Bank Math

PrepBank · পাতা ৬৭ / ১৬১ · ৬,৬০১–৬,৭০০ / ১৬,১২৪

৬,৬০১.

A bike originally cost Tk. 1000 and was discounted 10%. After three months it was sold after being discounted another 15%. How much was the bike sold for?

ক
Tk. 820
খ
Tk. 805
গ
Tk. 790
ঘ
Tk. 765

ব্যাখ্যা

Question: A bike originally cost Tk. 1000 and was discounted 10%. After three months it was sold after being discounted another 15%. How much was the bike sold for?

Solution:
Given
Original price = Tk. 1000
10% discounted price = 1000 - (10% of 1000)
= 1000 - (10/100 of 1000)
= Tk. 900

The bike was sold after another 15% discount.
∴ 15% of Tk. 900 = Tk. (15/100 of 900)
= Tk. 135

∴ Selling price= Tk. (900 - 135)
= Tk. 765

৬,৬০২.

There are 8 points in a plane, out of which 4 are collinear and the remaining 4 are non-collinear. How many distinct triangles can be formed by joining any 3 of these points?

ক
72 ways
খ
56 ways
গ
52 ways
ঘ
42 ways
ঙ
None

ব্যাখ্যা

Question: There are 8 points in a plane, out of which 4 are collinear and the remaining 4 are non-collinear. How many distinct triangles can be formed by joining any 3 of these points?

Solution:
Total combinations of 3 points from 8 = ⁸C₃
= 56 ways

Given,
there are 4 collinear points

From the 4 collinear points, no triangle can be formed using any 3 of them (since they lie on the same line).
Total combinations of 3 points from the 4 collinear points = ⁴C₃ = 4 ways

So the valid triangles = 56 − 4 = 52 ways

৬,৬০৩.

A vessel goes 6 km in an hour in still water. It requires thrice as much investment in covering the same separation against the current. Velocity of the current is:

ক
ক) 2 km/hr
খ
খ) 3 km/hr
গ
গ) 4 km/hr
ঘ
ঘ) 5 km/hr.

ব্যাখ্যা

Speed in still water =6 km/hr.
Speed against the current =6/3 km/hr = 2 km/hr
Let the speed of the current be x km/hr
so, 6-x = 2
=> x = 4 km/hr.

৬,৬০৪.

How many distinct arrangements can be made using all the letters of the word "MAMMAL" such that no two M's appear together?

ক
6
খ
8
গ
10
ঘ
12
ঙ
None of the above

ব্যাখ্যা

Question: How many distinct arrangements can be made using all the letters of the word "MAMMAL" such that no two M's appear together?

Solution:
MAMMAL has 6 letters, where M = 3 times, A = 2 times and L = 1 time.
First arrange the letters other than M.
Number of arrangements of A, A, L = 3!/2! = 3

Now place the three M's in the gaps of these letters.
For example: _ A _ A _ L _
Total gaps = 4

Number of ways to choose 3 gaps for M = ⁴C₃ = 4

∴ Required number of arrangements = 3 × 4
= 12

৬,৬০৫.

A dishonest shopkeeper pretends to sell his goods at cost price but uses false weights and gains 11.11%. Find the false weight he is using instead of 1kg weight.

ক
900 gm
খ
950 gm
গ
850 gm
ঘ
800 gm

ব্যাখ্যা

Question: A dishonest shopkeeper pretends to sell his goods at cost price but uses false weights and gains 11.11%. Find the false weight he is using instead of 1kg weight.

Solution:

Let the false weight be x gm.

Gain % = [(True weight - False weight)/ False weight] × 100

⇒ 11.11 = 100/9 = [(1000 - x)/x] × 100

⇒ 9000 - 9x = x
⇒ 10x = 9000

⇒ x = 900

∴ the shopkeeper is using weights of 900 gm instead of 1kg.

৬,৬০৬.

In 1 -3x ≤ 4, then-

ক
ক) x ≤ -2
খ
খ) x ≥ -2
গ
গ) x ≤ -1
ঘ
ঘ) x ≥ -1

ব্যাখ্যা

প্রদত অসমতাটি হলো,
1 - 3x ≤ 4
⇒ 1 - 3x - 1 ≤ 4 - 1
⇒ -3x ≤ 3
⇒ 3x ≥ -3 [উভয়পক্ষে (-1) দ্বারা গুণ করে]
⇒ x ≥ -3/3
⇒ x ≥ -1
Answer: x ≥ -1.

৬,৬০৭.

A mother told her daughter, "I was your present age at the time of your birth." If the mother is currently 38 years old, what was the daughter’s age five years ago?

ক
20 years.
খ
15 years.
গ
14 years.
ঘ
25 years.

ব্যাখ্যা

Question: A mother told her daughter, "I was your present age at the time of your birth." If the mother is currently 38 years old, what was the daughter’s age five years ago?
Solution:
Let,
Daughter present age = x years
Mother's present age = (x + x) years
= 2x years

ATQ,
2x = 38
⇒ x = 19
So daughter age five years back was = 19 - 5
= 14 years.

৬,৬০৮.

Rakib and Kabila are working on an assignment. Rakib takes 6 hours to type 32 pages on a computer, while Kabila takes 5 hours to type 40 pages. How much time will they take, working together on two different computers to type an assignment of 220 pages?

ক
14 hour 15 minutes
খ
12 hour 30 minutes
গ
8 hour 15 minutes
ঘ
16 hour 30 minutes

ব্যাখ্যা

Question: Rakib and Kabila are working on an assignment. Rakib takes 6 hours to type 32 pages on a computer, while Kabila takes 5 hours to type 40 pages. How much time will they take, working together on two different computers to type an assignment of 220 pages?

Solution:
Number of pages typed by Rakib in 1 hour = 32/6 = 16/3
Number of pages typed by Kabila in 1 hour = 40/5 = 8
Number of pages typed by both in 1 hour = (16/3 + 8) = 40/3

Therefore Time taken by both to type 220 pages = 220 × (3/40) hours
= 33/4 hours
= 16.5 hours
= 16 hour 30 minutes

৬,৬০৯.

When 20% of a number is added to another number, the second number increased by 150%. What is the ratio between the first and the second number?

ক
2 : 15
খ
15 : 2
গ
15 : 1
ঘ
1 : 15

ব্যাখ্যা

Question: When 20% of a number is added to another number, the second number increased by 150%. What is the ratio between the first and the second number?

Solution:
Let,
The numbers be X and Y

ATQ,
Y + 20% of X = Y + 150% of Y
⇒ 20X/100 = 150Y/100
⇒ 20X = 150Y
⇒ X/Y = 150/20
∴ X : Y = 15 : 2

৬,৬১০.

In what ratio must two kinds of sugar at Tk. 115 and Tk. 124 per kg be mixed so that by selling at Tk. 150 per kg, 25% may be gained?

ক
4 : 5
খ
5 : 3
গ
1 : 1
ঘ
2 : 3

ব্যাখ্যা

Question: In what ratio must two kinds of sugar at Tk. 115 and Tk. 124 per kg be mixed so that by selling at Tk. 150 per kg, 25% may be gained?

Solution:
SP = 150
Profit = 25%.
∴ CP = 150 × (100/125) = 120

Therefore required ratio = 4 : 5

৬,৬১১.

5, 8, 14, 26, 50, 98, ..... What number should come next?

ক
192
খ
194
গ
200
ঘ
202

ব্যাখ্যা

Question: 5, 8, 14, 26, 50, 98, ..... What number should come next?

Solution: Given series: 5, 8, 14, 26, 50, 98...

First number: 5
Now,
5 + 3 = 8
8 + 6 = 14
14 + 12 = 26
26 + 24 = 50
50 + 48 = 98

∴ Next number = 98 + 96 = 194

৬,৬১২.

Find the roots of the quadratic equation 2x² - 9x - 35 = 0

ক
-7, - 5/2
খ
5, 7/2
গ
7, - 5/2
ঘ
7, 2/5

ব্যাখ্যা

Question: Find the roots of the quadratic equation 2x² - 9x - 35 = 0

Solution:
2x² - 9x - 35 = 0
⇒ 2x² - 14x + 5x -35 = 0
⇒ 2x(x - 7) + 5(x -7) = 0
⇒ (x - 7) (2x + 5) = 0
∴ x = 7, - 5/2

৬,৬১৩.

A train running at the speed of 60 km/hr crosses a pole in 9 seconds. What is the length of the train?

ক
80m
খ
100m
গ
120m
ঘ
150m

ব্যাখ্যা

Speed
= (60 × 5/18)m/sec
= 50/3m/sec
Length of the train
= (Speed × Time)
∴ Length of the train = (50/3 × 9)m = 150m

৬,৬১৪.

A bag contains 7 red, 9 yellow and 3 black balls. If a ball is picked at random, what is the probability that the ball drawn will be either a red or a black ball.

ক
ক) 7/19
খ
খ) 9/19
গ
গ) 10/19
ঘ
ঘ) 11/19

ব্যাখ্যা

Total number of balls = 7 + 9 +3 = 19
If a ball is picked at random,
the probability that the ball drawn will be a red = 7/19
the probability that the ball drawn will be a black = 3/19
the probability that the ball drawn will be either a red or a black ball = 7/19 + 3/19 = 10/19

৬,৬১৫.

If (x + y) = 3, xy = 2, then what is the value of x³ + y³?

ক
ক) 5
খ
খ) 3
গ
গ) 7
ঘ
ঘ) 9

ব্যাখ্যা

Given that
(x + y) = 3
xy = 2

x³ + y³ = (x + y)³ - 3xy(x + y)
= 3³ - 3 × 2 × 3
= 27 - 18
= 9

৬,৬১৬.

1.14 expressed as a percent of 1.9 is-

ক
6%
খ
10%
গ
60%
ঘ
90%
ঙ
None of these

ব্যাখ্যা

Question: 1.14 expressed as a percent of 1.9 is-

Solution:
Required Percentage = (1.14 ×100)/1.9
= (114 × 100 × 10)/(19 × 100)
= 60%

৬,৬১৭.

A boy was asked to multiply a number by 25 but by mistake he multiplied by 45 and the answer was 200 more than the correct answer. What was the number?

ক
7
খ
8
গ
10
ঘ
12
ঙ
None of these

ব্যাখ্যা

Question: A boy was asked to multiply a number by 25 but by mistake he multiplied by 45 and the answer was 200 more than the correct answer. What was the number?

Solution:
Let the correct number be x.
The boy was supposed to multiply the number by 25, so the correct answer would be 25x
Instead, he multiplied the number by 45, giving 45x

According to the problem, the wrong answer is 200 more than the correct answer:
45x = 25x + 200
⇒ 45x - 25x = 200
⇒ 20x = 200
∴ x = 10

৬,৬১৮.

Which one of the following numbers can be removed from the set S = {0, 2, 4, 5, 9, 10} without changing the average of set S?

ক
2
খ
4
গ
5
ঘ
9

ব্যাখ্যা

The average of the elements in the original set S is:
(0+2+4+5+9 +10)/6
= 30/6
= 5

If we remove an element that equals the average, then the average of the new set will remain unchanged.
The new set after removing 5 is {0, 2, 4, 9, 10}.
The average of the elements is,
(0+2+4+9+10)/5
= 25/5
= 5

৬,৬১৯.

In the given figure, ∠ONY = 50° and ∠OMY = 15°. Then the value of the ∠MON is-

ক
70°
খ
85°
গ
55°
ঘ
75°

ব্যাখ্যা

Question: In the given figure, ∠ONY = 50° and ∠OMY = 15°. Then the value of the ∠MON is-

Solution:
According to the figure.
OM = OY = ON
∴ In ΔOMY
∠OMY = ∠OYM = 15°
∴ ∠MOY = 180° - 15° - 15°
∠MOY = 150°
In ΔONY
∠ONY = ∠OYN = 50°
∴ ∠NOY = 180° - 50° - 50°
∠NOY = 80°
∴ ∠MON = 150° - 80°
∠MON = 70°

৬,৬২০.

Subhash can copy 50 pages in 10 hours; Subhash and Prakash together can copy 300 pages in 40 hours. In how much time can Prakash copy 30 pages?

ক
ক) 11 hours
খ
খ) 12 hours
গ
গ) 15 hours
ঘ
ঘ) 18 hours

ব্যাখ্যা

Number of page copied by Subhash and Prakash in 1 hour
= 300/40
= 7.5 pages;

Subhash copied pages in one hour
= 50/10
= 5 pages

Hence, Prakash copied pages in one hour
= 7.5 - 5
= 2.5

Thus, Prakash can copied 30 pages in
= 30/2.5
= 12 hours

৬,৬২১.

What is the area of a triangle with base 5 meters and height 10 meters?

ক
20 square meters
খ
35 square meters
গ
40 square meters
ঘ
25 square meters

ব্যাখ্যা

Question: What is the area of a triangle with base 5 meters and height 10 meters?

Solution:
Area of a triangle = (1/2) × base × height
So, the area = (1/2) × 5 × 10
= 25 square meters

৬,৬২২.

Rokeya bought 30 kg of wheat at the rate of Tk. 9.50 per kg and 40 kg of wheat at the rate of Tk. 8.50 per kg and mixed them. She sold the mixture at the rate of Tk. 8.90 per kg. Her total profit or loss in the transaction was -

ক
ক) Tk. 2 loss
খ
খ) Tk. 2 profit
গ
গ) Tk. 7 loss
ঘ
ঘ) Tk. 7 profit

ব্যাখ্যা

C.P. of 70 kg wheat
= Tk. (30 × 9.50 + 40 × 8.50)
= Tk. (285 + 340)
= Tk. 625
S.P. of 70 kg wheat
= Tk. (70 × 8.90)
= Tk. 623
∴ Loss = Tk. (625 - 623) = Tk. 2

৬,৬২৩.

A right triangle with sides 3 cm, 4 cm and 5 cm is rotated the side of 3 cm to form a cone. The volume of the cone so formed is:

ক
ক) 12π cm³
খ
খ) 8π cm³
গ
গ) 16π cm³
ঘ
ঘ) 15π cm³

ব্যাখ্যা

We have
r=3cm
h=4cm

∴Volume=(1/3)πr²h
(1/3) × π × 3²× 4)cm³= 12π cm³

৬,৬২৪.

One half of a pillar is deep into the soil under the earth, one third of it is above the soil under water and 2 meters is above the water level. What is the length of the pillar?

ক
ক) 8 meters
খ
খ) 12 meters
গ
গ) 18 meters
ঘ
ঘ) 14 meters

ব্যাখ্যা

Question: One half of a pillar is deep into the soil under the earth, one third of it is above the soil under water and 2 meters is above the water level. What is the length of the pillar?

Solution:
ধরি,
খুঁটির দৈর্ঘ্য x মিটার

প্রশ্নমতে,
x - {(1/2)x + (1/3)x} = 2
বা, x - {(3x + 2x)/6} = 2
বা, 6x - 5x = 2 × 6
∴ x = 12

খুঁটির দৈর্ঘ্য 12 মিটার।

৬,৬২৫.

Two pipes can fill a tank together in 8 minutes. Both pipes are opened, and after 6 minutes the first pipe is closed. It then takes 6 more minutes for the tank to be completely filled. How long would it take to fill the tank using only the second pipe?

ক
45 minutes
খ
28 minutes
গ
42 minutes
ঘ
24 minutes
ঙ
36 minutes

ব্যাখ্যা

Question: Two pipes can fill a tank together in 8 minutes. Both pipes are opened, and after 6 minutes the first pipe is closed. It then takes 6 more minutes for the tank to be completely filled. How long would it take to fill the tank using only the second pipe?

Solution:
Together, the two pipes fill the tank in 8 minutes = 1 full tank.
∴ In 1 minute, they fill = 1/8 of the tank.
∴ In 6 minutes, they fill = (1 × 6)/8 = 3/4 of the tank.

∴ Remaining part of the tank = 1 - (3/4) = 1/4.

The second pipe alone fills this 1/4 of the tank in 6 minutes.
∴ Time for the second pipe to fill the whole tank = 6 × 4 = 24 minutes.

So the second pipe alone would take 24 minutes to fill the tank.

৬,৬২৬.

A tank is filled using 20 buckets, each having a capacity of 10 liters. How many buckets would be required to fill the same tank if each bucket can hold 40 liters of water?

ক
8 buckets
খ
5 buckets
গ
10 buckets
ঘ
12 buckets

ব্যাখ্যা

Question: A tank is filled using 20 buckets, each having a capacity of 10 liters. How many buckets would be required to fill the same tank if each bucket can hold 40 liters of water?

Solution:

Total capacity of the tank = 20 × 10 = 200 liters

∴ Number of buckets needed with 40-liter buckets = 200 ÷ 40
= 5 buckets

৬,৬২৭.

The least number which is a perfect square and is divisible by each of the numbers 16, 20 and 24, is -

ক
ক) 14400
খ
খ) 6400
গ
গ) 3600
ঘ
ঘ) 1600

ব্যাখ্যা

Question: The least number which is a perfect square and is divisible by each of the numbers 16, 20 and 24, is -

Solution:
The least number divisible by 16, 20, 24
L.C.M of 16, 20, 24 = 240
Factors of 240 = 2×2×2×2×3×5
By making pair = (2×2)×(2×2)×3×5
Since 3 and 5 has no pair, So, to make it a perfect square, it must be multiplied by 3 × 5

∴ Required number = 240 × 3 × 5
= 3600

৬,৬২৮.

An observer 1.6m tall is 25√3m away from a tower. The angle of elevation from his eye to the top of the tower is 30° .The height of the tower is -

ক
ক) 24.6 m
খ
খ) 25.6 m
গ
গ) 26.6 m
ঘ
ঘ) 27.6 m

ব্যাখ্যা

Let AB be the observer and CD tower
Draw BE perpendicular to CD
Then CE = AB = 1.6 m

And BE = AC = 25√3 m
Now
∴tan 30°=DE/BE
1/√3 =DE/25√3
DE = 25

CD = CE + DE = 1.6 + 25= 26.6 m

৬,৬২৯.

A number when divided by 14 leaves reminder of 8, but when the same number is divided by 7, it will leave the remainder :

ক
ক) 3
খ
খ) 2
গ
গ) 1
ঘ
ঘ) 4

ব্যাখ্যা

When the number is divided by 14 it gives a remainder of 8,
The number = 14N + 8 (14N is divisible by 14)
When same number is divided by 7 it will give remainder 1

৬,৬৩০.

To gain 10% on selling sample of milk at the cost price of pure milk, the quantity of water to be mixed with 50 kg. of pure milk is -

ক
ক) 2.5 kg
খ
খ) 5 kg
গ
গ) 7.5 kg
ঘ
ঘ) 10 kg

ব্যাখ্যা

Question: To gain 10% on selling sample of milk at the cost price of pure milk, the quantity of water to be mixed with 50 kg of pure milk is -

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

Hence,
% gain = x × (100/50)
⇒ 10 = 100x/50
⇒ 100x = 500
⇒ x = 5

৬,৬৩১.

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

ক
ক) 90
খ
খ) 75
গ
গ) 60
ঘ
ঘ) 55

ব্যাখ্যা

Pass mark = (17 + 10) = 27
Let maximum marks be x
Then 36% of x = 27
Or, 36x/100 = 27
Or, 36x = 2700
Hence, x = 75

৬,৬৩২.

A father is 25 years older than his son. In 5 years, his age will be three times his son's age. What is the present age of the son?

ক
9 years
খ
5.5 years
গ
7.5 years
ঘ
12 years

ব্যাখ্যা

Question: A father is 25 years older than his son. In 5 years, his age will be three times his son's age. What is the present age of the son?

Solution:
Let the son’s present age be x years.
Then, the father’s present age is = x + 25 years.

ATQ
In 5 years, the father’s age will be three times his son's age.
So, (x + 25) + 5 = 3 (x + 5)
⇒ 3x + 15 = x + 30
⇒ 3x - x = 30 - 15
⇒ 2x = 15
∴ x = 7.5

Therefore, the present age of the son is 7.5 years.

৬,৬৩৩.

After being dropped a certain ball always bounces back to 2/5 of the height of its previous bounce. After the first bounce it reaches a height of 125 inches. How high (in inches) will it reach after its third bounce?

ক
ক) 60 inches
খ
খ) 50 inches
গ
গ) 25 inches
ঘ
ঘ) None of these

ব্যাখ্যা

প্রশ্ন: After being dropped a certain ball always bounces back to 2/5 of the height of its previous bounce. After the first bounce it reaches a height of 125 inches. How high (in inches) will it reach after its third bounce?

সমাধান:
প্রতি ড্রপে বলটি পূর্ববর্তী উচ্চতার ২/৫ অংশ উঠে।

১ম ড্রপে বলটি উঠে ১২৫ ইঞ্চি

২য় ড্রপে বলটি উঠে {১২৫ × (২/৫)} ইঞ্চি
= ৫০ ইঞ্চি

৩য় ড্রপে বলটি উঠে {৫০ × (২/৫)} ইঞ্চি
= ২০ ইঞ্চি

৬,৬৩৪.

In a dairy farm, 20 cows eat 20 bags in 20 days. In how many days one cow will eat one bag of husk?

ক
ক) 10 days
খ
খ) 1/10 days
গ
গ) 20 days
ঘ
ঘ) 40 days

ব্যাখ্যা

We know,
Indirect proportion: Less cows (↓) More days (↑)
Direct proportion: Less bags (↓) Less days (↓)

20 : x :: { 1 : 20 ------ (Cows); 20 : 1 -------(Bags)}
1 × 20 × x = 20 × 1 × 20
⇒ x = (20 × 1 × 20)/(1 × 20)
⇒ x = 20 Days.

৬,৬৩৫.

Today, Jakir is twice as old as Moin. In 5 years, Jakir will be 1.5 times as old as Moin. How old is Jakir today?

ক
12
খ
10
গ
15
ঘ
20

ব্যাখ্যা

Question: Today, Jakir is twice as old as Moin. In 5 years, Jakir will be 1.5 times as old as Moin. How old is Jakir today?

Solution:
Let Jakir’s age = J, Moin’s age = M.
Given:
⇒ J=2M --------------------(Equation 1)
In 5 years:
⇒ J+5=1.5(M+5)J+5=1.5(M+5) ---(Equation 2)

Substitute J=2M (equation 1) in equation (2),
⇒ 2M+5=1.5M+7.5 ⟹ 0.5M=2.5
⟹ M=5
Then, J=2×5=10
Answer: 10

৬,৬৩৬.

An accurate clock shows 2:00 PM. Through how many degrees will the hour hand rotate by the time the clock shows 8:00 PM?

ক
180°
খ
150°
গ
210°
ঘ
240°
ঙ
120°

ব্যাখ্যা

Question: An accurate clock shows 2:00 PM. Through how many degrees will the hour hand rotate by the time the clock shows 8:00 PM?

Solution:
We know,
A clock is a full circle = 360°
There are 12 hours on the clock face
So the hour hand moves = 360° ÷ 12 = 30° per hour

From 2 : 00 PM to 8 : 00 PM is exactly 6 hours later.
Therefore, the hour hand rotates,
6 hours × 30° per hour = 180°

So the hour hand will rotate through 180°.

৬,৬৩৭.

Two trains moving in same direction run at a speed of 60 km/hr and 40 km/hr respectively. If a man sitting in slow train is passed by the fast train in 10 seconds, then what is the length of the faster train?

ক
ক) 53.7 m
খ
খ) 53.2 m
গ
গ) 55.6 m
ঘ
ঘ) 57.2 m

ব্যাখ্যা

Given: Speed of slow train = 60 km/hr, speed of fast train = 40 km/hr
Here both the trains move in same direction. Hence their relative speed is obtained by subtracting the individual speeds of trains.
Relative speed = 60–40 = 20 km/hr
1) Convert km/hr into m/s
20x(5/18)=100/18= 5.56 m/s
2) Distance (Length of faster train) = Speed x Time
Length of faster train = 5.56 x 10 m = 55.6 m

৬,৬৩৮.

Present population of a city is 20 lac. What will be the population of the city after 3 years if the growth rate of population of that city is 50 per thousand ?

ক
21,15,250
খ
23,15,250
গ
22,05,000
ঘ
21,00,000

ব্যাখ্যা

Question : Present population of a city is 20 lac. What will be the population of the city after 3 years if the growth rate of population of that city is 50 per thousand ?

Solution :
Given that,
The present population of the city is P = 20,00,000
Groth rate of population r = 50/1000
= 5/100
= 1/20
Time n= 3 years

So the population of the city after 3 years will be = P(1 + r)ⁿ
= 20,00,000 × {1 + (1/20)}³
= 20,00,000 × (21/20)³
= 20,00,000 × 9,261/8,000
= 23,15,250

∴ After 3 years, population of the city will be 23,15,250.

৬,৬৩৯.

A wholesaler bought 1200 radios for Tk. 18 each. The wholesaler sold 60 percent of the radios for Tk. 30 each and the rest for Tk. 15 each. What is the wholesaler's average (arithmetic mean) profit per radio?

ক
ক) Tk. 6
খ
খ) Tk. 5
গ
গ) Tk. 4
ঘ
ঘ) Tk. 3

ব্যাখ্যা

Question: A wholesaler bought 1200 radios for Tk. 18 each. The wholesaler sold 60 percent of the radios for Tk. 30 each and the rest for Tk. 15 each. What is the wholesaler's average (arithmetic mean) profit per radio?

Solution:
৬০% রেডিও বিক্রি করে ৩০ টাকায় এবং ৪০% বিক্রি হয় ১৫ টাকায়

মোট বিক্রয়মূল্য = {১২০০ × (৬০/১০০) × ৩০} + {১২০০ × (৪০/১০০) × ১৫}
= ২১৬০০ + ৭২০০ টাকা
= ২৮৮০০ টাকা

গড় বিক্রয়মূল্য = ২৮৮০০/১২০০ টাকা
= ২৪ টাকা

গড় ক্রয়মূল্য ১৮ টাকা

গড় লাভ = (২৪ - ১৮) টাকা
= ৬ টাকা

৬,৬৪০.

How much time will it take for an amount of Tk. 450 to yield Tk. 81 as interest at 9% per annum of simple interest?

ক
2 years
খ
3 years
গ
4 years
ঘ
None of these

ব্যাখ্যা

Question: How much time will it take for an amount of Tk. 450 to yield Tk. 81 as interest at 9% per annum of simple interest?

Solution:
I = Pnr
⇒ 81 = 450 × n × 9/100
∴ n = (81 × 100)/(450 ×9)
= 2 years

৬,৬৪১.

The product of two positive numbers is 11520 and their quotient is 9/5. Find the difference of two numbers.

ক
58
খ
64
গ
70
ঘ
75

ব্যাখ্যা

Question: The product of two positive numbers is 11520 and their quotient is 9/5. Find the difference of two numbers.

Solution:
Let, the numbers be x and y
∴ xy = 11520
and x/y = 9/5

ATQ,
xy × (x/y) = 11520 × (9/5)
⇒ x² = 2304 × 9
⇒ x = √(2304 × 9)
⇒ x = 48 × 3 = 144

From x/y = 9/5 we have,
y = (5 × 144)/9
∴ y = 80

∴ Required difference = (144 - 80) = 64

৬,৬৪২.

Three numbers are in the ratio of 4 : 5 : 6 and their L.C.M is 1800. What is their H.C.F?

ক
ক) 25
খ
খ) 30
গ
গ) 35
ঘ
ঘ) 40

ব্যাখ্যা

Question: Three numbers are in the ratio of 4 : 5 : 6 and their L.C.M is 1800. What is their H.C.F?

Solution:
Let the numbers be 4x, 5x and 6x
Then, their L.C.M = 60x

So, 60x = 1800
∴ x = 30

The numbers are (4 × 30), (5 × 30) and (6 × 30)

Hence, required H.C.F = 30

৬,৬৪৩.

If A can do 1/4 of a work in 3 days and B can do 1/6 of the same work in 4 days, how much will A get if both work together and are paid Tk. 180 in all?

ক
ক) Tk. 60
খ
খ) Tk. 120
গ
গ) Tk. 90
ঘ
ঘ) Tk. 180

ব্যাখ্যা

A in 3 days can do = 1/4 of a work
∴ In 1 day = 1/12 of a work
B in 4 days can do = 1/6 of a work
∴ In 1 day = 1/24 of a work
In terms of ratio, A : B = 1/12 : 1/24 = 2 : 1
∴ A gets = 180 × 2/(2+1) = Tk. 120

৬,৬৪৪.

x = 2y + 3 and y = - 2; Quantity A = x and Quantity B = - 1

ক
ক) Quantity A is greater
খ
খ) Quantity B is greater
গ
গ) The two quantities are equal
ঘ
ঘ) The relationship cannot be determined from the information given
ঙ
ঙ) None of these

ব্যাখ্যা

Question: x = 2y + 3 and y = - 2; Quantity A = x and Quantity B = - 1

Solution:
Here,
y = - 2

∴ x = 2y + 3
= 2 ×(- 2) + 3
= - 4 + 3
= - 1

So,
A = x = -1
And B = - 1

∴ The two quantities are equal.

৬,৬৪৫.

A and B together can paint a wall in 3 days. A can do it alone in 5 days. How many days would it take B to do this job alone?

ক
0.2
খ
7.5
গ
5.0
ঘ
6.4
ঙ
None of these

ব্যাখ্যা

Question: A and B together can paint a wall in 3 days. A can do it alone in 5 days. How many days would it take B to do this job alone?

Solution:
A's 1 day's work 1/5
A and B's 1 day's work 1/3

∴ B's 1 day's work = 1/3 - 1/5 = (5 - 3)/15 = 2/15

2/15 part of job done by B in 1 day
∴ Full job done by B in (15/2) days = 7.5 days

৬,৬৪৬.

In how many ways can two men and five boys be seated in a linear arrangement so that all the men sit together?

ক
ক) 1220
খ
খ) 1440
গ
গ) 2420
ঘ
ঘ) 1660

ব্যাখ্যা

Men can be treated as one group.
So, five boys and 1 group of men can be arranged in 6! ways.
Two men can be arranged in 2! ways.
Total no. of cases = 6! × 2!
= 720 × 2
= 1440 ways

∴ Required no. of ways =1440

৬,৬৪৭.

A can complete a project in 20 days and B can complete the same project in 30 days. A and B start working on the project together and A quits 10 days before the project is expected to be completed. How many days in total will the project take to complete?

ক
ক) 16 days
খ
খ) 18 days
গ
গ) 23 days
ঘ
ঘ) 27 days

ব্যাখ্যা

A ও B একত্রে 1 দিনে করে = (1/20 + 1/30) = 1/12 অংশ
B একা 10 দিনে করে = 10/30 অংশ = 1/3 অংশ
অবশিষ্ট কাজ = (1 - 1/3) = 2/3 অংশ

এখন, A ও B একত্রে 1/12 অংশ করে 1 দিনে
A ও B একত্রে 1 অংশ করে = (12 × 1) দিনে
A ও B একত্রে 2/3 অংশ করে = (12 × 2) /3 = 8 দিনে
∴ মোট সময় লাগে = (10 + 8) দিন = 18 দিন

৬,৬৪৮.

The ratio of syrup and water in a mixture is 3 : 1, the percentage of water in this mixture is -

ক
ক) 75%
খ
খ) 20%
গ
গ) 25%
ঘ
ঘ) 80%

ব্যাখ্যা

Question: The ratio of syrup and water in a mixture is 3 : 1, the percentage of water in this mixture is -

Solution:
Percentage of water = (1/4) × 100% = 25%

৬,৬৪৯.

A trader mixes 26 kg of fertilizer at tk 20 per kg with 30 kg of fertilizer of other variety at tk 36 per kg and sells the mixture at tk 30 per kg. His profit percent is:

ক
4%
খ
5%
গ
6%
ঘ
8%

ব্যাখ্যা

Question: A trader mixes 26 kg of fertilizer at tk 20 per kg with 30 kg of fertilizer of other variety at tk 36 per kg and sells the mixture at tk 30 per kg. His profit percent is:

Solution:
Cost price of 56 kg fertilizer = (26 × 20) + (30 × 36) tk
= 520 + 1080 tk
= 1600 tk

Selling price of 56 kg rice = 56 × 30 tk
= 1680 tk

Profit = 1680 - 1600 tk = 80 tk
∴ Gain = (80/1600) × 100
= 5%

৬,৬৫০.

Pipe A usually fills a tank in 2 hours. On account of a leak at the bottom of the tank, it takes pipe A 30 more minutes to fill the tank. How long will the leak take to empty a full tank if pipe A is shut?

ক
ক) 5 hr
খ
খ) `8 hr
গ
গ) 10 hr
ঘ
ঘ) 12 hr

ব্যাখ্যা

Question: Pipe A usually fills a tank in 2 hours. On account of a leak at the bottom of the tank, it takes pipe A 30 more minutes to fill the tank. How long will the leak take to empty a full tank if pipe A is shut?

Solution:
Pipe A ২ ঘণ্টা বা ১২০ মিনিটে পূর্ণ করে ১ অংশ
১ মিনিটে পূর্ণ করে ১/১২০ অংশ

পাইপের ছিদ্র দিয়ে পানি বের হওয়ার ফলে,
১৫০ মিনিটে পূর্ণ হয় ১ অংশ
১ মিনিটে পূর্ণ হয় ১/১৫০ অংশ

১ মিনিটে খালি করে (১/১২০) - (১/১৫০) অংশ
= (৫ - ৪)/৬০০
= ১/৬০০ অংশ

সম্পূর্ণ অংশ খালি করতে সময় লাগে = ১/১/৬০০ মিনিট
= ৬০০ মিনিট
= ৬০০/৬০ ঘণ্টা
= ১০ ঘণ্টা

৬,৬৫১.

The volume of a right circular cylinder, 14 cm in height is equal to that of a cube whose edge is 11 cm. What is the radius of the base of the cylinder?

ক
ক) 3.5 cm
খ
খ) 4 cm
গ
গ) 5.5 cm
ঘ
ঘ) 6 cm

ব্যাখ্যা

Question: The volume of a right circular cylinder, 14 cm in height is equal to that of a cube whose edge is 11 cm. What is the radius of the base of the cylinder?

Solution:
Volume of the cylinder = Volume of the cube = (11)³ cm³ = 1331 cm³
Let the radius of the base be r cm
Then,
πr²h = (22/7) × r² × 14 = 1331
⇒ r² = 1331/44
⇒ r² = 121/4
⇒ r = 11/2 = 5.5 cm

৬,৬৫২.

In a triangle the lengths of two sides are 5 and 9 and the length of the third side is represented by x. Which statement is always true?

ক
x > 5
খ
x < 9
গ
5 ≤ x ≤ 9
ঘ
4 < x < 14
ঙ
None of these

ব্যাখ্যা

Question: In a triangle the lengths of two sides are 5 and 9 and the length of the third side is represented by x. Which statement is always true?

Solution:
In a triangle,
The sum of the lengths of any two sides must be greater than the length of the third side,
- And the difference between the lengths of any two sides must be less than the length of the third side.
This is known as the Triangle Inequality Theorem.

Given that two sides of the triangle have lengths of 5 and 9, and the third side is represented by x, the theorem gives us the following inequalities:
x + 5 > 9, which simplifies to x > 4
x + 9 > 5, which is always true since x > - 4
5 + 9 > x, which simplifies to x < 14

Thus, the third side x must satisfy the condition 4 < x < 14.

৬,৬৫৩.

A man bought some sugar and salt for Tk. 440. The ratio of the weights of sugar and salt is 5 : 2, and the price per unit weight of sugar and salt is in the ratio 6 : 7. What is the price of the total sugar?

ক
Tk. 280
খ
Tk. 300
গ
Tk. 320
ঘ
Tk. 340

ব্যাখ্যা

Question: A man bought some sugar and salt for Tk. 440. The ratio of the weights of sugar and salt is 5 : 2, and the price per unit weight of sugar and salt is in the ratio 6 : 7. What is the price of the total sugar?

Solution:
Given,
Weight ratio (Sugar : Salt) = 5 : 2
Price ratio per unit weight (Sugar : Salt) = 6 : 7

Total price ratio of sugar and salt = (Weight × Price per unit)
= 5 × 6 : 2 × 7
= 30 : 14
= 15 : 7

Total units = 15 + 7 = 22 units

Accordint to the question,
22 units = Tk. 440
⇒ 1 unit = 440/22
∴ 1 unit= Tk. 20

∴ Price of total sugar = 15 × 20 = Tk. 300

৬,৬৫৪.

400 students took a mock test: 60% of the boys and 80% of the girls cleared the cut off in the test. If the total percentage of students clearing the cut off in the test is 65%, then how many girls appeared in the test?

ক
100
খ
120
গ
150
ঘ
300

ব্যাখ্যা

Question: 400 students took a mock test: 60% of the boys and 80% of the girls cleared the cut off in the test. If the total percentage of students clearing the cut off in the test is 65%, then how many girls appeared in the test?

Solution:
400 students took a mock test:
60% of the boys and 80% of the girls cleared the cut off in the test.
the total percentage of students clearing the cut off in the test is 65%

Let's assume there were b boys and g girls in the test.

According To question:
the total number of students who cleared the cut off = (60/100)b + (80/100)g = 0.65 × 400 .........(1)
the total number of students who took the test = b + g = 400
⇒ b = 400 - g

Substituting this value of b in the first equation: (60/100)(400 - g) + (80/100)g = 0.65 × 400
⇒ (24000 - 60g)/100 + 80g/100 = 260
⇒ 24000 - 60g + 80g = 26000
⇒ 20g = 2000
∴ g = 100
So, 100 girls appeared in the test.

৬,৬৫৫.

If the area of a small pizza is 154 inch², what size pizza box would best fit the small pizza?

ক
ক) 16 inch
খ
খ) 14 inch
গ
গ) 12 inch
ঘ
ঘ) 10 inch

ব্যাখ্যা

প্রশ্ন : If the area of a small pizza is 154 inch², what size pizza box would best fit the small pizza?

সমাধান :

Area of a pizza (circle) = πr² = 154
Or, r² =154/π = 154/(22/7) = 49
Or, r = 7
So, the size is 2r = 2 × 7 = 14 inch

৬,৬৫৬.

If x + 1/x = 2, then what is the value of x¹⁰ + x¹⁰⁰?

ক
1
খ
0
গ
2
ঘ
100000

ব্যাখ্যা

Question: If x + 1/x = 2, then what is the value of x¹⁰ + x¹⁰⁰?

Solution:
দেওয়া আছে,
x + 1/x = 2
⇒ x² + 1 = 2x [উভয় পক্ষকে x দ্বারা গুণ]
⇒ x² - 2x + 1 = 0
⇒ (x - 1)² = 0
⇒ x - 1 = 0
⇒ x = 1

এখন,
x¹⁰ + x¹⁰⁰
= (1)¹⁰ + (1)¹⁰⁰
= 1 + 1
= 2

সুতরাং, নির্ণেয় মান হলো 2।

৬,৬৫৭.

Ten years ago, the age of mother was three times the age of her son. After ten years, mother’s age will be twice that of his son. Find the ratio of their present ages.

ক
8 : 3
খ
7 : 3
গ
9 : 4
ঘ
9 : 2

ব্যাখ্যা

Question: Ten years ago, the age of mother was three times the age of her son. After ten years, mother’s age will be twice that of his son. Find the ratio of their present ages.

Solution:
Let, the age of son was x and mother's age was 3x.
At present,
Mother's age is (3x + 10)
and son’s age is (x + 10)

After ten years,
Mother's age will be (3x + 10) +10
and son’s age will be (x + 10) + 10.

ATQ,
(3x + 10) +10 = 2 [(x + 10) + 10]
⇒ (3x + 20) = 2[x + 20]
⇒ 3x + 20 = 2x + 40
⇒ 3x - 2x = 40 - 20
⇒ x = 20

∴ (3x + 10) : (x + 10) = 70 : 30 = 7 : 3

৬,৬৫৮.

If Rafi and Suman together can complete a job in 12 days. If Rafi alone can finish the job in 20 days, in how many days can Suman alone complete the work?

ক
25 days
খ
60 days
গ
45 days
ঘ
30 days

ব্যাখ্যা

Question: If Rafi and Suman together can complete a job in 12 days. If Rafi alone can finish the job in 20 days, in how many days can Suman alone complete the work?

Solution:
Rafi and Suman together can complete a piece of work in 12 days
Rafi and Suman together can complete in 1 day = 1/12

Rafi alone in 20 days
Rafi alone in one day = 1/20

∴ Suman complete in one day = (1/12) - (1/20)
= (5 - 3)/60
= 2/60
= 1/30

∴ Suman can complete the work in 30 days

৬,৬৫৯.

The diameters of two cones are equal, If their slant heights be in the ratio of 5 : 7 then find the ratio of their Curved surface areas.

ক
3 : 7
খ
5 : 2
গ
5 : 7
ঘ
5 : 1

ব্যাখ্যা

Question: The diameters of two cones are equal, If their slant heights be in the ratio of 5 : 7 then find the ratio of their Curved surface areas.

SolutioN:
Given,
l₁/l₂ = 5/7

Now,
curved surface area of first cone
= πrl₁

curved surface area of second cone
= πrl₂

Therefore, Ratio
= πrl₁ / πrl₂
= l₁ / l₂
= 5 : 7

৬,৬৬০.

After paying 10% tax on all income over 2000. Rahman had a net income of Tk 20000. Rahman's income before tax was?

ক
Tk 25000
খ
Tk 22450
গ
Tk 22000
ঘ
Tk 21990

ব্যাখ্যা

Question: After paying 10% tax on all income over 2000. Rahman had a net income of Tk 20000. Rahman's income before tax was?

Solution:
Let, Rahim’s income was = x
Taxable Income = x - 2000
Tax paid = 10% × (x - 2000)

According to the question,
⇒ x - 10% × (x - 2000) = 20000
⇒ x - {10 × (x - 2000)}/100 = 20000
⇒ x - (x - 2000)/10 = 20000
⇒ 10x - (x - 2000) = 200000
⇒ 10x - x + 2000 = 200000
⇒ 9x = 200000 - 2000
⇒ x = 198000/9
⇒ x = 22000

∴ Rahim's income before tax was Tk 22,000

৬,৬৬১.

Find the value of sin²130° + cos²130°

ক
2
খ
1/2
গ
1/√2
ঘ
1

ব্যাখ্যা

Question: Find the value of sin²130° + cos²130°

Solution:
We know, sin²θ + cos²θ = 1

∴ sin²130° + cos²130° = 1

৬,৬৬২.

A right triangle has sides in the ratio of 5 : 12 : 13. What is the measure of the smallest angle in the triangle, in degree?

ক
13.34
খ
22.62
গ
34.14
ঘ
42.71

ব্যাখ্যা

Question: A right triangle has sides in the ratio of 5 : 12 : 13. What is the measure of the smallest angle in the triangle, in degree?

Solution:

We know that,
sinθ = AB/AC
⇒ sin∠ACB = 5/13
⇒ θ = sin^-1(5/13)
∴ θ = 22.62°

৬,৬৬৩.

The sum of four consecutive even numbers is 180. What is the sum of the set of next four consecutive even numbers?

ক
ক) 214
খ
খ) 210
গ
গ) 212
ঘ
ঘ) 204

ব্যাখ্যা

Question: The sum of four consecutive even numbers is 180. What is the sum of the set of next four consecutive even numbers?

Solution:
Let the four consecutive even numbers be a, a + 2, a + 4 and a + 6.

ATQ,
a + a + 2 + a + 4 + a + 6 = 180
Or, 4a + 12 = 180
Or, 4a = 180 - 12
Or, 4a = 168
∴ a = 42

So, these numbers are 42, 44, 46, 48.
Sum of the next four consecutive even numbers = (50 + 52 + 54 + 56)
= 212

৬,৬৬৪.

A merchant has 100 kg of sugar, part of which he sells at 8% profit and the rest at 18% profit. He gains 14% on the whole. The quantity sold at 18% profit is-

ক
60 kg
খ
50 kg
গ
75 kg
ঘ
80 kg

ব্যাখ্যা

Question: A merchant has 100 kg of sugar, part of which he sells at 8% profit and the rest at 18% profit. He gains 14% on the whole. The quantity sold at 18% profit is-

Solution:
By the rule of alligation, we have:

⇒ Quantity of cheaper : Quantity of Dearer = (CP of of Dearer - Mean Price) : (Mean Price - CP of Cheaper)
⇒ Quantity of cheaper : Quantity of Dearer = (18-14) : (14-8)
= 4 : 6
= 2 : 3

∴ Quantity of 2nd kind = (3/5 × 100) kg
= 60 kg

৬,৬৬৫.

4 men and 6 women can complete a work in 8 days, while 3 men and 7 women can complete it in 10 days. In how many days will 10 women complete it?

ক
40
খ
35
গ
45
ঘ
50
ঙ
None of these

ব্যাখ্যা

Question: 4 men and 6 women can complete a work in 8 days, while 3 men and 7 women can complete it in 10 days. In how many days will 10 women complete it?

Solution:
Let,
1 man's 1 day's work = x and 1 woman's 1 day's work = y.
Then,
4x + 6y = 1/8 and 3x + 7y = 1/10
Solving the two equations, we get: x = 11/400 , y = 1/400

∴ 1 woman's 1 day's work = 1/400
10 women's 1 day's work = (1/400) × 10 = 1/40
Hence, 10 women will complete the work in 40 days.

৬,৬৬৬.

If 3x - 7y = 0 and x + 2y = 13 then y is –

ক
ক) 2
খ
খ) 3
গ
গ) 4
ঘ
ঘ) 7

ব্যাখ্যা

Given, 3x - 7y = 0 .... (i)
⇒ 3x = 7y
and x + 2y = 13 .... (ii)
(ii)×3 ⇔ 3x + 6y = 39
⇒ 7y + 6y = 39
⇒ 13y = 39
∴ y = 3

৬,৬৬৭.

What is the least number which when doubled will be exactly divisible by 12, 14, 18 and 22 ?

ক
1386
খ
1216
গ
1286
ঘ
1436

ব্যাখ্যা

LCM of 12, 14, 18 and 22 = 2772
Hence the least number which will be exactly divisible by 12, 14, 18, and 22 = 2772
2772/2 = 1386
1386 is the number which when doubled, we get 2772
Hence, 1386 is the least number which when doubled will be exactly divisible by 12, 14, 18, and 22.

৬,৬৬৮.

An amount of Tk. 10000 becomes Tk. 14641 in 2 years if the interest is compounded half-yearly. What is the rate of compound interest per annum?

ক
10%
খ
15%
গ
18%
ঘ
20%

ব্যাখ্যা

Question: An amount of Tk. 10000 becomes Tk. 14641 in 2 years if the interest is compounded half-yearly. What is the rate of compound interest per annum?

Solution:
Let
the rate be r% per annum

Now
10000 × {1 + r/(2 × 100)}^{2 × 2} = 14641
⇒ {1 + r/(2 × 100)}⁴ = 14641/10000
⇒ (1 + r/200)⁴ = (11/10)⁴
⇒ 1 + r/200 = 11/10
⇒ r/200 = (11/10) - 1
⇒ r/200 = (11 - 10)/10
⇒ r/200 = 1/10
⇒ r = 200/10
∴ r = 20

The rate be 20% per annum.

৬,৬৬৯.

A person's present age is two-fifth of the age of his mother. After 8 years, he will be one half of the age of his mother. What is the present age of the mother?

ক
30
খ
35
গ
40
ঘ
50

ব্যাখ্যা

Question: A person's present age is two-fifth of the age of his mother. After 8 years, he will be one half of the age of his mother. What is the present age of the mother?

Solution:
Let the age of the person’s mother be ‘x years
Age of the person = (2x/5)

After 8 years, age of the person’s mother = ‘x + 8’ years
Age of the person after 8 years = (2x/5) + 8 years

According to question:
(2x/5) + 8 = (1/2)(x + 8)
⇒ (2x/5) + 8 = (x/2) + 4
⇒ (x/2) - (2x/5) = 8 - 4
⇒ (5x - 4x)/10 = 4
⇒ x/10 = 4
∴ x = 40

So, the present age of his mother is 40 years

৬,৬৭০.

10 pens costs Tk. 100 each. If half of the pens are sold at 10% loss then find at what price remaining each pen should be sold for making no loss and no profit.

ক
Tk. 90
খ
Tk. 130
গ
Tk. 100
ঘ
Tk. 120
ঙ
None of the above

ব্যাখ্যা

Question: 10 pens costs Tk. 100 each. If half of the pens are sold at 10% loss then find at what price remaining each pen should be sold for making no loss and no profit.

Solution:
The total cost price of 10 pens = 10 × 100 = Tk. 1000

Selling price of 1 pen = 100 - (100 × 10%) = Tk. 90
Hence, the selling price of 5 pens = Tk. 450

Now, the selling price of the remaining 5 pens = 1000 - 450 = Tk. 550

Hence, the selling price of 1 pen = Tk. 110

৬,৬৭১.

Two ships are sailing in the sea on the two sides of a lighthouse. The angles of elevation of the top of the lighthouse as observed from the two ships are 30° and 45° respectively. If the lighthouse is 100m high, the distance between the two ships is :

ক
ক) 173m
খ
খ) 200m
গ
গ) 273m
ঘ
ঘ) 300m

ব্যাখ্যা

Question: Two ships are sailing in the sea on the two sides of a lighthouse. The angles of elevation of the top of the lighthouse as observed from the two ships are 30° and 45° respectively. If the lighthouse is 100m high, the distance between the two ships is :

Solution:

ধরি
বাতিঘরের উচ্চতা AB = 100 মিটার
C ও D হলো জাহাজের অবস্থান

ΔABC এ
tan∠ACB = AB/BC
tan 30° = 100/BC
1/√3 = 100/BC
BC = 100√3

ΔABC এ
tan∠ADB = AB/BD
tan45° = 100/BD
1 = 100/BD
BD = 100

CD = 100√3 + 100
= 173.205 + 100
= 273.205
≈ 273

৬,৬৭২.

The area of a parallelogram is 72 square centimetre and its altitude is twice the corresponding base. What is the length of the base?

ক
9 c.m.
খ
4 c.m.
গ
5 c.m.
ঘ
6 c.m.
ঙ
15 c.m.

ব্যাখ্যা

Let, base = x
Then, height = 2x
Area = base × height
= x × 2x
= 2x²
Area is given as 72 cm²
2x² = 72 cm²
⇒ x² = 36 cm²
⇒ x = 6 cm
Hence, the length of the base is 6 cm.

৬,৬৭৩.

A takes three times as much time as B or twice as much time as C to finish a piece of work. Working together, they can finish the work in 3 days. B can do the work alone in how many days?

ক
14 days
খ
8 days
গ
6 days
ঘ
10 days

ব্যাখ্যা

Question: A takes three times as much time as B or twice as much time as C to finish a piece of work. Working together, they can finish the work in 3 days. B can do the work alone in how many days?

Solution:
ধরা যাক,
A, B ও C যথাক্রমে কাজটি শেষ করতে সময় লাগে x, x/3, এবং x/2 দিন।

তারা একসাথে ৩ দিনে কাজ শেষ করে।
অর্থাৎ তাদের একদিনের কাজ হলো 1/3।

∴ A + B + C এর একদিনের কাজ = (1/x) + (3/x) + (2/x)
= (1 + 3 + 2)/x
= 6/x

এখন,
6/x = 1/3
∴ x = 18

∴ B একা কাজ শেষ করতে সময় নেবে = 18/3 = 6 দিন।

৬,৬৭৪.

Ali can type 60 pages in 20 minutes. Sara can type 12 pages in 12 minutes. Working together, how many pages can they type in 30 minutes?

ক
100 pages
খ
220 pages
গ
120 pages
ঘ
130 pages
ঙ
110 pages

ব্যাখ্যা

Question: Ali can type 60 pages in 20 minutes. Sara can type 12 pages in 12 minutes. Working together, how many pages can they type in 30 minutes?

Solution:
Ali can type in 1 min = 60/20 = 3 pages
Sara can type in 1 min = 12/12 = 1 page

∴ Working together they can type in 1 min = (3 + 1) pages = 4 pages

∴ They can type in 30 min = 4 × 30 pages = 120 pages

৬,৬৭৫.

A is 50 years old and B is 35 years old. How many years ago was the ratio of their ages 3 : 2?

ক
5 Years
খ
3 Years
গ
6 Years
ঘ
7 Years

ব্যাখ্যা

Question: A is 50 years old and B is 35 years old. How many years ago was the ratio of their ages 3 : 2?

Solution:
Let,
'x' years ago, the ratio of their ages was 3 : 2

ATQ,
(50 - x) : (35 - x) = 3 : 2
⇒ (50 - x)/(35 - x) = 3/2
⇒ 105 - 3x = 100 - 2x
⇒ 3x - 2x = 105 - 100
∴ x = 5

৬,৬৭৬.

ক
0
খ
1
গ
e
ঘ
1/2

ব্যাখ্যা

প্রশ্ন:

সমাধান:

৬,৬৭৭.

If (2a + b)/(a + 4b) = 3 then find the value of (a + b)/(a + 2b)?

ক
ক) 5/9
খ
খ) 2/7
গ
গ) 10/9
ঘ
ঘ) 10/7

ব্যাখ্যা

Given,
(2a + b)/(a + 4b) = 3
2a + b = 3a + 12b
-a = 11b
a = -11b

∴ (a + b)/(a + 2b)
= (-11b + b)/(-11b + 2b)
= -10b/-9b
= 10/9.

৬,৬৭৮.

Find the value of cos(5π/4)

ক
1/2
খ
- √3/2
গ
- 1/2
ঘ
- 1/√2

ব্যাখ্যা

Question: Find the value of cos(5π/4)

Solution:
cos(5π/4)
= cos(π + π/4) [যেহেতু (π + π/4) তৃতীয় চতুর্ভাগে পড়ে এবং তৃতীয় চতুর্ভাগে cos ঋণাত্মক, তাই cos(π + θ) = -cosθ]
= - cos(π/4)
= - cos(45°)
= - 1/√2

৬,৬৭৯.

A person sells an item for Tk. 7,600 and incurs a 5% loss. At what price should the item be sold to gain a 15% profit?

ক
Tk. 9200
খ
Tk. 8740
গ
Tk. 9600
ঘ
Tk. 9100

ব্যাখ্যা

Question: A person sells an item for Tk. 7,600 and incurs a 5% loss. At what price should the item be sold to gain a 15% profit?

Solution:
5% ক্ষতিতে বিক্রয়মূল্য = 100 - 5 = 95 টাকা।

বিক্রয়মূল্য 95 টাকা হলে ক্রয়মূল্য 100 টাকা।
∴ বিক্রয়মূল্য 1 টাকা হলে ক্রয়মূল্য 100/95 টাকা।
∴ বিক্রয়মূল্য 7,600 টাকা হলে ক্রয়মূল্য (100/95) × 7,600 টাকা
= 8,000 টাকা।

আবার, 15% লাভে বিক্রয়মূল্য = 100 + 15 = 115 টাকা।

ক্রয়মূল্য 100 টাকা হলে বিক্রয়মূল্য 115 টাকা।
∴ ক্রয়মূল্য 1 টাকা হলে বিক্রয়মূল্য 115/100 টাকা।
∴ ক্রয়মূল্য 8,000 টাকা হলে বিক্রয়মূল্য (115/100) × 8,000 টাকা
= 9,200 টাকা।

∴ জিনিসটি 9,200 টাকায় বিক্রি করতে হবে।

৬,৬৮০.

When a certain number is divided by 7, the remainder is 0, if the remainder is not 0 when the number is divided by 14, then the remainder must be -

ক
8
খ
5
গ
2
ঘ
3
ঙ
7

ব্যাখ্যা

Numbers can be divided by 7 are: 7, 14, 21, 28, 35
Among them, 14, 28 are divided by 14 with no remainder
But when 7, 21, 35 these numbers are divided by 14, the remainder is 7

৬,৬৮১.

Before anybody could notice, Rifat took one-fourth of the chocolates from a box. Later, his four cousins arrived, and the remaining chocolates were distributed equally among the five of them. Rifat received a total of 40 chocolates. How many did each of his cousins receive?

ক
12
খ
15
গ
9
ঘ
10
ঙ
18

ব্যাখ্যা

Question: Before anybody could notice, Rifat took one-fourth of the chocolates from a box. Later, his four cousins arrived, and the remaining chocolates were distributed equally among the five of them. Rifat received a total of 40 chocolates. How many did each of his cousins receive?

Solution:
ধরি, মোট চকলেট ছিল x টি।
রিফাত প্রথমবার নিয়েছিল = x এর 1/4 অংশ = x/4 টি
বাকী চকলেট = x - x/4 = (4x - x)/4 = 3x/4 টি
এই বাকী চকলেট রিফাত এবং তার চারজন কাজিনের মধ্যে, অর্থাৎ মোট পাঁচজনের মধ্যে সমানভাবে ভাগ করা হয়েছিল।
∴ প্রত্যেকে পেয়েছে = (3x/4) × (1/5) = 3x/20 টি

প্রশ্ন অনুযায়ী, রিফাত মোট 40 টি চকলেট পেয়েছে।
∴ রিফাতের মোট চকলেট = (প্রথমবারের অংশ) + (সমানভাবে ভাগের অংশ)
⇒ (x/4) + (3x/20) = 40
⇒ (5x + 3x)/20 = 40
⇒ 8x/20 = 40
⇒ 2x/5 = 40
⇒ x = 40 × (5/2)
⇒ x = 100

সুতরাং, বাক্সে মোট চকলেট ছিল 100 টি।
প্রত্যেক কাজিন পেয়েছে = 3x/20 = (3 × 100)/20 = 300/20 = 15 টি
অতএব, তার প্রত্যেক কাজিন 15টি করে চকলেট পেয়েছিল।

৬,৬৮২.

If 18 pumps can raise 2170 tonnes of water in 10 days, working 7 hours a day; in how many days will 16 pumps raise 1736 tonnes of water, working 9 hours a day?

ক
ক) 6
খ
খ) 7
গ
গ) 8
ঘ
ঘ) 9

ব্যাখ্যা

Let,
The required number of days be x.
Less pumps, More days[Indirect proportion]
Less weight, Less days [Direct proportion]
More hours/days, Less days [Indirect proportion]
{Pumps(16 : 18) , Weight (2170 : 1736), Hours/Day(9 :7)} :: 10 : x
∴ (16 × 2170 × 9 × x) = (18 × 1736 × 7 × 10)
⇒ x = (18 × 1736 × 7 × 10)/(16 × 2170 × 9)
= 7.

৬,৬৮৩.

The average speed of a bus is half the speed of a train. The train covers 1000 km in 20 hours. How much distance will the bus cover in 48 minutes?

ক
15 km
খ
20 km
গ
24 km
ঘ
30 km

ব্যাখ্যা

Question: The average speed of a bus is half the speed of a train. The train covers 1000 km in 20 hours. How much distance will the bus cover in 48 minutes?

Solution:
ট্রেনের গতিবেগ = অতিক্রান্ত দূরত্ব/সময়
= 1000 কিমি/20 ঘন্টা
= 50 কিমি/ঘন্টা

এখন, বাসের গতিবেগ ট্রেনের গতিবেগের অর্ধেক।
∴ বাসের গতিবেগ = ট্রেনের গতিবেগ × 1/2
= 50 কিমি/ঘন্টা × 1/2
= 25 কিমি/ঘন্টা

∴ বাসের অতিক্রান্ত দূরত্ব = বাসের গতিবেগ × সময়
= 25 কিমি/ঘন্টা × 48 মিনিট
= 25 কিমি/ঘন্টা × (48/60) ঘন্টা
= 20 কিমি

সুতরাং, বাসটি 48 মিনিটে 20 কিমি দূরত্ব অতিক্রম করবে।

৬,৬৮৪.

The time taken by a swimmer to swim upstream is 4 hours more than the time he takes to swim downstream. He swims at a speed of 10 km/hr in still water.The stream is flowing gently at 2 km/hr. What is the swimming distance on one side?

ক
20 km
খ
72 km
গ
80 km
ঘ
96 km

ব্যাখ্যা

We know,
Man's/Boat's Speed = X
Stream/Current/River Speed = Y

∴ Downstream speed = X + Y
Upstream speed = X - Y

X+Y = 10 + 2 = 12 km/hr and X-Y = 10 - 2 = 8 km/hr
Let Time be T hours for downstream
Distance is same
∴ D = D
∴ 12 × T = 8 × (T + 4)
∴ T = 8 hours = Time for downstream
Distance = 12km/hr × 8 hours = 96 km

৬,৬৮৫.

12.1212 + 17.0005 - 9.1102 = ?

ক
20.0015
খ
20.0105
গ
20.0115
ঘ
20.1015

ব্যাখ্যা

Question: 12.1212 + 17.0005 - 9.1102 = ?

Solution:
12.1212 + 17.0005 - 9.1102
= 29.1217 - 9.1102
= 20.0115

৬,৬৮৬.

If 2a = 3b = 4c = 72, then what is the average (arithmetic mean) of a, b and c?

ক
ক) 39
খ
খ) 26
গ
গ) 24
ঘ
ঘ) 18

ব্যাখ্যা

Question: If 2a = 3b = 4c = 72, then what is the average (arithmetic mean) of a, b and c?

Solution:

2a = 72

বা, a = 36

একইভাবে, b = 24 এবং c = 18

সুতরাং, গাণিতিক গড় = (36+24+18) / 3 = 26

৬,৬৮৭.

A person makes a profit of 20% on 25% of the quantity, and a loss of 20% on the rest. What is the percentage gain or loss on the whole?

ক
12.5%
খ
7.5%
গ
9%
ঘ
10%
ঙ
8%

ব্যাখ্যা

Let Cost Price = 100
Profit = 25 x 20% = 5
Loss = 75 x 20% = 15
Net loss = 10
Net Loss as % = 10/100 x 100 = 10%

৬,৬৮৮.

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

ক
ক) 8
খ
খ) 15
গ
গ) 27
ঘ
ঘ) 35

ব্যাখ্যা

Students passed in English = 80%
Students passed in Math's = 85%
Students passed in both subjects = 73%
Then, number of students passed in at least one subject
= (80+85)-73
= 92%.
Thus, students failed in both subjects = 100-92 = 8%

৬,৬৮৯.

If 2^a = 3^b and 2^{a + 2} - 3^{b + 1} = √3, then find the value of b.

ক
1
খ
1/2
গ
3
ঘ
√3

ব্যাখ্যা

Question: If 2^a = 3^b and 2^{a + 2} - 3^{b + 1} = √3, then find the value of b.

Solution:
Given,
2^{a + 2} - 3^{b + 1} = √3
⇒ (2^a × 2²) - (3^b × 3¹) = √3
⇒ (3^b × 4) - (3^b × 3) = √3
⇒ 3^b(4 - 3) = √3
⇒ 3^b = 3^1/2
∴ b = 1/2

৬,৬৯০.

If x = 0.5 and y = 0.2, then the value of √0.8 × (4y)^x is equal to?

ক
ক) 0.64
খ
খ) 0.8
গ
গ) 0.064
ঘ
ঘ) 0.88

ব্যাখ্যা

Question: If x = 0.5 and y = 0.2, then the value of √0.8 × (4y)^x is equal to?

Solution:
x = 0.5
y = 0.2

√0.8 × (4y)^x
= √0.8 × (4 × 0.2)^0.5
= √0.8 × √0.8
= 0.8

৬,৬৯১.

If x and y are negative, then which of the following statements is always true?

ক
ক) xy is positive
খ
খ) (x + y) is positive
গ
গ) 2(x + y) is positive
ঘ
ঘ) None of the above

ব্যাখ্যা

Question: If x and y are negative, then which of the following statements is always true?

Solution:
If x < 0 and y < 0 then xy > 0.
So, whenever x and y are negative, then xy is positive.

Example: If x = - 1 and y = - 1 then xy = (- 1) × (- 1) = 1 > 0

৬,৬৯২.

Ricky purchased a shirt for Tk. 2000. The discount offered for the shirt is 10%. Find how many takas has he given to the cashier?

ক
Tk. 1900
খ
Tk. 2200
গ
Tk. 1600
ঘ
Tk. 1800

ব্যাখ্যা

Question: Ricky purchased a shirt for Tk. 2000. The discount offered for the shirt is 10%. Find how many takas has he given to the cashier?

Solution:
Here,
Principal amount, p = Tk. 2000
Discount rate, r = 10%

Total Discount = p × r
= 2000 × 10% = 200

Dress price after discount = Principal amount - Total Discount
= 2000 - 200 = 1800

Ricky has given Tk. 1800 to the cashier.

৬,৬৯৩.

If cosθ.cosec23° = 1, the value of θ is:

ক
37°
খ
47°
গ
57°
ঘ
67°

ব্যাখ্যা

Question: If cosθ.cosec23° = 1, the value of θ is:

Solution:
Here,
cosθ.cosec23° = 1
⇒ 1/cos θ = secθ
⇒ cosec23° = cosec(90° – θ)
⇒ 23° = 90° – θ
⇒ θ = 90° – 23° = 67°

Alternative way,
cosθ.cosec23° = 1
If cosA.cosecB = 1

Then, A + B = 90°
So, θ + 23° = 90°
∴ θ = 67°

৬,৬৯৪.

What is the value of the following expression?

ক
1
খ
5
গ
60
ঘ
1/60
ঙ
1/2

ব্যাখ্যা

Question: What is the value of the following expression?

Solution:

= log₆₀3 + log₆₀4 + log₆₀5 [∵ 1/log_ba = log_ab]
= log₆₀(3 × 4 × 5) [∵ log_b(m) + log_b(n) = log_b(m × n)]
= log₆₀60
= 1

৬,৬৯৫.

The total surface area of solid hemisphere of radius 14 cm, is -

ক
ক) 1650 cm²
খ
খ) 1728 cm²
গ
গ) 1848 cm²
ঘ
ঘ) 1935 cm²

ব্যাখ্যা

Question: The total surface area of solid hemisphere of radius 14 cm, is -

Solution:
Total surface area = 3πR²
= {3 × (22/7) × 14 × 14} cm²
= 1848 cm²

৬,৬৯৬.

The value of p, for which the equation x²+ (p - 3)x + p = 0 has real and equal roots is

ক
ক) 9
খ
খ) 3
গ
গ) 4
ঘ
ঘ) 0

ব্যাখ্যা

দেয়া আছে,
x² + (p - 3)x + p = 0

x² + (p - 3)x + p = 0 কে ax² + bx + c = 0 সমীকরণের সাথে তুলনা করে পাই
a = 1 , b = p - 3, c = p

সমীকরণের মূলদ্বয় বাস্তব ও সমান হলে
নিশ্চায়ক = 0 হবে

b² - 4ac = 0
(p - 3)² - 4 × 1 × p = 0
p² - 2 .p.3 + 9 - 4p = 0
p² - 6p + 9 - 4p = 0
p² - 10p + 9 = 0
p² - 9p - p + 9 = 0
p(p - 9) - 1(p - 9) = 0
(p - 9)(p - 1) = 0

∴ p = 1, 9

৬,৬৯৭.

ক
5
খ
9
গ
11/2
ঘ
7/2

ব্যাখ্যা

Question:

Solution:

৬,৬৯৮.

Hemal and Titu start walking towards each other at 4 AM at the speed of 5 kmph and 3 kmph respectively. They were initially 28 km apart. At what time do they meet?

ক
8 AM
খ
5 : 30 AM
গ
6 : 30 AM
ঘ
7 : 30 AM
ঙ
None of these

ব্যাখ্যা

Question: Hemal and Titu start walking towards each other at 4 AM at the speed of 5 kmph and 3 kmph respectively. They were initially 28 km apart. At what time do they meet?

Solution:
Let both will meet after t hours
Distance covered by Hemal in t hours at a speed of 5kmph = 5t
Distance covered by Titu in t hours at a speed of 3kmph = 3t
Total Distance = 28 km

According to question,
5t + 3t = 28
⇒ 8t = 28
⇒ t = 28/8 = 3.5 = 3hours 30mins

∴ Hemal and Titu started at 4 am so they will meet after 3 hours 30mins at 7 : 30 AM

৬,৬৯৯.

Akmal multiplied a number by 2/5 instead of 5/2. What is the percentage error in the calculation?

ক
54%
খ
80%
গ
44%
ঘ
84%

ব্যাখ্যা

Question: Akmal multiplied a number by 2/5 instead of 5/2. What is the percentage error in the calculation?

Solution:
Let the number be x

∴ Error = (5x/2) - (2x/5)
= (25x - 4x)/10
= 21x/10

∴ percentage Error = {(21x/10)/(5x/2) × 100}%
= 84%

৬,৭০০.

How many permutations of nine different digits may be made?

ক
9!
খ
8!
গ
120
ঘ
6!

ব্যাখ্যা

Question: How many permutations of nine different digits may be made?

Solution:
আমরা জানি,
n সংখ্যক ভিন্ন জিনিস বা অক্ষর থেকে সবগুলি নিয়ে বিন্যাস সংখ্যা (Permutation) হলো n!
এখানে মোট অক্ষর সংখ্যা, n = 9
∴ নির্ণেয় বিন্যাস সংখ্যা = 9!