Bank Math

PrepBank · পাতা ১২৮ / ১৬১ · ১২,৭০১–১২,৮০০ / ১৬,১২৪

১২,৭০১.

In a Bank, 30% of the applicants are from Rajshahi district and 20% of them from Rajshahi city. What percent of the total applicants are from Rajshahi city?

ক
ক) 4%
খ
খ) 6%
গ
গ) 8%
ঘ
ঘ) 10%

ব্যাখ্যা

Question: In a Bank, 30% of the applicants are from Rajshahi district and 20% of them from Rajshahi city. What percent of the total applicants are from Rajshahi city?

Solution:
ধরি,
মোট আবেদনকারী = ১০০ জন
রাজশাহী থেকে আছে = ১০০ এর ৩০% = ৩০ জন

রাজশাহী শহর থেকে আছে = ৩০ এর ২০% = ৬%

১২,৭০২.

Three pipes A, B, and C can fill the tank in 10 hours, 20 hours and 40 hours respectively. In the beginning all of them are opened simultaneously. After 2 hours, tap C is closed and A and B are kept running. After the 4th hour, tap B is also closed. The remaining work is done by tap A alone. What is the percentage of the work done by tap A alone?

ক
ক) 30%
খ
খ) 35%
গ
গ) 40%
ঘ
ঘ) 55%

ব্যাখ্যা

Pipe A's work in % = 100/10 = 10%
Pipe B's work in % = 100/20 = 5%
Pipe C's work in % = 100/40 = 2.5%
All of them are opened for 2 hours + after 2 hours,
tap C is closed + After the 4th hour, tap B is also closed = 100

According to the question,
(10 + 5 + 2.5) × 2 + (10 + 5) × 2 + work by tap A alone = 100
⇒ 17.5 × 2 + 15 × 2 + work by tap A alone = 100
⇒ 35 + 30 + work by tap A alone = 100
⇒ work by tap A alone = 100 - 65 = 35%.

১২,৭০৩.

Two trains, each 100 meters long, are running on parallel tracks in opposite directions. If their speeds are 72 km/h and 54 km/h, how long will it take them to cross each other completely?

ক
5 seconds
খ
5.2 seconds
গ
5.5 seconds
ঘ
5.7 seconds

ব্যাখ্যা

Question: Two trains, each 100 meters long, are running on parallel tracks in opposite directions. If their speeds are 72 km/h and 54 km/h, how long will it take them to cross each other completely?

Solution:
Train 1 length = 100 meters
Train 2 length = 100 meters
Total distance = (100 + 100) = 200 meters

Speed of Train 1 = 72 km/h = 72 × (5/18) m/s = 20 m/s
Speed of Train 2 = 54 km/h = 54 × (5/18) m/s = 15 m/s

Relative speed = (20 + 15) = 35 m/s

Time = (200/35) = 5.71 seconds
= 5.7 seconds

১২,৭০৪.

If ABC and PQR are similar triangles in which ∠A = 46° and ∠B = 82°, then ∠C is:

ক
50°
খ
52°
গ
56°
ঘ
60°

ব্যাখ্যা

Question: If ABC and PQR are similar triangles in which ∠A = 46° and ∠B = 82°, then ∠C is:

Solution:
Since ΔABC and ΔPQR are similar triangles.
Then, ∠B = ∠Q = 82° [জ্যামিতির নিয়ম অনুযায়ী, দুটি ত্রিভুজ সদৃশ হলে তাদের অনুরূপ কোণগুলো সমান হয়।]

আমরা জানি, কোনো ত্রিভুজের তিনটি কোণের সমষ্টি 180°। অর্থাৎ, ΔABC-এর ক্ষেত্রে, ∠A + ∠B + ∠C = 180°।

Thus, in ∆ABC,
∠C= 180° - (∠A + ∠ B)
or, ∠C= 180° - (46° + 82°)
∴ ∠C = 52°

১২,৭০৫.

If α, β are the roots of the equation x² - 7x + 12 = 0, then α² + β² equals to:

ক
14
খ
19
গ
24
ঘ
25

ব্যাখ্যা

Question: If α, β are the roots of the equation x² - 7x + 12 = 0, then α² + β² equals to:

Solution:
x² - 7x + 12 = 0
⇒ x² - 3x - 4x + 12 = 0
⇒ x(x - 3) - 4(x - 3) = 0
⇒ (x - 3)(x - 4) = 0
∴ x = 3, 4

Hence, α = 3, β = 4
Hence, The value of α² + β² = 3² + 4² = 9 + 16 = 25

১২,৭০৬.

A box contains 7 red balls, 8 black balls, and 5 white balls. One ball is drawn at random. What is the probability that the ball drawn is neither red nor white?

ক
2/5
খ
3/8
গ
1/3
ঘ
2/7

ব্যাখ্যা

Question: A box contains 7 red balls, 8 black balls, and 5 white balls. One ball is drawn at random. What is the probability that the ball drawn is neither red nor white?

Solution:
Number of red balls = 7
Number of black balls = 8
Number of white balls = 5

∴ Total balls = 7 + 8 + 5 = 20

Let, event E = The ball drawn is neither red nor white, so it must be black.
∴ Number of favorable outcomes = 8

∴ P(E) = 8/20
= 2/5

১২,৭০৭.

50% of a number is 1200. What is 6% of that number?

ক
216
খ
242
গ
224
ঘ
144

ব্যাখ্যা

Question: 50% of a number is 1200. What is 6% of that number?

Solution:
Let the number be x

Then,
50% of x = 1200
⇒ 50x/100 = 1200
⇒ x = (1200 × 100)/50
⇒ x = 2400

∴ 6% of 2400
= 2400 × (6/100)
= 144

১২,৭০৮.

By selling a shirt for Tk. 300, a shopkeeper gains 20%. During a clearance sale, the shopkeeper allows a discount of 10% on the marked price. His gain percent during the sale is-

ক
10%
খ
9%
গ
8%
ঘ
5%

ব্যাখ্যা

Question: By selling a shirt for Tk. 300, a shopkeeper gains 20%. During a clearance sale, the shopkeeper allows a discount of 10% on the marked price. His gain percent during the sale is-

Solution:
Marked price = Tk. 300
C.P = (100/120) × 300 = Tk. 250

Sale price = 90% of Tk. 300
= 300 × (90/100)
= Tk. 270

Required gain% = {(20/250) × 100}% = 8%

১২,৭০৯.

A train moving at 54 km/hr crosses a bridge in 40 seconds. The length of train is 150 meters. Find the length of the bridge.

ক
425 m
খ
445 m
গ
475 m
ঘ
450 m

ব্যাখ্যা

Question: A train moving at 54 km/hr crosses a bridge in 40 seconds. The length of train is 150 meters. Find the length of the bridge.

Solution:
Speed of train= 54 km/h= 54 × (5/18) = 3 × 5 = 15m/second
Time taken = 40 seconds
Distance covered= speed × Time
Distance = 15× 40= 600 m

Distance covered = Length of Train + Length of Bridge
600 = 150 + Length of Bridge
Length of Bridge = 600 - 150 = 450 m

১২,৭১০.

The side length of a square inscribed in a circle is 2. What is the area of the circle shown at the right?

ক
ক) 2π
খ
খ) π
গ
গ) 2πr²
ঘ
ঘ) 2√2

ব্যাখ্যা

প্রশ্ন: The side length of a square inscribed in a circle is 2. What is the area of the circle shown at the right?

সমাধান:
বর্গক্ষেত্রের কর্ণ = √2 × এক বাহুর দৈর্ঘ্য
= 2√2 একক

চিত্রে,
বর্গক্ষেত্রটির কর্ণ = বৃত্তের ব্যস

বৃত্তের ব্যস = 2√2 একক
বৃত্তের ব্যাসার্ধ = 2√2/2 একক
= √2 একক

∴ বৃত্তের ক্ষেত্রফল = π ব্যাসার্ধ^২
= π (√2)²
= 2π বর্গএকক

১২,৭১১.

A does 20% less work than B. If A can complete a piece of work in 15/2 hours, then B can do it in-

ক
ক) 5 hours
খ
খ) 7 hours
গ
গ) 6 hours
ঘ
ঘ) 8 hours

ব্যাখ্যা

Question: A does 20% less work than B. If A can complete a piece of work in 15/2 hours , then B can do it in-

Solution:
Ratio of times taken by A and B = 100 : 80 = 5 : 4
Let
B takes x days to do the work

Here
5 : 4 = (15/2) : x
5/4 = (15/2)/x
5x = (15 × 4/2)
5x = 30
x = 6 hours

১২,৭১২.

If A : B : C = 2 : 3 : 4, then A/B : B/C : C/A = ?

ক
7 : 10 : 24
খ
8 : 9 : 24
গ
11 : 14 : 23
ঘ
8 : 13 : 28
ঙ
None

ব্যাখ্যা

Question: If A : B : C = 2 : 3 : 4, then A/B : B/C : C/A = ?

Solution:
A/B : B/C : C/A
= 2/3 : 3/4 : 4/2
= (2/3) × 12 : (3/4) × 12 : (4/2) × 12
= 8 : 9 : 24

১২,৭১৩.

What is the weight of 1 cubic meter of water?

ক
10 Kg
খ
100 Kg
গ
789 Kg
ঘ
1000 Kg
ঙ
10000 Kg

ব্যাখ্যা

Weight of 1 cubic meter of water is = 1000 Kg

১২,৭১৪.

What is the LCM if the HCF is 48 and the product of two numbers is 16128?

ক
ক) 112
খ
খ) 224
গ
গ) 336
ঘ
ঘ) 448

ব্যাখ্যা

L.C.M. (a,b) × H.C.F. (a,b) = product of a and b
Here, a and b are two numbers.
Therefore,
L.C.M.=16128/48
⇒ L.C.M. = 336.
Hence, the L.C.M. is 336.
=========================
লসাগু × গসাগু = সংখ্যা দুইটির গুণফল
লসাগু = সংখ্যা দুইটির গুণফল /গসাগু
লসাগু = ১৬১২৮/৪৮ = ৩৩৬

১২,৭১৫.

Father is aged four times more than his son. After 8 years, he would be two and a half times of son's age. What is the current age of the father?

ক
ক) 30 years
খ
খ) 22 years
গ
গ) 40 years
ঘ
ঘ) 32 years

ব্যাখ্যা

Question: Father is aged four times more than his son. After 8 years, he would be two and a half times of son's age. What is the current age of the father?

Solution:
Let,
The current age of son is x years
∴ The current age of the father is 4x years

ATQ,
4x + 8 = (5/2)(x + 8)
⇒ 8x + 16 = 5x + 40
⇒ 3x = 24
∴ x = 8

∴ The current age of the father is (4 × 8) years
= 32 years

১২,৭১৬.

A natural number n is such that 120 ≤ n ≤ 240. If HCF of n and 240 is 1, how many values of n are possible?

ক
48
খ
40
গ
32
ঘ
28
ঙ
24

ব্যাখ্যা

Question: A natural number n is such that 120 ≤ n ≤ 240. If HCF of n and 240 is 1, how many values of n are possible?

Solution:
240 = 2⁴× 3 × 5, Therefore n must not be multiple of 2, 3 or 5

Lets calculate the number of multiples of 2,3 & 5 and subtract from total
Number of multiples of, 2 from 120 to 240 (both inclusive) = [{240 - 120}/2] + 1 = 61
3 from 120 to 240 (both inclusive) =[{240 - 120}/]3 + 1 = 41
5 from 120 to 240 (both inclusive) =[{240 - 120}/5] + 1 = 25

Multiples common to 2 & 3, 2 & 5, 3 & 5 twice. The same needs to be subtracted
6 from 120 to 240 (both inclusive) = [{240 - 120}/6] + 1 = 21
10 from 120 to 240 (both inclusive) = [{240 - 120}/10] + 1 = 13
15 from 120 to 240 (both inclusive) = [{240 - 120}/15] + 1 = 9

We need to add the multiple of 2 & 3 & 5 once as the subtraction of cases 2 & 3, 3 & 5, 2 & 5 from 2, 3 & 5 removes all the cases of 2 & 3 & 5.
30 from 120 to 240 (both inclusive) = [{240 - 120}/30] + 1 = 5

Total number of multiples of 2, 3 or 5 from 120 to 240 = 61 + 41 + 25 - 21 - 13 - 9 + 5 = 89
Possible values of n = 121 - 89 = 32

১২,৭১৭.

The profit-principal of some principal is Tk. 8,000 in 3 years. If the profit is 3/5 part of the principal, find the rate of profit ?

ক
12.5%
খ
15%
গ
20%
ঘ
17.5%

ব্যাখ্যা

Question : The profit-principal of some principal is Tk. 8,000 in 3 years. If the profit is 3/5 part of the principal, find the rate of profit ?

Solution :
Given,
The profit-principal A = Tk. 8000
Time n = 3 years

Let,
the principal = P
So the profit = (3/5 × P)
= 3P/5

We know
Principal + profit = profit-principal
⇒ P + (3P/5) = 8000
⇒ 8P/5 = 8000
⇒ 8P = 8000 × 5
⇒ 8P = 40000
⇒ P = 40000/8
∴ P = 5000

So the principal P = Tk. 5000

∴ the profit I = (3/5 × 5000)
= Tk. 3000

We know,
r = (I × 100) ÷ Pn
= (3000 × 100) ÷ (5000 × 3 )
= 300000 ÷ 15000
= 20%

১২,৭১৮.

A cube has a total surface area of 150 square units. What is the volume of the cube?

ক
25
খ
64
গ
100
ঘ
125

ব্যাখ্যা

Question: A cube has a total surface area of 150 square units. What is the volume of the cube?

Solution:
Given that,
Total surface area of a cube, S = 150 square units.

We know,
Surface area of a cube, S = 6a²
⇒ 6a² = 150
⇒ a² = 150/6
⇒ a² = 25
⇒ a² = 5²
∴ a = 5

And we know,
Volume of a cube, V = a³
= 5³
= 125

So the volume of the cube is 125 cubic units.

১২,৭১৯.

ক
30°
খ
45°
গ
60°
ঘ
90°

ব্যাখ্যা

Question: If

Solution:

১২,৭২০.

ক
1/2
খ
1/3
গ
- 1/2
ঘ
2/3

ব্যাখ্যা

Question:

Solution:

১২,৭২১.

If x is an integer then solve (log₂ x)² - log₂ x⁴ - 32 = 0.

ক
125
খ
256
গ
375
ঘ
265
ঙ
None of these

ব্যাখ্যা

Question: If x is an integer then solve (log₂ x)² - log₂ x⁴ - 32 = 0.

Solution:
We have (log₂x)² - log₂x⁴ - 32 = 0.
⇒ (log₂x)² - 4log₂x - 32 = 0 ......(1)
Let log₂x = y
(i) ⇒ y² - 4y - 32 = 0
⇒ y² - 8y + 4y - 32 = 0
⇒ y(y - 8) + 4(y - 8) = 0
⇒ (y - 8)(y + 4) = 0
⇒ y = 8, - 4
⇒ log₂x = 8 or log₂x = - 4
⇒ x = 2⁸ = 256 or x = 2^{- 4} = 1/16
Since ‘x’ is an integer so x = 256.

১২,৭২২.

A 207m long train running at the speed of 62 km/hour crosses another train running in opposite direction at the speed of 38 km/hour in 18 seconds. The length of the other train -

ক
230
খ
267
গ
279
ঘ
293
ঙ
300

ব্যাখ্যা

Let the length of the other train be x meter.
We have,
t = ( s₁ + s₂ ) / ( v₁ + v₂ )
or, 18 = 18 ( 207 + x ) / { 5 ( 62 + 38 ) }
or, 207 + x = ( 100 × 5 × 18 ) 18
or, x = 500 - 207 = 293

১২,৭২৩.

A number when divided by a divisor leaves a remainder of 24. When twice the original number is divided by the same divisor, the remainder is 11. What is the value of the divisor?

ক
13
খ
59
গ
35
ঘ
37

ব্যাখ্যা

Question: A number when divided by a divisor leaves a remainder of 24. When twice the original number is divided by the same divisor, the remainder is 11. What is the value of the divisor?

Solution:
Let the original number be 'a'.
Let the divisor be 'd'.
Let the quotient of dividing 'a' by 'd' be 'x'.
and the remainder is 24.
∴ a = dx + 24
⇒ 2a = 2(dx + 24)
⇒ 2a = 2dx + 48

Twice the original number is divided by 'd' means 2a is divided by d.
When (2dx + 48) is divided by 'd' the remainder is 11.
2dx is divisible by 'd' and will therefore, not leave a remainder.
The remainder of 11 would be the remainder of dividing 48 by d.

The question essentially becomes "What number will leave a remainder of 11 when it divides 48?"
When 37 divides 48, the remainder is 11.

Hence, the divisor is 37.

১২,৭২৪.

A batsman makes a score of 87 runs in the 17th innings and thus increases his average by 3. What is his average after 17th innings?

ক
35
খ
40.5
গ
39
ঘ
42

ব্যাখ্যা

Let average of 17 innings = x
Total runs scored in 17 innings = 17x
Average of 16 innings = (x - 3)
Total runs scored in 16 innings = 16 (x -3)
Total runs scored in 16 innings + 87 = Total runs scored in 17 innings
⇒ 16(x - 3) + 87 = 17x
⇒ 16x - 48 + 87 = 17x
⇒ x = 39.

১২,৭২৫.

A two member committee comprising of one male and one female member is to be constituted out of six males and four females. Amongst the females, Ms. C refuses to be a member of the committee in which Mr. D is taken as the member. In how many different ways can the committee be constituted?

ক
21
খ
22
গ
23
ঘ
24
ঙ
None of the above

ব্যাখ্যা

Question: A two member committee comprising of one male and one female member is to be constituted out of six males and four females. Amongst the females, Ms. C refuses to be a member of the committee in which Mr. D is taken as the member. In how many different ways can the committee be constituted?

Solution:
Total number of committees without restriction
= ⁶C₁ × ⁴C₁= 6 × 4
= 24

Now,
the committee containing Mr. D and Ms. C is not allowed.
Number of such committees = 1

∴ Required number of committees
= 24 - 1
= 23

১২,৭২৬.

If a man goes 32 km downstream in 4 hours and returns against the stream in 16 hours, then the speed of the stream in km/h is -

ক
6 km/h
খ
5 km/h
গ
4 km/h
ঘ
3 km/h

ব্যাখ্যা

Question: If a man goes 32 km downstream in 4 hours and returns against the stream in 16 hours, then the speed of the stream in km/h is -

Solution:
Downstream speed = 32/4 = 8 km/h
Upstream speed = 32/16 = 2 km/h

∴ Speed of the stream = (8 - 2)/2 = 3 km/h

১২,৭২৭.

Find the number of divisors of 360.

ক
16
খ
28
গ
18
ঘ
24

ব্যাখ্যা

Question: Find the number of divisors of 360.

Solution:
Prime Factorization,
360 = 36 × 10 = (9 × 4) × (2 × 5) = 3² × 2² × 2 × 5 = 2³ × 3² × 5¹

Apply Divisor Formula:
(3 + 1)(2 + 1)(1 + 1) = 4 × 3 × 2 = 24

∴ 360 has 24 divisors.

১২,৭২৮.

1/log_ab × 1/log_bc × 1/log_ca =কত?

ক
ক) 0
খ
খ) 1
গ
গ) 2
ঘ
ঘ) abc

ব্যাখ্যা

প্রশ্ন: 1/log_ab × 1/log_bc × 1/log_ca = কত?

সমাধান:
1/log_ab × 1/log_bc × 1/log_ca
= log_ba × log_cb × log_ac
= (log_ba × log_cb) × log_ac
= log_ca × log_ac
= 1

১২,৭২৯.

If the number of quantities in group A is 10 and in group B is 8, and their individual average is 24 and 16 respectively, find the combined average of the two groups.

ক
18.22
খ
20.44
গ
16.22
ঘ
18.66

ব্যাখ্যা

Question: If the number of quantities in group A is 10 and in group B is 8, and their individual average is 24 and 16 respectively, find the combined average of the two groups.

Solution:
The sum of group A = 10 × 24 = 240
The sum of group B = 8 × 16 = 128

∴ Combined Average of the two groups = (240 + 128)/(10 + 8)
= 368/18
= 20.44

১২,৭৩০.

A 100 m long train is going at a speed of 108 km/h. It will cross a 140 m long tunnel in-

ক
15 sec
খ
12 sec
গ
8 sec
ঘ
18 sec

ব্যাখ্যা

Question: A 100 m long train is going at a speed of 108 km/h. It will cross a 140 m long tunnel in-

Solution:
Given,
Speed of the train = 108 km/h
= (108 × 5/18) m/sec
= 30 m/sec

∴ Total distance covered = (100 + 140) m
= 240 m

∴ Required time = (240 ÷ 30) sec
= 8 sec

১২,৭৩১.

An observer 2.8 m tall is 14√3 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
25m
খ
16.8 m
গ
21.8 m
ঘ
18.5 m

ব্যাখ্যা

Question: An observer 2.8 m tall is 14√3 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-

Solution:

Here, Height = AB
Now,
tan∠c = AE/CE
⇒ tan30° = AE/14√3
⇒ 1/√3 = AE/14√3
∴ AE = 14

∴ AB = AE + BE = 14 + 2.8 = 16.8m

১২,৭৩২.

After a 25% discount, the price of a refrigerator is Tk. 18,000. What was the price before the discount?

ক
Tk. 21,000
খ
Tk. 24,000
গ
Tk. 25,000
ঘ
Tk. 30,000
ঙ
None

ব্যাখ্যা

Question: After a 25% discount, the price of a refrigerator is Tk. 18,000. What was the price before the discount?

Solution:
In 25% discount,
Discount price = 75 when original price = 100

∴ Discount price 1 = 100/75 of original price

∴ Discount price 18,000 = (18,000 × 100)/75
= 24,000

∴ Original price = Tk. 24,000

১২,৭৩৩.

An error 2% in excess in made while measuring the side of a square. The percentage of error in the calculated area of the square is:

ক
ক) 2.00%
খ
খ) 4.04%
গ
গ) 2.02%
ঘ
ঘ) 4.00%

ব্যাখ্যা

Let the correct side of the square be x cm

Then, incorrect side =x + 2x/100 =1.02x

Correct area =x²
Incorrect area =(1.02x)²=1.0404x²

% Error ={(1.0404x² - x²/x²) × 100}%
=(0.0404×100)%
= 4.04%

১২,৭৩৪.

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

ক
- 19
খ
35
গ
20
ঘ
- 29

ব্যাখ্যা

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

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

১২,৭৩৫.

At what rate percent per annum will a sum of Tk. 15,625 amount to Tk. 21,952 in three years, if the interest is compounded annually?

ক
12%
খ
9%
গ
6%
ঘ
3%

ব্যাখ্যা

Question : At what rate percent per annum will a sum of Tk. 15,625 amount to Tk. 21,952 in three years, if the interest is compounded annually?

Solution :
Given,
Principal P = Tk. 15,625
Amount C = Rs. 21,952
Time n = 3 years
Rate of Interest = r%
= r/100

Compound interest when compounded annually is:
C = P (1 + r / 100)ⁿ
⇒ 21,952 = 15,625 (1 + r / 100)³
⇒ 21,952 / 15,625 = (1 + r/100)³
⇒ (1 + r/100)³ = 1.404928
⇒ (1 + r/100) = ³√1.404928 [ taking the cube root on both sides ]
⇒ (1 + r/100) ≈ 1.12
⇒ r/100 = 1.12 - 1
⇒ r/100 = 0.12
⇒ r = 0.12 × 100
∴ r = 12

১২,৭৩৬.

A woman spends a part of her monthly income and saves the rest. The ratio of her expenditure to her savings is 7 : 3. If her monthly income is Tk. 25,000, what is the amount of her monthly savings?

ক
Tk. 7000
খ
Tk. 7500
গ
Tk. 8000
ঘ
Tk. 8200

ব্যাখ্যা

Question: A woman spends a part of her monthly income and saves the rest. The ratio of her expenditure to her savings is 7 : 3. If her monthly income is Tk. 25,000, what is the amount of her monthly savings?

Solution:
Given,
Expenditure : Savings = 7 : 3

∴ Sum of the terms of ratio = (7 + 3) = 10

Given,
Total monthly salary = Tk. 25000

∴ Monthly savings = Tk.{(3/10) × 25000}
= Tk. 7500

১২,৭৩৭.

log₃√6 + log₃√(3/2) = ?

ক
0
খ
1
গ
2
ঘ
3
ঙ
None

ব্যাখ্যা

Question: log₃√6 + log₃√(3/2) = ?

Solution:
log₃√6 + log₃√(3/2)
= log₃[√{6 · (3/2)}]
= log₃√(3 · 3)
= log₃√(3²)
= log₃3
= 1

১২,৭৩৮.

A number, when divided by 275, leaves a remainder of 38. If the same number is divided by 25, what remainder will it leave?

ক
15
খ
23
গ
13
ঘ
22
ঙ
None

ব্যাখ্যা

Question: A number, when divided by 275, leaves a remainder of 38. If the same number is divided by 25, what remainder will it leave?

Solution:
Let the number be x and the quotient be q.

Then,
x = 275q + 38
⇒ x = (25 × 11q) + 38
⇒ x = (25 × 11q) + (25 × 1) + 13
⇒ x = 25(11q + 1) + 13

∴ When the number is divided by 25, the remainder is 13.

১২,৭৩৯.

If log_xy = 100 and log₂x = 10, then the value of y is-

ক
2¹⁰
খ
2¹⁰⁰
গ
2¹⁰⁰⁰
ঘ
2¹⁰⁰⁰⁰

ব্যাখ্যা

Question: If log_xy = 100 and log₂x = 10, then the value of y is-

Solution:
log₂x = 10
⇒ x = 2¹⁰

log_xy = 100
⇒ y = x¹⁰⁰
= (2¹⁰)¹⁰⁰
= 2¹⁰⁰⁰

১২,৭৪০.

What is the purchase price of an article if the profit by selling it for tk 250 is 25%

ক
ক) Tk. 150
খ
খ) Tk. 180
গ
গ) Tk. 200
ঘ
ঘ) Tk. 210

ব্যাখ্যা

For 25% profit,
If the selling price is 125 tk, then buying price is 100 tk
So, when the selling price is 250 tk, buying price is (100×250)/125 = 200 tk

১২,৭৪১.

If 2/3 of X = 75% of Y = 0.4 of Z, then find the ratio X : Y : Z.

ক
9 : 8 : 15
খ
8 : 10 : 17
গ
16 : 7 : 21
ঘ
9 : 13 : 17

ব্যাখ্যা

Question: If 2/3 of X = 75% of Y = 0.4 of Z, then find the ratio X : Y : Z.

Solution:
According to the question,
(2/3) × X = 75% of Y
⇒ 2X/3 = 75Y/100
⇒ 2X/3 = 3Y/4
⇒ X/Y = 9/8

Now,
75% of Y = 0.4 of Z
⇒(75Y/100) = (2Z/5)
⇒ (3Y/4) = (2Z/5)
⇒ 15Y = 8Z
⇒ Y/Z = 8/15

X : Y : Z = (9/8) : 1 : (15/8)
Multiply all terms by 8:
X = 9 × 1 = 9
Y = 1 × 8 = 8
Z = (15/8) × 8 = 15

Therefore,
X : Y : Z = 9 : 8 : 15

১২,৭৪২.

Eight year ago, Shanta's age was equal to the sum of the present ages of her one son and one daughter. Five years hence, the respective ratio between the ages of her daughter and her son that time will be 7 : 6. If Shanta's husband is 7 years elder to her and his present age is three times the present age of their son, what is the present age of the daughter ? (in year)

ক
18 years
খ
20 years
গ
23 years
ঘ
25 years

ব্যাখ্যা

Question: Eight year ago, Shanta's age was equal to the sum of the present ages of her one son and one daughter. Five years hence, the respective ratio between the ages of her daughter and her son that time will be 7 : 6. If Shanta's husband is 7 years elder to her and his present age is three times the present age of their son, what is the present age of the daughter ? (in year)

Solution:
Let, the age of the son and the daughter of Shanta be 6a years and 7a years respectively.
5 years hence, present age of son = (6a - 5)
and present age of daughter = (7a - 5)

Eight years ago, the age of Shanta = (6a - 5 + 7a - 5) = 13a - 10
So, present age of Shanta = (13a - 10 + 8) = 13a - 2
Since, present age of Shanta's husband = 3(6a - 5)

The difference of present age of Shanta's husband and Shanta = 7 (Given)

ATQ,
3(6a - 5) - (13a - 2) = 7
⇒ 18a - 15 - 13a + 2 = 7
⇒ 5a = 20

∴ The present age of daughter = (7a - 5)
= 7 × 4 - 5
= 23 years

১২,৭৪৩.

ক
13/5
খ
17/5
গ
9/5
ঘ
11/5

১২,৭৪৪.

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

ক
ক) 9
খ
খ) 10
গ
গ) 11
ঘ
ঘ) 12

ব্যাখ্যা

Let the required number of working hours per day be x.
More pumps , Less working hours per day (Indirect Proportion)
Less days, More working hours per day (Indirect Proportion)
Pumps 4 : 3 and Days 1 : 2 } : : 8 : x
Or, (4 × 1 × x) = (3 × 2 × 8)
Or, x = 12

১২,৭৪৫.

In what ratio salt costing 13 Tk/kg mixed with another salt costing 7 Tk/ kg to get a mixture costing 9 Tk/kg?

ক
3 : 2
খ
1 : 3
গ
1 : 2
ঘ
1 : 4

ব্যাখ্যা

Question: In what ratio salt costing 13 Tk/kg mixed with another salt costing 7 Tk/ kg to get a mixture costing 9 Tk/kg?

Solution:
ধরি,
13 টাকা কেজি লবণের পরিমাণ x কেজি এবং 7 টাকা কেজি লবণের পরিমাণ y কেজি।

(13x + 7y)/(x + y) = 9
⇒ (13x + 7y) = 9 (x + y)
⇒ 13x + 7y = 9x + 9y
⇒ 13x - 9x = 9y - 7y
⇒ 4x = 2y
∴ x / y = 2/4 = 1/2
x : y = 1 : 2

১২,৭৪৬.

In a simultaneous throw of two dice, what is the probability of getting a total of 9?

ক
ক) 1/9
খ
খ) 1/4
গ
গ) 5/36
ঘ
ঘ) 1/6

ব্যাখ্যা

Question: In a simultaneous throw of two dice, what is the probability of getting a total of 9?

Solution:

The total number of events ⇒ 36.
The number of events of getting total 9 ⇒ 4.

∴ The probability of getting a total of 9 is 4/36 = 1/9.

১২,৭৪৭.

The 2^nd angle of a right triangle is 30 degrees. Then how many degrees is the 3rd angle?

ক
150°
খ
120°
গ
60°
ঘ
30°
ঙ
None

ব্যাখ্যা

Question: The 2^nd angle of a right triangle is 30 degrees. Then how many degrees is the 3rd angle?

Solution:
আমরা জানি,
সমকোণী ত্রিভুজের একটি কোণ 90°
এখন, ২য় কোণ 30° হলে,

তিন কোণের সমষ্টি = 180°
বা, 90° + 30° + ৩য় কোণ = 180°
বা, ৩য় কোণ = 180° - 120°
∴ ৩য় কোণ = 60°

১২,৭৪৮.

The ratio of income in two consecutive years is 2 : 3 respectively. The ratio of their expenditure is 5 : 9. Income of second-year is Tk. 45000 and Expenditure of first-year is Tk. 25000. Find the Savings in both years together.

ক
5000
খ
7000
গ
6075
ঘ
8025

ব্যাখ্যা

Question: The ratio of income in two consecutive years is 2 : 3 respectively. The ratio of their expenditure is 5 : 9. Income of second-year is Tk. 45000 and Expenditure of first-year is Tk. 25000. Find the Savings in both years together.

Solution:
Let the First-year income = 2x
And, Second-year income = 3x
But ATQ, the second-year income = 45000
So, x = 45000/3 = 15000
Then, first-year income = 2 × 15000 = 30000

Similarly,
Let the first-year expenditure = 5y
And, second-year expenditure = 9y
But ATQ, the first-year expenditure = 25000
So, y = 25000/5 = 5000
Second-year expenditure = 9y = 9 × 5000 = 45000

Income = expenditure + savings
Or, savings = income - expenditure

∴ first-year savings = 30000 - 25000 = 5000
Similarly,
Second-year savings = 45000 - 45000 = 0

Now, total saving in two years = first-year savings + second-year savings
= 5000 + 0 = 5000.

১২,৭৪৯.

The sum and products of two numbers is 20 and 96 respectively. what is the sum of their reciprocals?

ক
ক) 3/4
খ
খ) 1/6
গ
গ) 5/24
ঘ
ঘ) 1/8

ব্যাখ্যা

Question: The sum and products of two numbers is 20 and 96 respectively. what is the sum of their reciprocals?

Solution:
ধরি,
বড় সংখ্যাটি a ও ছোট সংখ্যাটি b

প্রশ্নমতে,
a + b = 20 ................. (1)
ab = 96 ..................... (2)

আমরা জানি,
(a - b)² = (a + b)² - 4ab
= (20)² - 4 × 96
= 400 - 384
= 16
∴ a - b = 4 ........................ (3)

(1) + (2) হতে পাই,
a + b = 20
a - b = 4
2a = 24
∴ a = 12

a এর মান (1) নং বসিয়ে পাই,
12 + b = 20
বা, b = 8

12 ও 8 এর ব্যস্তানুপাত = (1/12) + (1/8)
= (2 + 3)/24
= 5/24

১২,৭৫০.

21 binders can bind 1400 books in 15 days. How many binders will be required to bind 800 books in 20 days?

ক
ক) 7
খ
খ) 9
গ
গ) 12
ঘ
ঘ) 14

ব্যাখ্যা

Let,
The required number of binders is x.
Less books, Less binders [Direct proportion]
More days, Less binders [Indirect proportion]
Books {(1400 : 800) , Days (20 : 15)} :: 21 : x
∴ 1400 × 20 × x = 800 × 15 × 21
⇒ x = (800 × 15 × 21)/(1400 × 20)
= 9.

১২,৭৫১.

Sum of 10 integer is 462. The average of first 4 number is 52 and the average of last 5 number is 38. What is the fifth number?

ক
ক) 64
খ
খ) 60
গ
গ) 50
ঘ
ঘ) 62

ব্যাখ্যা

Question: Sum of 10 integer is 462. The average of first 4 number is 52 and the average of last 5 number is 38. What is the fifth number?

Solution:
১ টি পূর্ণসংখ্যার সমষ্টি = ৪৬২

প্রথম ৪ টি সংখ্যার সমষ্টি = (৪ × ৫২) = ২০৮
শেষ ৫ টি সংখ্যার সমষ্টি = (৫ × ৩৮) = ১৯০

∴ ৫ম সংখ্যাটি = (৪৬২ - ২০৮ - ১৯০) = ৬৪

১২,৭৫২.

A basket contains 2 red, 4 blue and 3 green marbles. If three marbles are picked up at random what is the probability that at least one is blue?

ক
ক) 5/42
খ
খ) 4/9
গ
গ) 37/42
ঘ
ঘ) 5/9

ব্যাখ্যা

Total number of marbles = (2 + 4 + 3) = 9
Let E be the event of drawing 3 marbles such that none is blue.
Then, n (E) = number of ways of drawing 3 marbles out of 5 = ⁵C₃ = 10
n(S)=⁹C₃
= 84
∴P(E)=n(E)/n(S)
= 10/84
= 5/42

∴ Required probability
= 1 - P(E)
= 1 - (5/42)
= (42 - 5)/42
= 37/42

১২,৭৫৩.

The population of a village decreases at the rate of 20% per annum. If its population 2 years ago was 10000, the present population is = ?

ক
ক) 4600
খ
খ) 6400
গ
গ) 7600
ঘ
ঘ) 600

ব্যাখ্যা

Given that
Population of a village 2 years ago = 10,000
Rate of growth = 20 % p.a.
Time period = 2 years
Solution
Present population = P(1-R/100)ⁿ
= 10,000(1-20/100)²
= 10,000(100-20/100)²
= 10,000(80/100)²
= 10,000(4/5)²
= 10,000 × 16/25
= 6,400

১২,৭৫৪.

A box contains 4 tennis ball, 6 season and 8 dues balls. 3 balls are randomly drawn from the box. What is the probability that the balls are different?

ক
ক) 2/17
খ
খ) 4/17
গ
গ) 4/11
ঘ
ঘ) 3/13

ব্যাখ্যা

There are a total of 18 balls.
∴ Probability of the balls to be different is,
= (⁴C₁ × ⁶C₁ × ⁸C₁)/(¹⁸C3)
= (4 × 6 × 8)/(18×17×16 / 6)
= (4 × 6 × 8 × 6)/(18 × 17 × 16)
= 4/17

১২,৭৫৫.

How many cubes of 8 cm edge can be put in a cubical box of 80 cm edge?

ক
700 cubes
খ
500 cubes
গ
2000 cubes
ঘ
1000 cubes

ব্যাখ্যা

Question: How many cubes of 8 cm edge can be put in a cubical box of 80 cm edge?

Solution:
Number of cubes = (80 × 80 × 80)/(8 × 8 × 8)
= 512000/512
= 1000 cubes

[Note: 1 m = 100 cm, here all dimensions are in cm]

১২,৭৫৬.

If tanA = 3/4, then secA = ?

ক
3/4
খ
5/4
গ
5/6
ঘ
1/8

ব্যাখ্যা

Question: If tanA = 3/4, then secA =?

Solution:
tanA = 3/4
লম্ব/ভূমি = 3/4
অতিভুজ = √{(4)² + (3)²} = 5

secA = অতিভুজ/ভূমি
= 5/4

১২,৭৫৭.

A train running at a speed of 72 km/hr crosses a platform double its length in 45 seconds. What is the length of the platform in metres?

ক
400m
খ
650m
গ
600m
ঘ
800m

ব্যাখ্যা

Question: A train running at a speed of 72 km/hr crosses a platform double its length in 45 seconds. What is the length of the platform in metres?

Solution:
72 কি.মি = 72 × 1000 = 72000 মিটার
1 ঘণ্টা = 60 × 60 = 3600 সেকেন্ড

∴ 3600 সেকেন্ড যায় = 72000 মিটার
3600 সেকেন্ড যায় = 72000/3600 মিটার
45 সেকেন্ড যায় = {(72000 × 45)/3600} = 900 মিটার

ধরি, ট্রেনের দৈর্ঘ্য = x
প্লাটফর্মের দৈর্ঘ্য = 2x

প্রশ্নমতে,
⇒ x + 2x = 900
⇒ 3x = 900
⇒ x = 900/3
∴ x = 300

∴ প্লাটফর্মের দৈর্ঘ্য = 2x = 2 × 300 = 600 মিটার

১২,৭৫৮.

The vessels containing water and milk in the ratio of 1 : 2 and 2 : 5 are mixed is 1 : 4 then what will be the resulting ratio?

ক
ক) 31 : 25
খ
খ) 22 : 31
গ
গ) 47 : 31
ঘ
ঘ) 31 : 74

ব্যাখ্যা

Question: The vessels containing water and milk in the ratio of 1 : 2 and 2 : 5 are mixed is 1 : 4 then what will be the resulting ratio?

Solution:
We can create the following ratios:
Vessel 1: water : milk = x : 2x
Vessel 2: water : milk = 2y : 5y

We can create the equation:
(x + 2x)/(2y + 5y) = 1/4
3x/7y = 1/4
12x = 7y

We see that we can let x = 7 and y = 12.
So we have:
Vessel 1: water = 7 and milk = 14
Vessel 2: water = 24 and milk = 60

Therefore, in the final mixture, the ratio of water to milk is:
(7 + 24)/(14 + 60) = 31/74

১২,৭৫৯.

The sum of the ages of the son and father is 120 years. The father age is 3 times than the son. What is the age of father?

ক
ক) 75
খ
খ) 80
গ
গ) 90
ঘ
ঘ) 85

ব্যাখ্যা

ধরি,
পুত্রের বয়স x বছর
পিতার বয়স 3x
প্রশ্নমতে,
x + 3x = 120
4x = 120
x=30

পিতার বয়স = 3 × 30 = 90 বছর

১২,৭৬০.

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

ক
4%
খ
2%
গ
6%
ঘ
3%

ব্যাখ্যা

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

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

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

১২,৭৬১.

Time taken by A to finish a piece of work is twice the time taken B and thrice the time taken by C. If all three of them work together, it takes them 2 days to complete the entire work. How much work was done by B alone?

ক
ক) 2 days
খ
খ) 6 days
গ
গ) 3 days
ঘ
ঘ) 5 days

ব্যাখ্যা

Time taken by A = x days

Time taken by B = x/2 days

Time Taken by C = x/3 days

⇒ {(1/x) + (2/x) + (3/x) = 1/2

⇒ 6/x = 1/2

⇒ x = 12

Time taken by B = x/2 = 12/2 = 6 days.

১২,৭৬২.

ক
ক) 2
খ
খ) 3
গ
গ) 4
ঘ
ঘ) 5

ব্যাখ্যা

প্রশ্ন:

সমাধান:

১২,৭৬৩.

৭ + ২ [- ৮ - {- ৩ - (- ২ - ৩)} - ৪] = কত?

ক
ক) - ২১
খ
খ) - ২০
গ
গ) - ১৪
ঘ
ঘ) - ১৬

ব্যাখ্যা

৭ + ২ [- ৮ - {- ৩ - (- ২ - ৩)} - ৪]
= ৭ + ২ [- ৮ - {- ৩ - (- ৫)} - ৪]
= ৭ + ২ [- ৮ - {- ৩ + ৫} - ৪]
= ৭ + ২ [- ৮ - { + ২} - ৪]
= ৭ + ২ [- ৮ - ২ - ৪]
= ৭ + ২ [- ১ ৪]
= ৭ - ২৮
= - ২১

১২,৭৬৪.

What is the reflex angle between the hands of a clock at 9 : 35?

ক
277.5°
খ
280°
গ
282.5°
ঘ
285°
ঙ
290°

ব্যাখ্যা

Question: What is the reflex angle between the hands of a clock at 9 : 35?

Solution:
The angle between the hands of the clock is |11M - 60H|°/2
= |(11 × 35) - (60 × 9)|°/2
= |385 - 540|°/2
= |- 155|°/2
= 155°/2
= 77.5°

∴ Reflex angle = 360° - 77.5°
= 282.5°

১২,৭৬৫.

a + b = √8, a - b = √6. Find the value of 14ab(a² + b²) = ?

ক
76
খ
49
গ
196
ঘ
92

ব্যাখ্যা

Question: a + b = √8, a - b = √6. Find the value of 14ab(a² + b²) = ?

Solution:
Given that,
a + b = √8
a - b = √6

ATQ,
14ab(a² + b²)
= (14/8) × 8ab(a² + b²)
= (14/8) × 4ab × 2(a² + b²)
= (14/8) × {(a + b)²- (a - b)²)} {(a + b)²+(a - b)²)}
= (14/8) × {(√8)²- (√6)²)} {(√8)²+(√6)²)}
= (14/8) × (8 - 6) × (8 + 6)
= (14/8) × 2 × 14
= (14/4) × 14
= 7 × 7
= 49

১২,৭৬৬.

If the sum of two digits of a two-digit number is 9, the number obtained by interchanging two digits is 45 less than the given number. What will the number be?

ক
ক) 27
খ
খ) 72
গ
গ) 63
ঘ
ঘ) 54

ব্যাখ্যা

Question: If the sum of two digits of a two-digit number is 9, the number obtained by interchanging two digits is 45 less than the given number. What will the number be?

Solution:
ধরি, একক স্থানীয় অংক = x
দশক স্থানীয় অংক = y
সংখ্যাটি = 10y + x
স্থান বিনিময়কৃত সংখ্যাটি = 10x + y

প্রশ্নমতে,
x + y = 9
y = 9 - x -----------(1)

প্রশ্নমতে,
10x + y + 45 = 10y + x
বা, 10x + 9 - x + 45 = 10(9 - x) + x
বা, 9x + 9x = 90 - 54
বা, 18x = 36
∴ x = 2
এবং y = 9 - 2 = 7

∴ নির্ণেয় সংখ্যাটি = (10 × 7) + 2 = 72

১২,৭৬৭.

5x² - (4x - 3)(3x + 2) = ?

ক
- 6x² + 2x + 4
খ
7x² + x + 6
গ
- 7x² + x + 6
ঘ
- 7x² + 4x + 7

ব্যাখ্যা

Question: 5x² - (4x - 3)(3x + 2) = ?

Solution:

১২,৭৬৮.

A dishonest milkman professes to sell his milk at cost price but he mixes it with water and thereby gains 25%. The percentage of water in the mixture is:

ক
ক) 10%
খ
খ) 15%
গ
গ) 20%
ঘ
ঘ) 27%

ব্যাখ্যা

Let the cost of 1 litre of milk be Tk.100.
He gains 25% profit if he sells the same milk at Tk.125.

Now he has to sell 1.25 litres of milk at Tk.125 (as the price of milk should be proportional to the quantity).
To accommodate for the additional 250 ml of milk, he uses water.

So, 250 ml of water is used in 1.25 litres of the mixture.
∴ Percentage of water in mixture = (.25/1.25) × 100 = 20%

Alternative Method:

Let C.P. of 1 litre milk be Tk.1
Then, S.P. of 1 litre of mixture = Tk.1, Gain = 25%
C.P. of 1 litre mixture = Tk. (100/125×1) = 4/5

By the rule of allegation, we have:

∴ Ratio of milk to water = 4/5 : 1/5 = 4 : 1
Hence, percentage of water in the mixture = (1/5×100)% = 20%

১২,৭৬৯.

A school has a total of 90 students. There are 30 students taking Physics, 25 taking English, and 13 taking both. How many students are taking either Physics or English?

ক
40
খ
42
গ
44
ঘ
46

ব্যাখ্যা

Question: A school has a total of 90 students. There are 30 students taking Physics, 25 taking English, and 13 taking both. How many students are taking either Physics or English?

Solution:
Students taking physics n(P) = 30 (these 30 include those 13 that take both)
Students taking english n(E) = 25 (these 25 also include those 13)
Students taking both n(P ∩ E) = 13
Students taking either Physics or English n(P ∪ E) = ?

We know
n(P ∪ E) = n(P) + n(E) - n(P ∩ E)
= 30 + 25 - 13 = 42

১২,৭৭০.

In a room, if 4 students sit in each bench, there are 3 empty benches, but 6 students have to stand if 3 students sit each bench. How many students are there in that room?

ক
80
খ
70
গ
60
ঘ
50
ঙ
None

ব্যাখ্যা

Question: In a room, if 4 students sit in each bench, there are 3 empty benches, but 6 students have to stand if 3 students sit each bench. How many students are there in that room?

Solution:
ধরি,
বেঞ্চ সংখ্যা ক টি

একটি রুমের প্রতি বেঞ্চে ৪ জন করে ছাত্র বসলে ৩ টি বেঞ্চ খালি থাকে।
∴ ছাত্রসংখ্যা= (ক - ৩) × ৪ জন

প্রতি বেঞ্চে ৩ জন করে ছাত্র বসালে ৬ জন ছাত্রকে দাঁড়িয়ে থাকতে হয়।
∴ ছাত্রসংখ্যা = ৩ক + ৬ জন

প্রশ্নমতে,
(ক - ৩) × ৪ = ৩ক + ৬
⇒ ৪ক - ১২ = ৩ক + ৬
∴ ক = ১৮

ছাত্রসংখ্যা = (ক - ৩) × ৪ জন
= (১৮ - ৩) × ৪ জন
= ১৫ × ৪ জন
= ৬০ জন

১২,৭৭১.

In the figure, DE is parallel to BC. If the area of a triangle ADE is half that of trapezium DECB, what is the ratio of ΔADE to ΔABC?

ক
2 : 1
খ
1 : 2
গ
2 : 3
ঘ
1 : 3

ব্যাখ্যা

Question: In the figure, DE is parallel to BC. If the area of a triangle ADE is half that of trapezium DECB, what is the ratio of ΔADE to ΔABC?

Solution:
দেওয়া আছে,
ADE ত্রিভুজের ক্ষেত্রফল = (1/2) × DECB ট্রাপিজিয়ামের ক্ষেত্রফল
∴ DECB ট্রাপিজিয়ামের ক্ষেত্রফল = 2 × ADE ত্রিভুজের ক্ষেত্রফল ........................(১)

এখানে,
ABC ত্রিভুজের ক্ষেত্রফল = ADE ত্রিভুজের ক্ষেত্রফল + DECB ট্রাপিজিয়ামের ক্ষেত্রফল
⇒ ABC ত্রিভুজের ক্ষেত্রফল = ADE ত্রিভুজের ক্ষেত্রফল + 2 × ADE ত্রিভুজের ক্ষেত্রফল [(১) নং এর সাহায্যে]
∴ ABC ত্রিভুজের ক্ষেত্রফল = 3 × ADE ত্রিভুজের ক্ষেত্রফল

অতএব,
ADE ত্রিভুজের ক্ষেত্রফল/ABC ত্রিভুজের ক্ষেত্রফল = 1/3
∴ ΔADE : ΔABC = 1 : 3

১২,৭৭২.

A laborer is paid Tk. 30 per hour for an 8-hour day. If he works overtime (above 8 hours) in a single day, he is paid 1.5 times the rate he is paid for regular hours. If the laborer receives Tk. 330 for a single day's work. How long did he work on that day?

ক
9 hours 30 minutes
খ
10 hours
গ
9 hours
ঘ
10 hours 30 minutes

ব্যাখ্যা

Question: A laborer is paid Tk. 30 per hour for an 8-hour day. If he works overtime (above 8 hours) in a single day, he is paid 1.5 times the rate he is paid for regular hours. If the laborer receives Tk. 330 for a single day's work. How long did he work on that day?

Solution:
শ্রমিকের নিয়মিত বেতন (8 ঘণ্টার জন্য) = 8 ঘণ্টা × 30 টাকা/ঘণ্টা = 240 টাকা।

শ্রমিকের অতিরিক্ত কাজের (ওভারটাইম) আয় = 330 টাকা - 240 টাকা
= 90 টাকা।

শ্রমিকের প্রতি ঘণ্টার ওভারটাইম বেতন = 30 টাকা × 1.5 = 45 টাকা।

শ্রমিকের ওভারটাইম কাজের সময় = 90 টাকা ÷ 45 টাকা/ঘণ্টা
= 2 ঘণ্টা।

শ্রমিকের মোট কাজের সময় = 8 ঘণ্টা (নিয়মিত) + 2 ঘণ্টা (ওভারটাইম)
= 10 ঘণ্টা।

১২,৭৭৩.

An article is sold for Tk. 2400 at a profit of 25%. What would have been the actual profit or loss if it had been sold at Tk. 1800?

ক
Loss Tk. 120
খ
Profit Tk. 120
গ
Loss Tk. 80
ঘ
Profit Tk. 80

ব্যাখ্যা

Question: An article is sold for Tk. 2400 at a profit of 25%. What would have been the actual profit or loss if it had been sold at Tk. 1800?

Solution:
Firstly let us find the cost price of the article. C.P. = 2400 × (100/125) = 1920.

New selling price = Tk. 1800
∴ Loss = 1920 - 1800 = 120

১২,৭৭৪.

Two pipes P and Q, when opened alone can fill the tank in 20 and 30 hours respectively. If both pipes are opened together, then in how many hours will the tank be filled?

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

ব্যাখ্যা

Question: Two pipes P and Q, when opened alone can fill the tank in 20 and 30 hours respectively. If both pipes are opened together, then in how many hours will the tank be filled?

Solution:
Part of tank filled by pipe P in 1 hour = 1/20
Part of tank filled by pipe Q in 1 hour = 1/30

Tank filled by both pipes in 1 hour = 1/20 + 1/30
= 5/60
= 1/12

∴ Complete tank will be filled by both in 1/1/12 hours
= 12 hours

১২,৭৭৫.

There is a total of 120 marbles in a box, each of which is red, green, blue, or white. If one marble is drawn from the box at random, the probability that it will be white is 1/4 and the probability that it will be green is 1/3. What is the probability that the marble will be either red or blue?

ক
1/4
খ
2/7
গ
1/3
ঘ
5/12

ব্যাখ্যা

Question: There is a total of 120 marbles in a box, each of which is red, green, blue, or white. If one marble is drawn from the box at random, the probability that it will be white is 1/4 and the probability that it will be green is 1/3. What is the probability that the marble will be either red or blue?

Solution:
Probability(white )= 1/4
Probability(green) = 1/3
Probability (red or blue marbles) = 1 - (1/4 + 1/3)
= 1 - 7/12
= 5/12

১২,৭৭৬.

A student took five papers in an examination, where full marks were the same on each paper. Her marks in these papers were in the proportion of 6 : 7 : 8 : 9 : 10. In all these papers together, the candidate obtained 60% of the total marks. Then the number of papers in which she got more than 50% marks is -

ক
ক) 2
খ
খ) 3
গ
গ) 4
ঘ
ঘ) 5

ব্যাখ্যা

Let,
the full marks for each paper be x.
Let the marks obtained in the five papers be 6y, 7y, 8y, 9y and 10y respectively.
Then, (6y + 7y + 8y + 9y + 10y)/5x = 60/100
⇒ 40y/5x = 3/5
⇒ 40y = 3x
⇒ x = (40/3)y
50% of x = (50/100) × (40/3)y
= 20y/3
= 6(2/3)y.
Clearly, the student got more than 50% marks in each of the last 4 papers.

১২,৭৭৭.

Reena took a loan of Tk. 1200 with simple interest for as many years as the rate of interest. If she paid Tk. 432 as interest at the end of the loan period, what was the rate of interest?

ক
3.6
খ
6
গ
18
ঘ
8
ঙ
24

ব্যাখ্যা

Cannot be determined
Let rate = R% and time = R years.
Then, (1200 × R × R/100) = 432
12R² = 432
=> R² = 36
=> R = 6

১২,৭৭৮.

A takes 3 hours more than B to walk d km, but if A doubles his speed, then he can make it in 1 hour less than B. How much time does A require for walking d km?

ক
5 hours
খ
6 hours
গ
7 hours
ঘ
8 hours

ব্যাখ্যা

Question: A takes 3 hours more than B to walk d km, but if A doubles his speed, then he can make it in 1 hour less than B. How much time does A require for walking d km?

Solution:
Let,
B takes "t" hours to walk "d" km.

Then, A takes (t + 3) hours to walk d km.
With double speed, A will take = (1/2)(t + 3) hours.

ATQ,
t - (1/2)(t + 3) = 1
⇒ {2t - (t + 3)}/2 = 1
⇒ 2t - t - 3 = 2
∴ t = 5

∴ B takes 5 hours to walk d km.

∴ A takes (5 + 3) = 8 hours to walk d km.

১২,৭৭৯.

ক
- 3, 3
খ
- 1/3, 3
গ
1/3, 3
ঘ
- 5, 2

ব্যাখ্যা

Question:

Solution:
(2x - 1)/(x - 1) = (7x - 1)/(2x + 2)
⇒ (2x - 1)(2x + 2) = (7x - 1)(x - 1)
⇒ 4x² + 4x - 2x - 2 = 7x² - 7x - x + 1
⇒ 4x² + 2x - 2 = 7x²- 8x + 1
⇒ 3x² - 10x + 3 = 0
⇒ 3x² - 9x - x + 3 = 0
⇒ 3x(x - 3) - 1(x - 3) = 0
⇒ (x - 3)(3x - 1) = 0
∴ x = 3 or x = 1/3

১২,৭৮০.

If the cost price of 15 books is equal to the selling price of 20 books, the loss percentage is-

ক
ক) 20%
খ
খ) 25%
গ
গ) 30%
ঘ
ঘ) 15%

ব্যাখ্যা

Let C.P. of each books Tk. 1
C.P. of 20 books Tk. 20
S.P. of 20 books Tk. 15
Loss = TK. (20 - 15) = Tk.5

Loss% = {(5/20) × 100}% = 25%

১২,৭৮১.

If sec(x − 30°) = 2, then cot x = ?

ক
ক) 1/2
খ
খ) 0
গ
গ) 1
ঘ
ঘ) None of the above

ব্যাখ্যা

sec (x − 30°) = 2
⇒ sec (x - 30°) = sec 60°
⇒ x - 30° = 60°
⇒ x = 90°
∴ cot 90° = 0

১২,৭৮২.

A merchant sells one table for Tk. 1200 at a profit of 25% and another for Tk. 1500 at a loss of 10%. His total gain or loss percentage is-

ক
3.14% gain
খ
2.78% gain
গ
4.82% loss
ঘ
1.73% loss

ব্যাখ্যা

Question: A merchant sells one table for Tk. 1200 at a profit of 25% and another for Tk. 1500 at a loss of 10%. His total gain or loss percentage is-

Solution:
C.P. of 1st table = Tk.(100 × 1200)/125 = Tk. 960
C.P. of 2nd table = Tk.(100 × 1500)/90 = Tk. 1667
So, total C.P. = Tk.(960 + 1667) = Tk. 2627
Total S.P. = Tk.(1200 + 1500) = Tk. 2700

∴ Gain% = {(2700 - 2627) × 100}/2627%
= (73 × 100)/2627%
= 2.78%

১২,৭৮৩.

Fahim's collection contains US, Bangladeshi and British stamps. If the ration of US to Bangladeshi stamps is 5 : 2 and the ration of Bangladeshi of British stamps is 5 : 1, what is the ration of US to British stamps?

ক
ক) 5 : 1
খ
খ) 10 : 5
গ
গ) 15 : 2
ঘ
ঘ) 20 : 2
ঙ
ঙ) 25 : 2

ব্যাখ্যা

Question: Fahim's collection contains US, Bangladeshi and British stamps. If the ration of US to Bangladeshi stamps is 5 : 2 and the ration of Bangladeshi of British stamps is 5 : 1, what is the ration of US to British stamps?

Solution:
US : Bangladesh = 5 : 2 = 25 : 10
Bangladeshi : British = 5 : 1 = 10 : 2

US : British = 25 : 2

১২,৭৮৪.

Abir scored 30% marks and failed by 15 marks. Sunny scored 40% marks and obtained 35 marks more than those required to pass. The pass percentage is-

ক
25%
খ
30%
গ
32%
ঘ
33%

ব্যাখ্যা

Question: Abir scored 30% marks and failed by 15 marks. Sunny scored 40% marks and obtained 35 marks more than those required to pass. The pass percentage is-

Solution:
Let, total mark is x

Now
0.3x + 15 = 0.4x - 35
⇒ 0.4x - 0.3x = 35 + 15
⇒ 0.1x = 50
⇒ x = 50/0.1
∴ x = 500

Pass mark = 0.3 × 500 + 15
= 150 + 15
= 165

Pass percentage = (165/500) × 100%
= 33%

১২,৭৮৫.

7 litres mixture of milk and water contains 30% water. 3.5 litres of milk is added to this mixture. What is the percentage of water in the new mixture?

ক
10
খ
15
গ
20
ঘ
25

ব্যাখ্যা

Question: 7 litres mixture of milk and water contains 30% water. 3.5 litres of milk is added to this mixture. What is the percentage of water in the new mixture?

Solution:
Quantity of water in the mixture = 30% of 7 litres = 2.1 litre
Quantity of milk in the mixture = 7 - 2.1 = 4.9 litres

New quantity of milk in the mixture after adding 3.5 litres of milk = 4.9 + 3.5 = 8.4 litres
Quantity of New mixture = 8.4 + 2.1 = 10.5 litres

∴ Percentage of water in new mixture = (2.1/10.5) × 100 % = 20%

১২,৭৮৬.

If log₅(a²+ a) − log₅(a + 1) = 2, then the value of a is -

ক
5
খ
15
গ
20
ঘ
25

ব্যাখ্যা

Question: If log₅(a²+ a) − log₅(a + 1) = 2, then the value of a is -

Solution:
log₅(a²+ a) − log₅(a + 1) = 2
⇒ log₅{(a²+ a)/(a + 1)} = 2
⇒ log₅{a(a + 1)/(a + 1)} = 2
⇒ log₅a = 2
⇒ a = 5²
∴ a = 25

১২,৭৮৭.

এক দোকানদারের দাবি, তিনি একটি পণ্য ২৩ টাকা প্রতি কেজি বিক্রি করেন, যদিও তার জন্য এটি খরচ হচ্ছে ২৫ টাকা প্রতি কেজি। তিনি বলেন, গ্রাহকদের ভালো সুবিধা দেয়ার জন্য তিনি এই ক্ষতি করছেন। তবে, তদন্ত করে দেখা যায় যে দোকানদার ১ কেজির জায়গায় ৮০০ গ্রাম ওজন ব্যবহার করছেন। তাহলে, দোকানদার কতটা লাভ বা ক্ষতি করছেন?

ক
১৫% লাভ
খ
১০% লাভ
গ
১০% ক্ষতি
ঘ
৮% ক্ষতি
ঙ
কোনটি নয়

ব্যাখ্যা

প্রশ্ন: এক দোকানদারের দাবি, তিনি একটি পণ্য ২৩ টাকা প্রতি কেজি বিক্রি করেন, যদিও তার জন্য এটি খরচ হচ্ছে ২৫ টাকা প্রতি কেজি। তিনি বলেন, গ্রাহকদের ভালো সুবিধা দেয়ার জন্য তিনি এই ক্ষতি করছেন। তবে, তদন্ত করে দেখা যায় যে দোকানদার ১ কেজির জায়গায় ৮০০ গ্রাম ওজন ব্যবহার করছেন। তাহলে, দোকানদার কতটা লাভ বা ক্ষতি করছেন?

সমাধান:
০.৮ কেজির বিক্রয় মূল্য = ২৩ টাকা

এখন,
১ কেজির বিক্রয় মূল্য = ২৩/০.৮ টাকা
= ১১৫/৪ টাকা
= ২৮.৭৫ টাকা

∴ শতকরা লাভ = {(২৮.৭৫ - ২৫)/২৫} × ১০০%
= (৩.৭৫/২৫) × ১০০%
= ১৫%

১২,৭৮৮.

Four people are running around a circular ground from a point on the circumference at 9.00 am. For one round, these four persons take respectively 40, 50, 60 and 30 minutes. At what time will they meet together again?

ক
ক) 6.00 pm
খ
খ) 7.00 pm
গ
গ) 5.30 pm
ঘ
ঘ) 7.30 pm

ব্যাখ্যা

Question: Four people are running around a circular ground from a point on the circumference at 9.00 am. For one round, these four persons take respectively 40, 50, 60 and 30 minutes. At what time will they meet together again?

Solution:
L.C.M. of 40, 50, 60 and 30
= 600 minutes
= 10 hours
So, they meet again 10 hours after they start.
They meet together again = 9.00 am + 10 hours
= 7.00 pm

১২,৭৮৯.

The value of the following is:

ক
0
খ
2
গ
7
ঘ
1

ব্যাখ্যা

Question: The value of the following is:

Solution:

১২,৭৯০.

How many times are the hands of a clock at right angle in a day?

ক
22
খ
24
গ
44
ঘ
48
ঙ
None

ব্যাখ্যা

Question: How many times are the hands of a clock at right angle in a day?

Solution:
In 12 hours, they are at right angles 22 times.
∴ In 24 hours, they are at right angles 44 times.

১২,৭৯১.

A boat's speed with the current is 17 km/hr and the speed of the current is 3.5 km/hr. What is the boat's speed against the current?

ক
10 km/h
খ
7 km/h
গ
13.5 km/h
ঘ
17 km/h

ব্যাখ্যা

Question: A boat's speed with the current is 17 km/hr and the speed of the current is 3.5 km/hr. What is the boat's speed against the current?

Solution:
Given that,
Boat's speed with the current = 17 km/hr
Speed of the current = 3.5 km/hr

∴ Boat's speed in still water = speed with the current - speed of the current
= (17 - 3.5) km/hr
= 13.5 km/hr

∴ Boat's speed against the current = speed in still water - speed of the current
= (13.5 - 3.5) km/hr
= 10 km/hr

Therefore, the boat's speed against the current is 10 km/hr.

১২,৭৯২.

Which of the following angle can be constructed with the help of a ruler and a pair of compasses?

ক
ক) 35°
খ
খ) 40°
গ
গ) 37.5°
ঘ
ঘ) 47.5°

ব্যাখ্যা

কম্পাস এবং রুলার ব্যবহার করে 15° দ্বারা ভাগ করা যায় এমন কোণ এবং ওই সব কোণের অর্ধেক অঙ্কন করা যায়।
একটি কম্পাসের সাহায্যে 60° এবং 90° কোণ অঙ্কন করার পর ওই দুই কোণের ছেদবিন্দু থেকে 75° কোণ অঙ্কন করা যায়।
এরপর 0° এবং 75° এর ছেদবিন্দু ব্যবহার করে 37.5° কোণ অঙ্কন করা যাবে।

১২,৭৯৩.

The sum of the ages of X, Y, and Z is 120 years. Ten years ago, the ratio of their ages was 2 : 3 : 4. What is the present age of Y?

ক
30 years
খ
40 years
গ
50 years
ঘ
45 years

ব্যাখ্যা

Question: The sum of the ages of X, Y, and Z is 120 years. Ten years ago, the ratio of their ages was 2 : 3 : 4. What is the present age of Y?

Solution:
Let, X, Y, and Z's ages 10 years ago be 2a, 3a, and 4a years respectively

ATQ,
(2a + 10) + (3a + 10) + (4a + 10) = 120
⇒ 9a + 30 = 120
⇒ 9a = 90
∴ a = 10

So, the age of Y = 3 × 10 + 10 = 40 years.

১২,৭৯৪.

There are four hotels in a town. If 3 men check into the hotels in a day then what is the probability that each checks into a different hotel?

ক
5/9
খ
2/9
গ
3/8
ঘ
3/7

ব্যাখ্যা

Question: There are four hotels in a town. If 3 men check into the hotels in a day then what is the probability that each checks into a different hotel?

Solution:
Total cases of checking in the hotels = 4 × 4 × 4 = 64 ways.
Cases when 3 men are checking in different hotels = 4 × 3 × 2 = 24 ways.

Required probability = 24/64
= 3/8

১২,৭৯৫.

Two trains are running in the same direction at 90 km/h and 60 km/h. The faster train crosses a man in the slower train in 18 seconds. What is the length of the faster train?

ক
120 meters
খ
135 meters
গ
150 meters
ঘ
180 meters

ব্যাখ্যা

Question: Two trains are running in the same direction at 90 km/h and 60 km/h. The faster train crosses a man in the slower train in 18 seconds. What is the length of the faster train?

Solution:
Relative speed of train = (90 - 60) km/h
= 30 km/h
= (30 × 1000)/3600
= 25/3 m/s

∴ Distance covered = Speed × Time
= (25/3) × 18
= 150 meters

To overtake the person, the train has to travel a distance equal only to its own length.

So, Length of faster train = Distance covered = 150 meters.

১২,৭৯৬.

Rana and Rahim invest TK 30,000 and 40,000 respectively in a partnership. They earned a profit of TK 7000. Find Rana's share of the profit

ক
TK 3500
খ
TK 4000
গ
TK 3000
ঘ
TK 4500

ব্যাখ্যা

Question: Rana and Rahim invest TK 30,000 and 40,000 respectively in a partnership. They earned a profit of TK 7000. Find Rana's share of the profit.

Solution:
Given that,
Investment ratio = 30,000 : 40,000
= 3 : 4

sum of the ratio's = 3 + 4 = 7
∴ Rana's share of the profit = 7000 × (3/7)
= 1000 × 3
= 3000 tk

১২,৭৯৭.

A sum of Tk. 4000 amounts to Tk. 4840 in 2 years at compound interest. Find the rate of interest per annum.

ক
5%
খ
6%
গ
8%
ঘ
10%

ব্যাখ্যা

Question: A sum of Tk. 4000 amounts to Tk. 4840 in 2 years at compound interest. Find the rate of interest per annum.

Solution:
Here, Principal, P = 4000 Tk.
Final amount, A = 4840 Tk.
Time, n = 2 years
Interest rate, r = ?

Now,
A = P × (1 + r/100)ⁿ⇒ 4840 = 4000 × (1 + r/100)²
⇒ (1 + r/100)² = 4840/4000
⇒ (1 + r/100)² = 121/100
⇒ 1 + r/100 = √(121/100)
⇒ 1 + r/100 = 11/10
⇒ r/100 = 11/10 - 1
⇒ r/100 = 1/10
⇒ r = (1/10) × 100
⇒ r = 10

∴ The annual rate of interest is 10%.

১২,৭৯৮.

A train running at the speed of 72 km/h crosses a pole in 20 seconds. What is the length of the train?

ক
400 m
খ
360 m
গ
380 m
ঘ
420 m

ব্যাখ্যা

Question: A train running at the speed of 72 km/h crosses a pole in 20 seconds. What is the length of the train?

Solution:
Given,
Speed = 72 km/h
= (72 × 5/18) m/sec
= 20 m/sec

Length of the train = Speed × Time
= 20 × 20 m
= 400 m

∴ Length of the train = 400 m

১২,৭৯৯.

Four numbers are in the ratio 1 : 2 : 3 : 4 and their HCF 12. The average of four numbers is -

ক
28
খ
25
গ
24
ঘ
30

ব্যাখ্যা

Question: Four numbers are in the ratio 1 : 2 : 3 : 4 and their HCF 12. The average of four numbers is -

Solution:
Let the four numbers are x, 2x, 3x, 4x respectively
Their HCF = x

ATQ,
x = 12

So, the four numbers are 12, 24, 36 and 48 respectively
The average of the four number = (12 + 24 + 36 + 48)/4 = 30

১২,৮০০.

A man’s speed with the current is 15 km/hr and the speed of the current is 2.5 km/hr. The man’s speed against the current is -

ক
ক) 8.5 km/hr
খ
খ) 9 km/hr
গ
গ) 10 km/hr
ঘ
ঘ) 12.5 km/hr

ব্যাখ্যা

Man's speed with the current = 15 km/hr
Speed of the man + Speed of the current = 15 km/hr
Speed of the current is 2.5 km/hr
Hence, Speed of the man = 15 - 2.5 = 12.5 km/hr
Man's speed against the current = (Speed of the man - Speed of the current)
= 12.5 - 2.5
= 10 km/hr.