Bank Math

PrepBank · পাতা ৭৩ / ১৬১ · ৭,২০১–৭,৩০০ / ১৬,১২৪

৭,২০১.

A is thrice as good a workman as B and so takes 40 days less than B for doing a job. The time in which they can do the job together is -

ক
ক) 20 days
খ
খ) 15 days
গ
গ) 30 days
ঘ
ঘ) 24 days

সঠিক উত্তর: খ

খ) 15 days

উত্তর

সঠিক উত্তর: খ

খ) 15 days

খ

ব্যাখ্যা

Question: A is thrice as good a workman as B and so takes 40 days less than B for doing a job. The time in which they can do the job together is -

Solution:
Let A alone takes x days
and B alone takes 3x days to complete the job.

ATQ,
3x - x = 40
⇒ 2x = 40
⇒ x = 20

So, A alone takes 20 days and B alone takes 3 × 20 = 60 days to complete the job.

(A + B)'s 1 days work = (1/20 + 1/60) = 1/15 part

∴ A and B together can do the work in 15 days.

৭,২০২.

৮ জন ছাত্রের গড় বয়স ২০ বছর। প্রথম ২ জন ছাত্রের গড় বয়স ১৯ বছর এবং পরবর্তী ৩ জন ছাত্রের গড় বয়স ২১ বছর। যদি ৬ষ্ঠ ছাত্রের বয়স ৭ম ছাত্রের চেয়ে ২ বছর কম এবং ৮ম ছাত্রের চেয়ে ৩ বছর কম হয়, তবে ৮ম ছাত্রের বয়স কত?

ক
২১ বছর
খ
১৮ বছর
গ
২৪ বছর
ঘ
৩০ বছর
ঙ
কোনোটিই নয়

সঠিক উত্তর: ক

২১ বছর

উত্তর

সঠিক উত্তর: ক

২১ বছর

ক

ব্যাখ্যা

প্রশ্ন: ৮ জন ছাত্রের গড় বয়স ২০ বছর। প্রথম ২ জন ছাত্রের গড় বয়স ১৯ বছর এবং পরবর্তী ৩ জন ছাত্রের গড় বয়স ২১ বছর। যদি ৬ষ্ঠ ছাত্রের বয়স ৭ম ছাত্রের চেয়ে ২ বছর কম এবং ৮ম ছাত্রের চেয়ে ৩ বছর কম হয়, তবে ৮ম ছাত্রের বয়স কত?

সমাধান:
৮ জনের গড় বয়স = ২০ বছর
∴৮ জনের মোট বয়স = ৮ × ২০ = ১৬০ বছর

প্রথম ২ জন ছাত্রের গড় বয়স ১৯ বছর
∴প্রথম ২ জনের মোট বয়স = ২ × ১৯ = ৩৮ বছর

পরবর্তী ৩ জন ছাত্রের গড় বয়স ২১ বছর
∴ পরবর্তী ৩ জনের মোট বয়স = ৩ × ২১ = ৬৩ বছর

ধরি, ৬ষ্ঠ ছাত্রের বয়স = x বছর
∴ ৭ম ছাত্রের বয়স = x + ২ বছর
∴ ৮ম ছাত্রের বয়স = x + ৩ বছর

প্রশ্নমতে,
মোট বয়স = প্রথম ৫ জনের বয়স + শেষ ৩ জনের বয়স
⇒ ১৬০ = ৩৮ + ৬৩ + x + (x + ২) + (x + ৩)
⇒ ১৬০ = ১০১ + ৩x + ৫
⇒ ৩x + ১০৬ = ১৬০
⇒ ৩x = ১৬০ - ১০৬
⇒ ৩x = ৫৪
∴ x = ১৮

∴ ৮ম ছাত্রের বয়স = x + ৩ বছর
= ১৮ + ৩ = ২১ বছর

৭,২০৩.

There are 15 boys and 10 girls in a class. If three students are selected at random, what she probability that 1 girl and 2 boys are selected?

ক
ক) 1/40
খ
খ) 1/2
গ
গ) 21/46
ঘ
ঘ) 7/41

সঠিক উত্তর: গ

গ) 21/46

উত্তর

সঠিক উত্তর: গ

গ) 21/46

গ

ব্যাখ্যা

Let
S be the sample space
E be the event of selecting 1 girl and 2 boys.

Then, n(S) = Number ways of selecting 3 students out of 25 = ²⁵C₃ = 2300
Number ways of selecting 1 girl out of 10 girls and 2 boys out of 15 boys
n(E) = (¹⁰C₁×¹⁵C₂)= 10 × 105 = 1050

Hence the probability that 1 girl and 2 boys are selected, is,
P(E) =1050/2300
= 21/46

৭,২০৪.

The current birth rate per thousand is 30, whereas the corresponding death rate is 10 per thousand. Find the net growth rate in terms of population increase in percent.

ক
1.5%
খ
2%
গ
2.5%
ঘ
3%

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

We are given that,
1) Current birth rate per thousand is 30
2) Corresponding death rate is 10 per thousand

Hence, net growth on 1000 = Current birth rate - the death rate
= 30 – 10 =20

We are asked to find,
the net growth rate in terms of population increase in percent (which means net growth on 100)
Net worth on 100 = {(Net worth of 1000)/1000} × 100
Net worth on 100 = (20/1000) × 100
= 2%.

৭,২০৫.

The difference between two numbers is 5 and the difference between their squares is 65. What is the sum of two numbers?

ক
ক) 12
খ
খ) 13
গ
গ) 14
ঘ
ঘ) 15

সঠিক উত্তর: খ

খ) 13

উত্তর

সঠিক উত্তর: খ

খ) 13

খ

ব্যাখ্যা

প্রশ্ন: The difference between two numbers is 5 and the difference between their squares is 65. What is the sum of two numbers?

সমাধান:
ধরি,
বৃহত্তম সংখ্যাটি = x
ক্ষুদ্রতম সংখ্যাটি = y

প্রশ্নমতে,
বা, x - y = 5 ..............(1)
বা, x² - y² = 65 ..............(2)

(2)নং সমীকরণ হতে পাই,
x² - y² = 65
বা, (x + y)(x - y) = 65
বা, (x + y) × 5 = 65
বা, x + y = 13.................(3)

৭,২০৬.

What is the largest number that divides 84, 144 or 36 without any remainder?

ক
ক) 6
খ
খ) 12
গ
গ) 18
ঘ
ঘ) 24

সঠিক উত্তর: খ

খ) 12

উত্তর

সঠিক উত্তর: খ

খ) 12

খ

ব্যাখ্যা

HCF of 84, 144 and 36 is 12
So, 12 is the largest number that divides 84, 144 or 36 without any remainder

৭,২০৭.

Abir was both the 23rd highest and the 23rd lowest in a tennis tournament. How many participants were in the tournament?

ক
ক) 28
খ
খ) 30
গ
গ) 38
ঘ
ঘ) 45

সঠিক উত্তর: ঘ

ঘ) 45

উত্তর

সঠিক উত্তর: ঘ

ঘ) 45

ঘ

ব্যাখ্যা

Number of participants are = 23 + 23 - 1 = 45

৭,২০৮.

If 32 × 4^{a + 2} = 2^b, then b =?

ক
ক) 2a + 8
খ
খ) a + 9
গ
গ) 2a + 9
ঘ
ঘ) 2a + 5

সঠিক উত্তর: গ

গ) 2a + 9

উত্তর

সঠিক উত্তর: গ

গ) 2a + 9

গ

ব্যাখ্যা

Question: If 32 × 4 ^{a + 2} = 2 ^b, then b =?

Solution:
32 × 4 ^{a +} ² = 2 ^b
⇒ 2⁵ × 2^{2(a + 2)} = 2^b
⇒ 2⁵ × 2^{2a + 4} = 2b
⇒ 2 ^{5 + 2a + 4} = 2^b
⇒ 2 ^{2a + 9} = 2^b
⇒ b = 2a + 9

৭,২০৯.

A thief is noticed by a policeman from a distance of 300 meters. The thief starts running and the policeman chases him. The thief and the policeman run at the rate of 10 km/hr and 12 km/hr respectively. What is the distance between them after 6 minutes?

ক
50 meters
খ
100 meters
গ
150 meters
ঘ
200 meters

সঠিক উত্তর: খ

100 meters

উত্তর

সঠিক উত্তর: খ

100 meters

খ

ব্যাখ্যা

Question: A thief is noticed by a policeman from a distance of 300 meters. The thief starts running and the policeman chases him. The thief and the policeman run at the rate of 10 km/hr and 12 km/hr respectively. What is the distance between them after 6 minutes?

সমাধান:
যেহেতু পুলিশ ও চোর একই দিকে দৌড়াচ্ছে, তাই তাদের আপেক্ষিক গতিবেগ (relative speed) হবে তাদের গতিবেগের পার্থক্য।
∴ আপেক্ষিক গতিবেগ = (12 - 10) কিমি/ঘন্টা
= 2 কিমি/ঘন্টা।
= (2 × 1000)/60 মিটার/মিনিট
= 200/6 মিটার/মিনিট

6 মিনিটে পুলিশ চোরের থেকে যে দূরত্ব কমিয়ে আনবে, তা হলো:
অতিক্রান্ত দূরত্ব = আপেক্ষিক গতিবেগ × সময়
= (200/6)× 6
= 200 মিটার

সুতরাং, 6 মিনিট পরে তাদের মধ্যে অবশিষ্ট দূরত্ব = প্রাথমিক দূরত্ব - অতিক্রান্ত দূরত্ব
= 300 মিটার - 200 মিটার
= 100 মিটার

৭,২১০.

If n is even, (6ⁿ - 1) is divisible by -

ক
35
খ
6
গ
13
ঘ
25

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: If n is even, (6ⁿ - 1) is divisible by -

Solution:
if n = 2 then, (6² - 1) = 35
if n = 4 then, (6⁴ - 1) = 1295
if n = 6 then, (6⁶ - 1) = 46655

the HCF of 35, 1295 and 46655 is 35

so, (6ⁿ - 1) is always divisible by 35

৭,২১১.

Chocolates are bought at 10 for Tk 20 and sold at 12 for Tk 30. The gain percent is-

ক
22%
খ
25%
গ
30%
ঘ
35%

সঠিক উত্তর: খ

25%

উত্তর

সঠিক উত্তর: খ

25%

খ

ব্যাখ্যা

Question: Chocolates are bought at 10 for Tk 20 and sold at 12 for Tk 30. The gain percent is-

Solution:
Cost per chocolate = 20/10 Tk.
= 2 Tk.

Sell per chocolate = 30/12 Tk.
= 2.5 Tk.

So, gain percentage = {(2.5 - 2)/2} × 100
= 25 Tk. or 25%

৭,২১২.

What is the probability of getting a sum 5 from two throws of a dice?

ক
1/6
খ
2/9
গ
1/9
ঘ
2/7
ঙ
5/17

সঠিক উত্তর: গ

1/9

উত্তর

সঠিক উত্তর: গ

1/9

গ

ব্যাখ্যা

Question: What is the probability of getting a sum 5 from two throws of a dice?

Solution:
In two throws a dice, n(S) = 6 × 6 = 36
Let E is the event of getting a sum of five.
E = (1, 4), (4, 1), (2, 3), (3, 2)
So, n(E) = 4

∴ P(E) = n(E)/n(S)
= 4/36
= 1/9

৭,২১৩.

An employee’s monthly salary was Tk. 12,000 and his monthly expenditure was Tk. 8,000. The next year, his salary increased by 25% and his expenditure increased by 10%. Find the difference of his savings.

ক
Tk. 1200
খ
Tk. 1500
গ
Tk. 1800
ঘ
Tk. 2000
ঙ
Tk. 2200

সঠিক উত্তর: ঙ

Tk. 2200

উত্তর

সঠিক উত্তর: ঙ

Tk. 2200

ঙ

ব্যাখ্যা

Question: An employee’s monthly salary was Tk. 12,000 and his monthly expenditure was Tk. 8,000. The next year, his salary increased by 25% and his expenditure increased by 10%. Find the difference of his savings.

Solution:
Savings before increased income = Tk. (12000 - 8000)
= Tk. 4000

New salary = 125% of 12000 = Tk. (125/100) × 12000
= Tk. 15000

New expenditure = 110% of 8000 = Tk. (110/100) × 8000
= Tk. 8800

Saving after Increased income = Tk. (15000 - 8800)
= Tk. 6200

So the difference of his savings = Tk. (6200 - 4000)
= Tk. 2200

৭,২১৪.

A bus is running with a uniform speed of 37 km/h. What distance will be covered by bus at 8 h?

ক
ক) 246 km
খ
খ) 289 km
গ
গ) 296 km
ঘ
ঘ) 276 km
ঙ
ঙ) 250 km

সঠিক উত্তর: গ

গ) 296 km

উত্তর

সঠিক উত্তর: গ

গ) 296 km

গ

ব্যাখ্যা

We know that, speed = Distance/Time
Therefore, Required distance = Speed × Time
=37 × 8
= 296km

৭,২১৫.

What is 10% of y/3, if 2y/3 is 10% of 400?

ক
ক) 3
খ
খ) 2
গ
গ) 1
ঘ
ঘ) 5

সঠিক উত্তর: খ

খ) 2

উত্তর

সঠিক উত্তর: খ

খ) 2

খ

ব্যাখ্যা

Question: What is 10% of y/3, if 2y/3 is 10% of 400?

Solution:
2y/3 = 10% of 400
⇒ 2y/3 = 400 ×10/100
⇒ 2y/3 = 40
⇒ y = 40 × 3/2
⇒ y = 60

10% of y/3 is = 10% of 60/3
= 10% of 20
= 20 × 10/100
= 2

৭,২১৬.

If the difference between the circumference and diameter of a circle is 90 cm, then the diameter of the circle is -

ক
ক) 21 cm
খ
খ) 28 cm
গ
গ) 42 cm
ঘ
ঘ) 24 cm

সঠিক উত্তর: গ

গ) 42 cm

উত্তর

সঠিক উত্তর: গ

গ) 42 cm

গ

ব্যাখ্যা

Question: If the difference between the circumference and diameter of a circle is 90 cm, then the diameter of the circle is -

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

প্রশ্নমতে,
2πr - 2r = 90
⇒ 2r(π - 1) = 90
⇒ r = (90/2){(22/7) - 1}
⇒ r = 45/(22 - 7)/7
⇒ r = (45 × 7)/15
∴ r = 21

∴ বৃত্তের ব্যাস = 2r = 2 × 21 = 42 সে.মি.

৭,২১৭.

a ⊕ b = (a × b) + b, then 5 ⊕ 7 = ?

ক
18
খ
28
গ
34
ঘ
42

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

a ⊕ b = (a × b) + b
5 ⊕ 7 = (5 × 7) + 7 = 35 + 7 = 42

৭,২১৮.

A pole is broken such that the broken part makes an angle of 45° with the ground and touches it at a point 10 meters from the base. What is the length of the broken part?

ক
10√2 m
খ
20 m
গ
5√2 m
ঘ
15 m
ঙ
None of these

সঠিক উত্তর: ক

10√2 m

উত্তর

সঠিক উত্তর: ক

10√2 m

ক

ব্যাখ্যা

Question: A pole is broken such that the broken part makes an angle of 45° with the ground and touches it at a point 10 meters from the base. What is the length of the broken part?

Solution:

To find the length of the broken part of the pole, we can model this as a right-angled triangle:

The broken part of the pole is the hypotenuse.
The horizontal distance from the base of the pole to where the top touches the ground is 10 meters.
The angle between the broken part and the ground is 45°.
let the hypotenuse = x

Using the cosine function:
cos(45°) = Adjacent(base)/hypotenuse
Or, 1/√2 = 10/x
Or, x = 10√2

৭,২১৯.

If 2x + y = 15, 2y + z = 25 and 2z + x = 26, what is the value of z?

ক
ক) 9
খ
খ) 11
গ
গ) 13
ঘ
ঘ) 15

সঠিক উত্তর: খ

খ) 11

উত্তর

সঠিক উত্তর: খ

খ) 11

খ

ব্যাখ্যা

Question: If 2x + y = 15, 2y + z = 25 and 2z + x = 26, what is the value of z?

Solution:
2x + y = 15..............(1)
2y + z = 25 ..............(2)
2z + x = 26..............(3)

(1) + (2) + (3)⇒
2x + y + 2y + z + 2z + x = 15 + 25 + 26
3x + 3y + 3z = 66
3(x + y + z) = 66
x + y + z = 22..................(4)

From (2)
2y + z = 25
2y = 25 - z
y = (25 - z)/2

From (3)
2z + x = 26
x = 26 - 2z

From (4)
x + y + z = 22
26 - 2z + {(25 - z)/2} + z = 22
52 - 4z + 25 - z + 2z = 44
77 - 3z = 44
- 3z = 44 - 77
- 3z = - 33
z = 11

৭,২২০.

ক
ক) 9 : 11 : 13
খ
খ) 9 : 8 : 13
গ
গ) 9 : 8 : 10
ঘ
ঘ) 7 : 8 : 12

সঠিক উত্তর: গ

গ) 9 : 8 : 10

উত্তর

সঠিক উত্তর: গ

গ) 9 : 8 : 10

গ

ব্যাখ্যা

2A/3 = 75B/100 = 6C/10
A/B = 9/8
B/C = 8/10
A : B : C = 9 : 8 : 10

৭,২২১.

Mamun buys 1,000 shares of stock for the current stock price of 20 tk. per share. If the stock price goes up to 25 tk. per share, by what percentage does Mamun increase his money?

ক
ক) 20%
খ
খ) 25%
গ
গ) 12.5%
ঘ
ঘ) None of these

সঠিক উত্তর: খ

খ) 25%

উত্তর

সঠিক উত্তর: খ

খ) 25%

খ

ব্যাখ্যা

Question: Mamun buys 1,000 shares of stock for the current stock price of 20 tk. per share. If the stock price goes up to 25 tk. per share, by what percentage does Mamun increase his money?

Solution:
প্রতি শেয়ার ২০ টাকা হলে, মোট শেয়ার = ১০০০ × ২০ টাকা
= ২০০০০ টাকা

২৫ টাকা হলে মোট শেয়ার = ১০০০ × ২৫ টাকা
= ২৫০০০ টাকা

টাকা বৃদ্ধি পাবে = ২৫০০০ - ২০০০০ টাকা
= ৫০০০ টাকা

শতকরা বৃদ্ধি = (৫০০০/২০০০০) × ১০০%
= ২৫%

৭,২২২.

ক
a + b
খ
(1 + b)²
গ
1 + b
ঘ
1 - b

সঠিক উত্তর: গ

1 + b

উত্তর

সঠিক উত্তর: গ

1 + b

গ

ব্যাখ্যা

Question:

Solution:

৭,২২৩.

The daily rate for a hotel room that sleeps 4 people is Tk. 350 each for first two person and X taka for each additional person. If 4 people take the room for one day and each pays Tk 250 for the room, then what is the value of X?

ক
180
খ
150
গ
120
ঘ
110

সঠিক উত্তর: খ

150

উত্তর

সঠিক উত্তর: খ

150

খ

ব্যাখ্যা

Question: The daily rate for a hotel room that sleeps 4 people is Tk. 350 each for first two person and X taka for each additional person. If 4 people take the room for one day and each pays Tk 250 for the room, then what is the value of X?

Solution:
The daily rate for 1 person is Tk. 350
for two persons = 700 Tk.
For each additional person daily rate X taka
The total cost for 4 people is = 700 + 2X

If 4 people take the room for one day and each pays Tk 250 for the room
Total cost = 250 × 4 = Tk. 1000

According to the question
700 + 2X = 1000
⇒ 2X = 1000 - 700
⇒ 2X = 300
∴ X = 150

৭,২২৪.

P and Q started a business investing Tk 85000 and Tk 15000 respectively. In what ratio the profit earned after 2 years be divided between P and Q respectively?

ক
ক) 3 : 4
খ
খ) 17 : 3
গ
গ) 15 : 23
ঘ
ঘ) 3 : 5

সঠিক উত্তর: খ

খ) 17 : 3

উত্তর

সঠিক উত্তর: খ

খ) 17 : 3

খ

ব্যাখ্যা

Question: P and Q started a business investing Tk 85000 and Tk 15000 respectively. In what ratio the profit earned after 2 years be divided between P and Q respectively?

Solution:
In this type of question as time frame for both investors is equal then just get the ratio of their investments.

ATQ,
P : Q = 2 × 85000 : 2 ×15000
= 85 : 15
= 17 : 3

৭,২২৫.

A Company employs 15 persons, each working 44 hours a week. If 4 persons are absent, how many hours a week would the rest of the persons have to work to make up the time lost?

ক
ক) 45
খ
খ) 48
গ
গ) 56
ঘ
ঘ) 60

সঠিক উত্তর: ঘ

ঘ) 60

উত্তর

সঠিক উত্তর: ঘ

ঘ) 60

ঘ

ব্যাখ্যা

Question: A Company employs 15 persons, each working 44 hours a week. If 4 persons are absent, how many hours a week would the rest of the persons have to work to make up the time lost?

Solution:
একটি কোম্পানি ১৫ জন কর্মচারী নিয়োগ দেয়। প্রত্যেকে ৪৪ ঘণ্টা কাজ করে।
মোট কাজ হয় = (১৫ × ৪৪)
= ৬৬০ ঘণ্টা

৪ জন অনুপস্থিত থাকলে, বাকি থাকে = ১৫ - ৪ জন
= ১১ জন

∴ প্রত্যেকের কাজ করতে হবে = ৬৬০/১১ ঘণ্টা
= ৬০ ঘণ্টা

৭,২২৬.

Seventy-two percent of a number is 30 less than three-fourths of that number. What is the number?

ক
1120
খ
980
গ
1240
ঘ
1000

সঠিক উত্তর: ঘ

1000

উত্তর

সঠিক উত্তর: ঘ

1000

ঘ

ব্যাখ্যা

Question: Seventy-two percent of a number is 30 less than three-fourths of that number. What is the number?

Solution:
Let, the number = x.

ATQ,
72% of x = (3/4) of x - 30
⇒ 72x/100 = (3x/4) - 30
⇒ 18x/25 = (3x/4) - 30
⇒ (3x/4) - (18x/25) = 30
⇒ (75x - 72x)/100 = 30
⇒ 3x/100 = 30
⇒ 3x = 3000
⇒ x = 3000/3
∴ x = 1000

So, the number is 1000

৭,২২৭.

You visit a fruit market that has 10 different types of fruits. But the Pineapple and Durian are too spiky and smelly for your taste, so you decide to never choose them. You want to make a fruit basket with 6 different fruits from the ones you like. How many ways can you make your basket?

ক
70 ways
খ
56 ways
গ
28 ways
ঘ
22 ways

সঠিক উত্তর: গ

28 ways

উত্তর

সঠিক উত্তর: গ

28 ways

গ

ব্যাখ্যা

Question: You visit a fruit market that has 10 different types of fruits. But the Pineapple and Durian are too spiky and smelly for your taste, so you decide to never choose them. You want to make a fruit basket with 6 different fruits from the ones you like. How many ways can you make your basket?

Solution:
Number of different types of fruits =10
Number of fruits available to pick from = 10 - 2 = 8
You are choosing 6 different fruits out of 8.

∴ Number of ways = ⁸C₆
=8!/6!(8 - 6)!
= (8 × 7 × 6!)/(6! × 2!)
= 56/2
= 28 ways

৭,২২৮.

Train A, 600 metres long is running at 80 kmph will take how much time to cross a man sitting in another train which is 400 metres long, running at 64 kmph in the opposite direction?

ক
ক) 5 seconds
খ
খ) 10 seconds
গ
গ) 15 seconds
ঘ
ঘ) 20 seconds
ঙ
ঙ) 25 seconds

সঠিক উত্তর: গ

গ) 15 seconds

উত্তর

সঠিক উত্তর: গ

গ) 15 seconds

গ

ব্যাখ্যা

Distance = 600 metres
Total Speed = 64 + 80 = 144 kmph (added because they are travelling in opposite directions)
In m/sec, speed = 144×(5/18) = 40 m/sec
Distance = Speed×Time
600 = 40×Time
Therefore, Time = 15 seconds

৭,২২৯.

What is the maximum value of cos θ?

ক
1/2
খ
90
গ
1
ঘ
180

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: What is the maximum value of cos θ?

Solution:
cosθ এর সর্বনিম্ন মান −1 এবং সর্বোচ্চ মান 1
∴ cosθ এর সর্বোচ্চ মান = 1

৭,২৩০.

If sinθ - cosθ = 0, then what is the value of sin⁴θ + cos⁴θ?

ক
1
খ
1/2
গ
1/4
ঘ
0

সঠিক উত্তর: খ

1/2

উত্তর

সঠিক উত্তর: খ

1/2

খ

ব্যাখ্যা

Question: If sinθ - cosθ = 0, then what is the value of sin⁴θ + cos⁴θ?

Solution:
Given,
sinθ - cosθ = 0
⇒ sinθ = cosθ

Hence, θ = 45° [∵ Sin45° = cos45° = 1/√2]

Now,
sin⁴θ + cos⁴θ
= sin⁴45° + cos⁴45°
= 2 × (1/√2)⁴
= 2 × (1/4)
= 1/2

৭,২৩১.

A, B and C enter into a partnership with a certain capital in which A's contribution is Tk.10,000. If out of a total profit of Tk. 1000, A gets Tk. 500, B gets Tk. 300 then C's capital is -

ক
ক) Tk. 5,000
খ
খ) Tk. 2,000
গ
গ) Tk. 4,000
ঘ
ঘ) Tk. 6,000

সঠিক উত্তর: গ

গ) Tk. 4,000

উত্তর

সঠিক উত্তর: গ

গ) Tk. 4,000

গ

ব্যাখ্যা

According to the problem,
Total profit is = 1000 TK,
A’s profit = 500 TK
B’s profit = 300 TK
∴ C’s profit is = (Total profit – (A + B)’ profit) = (1000 − 800) = 200

∴ Profit Ratio between A, B and C is
⇒ 500 : 300 : 200
⇒ 5 : 3 : 2
Let their capitals be 5x, 3x, and 2x respectively.

According to the problem,
A’s contribution is TK. 10000.
∴ 5x = 10000
⇒ x = 2000

∴ C’s share = 2x
= 2 × 2000
= 4000Tk

৭,২৩২.

If a = 7, b = 5, c = 3, then the value of a2 + b2 + c2 - ab - bc - ca is -

ক
-12
খ
0
গ
8
ঘ
12

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

a² + b² + c² - ab - bc - ca = (a + b + c)² - 3(ab + bc + ca)
= (7 + 5 + 3)² - 3{(7 × 5) + (5 × 3) + (3 × 7)}
= 15² + 3(35 + 15 + 21)
= 225 - 3 × 71
=225 - 213
= 12.

৭,২৩৩.

The ages of Jamir and Amir are in the ratio of 8 : 7 respectively. After 10 years, the ratio of their ages will be 13 : 12. What is the difference between their ages?

ক
8 Years
খ
6 Years
গ
4 Years
ঘ
2 Years
ঙ
None

সঠিক উত্তর: ঘ

2 Years

উত্তর

সঠিক উত্তর: ঘ

2 Years

ঘ

ব্যাখ্যা

Question: The ages of Jamir and Amir are in the ratio of 8 : 7 respectively. After 10 years, the ratio of their ages will be 13 : 12. What is the difference between their ages?

Solution:
Let, Jamir's age be = 8x years,
Then Amir's age = 7x years

According to the question,
∴ (8x + 10)/(7x + 10) = 13/12
⇒ 12(8x + 10) = 13(7x + 10)
⇒ 96x + 120 = 91x + 130
⇒ 5x = 10
⇒ x = 2

Difference between their ages
= (8x - 7x)
= x years
= 2years

৭,২৩৪.

From a pack of cards two cards are drawn one after the other, with replacement. The probability that the first is a red card and the second is a king is -

ক
3/52
খ
15/26
গ
1/26
ঘ
11/26

সঠিক উত্তর: গ

1/26

উত্তর

সঠিক উত্তর: গ

1/26

গ

ব্যাখ্যা

Question: From a pack of cards two cards are drawn one after the other, with replacement. The probability that the first is a red card and the second is a king is -

Solution:
Total cards in a pack = 52
Red cards = 26 (13 hearts + 13 diamonds)
Kings = 4 (2 red, 2 black)
Red kings = 2

Now,
Probability of first card being red,
∴ P(red) = 26/52 = 1/2

And,
Probability of second card being a king (after replacement),
∴ P(king) = 4/52 = 1/13

∴ P(first red and second king) = P(first red) × P(second king) = (1/2) × (1/13) = 1/26

৭,২৩৫.

The volume of a sphere with radius r is (4/3)πr³ and the surface area is 4πr². If a spherical ball has a surface area of 324π square centimeters. Find its volume.

ক
620π cm³
খ
578π cm³
গ
516π cm³
ঘ
972π cm³

সঠিক উত্তর: ঘ

972π cm³

উত্তর

সঠিক উত্তর: ঘ

972π cm³

ঘ

ব্যাখ্যা

Question: The volume of a sphere with radius r is (4/3)πr³ and the surface area is 4πr². If a spherical ball has a surface area of 324π square centimeters. Find its volume.

Solution:
surface area = 4πr²

ATQ,
⇒ 4πr² = 324π
⇒ r² = 324π/4π
⇒ r² = 81 = 9²
∴ r = 9

Now,
volume = (4/3)πr³
= (4/3)π × (9)³
= (4/3)π × 729
= 972π

So, the volume would be 972π cm³

৭,২৩৬.

A person buys certain number of marbles at 20 per taka and an equal number at 30 per taka. He mixes them and sells them at 25 per taka. His gain or loss in the transaction is-

ক
2% loss
খ
4% loss
গ
2% gain
ঘ
4% gain

সঠিক উত্তর: খ

4% loss

উত্তর

সঠিক উত্তর: খ

4% loss

খ

ব্যাখ্যা

Question: A person buys certain number of marbles at 20 per taka and an equal number at 30 per taka. He mixes them and sells them at 25 per taka. His gain or loss in the transaction is-

Solution:
Cost price of a each marble at 20 per Tk. = Tk. 1/20
Cost price of a each marble at 30 per Tk. = Tk. 1/30
∴ Average cost of marble = {(1/20) + (1/30)}/2
= (3 + 2)/(2 × 60)
= 5/120
= 1/24

Selling price of mixed marble = Tk. 1/25

∴ Loss = (1/24) - (1/25) = 1/600
Cost price 1/24 then loss Tk. 1/600
Cost price 1 then loss Tk. (1/600)/(1/24)
∴ Cost price 1 then loss Tk. {(1/600)/(1/24)} × 100
= 4%

∴ There is loss of 4%

৭,২৩৭.

The average age of father and son is 30 years. After 6 years, the ratio of the age of them is 5 : 1. The present age of son is

ক
ক) 4
খ
খ) 6
গ
গ) 12
ঘ
ঘ) 14

সঠিক উত্তর: খ

খ) 6

উত্তর

সঠিক উত্তর: খ

খ) 6

খ

ব্যাখ্যা

The average age of father and son is 30 years.
The total age of father and son is 30 × 2 = 60 years.
Suppose, after 6 years, the age of father and son are 5x and x years respectively.
5x - 6 + x - 6 = 60
⇒ 6x = 72
∴ x = 12
present age of son is 12 - 6 = 6 years.
------------------------------------------------------
পিতা ও পুত্রের বয়সের গড় ৩০ বছর। ৬ বছর পরে তাদের বয়সের অনুপাত ৫ঃ১ হলে, পুত্রের বর্তমান বয়স কত?

পিতা ও পুত্রের বয়সের গড় ৩০ বছর হলে, তাদের মোট বয়স = ৩০ × ২ = ৬০ বছর
মনে করি, ৬ বছর পরে, পিতার বয়স 5x ও পুত্রের বয়স x বছর
5x - 6 + x - 6 = 60
⇒ 6x = 72
∴ x = 12
পুত্রের বর্তমান বয়স ১২ - ৬ = ৬ বছর

৭,২৩৮.

The length of two sides of the right angled triangle is 13cm and 5cm respectively. The length of the third side is -

ক
ক) greater than 15cm
খ
খ) less than 10 cm
গ
গ) equal to 3cm
ঘ
ঘ) equal to 12cm

সঠিক উত্তর: ঘ

ঘ) equal to 12cm

উত্তর

সঠিক উত্তর: ঘ

ঘ) equal to 12cm

ঘ

ব্যাখ্যা

c² = a² + b²
⇒ 13² = 5² + b²
⇒ 169 = 25 + b²
⇒ b² = 144
⇒ b = 12
So, Length of the third side is 12 cm

৭,২৩৯.

A bicycle originally cost Tk. 100 and was discounted 10%. After three months it was sold after being discounted another 15%. How much was the bicycle sold for?

ক
ক) Tk. 76.50
খ
খ) Tk. 75
গ
গ) Tk. 73.50
ঘ
ঘ) Tk. none of these

সঠিক উত্তর: ক

ক) Tk. 76.50

উত্তর

সঠিক উত্তর: ক

ক) Tk. 76.50

ক

ব্যাখ্যা

According to the question,
Original price = Tk. 100
10% discounted price = Tk. 90
The bicycle was sold after another 15% discount.
Meaning, the bicycle was sold at 85% of the discounted price.
Selling price= Tk. 90 × 85%
= Tk. 76.50

৭,২৪০.

If the volume of a cube is 1728 cm³, then the surface area of the cube will be -

ক
ক) 144 cm²
খ
খ) 432 cm²
গ
গ) 512 cm²
ঘ
ঘ) 864 cm²

সঠিক উত্তর: ঘ

ঘ) 864 cm²

উত্তর

সঠিক উত্তর: ঘ

ঘ) 864 cm²

ঘ

ব্যাখ্যা

Question: If the volume of a cube is 1728 cm³, then the surface area of the cube will be -

Solution:
দেওয়া আছে,
আয়তন, a³ = 1728
⇒ a³ = 12³
⇒ a = 12

পৃষ্ঠের ক্ষেত্রফল = 6a²
= 6 × 12²
= 6 × 144
= 864 cm²

৭,২৪১.

If x≥ 8 and y ≤ 3, then which of the following must be true?

ক
ক) x+y ≥ 5
খ
খ) x+y ≤ 11
গ
গ) x-y ≤ 5
ঘ
ঘ) x−y ≥ 5
ঙ
ঙ) None

সঠিক উত্তর: ঘ

ঘ) x−y ≥ 5

উত্তর

সঠিক উত্তর: ঘ

ঘ) x−y ≥ 5

ঘ

ব্যাখ্যা

x−y ≥ 5 is the correct answer here.

৭,২৪২.

5 - [4 - {3 - (3 - 3 - 6)}] is equal to:

ক
10
খ
18
গ
6
ঘ
0

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: 5 - [4 - {3 - (3 - 3 - 6)}] is equal to:

Solution:
Given that,
5 - [4 - {3 - (3 - 3 - 6)}]
= 5 - [4 - {3 - (-6)}]
= 5 - [4 - {3 +6}]
= 5 - [4 - {9}]
= 5 - [4 - 9]
= 5 - [-5]
= 5 + 5
= 10

৭,২৪৩.

Painting a wall costs BDT. 50 per sq. m. If the 20m long wall has a width of 10m, how much would it cost to paint 7/10th of the wall?

ক
ক) 5000
খ
খ) 10000
গ
গ) 8000
ঘ
ঘ) 7000

সঠিক উত্তর: ঘ

ঘ) 7000

উত্তর

সঠিক উত্তর: ঘ

ঘ) 7000

ঘ

ব্যাখ্যা

Walls area = 20 × 10 = 200 sq. m
∴ Cost of painting of 7/10th of the wall = (200 × 7/10 × 50) = 7000 taka

৭,২৪৪.

Anik dey invested in two accounts, which pay 5% and 10% interest annually. The amount invested at 10% rate is 110% of the amount invested at 5% rate. After three years, he earns 5760 taka in interest. How much did he invest at the 10% rate?

ক
18285 tk.
খ
13200 tk.
গ
12000 tk.
ঘ
10000 tk.

সঠিক উত্তর: খ

13200 tk.

উত্তর

সঠিক উত্তর: খ

13200 tk.

খ

ব্যাখ্যা

Question: Anik dey invested in two accounts, which pay 5 % and 10% interest annually. The amount invested at 10% rate is 110% of the amount invested at 5% rate. After three years, he earns 5760 taka in interest. How much did he invest at the 10% rate?

Solution:
5% সুদের জন্য আসল x টাকা
10% সুদের জন্য আসল 110%x = 1.1x টাকা

5% হারে সুদ = x × 3 × 5/100 = 3x/20 টাকা
10% হারে সুদ = 1.1x × 3 × 10/100 = 3.3x/10 টাকা

(3x/20) + (3.3x/10) = 5760
⇒ (3x + 6.6x)/20 = 5760
⇒ 9.6x = 5760 × 20
⇒ x = (5760 × 20)/9.6
= 12000 tk.

10% সুদের হারের জন্য আসল = 1.1 × 12000
= 13200 tk.

৭,২৪৫.

A man rows to a place 48 km distant and back in 14 hours. He finds that he can row 4 km with the stream at the same time as 3 km against the stream. The rate of the stream is:

ক
ক) 1 km/hr
খ
খ) 1.5 km/hr
গ
গ) 1.8 km/hr
ঘ
ঘ) 3.5 km/hr

সঠিক উত্তর: ক

ক) 1 km/hr

উত্তর

সঠিক উত্তর: ক

ক) 1 km/hr

ক

ব্যাখ্যা

Suppose he moves 4 km downstream in x hours.
Then, Speed downstream = (4/x) km/hr
Speed upstream = (3/x) km/hr

∴ 48/(4/x) + 48/(3/x) = 14
⇒ 48x/4 + 48x/3 = 14
⇒ 48{(x/4) + (x/3)} = 14
⇒ 7x/12 = 14/48
⇒ 7x/12 = 7/24
⇒ 7x = (7/24) × 12
⇒ 7x = 7/2
⇒ 14x = 7
⇒ x = 7/14
⇒ x = 1/2

So, speed downstream = 7 km/hr
Rate of the stream = 1/2(8 - 6) km/hr = 1 km/hr.

৭,২৪৬.

The average of the first six multiples of 4 is-

ক
14
খ
12
গ
11
ঘ
16
ঙ
18

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: The average of the first six multiples of 4 is-

Solution:
The first six multiples of 4 are = 4, 8, 12, 16, 20, 24

Average = (4 + 8 + 12 + 16 + 20 + 24)/6
= 14

৭,২৪৭.

In a family the average age of the father and mother is 38 years whereas the average age of father, mother and their only daughter is 28 years. Then the age of the daughter is -

ক
7 years
খ
6 years
গ
5 years
ঘ
4 years
ঙ
8 years

সঠিক উত্তর: ঙ

8 years

উত্তর

সঠিক উত্তর: ঙ

8 years

ঙ

ব্যাখ্যা

The age of the daughter = (3 × 28) - (2 × 38)
= 84 - 76
= 8 years

৭,২৪৮.

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
45 min
খ
1 hour
গ
30 min
ঘ
2 hour

সঠিক উত্তর: খ

1 hour

উত্তর

সঠিক উত্তর: খ

1 hour

খ

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:
5 books can be arranged in = 5! ways
= 5 × 4 × 3 × 2 × 1 = 120 ways

Now,
Time per arrangement 1 move = 1/2 minute

So, total time required is = 120 × (1/2) = 60 minute = 1 hour

৭,২৪৯.

If θ be a positive acute angle satisfying cos²θ + cos⁴θ = 1, then the value of tan²θ + tan⁴θ is?

ক
3/2
খ
1
গ
1/2
ঘ
2

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: If θ be a positive acute angle satisfying cos²θ + cos⁴θ = 1, then the value of tan²θ + tan⁴θ is?

Solution:
Given that,
cos²θ + cos⁴θ = 1 ..........(1)
⇒ cos⁴θ = 1 - cos²θ
⇒ cos⁴θ = sin²θ
⇒ cos²θ.cos²θ = sin²θ
⇒ cos²θ = sin²θ/cos²θ
⇒ cos²θ = tan²θ

Now, putting cos²θ = tan²θ in equation (1) than we get,
⇒ tan²θ + tan⁴θ = 1

৭,২৫০.

The selling price of an article is tk 39. If its cost price is numerically equal to its profit percent, then its cost price is -

ক
29
খ
27
গ
33
ঘ
30
ঙ
32

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Let, cost price be x, sales price is 39. Then gain percentage = x%.

According to question,

Sales Price = (100 + profit%)/100 * cost price

Or, 39 = {(100 + x)/100} * x

Or, 3900 = x² + 100x

Or, x² + 100x - 3900 = 0

Or, x² + 130x -30x - 3900 = 0

Or, x( x + 130) - 30 ( x + 130) = 0

Or, ( x + 130) (x - 30) = 0

So, x = 30, x = -130 ( Cost price can not be negative)

৭,২৫১.

Tomal is younger than Jamal by 7 years. If their ages are in the respective ratio of 7 : 9, how old is Jamal?

ক
24.5 years
খ
30.5 years
গ
31.5 years
ঘ
32 years

সঠিক উত্তর: গ

31.5 years

উত্তর

সঠিক উত্তর: গ

31.5 years

গ

ব্যাখ্যা

Question: Tomal is younger than Jamal by 7 years. If their ages are in the respective ratio of 7 : 9, how old is Jamal?

Solution:
Let Jamal's age be x years.
Then, Tomal's age = (x - 7) years.

ATQ,
(x - 7)/x = 7/9
⇒ 9((x - 7) = 7x
⇒ 9x - 63 = 7x
⇒ 9x - 7x = 63
⇒ 2x = 63
∴ x = 31.5

Jamal's age is 31.5 years.

৭,২৫২.

In how many ways can seven books be arranged on a shelf?

ক
5040
খ
720
গ
7
ঘ
1

সঠিক উত্তর: ক

5040

উত্তর

সঠিক উত্তর: ক

5040

ক

ব্যাখ্যা

Question: In how many ways can seven books be arranged on a shelf?

Solution:
Number of ways in which the first book can be placed = 7
Number of ways in which the second book can be placed = 6
Similarly,
The total number of ways in which seven books can be arranged on a shelf = 7 × 6 × 5 × 4 × 3 × 2 × 1
= 7!
= 5040

৭,২৫৩.

If 8^x + 2^x = 68, then what is the value of x?

ক
2
খ
3
গ
1/2
ঘ
1/3
ঙ
4

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: If 8^x + 2^x = 68, then what is the value of x?

Solution:
8^x + 2^x
Let x = 2
8² + 2² = 64 + 4 = 68

৭,২৫৪.

If 2log₄x = 1 + log₄(x - 1), find the value of x?

ক
1
খ
4
গ
2
ঘ
3

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: If 2log₄x = 1 + log₄(x - 1), find the value of x?

Solution:
2log₄x = 1 + log₄(x - 1)
⇒ log₄x² = log₄4 + log₄(x - 1)
⇒ x² = 4(x - 1)
⇒ x² - 4x + 4 = 0
⇒ (x - 2)² = 0
⇒ x - 2 = 0
∴ x = 2

৭,২৫৫.

If m² + (1/m²) = 47, what is the value of m + (1/m)?

ক
5
খ
7
গ
9
ঘ
11

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: If m² + (1/m²) = 47, what is the value of m + (1/m)?

Solution:
m² + (1/m²) = 47
⇒ (m + 1/m)² - 2 . m. (1/m) = 47
⇒ (m + 1/m)² - 2 = 47
⇒ (m + 1/m)² = 47 + 2 = 49
∴ (m + 1/m) = √49 = 7

৭,২৫৬.

(1000)¹³÷ (10)³⁷ = ?

ক
ক) 1
খ
খ) 100
গ
গ) 10
ঘ
ঘ) 1/10

সঠিক উত্তর: খ

খ) 100

উত্তর

সঠিক উত্তর: খ

খ) 100

খ

ব্যাখ্যা

(1000)¹³÷ (10)³⁷
(10³)¹³ ÷ (10)³⁷
10³⁹ ÷ (10)³⁷
10^{39 - 37}
10²
100

৭,২৫৭.

The sum of three consecutive odd natural numbers each divisible by 3 is 63. Find the largest number among them.

ক
32
খ
28
গ
27
ঘ
25

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: The sum of three consecutive odd natural numbers each divisible by 3 is 63. Find the largest number among them.

Solution:
let, the numbers 3x, 3x + 6, 3x + 12

3x + 3x + 6 + 3x + 12 = 63
⇒ 9x + 18 = 63
⇒ 9x = 45
∴ x = 5

The largest among them is = (3 × 5) + 12
= 15 + 12
= 27

৭,২৫৮.

ACZ, CEY, EGX, GIW, _——, KMU.

ক
IKV
খ
ILT
গ
LVX
ঘ
HKU

সঠিক উত্তর: ক

IKV

উত্তর

সঠিক উত্তর: ক

IKV

ক

ব্যাখ্যা

Question: ACZ, CEY, EGX, GIW, _——, KMU.

Solution:
There are two independent alphabetical series here.

First series (first two letters as a pair):
Each pair advances by +2 in the alphabet for each letter :
A (+2) → C (+2) → E (+2) → G (+2) → I (+2) → K
C (+2) → E (+2) → G (+2) → I (+2) → K (+2) → M
∴ (AC, CE, EG, GI, IK, and KM).

Second series (third letter):
This is a reverse alphabetical sequence (decreasing by 1).
Z (-1) → Y (-1) → X (-1) → W (-1) → V (-1) → U
∴ Z, Y, X, W, V, U.

So the missing term after GIW is IKV.

৭,২৫৯.

The probability that a card drawn from a pack of 52 cards will be a diamond or a king is -

ক
2/13
খ
1/13
গ
4/13
ঘ
1/26

সঠিক উত্তর: গ

4/13

উত্তর

সঠিক উত্তর: গ

4/13

গ

ব্যাখ্যা

Question: The probability that a card drawn from a pack of 52 cards will be a diamond or a king is -

Solution:
Here, n(S) = 52
There are 13 cards of diamonds (including one king) and there are three more kings.
Let E = the event of getting a diamond or a king
Then, n(E) = (13 + 3) = 16

∴ P(E) = n(E)/n(S)
= 16/52
= 4/13

৭,২৬০.

The average of 7 consecutive numbers is a. If the next two numbers are included, the average will be-

ক
ক) increase by 1
খ
খ) increase by 2
গ
গ) increase by 3
ঘ
ঘ) remain same

সঠিক উত্তর: ক

ক) increase by 1

উত্তর

সঠিক উত্তর: ক

ক) increase by 1

ক

ব্যাখ্যা

Question: The average of 7 consecutive numbers is a. If the next two numbers are included, the average will be-

Answer:
let first number is n
so, the numbers are n, n+1, n+2,......, n+6
(n + n + 1 + .... + n + 6)/7 = a
⇒ 7n + (1 + 2 + 3 +....+ 6) = 7a
⇒ 7n + (6 × 7)/2 = 7a
⇒ 7n + 21 = 7a
⇒ 7n = 7a - 21
⇒ n = (7a - 21)/7
∴ n = 7(a - 3)/7 = a - 3

If the next two numbers are included, the average will be
= (n + n + 1 + .... + n + 8)/9
= {9n +(1 + 2 + 3 +.....+8)}/9
= (9n + 36)/9
= n + 4
= a - 3 + 4
= a + 1

So, the average will increase by 1

৭,২৬১.

While working 8 hours a day, A alone can complete a piece of work in 8 days and B alone in 10 days. At what time would they complete it together, 5 hours a day?

ক
128/9 days
খ
7 days
গ
55/9 days
ঘ
64/9 days

সঠিক উত্তর: ঘ

64/9 days

উত্তর

সঠিক উত্তর: ঘ

64/9 days

ঘ

ব্যাখ্যা

Question: While working 8 hours a day, A alone can complete a piece of work in 8 days and B alone in 10 days. In what time would they complete it together, 5 hours a day?

Solution:
A can complete the work in 8 × 8 = 64 hours
1 hour's work of A = 1/64

B can complete the work in 10 × 8 = 80 hours
1 hour's work of B = 1/80

(A + B)'s 1 hour's work
= 1/64 + 1/80
= (5 + 4)/320
= 9/320

they can complete the work in = 320/9 hours

∴ as they will work 5 hours a day, it will take = 320/(9 × 5) = 64/9 days

৭,২৬২.

A leak in the bottom of a tank can empty the full tank in 5 hours. An inlet pipe fills water are at the rate 4 liters a minute. When the tank is full, the inlet is opened and due to the leak, the tank is empty in 15 hours. how many liters does the tank hold?

ক
ক) 1800
খ
খ) 1200
গ
গ) 1400
ঘ
ঘ) None of these

সঠিক উত্তর: ক

ক) 1800

উত্তর

সঠিক উত্তর: ক

ক) 1800

ক

ব্যাখ্যা

Question: A leak in the bottom of a tank can empty the full tank in 5 hours. An inlet pipe fills water are at the rate 4 liters a minute. When the tank is full, the inlet is opened and due to the leak, the tank is empty in 15 hours. how many liters does the tank hold?

Solution:
ছিদ্র দ্বারা ৫ ঘণ্টায় খালি হয় ১ অংশ
ছিদ্র দ্বারা ১ ঘণ্টায় খালি হয় ১/৫ অংশ

আবার, ট্যাঙ্কটি পূর্ণ থাকা অবস্থায় ইনলেট পাইপ এবং ছিদ্র দ্বারা,
১৫ ঘণ্টায় খালি হয় সম্পূর্ণ বা ১ অংশ
১ ঘণ্টায় খালি হয় ১/১৫ অংশ

[শুধু ছিদ্র দ্বারা খালি হওয়ার অংশ, ইনলেট পাইপ এবং ছিদ্র খোলা অবস্থায় খালি হওয়ার অংশের তুলনায় বেশি। তাই বড় অংশ থেকে ছোট অংশ বিয়োগ করলে শুধু ইনলেট পাইপ দ্বারা ১ ঘণ্টায় ট্যাঙ্কটি কত অংশ পূর্ণ তা পাওয়া যাবে।]

সুতরাং, ইনলেট পাইপ দ্বারা ১ ঘণ্টায় নেট পূর্ণ হয় = ১/৫ - ১/১৫ অংশ
= (৩ - ১)/১৫ অংশ
= ২/১৫ অংশ

ইনলেট পাইপ দ্বারা ২/১৫ অংশ পূর্ণ হয় ১ ঘণ্টায়
ইনলেট পাইপ দ্বারা ১ অংশ পূর্ণ হয় ১৫/২ ঘণ্টায়

এখন, ১৫/২ ঘণ্টা = (১৫/২) × ৬০ মিনিট = ৪৫০ মিনিট

আবার,
১ মিনিটে পূর্ণ হয় ৪ লিটার
৪৫০ মিনিটে পূর্ণ হয় ৪ × ৪৫০ লিটার
= ১৮০০ লিটার

∴ ট্যাঙ্কটি ধারণক্ষমতা ১৮০০ লিটার।

৭,২৬৩.

The sum of five consecutive numbers is 85. What is the difference between thrice the smallest number and twice the largest number?

ক
9
খ
8
গ
7
ঘ
6

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: The sum of five consecutive numbers is 85. What is the difference between thrice the smallest number and twice the largest number?

Solution:
Let the smallest number be x
Then the five consecutive numbers are: x, x + 1, x + 2, x + 3, x + 4

ATQ,
x + (x + 1) + (x + 2) + (x + 3) + (x + 4) = 85
⇒ 5x + 10 = 85
⇒ 5x = 75
⇒ x = 15

∴ The difference between thrice the smallest number and twice the largest number
= (x × 3) - {(x + 4) × 2}
= (15 × 3) - {(15 + 4) × 2}
= 7

৭,২৬৪.

A shopkeeper offers two successive discounts of 10% and 20% on an item marked at 1,000 Taka. What is the final selling price?

ক
680 Taka
খ
720 Taka
গ
750 Taka
ঘ
780 Taka

সঠিক উত্তর: খ

720 Taka

উত্তর

সঠিক উত্তর: খ

720 Taka

খ

ব্যাখ্যা

Question: A shopkeeper offers two successive discounts of 10% and 20% on an item marked at 1,000 Taka. What is the final selling price?

Solution:
New Price after First Discount:
1000 - 10% of 1000
= 1000 - 100
= 900 Taka

∴ Final selling price = 900 - 20% of 900
= 900 - 180
= 720 Taka

৭,২৬৫.

What is the angle between the hour and minute hand of a clock when it is 2 : 30 pm?

ক
100°
খ
105°
গ
108°
ঘ
110°

সঠিক উত্তর: খ

105°

উত্তর

সঠিক উত্তর: খ

105°

খ

ব্যাখ্যা

Question: What is the angle between the hour and minute hand of a clock when it is 2 : 30 pm?

Solution:
2টা 30 মিনিট = 2 + (30/60) ঘন্টা
= 2 + 1/2 = 5/2 ঘন্টা

আমরা জানি,
ঘণ্টার কাঁটা 12 ঘণ্টায় 360° ঘোরে।
∴ 1 ঘণ্টায় ঘোরে = 360°/12 = 30°
∴ 5/2 ঘণ্টায় ঘোরে = (30° × 5)/2
= 75°

আবার,
মিনিটের কাঁটা 60 মিনিটে 360° ঘোরে।
∴ 1 মিনিটে ঘোরে = 360°/60 = 6°
∴ 30 মিনিটে ঘোরে = 30 × 6° = 180°

∴ ঘড়ির কাঁটা দুটির মধ্যবর্তী কোণ = |180° - 75°| = 105°

৭,২৬৬.

A wheel that has 6 cogs has meshed with a larger wheel of 14 cogs. When the smaller wheel has made 21 revolutions, the number of revolutions made by the larger wheel will be-

ক
ক) 4
খ
খ) 9
গ
গ) 12
ঘ
ঘ) 49

সঠিক উত্তর: খ

খ) 9

উত্তর

সঠিক উত্তর: খ

খ) 9

খ

ব্যাখ্যা

Question: A wheel that has 6 cogs has meshed with a larger wheel of 14 cogs. When the smaller wheel has made 21 revolutions, the number of revolutions made by the larger wheel will be-

Solution:
Let the required number of revolutions made by the larger wheel be x.
We know, More cogs ⟹ Less revolutions

So, it is Indirect Proportion.
∴ 14 : 6 : : 21 : x
⇒ 14 × x = 6 × 21
⇒ x = (6 × 21)/14
⇒ x = 9

Hence, the number of revolutions made by the larger wheel is 9

৭,২৬৭.

A boat takes 90 minutes less to travel 36 miles downstream than to travel the same distance upstream. If the speed of the boat in still water is 10 mph, the speed of the stream is-

ক
2mph
খ
3mph
গ
4mph
ঘ
5mph

সঠিক উত্তর: ক

2mph

উত্তর

সঠিক উত্তর: ক

2mph

ক

ব্যাখ্যা

Question: A boat takes 90 minutes less to travel 36 miles downstream than to travel the same distance upstream. If the speed of the boat in still water is 10 mph, the speed of the stream is-

Solution:
Let,
the speed of the stream = b mph

Then,
Speed downstream = (10 + b) mph
Speed upstream = (10 - b) mph

ATQ,
{36/(10 - b) - 36/(10 + b)} = 90/60
⇒ (72b × 60) = 90(100 - b²)
⇒ b²+ 48b - 100 = 0
⇒ (b + 50)(b - 2) = 0
⇒ b = 2mph

৭,২৬৮.

What is the remainder when 3³⁷ is divided by 10?

ক
1
খ
3
গ
6
ঘ
7

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: What is the remainder when 3³⁷ is divided by 10?

Solution:

When dividing a positive integer by 10, the remainder is always the units digit of that integer. For instance, 123 divided by 10 yields the remainder of 3. Hence, essentially we need to find the units digit of 3³⁷.

For that,
we can use the cyclicity of 3 in positive integer power, which is four, meaning that the units digit of 3 in positive integer power repeats in blocks of four {3, 9, 7, 1}{3, 9, 7, 1}...

3¹ = 3

3² = 9

3³ = 27

3⁴ = 81

3⁵ = 243

...

The power, 37, is 1 greater than a multiple of 4, so the units digit of 3³⁷ will be the first number in the cyclicity block, which is 3, giving the remainder of 3 when divided by 10.

৭,২৬৯.

Which number replaces the question mark?

ক
12
খ
6
গ
9
ঘ
8

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: Which number replaces the question mark?

Solution:
Here,
6 - 4 = 2
3 - 2 = 1
7 - 1 = 6

∴ The question mark is replaced by 6

৭,২৭০.

A cistern can be filled by two pipes in 20 and 30 min respectively. Both pipes being open, when must the first pipe be turned off so that the cistern may be filled in 10 min more?

ক
5 min
খ
6 min
গ
8 min
ঘ
9 min

সঠিক উত্তর: গ

8 min

উত্তর

সঠিক উত্তর: গ

8 min

গ

ব্যাখ্যা

Question: A cistern can be filled by two pipes in 20 and 30 min respectively. Both pipes being open, when must the first pipe be turned off so that the cistern may be filled in 10 min more?

Solution:
In 1 min both pipes can fill = (1/20) + (1/30)
= (3 + 2)/60
=5/60
= 1/12
In 10 min second pipe can fill = (1/30) × 10 = 1/3 part

Part of cistern filled by both the pipes = 1 - 1/3
= (3 - 1)/3
= 2/3

1/12 part is filled in 1 min
∴ 2/3 part will be filled in (12 × 2)/3 = 8 min

Hence, first pipe should be turned off after 8 min.

৭,২৭১.

The B.D. and T.D. on a certain sum is Tk. 200 and Tk. 100 respectively. Find out the sum.

ক
ক) 400
খ
খ) 300
গ
গ) 100
ঘ
ঘ) 200

সঠিক উত্তর: ঘ

ঘ) 200

উত্তর

সঠিক উত্তর: ঘ

ঘ) 200

ঘ

ব্যাখ্যা

F = (BD×TD)/(BD−TD)
= (200 × 100)/ (200−100)
= 200−100/ 100
= Tk.200

৭,২৭২.

Age of a mother 10 years ago was 3 times the age of her son. After 10 years, the mother's age will be twice that of his son. Find the ratio of their present ages?

ক
11 : 7
খ
9 : 5
গ
7 : 4
ঘ
7 : 3

সঠিক উত্তর: ঘ

7 : 3

উত্তর

সঠিক উত্তর: ঘ

7 : 3

ঘ

ব্যাখ্যা

We are given that,
age of mother 10 years ago was 3 times the age of her son
So,
let the age of the son be x and as the mother's age is 3 times the age of her son, let it be 3x, three years ago.
At present,
Mother's age will be (3x + 10) and son's age will be (x + 10)
After 10 years,
Mother's age will be (3x + 10) +10 and son’s age will be (x + 10) + 10

Mother's age is twice that of son
(3x + 10) +10 = 2 [(x + 10) + 10]
(3x + 20) = 2[x + 20]
Solving the equation, we get x = 20
We are asked to find the present ratio.
(3x + 10) : (x + 10) = 70 : 30
(3x + 10) : (x + 10) = 7 : 3.

৭,২৭৩.

A, B, C enter into partnership. A invests 3 times as much as B invests and B invests two third of what C invests. At the end of the year, profit earned is TK. 6600. What is the share of B?

ক
1100
খ
1150
গ
1175
ঘ
1200

সঠিক উত্তর: ঘ

1200

উত্তর

সঠিক উত্তর: ঘ

1200

ঘ

ব্যাখ্যা

Question: A, B, C enter into partnership. A invests 3 times as much as B invests and B invests two third of what C invests. At the end of the year, profit earned is TK. 6600. What is the share of B?

Solution:
A invests 3 times as much as B invests and B invests two third of what C invests.
Let,
C invests = x
B invests = 2x/3
A invests = 3 × (2x/3) = 2x

∴ The ratio of invests or profit = 2x : 2x/3 : x
= 6x : 2x : 3x
= 6 : 2 : 3

The share of B = (2/11) × 6600
= 1200

৭,২৭৪.

A cricket group of 6 members is to be formed from Academy A. The members of the group are to be from 3 bowlers, 4 batsman and 5 coaches. How many ways are possible to form the group so that it has either 3 coaches and 3 batsman or 2 bowlers and 4 coaches.

ক
22
খ
55
গ
90
ঘ
144

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

3 coaches and 3 batsman or 2 bowlers and 4 coaches means
(3 coaches x 3 batsman) + (2 bowlers x 4 coaches)

Select 3 coaches out of 5 = ⁵C₃ = 5!/(3! × 2!) = 10

Select 3 batsman out of 4 = ⁴C₃ = 4!/(3! × 1!) = 4

Select 2 bowlers out of 3 = ³C₂ = 3!/(2! × 1!) = 3

Select 4 coaches out of 5 = ⁵C₄ = 5!/(4! × 1!) = 5

Total ways to form the group = (10 × 4) + (3 × 5) = 40 + 15 = 55.

৭,২৭৫.

Two pipes can fill a tank in 20 and 24 minutes respectively and a waste pipe can empty 3 gallons per minute. All the three pipes working together can fill the tank in 15 minutes. The capacity of the tank is-

ক
60 gallons
খ
100 gallons
গ
120 gallons
ঘ
180 gallons

সঠিক উত্তর: গ

120 gallons

উত্তর

সঠিক উত্তর: গ

120 gallons

গ

ব্যাখ্যা

Question: Two pipes can fill a tank in 20 and 24 minutes respectively and a waste pipe can empty 3 gallons per minute. All the three pipes working together can fill the tank in 15 minutes. The capacity of the tank is-

Solution:
1^st pipe fill in 1 minutes = 1/20 part
2nd pipe fill in 1 minutes = 1/24 part

∴ Work done by the waste pipe in 1 minute = {1/15 - (1/20 + 1/24)} part
= (1/15 - 11/120) part
= (8 - 11)/120 part
= - (3/120) part
= - (1/40) [ −ve sign means emptying]

ATQ,
1/40 part = 3 gallons

∴ Volume of whole=(3 × 40) gallons
= 120 gallons

৭,২৭৬.

Pervez and Sunny can complete a piece of work in 20 and 15 days respectively. They worked together for 6 days, after which Sunny was replaced by Ashu. If the work would be finished in next 4 days, find the number of days in which Ashu alone could complete the work.

ক
36
খ
40
গ
45
ঘ
56

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: Pervez and Sunny can complete a piece of work in 20 and 15 days respectively. They worked together for 6 days, after which Sunny was replaced by Ashu. If the work would be finished in next 4 days, find the number of days in which Ashu alone could complete the work.

Solution:
Pervez can complete the work in 20 days
Sunny can complete the same job in 15 days
Let the total work = LCM of both
Therefore, the LCM of 20 and 15 is 60, that means total work = 60 units
Now, Pervez's one-day work efficiency = 60/20 = 3 units
Sunny's one-day work efficiency = 60/15 = 4 units
ATQ,
(Pervez + Sunny) together works for 6 days
That means (Pervez+Sunny) have done 7 × 6 = 42 units work in 6 days
Now, the remaining work will be 60 - 42 = 18 units, and it will finish in 4 days.
Now, Sunny is replaced with Ashu:
So, per days work required to be finished = 18 units / 4 days = 4.5 units
But, we know that Pervez's one-day work = 3 units, so Ashu's one-day work = 4.5 - 3 = 1.5 units
Therefore, Ashu alone can finish the total work in 60 units /1.5 days = 40 days

৭,২৭৭.

After 15 years, Rahul’s age will be five times what it was 5 years before. How old is Rahul now?

ক
12 years
খ
10 years
গ
9 years
ঘ
16 years

সঠিক উত্তর: খ

10 years

উত্তর

সঠিক উত্তর: খ

10 years

খ

ব্যাখ্যা

Question: After 15 years, Rahul’s age will be five times what it was 5 years before. How old is Rahul now?

Solution:
Given that,
Rahul’s age after 15 years will be five times his age 5 years ago.

Let Rahul’s present age = x years

According to the question:
⇒ x + 15 = 5(x - 5)
⇒ x + 15 = 5x - 25
⇒ 5x - x = 15 + 25
⇒ 4x = 40
∴ x = 10

∴ Rahul is 10 years old now.

৭,২৭৮.

180 fazley mangoes were bought from a fruits shop. After two days 9 mangoes rotted. How many mangoes are quite good?

ক
80%
খ
85%
গ
90%
ঘ
95%

সঠিক উত্তর: ঘ

95%

উত্তর

সঠিক উত্তর: ঘ

95%

ঘ

ব্যাখ্যা

Question: 180 fazley mangoes were bought from a fruits shop. After two days 9 mangoes rotted. How many mangoes are quite good?

Solution:
The total number of mangoes which were bought = 180
The number of rotten mangoes = 9.

The number of good mangoes = 180 - 9 = 171.

The percentage of good mangoes is (171 × 100)/180
= 95%.

৭,২৭৯.

If A = 45°, What is the value of 3 - sec²A?

ক
- 1
খ
1
গ
2
ঘ
- 2

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: If A = 45°, What is the value of 3 - sec²A?

Solution:
A = 30°

3 - sec²A
= 3 - (sec45°)²
= 3 - 2
= 1

৭,২৮০.

A room has dimensions 6 m × 4 m × 3 m (length × width × height). If only the walls and the ceiling are to be painted, what is the total area to be painted?

ক
74m³
খ
64m²
গ
60m³
ঘ
84m²

সঠিক উত্তর: ঘ

84m²

উত্তর

সঠিক উত্তর: ঘ

84m²

ঘ

ব্যাখ্যা

Question: A room has dimensions 6 m × 4 m × 3 m (length × width × height). If only the walls and the ceiling are to be painted, what is the total area to be painted?

Solution:
Here,
Length, L = 6 m
Width, W = 4 m
Height, H = 3 m

We know,
Wall Area = 2 × (L + W) × H
= 2 × (6 + 4) × 3
= 2 × 10 × 3
= 60 m²

And,
Ceiling is just the top surface of the floor (rectangle).
Ceiling Area = L × W = 6 × 4 = 24 m²

∴ Total Area = Wall Area + Ceiling Area = (60 + 24) m² = 84m²

৭,২৮১.

The banker's gain on a sum due 2 years hence at 10% per annum is 320 Tk. The banker's discount is:

ক
1600 Tk
খ
1920 Tk
গ
2100 Tk
ঘ
2550 Tk
ঙ
2820 Tk

সঠিক উত্তর: খ

1920 Tk

উত্তর

সঠিক উত্তর: খ

1920 Tk

খ

ব্যাখ্যা

Question: The banker's gain on a sum due 2 years hence at 10% per annum is 320 Tk. The banker's discount is:

Solution:
Here,
Banker’s Gain (B.G) = 320 Tk
Rate (R) = 10%
Time (T) = 2 years

We know,
True Discount (T.D) = B.G/(R × T)
= (320 × 100)/(10 × 2)
= 3200/2
= 1600 Tk

∴ Banker’s Discount (B.D) = 1600 + 320
= 1920 Tk

৭,২৮২.

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

ক
ক) 6
খ
খ) 12
গ
গ) 18
ঘ
ঘ) 24

সঠিক উত্তর: ক

ক) 6

উত্তর

সঠিক উত্তর: ক

ক) 6

ক

ব্যাখ্যা

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

৭,২৮৩.

D is any point on side AC of ΔABC. If P, Q, X, Y are the mid-point of AB, BC, AD and DC respectively, then the ratio of PX and QY is

ক
2 : 1
খ
3 : 2
গ
1 : 1
ঘ
2 : 5

সঠিক উত্তর: গ

1 : 1

উত্তর

সঠিক উত্তর: গ

1 : 1

গ

ব্যাখ্যা

Question: D is any point on side AC of ΔABC. If P, Q, X, Y are the mid-point of AB, BC, AD and DC respectively, then the ratio of PX and QY is

Solution:
ATQ,

PX || BD
∴ PX = (1/2)BD
Similarly, QY || BD
∴ QY = (1/2)BD

∴ PX : QY = (1/2)BD : (1/2)BD = 1 : 1

৭,২৮৪.

The speed ratio of car, train and bus are 5 : 9 : 4. If the average speed of the 3 vehicles is 72 km per hour, what is the average speed of the car and the bus?

ক
ক) 76 km/hr.
খ
খ) 54 km/hr.
গ
গ) 67 km/hr.
ঘ
ঘ) 86 km/hr.

সঠিক উত্তর: খ

খ) 54 km/hr.

উত্তর

সঠিক উত্তর: খ

খ) 54 km/hr.

খ

ব্যাখ্যা

Question: The speed ratio of car, train and bus are 5 : 9 : 4. If the average speed of the 3 vehicles is 72 km per hour, what is the average speed of the car and the bus?

Solution:
ধরি
গাড়ির গতিবেগ = ৫ক কি.মি./ ঘণ্টা
ট্রেনের গতিবেগ = ৯ক কি.মি./ ঘণ্টা
বাসের গতিবেগ = ৪ক কি.মি./ ঘণ্টা

প্রশ্নমতে,
(৫ক + ৯ক + ৪ক)/৩ = ৭২
বা, ১৮ক/৩ = ৭২
বা, ৬ক = ৭২
ক = ১২

গাড়ির গতিবেগ = ৫ × ১২ = ৬০ কি.মি./ ঘণ্টা
বাসের গতিবেগ = ৪ × ১২ = ৪৮ কি.মি./ ঘণ্টা

গাড়ি ও বাসের গড় গতিবেগ = (৬০ + ৪৮)/২ = ৫৪ কি.মি./ঘণ্টা

৭,২৮৫.

Apple is producing the iPhone at a cost of 200 dollars and selling it at 600 dollars. In the production of 100 phones, 40 were defective phones. What is the percentage of loss or profit?

ক
80% loss
খ
20% profit
গ
80% profit
ঘ
10% loss

সঠিক উত্তর: গ

80% profit

উত্তর

সঠিক উত্তর: গ

80% profit

গ

ব্যাখ্যা

Question: Apple is producing the iPhone at a cost of 200 dollars and selling it at 600 dollars. In the production of 100 phones, 40 were defective phones. What is the percentage of loss or profit?

Solution:
100 phones making cost = (100 × 200) = 20000 dollars.
Selling price of 60 phones = (60 × 600) = 36000 dollars

profit = (36000 - 20000) = 16000

profit percentage = (16000/20000)100%
= 80%

৭,২৮৬.

The average age of a husband and his wife was 25 years at the time of their marriage. After five years they have a three-year old child. What is the average age of the family now?

ক
ক) 20 years
খ
খ) 21 years
গ
গ) 22 years
ঘ
ঘ) 23 years

সঠিক উত্তর: খ

খ) 21 years

উত্তর

সঠিক উত্তর: খ

খ) 21 years

খ

ব্যাখ্যা

Question: The average age of a husband and his wife was 25 years at the time of their marriage. After five years they have a three-year old child. What is the average age of the family now?

solution:
The average age of a husband and his wife was 25 years at the time of their marriage
Sum of age = 25 × 2 = 50

After five years, sum of their age = 50 + 5 + 5 = 60
Sum of age of the family = 60 + 3 = 63 years

∴the average age of the family is = 63/3 = 21 years

৭,২৮৭.

Which of the following fractions is equal to 0.16?

ক
1/4
খ
4/25
গ
5/8
ঘ
8/5

সঠিক উত্তর: খ

4/25

উত্তর

সঠিক উত্তর: খ

4/25

খ

ব্যাখ্যা

Question: Which of the following fractions is equal to 0.16?

Solution:
1/4 = 0.25
4/25 = 0.16
5/8 = 0.625
8/5 = 1.6

৭,২৮৮.

A boat covers a certain distance in half an hour downstream with the speed of 40km/hr in still water and the speed of the current is 8km/hr. Then the distance travelled by the boat is:

ক
ক) 6km
খ
খ) 7km
গ
গ) 12km
ঘ
ঘ) 24km

সঠিক উত্তর: ঘ

ঘ) 24km

উত্তর

সঠিক উত্তর: ঘ

ঘ) 24km

ঘ

ব্যাখ্যা

Given that,
The speed of a boat in still water is 40 km/hr
The speed of the stream is 8 km/hr.
Then,
The speed downstream = (40 + 8) km/hr
= 48 km/hr
The distance travelled in half an hour = 48 x 1/2
= 24 km
Hence the distance travelled by the boat is 24 km.

৭,২৮৯.

A train with a length of 350 m long, takes 40 seconds to cross a 450m long bridge. How much time will the train take to cross a 550 m long platform?

ক
35 sec
খ
45 sec
গ
50 sec
ঘ
55 sec
ঙ
None

সঠিক উত্তর: খ

45 sec

উত্তর

সঠিক উত্তর: খ

45 sec

খ

ব্যাখ্যা

Question: A train with a length of 350 m long, takes 40 seconds to cross a 450m long bridge. How much time will the train take to cross a 550 m long platform?

Solution:
Length of the train = 350 m
Length of the bridge = 450 m
∴ Total length = (450 + 350) m
= 800m

∴ Speed of the train = 800/40 m/sec
= 20 m/sec

Now total length of the train and new bridge = (550 + 350) m
= 900 m

∴ Time taken = (900/20) sec
= 45 sec

৭,২৯০.

If 3^x^{+ 3} + 7 = 250, then x is equal to?

ক
ক) 5
খ
খ) 3
গ
গ) 2
ঘ
ঘ) 1

সঠিক উত্তর: গ

গ) 2

উত্তর

সঠিক উত্তর: গ

গ) 2

গ

ব্যাখ্যা

প্রশ্ন: If 3^x^{+ 3} + 7 = 250, then x is equal to?

সমাধান:
3^{x + 3} + 7= 250
⇒ 3^{x + 3} = 250 - 7
⇒ 3^{x + 3} = 243
⇒ 3^{x + 3} = 3⁵
∴ x + 3 = 5
∴ x = 2

৭,২৯১.

Six bells commence tolling together and toll at intervals of 3, 5, 6, 9, 10, and 15 seconds respectively. In 45 minutes, how many times do they toll together?

ক
29
খ
30
গ
31
ঘ
34

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: Six bells commence tolling together and toll at intervals of 3, 5, 6, 9, 10, and 15 seconds respectively. In 45 minutes, how many times do they toll together?

Solution:
3 = 3¹
5 = 5¹
6 = 2 × 3
9 = 3²10 = 2 × 5
15 = 3 × 5

∴ ল.সা.গু. = 2¹ × 3² × 5¹ = 2 × 9 × 5 = 90।

সুতরাং, ঘণ্টাগুলো প্রতি 90 সেকেন্ড পর পর একসাথে বাজবে।

এখন, 45 মিনিট = 45 × 60 = 2700 সেকেন্ড।

মোট 2700 সেকেন্ডে ঘণ্টাগুলো যতবার একসাথে বাজবে তার সংখ্যা হলো = 2700/90 = 30 বার।

যেহেতু ঘণ্টাগুলো প্রথমে একবার একসাথে বাজা শুরু করেছিল, তাই মোট সংখ্যাটি হবে 30 এর সাথে সেই প্রথমবারটি যোগ করে।
∴ মোট সংখ্যা = 30 + 1 = 31 বার।

সুতরাং, 45 মিনিটে ঘণ্টাগুলো মোট 31 বার একসাথে বাজবে।

৭,২৯২.

If a = 2 then the value of a³ + 27a² + 243a + 631 is -

ক
1232
খ
1233
গ
1133
ঘ
1234

সঠিক উত্তর: খ

1233

উত্তর

সঠিক উত্তর: খ

1233

খ

ব্যাখ্যা

Question: If a = 2 then the value of a³ + 27a² + 243a + 631 is -

Solution:
Given,
a = 2

Now,
a³ + 27a² + 243a + 631
= a³ + 3 . a² . 9 + 3 . a . 9² + 9³ - 98
= (a + 9)³ - 98
= (2 + 9)³ - 98
= 1331 - 98
= 1233

৭,২৯৩.

In a cricket championship, there are 21 matches. If each team plays one match with every other team, the number of teams is-

ক
6
খ
10
গ
9
ঘ
8
ঙ
7

সঠিক উত্তর: ঙ

উত্তর

সঠিক উত্তর: ঙ

ঙ

ব্যাখ্যা

Question: In a cricket championship, there are 21 matches. If each team plays one match with every other team, the number of teams is-

Solution:
Let n be the number of teams.
ⁿC₂ = 21
⇒ n!/{2! × (n - 2)!} = 21
⇒ {n(n - 1)(n - 2)!}/{2! × (n - 2)!} = 21
⇒ n(n - 1)/2 = 21
⇒ n(n - 1) = 42
⇒ n² - n - 42 = 0
⇒ n - 7n + 6n - 42 = 0
⇒ n(n - 7) + 6(n - 7) = 0
⇒ (n - 7)(n + 6)
∴ n = 7 [Negative value is not acceptable]

৭,২৯৪.

The total age of A and B is 12 years more than the total age of B and C. C is how many years younger than A?

ক
12
খ
13
গ
14
ঘ
15

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: The total age of A and B is 12 years more than the total age of B and C. C is how many years younger than A?

Solution:
Here,
(A + B) - (B + C) = 12
∴ A - C = 12.

∴ C is younger than A by 12 years.

৭,২৯৫.

What is the maximum value of 3sec²A - 4tan²A?

ক
0
খ
6
গ
2
ঘ
3

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: What is the maximum value of 3sec²A - 4tan²A?

Solution:
3sec²A - 4tan²A
= 3sec²A - 3tan²A - tan²A
= 3(sec²A - tan²A) - tan²A
= 3 - tan²A
= 3 - 0 [the maximum value of the equation depends on the minimum value of tan²A]
= 3

৭,২৯৬.

The base of a rectangle is three times as long as the height. If the perimeter is 64, what is the area of the rectangle?

ক
ক) 24
খ
খ) 64
গ
গ) 96
ঘ
ঘ) 192

সঠিক উত্তর: ঘ

ঘ) 192

উত্তর

সঠিক উত্তর: ঘ

ঘ) 192

ঘ

ব্যাখ্যা

Question: The base of a rectangle is three times as long as the height. If the perimeter is 64, what is the area of the rectangle?

Solution:
: মনেকরি,
আয়তক্ষেত্রের উচ্চতা = x একক
আয়তক্ষেত্রের ভূমি = 3x একক
আয়তক্ষেত্রের পরিসীমা = 2(x + 3x) একক

প্রশ্নমতে,
2(x + 3x) = 64
2 × 4x = 64
8x = 64
x = 8

আয়তক্ষেত্রের উচ্চতা = ৪ একক
এবং আয়তক্ষেত্রের ভূমি = 3 × 8 = 24 একক
আয়তক্ষেত্রের ক্ষেত্রফল = ভূমি × উচ্চতা
= 24 × 8
= 192 বর্গ একক

৭,২৯৭.

In an arithmetic progression, the first term a = 5 and the common difference d = 3. What is the 10th term?

ক
38
খ
37
গ
35
ঘ
32

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: In an arithmetic progression, the first term a = 5 and the common difference d = 3. What is the 10th term?

Solution:
Here,
a = 5
d = 3

10th trem = a + (10 - 1)d
= 5 + 9 × 3
= 5 + 27
= 32

৭,২৯৮.

If A : B = 4 : 5, B: C = 7 : 9 and C : D = 3 : 4, and if A’s share is Tk. 1680, the share of D is -

ক
ক) Tk. 2100
খ
খ) Tk. 2700
গ
গ) Tk. 2900
ঘ
ঘ) Tk. 3600

সঠিক উত্তর: ঘ

ঘ) Tk. 3600

উত্তর

সঠিক উত্তর: ঘ

ঘ) Tk. 3600

ঘ

ব্যাখ্যা

A : B = 4 : 5;

B : C = 7 : 9
= (7 × 5/7) : (9 × 5/7)
= 5 : 45/7

C : D = 3 : 4
= (3 × 15/7) : (4 × 15/7)
= 45/7 : 60/7.

A : B : C : D
= 4 : 5 : 45/7 : 60/7
= 28 : 35 : 45 : 60.

Let the shares of A, B, C, D be 28x, 35x, 45x, 60x respectively.

Then,
28x = 1680
⇒ x = 1680/28
= 60

∴ D's Share = 60x = Tk. (60 × 60)
= Tk. 3600.

৭,২৯৯.

Jafar paid Tk. 100 tax. If X was his income then which of the following statements is true?

ক
ক) 6,000 < X < 8,000
খ
খ) 4,000 < X < 6,000
গ
গ) 0 < X < 4,000
ঘ
ঘ) All of the above

সঠিক উত্তর: ক

ক) 6,000 < X < 8,000

উত্তর

সঠিক উত্তর: ক

ক) 6,000 < X < 8,000

ক

ব্যাখ্যা

Question: Jafar paid Tk. 100 tax. If X was his income then which of the following statements is true?

Solution:
ধরি জাফরের ইনকাম = X টাকা

৬০০০ টাকায় টেক্স দিতে হয় = ৪০ + (৪০০০ টাকার বেশি ইনকামের ২%)
= ৪০ + (৬০০০ -৪০০০) এর ২%
= ৮০ টাকা

৮০০০ টাকায় টেক্স দিতে হয় = ৮০ + (৬০০০ এর বেশি ইনকামের ৩%)
= ৮০ + (৮০০০ - ৬০০০) এর ৩%
= ৮০ + ৬০
= ১৪০ টাকা

অর্থাৎ তার ইনকাম ৬০০০ থেকে ৮০০০ এর মধ্যে।

∴ 6000 < X < 8000

৭,৩০০.

A student was asked to divide a number by 8/17. Instead, he actually multiplied it by 8/17 and hence got 225 less than the expected answer. The given number was?

ক
ক) 64
খ
খ) 136
গ
গ) 139
ঘ
ঘ) 224

সঠিক উত্তর: খ

খ) 136

উত্তর

সঠিক উত্তর: খ

খ) 136

খ

ব্যাখ্যা

Let
The number be x.

According to the question,
or, (17x/8) - (8x/17) = 225
or, (289x - 64x)/136 = 225
or, 225x/136 = 225
or, x/136 = 1
or, x = 136