Bank Math

PrepBank · পাতা ৩০ / ১৬১ · ২,৯০১–৩,০০০ / ১৬,১২৪

২,৯০১.

The cost of a bicycle was Tk. 6000. The cost was reduced by 15% after it was increased by 20%. How much profit or loss percentage does it remain?

ক
5% loss
খ
2% profit
গ
2.5% loss
ঘ
3% profit

ব্যাখ্যা

Question: The cost of a bicycle was Tk. 6000. The cost was reduced by 15% after it was increased by 20%. How much profit or loss percentage does it remain?

Solution:
first the price was increased by 20% and then reduced by 15%
after increase the price is = (6000 + 20% of 6000)
= 7200

after reducing the price the price is = (7200 - 15% of 7200)
= 6120

profit = 6120 - 6000
= 120

percentage = (120/6000)100%
= 2%

২,৯০২.

If x^a = y^b, then

ক
ক) logx/logy = a/b
খ
খ) logx/logy = b/a
গ
গ) logx/logy = ab
ঘ
ঘ) None of these

ব্যাখ্যা

Question: If x^a = y^b, then

Solution:
x^a = y^b
Take log on both the sides, we get
logx^a = logy^b
⇒ a(logx) = b(logy)
∴ logx/logy = b/a

২,৯০৩.

Find the HCF of: x³ - 1, x² - 1, x² - 2x + 1

ক
x - 3
খ
0
গ
x
ঘ
x - 1

ব্যাখ্যা

Question: Find the HCF of: x³ - 1, x² - 1, x² - 2x + 1

Solution:
x³ - 1
= (x - 1)(x² + 2x + 1)
= (x - 1)(x + 1)²
= (x - 1)(x + 1)(x + 1)

x² - 1
= (x + 1)(x - 1)

x² - 2x + 1
= (x - 1)²
= (x - 1)(x - 1)

∴ HCF = (x - 1)

২,৯০৪.

A father and a son started for a shop at the same time. In one minute, the son moved 20 steps forward and in the same time, the father moved 30 steps forward. In one step the son covered 1 ft. and the father covered 1.5 ft. If the son reached the store 10 minutes after his father, what was the distance of the store in ft?

ক
ক) 280
খ
খ) 240
গ
গ) 360
ঘ
ঘ) 320

ব্যাখ্যা

ধরি,
Distance of store = x feet
Now, Distance covered by son in one minute = 1 × 20
= 20 ft. and
Distance covered by father = 30 × 1.5
= 45 ft.
∴ Time taken by son = x/20 minutes = Time taken by father = x/45 minutes.
∴ x/20 = x/45 + 10
⇒ (x/20 - x/45) = 10
⇒ (9x - 4x)/180 = 10
⇒ 5x = 1800
⇒ x = 1800/5
⇒ x = 360 ft.

২,৯০৫.

If 9 workers can assemble a truck in 12 hours, how long would it take 6 workers to assemble the same truck?

ক
3 hours
খ
6 hours
গ
9 hours
ঘ
18 hours
ঙ
24 hours

ব্যাখ্যা

Question: If 9 workers can assemble a truck in 12 hours, how long would it take 6 workers to assemble the same truck?

Solution:

Here, M₁ = 9, M₂ = 6, W₁ = W₂ = 1, T₁ = 12, T₂ = ?

∴ (M₁ × T₁)/(M₂ × T₂) = W₁/W₂
⇒ (9 × 12)/ (6 × T₂) = 1
⇒ T₂ = (9 × 12)/6
⇒ T₂ = 18

২,৯০৬.

The sum of five consecutive multiples of 6 is 150. What is the second largest number?

ক
48
খ
36
গ
42
ঘ
30

ব্যাখ্যা

Question: The sum of five consecutive multiples of 6 is 150. What is the second largest number?

Solution:
ধরি, ৬ এর পাঁচটি ক্রমিক গুণিতক হলো যথাক্রমে (x - 12), (x - 6), x, (x + 6) এবং (x + 12)

প্রশ্নমতে,
(x - 12) + (x - 6) + x + (x + 6) + (x + 12) = 150
⇒ 5x = 150
⇒ x = 150/5
⇒ x = 30

সুতরাং, সংখ্যাগুলো হলো 18, 24, 30, 36, 42।
এদের মধ্যে দ্বিতীয় বৃহত্তম সংখ্যাটি হলো 36।

২,৯০৭.

If the sum of three consecutive odd integers is 141, what is the largest number?

ক
41
খ
43
গ
49
ঘ
None

ব্যাখ্যা

Question: If the sum of three consecutive odd integers is 141, what is the largest number?

Solution:
Let the three consecutive odd integers be:
x, x + 2, x + 4

Accordingly:
x + (x + 2) + (x + 4) = 141
⇒ 3x + 6 = 141
⇒ 3x = 141 - 6
⇒ 3x = 135
⇒ x = 135/3
⇒ x = 45

So the numbers are:
45, 47, 49(Largest)

∴ The largest number is 49.

২,৯০৮.

A number when divided by the sum of 555 and 445 gives two times their difference as quotient and 30 as remainder. The number is:

ক
ক) 22030
খ
খ) 220030
গ
গ) 23030
ঘ
ঘ) 24030

ব্যাখ্যা

(550 + 445) × 2 × 110 + 30 = 220030

২,৯০৯.

How many pairs of natural numbers is there the difference of whose squares are 45?

ক
6
খ
3
গ
1
ঘ
None of these

ব্যাখ্যা

Question: How many pairs of natural numbers is there the difference of whose squares are 45?

Solution:
ধরি, 45 কে x ও y এর বর্গের বিয়োগফল আকারে প্রকাশ করা যায়।
45 = x² - y² = (x - y) (x + y)

45
= 1 × 45
= 5 × 9
= 3 × 15

♦ x + y = 45, x - y = 1 হলে,
x + y + x - y = 45 + 1
⇒ 2x = 46
⇒ x = 23∴ y = 45 - 23 = 22

♦ x + y = 9, x - y = 5 হলে,
x + y + x - y = 5 + 9
⇒ 2x = 14
⇒ x = 7∴ y = 9 - 7 = 2

♦ x + y = 15, x - y = 3 হলে,
x + y + x - y = 15 + 3
⇒ 2x = 18
⇒ x = 9

∴ y = 15 - 9 = 6

অতএব, 45 কে দুটি স্বাভাবিক সংখ্যার বর্গের বিয়োগফল আকারে প্রকাশ করা যায় এমন 3 টি ক্রমজোড় হল (x, y) = (23, 22), (7, 2), (9, 6)

২,৯১০.

Today is Sunday. After 87 days, what day of the week will it be?

ক
Monday
খ
Tuesday
গ
Wednesday
ঘ
Thursday

ব্যাখ্যা

Question: Today is Sunday. After 87 days, what day of the week will it be?

Solution:
Each day of the week is repeated after 7 days.

So, after (7 × 12) = 84 days, it will be Sunday.

After 85 days, it will be Monday.
After 86 days, it will be Tuesday.
∴ After 87 days, it will be Wednesday.

২,৯১১.

A lends Tk. 2500 to B and a certain sum to C at the same time at 7% p.a. simple interest. If after 4 years, A altogether receives Tk. 1120 as interest from B and C, then the sum lent to C is -

ক
Tk. 1000
খ
Tk. 1200
গ
Tk. 1500
ঘ
Tk. 1600

ব্যাখ্যা

Question: A lends Tk. 2500 to B and a certain sum to C at the same time at 7% p.a. simple interest. If after 4 years, A altogether receives Tk. 1120 as interest from B and C, then the sum lent to C is -

Solution:
ধরি,
C এর মুলধন x টাকা

প্রশ্নমতে,
(2500 × 7 × 4)/100) + (x × 7 × 4/100) = 1120
⇒ 7x/25 = (1120 - 700)
⇒ x = (420 × 25)/7
∴ x = 1500

C এর মুলধন 1500 টাকা।

২,৯১২.

If f(x) = 3x² + bx + 4 and f(- 1) = 0; Then b = ?

ক
ক) 3
খ
খ) 4
গ
গ) 6
ঘ
ঘ) 7

ব্যাখ্যা

Question: If f(x) = 3x² + bx + 4 and f(- 1) = 0; Then b = ?

Solution:
f(x) = 3x² + bx + 4
∴ f(- 1) = 3(- 1)² + b(- 1) + 4
= 3 - b +4
= 7 - b

A.T.Q,
7 - b = 0
∴ b = 7

২,৯১৩.

Today is Sabit's birthday. One year, from today he will be twice as old as he was 12 years ago. How old is Sabit today?

ক
ক) 20 years
খ
খ) 25 years
গ
গ) 22 years
ঘ
ঘ) 27 year

ব্যাখ্যা

Question: Today is Sabit's birthday. One year, from today he will be twice as old as he was 12 years ago. How old is Sabit today?

Solution:
আজকে সাবিতের বয়স = x বছর
১ বছর পর সাবিতের বয়স হবে = (x + 1) বছর

শর্তমতে,
x + 1 = 2(x - 12)
⇒ x + 1 = 2x - 24
⇒ x = 25

২,৯১৪.

A man's present age is two-fifth of the age of his mother. After 9 years, he will be one-half of the age of his mother. How old is the mother at present?

ক
ক) 40 years
খ
খ) 43 years
গ
গ) 45 years
ঘ
ঘ) 48 years

ব্যাখ্যা

Question: A man's present age is two-fifth of the age of his mother. After 9 years, he will be one-half of the age of his mother. How old is the mother at present?

Solution:
Let mother's age be x years
Then, man's age = 2x/5 years

ATQ,
(2x/5) + 9 = (x + 9)/2
⇒ (x/2) - (2x/5) = 9 - (9/2)
⇒ (5x - 4x)/10 = (18 - 9)/2
⇒ x/10 = 9/2
⇒ x = (9 × 10)/2
⇒ x = 9 × 5
∴ x = 45

∴ Mother's present age is 45 years

২,৯১৫.

A man invested a sum of money at compound interest. It amounted to Tk 2420 in 2 years and to Tk 2662 in 3 years. Find the sum.

ক
1800
খ
1900
গ
2000
ঘ
2100
ঙ
None

ব্যাখ্যা

Question: A man invested a sum of money at compound interest. It amounted to Tk 2420 in 2 years and to Tk 2662 in 3 years. Find the sum.

Solution:

২,৯১৬.

A trader marks his goods at 40% above the cost price but allows a discount of 20% on the marked price. His profit percentage is-

ক
8%
খ
10%
গ
12%
ঘ
15%

ব্যাখ্যা

Question: A trader marks his goods at 40% above the cost price but allows a discount of 20% on the marked price. His profit percentage is-

Solution:
Let
the Cost Price of the goods Tk.100

∴ Marked price of the goods = 100 + 40% of 100
= 100 + (40 × 100)/100
= 100 + 40
= 140

∴ Selling Price of the goods = 80% of 140
= (140 × 80)/100
= 112

∴ profit = 112 - 100 = 12 Tk.

∴ profit percentage is = (12/100) × 100%
= 12%

২,৯১৭.

A man and a boy received 6000 Tk. as wages for 5 days for the work they did together. The man's efficiency in the work was double that of the boy. What are the daily wages of the boy?

ক
400 Tk.
খ
300 Tk.
গ
600 Tk.
ঘ
800 Tk.

ব্যাখ্যা

Question: A man and a boy received 6000 Tk. as wages for 5 days for the work they did together. The man's efficiency in the work was double that of the boy. What are the daily wages of the boy?

Solution:
Ratio of 1 day's work of man and boy = 2 : 1
Total wages of the boy = (6000 × 1/3)
= 2000 Tk.

∴ Daily wages of the boy= 2000/5 = 400Tk.

২,৯১৮.

The compound interest on Tk. 2600 for 18 months at 10% per annum is-

ক
Tk. 403
খ
Tk. 395
গ
Tk. 465
ঘ
Tk. 420

ব্যাখ্যা

Question: The compound interest on Tk. 2600 for 18 months at 10% per annum is-

Solution:
Interest after 1 year or 12 month = 2600 × 1 × (10/100)
= 260
New principal = 2600 + 260
= 2860

6 months = 1/2 year

Now,
I = Pnr
= 2860 × (1/2) × (10/100)
= 143

∴ Total interest = 260 + 143 = Tk. 403

২,৯১৯.

The angle of elevation of a ladder leaning against a wall is 60° and the foot of the ladder is 4.6 m away from the wall. The length of the ladder is:

ক
9.2m
খ
8.8m
গ
7.6m
ঘ
6.2m

ব্যাখ্যা

Question: The angle of elevation of a ladder leaning against a wall is 60° and the foot of the ladder is 4.6 m away from the wall. The length of the ladder is:

Solution:

Let AB be the wall and AC be the ladder.

Then, ∠ACB = 60° and BC = 4.6 m

BC/AC = cos60° [ cosθ = ভূমি/অতিভুজ ]
⇒ BC/AC = 1/2
⇒ AC = 2BC
⇒ AC = 2 × 4.6
∴ AC = 9.2m

∴ The length of the ladder is 9.2m

২,৯২০.

If the radius of a sphere is 3 cm, what is its volume?

ক
18π cm³
খ
52π cm³
গ
27π cm³
ঘ
36π cm³

ব্যাখ্যা

Question: If the radius of a sphere is 3 cm, what is its volume?

Solution:
Given that,
Radius of sphere = 3 cm

We know,
Volume of a sphere = (4/3) × πr³= (4/3) × π(3)³= (4/3) × π × 27
= 36π cm³

২,৯২১.

The speed of the boat is still water at 12 kmph. It can travel downstream through 45 kms in 3 hrs. In what time would it cover the same distance upstream?

ক
ক) 4 hours
খ
খ) 8 hours
গ
গ) 6 hours
ঘ
ঘ) 5 hours

ব্যাখ্যা

Speed of the boat in still water = 12 km/hr.
Speed downstream = 45/3 = 15 km/hr.
Speed of the stream = (15 -12) km/hr.
= 3 km/hr.
Speed upstream = (12 - 3)
= 9 km/hr.
Time is taken to cover 45 km upstream = 45/9 hr.
= 5 hrs.

২,৯২২.

The perimeter of one face of a cube is 16 cm. Its volume must be-

ক
ক) 25 cm³
খ
খ) 64 cm³
গ
গ) 75 cm³
ঘ
ঘ) 125 cm³

ব্যাখ্যা

Question: The perimeter of one face of a cube is 16 cm. Its volume must be-

Solution:
the perimeter of one face is 16 cm

let, the length of one side is a cm
perimeter = 4a cm

⇒ 4a = 16
⇒ a = 16/4
= 4 cm

volume = a³
= 4³
= 64 cm³

২,৯২৩.

What is the probability of getting 53 Mondays in a leap year?

ক
3/8
খ
5/9
গ
2/5
ঘ
2/7
ঙ
3/13

ব্যাখ্যা

Question: What is the probability of getting 53 Mondays in a leap year?

Solution:
1 year = 365 days . A leap year has 366 days
A year has 52 weeks. Hence there will be 52 Sundays for sure.
52 weeks = 52 × 7 = 364days
366 - 364 = 2 days

In a leap year there will be 52 Sundays and 2 days will be left.
These 2 days can be:
1. Sunday, Monday
2. Monday, Tuesday
3. Tuesday, Wednesday
4. Wednesday, Thursday
5. Thursday, Friday
6. Friday, Saturday
7. Saturday, Sunday
Of these total 7 outcomes, the favourable outcomes are 2.

Hence the probability of getting 53 days = 2/7

২,৯২৪.

√{(16)(20) + (8)(32)} = ?

ক
ক) 4√20
খ
খ) 24
গ
গ) 25
ঘ
ঘ) 32

ব্যাখ্যা

√{(16)(20) + (8)(32)}
= √{(320) + (256)}
= √(576)
= 24

২,৯২৫.

Two friends Roman and Shimul invest in a grocery shop. Shimul invests Tk. 25000/- while Roman invests Tk. 35000. Third friend Salim, joins them with the condition that all of them must get equal share of profit. To do so, he gives Tk. 400000 to Roman and Shimul to share between themselves. Find the ratio in which Roman and Shimul should share the money given by Salim?

ক
ক) 5:7
খ
খ) 7:5
গ
গ) 7:13
ঘ
ঘ) 13:7

ব্যাখ্যা

The total value of investment of Roman and Shimul after 12 months is =
Tk. 35000 x 12 months : Tk. 25000 x 12 months = 420000 : 300000

Profits need to be the same so investment share must be the same too.

Now 400000 given by Salim needs to be shared by Roman and Shimul so that their investment value becomes the same.
∴ If Tk. X must be given to Roman,
Then,
420000 + X = 300000 + (400000-X)
∴ X = Tk. 140000 = Roman should get this much
Required ratio = Share of Roman: Share of Shimul
= 140000 : (400000 - 140000)
= 7 : 13.

২,৯২৬.

Two pipes A and B can fill a tank in 15 and 30 hours respectively. If both the pipes are used together, then how long will it take to fill the tank?

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

ব্যাখ্যা

Question: Two pipes A and B can fill a tank in 15 and 30 hours respectively. If both the pipes are used together, then how long will it take to fill the tank?

Solution:
Part filled by A in 1 hour = 1/15

Part filled by B in 1 hour = 1/30

Part filled by (A + B) in 1 hour
= (1/15) + (1/30)
= (2 + 1)/30
= 3/30
= 1/10

∴ Both pipes can fill the tank in 10 hours

২,৯২৭.

There are 6 true-false questions in an examination. In how many ways can these questions be answered?

ক
64 ways
খ
24 ways
গ
56 ways
ঘ
42 ways

ব্যাখ্যা

Question: There are 6 true-false questions in an examination. In how many ways can these questions be answered?

Solution:
Total number of questions = 6
Each question has 2 possible answers (True or False).

∴ Total number of ways = 2⁶ = 64 ways

২,৯২৮.

A sport club has 50 members. Of these, 35 play golf, 30 play soccer and 18 play both golf and soccer. How members do play neither golf nor soccer?

ক
ক) 0
খ
খ) 5
গ
গ) 3
ঘ
ঘ) 17

ব্যাখ্যা

Players who play at least one sport = 35 + 30 - 18 = 47
Players who play neither golf nor soccer = 50 - 47 = 3

২,৯২৯.

Monir and Mohin bought the same item. Monir paid Tk 15,000. Mohin got discount and paid Tk. 3,000 less than Monir. How much discount did Mohin get?

ক
10%
খ
15%
গ
18%
ঘ
20%

ব্যাখ্যা

Question: Monir and Mohin bought the same item. Monir paid Tk 15,000. Mohin got discount and paid Tk. 3,000 less than Monir. How much discount did Mohin get?

Solution:
In Tk. 15000 Mohin get discount Tk. 3000
In Tk. 1 Mohin get discount Tk. (3000/15000)
In Tk. 100 Mohin get discount Tk. (3000 × 100)/15000
= Tk. 20

∴ Mohin get 20% discount.

২,৯৩০.

If 10⁵⁰ - 74 is written as an integer in base 10 notation, what is the sum of the digits in that integer?

ক
433
খ
467
গ
424
ঘ
440
ঙ
449

ব্যাখ্যা

Question: If 10⁵⁰ - 74 is written as an integer in base 10 notation, what is the sum of the digits in that integer?

Solution:
10⁵⁰ has 51 digits (1 followed by 50 zeros).
10⁵⁰ - 74 has 50 digits: the last 2 digits are 2 and 6 [100 - 74 = 26] and the first 48 digits are 9's
So the sum of the digits is (9 × 48) + 2 + 6 = 440

২,৯৩১.

In how many different ways can the letters of the word "DESIGN" be arranged so that the vowels are at the two ends?

ক
24
খ
32
গ
48
ঘ
60

ব্যাখ্যা

Question: In how many different ways can the letters of the word "DESIGN" be arranged so that the vowels are at the two ends?

Solution:
The given word "DESIGN" contains 4 consonants and 2 vowel.
At the two ends the two vowels can be arranged in 2! = 2 ways.
Remaining 4 letters can be arranged in = 4!
= 24

∴ Total number of ways = (24 × 2) = 48

২,৯৩২.

Q is as much younger than R as he is older than T. If the sum of the ages of R and T is 50 years, what is definitely the difference between R and Q's age?

ক
1 year
খ
2 years
গ
25 years
ঘ
Data inadequate
ঙ
None of these

ব্যাখ্যা

Question: Q is as much younger than R as he is older than T. If the sum of the ages of R and T is 50 years, what is definitely the difference between R and Q's age?

Solution:

The difference of age between R and Q = The difference of age between Q and T.

Sum of age of R and T is 50
∴ (R + T) = 50.

R - Q = ?

ATQ,

R - Q = Q - T

∴ R + T = 2Q

Now given that, (R + T) = 50

50 = 2Q
∴ Q = 25.

Question is (R - Q) = ?

Here we know the value(age) of Q (25), but we don't know the age of R.

∴ (R - Q) cannot be determined.

২,৯৩৩.

The ratio of the present ages of Pranto and Qureshi is 4 : 5. Five years ago, the ratio of their ages was 7 : 9. Find their present ages? (In years)

ক
40, 50
খ
18, 25
গ
40, 60
ঘ
20, 25
ঙ
None of these

ব্যাখ্যা

Question: The ratio of the present ages of Pranto and Qureshi is 4 : 5. Five years ago, the ratio of their ages was 7 : 9. Find their present ages? (In years)

Solution:

Their present ages be 4x and 5x.

5 years ago, the ratio of their ages was 7 : 9, then
(4x - 5) : (5x - 5) = 7 : 9

⇒ 36x - 45 = 35x - 35
⇒ x = 45 - 35
∴ x = 10.

Their present ages are: 40, 50.

২,৯৩৪.

What will come at the place of the question mark? 1, 9, 25, 49, 81,?

ক
100
খ
121
গ
144
ঘ
150

ব্যাখ্যা

Question: What will come at the place of the question mark?
1, 9, 25, 49, 81,?

Solution:
1 = 1²
9 = 3²
25 = 5²
49 = 7²
81 = 9²
121 = 11²

২,৯৩৫.

On dividing a number by 56, we get 29 as remainder. On dividing the same number by 8, what will be the remainder?

ক
3
খ
4
গ
5
ঘ
6

ব্যাখ্যা

Question: On dividing a number by 56, we get 29 as remainder. On dividing the same number by 8, what will be the remainder?

Solution:
If a number divided by 56 leaves a remainder of 29, we can express it as:
Number = 56 × Quotient + 29

Now, to find the remainder when this number is divided by 8, let's try to simplify the expression.
Number = 56 × Quotient + 29

We know that 56 is divisible by 8, so let's express 56 in terms of 8:
56 = 8 × 7

Therefore, we can rewrite the expression for the number:
Number = 8 × 7 × Quotient + 29
Number = 8 × (7 × Quotient) + 29

Now, when we divide this expression by 8, we'll focus on the remainder:

8 × (7 × Quotient) leaves no remainder when divided by 8 because it's a multiple of 8.

So, the remainder when 29 is divided by 8 is 5.

২,৯৩৬.

x² - (a + b)x + ab = 0; x = ?

ক
a, b
খ
a
গ
b
ঘ
ab

ব্যাখ্যা

Question: x² - (a + b)x + ab = 0; x = ?

Solution:
Given that
x² - (a + b)x + ab = 0
x² - ax - bx + ab = 0
x(x - a) - b(x - a) = 0
(x - a)(x - b) = 0

হয়
x - a = 0
x = a

অথবা
x - b = 0
x = b

নির্ণেয় সমধান x = a, b

২,৯৩৭.

In how many ways can 12 identical balls be distributed among 4 distinct boxes such that each box contains at least one ball?

ক
165
খ
220
গ
286
ঘ
330

ব্যাখ্যা

Question: In how many ways can 12 identical balls be distributed among 4 distinct boxes such that each box contains at least one ball?

Solution:

Since each box must get at least one ball, give 1 ball to each box first.

Balls used = 4
Remaining balls = 12 - 4 = 8

Now, distributing 8 identical balls among 4 boxes with no restrictions.
Formula = ^{n + k - 1}C_{k - 1} ;where n = 8, k = 4
= ¹¹C₃
= 11!/(3! × 8!)
= (11 × 10 × 9)/(3 × 2 × 1)
= 990/6
= 165

২,৯৩৮.

The average of 10 integers is 16. If the sum of 6 of them is 100. What is the average of other 4?

ক
ক) 12
খ
খ) 13
গ
গ) 15
ঘ
ঘ) 17

ব্যাখ্যা

The average of 10 integers is 16
Therefore, the sum of 10 integers is 16 × 10 = 160
The sum of 6 integers of them(10 integers) is 100
The sum of other 4 integers of them(10 integers) is 160 - 100 = 60
The average of other 4 integers of them(10 integers) is 60/4 = 15
-------------------------------------------------------------------------
১০ টি পূর্ণ সংখ্যার গড় ১৬। ১০ টি পূর্ণ সংখ্যার মধ্যে ৬ টি পূর্ণ সংখ্যার যোগফল ১০০। অবশিষ্ট ৪ টি পূর্ণ সংখ্যার যোগফল কত?

১০ টি পূর্ণ সংখ্যার গড় ১৬ হলে যোগফল ১০ × ১৬ = ১৬০
১০ টি পূর্ণ সংখ্যার মধ্যে ৬ টি পূর্ণ সংখ্যার যোগফল ১০০।
সুতরাং অবশিষ্ট ৪ টি পূর্ণ সংখ্যার যোগফল ১৬০ - ১০০ = ৬০
অবশিষ্ট ৪ টি পূর্ণ সংখ্যার গড় ৬০/৪ = ১৫

২,৯৩৯.

What reminder of any perfect square is divided by 3?

ক
0
খ
1
গ
0 or 1
ঘ
0 and 1
ঙ
None

ব্যাখ্যা

When a perfect square is divided by 3, the remainder is always 0 or 1.
It is zero when the base number (a in case of a^2) is divisible by 3.
For example - 3² is divided by 3. So, we get 0 as a remainder.
For all other numbers, the square when divided by 3, the remainder is always 1.
When the number is divisible by 3, the remainder is obviously going to be zero.
Let’s say the number is not divisible by 3.
Such a number is always in the form of 3a + 1 or 3a - 1,
for example 4 = 3 × 2 + 1 and 5 = 3× 2 - 1.
Now, when you square 3a + 1,
you get 9a² + 6a + 1.
Now 9a² + 6a = 3×(3a²+2),
therefore that portion is divisible by 3 and what remains (remainder) is 1.
Similarly, when you square 3a - 1, you get 9a² - 6a + 1.
Now 9a² - 6a = 3×(3a²-2),
therefore that portion is divisible by 3 and what remains (remainder) is 1.
Thus, in all situations, we find that the remainder is 1 when the starting number is not divisible by 3.

২,৯৪০.

Find the first term of an arithmetic progression(A.P.) whose 8^th and 12^th terms are respectively 39 and 59.

ক
5
খ
4
গ
6
ঘ
3

ব্যাখ্যা

Question: Find the first term of an arithmetic progression(A.P.) whose 8^th and 12^th terms are respectively 39 and 59.

Solution:
We know,
Arithmetic progression, n^th term is = a + (n - 1)d,
where a is the first term and d is the common difference.
Given that,
8^th term = 39 and 12^th term = 59

Now,
8^th term = a + 7d = 39 ........... (i)
12^th term = a + 11d = 59 ........... (ii)
(i) - (ii) ⇒ a + 7d - a - 11d = 39 - 59
⇒ 4d = 20
∴ d = 5
Hence, a + 7 × 5 = 39
Thus, a = 39 - 35 = 4

so the first term of the A.P. is 4.

২,৯৪১.

If the length of the shorter diagonal is four, what is the length of the longer diagonal of this kite?

ক
3√5
খ
4√5
গ
5√3
ঘ
5√4

ব্যাখ্যা

Question: If the length of the shorter diagonal is four, what is the length of the longer diagonal of this kite?

Solution:
We can find the longer diagonal by adding together the altitude of the top triangle and the altitude of the bottom triangle. To find these, use Pythagorean Theorem. We can use Pythagorean Theorem because one of the properties of a kite is that the two diagonals are perpendicular.

The top triangle has two sides of length 3 [labeled in the picture], and a base of 4 [provided in the written directions]. To figure out the altitude, split this triangle into 2 right triangles. The two legs are x [the altitude] and 2 [half of the base 4], and the hypotenuse is 3:
x² + 2² = 3²
⇒ x² + 4 = 9
⇒ x² = 5
∴ x = √5

We will do something similar for the bottom triangle. Consider one of the right triangles. It will have a hypotenuse of 7, one leg that we don't know, x [the altitude], and one leg 2 [half the shorter diagonal]. Set up the equation using the Pythagorean Theorem:
x² + 2² = 7²
⇒ x² + 4 = 49
⇒ x² = 45
∴ x = √45 = 3√5

∴ The length of the longer diagonal of this kite = 3√5 + √5 = 4√5

২,৯৪২.

Sum of two numbers is thrice their difference. Their ratio is :

ক
ক) 2 : 1
খ
খ) 3 : 1
গ
গ) 4 : 1
ঘ
ঘ) 2 : 3

ব্যাখ্যা

Let the numbers = a, b
According to the question,
a + b = 3 (a - b)
⇒ a/b = 2
⇒ a : b = 2 : 1

২,৯৪৩.

If the roots of px² + qx + 2 = 0 are reciprocal of each other, then

ক
P = 0
খ
p = - 2
গ
p = ± 2
ঘ
p = 2

ব্যাখ্যা

Question: If the roots of px² + qx + 2 = 0 are reciprocal of each other, then

Solution:
Let,
α and β are the roots
here
α = 1/β
∴ αβ = 1
⇒ 2/p = 1
∴ p = 2

২,৯৪৪.

The price of a coat in a certain store in Tk. 500. If the price of the coat is to be reduced by Tk. 150, by what percent is the price to be reduced?

ক
ক) 10%
খ
খ) 15%
গ
গ) 20%
ঘ
ঘ) 30%

ব্যাখ্যা

x% of 500 = 150
x/100 of 500 = 150
5x = 150
x = 30

২,৯৪৫.

The angle of elevation of the sun, when the length of the shadow of a tree 1/√3 times the height of the tree, is:

ক
30°
খ
45°
গ
60°
ঘ
90°

ব্যাখ্যা

Question: The angle of elevation of the sun, when the length of the shadow of a tree 1/√3 times the height of the tree, is:

Solution:

Let,
Height of the tree BC = x
Length of the shadow AB = (1/√3)x
The angle of elevation of the sun ∠A = θ

Now,
tanθ = BC/AB
⇒ tanθ = x/{(1/√3)x}
⇒ tanθ = √3
⇒ tanθ = tan60°
∴ θ = 60°

২,৯৪৬.

কোনো আসল ৩ বছরে মুনাফাসহ ৫৫০০ টাকা হয় । মুনাফা আসলের ৩/৮ অংশ হলে মুনাফার হার কত?

ক
ক) ১২.৪০%
খ
খ) ১২.৫০%
গ
গ) ১২%
ঘ
ঘ) ১০%

ব্যাখ্যা

প্রশ্ন: কোনো আসল ৩ বছরে মুনাফাসহ ৫৫০০ টাকা হয় । মুনাফা আসলের ৩/৮ অংশ হলে মুনাফার হার কত?

সমাধান:
ধরি,
আসল, P টাকা
∴ মুনাফা, I = (৩P)/৮ টাকা
মুনাফার হার, r
সময়, n = ৩ বছর

শর্তমতে,
P + (৩P)/৮ = ৫৫০০
বা, ৮P + ৩P = ৫৫০০ × ৮
বা, ১১P = ৫৫০০ × ৮
বা, P = ৫০০ × ৮
∴ P = ৪০০০

আসল ৪০০০ টাকা

মুনাফা (৩ × ৪০০০)/৮ টাকা
= ১৫০০ টাকা

মুনাফার হার, r = I/(Pn) = ১৫০০/(৪০০০ × ৩)
= (১/৮) × ১০০ %
= ১২.৫০%

২,৯৪৭.

A box contains 2 white balls, 3 black balls and 4 red balls. In how many ways can 3 balls be drawn from the box, if at least one black ball is to be included in the draw?

ক
32
খ
48
গ
64
ঘ
96
ঙ
69

ব্যাখ্যা

We may have (1 black and 2 non-black) or (2 black and 1 non-black) or (3 black).
required number of ways = (³C₁ x ⁶C₂) + (³C₂ x ⁶C₁) + (³C₃)
= 3 x (6 x 5)/(1 x 2) + (3 x 2)/(1 x 2) x 6 + 1
= (45 + 18 + 1)
= 64.

২,৯৪৮.

Two trains travel in opposite directions at 36 km/hr and 45 km/hr and a man sitting in slower train passes the faster train in 8 seconds. The length of the faster train is:

ক
240 m
খ
90 m
গ
120 m
ঘ
180 m

ব্যাখ্যা

Question: Two trains travel in opposite directions at 36 km/hr and 45 km/hr and a man sitting in slower train passes the faster train in 8 seconds. The length of the faster train is:

Solution:
As the trains travel in opposite directions then
Relative speed = (45 + 36)km/hr.
= 81 km/hr.
= (81 × 1000)/3600
= 45/2

The length of the faster train = (45/2) × 8
= 180 m

২,৯৪৯.

A salesman receives daily wage of Tk. 150 and earns a commission of 25% on all sales he makes. How much taka worth of sales does he need to make in order to bring his total daily income of Tk. 1000?

ক
ক) Tk. 3800
খ
খ) Tk. 3600
গ
গ) Tk. 3400
ঘ
ঘ) Tk. 3200

ব্যাখ্যা

Question: A salesman receives daily wage of Tk. 150 and earns a commission of 25% on all sales he makes. How much taka worth of sales does he need to make in order to bring his total daily income of Tk. 1000?

Solution:
দৈনিক মোট আয় করতে হবে = 1000 টাকা
দৈনিক মজুরি = 150 টাকা

∴ অবশিষ্ট = (1000 - 150) টাকা
= 850 টাকা

25% সেলস কমিশন = 850 টাকা
1% সেলস কমিশন = 850/25 টাকা
∴ 100% সেলস কমিশন = (850 × 100)/25 টাকা
= 3400 টাকা

২,৯৫০.

What should come in place of the question mark '?' in the following number series.
10, 30, 68, 130, ?

ক
210
খ
222
গ
218
ঘ
224
ঙ
230

ব্যাখ্যা

Question: What should come in place of the question mark '?' in the following number series.
10, 30, 68, 130, ?

Solution:
Each term is n³ + n
1^st term = 2³ + 2 = 8 + 2 = 10
2^nd term = 3³ + 3 = 27 + 3 = 30
3^rd term = 4³ + 4 = 64 + 4 = 68
4^th term = 5³ + 5 = 125 + 5 = 130
5^th term = 6³ + 6 = 216 + 6 = 222

∴ The missing number is 222.

২,৯৫১.

Solve the following equation.

ক
1
খ
2
গ
3
ঘ
4

ব্যাখ্যা

Question: Solve the following equation.

Solution:

২,৯৫২.

Two equal glasses are respectively two-third and three-fourth full of milk. They are then filled up with water and the contents are mixed in a tumbler. Ratio of milk and water in tumbler is

ক
3 : 19
খ
5 : 17
গ
9 : 13
ঘ
17 : 7

ব্যাখ্যা

Question: Two equal glasses are respectively two-third and three-fourth full of milk. They are then filled up with water and the contents are mixed in a tumbler. Ratio of milk and water in tumbler is

Solution:
In 1st glass milk = 2/3
Water = (1 - 2/3)
= 1/3

In 2nd glass milk = 3/4
Water = (1 - 3/4)
= 1/4

Ratio of milk and water = (2/3 + 3/4) : (1/3 + 1/4)
= {(8 + 9)/12} : {(4 + 3)/12}
= (17/12) : (7/12)
= 17 : 7

২,৯৫৩.

Akash lends a part of Tk. 20000 at 8% simple interest and remaining at (4/3)% simple interest. His total income after a year was Tk. 800. Find the sum lent at 8%.

ক
Tk. 5000
খ
Tk. 8000
গ
Tk.10000
ঘ
Tk. 12000

ব্যাখ্যা

Question: Akash lends a part of Tk. 20000 at 8% simple interest and remaining at (4/3)% simple interest. His total income after a year was Tk. 800. Find the sum lent at 8%.

Solution:
Amount lent at 8% rate of interest = Tk. x
Amount lent at 4/3 % rate of interest = Tk. (20000 - x)

∴ S.I. = (Principal × Rate × Time)/100
{(x × 8 × 1)/100} + [{(20000 - x) × (4/3) × 1}/100] = 800
⇒ (2x/25) + {(20000 - x)/75} = 800
⇒ (6x + 20000 - x)/75 = 800
⇒ 5x + 20,000 = 75 × 800 = 60000
⇒ 5x = 60,000 - 20000 = 4000
⇒ x = 40000/5
∴ x = Tk. 8000

২,৯৫৪.

If 1/Q > 1, which of the following must be true?

ক
1 < Q²
খ
Q² > 2
গ
1 > Q²
ঘ
Q² > 1
ঙ
Q < Q²

ব্যাখ্যা

Question: If 1/Q > 1, which of the following must be true?

Solution:
Given, 1/Q > 1
Since 1/Q > 1 and 1 > 0, we know that 1/Q is positive, hence Q must also be positive.

Now, multiply both sides of the inequality by Q :
Q × (1/Q) > Q × 1
⇒ 1 > Q
⇒ 1 × Q > Q × Q
⇒ Q > Q²
⇒ 1 > Q² [Since 1 > Q and Q > Q²]

Therefore, 1 > Q² must be true.

২,৯৫৫.

M men agree to purchase a gift for Taka D. If three men drop out how much more will each have to contribute towards the purchase of the gift?

ক
ক) D/(M - 3)
খ
খ) MD/3
গ
গ) M/(D - 3)
ঘ
ঘ) 3D/(M² - 3M)
ঙ
ঙ) None of these

ব্যাখ্যা

Question: M men agree to purchase a gift for Taka D. If three men drop out how much more will each have to contribute towards the purchase of the gift?

Solution:
M men agree to purchase a gift for Taka D.
∴ 1 man purchase for the gift = D/M Taka

If three men drop out then remaining = M - 3 men
∴ 1 man has to purchase D/(M - 3) Taka

Now,
D/(M - 3) - D/M
⇒ {DM - D(M - 3)}/M(M - 3)
⇒ {DM - DM + 3D}/(M² - 3M)
∴ 3D/(M² - 3M)

২,৯৫৬.

If tan(θ + 15°) = √3, what is the value of sinθ?

ক
0
খ
1/2
গ
1/√2
ঘ
√3/2

ব্যাখ্যা

প্রশ্ন: If tan(θ + 15°) = √3, what is the value of sinθ?

সমাধান:
দেওয়া আছে,
tan(θ + 15°) = √3
⇒ tan(θ + 15°) = tan 60°
⇒ θ + 15° = 60°
⇒ θ = 60° - 15°
⇒ θ = 45°

এখন,
sinθ
= sin45°
= 1/√2

২,৯৫৭.

A factory produces 500 bottles of soda in 2 hours. How many bottles will it produce in 6 hours, working at the same rate?

ক
ক) 500 bottles
খ
খ) 1000 bottles
গ
গ) 1200 bottles
ঘ
ঘ) 1500 bottles

ব্যাখ্যা

Question: A factory produces 500 bottles of soda in 2 hours. How many bottles will it produce in 6 hours, working at the same rate?

Solution:
Production rate = Number of items produced / Time
Production rate = 500 bottles / 2 hours = 250 bottles per hour

Bottles produced in 6 hours = Production rate × Time
Bottles produced in 6 hours = 250 bottles/hour × 6 hours = 1500 bottles

২,৯৫৮.

A 60g silver-copper alloy contains 70% silver. How much additional silver is needed to raise the silver percentage to 85%?

ক
80g
খ
70g
গ
60g
ঘ
100g
ঙ
None

ব্যাখ্যা

Question: A 60 g silver-copper alloy contains 70% silver. How much additional silver is needed to raise the silver percentage to 85%?

Solution:
Silver in alloy = 60 × 70% = 42 g
Copper in alloy = 60 × 30% = 18 g

Let the additional silver be x g.

Then, total weight after adding silver = 42 + x + 18 = 60 + x

ATQ,
(42 + x)/(60 + x) = 85/100
⇒ 100(42 + x) = 85(60 + x)
⇒ 4200 + 100x = 5100 + 85x
⇒ 100x - 85x = 5100 - 4200
⇒ 15x = 900
∴ x = 60 g

২,৯৫৯.

Given that the diagonal of a square measures 10√6 units, find the area of the square in units.

ক
600 square units
খ
300 square units
গ
100√3 square units
ঘ
500 square units

ব্যাখ্যা

Question: Given that the diagonal of a square measures 10√6 units, find the area of the square in units.

Solution:
দেয়া আছে,
বর্গক্ষেত্রের কর্ণের দৈর্ঘ্য = 10√6 একক

আমরা জানি,
বর্গক্ষেত্রের কর্ণের দৈর্ঘ্য = √2 × বাহু

প্রশ্নমতে,
√2 × বাহু = 10√6
⇒ বাহু = 10√6/√2
⇒ বাহু = 10√3 একক

এখন, বর্গক্ষেত্রের ক্ষেত্রফল = বাহু²
= (10√3)²
= 300 বর্গ একক

∴ বর্গক্ষেত্রের ক্ষেত্রফল 300 বর্গ একক

২,৯৬০.

There is a group of 5 men, 6 women and 8 children. 1 man, 1 woman and one child are going to be selected to play a game. In how many ways can the selection be done?

ক
ক) 240 ways
খ
খ) 480 ways
গ
গ) 120 ways
ঘ
ঘ) None of these

ব্যাখ্যা

5 জন পুরুষ থেকে 1 জন পুরুষ বাছাই করার উপায় = ⁵c₁ = 5
6 জন মহিলা থেকে 1 জন মহিলা বাছাই করার উপায় =⁶c₁ = 6
8 শিশু থেকে 1 জন শিশু বাছাই করার উপায় = ⁸c₁ = 8
মোট বাছাই করার উপায় = 5 × 6 × 8 = 240

২,৯৬১.

(52.02² - 34.01²)÷17.99 × √? = 1,720

ক
ক) 25
খ
খ) 20
গ
গ) 400
ঘ
ঘ) 625

ব্যাখ্যা

(52.02² - 34.01²)÷17.99 ×√x =1720
(2706.0804 - 1156.6801) ÷ 17.99 ×√x = 1720
1549.4 ÷ 17.99 ×√x = 1720
86.1256253 ×√x = 1720
√x =1720÷86.1256253
√x =19.9708
(√x)² = (19.9708)²x= 398.83465
≈400

২,৯৬২.

A man invests some money partly in 9% stock at 96 and partly in 12% stock at 120. To obtain equal dividends from both, he must invest the money in the ratio -

ক
ক) 17 : 14
খ
খ) 19 : 16
গ
গ) 16 : 15
ঘ
ঘ) 18 : 15

ব্যাখ্যা

For an income of Tk. 1 in 9% stock at 96,
investment = Tk. (96/9)
= Tk. 32/3

For an income of Tk. 1 in 12% stock at 120,
investment = Tk. (120/12)
= Tk. 10

Ratio of investments = (32/3) : 10
= 32 : 30
= 16 : 15.

২,৯৬৩.

Four girls are sitting in a line for photograph. Tania is to the right of Jannat. Lamia is to the right of Mania. Mania is between Tania and Lamia. Jannat is at the extreme left end. Who is third from the right in the line?

ক
Mania
খ
Tania
গ
Lamia
ঘ
Jannat

ব্যাখ্যা

Question: Four girls are sitting in a line for photograph. Tania is to the right of Jannat. Lamia is to the right of Mania. Mania is between Tania and Lamia. Jannat is at the extreme left end. Who is third from the right in the line?

Solution:
তানিয়া জান্নাতের ডান দিকে। জান্নাত ⇔ তানিয়া
মানিয়া, তানিয়া এবং লামিয়ার মাঝে। তানিয়া ⇔ মানিয়া ⇔ লামিয়া
জান্নাত একদম বামপ্রান্তে। জান্নাত ⇔ তানিয়া ⇔ মানিয়া ⇔ লামিয়া

∴ জান্নাত ⇔ তানিয়া ⇔ মানিয়া ⇔ লামিয়া
∴ তানিয়া সারিটির ডানদিক থেকে তৃতীয় হবে।

২,৯৬৪.

After successive discount of 12% and 5% an article was sold for Tk.209. What was the original price of the article?

ক
ক) Tk. 260
খ
খ) Tk. 270
গ
গ) Tk. 280
ঘ
ঘ) Tk. 250

ব্যাখ্যা

Question: After successive discount of 12% and 5% an article was sold for Tk.209. What was the original price of the article?

Solution:
Let the original price x
Now
95% of 88% of x = 209
(95/100) × (88/100) × x = 209
x = (209 × 100 × 100)/(95 × 88)
x = 250

২,৯৬৫.

An iron rod that weighs 24 kg is cut into two pieces so that one of these pieces weighs 16 kg and is 34 m long. If the weight of each piece is proportional to its length, how long is the other piece?

ক
ক) 64 m
খ
খ) 17 m
গ
গ) 34 m
ঘ
ঘ) 11 m

ব্যাখ্যা

Given, 16 kg rod = 34m
ATQ, 24 kg rod = (34×24)/16 = 51m
∴ Length of the other part is = 51 - 34 = 17m

২,৯৬৬.

If x - y = 15 and xy = 54, then what is the value of x + y?

ক
21
খ
23
গ
18
ঘ
27

ব্যাখ্যা

Question: If x - y = 15 and xy = 54, then what is the value of x + y?

Solution:
দেওয়া আছে,
x - y = 15
xy = 54

আমরা জানি,
(x + y)² = (x - y)² + 4xy
⇒ (x + y)² = (15)² + 4 × 54
⇒ (x + y)² = 225 + 216
⇒ (x + y)² = 441
⇒ x + y = √441
⇒ x + y = 21

সুতরাং, x + y এর মান হলো 21।

২,৯৬৭.

If a sum triples in 15 years at simple interest, how much will it be in 10 years?

ক
1/3 times
খ
2/3 times
গ
5/3 times
ঘ
7/3 times

ব্যাখ্যা

Question: If a sum triples in 15 years at simple interest, how much will it be in 10 years?

Solution:
Let, Principal = P

If the sum triples in 15 years,
Amount after 15 years = 3P

∴ Simple interest for 15 years = 3P - P
= 2P

So, Simple interest for 1 year = 2P/15

∴ Simple interest for 10 years = (2P/15) × 10
= 20P/15
= 4P/3

Amount after 10 years = P + 4P/3
= 7P/3

Hence, the sum will be = 7/3 times the principal.

২,৯৬৮.

A bus trip of 450 miles would have taken 1 hour less if the average speed S for the trip had been greater by 5 miles per hour. What was the average speed S, in miles per hour, for the trip?

ক
40
খ
45
গ
50
ঘ
55

ব্যাখ্যা

Question: A bus trip of 450 miles would have taken 1 hour less if the average speed S for the trip had been greater by 5 miles per hour. What was the average speed S, in miles per hour, for the trip?

Solution:
Average Speed = S
∴ Time reqired in Average Speed = 450/S

Time required in speed S for the trip had been greater by 5 miles per hour = 450/(S + 5)

So, we get:
450/S - 450/(S + 5) = 1
⇒ 450/S = 450/(S + 5) + 1
⇒ 450 = 450S/(S + 5) + S
⇒ 450(S + 5) = 450S + S(S + 5)
⇒ 450S + 2250 = 450S + S² + 5S
⇒ 2250 = S² + 5S
⇒ S² + 5S - 2250 = 0
⇒ (S + 50)(S - 45) = 0
∴ S = - 50, OR S = 45
Since the speed can't be negative, the correct answer must be S = 45

২,৯৬৯.

In the beginning, Rakib works at a rate such that he can finish a piece of work in 24 hrs, but he only works at this rate for 16 hrs. After that, he works at a rate such that he can do the whole work in 18 hrs. If Rakib is to finish this work at a stretch, how many hours will he take to finish this work?

ক
12 hrs
খ
18 hrs
গ
11.5 hrs
ঘ
15 hrs
ঙ
22 hrs

ব্যাখ্যা

Question: In the beginning, Rakib works at a rate such that he can finish a piece of work in 24 hrs, but he only works at this rate for 16 hrs. After that, he works at a rate such that he can do the whole work in 18 hrs. If Rakib is to finish this work at a stretch, how many hours will he take to finish this work?

Solution:
Rakib’s 16 hr work = 16/24 = 2/3.
Remaining work = 1 - 2/3
= 1/3.
Using work and time formula:
This will be completed in (1/3) ×18 = 6 hrs.
So, total time taken to complete work = 16 + 6= 22 hrs.

২,৯৭০.

If xy = 60 and x² + y² = 169, then x + y =?

ক
15
খ
17
গ
19
ঘ
21

ব্যাখ্যা

Question: If xy = 60 and x² + y² = 169, then x + y =?

Solution:
xy = 60 ............(1)
x² + y² = 169 ...............(2)

(x + y)² = x² + y² + 2xy
⇒ (x + y)² = 169 + (2 × 60)
⇒ (x + y)² = 289
∴ x + y = 17

২,৯৭১.

If a number is in the form of 8¹⁰ × 9⁷ × 7⁸, find the total number of prime factors of the given number.

ক
52
খ
560
গ
3360
ঘ
25

ব্যাখ্যা

Question: If a number is in the form of 8¹⁰ × 9⁷ × 7⁸, find the total number of prime factors of the given number.

Solution:
If a number of the form x^a × y^b × z^c ...... and so on,
then total prime factors = a + b + c ..... and so on Where x, y, z, ... are prime numbers

The number 8¹⁰ × 9⁷ × 7⁸ can be written as (2³)¹⁰ × (3²)⁷ × 7⁸
= 2³⁰ × 3¹⁴ × 7⁸

Total number of prime factors = 30 + 14 + 8 = 52

∴ The total number of prime factors are 52

২,৯৭২.

The contents of a certain box consist of 14 apples and 23 oranges. How many oranges must be removed from the box so that 70% of the pieces of fruit in the box will be apples ?

ক
ক) 6
খ
খ) 12
গ
গ) 17
ঘ
ঘ) 36

ব্যাখ্যা

Total number of fruits = (14 + 23) = 37
Let x oranges be removed.
Then,
70% of (37 - x) = 14
⇒ 7 (37 - x) = 140
⇒ 37 - x = 20
⇒ x = 17

২,৯৭৩.

A player's average test score on 4^th test is 78. What must be his score on a 5^th test for average score on the 5^th tests to be 80?

ক
ক) 82
খ
খ) 84
গ
গ) 86
ঘ
ঘ) 88

ব্যাখ্যা

After 5 test player's total score will be = 5×80 = 400
Total score after 4 test was = 4×78 = 312
So, required score is = 400 - 312 = 88

২,৯৭৪.

A man sold his watch at a loss of 5 %. Had he sold it for Tk 56.25 more he would have gained 10%. What was the cost of the watch?

ক
Taka 375
খ
Taka 306
গ
Taka 366
ঘ
Taka 300

ব্যাখ্যা

Question: A man sold his watch at a loss of 5 %. Had he sold it for Tk 56.25 more he would have gained 10%. What was the cost of the watch?

Solution:
Let, the price of watch x taka
According to the question,
x(100% - 5%) + 56.25 = x(100% + 10 %)
→ 95%x + 56.25 = 110% x
→ 15%x = 56.25
→ x = (5625/15)
→ x = 375 taka

২,৯৭৫.

The average of a set of 12 numbers, which includes 34 is A. If 34 is removed from the set and 38 is included to the set. What is the average of new set of numbers in terms of A?

ক
A + 4
খ
(A + 38)/12
গ
12A + 4
ঘ
A + (1/3)
ঙ
None of these

ব্যাখ্যা

প্রশ্ন: The average of a set of 12 numbers, which includes 34 is A. If 34 is removed from the set and 38 is included to the set. What is the average of new set of numbers in terms of A?

সমাধান:
12টি সংখ্যার সমষ্টি = 12A - 34 + 38
= 12A + 4

12টি সংখ্যার গড় = (12A + 4)/12
= (12A/12) + (4/12)
= A + (1/3)

২,৯৭৬.

If 3sec²θ - 2 = 2, find the value of θ

ক
30°
খ
0°
গ
45°
ঘ
60°

ব্যাখ্যা

Question: If 3sec²θ - 2 = 2, find the value of θ.

Solution:
3sec²θ - 2 = 2
3sec²θ = 4
sec²θ = 4/3
secθ = 2/√3
secθ = sec30°
∴ θ = 30°

২,৯৭৭.

A and B invested in the ratio of 11 : 12. B invested for 11 month. If the profit ratio of A and B is 2 : 3, for how much time A invested the money?

ক
ক) 8 month
খ
খ) 12 month
গ
গ) 10 month
ঘ
ঘ) 9 month

ব্যাখ্যা

Question: A and B invested in the ratio of 11 : 12. B invested for 11 month. If the profit ratio of A and B is 2 : 3, for how much time A invested the money?

Solution:
Let A and B invested 11p and 12p,
A invested for Y month

Then,
(11p × Y) : (12p × 11) = 2 : 3
11Y : 132 = 2 : 3
33Y = 264
Y = 8

২,৯৭৮.

A person crosses a 600 m long street in 5 minutes, What is his speed in km per hour?

ক
3.6
খ
7.2
গ
8.4
ঘ
10

ব্যাখ্যা

Question: A person crosses a 600 m long street in 5 minutes, What is his speed in km per hour?

Solution:
Speed = 600/(5 × 60) m/sec
= 2 m/sec
= 2 × (18/5) km/hr
= 7.2 km/hr

২,৯৭৯.

Reena and Shaloo are partners in a business, Reena invest Tk. 35,000 for 8 months and Shallo invest Tk. 42,000 for 10 months. Out of a profit of Tk. 31,570, Reena's share is:

ক
Tk. 9471
খ
Tk. 12,628
গ
Tk. 18,040
ঘ
Tk. 18,942

ব্যাখ্যা

Question: Reena and Shaloo are partners in a business, Reena invest Tk. 35,000 for 8 months and Shallo invest Tk. 42,000 for 10 months. Out of a profit of Tk. 31,570, Reena's share is:

Solution:
Ratio of Reena and Shaloo shares = (35000 × 8) : (42000 × 10)
= 280000 : 420000
= 2 : 3.

Reena's share Tk. 31570 × (2/5)
= Tk. 12628

২,৯৮০.

The average salary of all the workers in a workshop is Tk. 6000. The average salary of 7 technicians is Tk. 12000 and the average salary of the rest is Tk. 5000. How many workers are there?

ক
ক) 59
খ
খ) 49
গ
গ) 39
ঘ
ঘ) 29

ব্যাখ্যা

Question: The average salary of all the workers in a workshop is Tk. 6000. The average salary of 7 technicians is Tk. 12000 and the average salary of the rest is Tk. 5000. How many workers are there?

Soluion:
let, there are x number of workers
The average salary of all the workers in a workshop is Tk. 6000
∴ total salary = 6000 × x = 6000x

The average salary of 7 technicians is Tk. 12000
∴ salary of 7 technicians is = (12000 × 7) = 84000 tk

the average salary of the rest is Tk. 5000
∴ salary of the rest = 5000 × (x - 7) tk

5000 × (x - 7) + 84000 = 6000x
⇒ 5000x - 35000 + 84000 = 6000x
⇒ 1000x = 49000
∴ x = 49

so, there are 49 workers.

২,৯৮১.

Two pipes A and B can fill a tank in 20 and 30 minutes respectively. If both the pipes are used together, then how long will it take to fill the tank?

ক
15 minutes
খ
25 minutes
গ
50 minutes
ঘ
12 minutes
ঙ
9 minutes

ব্যাখ্যা

Part filled by A in 1 minute = 1/20
Part filled by B in 1 minute = 1/30

Part filled by (A + B) in 1 minute
= 1/20 + 1/30
= 1/12

∴ Both pipes can fill the tank in 12 minutes

২,৯৮২.

What is the difference between the smallest six-digit number and the largest five-digit number?

ক
ক) - 1
খ
খ) 1
গ
গ) 10
ঘ
ঘ) 99

ব্যাখ্যা

Question: What is the difference between the smallest six-digit number and the largest five-digit number?

Solution:
ছয় অঙ্কের ক্ষুদ্রতম সংখ্যা = ১০০০০০
পাঁচ অঙ্কের বৃহত্তম সংখ্যা = ৯৯৯৯৯

∴ অন্তর = ১০০০০০ - ৯৯৯৯৯ = ১

২,৯৮৩.

A manufacturer marked an article at Tk.50 and sold it allowing 20% discount. If his profit was 25%, then the cost price of the article was-

ক
Tk.32
খ
Tk.35
গ
Tk.37
ঘ
Tk.40
ঙ
Tk.35.8

ব্যাখ্যা

Selling price = 50 - (50 × 20%)
= 50 - 10
= 40
Cost Price = (100/100+25) × 40
= (100/125) × 40
= 32

২,৯৮৪.

3log2 + log5 =?

ক
ক) log20
খ
খ) log10
গ
গ) log40
ঘ
ঘ) log30

ব্যাখ্যা

3log2 + log5
= log2³ + log5
= log8 + log5
= log (8 × 5)
= log40

২,৯৮৫.

নল 'ক' দ্বারা একটি ট্যাংক ২৮ মিনিটে পূ‍র্ণ হয়। নল 'খ' দ্বারা ট্যাংকটি ১৪ মিনিটে পূ‍র্ণ হয়। নল 'গ' দ্বারা ট্যাংকটি ৪২ মিনিটে খালি হয়। তিনটি নল একত্রে খুলে দেয়া হলে, ট্যাংকটি পূ‍র্ণ হতে কত মিনিট লাগবে?

ক
২১ মিনিট
খ
১৮ মিনিট
গ
১২ মিনিট
ঘ
৯ মিনিট
ঙ
কোনটিই নয়

ব্যাখ্যা

প্রশ্ন: নল 'ক' দ্বারা একটি ট্যাংক ২৮ মিনিটে পূ‍র্ণ হয়। নল 'খ' দ্বারা ট্যাংকটি ১৪ মিনিটে পূ‍র্ণ হয়। নল 'গ' দ্বারা ট্যাংকটি ৪২ মিনিটে খালি হয়। তিনটি নল একত্রে খুলে দেয়া হলে, ট্যাংকটি পূ‍র্ণ হতে কত মিনিট লাগবে?

সমাধান:
নল ৩টি একসাথে খুলে দিলে ১ মিনিটে পূ‍র্ণ হয় = (১/১৪ + ১/২৮ - ১/৪২)
= (৬ + ৩ - ২)/৮৪
= ৭/৮৪
= ১/১২

নল তিনটি দ্বারা ১/১২ অংশ পূ‍র্ণ হয় ১ মিনিটে
নল তিনটি দ্বারা ১ বা সম্পূ‍র্ণ অংশ পূ‍র্ণ হয় ১২ মিনিটে
= ১২ মিনিট

২,৯৮৬.

A and B are business partners. A invests Tk. 20,000 for 4 months, while B invests Tk. 30,000 for 6 months. If the total profit at the end of the year is Tk. 65,000, what is B’s share of the profit?

ক
45000 Taka
খ
52000 Taka
গ
62000 Taka
ঘ
20000 Taka

ব্যাখ্যা

Question: A and B are business partners. A invests Tk. 20,000 for 4 months, while B invests Tk. 30,000 for 6 months. If the total profit at the end of the year is Tk. 65,000, what is B’s share of the profit?

Solution:
A’s Contribution = 20000 × 4 = 80000 Taka
B’s Contribution = 30000 × 6 = 180000 Taka

The ratio of A's and B's investment = 80000 : 180000
= 4 : 9

So, B's share in profit = {9/(4+9)} × 65000
= 45000 Taka

২,৯৮৭.

Roni is as much younger than Sagar as he is older than Piyal. If the sum of the ages of Piyal and Sagar is 66 years, and Sagar's age is 48 years, then what is the difference between Roni and Piyal's age?

ক
11 years
খ
15 years
গ
25 years
ঘ
30 years

ব্যাখ্যা

Question: Roni is as much younger than Sagar as he is older than Piyal. If the sum of the ages of Piyal and Sagar is 66 years, and Sagar's age is 48 years, then what is the difference between Roni and Piyal's age?

Solution:
Sagar's age is 48 years
the sum of the ages of Piyal and Sagar is 66 years
∴ The age of Piyal is = 66 - 48 year
= 18 year

let, age of Roni is x

48 - x = x - 18
⇒ 2x = 66
∴ x = 33
the difference between Roni and Piyal's age is = 33 - 18
= 15 years

২,৯৮৮.

Find the equation of the vertical line passing through the point (7, - 4).

ক
y = - 4
খ
x = 7
গ
y = 7
ঘ
x = - 4

ব্যাখ্যা

Question: Find the equation of the vertical line passing through the point (7, - 4).

Solution:
একটি উল্লম্ব রেখা (vertical line) হলো এমন একটি সরলরেখা যা Y-অক্ষের সমান্তরাল। এই ধরনের রেখার একটি বিশেষ বৈশিষ্ট্য হলো, এই রেখার উপর অবস্থিত প্রতিটি বিন্দুর x-স্থানাঙ্ক (x-coordinate) একই থাকে, কিন্তু y-স্থানাঙ্ক (y-coordinate) পরিবর্তিত হতে পারে।

উল্লম্ব রেখার (vertical line) সাধারণ সমীকরণ হলো: x = a
যেখানে a একটি ধ্রুবক সংখ্যা এবং রেখার প্রতিটি বিন্দুর x এর মান একই থাকে।

প্রশ্নে বলা হয়েছে রেখাটি (7, - 4) বিন্দুর মধ্য দিয়ে যায়।
যেহেতু, আমরা জানি, একটি উল্লম্ব রেখার প্রতিটি বিন্দুর x-স্থানাঙ্ক একই থাকে, এবং এই বিন্দুর x-স্থানাঙ্ক হলো 7, সুতরাং, রেখাটির সমীকরণ হবে:
x = 7

২,৯৮৯.

If log₁₀(2^m + m - 4) = m(1 - log₁₀5), then m =?

ক
ক) 0
খ
খ) 1
গ
গ) 3
ঘ
ঘ) 4

ব্যাখ্যা

Question: If log₁₀(2^m + m - 4) = m(1 - log₁₀5), then m =?

Solution:
log₁₀(2^m + m - 4) = m(1 - log₁₀5)
⇒ log₁₀(2^m + m - 4) = m (log₁₀10 - log₁₀5)
⇒ log₁₀(2^m + m - 4) = m log₁₀(10/5)
⇒ log₁₀(2^m + m - 4) = m log₁₀2
⇒ log₁₀(2^m+ m - 4) = log₁₀2^m
⇒ 2^m + m - 4 = 2^m
⇒ m = 4

২,৯৯০.

A man sells a watch for Tk. 550 and makes a profit of 10%. What was the cost price of the watch?

ক
600 Taka
খ
670 Taka
গ
700 Taka
ঘ
500 Taka

ব্যাখ্যা

Question: A man sells a watch for Tk. 550 and makes a profit of 10%. What was the cost price of the watch?

Answer:
Selling Price = 110% of Cost Price
⇒ Cost Price = 550 ÷ 1.1
= 500

Another Method,
With 10% gain,
The selling price is 110 Taka when the cost price is 100 Taka
∴ The selling price is 1 Taka when the cost price is = 100/110 Taka
∴ The selling price is 550 Taka when the cost price is (100 × 550)/110 Taka
= 500 Taka

২,৯৯১.

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

ক
ক) 3√15 cm²
খ
খ) 2√14 cm²
গ
গ) 5√7 cm²
ঘ
ঘ) 7√11 cm²
ঙ
ঙ) 4√3 cm²

ব্যাখ্যা

অর্ধপরিসীমা, s = (3 + 5 + 6)/2
= 7 সে.মি
∴ ক্ষেত্রফল = √{s(s - a)(s - b)(s - c)} বর্গএকক
= √{7 (7 - 3) (7 - 5) (7 - 6)} বর্গসে.মি
= √(7 × 4 × 2 × 1)
= 2√14 বর্গসে.মি

২,৯৯২.

3x² + kx + 4 is divisible by x - 1. The expression is also divisible by-

ক
ক) 4x - 3
খ
খ) 2x - 3
গ
গ) 3x - 2
ঘ
ঘ) 3x - 4

ব্যাখ্যা

Question: 3x² + kx + 4 is divisible by x - 1. The expression is also divisible by-

Solution:
Given that
The expression 3x² + kx + 4 divisible by x - 1,
⇒ 3x² + kx + 4 = 0............(1)

Put x = 1 in equation (1)
⇒ 3 × 1² + k × 1 + 4 = 0
⇒ 3 + k + 4 = 0
⇒ k = - 7

Put the value k in equation (1)
⇒ 3x² - 7x + 4 = 0
⇒ 3x² - 3x - 4x + 4 = 0
⇒ 3x (x - 1) - 4(x – 1) = 0
⇒ (x - 1) (3x - 4) = 0

So, expression divisible by (x - 1) and (3x - 4).

২,৯৯৩.

If n is an even integer, which of the following must be an odd integer?

ক
n² - n
খ
n + 2
গ
3n - 1
ঘ
3n³

ব্যাখ্যা

Question: If n is an even integer, which of the following must be an odd integer?

Solution:
ধরি
n = 2

ক) n² - n = 2² - 2 = 4 - 2 = 2 [যা জোড়]
খ) n + 2 = 2 + 2 = 4 [যা জোড়]
গ) 3n - 1 = 3 × 2 - 1 = 5 [যা বিজোড়]
ঘ) 3n³ = 3 × 2³ = 24 [যা জোড়]

২,৯৯৪.

2% of 2 = ?

ক
ক) 0.0004
খ
খ) 0.4
গ
গ) 0.02
ঘ
ঘ) 0.04

ব্যাখ্যা

2 এর 2% = 2 এর 2/100 = 4/100 = 0.04

২,৯৯৫.

A boat goes 15 km downstream in 45 minutes. The speed of stream is 3km/hr. The speed of boat in still water is-

ক
15 km/hr
খ
17 km/hr
গ
12 km/hr
ঘ
23 km/hr

ব্যাখ্যা

Question: A boat goes 15 km downstream in 45 minutes. The speed of stream is 3km/hr. The speed of boat in still water is-

Solution:
স্রোতের প্রতিকূলে 45 মিনিটে যায় 15 কি.মি.
স্রোতের প্রতিকূলে 1 মিনিটে যায় 15/45 কি.মি.
স্রোতের প্রতিকূলে 1 ঘণ্টা বা 60 মিনিটে যায় (15 × 60)/45 কি.মি./ঘণ্টা
= 20 কি.মি./ঘণ্টা

স্রোতের বেগ = 3 কি.মি./ঘণ্টা
স্থির পানিতে নৌকার বেগ = (20 - 3) কি.মি./ঘণ্টা
= 17 কি.মি./ঘণ্টা

২,৯৯৬.

A horse covers a distance of 1500 meters in 1 minute 20 seconds. At what speed the horse is running?

ক
67.2 km/hr
খ
67.7 km/hr
গ
67.5 km/hr
ঘ
67.9 km/hr

ব্যাখ্যা

Question: A horse covers a distance of 1500 meters in 1 minute 20 seconds. At what speed the horse is running?

Solution:
Distance = 1500 meters
Time = 1 minute 20 seconds = 60 + 20 = 80 seconds

So, Required Speed = 1500/80 = 75/4 m/s

We need answer in km/hr:
So, Speed in km/hr = (75/4) × (18/5) = 67.5 km/hr

২,৯৯৭.

A store runs a "Buy 3, Get 1 Free" offer. What is the equivalent discount percentage?

ক
20%
খ
25%
গ
30%
ঘ
16%

ব্যাখ্যা

Question: A store runs a "Buy 3, Get 1 Free" offer. What is the equivalent discount percentage?

Solution:
Assume the price of each item = 1 Taka.
Then,
Cost of 3 items = 3 Taka
Since 1 item is received for free, total items = 3 + 1 = 4

After the discount,
Cost of 4 items = 3 Taka
∴ Cost per item = 3/4 Taka

Actual price per item = 1 Taka
∴ Discount per item = 1 − ( 3/4 ) = 1/4 Taka

So, discount on 1 Taka = 1/4 Taka
∴ Discount on 100 Taka = 100 × ( 1/4 ) = 25 Taka

Therefore, in a “Buy 3, Get 1 Free” offer, the equivalent discount is 25%.

২,৯৯৮.

If x + 4/x = 4, then find the value of x?

ক
5
খ
2
গ
6
ঘ
4

ব্যাখ্যা

x + 4/x = 4
বা, (x² + 4)/x = 4
বা, x² + 4 = 4x
বা, x² - 4x + 4 = 0
বা, x² - 2.2x.1 + 2² = 0
বা, (x - 2)² = 0
বা, x = 2

২,৯৯৯.

A new square is formed by joining the midpoints of the sides of a square. The same process is repeated indefinitely. If the side of the first square is 4 cm, find the sum of the areas of all the squares

ক
16 cm2
খ
32 cm²
গ
48 cm²
ঘ
24 cm²
ঙ
None

ব্যাখ্যা

Question: A new square is formed by joining the midpoints of the sides of a square. The same process is repeated indefinitely. If the side of the first square is 4 cm, find the sum of the areas of all the squares

Solution:
Side of the first square is 4 cm.
side of second square = 2√2 cm.
Side of third square = 2 cm.
Side of fourth square = √2 cm.
...............................
...............................

Area of these squares will be = 16, 8, 4, 2, ........................
the sum of the areas of all the squares = 16 + 8 + 4 + 2, ........................
= 16/{1 - (1/2)}
= 16/(1/2)
= 32 cm²

৩,০০০.

If 2^{n - 1} + 2^{n + 1} = 640, then n is equal to-

ক
ক) 2
খ
খ) 3
গ
গ) 7
ঘ
ঘ) 8

ব্যাখ্যা

2^{n - 1} + 2^{n + 1} = 640
⇒ 2ⁿ.2^{- 1} + 2ⁿ. 2¹ = 640
⇒ 2ⁿ(1/2) + 2ⁿ .2 = 640
⇒ 2ⁿ(2 + 1/2) = 640
⇒ 2ⁿ (5/2) = 640
⇒ 2ⁿ = (640 × 2)/5
⇒ 2ⁿ = 128 × 2
⇒ 2ⁿ = 256
⇒ 2ⁿ = 2⁸
∴ n = 8