The ratio of lunch breaks to the total period in the office = 17/{(8 × 60) + 30}
= 17/510
= 1/30
= 1 : 30

৬,০৭২.

The difference between simple and compound interest at 5% annually on a sum of TK. 10000 at the end of 2 years -

ক
ক) 25
খ
খ) 50
গ
গ) 100
ঘ
ঘ) 250

ব্যাখ্যা

SI = 10000 × 2 × 5/100 = 1000
CI = 10000(1 + 5/100)² – 10000 = 1025
∴ Difference = 1025 - 1000 = 25

৬,০৭৩.

Which number replace the question mark?

ক
44
খ
48
গ
54
ঘ
64

ব্যাখ্যা

Question: Which number replace the question mark?

Solution:
In the first figure
(12 + 18 + 30) ÷ 2 = 30

In the second figure
(16 + 32 + 40) ÷ 2 = 44

Therefore, the third figure
(36 + 18 + 34) ÷ 2 = 44

৬,০৭৪.

সুদের হার দশমিক ৭৫ শতাংশ হ্রাস পাওয়াতে একজন আমানতকারীর আমানতের উপর ৪ বছরের প্রাপ্ত আয় ৭৫০ টাকা কমে যায়। তার আমানতের মোট পরিমাণ কত?

ক
২৫,০০০ টাকা
খ
১৮,৭৫০ টাকা
গ
১,০০,০০০ টাকা
ঘ
৩০,০০০ টাকা

ব্যাখ্যা

প্রশ্ন: সুদের হার দশমিক ৭৫ শতাংশ হ্রাস পাওয়াতে একজন আমানতকারীর আমানতের উপর ৪ বছরের প্রাপ্ত আয় ৭৫০ টাকা কমে যায়। তার আমানতের মোট পরিমাণ কত?

সমাধান:
৪ বছরে কমে = ৭৫০ টাকা
১ বছরে কমে = ৭৫০/৪ = ১৮৭.৫ টাকা

০.৭৫ টাকা হ্রাস পেলে আমানত = ১০০ টাকা
১ টাকা হ্রাস পেলে আমানত = ১০০/০.৭৫ টাকা
∴ ১৮৭.৫ টাকা হ্রাস পেলে আমানত = (১০০ × ১৮৭.৫)/০.৭৫ টাকা
= ২৫,০০০ টাকা

৬,০৭৫.

At what rate percent per annum will the simple interest on a sum of money 2/5 of the amount in 10 years?

ক
ক) 4%
খ
খ) 5%
গ
গ) 6%
ঘ
ঘ) 10%

ব্যাখ্যা

Question: At what rate percent per annum will the simple interest on a sum of money 2/5 of the amount in 10 years?

Solution:
Let the principal be p Tk
So, interest = 2p/5 Tk
Time = 10 years
rate of interest, r =?

Now,
I = pnr
r = I / pn
= (2p/5) / (p × 10)
= (1/25) × 100%
= 4%

৬,০৭৬.

Two inlet pipes can fill a tank in 10 hours and 20 hours, respectively. An outlet pipe is attached to these two pipes, and thus, the tank was filled in 12 hours. In 90 hours, the outlet pipe alone can empty how many tanks?

ক
5 tanks
খ
6 tanks
গ
8 tanks
ঘ
4 tanks

ব্যাখ্যা

Question: Two inlet pipes can fill a tank in 10 hours and 20 hours, respectively. An outlet pipe is attached to these two pipes, and thus, the tank was filled in 12 hours. In 90 hours, the outlet pipe alone can empty how many tanks?

সমাধান:
ধরি,
ছিদ্র নলটি (Outlet Pipe) একা ট্যাঙ্কটি খালি করতে P ঘন্টা সময় নেয়।

তিনটি নল একত্রে 1 ঘন্টায় পূর্ণ করে = 1/10 + 1/20 - 1/P অংশ।
প্রশ্নমতে, তিনটি নল একত্রে 12 ঘন্টায় পূর্ণ করে।
∴ 1/10 + 1/20 - 1/P = 1/12

১. ছিদ্র নলটির সময় (P) নির্ণয়:
⇒ 1/P = 1/10 + 1/20 - 1/12
হরগুলির (Denominator) ল.সা.গু. (LCM) হলো 60।
⇒ 1/P = (6 + 3 - 5)/60
⇒ 1/P = 4/60
⇒ 1/P = 1/15
⇒ P = 15 ঘন্টা।

90 ঘন্টায় যতগুলি ট্যাঙ্ক খালি করতে পারে = 90 / P
= 90/15
= 6 টি ট্যাঙ্ক।
∴ 90 ঘন্টায় ছিদ্র নলটি একা 6 টি ট্যাঙ্ক খালি করতে পারে।

৬,০৭৭.

If a + b = 12 and a - b = 4, then a² - b² =

ক
ক) 24
খ
খ) 36
গ
গ) 48
ঘ
ঘ) 56

ব্যাখ্যা

a² - b²
= (a + b)(a - b)
= 12 × 4
= 48

৬,০৭৮.

In an examination, 80% of candidates passed in English, and 85% of candidates passed in Mathematics. If 73% of candidates passed in both these subjects, then what percent of candidates failed in both subjects?

ক
5%
খ
7%
গ
8%
ঘ
10%

ব্যাখ্যা

Question: In an examination, 80% of candidates passed in English, and 85% of candidates passed in Mathematics. If 73% of candidates passed in both these subjects, then what percent of candidates failed in both subjects?

Solution:
Students passed in English = 80%
Students passed in Math = 85%
Students passed in both subjects = 73%
Then, the number of students who passed in at least one subject
= (80 + 85) - 73
= 92%

Thus, students failed in both subjects = 100 - 92
= 8%

৬,০৭৯.

P and Q start a hotel. P left after 6 months. After the years ends P gets Tk. 6,000 as profit out of total profit of Tk. 9,000. What will be amount invested by Q if P had invested Tk. 20,000?

ক
ক) Tk. 6,000
খ
খ) Tk. 5,000
গ
গ) Tk. 4,500
ঘ
ঘ) Tk. 5,500

ব্যাখ্যা

মোট মুনাফা =৯০০০ টাকা
P মুনাফা পেল = ৬০০০ টাকা
Q মুনাফা পাবে = (৯০০০ - ৬০০০) টাকা
= ৩০০০ টাকা

P এবং Q এর মুনাফার অনুপাত = ৬০০০ : ৩০০০ = ২ : ১

Q বিনিয়োগ করেছিল x টাকা
প্রশ্নমতে,
২০০০০ × ৬ : x × ১২ = ২ :১
২০০০০ : ২x = ২ : ১
২০০০০/২x = ২/১
৪x = ২০০০০
x = ২০০০০/৪
x = ৫০০০

৬,০৮০.

A square is inscribed in a circle of diameter 2a and another square is circumscribing circle. The difference between the areas of outer and inner squares is-

ক
ক) a²
খ
খ) 2a²
গ
গ) 3a²
ঘ
ঘ) 4a²

ব্যাখ্যা

Question: A square is inscribed in a circle of diameter 2a and another square is circumscribing circle. The difference between the areas of outer and inner squares is-

Solution:

দেওয়া আছে
বৃত্তের ব্যাস = 2a
অভ্যন্তরীণ বর্গের কর্ণ = 2a
অভ্যন্তরীণ বর্গের এক বাহু = x

প্রশ্নমতে
√2x = 2a
x = 2a/√2
x = √2a

অভ্যন্তরীণ বর্গের ক্ষেত্রফল = (√2a)²
= 2a²

অভ্যন্তরীণ বর্গের কর্ণ = বহিঃস্থ বর্গের এক বাহু = 2a
বহিঃস্থ বর্গের ক্ষেত্রফল = (2a)² = 4a²

ক্ষেত্রফলের পার্থক্য = 4a² - 2a²
= 2a²

৬,০৮১.

A metal cube of edge 12 cm is melted and formed into three smaller cubes. If the edges of two smaller cubes are 6cm and 8 cm, find the edges of the third smaller cube.

ক
ক) 8 cm
খ
খ) 10 cm
গ
গ) 12 cm
ঘ
ঘ) 15 cm

ব্যাখ্যা

Let,
The edge of the third small cube be x cm
The volume of the cube = (edge)³
According to the question,
6³ + 8³ + x³ = 12³
⇒ 216 + 512 + x³ = 1728
⇒ x³ = 1728 - 728
= 1000
⇒ x = 1000^(1/3)
⇒ x = 10 cm.

৬,০৮২.

When the selling price is doubled, the profit becomes threefold. What is the percentage profit?

ক
65%
খ
100%
গ
120%
ঘ
250%

ব্যাখ্যা

Question: When the selling price is doubled, the profit becomes threefold. What is the percentage profit?

Solution:
ধরি,
ক্রয়মূল্য = 100 টাকা
বিক্রয়মূল্য = (100 + x) টাকা

∴ লাভ = (100 + x) - 100 টাকা = x টাকা

প্রশ্নমতে,
3x = 2(100 + x) - 100
⇒ 3x = 200 + 2x - 100
⇒ 3x = 100 + 2x
⇒ 3x - 2x = 100
⇒ x = 100

∴ লাভ = 100 টাকা

এখন,
লাভের হার = (লাভ/ক্রয়মূল্য) × 100%
= (100/100) × 100%
= 100%

∴ লাভের শতকরা হার = 100%

৬,০৮৩.

Four bells ring simultaneously at the start and then at intervals of 6 seconds, 12 seconds, 15 seconds, and 20 seconds respectively. How many times do they ring together in 2 hours?

ক
120 times
খ
119 times
গ
122 times
ঘ
121 times

ব্যাখ্যা

Question: Four bells ring simultaneously at the start and then at intervals of 6 seconds, 12 seconds, 15 seconds, and 20 seconds respectively. How many times do they ring together in 2 hours?

Solution:
Given that,
Four bells ring simultaneously at starting and an interval of 6 sec, 12 sec, 15 sec and 20 sec respectively.

∴ LCM of (6, 12, 15, 20) = 60
All 4 bells ring together again after every 60 seconds

Now, In 2 Hours, they ring together = [(2 × 60 × 60)/60] times + 1 (at the starting)
= (120 + 1) times
= 121 times

∴ In 2 hours they ring together for 121 times

৬,০৮৪.

If X is the difference of the squares of two consecutive even numbers. Which of the following numbers is a divisor of X?

ক
4
খ
7
গ
8
ঘ
9
ঙ
None of these

ব্যাখ্যা

Question: If X is the difference of the squares of two consecutive even numbers. Which of the following numbers is a divisor of X?

Solution:
ধরি
দুটি ক্রমিক জোড় সংখ্যা 2n এবং (2n + 2)

∴ x = (2n + 2)² - (2n)²
= (2n + 2 + 2n)(2n + 2 - 2n)
= 2(4n + 2)
= 2 × 2(2n + 1)
= 4(2n + 1) ;যা 4 দ্বারা বিভাজ্য।

৬,০৮৫.

A sector of a circle of radius 13 cm is recast into a right circular cone of height 12 cm. What is the volume of the resulting cone?

ক
ক) 14π cm³
খ
খ) 32π cm³
গ
গ) 33.33π cm³
ঘ
ঘ) 64.67π cm³

ব্যাখ্যা

r = √(13² - 12²)
= 5

So, the volume is V = 1/3∏r²h
= 1/3Π × 5² × 4
= 33.33Π cm³

৬,০৮৬.

If ⁿC₁₀ = ⁿC₈, what is the value of ⁿC₂ = ?

ক
81
খ
136
গ
153
ঘ
180

ব্যাখ্যা

Question: If ⁿC₁₀ = ⁿC₈, what is the value of ⁿC₂ = ?

Solution:
আমরা জানি,
যদি ⁿC_a = ⁿC_b হয়, তবে হয় a = b অথবা a + b = n হবে।

এখানে,
ⁿC₁₀ = ⁿC₈
⇒ 10 + 8 = n
⇒ n = 18

∴ ⁿC₂ = ¹⁸C₂
= 18!/(2!(18 - 2)!)
= 18!/(2! × 16!)
= (18 × 17 × 16!)/(2 × 1 × 16!)
= (18 × 17)/2
= 9 × 17
= 153

৬,০৮৭.

If dividing P(x) = 4x³- 7x² + bx - 5 by (x - 2) results in the remainder 13, then find the value of b.

ক
11
খ
8
গ
7
ঘ
- 5

ব্যাখ্যা

Question: If dividing P(x) = 4x³ - 7x² + bx - 5 by (x - 2) results in the remainder 13, then find the value of b.

Solution:
According to the Remainder Theorem, if a polynomial P(x) is divided by (x - c), then the remainder = P(c).
Here divisor is (x - 2).
So remainder = P(2).

Now,
P(2) = 4(2)³ - 7(2)² + b(2) - 5

= 4 × 8 - 7 × 4 + 2b - 5

= 32 - 28 + 2b - 5

= 32 + 2b - 33
= 2b - 1

According to the question, the remainder is 10.
So, 2b - 1 = 13

⇒ 2b = 13 + 1
⇒ 2b = 14
⇒ b = 7

৬,০৮৮.

What percentage of numbers from 1 to 50 has 3 or 7 in the unit's digit?

ক
25%
খ
12%
গ
15%
ঘ
20%

ব্যাখ্যা

Question: What percentage of numbers from 1 to 50 has 3 or 7 in the unit's digit?

Solution:
We can list the numbers from 1 to 50 and identify those that have 3 or 7 in the unit's place:

Numbers with 3 in the unit's digit:3, 13, 23, 33, 43
Numbers with 7 in the unit's digit: 7, 17, 27, 37, 47

So, the numbers with 3 or 7 in the unit's digit are:
3, 7, 13, 17, 23, 27, 33, 37, 43, 47

There are 10 such numbers.

There are 50 numbers in total from 1 to 50. The percentage of numbers that have 3 or 7 in the unit's digit is: (10/50) × 100%
= 20%

৬,০৮৯.

A trader marked the price of his commodity so as to include a profit of 25%. He allowed a discount of 16% on the marked price. His actual profit was:

ক
5%
খ
10%
গ
12%
ঘ
20%

ব্যাখ্যা

Question: A trader marked the price of his commodity so as to include a profit of 25%. He allowed a discount of 16% on the marked price. His actual profit was:

Solution:
let, cost price 100 taka
marked price = 100 + 25
= 125 taka

After 16% discount selling price = 125 - 125 × 16/100
= 125 - 20
= 105 taka

actual profit = (105 - 100) = 5 taka

৬,০৯০.

If x + y = 5 and x - y = 1 then what is the value of xy?

ক
6
খ
8
গ
10
ঘ
12

ব্যাখ্যা

Question: If x + y = 5 and x - y = 1 then what is the value of xy?

Solution:
Given that,
x + y = 5
x - y = 1

∴ xy = {(x + y)/2}² - {(x - y)/2}²
= (5/2)² - (1/2)²
= 25/4 - 1/4
= (25 - 1)/4
= 24/4
= 6

৬,০৯১.

There are 130 mangoes and guavas in a basket and their ratio is 3 : 2 respectively. To make the ratio of mango and guava 1 : 1 in that basket, how many new fruits should be added?

ক
২৬
খ
২১
গ
২২
ঘ
৩০
ঙ
৩২

ব্যাখ্যা

আম ও পেয়ারার অনুপাত = ৩:২
তাহলে আম আছে = ১৩০ এর ৩/ ৫ = ৭৮ টি
পেয়ারা আছে = ১৩০ এর ২/৫ = ৫২ টি
তাহলে অনুপাত ১:১ হতে হলে নতুন পেয়ারা যোগ করতে হবে = ৭৮-৫২ = ২৬ টি

৬,০৯২.

Which of the following fractions is greater than 1/2 and less than 3/4?

ক
ক) 9/10
খ
খ) 4/5
গ
গ) 2/3
ঘ
ঘ) None of the above

ব্যাখ্যা

প্রশ্ন : Which of the following fractions is greater than 1/2 and less than 3/4?

সমাধান :

3/4 = 0.75
1/2 = 0.50
2/3 = 0.66
4/5 = 0.80
9/10 = 0.9

৬,০৯৩.

The number 75 can be written as the sum of the squares of 3 different positive integers. What is the sum of these 3 integers?

ক
12
খ
13
গ
14
ঘ
15

ব্যাখ্যা

Question: The number 75 can be written as the sum of the squares of 3 different positive integers. What is the sum of these 3 integers?

Solution:
75
= 49 + 25 + 1
= 7² + 5² + 1²

sum = 7 + 5 + 1 = 13

৬,০৯৪.

The population of a town is 20,000, and it increases by 5% each year. What will the population be after 3 years?

ক
10,153
খ
13,153
গ
23,153
ঘ
25000
ঙ
None

ব্যাখ্যা

Question: The population of a town is 20,000, and it increases by 5% each year. What will the population be after 3 years?

Solution:
We can use the compound interest formula for population growth:

Population after n years = P × [1 + (r/100)]ⁿ

Here,
P = 20,000, r = 5%, n = 3

∴ Population after 3 years = 20,000 × [1 + (5/100)]³= 20,000 × (105/100)³
= 20,000 × 1.157625
= 23,152.5

∴ The population of the town after 3 years will be approximately 23,153.

৬,০৯৫.

Mr X will be the Chairman of the committee . In how many ways can a committee of 5 members be chosen from a total of 8 people given that Mr. X must be one them?

ক
ক) 35
খ
খ) 70
গ
গ) 120
ঘ
ঘ) 56

ব্যাখ্যা

As, Mr. x is always chosen, then to form a committee of 5 members from the pool of 8, we now have to choose 4 members from 7 people

So, the committee can be formed in ⁷c₄ = 35 ways

৬,০৯৬.

Find the difference between the compound interest (compounded annually) and the simple interest on Tk. 2000 for 3 years at the rate of 8% per annum.

ক
Tk. 39.42
খ
Tk. 19.42
গ
Tk. 29.42
ঘ
Tk. 52

ব্যাখ্যা

Simple interest = Pnr
= 2000 × 3 × 8%
= Tk. 480
A = P(1 + r)ⁿ
= 2000(1 + 8/100)³
= Tk. 2519.42
Compound interest = Tk. 2519.42 - Tk. 2000
= Tk. 519.42
Required difference = Tk. 519.42 - Tk. 480
= Tk. 39.42

৬,০৯৭.

For x = 10, determine which among the following expressions attains the minimum value.

ক
x - 2
খ
x/2
গ
2 - x
ঘ
x/5

ব্যাখ্যা

Question: For x = 10, determine which among the following expressions attains the minimum value.

Solution:
ক) x - 2 = 10 - 2 = 8

খ) x/2 = 10/2 = 5

গ) 2 - x = 2 - 10 = - 8

ঘ) x/5 = 10/5 = 2

সুতরাং,
2 - x এর মান সবচেয়ে কম।

৬,০৯৮.

If x : y = 15 : 4 then the value of (x - y)/(x + y) is-

ক
1/19
খ
11/17
গ
11/15
ঘ
11/19

ব্যাখ্যা

Question: If x : y = 15 : 4 then the value of (x - y)/(x + y) is-

Solution:
x : y = 15 : 4
⇒ x/y = 15/4
⇒ (x + y)/(x - y) = (4 + 15)/(15 - 4)
⇒ (x + y)/(x - y) = 19/11
⇒ (x - y)/(x + y) = 11/19

৬,০৯৯.

A committee of 5 members is to be selected from 7 men and 4 women. In how many ways can this be done if exactly 3 men must be selected?

ক
210
খ
360
গ
420
ঘ
280

ব্যাখ্যা

Question: A committee of 5 members is to be selected from 7 men and 4 women. In how many ways can this be done if exactly 3 men must be selected?

Solution:
এখানে, একটি কমিটি গঠন করতে হলে 3 জন পুরুষ এবং (5 - 3) = 2 জন মহিলা নির্বাচন করতে হবে।

7 জন পুরুষ থেকে 3 জন পুরুষ নির্বাচন করার উপায়:
⁷C₃ = 7!/{3! (7 - 3)!}
= (7 × 6 × 5)/(3 × 2 × 1)
= 35 টি

4 জন মহিলা থেকে 2 জন মহিলা নির্বাচন করার উপায়:
⁴C₂ = 4!/{2! (4 - 2)!}
= (4 × 3)/(2 × 1)
= 6 টি
সুতরাং, মোট সম্ভাব্য উপায় = 35 × 6 = 210 টি।

অতএব, কমিটি গঠনের মোট উপায় হলো 210 টি।

৬,১০০.

If the perimeter of an isosceles right-triangle is (6 + 3√2)m, then the area of the triangle is-

ক
ক) 9.0m²
খ
খ) 2.5m²
গ
গ) 4.5m²
ঘ
ঘ) 6.0m²

ব্যাখ্যা

Here
b²= a² +a²
b = √(a² + a²)
b = a√2

Now
a + a + b = 6 + 3√2
2a + a√2 = 6 + 3√2
a(2 + √2) = 3(2 + √2)
a = 3

Area = (1/2) × 3 × 3 = 9/2 = 4.5m²