Bank Math

PrepBank · পাতা ৯৬ / ১৬১ · ৯,৫০১–৯,৬০০ / ১৬,১২৪

৯,৫০১.

Find the ratio in which rice at Tk. 7.20 a kg be mixed with rice at Tk. 5.70 a kg to produce a mixture worth Tk. 6.30 a kg-

ক
ক) 6 : 8
খ
খ) 2 : 3
গ
গ) 3 : 2
ঘ
ঘ) 2 : 5

ব্যাখ্যা

Question: Find the ratio in which rice at Tk. 7.20 a kg be mixed with rice at Tk. 5.70 a kg to produce a mixture worth Tk. 6.30 a kg-

Solution:
Let, Tk. 7.20 per kg rice is X kg
and Tk. 5.70 per kg rice is Y kg

ATQ,
7.2X + 5.7Y = 6.3(X + Y)
7.2X + 5.7Y = 6.3X + 6.3Y
0.9X = 0.6Y

X : Y = 0.6 : 0.9 = 2 : 3

৯,৫০২.

How many shares of market value Tk. 25 each can be purchased for Tk. 20,400, brokerage being 2% ?

ক
500
খ
700
গ
600
ঘ
800

ব্যাখ্যা

Question: How many shares of market value Tk. 25 each can be purchased for Tk. 20,400, brokerage being 2% ?

Solution:
CP of each share = Tk. (25 + 2% of 25) = Tk. 25.50.

Number of shares= 20400/25.50
= 800

৯,৫০৩.

If x-1/x = 5, the value of (x+1/x)² is:

ক
ক) 25
খ
খ) 21
গ
গ) 19
ঘ
ঘ) 29

ব্যাখ্যা

আমরা জানি, (x+1/x)² = (x-1/x)²+4×X×(1/X
= 5²+4
= 29

৯,৫০৪.

How many terms are there in 20, 25, 30, 35, ….140?

ক
20
খ
22
গ
25
ঘ
27

ব্যাখ্যা

Question: How many terms are there in 20, 25, 30, 35, ….140?

Solution:
Number of terms, = (1st term - last term / common difference) + 1
= (140 − 20)/5 + 1
= 120/5 + 1
= 24 + 1
= 25

৯,৫০৫.

On dividing an individual number by 342, we get 47 as remainder. If the same number is divided by 18, what will be the remainder?

ক
ক) 19
খ
খ) 15
গ
গ) 13
ঘ
ঘ) 11

ব্যাখ্যা

Consider the number be 342 + 47 = 389.
Therefore 389 divided by 18 leave remainder 11

৯,৫০৬.

A man travelled a distance of 1260 km in 17 hours. He travelled partly by car at a speed of 40 km/h, and partly by train at a speed of 80 km/h. What is the distance traveled by the train?

ক
100 km
খ
340 km
গ
640 km
ঘ
1160 km
ঙ
1360 km

ব্যাখ্যা

Question: A man travelled a distance of 1260 km in 17 hours. He travelled partly by car at a speed of 40 km/h, and partly by train at a speed of 80 km/h. What is the distance traveled by the train?

Solution:
Given,
Total distance = 1260 km
Total time = 17 hours
Speed of car = 40 km/h
Speed of train = 80 km/h

Let,
Distance travelled by car = x km
Distance travelled by train = (1260 - x) km

ATQ,
(x/40) + [(1260 - x)/80] = 17
⇒ 2x + 1260 - x = 17 × 80
⇒ x = 1360 - 1260
⇒ x = 100

∴ Distance traveled by the train = (1260 - 100) km
= 1160 km

৯,৫০৭.

If one-third of one-fourth of a number is 15, then three-tenth of that number is-

ক
48
খ
54
গ
72
ঘ
60
ঙ
64

ব্যাখ্যা

Question: If one-third of one-fourth of a number is 15, then three-tenth of that number is-

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

So,required number is = (3/10) × 180 = 54

৯,৫০৮.

There are 71 members in group A, 18 members in group B and 53 members in group C. All the members of these groups went to a restaurant. The average amount spent on each member of group A, B and C is Tk.397, Tk.421 and Tk.137 respectively. The total average amount (in Tk.) spent per member is-

ক
Tk. 295
খ
Tk. 335
গ
Tk. 402
ঘ
Tk. 303

ব্যাখ্যা

Question: There are 71 members in group A, 18 members in group B and 53 members in group C. All the members of these groups went to a restaurant. The average amount spent on each member of group A, B and Cis Tk.397, Tk.421 and Tk.137 respectively. The total average amount (in Tk.) spent per member is-

Solution:
Given that,

Members in A = 71, B = 18, C = 53

Average spent per member: A = Tk. 397, B = Tk. 421 and C = Tk. 137

Now,
Total amount by A = 71 × 397 = 28187
Total amount by B = 18 × 421 = 7578
Total amount by C = 53 × 137 = 7261

∴ Total amount = 28187 + 7578 + 7261 = 43026

∴ Total members = 71 + 18 + 53 = 142

∴ Average = 43026/142 = 303

Total average amount spent per member = Tk. 303

৯,৫০৯.

How much should be added to each term of 4 : 7 so that it becomes 2 : 3?

ক
2
খ
5
গ
3
ঘ
6

ব্যাখ্যা

Question: How much should be added to each term of 4 : 7 so that it becomes 2 : 3?

Solution:
Given that,
Ratio of two numbers is 4 : 7
Let the number added to denominator and numerator be 'x'

Now according to the question,
(4 + x) : (7 + x) = 2 : 3
⇒ (4 + x)/(7 + x) = 2/3
⇒ 12 + 3x = 14 + 2x
∴ x = 2

∴ 2 will be added to make the term in the ratio of 2 : 3.

৯,৫১০.

An amount of Tk. 366 is to be divided among P, Q and R in such a manner that Q gets two-third as much as P and R together, and P may get half as much as Q and R together. What is the share of P?

ক
ক) 130
খ
খ) 122
গ
গ) 173
ঘ
ঘ) 146

ব্যাখ্যা

Given,
Q = 2/3(P + R)
Or, 3Q = 2(P + R)
Again,
P = (Q + R)/2
Or, 2P = Q + R
Now,
P + Q + R = 366
Or, P + 2P = 366
Or, 3P = 366
∴ P = 122

৯,৫১১.

A fruit seller buys lemons at a rate of 2 lemons for a Taka and sells them at a rate of 5 lemons for 3 Taka. What is her profit margin (based on cost price)?

ক
10%
খ
15%
গ
20%
ঘ
30%

ব্যাখ্যা

Question: A fruit seller buys lemons at a rate of 2 lemons for a Taka and sells them at a rate of 5 lemons for 3 Taka. What is her profit margin (based on cost price)?

Solution:
The cost price of 2 lemons = 1 Tk
The cost price of 1 lemon = 1/2 Tk

Then,
The selling price of 5 lemons = 3 Tk
The selling price of 1 lemon = 3/5 Tk

So, profit = 3/5 - 1/2 = 1/10 Tk

Hence, profit percentage = {(1/10)/(1/2)} × 100 = 20%

৯,৫১২.

The ratio of milk and water in a solution is 25 : 12 and after adding 9 liters of water in it the ratio of milk and water becomes 5 : 3, then find the final amount of water in the final solution-

ক
15 liters
খ
25 liters
গ
35 liters
ঘ
45 liters

ব্যাখ্যা

Question: The ratio of milk and water in a solution is 25 : 12 and after adding 9 liters of water in it the ratio of milk and water becomes 5 : 3, then find the final amount of water in the final solution.

Solution:
Let,
The initial amount of milk be 25x liters
and the amount of water 12x liters.

Ratio of milk and water after adding 5 liters
25x/(12x + 9) = 5/3
⇒ 75x = 60x + 45
⇒ 15x = 45
∴ x = 3

∴ Final amount of water in solution = 12x + 9 liters.
= 36 + 9 liters.
= 45 liters.

৯,৫১৩.

Two trains running in opposite directions cross a man standing on the platform in 17 seconds and 7 seconds respectively and they cross each other in 13 seconds. The ratio of their speeds is-

ক
2 ∶ 1
খ
3 ∶ 2
গ
4 ∶ 3
ঘ
5 : 4

ব্যাখ্যা

Question: Two trains running in opposite directions cross a man standing on the platform in 17 seconds and 7 seconds respectively and they cross each other in 13 seconds. The ratio of their speeds is-

Solution:
Let, the speed of the trains are x m/sec and y m/sec respectively

We know, Distance/Length = Speed × Time
∴ Length of first train = 17x meter
And, length of second train = 7y meter

Now, Total time = Total length/Total speed
(17x + 7y)/(x + y) = 13
⇒ 17x + 7y = 13x + 13y
⇒ 17x - 13x = 13y - 7y
⇒ 4x = 6y
⇒ x/y = 3/2
∴ x ∶ y = 3 ∶ 2

৯,৫১৪.

In a geometric progression, the 4^th term is 16 and the 7^th term is 128. Find the 10^th term.

ক
256
খ
512
গ
720
ঘ
1024

ব্যাখ্যা

Question: In a geometric progression, the 4^th term is 16 and the 7^th term is 128. Find the 10^th term.

Solution:
Let the first term = a
Common ratio = r
We know,
n-term = ar^{n - 1}

Then,
4^th term, ar³ = 16 ........(1)
7^th term, ar⁶ = 128 ........(2)

Now, divide equation (2) by equation (1) then we get,
(ar⁶)/(ar³) = 128/16
⇒ r³ = 8
⇒ r³ = 2³
∴ r = 2

Then substitute r = 2 into equation (1)
a (2)³ = 16
⇒ a × 8 = 16
∴ a = 2

Now, 10^th term
= ar⁹
= 2 × 2⁹
= 2 × 2⁹
= 2¹⁰
= 1024

∴ The 10th term is 1024

৯,৫১৫.

A train running at the speed of 60 kmph crosses a 200 m long platform in 27 seconds. What is the length of the train?

ক
250 meters
খ
180 meters
গ
320 meters
ঘ
220 meters

ব্যাখ্যা

Question: A train running at the speed of 60 kmph crosses a 200 m long platform in 27 seconds. What is the length of the train?

Solution:
Given that,
Time = 27 sec
∴ Speed = (60 × 5/18)m/sec = 50/3 m/sec

Let the length of the train be x metres.
Then,
(x + 200)/27 = 50/3
⇒ x + 200 = (50/3) × 27
⇒ x + 200 = 450
⇒ x = 450 - 200
∴ x = 250 metres

So the length of the train is 250 meters.

৯,৫১৬.

Shihab sells his goods 25% cheaper than Fahim and 25% dearer than Aftab. How much of the goods of Aftab are cheaper than Fahim's?

ক
33.33%
খ
66.66%
গ
40%
ঘ
50%

ব্যাখ্যা

Question: Shihab sells his goods 25% cheaper than Fahim and 25% dearer than Aftab. How much of the goods of Aftab are cheaper than Fahim's?

Solution:
Let, Fahim sells at 100 taka
Shihab sells at 75 taka

Aftab sells at 75/1.25
= 60 tka

the goods of Aftab are cheaper than Fahim's = (100 - 60)
= 40%

৯,৫১৭.

The ratio of three numbers is 5 : 6 : 7, and their L.C.M. is 210. Find their H.C.F. -

ক
12
খ
5
গ
3
ঘ
1

ব্যাখ্যা

Question: The ratio of three numbers is 5 : 6 : 7, and their L.C.M. is 210. Find their H.C.F. -

Solution:
Let the numbers be 5x, 6x and 7x
Then, their L.C.M. = 210x

So,
⇒ 210x = 210
∴ x = 1

∴ The HCF of the numbers is x = 1

৯,৫১৮.

The selling price of 40 apples is equal to cost price of 35 apples. Find the profit or loss obtained.

ক
ক) Gain of 5.5%
খ
খ) Gain of 12.5%
গ
গ) Loss of 5.5%
ঘ
ঘ) Loss of 12.5%

ব্যাখ্যা

Question: The selling price of 40 apples is equal to cost price of 35 apples. Find the profit or loss obtained.

Solution:
Let,
C.P. of each apple be Tk. 1

Therefore,
C.P. of 40 apples = Tk. 40
S.P. of 40 apples = Tk. 35

C.P. of 40 apples > S.P. of 40 apples

∴ Loss = 40 - 35 = 5 Taka

Loss % = {(5 × 100)/40}% = 12.5%

৯,৫১৯.

If (x/y) > 0, which of the following must be true?

ক
xy > 0
খ
(x - y) > 0
গ
(x + y) > 0
ঘ
None of these

ব্যাখ্যা

Question: If (x/y) > 0, which of the following must be true?

Solution:
যেহেতু (x/y) > 0 তাই
x ও y এর দুইটি একই সাথে ধণাত্মক বা ঋণাত্বক হবে।
আমরা জানি
দুইটি একই সাথে ধণাত্মক বা ঋণাত্বক সংখ্যার গুণফল ধণাত্মক হয়।
( - x) ( - y) = xy
x × y = xy

i) xy > 0 অবশ্যই ধণাত্মক

৯,৫২০.

Rima's height is 5.4". Eva is taller than Rima, but she is not taller than Promi. Promi is shorter than her cousin Rakib, but she is not shorter than Rima. Who is the tallest in the group?

ক
Promi
খ
Rima
গ
Rakib
ঘ
Eva

ব্যাখ্যা

Question: Rima's height is 5.4". Eva is taller than Rima, but she is not taller than Promi. Promi is shorter than her cousin Rakib, but she is not shorter than Rima. Who is the tallest in the group?

Solution:
Rima's height = 5'4"

Eva > Rima
Eva's height > 5'4"

Eva is not taller than Promi.
So, height of Promi > 5'4" + x

Let
Promi's height = (5'4" + x ) + y
Promi < Rakib.

Rakib's height >( 5'4" + x) + y

Rakib > Promi > Eva > Rima
Therefore, Rakib is the tallest.

৯,৫২১.

A man walking at the rate of 5 km/hr crosses a bridge in 15 minutes. The length of the bridge (in meters) is-

ক
ক) 600
খ
খ) 750
গ
গ) 1000
ঘ
ঘ) 1250

ব্যাখ্যা

15min = 1/4hrs
1 hr → 5 kms
1/4hr → 5/4 kms
So, length of the bridge=5/4 km = 1250 metres

৯,৫২২.

A, B and C enter into a partnership. A invests some money at the beginning, B invests double the amount after six months and C invest thrice the amount after eight months. If the annual profit be Tk. 27000; C's profit share is

ক
ক) 12000
খ
খ) 10000
গ
গ) 6000
ঘ
ঘ) 9000

ব্যাখ্যা

Question: A, B and C enter into a partnership. A invests some money at the beginning, B invests double the amount after six months and C invest thrice the amount after eight months. If the annual profit be Tk. 27000; C's profit share is

Solution:
ধরি,
A এর বিনিয়োগ X, লাভ পাবে 12 মাসের।
B এর বিনিয়োগ 2X, লাভ পাবে 6 মাসের।
C এর বিনিয়োগ 3X, লাভ পাবে 4 মাসের।

তাহলে,
A : B : C = (X × 12) : (2X × 6) : (3X × 4)
= 12X : 12X : 12X
= 1 : 1 : 1

∴ C এর লাভ হবে = (27000 এর 1/3)
= 9000 টাকা

৯,৫২৩.

A number is tripled, then 7 is subtracted from it. If the result is then doubled, it becomes 58. What is the number?

ক
15
খ
14
গ
12
ঘ
10

ব্যাখ্যা

Question: A number is tripled, then 7 is subtracted from it. If the result is then doubled, it becomes 58. What is the number?

Solution:
Let,
the number be x

ATQ,
2(3x - 7) = 58
⇒ 6x - 14 = 58
⇒ 6x = 58 + 14
⇒ 6x = 72
⇒ x = 72/6
∴ x = 12

So the number is 12.

৯,৫২৪.

In how many different ways can the letters of the word "JASHORE" be arranged so that the vowels always come together?

ক
120
খ
720
গ
114
ঘ
126

ব্যাখ্যা

Question: In how many different ways can the letters of the word "JASHORE" be arranged so that the vowels always come together?

Solution:
The word J A S H O R E = 7 letters
There, Vowels = A, O, E (3 vowels)
Consonants = J, S, H, R (4 consonants)

Now,
Treat the 3 vowels (A, O, E) as a single unit.
So now we have, [A O E] J S H R
Total units to arrange = 5 (1 vowel block + 4 consonants)

Number of ways to arrange these 5 units = 5! = 120 ways

And
The 3 vowels (A, O, E) can be arranged among themselves in = 3! = 6 ways

Total number of arrangements = (ways to arrange the 5 units) × (ways to arrange vowels inside the block)
= 120 × 6
= 720

So there are 720 different ways to arrange the letters of 'JASHORE' such that the vowels always come together.

৯,৫২৫.

Babu's income is Tk. 1,000 more than that of Selim. Their total salary is Tk. x. What is Selim's salary?

ক
(x/2) - 500
খ
x - 500
গ
(2x - 500)/2
ঘ
2x - 1000

ব্যাখ্যা

Question: Babu's income is Tk. 1,000 more than that of Selim. Their total salary is Tk. x. What is Selim's salary?

Solution:
Let,
Selim's salary is Tk. a
Babu's salary is Tk. (a + 1000)

ATQ,
a + a + 1000 = x
⇒ 2a = x - 1000
⇒ a = (x - 1000)/2
∴ a = x/2 - 500

৯,৫২৬.

The average of 20, 70 and x is 40. If the average of 20, 70, x and y is 50, then y =?

ক
100
খ
80
গ
70
ঘ
60

ব্যাখ্যা

Question: The average of 20, 70 and x is 40. If the average of 20, 70, x and y is 50, then y =?

Solution:
The average of 20, 70 and x is 40
(x + 20 + 70)/3 = 40
⇒ x + 20 + 70 = (3 × 40) = 120
⇒ x + 90 = 120
⇒ x = 120 - 90 = 30

the average of 20, 70, x and y is 50

(20 + 70 + x + y)/4 = 50
⇒ 90 + x + y = (4 × 50)
⇒ 90 + 30 + y = 200
⇒ 120 + y = 200
⇒ y = 200 - 120
∴ y = 80

৯,৫২৭.

Mr. Rabbani running at 150 m/min alongside a railway track in 250 meters ahead of the engine of a 120 meters long train running at 750 m/min in the same direction. In how much time will the train pass him?

ক
ক) 31 sec
খ
খ) 33 sec
গ
গ) 35 sec
ঘ
ঘ) 37 sec

ব্যাখ্যা

Question: Mr. Rabbani running at 150 m/min alongside a railway track in 250 meters ahead of the engine of a 120 meters long train running at 750 m/min in the same direction. In how much time will the train pass him?

Solution:
Speed of train relative to Mr. Rabbani = (750 - 150) m/min = 600 m/min = 10 m/s

Distance to be covered = (250 + 120) m = 370 m

Time taken = 370/10 = 37 sec

৯,৫২৮.

tan3A = 1/√3, then find the value of A = ?

ক
30°
খ
10°
গ
45°
ঘ
20°

ব্যাখ্যা

Question: tan3A = 1/√3, then find the value of A = ?

Solution:
Given that,
tan3A = 1/√3
⇒ tan3A = tan30°
⇒ 3A = 30°
⇒ A = 30°/3
∴ A = 10°

৯,৫২৯.

Out of 5 men and 6 women, how many ways can you form a 4-member committee with at least 2 women?

ক
252
খ
265
গ
240
ঘ
150

ব্যাখ্যা

Question: Out of 5 men and 6 women, how many ways can you form a 4-member committee with at least 2 women?

Solution:
6 জন মহিলা থেকে 2 জন ও 5 জন পুরুষ থেকে 2 জন নিয়ে 4 সদস্যের কমিটি গঠন করা যায়,
= ⁶C₂ × ⁵C2
= {6! / (2! × 4! )} × {5! / (2! × 3!)}
= {(6 × 5) / 2} × {(5 × 4) / 2}
= 15 × 10
= 150 উপায়ে

6 জন মহিলা থেকে 3 জন ও 5 জন পুরুষ থেকে 1 জন নিয়ে 4 সদস্যের কমিটি গঠন করা যায়,
= ⁶C₃ × ⁵C₁
= {6! / (3! × 3!)} × {5! / (1! × 4!)}
= {(6 × 5 × 4) / 6} × 5
= 20 × 5
= 100 উপায়ে

আবার,
6 জন মহিলা থেকে 4জন নিয়ে ( পুরুষ সদস্য ব্যতীত ) 4 সদস্যের কমিটি গঠন করা যায়,
= ⁶C₄
= 6! / (4! × 2!)
= (6 × 5) /2
= 15 উপায়ে

মোট উপায় = 150 + 100 + 15 = 265 টি

৯,৫৩০.

The average weight of 12 students in a class is 45.5 kg. What should be the weight of a 13th student so that the average weight of all 13 students becomes 47.2 kg?

ক
65 kg
খ
68 kg
গ
67.6 kg
ঘ
70 kg
ঙ
66.5 kg

ব্যাখ্যা

Question: The average weight of 12 students in a class is 45.5 kg. What should be the weight of a 13th student so that the average weight of all 13 students becomes 47.2 kg?

Solution:
Average weight of 12 students = 45.5 kg
∴ Total weight of 12 students = (45.5 × 12) kg
= 546 kg

Again,
Average weight of 13 students = 47.2 kg
Total weight of 13 students = (47.2 × 13) kg
= 613.6 kg

∴ Weight of the 13th student = (613.6 - 546) kg
= 67.6 kg

Therefore, the weight of the 13th student should be 67.6 kg.

৯,৫৩১.

In the figure above, V represents an observation point at one end of a pool. From V, an object that is actually located on the bottom of the pool at point R appears to be at point S. If VR = 10 feet, what is the distance RS, in feet, between the actual position and the perceived position of the object?

ক
10 - 5√3
খ
10 - 5√2
গ
2
ঘ
2.5
ঙ
4

ব্যাখ্যা

Question:

In the figure above, V represents an observation point at one end of a pool. From V, an object that is actually located on the bottom of the pool at point R appears to be at point S. If VR = 10 feet, what is the distance RS, in feet, between the actual position and the perceived position of the object?

Solution:

PR = √(VR² - VP²) = √(10² - 5²) = √(75) = 5√3

Thus, RS = PS - PR = 10 - 5√3

৯,৫৩২.

After getting two successive discount, a shirt with a list price of Tk.150 is available at Tk.105. If the second discount is 12.5%, find the first discount.

ক
ক) 15%
খ
খ) 20%
গ
গ) 22%
ঘ
ঘ) 25%

ব্যাখ্যা

Let the first discount be x%.
Now
87.5% of (100 - x)% of 150 = 105
(87.5/100) × {(100 - x)/100} × 150 = 105
100 - x = (105 × 100 × 100)/(150 × 87.5)
100 - x = 80
100 - 80 = x
x = 20

The first discount be 20%

৯,৫৩৩.

The product of the ages of A and B is 240. If twice the age of B is more than A’s age by 4 years, what was B’s age 2 years ago?

ক
12 years
খ
10 years
গ
8 years
ঘ
14 years

ব্যাখ্যা

Question: The product of the ages of A and B is 240. If twice the age of B is more than A’s age by 4 years, what was B’s age 2 years ago?

Solution:
Let A’s present age be x years.
Then, B’s present age = 240/x years

So, according to question
2(240/x ) - x = 4
⇒ 480 - x² = 4x
⇒ x² + 4x - 480 = 0
⇒ (x + 24) (x - 20) = 0
⇒ x = 20

B’s present age = 240/20 = 12 years
Thus, B’s age 2 years ago = 12 - 2 = 10 years

৯,৫৩৪.

In an examination, 36% are pass marks. If an examine gets 17 marks and fails by 10 marks, what are the maximum marks?

ক
45
খ
50
গ
60
ঘ
75

ব্যাখ্যা

• Question: In an examination, 36% are pass marks. If an examine gets 17 marks and fails by 10 marks, what is the maximum mark?

Solution:
Pass mark = (17 + 10) = 27

Let maximum marks be x

Then 36% of x = 27
Or, 36x/100 = 27
Or, 36x = 2700

Hence, x = 75

∴ The maximum mark is 75

৯,৫৩৫.

If x² + yz + zx + xy is divided by x + z, the result is-

ক
(x - y)
খ
(x - z)
গ
(x + z)
ঘ
(x + y)

ব্যাখ্যা

Question: If x² + yz + zx + xy is divided by x + z, the result is-

Solution:
x² + yz + zx + xy
= x² + xy + zx + yz
= x(x + y) + z(x + y)
= (x + y)(x + z)

∴ divided by (x + z) the result = (x + y)(x + z)/(x +z) = (x + y)

৯,৫৩৬.

The sum of the squares of three numbers is 120, and the sum of their products taken two at a time is 140. What is the sum of the three numbers?

ক
17
খ
19
গ
22
ঘ
20

ব্যাখ্যা

Question: The sum of the squares of three numbers is 120, and the sum of their products taken two at a time is 140. What is the sum of the three numbers?

Solution:
Let the numbers be a, b, c.
Given:
a2+ b2 + c2 = 120,
ab + bc + ca = 140,
⇒ (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca) = 120 + 2 × 140 = 120 + 280 = 400

∴ a + b + c = √400 = 20

৯,৫৩৭.

A train has a length of 150 metres. It is passing a man who is moving at 4 km/hr in the same direction of the train, in 3 seconds. Find out the speed of the train?

ক
ক) 169 km/hr
খ
খ) 152 km/hr
গ
গ) 184 km/hr
ঘ
ঘ) 180 km/hr

ব্যাখ্যা

Length of the train = 150 m
Speed of the man = 4 km/hr
Relative speed = 150/3 = 50 m/s
= 50 × 18/5
= 180 km/hr
Relative speed = Speed of train - Speed of the man (as both are moving in the same direction).
Therefore,
Speed of the train = Relative speed + Speed of the man
= 180 + 4
= 184 km/hr

৯,৫৩৮.

The sides of a triangle are in the ratio (1/2) : (1/3) : (1/4) and its perimeter is 104 cm . The length of the longest side(in cm ) is-

ক
ক) 42 cm
খ
খ) 44 cm
গ
গ) 24 cm
ঘ
ঘ) 48 cm

ব্যাখ্যা

Ratio of sides =(1/2) : (1/3) : (1/4)
= 6 : 4 : 3

Let
The sides is 6x, 4x and 3x

Now
6x + 4x + 3x = 104
13x = 104
x = 104/13
x = 8
The length of the longest side = 48 cm

৯,৫৩৯.

Mr. Kalam deposited a certain amount of money for a fixed period of time. On maturity, he received a total of Tk. 42,000 when the ratio of interest and investment became 2 : 5. If the simple interest rate was 4%, calculate the time period for which the money was invested.

ক
9.5 years
খ
15 years
গ
8.5 years
ঘ
10 years
ঙ
12 years

ব্যাখ্যা

Question: Mr. Kalam deposited a certain amount of money for a fixed period of time. On maturity, he received a total of Tk. 42,000 when the ratio of interest and investment became 2 : 5. If the simple interest rate was 4%, calculate the time period for which the money was invested.

Solution:
Let, time n.
Sum of these ratios = 2 + 5 = 7
We are given that, ratio of interest & Investment - 2 : 5.
So, interest is, 42000 × (2/7) = Tk. 12000
Investment is, 42000 × (5/7) = Tk. 30000

We know,
Principal × Interest rate × Time = Total Interest
⇒ 30000 × 4% × n = 12000
⇒ n = (12000 × 100)/(30000 × 4) = 10

So the money was invested for 10 years.

৯,৫৪০.

A median of a triangle divides it into two -

ক
ক) congruent triangles
খ
খ) triangles of equal area
গ
গ) isosceles triangles.
ঘ
ঘ) right triangles

ব্যাখ্যা

আমরা জানি,
যে কোন ত্রিভুজের মধ্যমা ত্রিভুজটিকে সমান ক্ষেত্রফল বিশিষ্ট দুটি নতুন ত্রিভুজে বিভক্ত করে।
তাই উত্তর হবে- B

৯,৫৪১.

X, Y and Z share Tk 1800 in such a way that X has 2.5 times as much as Y, and Y has 4 times as much as Z. How much money (in Taka) does Z receive?

ক
ক) 120
খ
খ) 132
গ
গ) 145
ঘ
ঘ) 200

ব্যাখ্যা

মনে করি,
z এর আয় = p টাকা
y এর আয় = 4p টাকা
x এর আয় = 2.5×4p টাকা = 10p টাকা

∴ x : y : z = 10p : 4p : p
= 10 : 4 : 1
তাহলে, z এর আয় = 1/(10 + 4 + 1) × 1800
= 1/15 × 1800
= 120 টাকা

৯,৫৪২.

A train takes 20 seconds to cross a pole. It takes 50 seconds to cross the platform. What is the ratio of the length of the platform to that of the train?

ক
2 : 3
খ
2 : 5
গ
5 : 2
ঘ
3 : 2

ব্যাখ্যা

Question: A train takes 20 seconds to cross a pole. It takes 50 seconds to cross the platform. What is the ratio of the length of the platform to that of the train?

Solution:
Let,
The speed of the train is x m/s

While cross the pole the train travels 20 × x meters = 20x meters, which is the length of the train

While cross the platform the train travels 50 × x meters = 50x meters
∴ Length of the platform 50x - 20x = 30x meters

∴ Length of the platform : Length of the train = 30x : 20x = 30 : 20 = 3 : 2

৯,৫৪৩.

The value of 12 ÷ (1/2) + {(35 ÷ 7) of 40} + 20 - 15 of 10 is-

ক
80
খ
88
গ
94
ঘ
99

ব্যাখ্যা

Question: The value of 12 ÷ (1/2) + {(35 ÷ 7) of 40} + 20 - 15 of 10 is-

Solution:
12 ÷ (1/2) + {(35 ÷ 7) of 40} + 20 - 15 of 10
= 12 ÷ (1/2) + {(5) of 40} + 20 - 15 of 10
= 12 ÷ (1/2) + 5 × 40 + 20 - 150
= 12 × 2 + 200 + 20 - 150
= 244 - 150
= 94

∴ The required answer is 94.

৯,৫৪৪.

Half of the pillar is under the earth, one third of it is within water and 12 feet is above water. What is the length of the pillar?

ক
ক) 72 feet
খ
খ) 80 feet
গ
গ) 60 feet
ঘ
ঘ) 84 feet
ঙ
ঙ) 70 feet

ব্যাখ্যা

Pillar above the water is 12 feet = 1 - (1/2 + 1/3) = 1/6
So, the length of the pillar is = 12 × 6 = 72 feet

৯,৫৪৫.

At the rate of 15% simple interest per annum, how much profit will be earned on Tk. 900 as principal in 7 years?

ক
Tk. 785
খ
Tk. 850
গ
Tk. 945
ঘ
Tk. 895

ব্যাখ্যা

Question: At the rate of 15% simple interest per annum, how much profit will be earned on Tk. 900 as principal in 7 years?

Solution:
Given,
Sum P = Tk. 900
Time n = 7 years
Rate r = 15%
= 15/100
= 3/20

We know,
Profit = Prn
= 900 × 3/20 × 7
= 945

৯,৫৪৬.

If 2^{(n + 3)} - 2^{(n + 1)} = 12. What is the value of n?

ক
0
খ
1
গ
2
ঘ
4

ব্যাখ্যা

Question: If 2^{(n + 3)} - 2^{(n + 1)} = 12. What is the value of n?

Solution:
Given,
2^{(n + 3)} - 2^{(n + 1)} = 12
⇒ (2ⁿ × 2³) - (2ⁿ + 2) = 12
⇒ 2ⁿ(2³ - 2) = 12
⇒ 2ⁿ(8 - 2) = 12
⇒ 2ⁿ × 6 = 12
⇒ 2ⁿ = 12/6
⇒ 2ⁿ = 2
∴ n = 1

৯,৫৪৭.

Moni and Mimi are partners in a business. Moni invests 36000 for 8 months and Mimi invests Tk 48000 for 10 months. Out of a profit of 7200, Moni's share is -

ক
Tk. 4500
খ
Tk. 2500
গ
Tk. 2700
ঘ
Tk. 3000

ব্যাখ্যা

Question: Moni and Mimi are partners in a business. Moni invests 36000 for 8 months and Mimi invests Tk 48000 for 10 months. Out of a profit of 7200, Moni's share is -

Solution:
Ratio of their shares = (36000 × 8) : (48000 × 10)
= 288 : 480
= 3 : 5

Moni's share = 7200 × (3/8) = Tk. 2700

৯,৫৪৮.

A takes twice as much time as B or thrice as much time as C to finish a piece of work. Working together, they can finish the work in 4 days. B can do the work alone in-

ক
8 days
খ
10 days
গ
12 days
ঘ
14 days

ব্যাখ্যা

Question: A takes twice as much time as B or thrice as much time as C to finish a piece of work. Working together, they can finish the work in 4 days. B can do the work alone in-

Solution:
Suppose A, B and C take x, x/2 and x/3 days respectively to finish the work.
Then,
1/x + 2/x + 3/x = 1/4
⇒ 6/x = 1/4
⇒ x/6 = 2
∴ x = 24

So, B takes (24/2) = 12 days to finish the work.

৯,৫৪৯.

A man invested Tk. 3,104 in a stock at 97 to obtain an income of Tk. 128. What is the dividend from the stock?

ক
ক) 12%
খ
খ) 9%
গ
গ) 4%
ঘ
ঘ) 10%

ব্যাখ্যা

By investing Tk. 3,104, income = Tk. 128
By investing Tk. 97, income = (128 × 97)/3104
= 4
Hence, the dividend is 4%.

৯,৫৫০.

The average of 60 numbers is 25. If two numbers, namely 30 and 40, are discarded, what is the average of the remaining numbers?

ক
22.84
খ
23.42
গ
24.66
ঘ
26.50

ব্যাখ্যা

Question: The average of 60 numbers is 25. If two numbers, namely 30 and 40, are discarded, what is the average of the remaining numbers?

Solution:
ATQ,
The average of 60 numbers is = 25
The sum of 60 numbers is = 25 × 60 = 1500
The two numbers discarded = 30 + 40 = 70
The sum of the remaining 58 numbers = 1500 - 70 = 1430

∴ The new average = 1430/58 = 24.66 (approximately)

৯,৫৫১.

In company there are 75% skilled workers and reaming are unskilled. 80% of skilled workers and 20% of unskilled workers are permanent. If number of temporary workers is 126, then what is the number of total workers?

ক
446
খ
420
গ
396
ঘ
360

ব্যাখ্যা

Question: In company there are 75% skilled workers and reaming are unskilled. 80% of skilled workers and 20% of unskilled workers are permanent. If number of temporary workers is 126, then what is the number of total workers?

Solution:
Let, the number of total workers = x
Number of skilled worker = 75% of x = 75x/100 = 3x/4
No. of unskilled workers = 25% of x = 25x/100 = x/4

∴ No. of permanent workers = {(80/100) × (3x/4) + (20/100) × (x/4)}
= (3x/5) + (x/20)
= 13x/20

∴ No .of temporary workers = {x - (13x/20)}
= 7x/20

ATQ,
7x/20 = 126
⇒ 7x = 2520
∴ x = 360

৯,৫৫২.

The average of a non-zero number and its square is 11 times the number. The number is-

ক
0
খ
9
গ
15
ঘ
21

ব্যাখ্যা

Question: The average of a non-zero number and its square is 11 times the number. The number is-

Solution:
Let the number be x (x ≠ 0).

According to the question,
The average of the number and its square is 11 times the number.
⇒ (x + x²)/2 = 11x
⇒ x + x² = 22x
⇒ x² + x - 22x = 0
⇒ x² - 21x = 0
⇒ x(x - 21) = 0
So, x = 0 or x = 21
But the number is non-zero, so we discard x = 0.

Therefore, the number is 21.

৯,৫৫৩.

Which among 2^1/2, 3^1/3, 4^1/4, 6^1/6, and 12^1/12 is the largest?

ক
ক) 2^1/2
খ
খ) 3^1/3
গ
গ) 4^1/4
ঘ
ঘ) 12^1/12

ব্যাখ্যা

Question: Which among 2^1/2, 3^1/3, 4^1/4, 6^1/6, and 12^1/12 is the largest?

Solution:
2^1/2, 3^1/3, 4^1/4, 6^1/6, and 12^1/12= 2^{(1/2) × 12}, 3^{(1/3) × 12}, 4^{(1/4) × 12}, 6^{(1/6) × 12}, and 12^{(1/12) × 12}
= 2⁶, 3⁴, 4³, 6² and 12¹
= 64, 81, 64, 36, and 12

Here, 3^1/3 is the largest number.

৯,৫৫৪.

How many 3-digit even numbers can be made using the digits 1, 2, 3, 4, 6, and 7 if no digit is repeated?

ক
360
খ
180
গ
120
ঘ
60

ব্যাখ্যা

Question: How many 3-digit even numbers can be made using the digits 1, 2, 3, 4, 6, and 7 if no digit is repeated?

Solution:
Given digits: 1, 2, 3, 4, 6, 7
Number of digits = 6

Number of possible digits at unit’s place = 3 (2, 4 and 6)
⇒ Number of permutations = ³P₁ = 3

When one of the digits is taken in units’ place, then the number of possible digits available = 5
⇒ Number of permutations = ⁵P₂ = 5!/(5 - 2)!
= 5!/3!
= 120/6
= 20

∴ The total number of permutations = 3 × 20 = 60.

৯,৫৫৫.

If the cost of gas on burning 5 burners for 5 hours a day for 7 days is Tk. 525, then how many burners can be used for 10 days at 5 hours a day for Tk. 750?

ক
7 burners
খ
5 burners
গ
8 burners
ঘ
10 burners

ব্যাখ্যা

Question: If the cost of gas on burning 5 burners for 5 hours a day for 7 days is Tk. 525, then how many burners can be used for 10 days at 5 hours a day for Tk. 750?

Solution:
Given that,
5 burners 5 hours a day for 7 days is Tk. 525
Total burner-hours = (5 × 5 × 7) burner-hours
= 175 burner-hours

Cost per burner-hour =(525 ÷ 175)
= 3 Tk

∴ 750 Tk. total burner-hour = (750 ÷ 3) burner-hours
= 250 burner-hours

Let, the number of burners = x

ATQ,
x × 5 × 10 = 250
⇒ x × 50 = 250
⇒ x = 250 ÷ 50
∴ x = 5

৯,৫৫৬.

A bag contains 4 red, 5 yellow and 6 pink balls. Two balls are drawn at random. What is the probability that none of the balls drawn is yellow in colour?

ক
ক) 1/7
খ
খ) 3/7
গ
গ) 2/7
ঘ
ঘ) 5/14

ব্যাখ্যা

Number of red balls = 4
Number of yellow ball = 5
Number of pink ball = 6
Total number of balls = 4 + 5 + 6 = 15
Total possible outcomes = selection of 2 balls out of 15 balls = ¹⁵C₂
= 15!/2!(15 - 2)!
= 15!/(2! × 13!)
= (15 × 14)/(1 × 2)
= 105.
Total favourable outcomes = selection of 2 balls out of 4 orange and 6 pink balls.
¹⁰C₂
= 10!/2!(10 - 2)!
= 10!/2!8!
= (10 × 9)/(1 × 2)
= 45.
∴ Required Probability = 45/105
= 3/7

৯,৫৫৭.

Namita borrowed Tk. 50000 for 3 years at the rate of 3.5% per annum. Find the interest accumulated at the end of 3 years.

ক
Tk. 5500
খ
Tk. 5550
গ
Tk. 5450
ঘ
Tk. 5350
ঙ
Tk. 5250

ব্যাখ্যা

Question: Namita borrowed Tk. 50000 for 3 years at the rate of 3.5% per annum. Find the interest accumulated at the end of 3 years.

Solution:
P = 50000
n = 3
r = 3.5%

∴ I = Pnr
= (50000 × 3 × 3.5)/100
= 5250

৯,৫৫৮.

If ⁿC₂ = 105, then n =?

ক
14
খ
15
গ
16
ঘ
17
ঙ
18

ব্যাখ্যা

Question: If ⁿC₂ = 105, then n =?

Solution:
ⁿC₂ = 105
⇒ n!/{2! ×(n - 2)!} = 105
⇒ n(n - 1) = 210
⇒ n² - n - 210 = 0
⇒ n² - 15n + 14n - 210 = 0
⇒ n(n - 15) + 14(n - 15) = 0
⇒ (n - 15)(n + 14) = 0
∴ n = 15, - 14 [- 14 is not acceptable]

৯,৫৫৯.

A jar contains a total of 400 coins of 5 cents and 10 cents that add up to Tk 30. Find the number of 10-cent coins.

ক
124
খ
140
গ
160
ঘ
200

ব্যাখ্যা

Question: A jar contains a total of 400 coins of 5 cents and 10 cents that add up to Tk 30. Find the number of 10-cent coins.

Solution:
Let,
the number of 10-cent coins be = x
So, the number of 5-cent coins is = (400 - x)

ATQ,
10x + {5 × (400 - x)} = (30 × 100)
⇒ 10x + 2000 - 5x = 3000
⇒ 5x = 1000
∴ x = 200

Therefore, there are 200 coins of 10 cents.

৯,৫৬০.

Rupa got married 6 years ago. Today her age is 5/4 times her age at the time of marriage. Her son's age is (1/10) times her age. Her son's age is-

ক
4 years
খ
1 year
গ
3 years
ঘ
7 years

ব্যাখ্যা

Question: Rupa got married 6 years ago. Today her age is 5/4 times her age at the time of marriage. Her son's age is (1/10) times her age. Her son's age is-

Solution:
Let
the age of Rupa before 6 years ago is x.
Then today her age is x + 6

Now
x + 6 = 5x/4
4x + 24 = 5x
5x - 4x = 24
x = 24

Now age of Rupa = 24 + 6 =30 years
Her son's age is = 30/10 ears
= 3 years

৯,৫৬১.

The least perfect square number divisible by 3, 4, 5, 6 and 8 is = ?

ক
ক) 900
খ
খ) 1200
গ
গ) 2500
ঘ
ঘ) 3600

ব্যাখ্যা

L.C.M. of 3, 4, 5, 6, 8 is 120
Now 120 = 2 × 2 × 2 × 3 × 5
To make it a perfect square, it must be multiplied by 2 × 3 × 5
So, required number
=22×22×32×52
=3600

৯,৫৬২.

A monkey climbs a 36 m high pole. In first minute he climbs 6 m and slips down 3 m in the next minute. How much time is required by it to reach the top?

ক
20 minutes.
খ
21 minutes.
গ
22 minutes.
ঘ
25 minutes.

ব্যাখ্যা

Question: A monkey climbs a 36 m high pole. In first minute he climbs 6 m and slips down 3 m in the next minute. How much time is required by it to reach the top?

Solution:
According to the question,
For 1^st min, he climbs 6 m and comes down 3 m in 2^nd
Effectively in 2 min, he climbs 3 m.

In 1 min, he climbs (3/2) m

So, in 20 mins he climbs = (3/2) × 20 = 30 m

So, in the 21^th minute, he climbs the next 6 m.

∴ The required value is 21 minutes.

৯,৫৬৩.

If sec²θ + tan²θ = 5/3, then what is the value of tan2θ?

ক
√3/2
খ
√2
গ
√3
ঘ
1/√2

ব্যাখ্যা

Question: If sec²θ + tan²θ = 5/3, then what is the value of tan2θ?

Solution:
We know,
sec²θ = 1 + tan²θ

Given that,
⇒ sec²θ + tan²θ = 5/3
⇒ 1 + tan²θ + tan²θ = 5/3
⇒ 2tan²θ = 2/3
⇒ tanθ = 1/√3
⇒ θ = 30°

∴ tan(2θ) = tan(2 × 30°) = tan60° = √3

৯,৫৬৪.

If log_x(64/125) = - 3, then x = ?

ক
4/5
খ
1/3
গ
5/4
ঘ
2/3

ব্যাখ্যা

Question: If log_x (64/125) = - 3, then x = ?
(Janata RC 2022 অনুযায়ী)

Solution:
log_x (64/125) = - 3
⇒ x^{- 3} = 64/125
⇒ x^{- 3} = (4/5)³
⇒ x^{- 3} = 1/(5/4)³
⇒ x^{- 3} = (5/4)^{- 3}
⇒ x = 5/4

৯,৫৬৫.

Make it compound: Rumi works hard so that she can complete the task.

ক
Rumi wants to complete the task and so she works hard.
খ
Works hard and Rumi can complete the task.
গ
She works hard in order to complete the task.
ঘ
So Rumi works hard and she complete the task.

ব্যাখ্যা

• Complex to compound
• Sentence এর Transformation এর ক্ষেত্রে যে কাজ গুলো করতে হয়।

• Add (যোগ করা),
• Deduct (বাদ দেয়া),
• Retain (যেভাবে আছে সেভাবে রাখা)।

• অর্থাৎ - বাক্যের ধরন অনুযায়ী Preposition/Conjunction একটা যোগ ও একটা বিয়োগ করতে হবে,
- এবং ২টি Clause এর মধ্যে ১টি Clause যেভাবে আছে সেভাবে রেখে দিতে হবে।

• Complex to compound এর সময়,
• so that দ্বারা গঠিত বাক্যকে Complex to compound করার নিয়ম:
• এ ধরনের বাক্যে Compound Sentence এ ৩টি কাজ করতে হয়। যথা:

1. Clause গুলোর অবস্থান উল্টে যাবে।
2. Want to অতিরিক্ত যোগ করতে হবে।
3. so that এর পরিবর্তে ‘And so/ therefore’ conjunction যোগ হবে।

Complex: Rumi works hard so that she can complete the task.
Compound: Rumi wants to complete the task and so she works hard.

Source: Live MCQ English Wizard Lecture.

৯,৫৬৬.

A boatman can row 2.5 km against the stream in 25 minutes and return in 12.5 minutes. Find the rate of flow of the current.

ক
ক) 1 km/hr
খ
খ) 3 km/hr
গ
গ) 4 km/hr
ঘ
ঘ) 5 km/hr

ব্যাখ্যা

Question: A boatman can row 2.5 km against the stream in 25 minutes and return in 12.5 minutes. Find the rate of flow of the current.

Solution:
Let,
x be the speed of man in still water
y be the speed of current

Speed of downstream = (2.5/12.5) × 60 = 12 km / hr
Speed of upstream = (2.5/ 25) × 60 = 6 km / hr

∴ rate of current = (12 - 6)/2 = 3 km/hr

৯,৫৬৭.

What is the value of n if (8-3) (8 - n) = 40?

ক
ক) -3
খ
খ) 8
গ
গ) 0
ঘ
ঘ) -5

ব্যাখ্যা

(8-3) (8 - n) = 40
=> 5(8 - n) = 40
=> 8 - n = 8
=> n = 0

৯,৫৬৮.

There are 4 candidates for the post of a lecturer in Mathematics and one is to be selected by votes of 5 men. The number of ways in which the votes can be given is

ক
1024
খ
560
গ
462
ঘ
None

ব্যাখ্যা

Question: There are 4 candidates for the post of a lecturer in Mathematics and one is to be selected by votes of 5 men. The number of ways in which the votes can be given is

Solution:
Since there are 4 candidates for the post of a lecturer and 5 men voting, each voter can choose one candidate from the 4. The number of ways in which the votes can be given can be calculated by considering that each of the 5 men can choose one candidate.

For each man, there are 4 possible choices of candidates. Therefore, for 5 men, the total number of ways the votes can be given = 4⁵ = 1024

৯,৫৬৯.

In how many different ways can 3 identical green shirts and 3 identical red shirts be distributed among 6 children such that each child receives a shirt?

ক
ক) 40
খ
খ) 216
গ
গ) 20
ঘ
ঘ) 720

ব্যাখ্যা

Question: In how many different ways can 3 identical green shirts and 3 identical red shirts be distributed among 6 children such that each child receives a shirt?

Solution:

6টি শিশু ৩টি সবুজ শার্ট গ্রহণ করার উপায় = ⁶c₃ = (6 × 5 × 4)/(3 × 2× 1) = 20

বাকি ৩টি শিশু ৩টি লাল শার্ট গ্রহণ করার উপায় = ³C₃ = (3 × 2 × 1)/(3 × 2 × 1) = 1
সুতরাং, মোট উপায় সংখ্যা = 20 ×1 = 20টি

৯,৫৭০.

The angle between the minute hand and the hour hand of a clock when the time is 4 : 20, is-

ক
10°
খ
12.5°
গ
8.5°
ঘ
12°

ব্যাখ্যা

Question: The angle between the minute hand and the hour hand of a clock when the time is 4 : 20, is-

Solution:
Angle = |(11M – 60H)/2|°
= |{(11 × 20) - (60 × 4)}/2|°
= |-20/2|°
= 10°

৯,৫৭১.

ক
1/2
খ
2
গ
0
ঘ
1/3

ব্যাখ্যা

Question:

Solution:

৯,৫৭২.

One-fifth of the light switches produced by a certain factory are defective. Four-fifths of the defective switches are rejected and 1/20 of the non defective switches are rejected by mistake. If all the switches not rejected are sold, what percent of the switches sold by the factory are defective?

ক
4%
খ
5%
গ
6.25%
ঘ
11%
ঙ
16%

ব্যাখ্যা

Question: One-fifth of the light switches produced by a certain factory are defective. Four-fifths of the defective switches are rejected and 1/20 of the non defective switches are rejected by mistake. If all the switches not rejected are sold, what percent of the switches sold by the factory are defective?

Solution:
Assume x as the number of switches and that x =100
Defective Switches = (1/5) × 100 = 20
Defective Switches that are rejected = (4/5) × 20 = 16
Defective Switches that are sold = 20 - 16 = 4

Non-defective Switches = 100 - 20 = 80
Non-defective Switches mistakenly rejected = 80 × (1/20) = 4
Non-defective Switches that are sold = 80 - 4 = 76

Total sold in the market = 76 + 4 = 80
% of defective = (4/80) × 100 = 5%

৯,৫৭৩.

From a group of 7 men and 6 women, five persons are to be selected to form a committee so that at least 3 men are there on the committee. In how many ways can it be done ?

ক
ক) 857
খ
খ) 896
গ
গ) 756
ঘ
ঘ) 675

ব্যাখ্যা

Ways in which at least 3 men are selected;
⇒ 3 men + 2 women
⇒ 4 men + 1 woman
⇒ 5 men + 0 woman

Number of ways = ⁷C₃ × ⁶C₂ + ⁷C₄ × ⁶C₁ + ⁷C₅ × ⁶C₀
= 35 × 15 + 35 × 6 + 21
= 756

∴ The required no of ways = 756

৯,৫৭৪.

A triangular garden has sides measuring 36 meters, 40 meters, and 32 meters. A fence is to be constructed around the garden, with pillars placed at intervals of 4 meters. How many pillars will be required to completely surround the garden?

ক
11 pillars
খ
21 pillars
গ
27 pillars
ঘ
40 pillars

ব্যাখ্যা

Question: A triangular garden has sides measuring 36 meters, 40 meters, and 32 meters. A fence is to be constructed around the garden, with pillars placed at intervals of 4 meters. How many pillars will be required to completely surround the garden?

Solution:
Perimeter of the triangle = 36 + 40 + 32 = 108 meters
Distance between pillars = 4 meters

Number of pillars required = Perimeter ÷ Distance
= 108 ÷ 4
= 27
= 27 pillars

∴ 27 pillars are needed to surround the triangular garden.

৯,৫৭৫.

Pipe B takes 12 hours to fill a tank by itself, while pipe A works three times faster. If both pipes are opened together, how many hours will they need to fill the tank?

ক
3 hours
খ
4 hours
গ
5 hours
ঘ
6 hours

ব্যাখ্যা

Question: Pipe B takes 12 hours to fill a tank by itself, while pipe A works three times faster. If both pipes are opened together, how many hours will they need to fill the tank?

Solution:
B নল দ্বারা চৌবাচ্চা পূর্ণ হয় = 12 ঘণ্টায়
∴ 1 ঘণ্টায় পূর্ণ হয় = 1/12 অংশ

A নল দ্বারা পূর্ণ হয় = 12/3 = 4 ঘণ্টায়
∴ 1 ঘণ্টায় পূর্ণ হয় = 1/4 অংশ

দুইটি নল দ্বারা একত্রে 1 ঘণ্টায় পূর্ণ হয় = (1/12) + (1/4)
= (1 + 3)/12
= 4/12
= 1/3 অংশ

দুইটি নল দ্বারা একত্রে,
1/3 অংশ পূর্ণ হয় = 1 ঘণ্টায়
∴ 1 অংশ পূর্ণ হয় = 1 × 3 = 3 ঘণ্টায়

৯,৫৭৬.

Two containers contain milk and water in the ratios 5 : 2 and 9 : 5. What ratio should the mixtures be combined in to achieve a final ratio of 2 : 1 milk to water?

ক
2 : 1
খ
1 : 2
গ
1 : 3
ঘ
1 : 4
ঙ
None

ব্যাখ্যা

Question: Two containers contain milk and water in the ratios 5 : 2 and 9 : 5. What ratio should the mixtures be combined in to achieve a final ratio of 2 : 1 milk to water?

Solution:
Let,
P unit of the first mixture is added to Q unit of the second mixture.

So, in the P unit of the first mixture,
Amount of milk present = (5/7) × P = 5P/7
Amount of water present = (2/7) × P = 2P/7

In the Q unit of the second mixture,
Amount of milk present = (9/14) × Q = 9Q/14
Amount of water present = (5/14) × Q = 5Q/14

ATQ,
{(5P/7) + (9Q/14)}/{(2P/7) + (5Q/14)} = 2/1
⇒ {(10P + 9Q)/14}/{(4P + 5Q)/14} = 2
⇒ 10P + 9Q = 8P + 10Q
⇒ 2P = Q

∴ P : Q = 1 : 2

৯,৫৭৭.

A rectangular field is to be fenced on three sides leaving a side of 20 feet uncovered. If the area of the field is 680 sq. feet, how many feet of fencing will be required?

ক
88
খ
92
গ
98
ঘ
102

ব্যাখ্যা

Question: A rectangular field is to be fenced on three sides leaving a side of 20 feet uncovered. If the area of the field is 680 sq. feet, how many feet of fencing will be required?

Solution:
Here,
l = 20 ft and
lb = 680 sq. ft.

So, b = 34 ft.

∴ Length of fencing = (l + 2b) = (20 + 68) ft = 88 ft.

৯,৫৭৮.

Find the value of: - 2 + (-2) - {-(2)} - 2

ক
ক) -6
খ
খ) 2
গ
গ) -2
ঘ
ঘ) 4
ঙ
ঙ) -4

ব্যাখ্যা

- 2 + (-2) - {-(2)} - 2
= - 2 - 2 + 2 - 2
= - 4

৯,৫৭৯.

A, B and C are partners in a business. Their shares are in the proportion of 1/3 : 1/4 : 1/5. A withdraws half of his capital after 15 months and after another 15 months, a profit of Tk. 4340 is divided. The share of C is?

ক
ক) Tk. 1550
খ
খ) Tk. 1360
গ
গ) Tk. 1240
ঘ
ঘ) Tk. 1245

ব্যাখ্যা

Quesation: A, B and C are partners in a business. Their shares are in the proportion of 1/3 : 1/4 : 1/5. A withdraws half of his capital after 15 months and after another 15 months, a profit of Tk. 4340 is divided. The share of C is?

Solution:
Ratio of initial investments = 1/3 : 1/4 : 1/5
= 20 : 15 : 12

Let their initial investments be 20x, 15x and 12x respectively.

A : B : C = (20x × 15) + (10x × 15): (15x × 30) : (12x × 30)
= 450x : 450x : 360x
= 5 : 5 : 4

Sum of the ratio = 5 + 5 + 4 = 14.

C's share = 4340 × (4/14)
= 1240 Tk.

৯,৫৮০.

A, B and C share the profit in the ratio of 2:3:7. If the average gain is Tk. 8000, then B's share is?

ক
ক) 2000
খ
খ) 1000
গ
গ) 1500
ঘ
ঘ) 6000

ব্যাখ্যা

Here, A : B : C
Ratio of Profit → 2 : 3 : 7
Average gain= (2+3+7)/3 = 4 units
According to the question,
4 units= Tk. 8000
1 unit= Tk. 2000
3 units=3×2000= Tk. 6000
∴Share of B= Tk. 6000

৯,৫৮১.

If the n^th terms of an arithmetic progression is 4n + 1, then the common difference is-

ক
4
খ
3
গ
5
ঘ
6

ব্যাখ্যা

Question: If the n^th terms of an arithmetic progression is 4n + 1, then the common difference is-

Solution:
The n^th terms of an arithmetic progression is 4n + 1
n = 1 then, T₁ = 4 . 1 + 1 = 5
n = 2 then, T₂ = 4 . 2 + 1 = 9
n = 3 then, T₃ = 4 . 3 + 1 = 13
n = 1 then, T₄ = 4 . 4 + 1 = 17
.....................................................
.....................................................

The common difference
T₂ - T₁ = 9 - 5 = 4
T₃ - T₂ = 13 - 9 = 4
T₄ - T₃ = 17 - 13 = 4

∴ The common difference is 4.

৯,৫৮২.

A wheel that has 5 cogs is meshed with a larger wheel of 15 cogs. When the smaller wheel has made 21 revolutions, the number of revolutions made by the larger wheel will be-

ক
ক) 9
খ
খ) 8
গ
গ) 6
ঘ
ঘ) 7

ব্যাখ্যা

Question: A wheel that has 5 cogs is meshed with a larger wheel of 15 cogs. When the smaller wheel has made 21 revolutions, the number of revolutions made by the larger wheel will be-

Solution:

smaller wheel cross = 5 × 21 = 105 cogs by 21 revolutions.
so,
larger wheel will made = 105/15 = 7 revolutions.

Shortcut:
cogs of A : cogs of B = reolution of B : revolution of A
hense,
5 : 15 = X : 21
X = 7

৯,৫৮৩.

In your bookshelf, you have five favorite books. If you decide to arrange these five books in every possible combination and moved just one book in every half a minute. How much time it will take you to arrange?

ক
ক) 3 hours
খ
খ) 1 hour
গ
গ) 2 hours
ঘ
ঘ) 30 hours

ব্যাখ্যা

Question: In your bookshelf, you have five favorite books. If you decide to arrange these five books in every possible combination and moved just one book in every half a minute. How much time it will take you to arrange?

Solution:
5টি বইকে সাজানো যায় মোট = 5! উপায়ে
= 120 উপায়ে

বই 1 বার সরাতে সময় লাগে 1/2 মিনিট
বই 120 বার সরাতে সময় লাগে 120/2 মিনিট
= 60 মিনিট
= 1 ঘণ্টা

৯,৫৮৪.

The sum of the squares of three numbers is 123, and the sum of their products taken two at a time is 119. What is the sum of the three numbers?

ক
17
খ
19
গ
21
ঘ
23

ব্যাখ্যা

Question: The sum of the squares of three numbers is 123, and the sum of their products taken two at a time is 119. What is the sum of the three numbers?

Solution:
Let the three numbers be a, b, and c
Then,
a² + b² + c²= 123
ab + bc + ca = 119

Now,
(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
⇒ (a + b + c)² = 123 + (2 × 119)
⇒ (a + b + c)² = 361
⇒ (a + b + c) = √361∴ (a + b + c) = 19

৯,৫৮৫.

The investment ratio of two partners, P and Q, is 7 : 9, while their profit ratio is 14 : 27. Given that P invested his funds for 6 months, determine the duration of Q's investment.

ক
9 months
খ
11 months
গ
12 months
ঘ
7 months

ব্যাখ্যা

Question: The investment ratio of two partners, P and Q, is 7 : 9, while their profit ratio is 14 : 27. Given that P invested his funds for 6 months, determine the duration of Q's investment.

Solution:
Let P invested Tk 7x for 6 months
Q invested Tk 9x for y months

Now,
(7x × 6) : (9x × y) = 14 : 27
⇒ (7x × 6)/(9x × y) = 14/27
⇒ 42x/9xy = 14/27
⇒ 42 × 27 = 14 × 9y
⇒ 1134 = 126y
⇒ y = 1134/126
∴ y = 9

So, Q invested for 9 months.

৯,৫৮৬.

The number of distinct permutations of the letters of the word "AMERICA" is how many times that of the word "CANADA"?

ক
21
খ
18
গ
20
ঘ
17
ঙ
None of these

ব্যাখ্যা

Question: The number of distinct permutations of the letters of the word "AMERICA" is how many times that of the word "CANADA"?

Solution:
In the word AMERICA there are 7 letters in total,
with A appearing 2 times and all other letters appearing 1 time each.

∴ Number of distinct permutations = 7!/2!
= (7 × 6 × 5 × 4 × 3 × 2 × 1)/2
= 7 × 6 × 5 × 4 × 3
= 2520

Again,
In the word CANADA there are 6 letters in total,
with A appearing 3 times and all other letters appearing 1 time each.

∴ Number of distinct permutations = 6!/3!
= (6 × 5 × 4 × 3 × 2 × 1)/(3 × 2 × 1)
= 6 × 5 × 4
= 120

Therefore, the number of arrangements of AMERICA is 2520/120 = 21 times the number of arrangements of CANADA.

So the number of distinct permutations of 'AMERICA' is 21 times the number of distinct permutations of 'CANADA'.

৯,৫৮৭.

If a + b = 11, a - b = 9, what is ab = ?

ক
ক) 8
খ
খ) 10
গ
গ) 12
ঘ
ঘ) 14

ব্যাখ্যা

Question: If a + b = 11, a - b = 9, what is ab = ?

Solution:
Given that
a + b = 13
a - b = 11

we know
4ab = (a + b)² - (a - b)²
4ab = 11² - 9²
4ab = 121 - 81
4ab = 40
ab = 40/4
ab = 10

৯,৫৮৮.

If a : b : c = 7 : 3 : 5, then (a + b + c) : (2a + b - c) is equal to-

ক
2 : 3
খ
4 : 3
গ
5 : 4
ঘ
6 : 5

ব্যাখ্যা

Question: If a : b : c = 7 : 3 : 5, then (a + b + c) : (2a + b - c) is equal to-

Solution:
Given,
a : b : c = 7 : 3 : 5

Let,
(a/7) = (b/3) = (c/5) = k
a = 7k, b = 3k, c = 5k

Now, (a + b + c) : (2a + b - c)
= (7k + 3k + 5k) : {(2 × 7k) + 3k - 5k)
= 15 k : 12 k
= 5 : 4

৯,৫৮৯.

A train traveled p miles in 40 minutes and completed the remaining 200 miles of the trip in q minutes. What was its average speed in miles per hour for the entire trip?

ক
ক) 60(p+200)/(40+q)
খ
খ) 240/(p+q)
গ
গ) 4(p+q)
ঘ
ঘ) None

ব্যাখ্যা

Speed = Distance/Time = (p + 200)/(40 + q) miles/min = (p + 200)/(40 + q)/60 miles/hr = 60(p + 200)/(40 + q).

৯,৫৯০.

1250 oranges were distributed among a group of girls of a class. Each girl got twice as many oranges as the number of girls in that group. The number of girls in the group was -

ক
22
খ
24
গ
27
ঘ
32
ঙ
25

ব্যাখ্যা

Let the number of girls in the group be x
Then, number of oranges given to each girl = 2x
∴x×2x=1250
⇔ 2x²=1250
⇔ x²=625
⇔ x=25

৯,৫৯১.

Income of A is 25% less than B. How much percent of B's income would be more than that of A?

ক
23.5%
খ
33.3%
গ
32.5%
ঘ
25%
ঙ
29%

ব্যাখ্যা

Let's say that B makes $100.
Then A makes $75. So,
Question then becomes 100 is what percent of 75
This can be solved by setting up the proportion
100 / 75 = x / 100
75x = 10000
x = 133.3
So, B's income is 33.3% more than A's income.

৯,৫৯২.

What is the length of the largest chord of a circle if its radius is 7 cm?

ক
14 cm
খ
7 cm
গ
21 cm
ঘ
28 cm

ব্যাখ্যা

প্রশ্ন: What is the length of the largest chord of a circle if its radius is 7 cm?
(কোনো বৃত্তের ব্যাসার্ধ ৭ সে.মি হলে বৃত্তের বৃহত্তম জ্যা -এর দৈর্ঘ্য কোনটি?)

সমাধান:
আমরা জানি,
বৃত্তের ব্যাসই বৃহত্তম জ্যা এবং ব্যাসার্ধের দ্বিগুণ হলো ব্যাস।
∴ বৃত্তের বৃহত্তম জ্যা = বৃত্তের ব্যাস
= ২ × ব্যাসার্ধ
= (২ × ৭) সে.মি
= ১৪ সে.মি

৯,৫৯৩.

The lengths of two sides of a triangle are 7 and 11. If the length of the third side is an integer, what is the least possible perimeter of the triangle?

ক
23
খ
18
গ
20
ঘ
22

ব্যাখ্যা

Question: The lengths of two sides of a triangle are 7 and 11. If the length of the third side is an integer, what is the least possible perimeter of the triangle?

Solution:
আমরা জানি,
ত্রিভুজের যে কোন দুই বাহুর সমষ্টি তৃতীয় বাহু অপেক্ষা বৃহত্তর।
যেহেতু, ৩য় বাহুটি একটি পূর্ণসংখ্যা।

ধরি, সংখ্যাটি 3 বা 4 বা 5 অথবা 6
এখন, 3 + 7 = 10 < 11 [গ্রহণযোগ্য নয়]
4 + 7 = 11 = 11 [গ্রহণযোগ্য নয়]
5 + 7 = 12 > 11 [যা অপর ২ বাহুর সমষ্টি থেকে বৃহত্তর]

৩য় বাহু = 5

∴ ত্রিভুজটির পরিসীমা = (7 + 11 + 5) = 23

৯,৫৯৪.

Mr. Ali is a trader. He mixes 26 kg of rice at Tk. 20 per kg with 30 kg of rice of other variety at Tk. 36 per kg and sells the mixture at Tk. 32 per kg. His profit percent is-

ক
12%
খ
7%
গ
9%
ঘ
6%

ব্যাখ্যা

Question: Mr. Ali is a trader. He mixes 26 kg of rice at Tk. 20 per kg with 30 kg of rice of other variety at Tk. 36 per kg and sells the mixture at Tk. 32 per kg. His profit percent is-

Solution:
Cost Price of 56 kg rice = {(26 × 20) + (30 × 36)}
= (520 + 1080)
= 1600 taka

Selling Price of 56 kg rice = (56 × 32)
= 1792 taka

∴ Profit = 1792 - 1600
= 192 taka

∴ Profit percentage = (192/1600) × 100%
= 12%

৯,৫৯৫.

A is two years older than B who is twice as old as C. If the total ages of A, B, and C is 27, how old is C?

ক
ক) 5 years
খ
খ) 8 years
গ
গ) 10 years
ঘ
ঘ) 15 years

ব্যাখ্যা

Question: A is two years older than B who is twice as old as C. If the total ages of A, B, and C is 27, how old is C?

Solution:
Let, C's age be x years.
Then, B's age = 2x years.
And A's age = (2x + 2) years.

ATQ,
2x + 2 + 2x + x = 27
⇒ 5x = 25
⇒ x = 5

৯,৫৯৬.

Find the number of lead balls, each with a diameter of 1 cm, that can be made from a sphere of diameter 12 cm.

ক
ক) 980
খ
খ) 1224
গ
গ) 1728
ঘ
ঘ) 1920

ব্যাখ্যা

Question: Find the number of lead balls, each with a diameter of 1 cm, that can be made from a sphere of diameter 12 cm.

Solution:
ব্যাসার্ধ, r = 6

গোলকটির আয়তন = (4/3) × π × r³
= (4/3) × π × 6³
= 288π

নতুন বলের ব্যাস 1 সেমি হলে ব্যাসার্ধ = 1/2 সেমি
নতুন বলের আয়তন = (4/3) × π × (1/2)³
= π/6

নতুন বলের সংখ্যা হবে = 288π/(π/6) = 1728

৯,৫৯৭.

Ten years ago, Alif was 1/3rd as old as Joy. If Alif is 18 years old now, how old is Joy now?

ক
32 years
খ
38 years
গ
36 years
ঘ
34 years

ব্যাখ্যা

Question: Ten years ago, Alif was 1/3rd as old as Joy. If Alif is 18 years old now, how old is Joy now?

Solution:
Let,
Age of Joy now x years

ATQ,
(x - 10)/3 = (18 - 10)
⇒ x - 10 = 24
⇒ x = 34

∴ The age of Joy now 34 years

৯,৫৯৮.

A batsman scored 86 runs which included 8 boundaries and 3 sixes. What percent of his total score did he make by running between the wickets?

ক
38.86%
খ
54.86%
গ
34.86%
ঘ
41.86%

ব্যাখ্যা

Question: A batsman scored 86 runs which included 8 boundaries and 3 sixes. What percent of his total score did he make by running between the wickets?

Solution:
Total score of batsman = 86 runs
runs from boundaries = 4 × 8 = 32
runs from sixes = 6 × 3 = 18
∴ Total runs from boundaries and sixes = 32 + 18 = 50 runs
Scores by running between the wickets = 86 - 50 = 36 runs

percentage of his score made by running between wickets = (36/86) × 100% = 41.86%

৯,৫৯৯.

A bus picks up a group of tourists at a hotel. The sightseeing bus travels 2 blocks north, 2 blocks east and 1 block south, 2 blocks east and 1 block south. Where is the bus in relation to the hotel?

ক
ক) 1 block west
খ
খ) 2 blocks north
গ
গ) 3 blocks south
ঘ
ঘ) 4 blocks east

ব্যাখ্যা

The bus is 4 block east in relation to the hotel.

৯,৬০০.

Noyon walking at a speed of 20 km/h reaches his college 10 minutes late. Next time he increases his speed by 5 km/h, but finds that he is still late by 4 minutes. What is the distance of his college from his house?

ক
ক) 10 km
খ
খ) 6 km
গ
গ) 12 km
ঘ
ঘ) 15 km
ঙ
ঙ) 20 km

ব্যাখ্যা

By increasing his speed by 25%, he will reduce his time by 20%. (This corresponds to a 6 minutes drop in his time.)
Hence, his time originally must have been 30 minutes.
Thus required distance = 20 kmph × 0.5 hours = 10 km.