Bank Math

PrepBank · পাতা ১১ / ১৬১ · ১,০০১–১,১০০ / ১৬,১২৪

১,০০১.

n is an integer between 40 and 90, then any of the following could be n + 8 except -

ক
52
খ
65
গ
75
ঘ
99

ব্যাখ্যা

Question: n is an integer between 40 and 90, then any of the following could be n + 8 except -

Solution:
if n is a positive integer, then 40 < n < 90
from option (4)
n + 8 = 99
n = 91

its not possible

১,০০২.

41 girls can complete a piece of work in 40 days. If x girls started the work and after 30 days 64 more girls joined them so that the whole work gets finished in the desired time, find the value of x.

ক
ক) 21
খ
খ) 25
গ
গ) 23
ঘ
ঘ) 27

ব্যাখ্যা

41 girls can complete a piece of work in 40 days.
x girls started the work
And after 30 days 64 more girls joined them

Concept Used:
Total work = Efficiency × Time
Here, we will take Efficiency = Number of workers
i.e Total work = Number of workers × Time

Calculation:
Total work = 41 × 40 = 1640 units

According to the question

For the second case
Total work = (x × 30) + (x + 64) × 10

So, we can write
(x × 30) + (x + 64) × 10 = 1640
⇒ 30x + 10x + 640 = 1640
⇒ 40x = 1000
⇒ x = 25

∴ The value of x is 25.

১,০০৩.

A circle has a number of tangents equal to-

ক
0
খ
1
গ
2
ঘ
3
ঙ
Infinite

ব্যাখ্যা

A circle has infinitely many tangents, touching the circle at infinite points on its circumference.

১,০০৪.

When 17 is divided by k, where K is a positive integer less than 17, the remainder is 3. What is the remainder when the sum of possible values of it is divided by 17?

ক
ক) 2
খ
খ) 3
গ
গ) 4
ঘ
ঘ) 5

ব্যাখ্যা

Question: When 17 is divided by k, where K is a positive integer less than 17, the remainder is 3. What is the remainder when the sum of possible values of it is divided by 17?

Solution:
যেহেতু k < 17, এটা বোঝায় যে k এর সম্ভাব্য মান অবশ্যই 1 থেকে 16 এর মধ্যে হতে হবে।
1 এবং 16 এর মধ্যে, শুধুমাত্র 7 বা 14 দ্বারা 17 কে ভাগ করতে ব্যবহৃত হয় অবশিষ্ট 3 দেবে
তাই, 7 এবং 14 হল k এর সম্ভাব্য মান।
k এর সম্ভাব্য মানের সমষ্টি = 7 + 14 = 21
k-এর সম্ভাব্য মানের সমষ্টি 21 কে 17 দ্বারা ভাগ করলে 4 ভাগশেষ থাকবে।

১,০০৫.

The ratio between the perimeter and the length of a rectangle is 5 : 2. If the area of the rectangle is 484 sq. cm, what is the length of the rectangle?

ক
11 cm
খ
16 cm
গ
23 cm
ঘ
32 cm
ঙ
44 cm

ব্যাখ্যা

Question: The ratio between the perimeter and the length of a rectangle is 5 : 2. If the area of the rectangle is 484 sq. cm, what is the length of the rectangle?

Solution:
Let Length = l
& Breadth = b
Perimeter of a rectangle = 2(l + b)

Now,
2(l + b)/l = 5/2
⇒ 4(l + b) = 5l
⇒ l = 4b

Area, l × b = 484
⇒ 4b × b = 484
⇒ b² = 121
⇒ b = 11

∴ l = 4 × 11 = 44
The length of the rectangle is 44 cm.

১,০০৬.

The length of two smaller sides of a right angled triangle are 5 cm and 12 cm respectively. The length of the third side is-

ক
ক) 16 cm
খ
খ) 17 cm
গ
গ) 19 cm
ঘ
ঘ) 13 cm

ব্যাখ্যা

আমরা জানি,
সমকোণী ত্রিভুজের ক্ষেত্রে অতিভূজ² = লম্ব² + ভূমি²
ধরি
অতিভূজ = x
লম্ব = 5
ভূমি = 12

এখন
x² = 5²+ 12²
x² = 25 + 144
x² = 169
x = 13

১,০০৭.

A sum of money amounts to 13675 taka after 5 years and 15880 taka after 8 years at the same rate of simple interest. What is the rate of interest per annum?

ক
7%
খ
8%
গ
7.85%
ঘ
7.35%
ঙ
12%

ব্যাখ্যা

Simple interest for 3 years = (15880 −13675) taka
= 2205 taka
Simple interest for 5 years = 2205 × 5/3 taka
= 3675 taka
Principle = (13675 - 3675) taka
= 10000 taka
Therefore, rate = 100 × 3675/(10000 × 5) %
= 7.35%

১,০০৮.

In a school election between two candidates, one candidate got 60% of the total valid votes. 10% of the votes were invalid. If the total votes were 20000, what is the number of valid votes the other candidate got?

ক
6660
খ
7200
গ
8000
ঘ
8060

ব্যাখ্যা

Question: In a school election between two candidates, one candidate got 60% of the total valid votes. 10% of the votes were invalid. If the total votes were 20000, what is the number of valid votes the other candidate got?

Solution:
Number of valid votes
= (100 - 10)% of 20000
= 90% of 20000
= (90/100) × 20000
= 18000

Valid votes polled by other candidates
= (100 - 60)% of 18000
= 40% of 18000
= (40/100) × 18000
= 7200

∴ The number of valid votes the other candidate got was 7200.

১,০০৯.

Rana sells 40 shares in a company via a stock broker who charges a flat Tk. 20 commission rate on all transactions under Tk. 1000. His bank account is credited with Tk. 692 from the sale of the shares. What price were his shares sold at?

ক
ক) Tk. 15.5
খ
খ) Tk. 21.5
গ
গ) Tk. 19.5
ঘ
ঘ) Tk. 17.8

ব্যাখ্যা

কমিশনসহ রেজার শেয়ারের বিক্রয়মূল্য ৬৯২+২০ = ৭১২ টাকা।
তাহলে ৪০ শেয়ারের বিক্রয়মূল্য ৭১২/৪০ = ১৭.৮।

১,০১০.

The diagonal of a rectangle is √41cm and its area is 20 sq. cm. The perimeter of the rectangle must be-

ক
18cm
খ
27cm
গ
20cm
ঘ
30cm
ঙ
24cm

ব্যাখ্যা

Question: The diagonal of a rectangle is √41cm and its area is 20 sq. cm. The perimeter of the rectangle must be-

Solution:
আমরা জানি, যদি আয়তক্ষেত্রের দৈর্ঘ্য x এবং প্রস্থ y হয়, তাহলে কর্ণ (diagonal) হবে,
√(x² + y²) = √41
⇒ x² + y² = 41
এবং ক্ষেত্রফল, xy = 20

আমরা জানি,
(x + y)²= x² + y² + 2xy
= 41 + (2 × 20)
= 81
⇒ x + y = √81 = 9
∴ x + y = 9

আয়তক্ষেত্রের পরিসীমার, Perimeter = 2(x + y) = 18cm

১,০১১.

The speed of a boat in still water is 10 km/hr. If it can travel 26 km downstream and 14 km upstream at the same time, the speed of the stream is-

ক
ক) 2 km/hr
খ
খ) 2.5 km/hr
গ
গ) 3 km/hr
ঘ
ঘ) 3.5 km/hr
ঙ
ঙ) 4 km/hr

ব্যাখ্যা

Let the speed of the stream be x km/hr
Then speed downstream = (10 + x) km/hr
Speed upstream
= (10−x)km/hr
∴ 26/(10+x) = 14/(10−x)
⇒ 260−26x = 140+14x
⇒ 40x = 120
⇒ x = 3km/hr

১,০১২.

cosec(90° - θ) = 3/2. What is the value of tanθ?

ক
√3/4
খ
√5/2
গ
√5/4
ঘ
5/4

ব্যাখ্যা

Question: cosec(90° - θ) = 3/2. What is the value of tanθ?

Solution:
cosec(90° - θ) = 3/2
or, secθ = 3/2
or, sec²θ = 9/4
or, 1 + tan²θ = 9/4
or, tan²θ = 9/4 - 1
or, tan²θ = 5/4
∴ tanθ = √5/2

১,০১৩.

There are peacock and horse in a zoo. The total number of their heads is 50 and the number of legs is 140. How many horses are there?

ক
20
খ
25
গ
15
ঘ
30

ব্যাখ্যা

Question: There are peacock and horse in a zoo. The total number of their heads is 50 and the number of legs is 140. How many horses are there?

Solution:
Let, there are x horses.

ATQ,
4x + (50 - x)2 = 140
4x + 100 - 2x = 140
2x = 40
x = 20

১,০১৪.

If log₂(a) = 3 and log₂(b) = 4, what is the value of log₂(ab) =?

ক
12
খ
28
গ
14
ঘ
7

ব্যাখ্যা

Question: If log₂(a) = 3 and log₂(b) = 4, what is the value of log₂(ab) =?

Solution:
given that,
log₂(a) = 3 and log₂(b) = 4

Now,
log₂(ab) = log₂(a) + log₂(b) ;[log_b(mn) = log_b(m) + log_b(n)]
= 3 + 4
= 7

১,০১৫.

The sum of the digits of two-digit number is 8. If the digits are reversed the number is decreased by 54. What is the number?

ক
53
খ
71
গ
37
ঘ
73

ব্যাখ্যা

Question: The sum of the digits of two-digit number is 8. If the digits are reversed the number is decreased by 54. What is the number?

Solution:
Let the two-digit number be 10x + y,
where, x = tens digit and y = ones digit.

Given,
1st conditions,
x + y = 8
x = 8 - y .......(1)

And 2nd conditions,
10x + y - (10y + x) = 54
⇒ 9x - 9y = 54
⇒ 9(8 - y) - 9y = 54
⇒ 72 - 9y - 9y = 54
⇒ - 18y = 54 - 72
⇒ y = - 18/- 18
∴ y = 1

From equation (1) we get,
x = 8 - y = 8 - 1 = 7
∴ x = 7

So the number is = 10 x + y = 10(7) + 1 = 71

১,০১৬.

In a particular business, X and Z invested the amounts in the ratio of 2 : 1, while the amount invested X and Y is 3 : 2. If the total profit was Tk. 1,57,300, then find the amount that Y received.

ক
Tk. 48,400
খ
Tk. 72,600
গ
Tk. 36,300
ঘ
Tk. 22,400

ব্যাখ্যা

Question: In a particular business, X and Z invested the amounts in the ratio of 2 : 1, while the amount invested X and Y is 3 : 2. If the total profit was Tk. 1,57,300, then find the amount that Y received.

Solution:
X : Z = 2 : 1
⇒ Z : X = 1 : 2
= 3 : 6

X : Y = 3 : 2
= 6 : 4

Z : X : Y = 3 : 6 : 4

the amount that Y received = 4 × 1,57,300/13
= Tk. 48400

১,০১৭.

A fort had provision of food for 150 men for 45 days. After 10 days, 25 men left the fort. The number of days for which the remaining food will last, is-

ক
29.5
খ
37.25
গ
42
ঘ
54
ঙ
None of these

ব্যাখ্যা

Question: A fort had provision of food for 150 men for 45 days. After 10 days, 25 men left the fort. The number of days for which the remaining food will last, is-

Solution:
fter 10 days : 150 men had food for 35 days.
Suppose 125 men had food for x days.

Now, Less men, More days (Indirect Proportion)
125 : 150 : : 35 : x
⇒ 125/150 = 35/x
⇒ 125 x = 150 × 35
⇒ x = (150 × 35)/125
∴ x = 42.

১,০১৮.

cosθ = (1/2){a + (1/a)}, then cos3θ =?

ক
3/2(a + (1/a))
খ
3/2(a³ + (1/a³))
গ
1/2(a + (1/a))
ঘ
1/2(a³ + (1/a³))

ব্যাখ্যা

১,০১৯.

The slope of a line perpendicular to one with slope 2 is:

ক
- 1/2
খ
- 3
গ
1/3
ঘ
2
ঙ
1/2

ব্যাখ্যা

Question: The slope of a line perpendicular to one with slope 2 is:

Solution:
আমরা জানি,
যেকোনো সরলরেখার ঢাল যদি m হয়, তাহলে তার উপর লম্ব রেখার ঢাল হবে: - 1/m

এখানে,
ঢাল m = 2
∴ লম্ব রেখার ঢাল = - 1/2

১,০২০.

If you divide 30 by half and add 10 with the resulting figure, then what is the final result?

ক
ক) 25
খ
খ) 70
গ
গ) 45
ঘ
ঘ) 55

ব্যাখ্যা

Question: If you divide 30 by half and add 10 with the resulting figure, then what is the final result?

Solution:
dividing 30 by half we get = 30/(1/2) = 60

adding 10 we get = 60 + 10 = 70

১,০২১.

In the first 15 overs of a cricket match, the run rate was 6.8 runs per over. What should be the required run rate in the remaining 35 overs to reach a target of 300 runs?

ক
5.66
খ
5.14
গ
4.86
ঘ
6.25

ব্যাখ্যা

Question: In the first 15 overs of a cricket match, the run rate was 6.8 runs per over. What should be the required run rate in the remaining 35 overs to reach a target of 300 runs?

Solution:
First 15 overs total run was = (6.8 × 15) = 102

∴ Required run rate = (300 - 102)/35
= 198/35
= 5.66

∴ Required run rate 5.66 runs per over

১,০২২.

The sum of four numbers is 64. If you add 3 to the first number, 3 is subtracted from the second number, the third is multiplied by 3 and the fourth is divided by 3, then all the results are equal. What is the difference between the largest and the smallest of the original numbers?

ক
ক) 21
খ
খ) 27
গ
গ) 32
ঘ
ঘ) 36

ব্যাখ্যা

Let the four numbers be, A, B, C and D
Let A + 3 = B - 3 = 3C = D/3 = x
Then,
A = x - 3
B = x + 3
C = x/3
D = 3x
⇒ (A + B + C + D) = 64
⇒ (x - 3) + (x + 3) + x/3 + 3x) = 64
⇒ 5x + x/3 = 64
⇒ 16x = 192
⇒ x = 12.
Thus the numbers are 9, 15, 4 and 36.
Required difference
= 36 - 4
= 32.

১,০২৩.

A alone can do a piece of work in 6 days, and B alone in 8 days. A and B undertook to do it for Tk. 3200. With the help of C, they completed the work in 3 days. How much is to be paid to C?

ক
320 Tk.
খ
400 Tk.
গ
450 Tk.
ঘ
560 Tk.

ব্যাখ্যা

Question: A alone can do a piece of work in 6 days, and B alone in 8 days. A and B undertook to do it for Tk. 3200. With the help of C, they completed the work in 3 days. How much is to be paid to C?

Solution:
C's 1 day's work = (1/3) - {(1/6) + (1/8)}
= (1/3) - (7/24)
= 1/24

Now,
A's wages : B's wages : C's wages = (1/6) : (1/8) : (1/24)
= (1/6) × 24 : (1/8) × 24 : (1/24) × 24
= 4 : 3 : 1

∴ C's share (for 3 days) = 3 × (1/24) × 3200
= 400 Tk.

১,০২৪.

If 8 × 2^x = (1/16), what is the value of x?

ক
ক) 1
খ
খ) - 7
গ
গ) 7
ঘ
ঘ) -8

ব্যাখ্যা

প্রশ্ন: If 8 × 2^x = (1/16), what is the value of x?

সমাধান:
8 × 2^x= (1/16)
⇒ 2^x = (1/16 × 8)
⇒ 2^x= 1/(2⁴ × 2³)
⇒ 2^x = 1/2⁷
⇒ 2^x= 2 ^{- 7}
∴ X = -7

১,০২৫.

A tree leaned due to storm. The stick with height of 7 meter from its foot was leaned against the tree to make it straight. If the angle of depression at the point of contacting with the stick on the ground is 30°, find the length of the stick.

ক
10 m
খ
12 m
গ
14 m
ঘ
16 m

ব্যাখ্যা

Question: A tree leaned due to storm. The stick with height of 7 meter from its foot was leaned against the tree to make it straight. If the angle of depression at the point of contacting with the stick on the ground is 30°, find the length of the stick.

Solution:

মনে করি,
খুঁটিটির দৈর্ঘ্য BC = x মিটার,
গাছের গোড়া থেকে AB = 7 মিটার উচ্চতায় খুঁটিটি ঠেস দিয়ে আছে এবং অবনতি ∠DBC = 30°
∠ACB = ∠DBC = 30° [একান্তর কোণ বলে]

সমকোণী ΔABC থেকে পাই,
sin∠ACB = AB/BC
বা, sin30° = 7/x
বা, 1/2 = 7/x
∴ x = 14

∴ খুঁটিটির দৈর্ঘ্য 14 মিটার।

১,০২৬.

{(0.6)⁴- (0.5)⁴}/{(0.6)² + (0.5)²} is equal to-

ক
ক) 0.11
খ
খ) 0.011
গ
গ) 1.11
ঘ
ঘ) 0.001

ব্যাখ্যা

{(0.6)⁴- (0.5)⁴}/{(0.6)² + (0.5)²}
= [{(0.6)²}² - {(0.5)²}²]/{(0.6)² + (0.5)²}
= {(0.6)² + (0.5)²}{(0.6)² - (0.5)²}/{(0.6)² + (0.5)²}
= {(0.6)² - (0.5)²}
= (0.6 + 0.5)(0.6 - 0.5)
= 1.1 × 0.1
= 0.11

১,০২৭.

A pipe can fill 1/4th of a tank in 30 minutes. How much time will it take to fill two tanks?

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

ব্যাখ্যা

Question: A pipe can fill 1/4th of a tank in 30 minutes. How much time will it take to fill two tanks?

Solution:
A pipe can fill 1/4th of a tank in 30 minutes
∴ A pipe can fill 1 part or full of a tank in = (4 × 30)
= 120 minutes = 2 hours

One tank is filled in 2 hours
∴ two tanks is filled in = 4 hours

১,০২৮.

A vessel is filled with liquid, 3 parts of which are water and 5 parts syrup. How much of mixture must be drawn off and replaced with water so that the mixture may be half water and half syrup?

ক
ক) 1/3
খ
খ) 1/4
গ
গ) 1/5
ঘ
ঘ) 1/7

ব্যাখ্যা

Let initially vessel have 8 litres of liquid and x litres of this liquid be replaced with water,
Then quantity of water in new mixture = 3 - 3x/8 + x
Quantity of syrup in new mixture = 5 - 5x/8
According to the question,
After replacement, the quantity water and syrup same,
(3 - 3x/8 + x) = (5 - 5x/8)
⇒ -3x/8 + x + 5x/8 = 5 - 3
⇒ (-3x + 8x + 5x)/8 = 2
⇒ 10x/8 = 2
⇒ x = 8/5
So, part of the mixture replaced, 8/5 × 1/8 ⇒ 1/5

∴ 1/5 of the mixture must be drawn off and replaced with water so that the mixture may be half water and half syrup.

১,০২৯.

A certain sum will amount to Tk. 12100 in 2 years at 10% per annum of compound interest, interest being compounded annually. The sum is = ?

ক
Tk. 12000
খ
Tk. 6000
গ
Tk. 8000
ঘ
Tk. 10000

ব্যাখ্যা

Question: A certain sum will amount to Tk. 12100 in 2 years at 10% per annum of compound interest, interest being compounded annually. The sum is = ?

Solution:
Given,
Amount = 12,100; r = 10%, t = 2 yrs

১,০৩০.

The average of five numbers is 7. If three new numbers would be added, then the new average comes out to be 8.5. What is the average of those three new numbers?

ক
6
খ
9
গ
8
ঘ
11
ঙ
13

ব্যাখ্যা

Question: The average of five numbers is 7. If three new numbers would be added, then the new average comes out to be 8.5. What is the average of those three new numbers?

Solution:
Total of five numbers = 7 × 5 = 35
Total of 8 numbers = 8 × 8.5 = 68
Total of three new number = 68 - 35 = 33

∴ Average of those three new numbers = 33/3 = 11

১,০৩১.

The cost price of a Tk 100 stock at 4 discount, when brokerage is 1/4% is:

ক
ক) 95.75
খ
খ) 96
গ
গ) 96.25
ঘ
ঘ) 104.25

ব্যাখ্যা

C.P. = (100 - 4 + 1/4 ) = tk 96.25

১,০৩২.

If a bookstore owner buys 15 books less for Tk. 900 when the price of each book goes up by Tk. 3, then find the original price of a book.

ক
ক) Tk. 20
খ
খ) Tk. 18
গ
গ) Tk. 15
ঘ
ঘ) Tk. 12

ব্যাখ্যা

Since the number of books is not given, I could not plug in the values. So this is solved using quadratic equations.
Let the number of books bought initially for Tk. 900 be 'x'. So the original price of the book was 900/x
Now price of the book is up by 3. i.e., (900/X) + 3. and number of books bought is reduced by 15. i.e., (x - 15)
Since new total amount spent is still same, the product of new price and new number of books is still 900

[(900/x) + 3] (x - 15) = 900
Or, (900 + 3x) (x - 15) = 900x
Or, 3x² + 855 - 13500 = 900x Now use quadratic equation to find the value of x.
Or, x² - 15x - 4500 = 0
Or, x² - 75x + 60x - 4500 = 0
Or, x(x - 75) + 60(x - 75) = 0
Or, (x - 75) (x + 60) = 0
So, x = 75 or x = - 60
since x cannot be negative, so, x = 75
∴ the original price of the book = 900/75 = 12

১,০৩৩.

If the average marks of three batches of 55,60 and 45 students respectively is 50,55,60, what is the average marks of all the students?

ক
50
খ
51.33
গ
53.23
ঘ
54.68

ব্যাখ্যা

Students in batch1 = 55
Average marks of batch 1 = 50
Total marks of batch 1 = 55 × 50 = 2750

Students in batch 2 = 60
Average marks of batch 2 = 55
Total marks of batch 2 = 60 × 55 = 3300

Students in batch 3 = 45
Average marks of batch3 = 60
Total marks of batch 3 = 45 × 60 = 2700

Total marks = 2750 + 3300 + 2700 = 8750
Total students = 55 + 60 + 45 = 160

Average marks of all students = 8750/160 = 54.68

১,০৩৪.

A football team has 15 players. In how many ways can a team of 11 players be chosen if the goalkeeper must always be included?

ক
455
খ
1001
গ
1365
ঘ
3003

ব্যাখ্যা

Question: A football team has 15 players. In how many ways can a team of 11 players be chosen if the goalkeeper must always be included?

সমাধান:
15 জন খেলোয়াড় থেকে 1 জনকে (গোলরক্ষক) ঠিক রেখে বাকি 14 জন থেকে (11 - 1) = 10 জনের টিম গঠন করা যাবে।

∴ অবশিষ্ট 14 জন থেকে 10 জনকে নির্বাচন করার উপায় = ¹⁴C₁₀= 14!/(14 - 10)! × 10!
= (14 × 13 × 12 × 11 × 10!)/(4! × 10!)
= (14 × 13 × 12 × 11)/(4 × 3 × 2 × 1)
= 1001

∴ 1001 উপায়ে দল গঠন করা যাবে।

১,০৩৫.

What are the solutions to the equation x² - 2x - 2 = 0

ক
ক) (1 + √2), (1 - √2)
খ
খ) (1 + √3), (1 - √3)
গ
গ) (1 + √5), (1 - √5)
ঘ
ঘ) (1 + √7), (1 - √7)

ব্যাখ্যা

Question: What are the solutions to the equation x² - 2x - 2 = 0

Solution:
Given that
x² - 2x - 2 = 0.........(1)
Comparing ax² + bx + c = 0 with (1) get, a = 1, b = - 2 and c = - 2

We know
x = {(- b) ± √(b² - 4ac)}/2a
= [{- (- 2)} ± √{(- 2)² - 4.1(- 2)]/2.1
= (2 ± √12)/2
=(2 ± 2√3)/2
= 1 ± √3
= (1 + √3), (1 - √3)

১,০৩৬.

If 4^x-1 + 4^x+1 = 4352, then find the value of x.

ক
ক) 0
খ
খ) 1
গ
গ) 2
ঘ
ঘ) 5

ব্যাখ্যা

Question: If 4^x-1 + 4^x+1 = 4352, then find the value of x.

Solution:
Given that,
4^x-1 + 4^x+1 = 4352
⇒ 4^x/4 + 4^x × 4 = 4352
⇒ 4^x(1/4 + 4) = 4532
⇒ 4^x × (17/4) = 4532
⇒ 4^x = (4532 × 4)/17
⇒ 4^x = 1024
⇒ 4^x = 4⁵
∴ x = 5

১,০৩৭.

The average age of 9 students and their teacher is 16 years. The average age of the first four students is 19 years and that of the last five is 10 years. The teacher's age is -

ক
34 years
খ
32 years
গ
30 years
ঘ
26 years

ব্যাখ্যা

Question: The average age of 9 students and their teacher is 16 years. The average age of the first four students is 19 years and that of the last five is 10 years. The teacher's age is -

Solution:
ATQ,
The average age of nine students and teachers = 16 years
Then, the total average age of students and teachers = 16 × 10 = 160
And, the average age of the first 4 students = 19 × 4 = 76
Average age of last 5 students = 10 × 5 = 50

∴ Teacher's age = 160 - 76 - 50 = 34 years

১,০৩৮.

X and Y start walking towards each other at 10 am at speeds of 3 km/hr and 4 km/hr respectively. They were initially 17.5 km apart. At what time do they meet?

ক
ক) 2:30 pm
খ
খ) 11 :30 pm
গ
গ) 1:30 pm
ঘ
ঘ) 12 :30 pm

ব্যাখ্যা

Let after T hours they meet
Then, 3T+4T=17.5
T=2.5
Time = 10:00 am + 2.5 hour = 12:30 pm

১,০৩৯.

The three angles of a triangle are x/3, x/3, and 4x/3 respectively. What is the sum of the two smallest angles?

ক
45°
খ
60°
গ
90°
ঘ
120°

ব্যাখ্যা

Question: The three angles of a triangle are x/3, x/3, and 4x/3 respectively. What is the sum of the two smallest angles?

Solution:
দেওয়া আছে,
ত্রিভুজের তিনটি কোণ যথাক্রমে x/3, x/3, এবং 4x/3

প্রশ্নমতে,
(x/3) + (x/3) + (4x/3) = 180°
⇒ (x + x + 4x)/3 = 180°
⇒ 6x/3 = 180°
⇒ 2x = 180°
⇒ x = 180°/2
⇒ x = 90°

এখন,
১ম কোণ = 90°/3 = 30°
২য় কোণ = 90°/3 = 30°
৩য় কোণ = (4 × 90°)/3 = 120°

∴ ক্ষুদ্রতম কোণ দুইটির সমষ্টি = 30° + 30° = 60°

১,০৪০.

A bike rider starts at 60 km/hr and he increases his speed every 2 hours by 3 km/hr. Then the maximum distance covered by him in 24 hours is -

ক
1000km
খ
918km
গ
899 km
ঘ
none of these

ব্যাখ্যা

বাইকারের যাত্রা শুরুর গতি = 60 কি.মি/ঘন্টা
প্রথম 2 ঘন্টায় যায় (60 × 2) = 120 কি.মি
পরের 2 ঘন্টায় যায় (63 × 2) = 126 কি.মি
আবার, পরের 2 ঘন্টায় যায় (66 × 2) = 132 কি.মি
এরকম 12 বার হবে।
সুতরাং, এটি একটি সমান্তর ধারা।

ধারাটি হবে, 120 + 126 + 132 + ------
এখানে ১ম পদ (a) = 120, সাধারণ অন্তর (d) = 6 ও পদসংখ্যা (n) = 12
∴ সমষ্টি (S) = (n/2) {2a + (n - 1)d}
= (12/2) { 2×120 + (12 - 1) × 6}
= 6 × 306
= 1836

সুতরাং, সঠিক উত্তর ঘ) none of these

১,০৪১.

ক
9
খ
5/4
গ
10/9
ঘ
7/8

ব্যাখ্যা

Question:

Solution:

১,০৪২.

A worker earns Tk. 250 on the first day and spends Tk. 200 on the second day, earns Tk. 250 on the third day and again spends Tk. 200 on the fourth day and so on. On which day would he have had Tk. 1000?

ক
40th day
খ
31th day
গ
30th day
ঘ
20th day

ব্যাখ্যা

Question: A worker earns Tk. 250 on the first day and spends Tk. 200 on the second day, earns Tk. 250 on the third day and again spends Tk. 200 on the fourth day and so on. On which day would he have had Tk. 1000?

Solution:
১ম দিনে আয় করে ২৫০ টাকা।
২য় দিনে ব্যয় করে ২০০ টাকা।

∴ ২ দিনে তার জমা থাকে (২৫০ - ২০০) = ৫০ টাকা।

এখন, (১০০০ - ২৫০) = ৭৫০ টাকা।

৫০ টাকা জমা থাকে ২ দিনে
১ টাকা জমা থাকে ২/৫০ দিনে
৭৫০ টাকা জমা থাকে (২ × ৭৫০)/৫০ দিনে
= ৩০ দিন।

৩০ দিন পর তার হাতে থাকে ৭৫০ টাকা
এবং ৩১ তম দিনে সে আয় করে ২৫০ টাকা।
তাহলে মোট টাকা হয় (৭৫০ + ২৫০) = ১০০০ টাকা,

সুতরাং ৩১ দিনে তার কাছে ১০০০ টাকা ছিল।

১,০৪৩.

(289)^0.17 × (17)^0.16= ?

ক
4
খ
√7
গ
√17
ঘ
√19

ব্যাখ্যা

Question: (289)^0.17 × (17)^0.16= ?

Solution:
Given that,
(289)^0.17 × (17)^0.16
= (17²)^0.17 × (17)^0.16
= (17)^0.34 ×(17)^0.16
= (17)^{0.34 + 0.16}
= (17)^0.50
= (17)^1/2
= √17

১,০৪৪.

What is the angle formed between the hour hand and the minute hand of a clock at 3:20 PM?

ক
20°
খ
30°
গ
40°
ঘ
50°
ঙ
None

ব্যাখ্যা

Question: What is the angle formed between the hour hand and the minute hand of a clock at 3:20 PM?

Solution:
3টা 20 মিনিট = 3 + (20/60) ঘন্টা
= 3 + 1/3 = 10/3 ঘন্টা

আমরা জানি,
ঘণ্টার কাঁটা 12 ঘণ্টায় 360° ঘোরে।
∴ 1 ঘণ্টায় ঘোরে = 360°/12 = 30°
∴ 10/3 ঘণ্টায় ঘোরে = 30 × 10/3 = 100°

আবার,
মিনিটের কাঁটা 60 মিনিটে 360° ঘোরে।
∴ 1 মিনিটে ঘোরে = 360°/60 = 6°
∴ 20 মিনিটে ঘোরে = 20 × 6 = 120°

∴ ঘড়ির কাঁটা দুটির মধ্যবর্তী কোণ = |120° - 100°| = 20°

১,০৪৫.

On 3 sales Shazan has received commissions of Tk. 240, Tk. 80, and Tk. 110, and he has 1 additional sale pending. If Shazan is to receive an average (arithmetic mean) commission of exactly Tk. 150 on the 4 sales, then the 4^th commission must be-

ক
Tk. 164
খ
Tk. 170
গ
Tk. 175
ঘ
Tk. 182
ঙ
Tk. 185

ব্যাখ্যা

Question: On 3 sales Shazan has received commissions of Tk. 240, Tk. 80, and Tk. 110, and he has 1 additional sale pending. If Shazan is to receive an average (arithmetic mean) commission of exactly Tk. 150 on the 4 sales, then the 4^th commission must be-

Solution:
Let,
The 4^th commission be x

(240 + 80 + 110 + x)/(4) =150
⇒ 430 + x = 600
⇒ x = 600 - 430
∴ x = 170

১,০৪৬.

In a school having roll strength 324, the ratio of boys and girls is 8 : 4. If 27 more girls get admitted into the school, the ratio of boys and girls becomes?

ক
8 : 5
খ
3 : 5
গ
7 : 3
ঘ
6 : 5

ব্যাখ্যা

Question: In a school having roll strength 324, the ratio of boys and girls is 8 : 4. If 27 more girls get admitted into the school, the ratio of boys and girls becomes?

Solution:
let the boys = 8x,
and the girls = 4x

ATQ,
8x + 4x = 324
⇒ 12x = 324
⇒ x = 324/12
∴ x = 27

Boys = 8 × 27 = 216
and girls = 4 × 27 = 108
27 more girls get admitted then number of girls become = 108 + 27 = 135
Now, new ratio of boys and girls = 216 : 135
= 8 : 5

১,০৪৭.

A can do a work in 8 days and B in 12 days. If they work on it together for 4 days, then the fraction of the work that is left is:

ক
1/12 part
খ
5/8 part
গ
3/8 part
ঘ
1/6 part

ব্যাখ্যা

Question: A can do a work in 8 days and B in 12 days. If they work on it together for 4 days, then the fraction of the work that is left is-

Solution:
A's 1 day's work = 1/8
B's 1 day's work = 1/12

∴ (A + B)'s 1 day's work = (1/8 + 1/12) part
= (3 + 2)/24 part
= 5/24 part

∴ (A + B)'s 4 day's work = (5/24 × 4) part
= 5/6 part

Therefore, Remaining work = (1 - 5/6) part
= 1/6 part

১,০৪৮.

An 80L solution of alcohol and water has 45% alcohol in it. If you want the mixture to be 75% alcohol, how much alcohol would you add to it?

ক
ক) 30 litres
খ
খ) 75 litres
গ
গ) 96 litres
ঘ
ঘ) 110 litres

ব্যাখ্যা

Current alcohol quantity = {(45/100) × 80}
= 36 Litres.

Let A be alcohol added.
So, 36 + A = (75/100) × (80 + A)
⇒ 36 + A = (3/4) × (80 + A)
⇒ 144 + 4A = 240 + 3A
∴ A = 96 Litres =

96 Litres is the additional quantity of alcohol to be added.

১,০৪৯.

If the radius of a circle is decreased to half of its previous radius. What is its present area compared to the previous area?

ক
1/2
খ
1/4
গ
1/8
ঘ
1/16

ব্যাখ্যা

Question: If the radius of a circle is decreased to half of its previous radius. What is its present area compared to the previous area?

Solution:
let the radius of the circle is r.
area = π(r)² = πr²

new radius = r/2

area = π(r/2)²
= πr²/4

so, the new area is = 1/4 of the previous area.

১,০৫০.

X is four years older than Y, who is three times as old as Z. If the sum of their ages is 39, then how old is Y?

ক
8 years
খ
12 years
গ
15 years
ঘ
18 years

ব্যাখ্যা

Question: X is four years older than Y, who is three times as old as Z. If the sum of their ages is 39, then how old is Y?

Solution:
Let the age of Z be = x years
Then, the age of Y = 3x years
and age of X = (3x + 4) years

According to the question,
(3x + 4) + 3x + x = 39
⇒ 7x + 4 = 39
⇒ 7x = 39 - 4
⇒ 7x = 35
⇒ x = 5

Hence, age of Y = 3x = (3 × 5) years = 15 years

১,০৫১.

There are 600 boys in a hostel. Each plays either hockey and football or both. If 75% play hockey and 45% play football, how many play both?

ক
ক) 340
খ
খ) 200
গ
গ) 120
ঘ
ঘ) 150

ব্যাখ্যা

The number of hockey players n(A) = 600 × 75%
= 450
The number of football players n(B) = 600 × 45%
= 270

n(A ∪ B) = 600
n(A ∩ B) = n(A) + n(B) - n(A ∪ B)
= 450 + 270 - 600
= 120

১,০৫২.

Which one is the acute angled triangle?

ক
50°, 40°, 90°
খ
30°, 30°, 120°
গ
50°, 70°, 60°
ঘ
50°, 30°, 100°
ঙ
45°, 45°, 90°

ব্যাখ্যা

৯০° অপেক্ষা ছোট কোণকে সূক্ষ্মকোণ বলে। আর যে ত্রিভুজের তিনটি কোণই সূক্ষ্মকোণ, তাকে সূক্ষ্মকোণী ত্রিভুজ বলে।

১,০৫৩.

What is the probability that an integer selected at random from those between 10 and 100 inclusive is a multiple of 5 or 11?

ক
ক) 26/91
খ
খ) 27/91
গ
গ) 29/91
ঘ
ঘ) None of the above

ব্যাখ্যা

Question: What is the probability that an integer selected at random from those between 10 and 100 inclusive is a multiple of 5 or 11?

Solution:
10 থেকে 100 এর মধ্যে 5 এর গুণিতক সংখ্যা গুলো হলো: 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100 = 19টি

10 থেকে 100 এর মধ্যে 11 এর গুণিতক সংখ্যা গুলো হলো: 11, 22, 33, 44, 55, 66, 77, 88, 99 = 9টি
মোট গুণিতক = (19 + 9)টি = 28

55 উভয়ের গুণিতক।
মোট অনুকূল ফলাফল = 28 - 1 = 27

10 থেকে 100 এর মধ্যে মোট সংখ্যা = 91টি

নির্ণেয় সম্ভাবনা = 27/91

১,০৫৪.

Two candidates contested an election. The losing candidate got 40% votes and lost by 2000 votes. Find the total number of votes cast.

ক
5,000
খ
8,000
গ
10,000
ঘ
20,000

ব্যাখ্যা

Let,
Total votes = a.
This means that Votes of candidate 1 + Votes of candidate 2 = a
We know that,
Votes of candidate 1 = 40% of a
= 40a/100
Hence, Votes of candidate 2 = (100% - 40%) of a
= 60% of a
= 60a/100

1^stcandidate lost by 1000 votes = difference of votes between both candidates
60a/100 - 40a/100 = 2000
∴ a = 10,000.

১,০৫৫.

In a crime, three suspects X, Y, and Z were caught and questioned. Each person said, ''One of the other two stole it. I did not do it. '' Later on the police found out that Z was lying and there was only one thief. Who was the thief?

ক
ক) X
খ
খ) Y
গ
গ) Z
ঘ
ঘ) Someone else

ব্যাখ্যা

এখানে, প্রত্যেকজন অন্য দুজনকে চোর বলছিলো।
যেহেতু Z মিথ্যা কথা বলেছিল এবং চোর একজনই তাই Z হচ্ছে প্রকৃত চোর।

১,০৫৬.

According to meteorological records, it rained on 14 days in the month of september last year. What is the probability that it will rain on fourth of september this year?

ক
7/15
খ
6/17
গ
5/13
ঘ
8/15

ব্যাখ্যা

Question: According to meteorological records, it rained on 14 days in the month of september last year. What is the probability that it will rain on fourth of september this year?

Solution:
September month has 30 days
favorable events = 14 days

∴ the probability that it will rain on fourth of september this year = 14/30
= 7/15

১,০৫৭.

A person takes a loan of Tk. 200 at 5% simple interest. He returns Tk.100 at the end of one year. In order clear his dues at the end of 2 year, he would pay :

ক
ক) Tk. 100
খ
খ) Tk. 105
গ
গ) Tk. 115
ঘ
ঘ) Tk. 110

ব্যাখ্যা

Amount to be paid = Tk.[100+(200×5×1)/100+(100×5×1)/100] = Tk. 115.

১,০৫৮.

In what ratio must a grocer mix two varieties of pulses costing Tk. 15 and Tk. 20 per kg respectively so as to get a mixture worth Tk. 16.50 kg?

ক
7 : 3
খ
5 : 7
গ
3 : 7
ঘ
7 : 5

ব্যাখ্যা

Question: In what ratio must a grocer mix two varieties of pulses costing Tk. 15 and Tk. 20 per kg respectively so as to get a mixture worth Tk. 16.50 kg?

Solution:
Let
15 Tk kg pulses = x kg and 20 Tk kg pulses = y kg
ATQ,
15x + 20y = 16.5(x + y)
⇒ 15x + 20y = 16.5x + 16.5y
⇒ 1.5x = 3.5y
⇒ x/y = 3.5/1.5
⇒ x/y = 7/3
⇒ x : y = 7 : 3

১,০৫৯.

A sales representative sells goods worth Tk. 22,000. If his commission rate is 12.5%, what is the amount of commission he earns?

ক
2750
খ
2820
গ
3560
ঘ
3550

ব্যাখ্যা

Question: A sales representative sells goods worth Tk. 22,000. If his commission rate is 12.5%, what is the amount of commission he earns?

Solution:
Commission = 12.5% of 22000
= (125/10) × (1/100) × 22000
= (125/1000) × 22000
= 2750

১,০৬০.

If (2a - 2b) = 2 and (3a + 2b) = 8 then find the value of (3a + 4).

ক
5
খ
6
গ
8
ঘ
10

ব্যাখ্যা

Question: If (2a - 2b) = 2 and (3a + 2b) = 8 then find the value of (3a + 4).

Solution:
2a - 2b = 2 ...........(1)
3a + 2b = 8 .............(2)

From (1) we get,
2a - 2b = 2
⇒ a - b = 1
⇒ a = b + 1 .......(3)

From (2) we get,
3(b + 1) + 2b = 8
⇒ 3b + 3 + 2b = 8
⇒ 5b = 5
∴ b = 1
From (3) we get,
a = 1 + 1
∴ a = 2

Now,
3a + 4
= (3 × 2) + 4
= 6 + 4
= 10

১,০৬১.

The simple interest on a certain sum of money for 5 years at 10% per annum is Tk. 150 less than the simple interest on the same sum for 7 years at 8% per annum. Find the sum.

ক
Tk. 7500
খ
Tk. 3500
গ
Tk. 2500
ঘ
Tk. 1500

ব্যাখ্যা

Question: The simple interest on a certain sum of money for 5 years at 10% per annum is Tk. 150 less than the simple interest on the same sum for 7 years at 8% per annum. Find the sum.

Solution:
Let,
sum = x taka

ATQ,
{x × (8/100) × 7} - {x × (10/100) × 5} = 150
⇒ (14x/25) - (x/2) = 150
⇒ (28x - 25x)/50 = 150
⇒ 3x/50 = 150
⇒ x/50 = 50
∴ x = 2500

১,০৬২.

A number consists of two digits such that the digit in the ten's place is less by 2 than the digit in the unit's place. Three times the number added to 6/7 times the number obtained by reversing the digits 108. The sum of the digits in the number is -

ক
ক) 4
খ
খ) 5
গ
গ) 8
ঘ
ঘ) 6

ব্যাখ্যা

Question: A number consists of two digits such that the digit in the ten's place is less by 2 than the digit in the unit's place. Three times the number added to 6/7 times the number obtained by reversing the digits 108. The sum of the digits in the number is -

Solution:
ধরি,
একক স্থানীয় অঙ্কটি x এবং দশক স্থানীয় অঙ্কটি (x - 2)

সংখ্যাটি = 10((x - 2) + x
= 11x - 20

স্থান বিনিময়কৃত সংখ্যাটি = 10x + x - 2
= 11x - 2

ATQ,
3(11x - 20) + (6/7) (11x - 2) = 108
⇒ 33x - 60 + (6/7) (11x - 2) = 108
⇒ 231x - 420 + 66x - 12 = 756
⇒ 297x = 756 + 432
⇒ 297x = 1188
⇒ x = 1188/297
∴ x = 4

∴ অঙ্কগুলোর সমষ্টি = x + x - 2
= 2x - 2
= (2 × 4) - 2
= 6

১,০৬৩.

If n = 3⁸ - 2⁸, which of the following is not a factor of n?

ক
ক) 97
খ
খ) 65
গ
গ) 35
ঘ
ঘ) 13

ব্যাখ্যা

n = 3⁸ - 2⁸
= (3⁴)² - (2⁴)²
= (3⁴ + 2⁴)(3⁴ - 2⁴)
= (81 + 16)(81 - 16)
= 97 × 65
= 97 × 13 × 5

So, we can see that here 35 can't be a factor of n

১,০৬৪.

The Cost Price of an article is 40% of the Selling Price. The percent that the Selling Price is of Cost Price is-

ক
250%
খ
100%
গ
60%
ঘ
160%

ব্যাখ্যা

Question: The Cost Price of an article is 40% of the Selling Price. The percent that the Selling Price is of Cost Price is-

Solution:
ধরি, বিক্রয়মূল্য = x টাকা

∴ ক্রয়মূল্য = x এর 40%
= 40x/100 = 2x/5

∴ ক্রয়মূল্য 2x/5 টাকা হলে বিক্রয়মূল্য = x টাকা
∴ ক্রয়মূল্য 1 টাকা হলে বিক্রয়মূল্য = 5x/2x = 5/2 টাকা
∴ ক্রয়মূল্য 100 টাকা হলে বিক্রয়মূল্য = (100 × 5)/2 = 250 টাকা

∴ বিক্রয়মূল্য 250% ক্রয়মূল্যের

১,০৬৫.

The ratio between the present ages of A and B is 5:3 respectively. The ratio between A’s age 4 years ago and B’s age 4 years hence is 1: 1. What is the ratio between A’s age 4 years hence and B’s age 4 years ago?

ক
ক) 1:3
খ
খ) 3:1
গ
গ) 2:1
ঘ
ঘ) 4:1

ব্যাখ্যা

Let A's age be 5x years.
Then, B's age = 3x years.
So,
(5x - 4)/3x + 4 = 1/1
⇒ 5x - 4 = 3x + 4
2x = 8
⇒ x = 4
∴ A's age 4 years hence/B's age 4 years ago = 5x + 4/3x - 4
= (5 × 4 + 4)/(3 × 4 - 4)
= 24/8
= 3/1
= 3:1
Answer: 3:1

১,০৬৬.

A train overtakes two persons who are walking in the same direction to that of the train at 2 kmph and 4 kmph and passes them completely in 9 and 10 seconds respectively. What is the length of the train?

ক
ক) 65 meter
খ
খ) 60 meter
গ
গ) 55 meter
ঘ
ঘ) 50 meter

ব্যাখ্যা

Let the speed of the train be v km/hr.
(v - 2) : (v - 4) = 10 : 9 [speed and time are inversely proportional]
⇒ 9v - 18 = 10v - 40
⇒ v = 22 km/hr.
Length of the train
= (22 - 2) × (5/18) × 9
= 50 meter.

১,০৬৭.

Four bells ringing together and ring at an interval of 12 sec, 15 sec, 20 sec, and 30 sec respectively. How many times will they ring together in 8 hours?

ক
481
খ
480
গ
482
ঘ
483

ব্যাখ্যা

Question: Four bells ringing together and ring at an interval of 12 sec, 15 sec, 20 sec, and 30 sec respectively. How many times will they ring together in 8 hours?

Solution:
Four bells ringing timing is 12 sec, 15 sec, 20 sec,30 sec
Now we have to take LCM of time interval ⇒ LCM of (12, 15, 20, 30) = 60
Total seconds in 8 hours = 8 × 3600 = 28800

Number of times bell rings = 28800/60
⇒ Number of times bell rings = 480

If four bells ring together in starting ⇒ 480 + 1
∴ The bell ringing 481 times in 8 hours.

Mistake Points: The bells start tolling together, the first toll also needs to be counted, that is the number of times of tolling since the first time.

১,০৬৮.

The banker's gain on a bill due 1 year hence at 12% per annum is Tk. 6. The true discount is:

ক
ক) 72
খ
খ) 36
গ
গ) 54
ঘ
ঘ) 50

ব্যাখ্যা

T.D. = (B.G.×100)/(R×T)
= Tk.(6×100 / 12×1)
= Tk.50

১,০৬৯.

Find the greatest number that exactly divides each of the numbers 36, 60, and 84.

ক
6
খ
8
গ
12
ঘ
18

ব্যাখ্যা

Question: Find the greatest number that exactly divides each of the numbers 36, 60, and 84.

Solution:
We know,
The HCF (Highest Common Factor) of two or more numbers is the greatest number that divides each of them exactly.

Now,
Prime factorization of 36 = 2 × 2 × 3 × 3

Prime factorization of 60 = 2 × 2 × 3 × 5

Prime factorization of 84 = 2 × 2 × 3 × 7

∴ HCF of 36, 60, and 84 = 2 × 2 × 3 = 12

Therefore, the greatest number is 12.

১,০৭০.

A box contains 12 red balls, 5 black balls, and 8 white balls. One ball is drawn at random. What is the probability that the ball drawn is neither red nor white?

ক
8/25
খ
1/5
গ
2/5
ঘ
13/25

ব্যাখ্যা

Question: A box contains 12 red balls, 5 black balls, and 8 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 = 12
Number of black balls = 5
Number of white balls = 8

∴ Total balls = 12 + 5 + 8 = 25

Let event E = The ball drawn is neither red nor white, so it must be black.

∴ Number of favorable outcomes = 5
∴ P(E) = 5/25 = 1/5

১,০৭১.

Naizel is twenty years older than Neketa. In three years Naizel will be twice as old as Neketa will be. How old is Neketa now?

ক
24
খ
17
গ
16
ঘ
8

ব্যাখ্যা

Question: Naizel is twenty years older than Neketa. In three years Naizel will be twice as old as Neketa will be. How old is Neketa now?

Solution:
Let,
The age of Neketa is x
∴ The age of Naizel is x + 20

ATQ,
x + 23 = 2(x + 3)
⇒ x + 23 = 2x + 6
⇒ x = 17

১,০৭২.

The simple interest on a sum of money in 5 years at 12% per annum is Tk. 500 less than the simple interest accrued on the same sum in 7 years at 10% per annum. Find the sum.

ক
Tk. 4000
খ
Tk. 4500
গ
Tk. 5000
ঘ
Tk. 5500

ব্যাখ্যা

Question: The simple interest on a sum of money in 5 years at 12% per annum is Tk. 500 less than the simple interest accrued on the same sum in 7 years at 10% per annum. Find the sum.

Solution:
Let the sum be P.
SI in 5 years at 12% per annum = P × 12 × (5/100) = 0.6 P
SI in 7 years at 10% per annum = P × 10 × (7/100) = 0.7 P

Now, according to the question,
0.7 P - 0.6 P = 500
⇒ 0.1 P = 500
⇒ P = 5000
Thus, the required sum is Tk. 5000

১,০৭৩.

A and B started a business by investing Tk. 4000 and Tk. 5000 respectively. Find the A's share out of a total profit of Tk. 1800.

ক
ক) Tk. 1200
খ
খ) Tk. 800
গ
গ) Tk. 600
ঘ
ঘ) Tk. 1800

ব্যাখ্যা

Ratio of A and B's investment = 4000 : 5000 = 4 : 5
A's share = 4/(4 + 5) × 1800 = Tk. 800

১,০৭৪.

X and Y invested in a business in a ratio of 4 ∶ 1. They donated 21% of a profit to a NGO and X’s share is Tk. 31,600. Find the total profit.

ক
ক) Tk. 52,000
খ
খ) Tk. 50,000
গ
গ) Tk. 45,000
ঘ
ঘ) Tk. 47,000

ব্যাখ্যা

X and Y invested in a business in a ratio of 4 ∶ 1.
They donated 21% of a profit to a NGO and X’s share is =Tk. 31,600

Let total profit is Tk. x
Donated to NGO = 21% of x
Remaining profit = (x - 21% of x)
= x - 21x/100
= (100x - 21x)/100
= 79x/100

According to question:
⇒ (79x/100) × (4/5) = 31,600
⇒ x = (31,600 × 100 × 5)/(79 × 4)
⇒ x = 50,000 Tk

∴ Required total profit is= 50,000

১,০৭৫.

The least number by which 108 must be multiplied to make it a perfect square is-

ক
2
খ
5
গ
3
ঘ
4

ব্যাখ্যা

Question: The least number by which 108 must be multiplied to make it a perfect square is-

Solution:
একটি সংখ্যা পূর্ণবর্গ সংখ্যা হতে হলে তার মৌলিক গুণনীয়কগুলোকে অবশ্যই জোড় সংখ্যায় (even power) থাকতে হবে।

108 = 2 × 2 × 3 × 3 × 3
= 2² × 3³
জোড়া গঠন করে পাই,(2 × 2) × (3 × 3) × 3
এখানে জোড়া বিহীন সংখ্যা 3

∴ 108 কে 3 দ্বারা গুণ করলে এটি পূর্ণবর্গ সংখ্যা হবে।

১,০৭৬.

A certain number of men can finish a piece of work in 80 days. If there were 15 men less, it would take 20 days more for the work to be finished. How many men were there originally?

ক
75
খ
82
গ
100
ঘ
110
ঙ
None of these

ব্যাখ্যা

Question: A certain number of men can finish a piece of work in 80 days. If there were 15 men less, it would take 20 days more for the work to be finished. How many men were there originally?

Solution:
Originally let there be x men.
Less men, More days (Indirect Proportion)
Therefore, (x - 15) : x : : 80 : 100
⇒ (x - 15)/x = 80/100
⇒ (x - 15) × 100 = x × 80
⇒ 100x - 1500 = 80x
⇒ 20x = 1500
∴ x = 75

১,০৭৭.

The ratio of P : Q is 3 : 4 and the ratio of Q : R is 5 : 6. If P is equal to 9, what is the value of R?

ক
14.4
খ
15.4
গ
16.5
ঘ
15.5

ব্যাখ্যা

Question: The ratio of P : Q is 3 : 4 and the ratio of Q : R is 5 : 6. If P is equal to 9, what is the value of R?
Solution:
Given that,
P : Q = 3 : 4
Q : R = 5 : 6
And P = 9

Now,
P : Q = 3 : 4
or, P/Q = 3/4
or, 9/Q = 3/4
or, 3Q = 36
or, Q = 36/3
∴ Q = 12

And,
Q : R = 5 : 6
or, Q/R = 5/6
or, 12/R = 5/6
or, 5R = 72
or, R = 72/5
∴ R = 14.4

so, the value of R is 14.4

১,০৭৮.

The average age of four brothers is 12 years. If the age of their mother is also included, the average is increased by 5 years. What is the age of the mother(in year)?

ক
50
খ
45
গ
74
ঘ
37
ঙ
85

ব্যাখ্যা

Sum of age of four brothers = 12 × 4 = 48
If the age of their mother is also included, the average is increased by 5 years.
So, the average age of four brothers and their mother is (12 + 5) years or 17 years.
Sum of age of four brothers and their mother = (17 × 5) years = 85 years
∴ Mother's age = (85 - 48) years = 37 years

১,০৭৯.

If the area of the trapezium, whose parallel sides are 6 cm and 10 cm is 64 sq. cm, what will be the distance between the parallel sides?

ক
12 cm
খ
10 cm
গ
9 cm
ঘ
8 cm

ব্যাখ্যা

Question: If the area of the trapezium, whose parallel sides are 6 cm and 10 cm is 64 sq. cm, what will be the distance between the parallel sides?

Solution:
Given,
Parallel sides of a trapezium = 6 cm, and 10 cm

We know,
Area of trapezium = (1/2)(sum of the parallel sides) × distance between the parallel sides
64 = (1/2)(6 + 10) × distance
⇒ 64 = 8 × distance
⇒ distance = 64/8
∴ distance = 8 cm

So, the distance between the parallel lines of trapezium = 8 cm.

১,০৮০.

An outgoing pipe is attached to a tank that can empty it in 10 hours. An ingoing pipe is connected to the tank that can fill it in 4 hours. How much time will it take to fill the half-full tank?

ক
10/3 hours
খ
10/6 hours
গ
20/3 hours
ঘ
3 hours

ব্যাখ্যা

Question: An outgoing pipe is attached to a tank that can empty it in 10 hours. An ingoing pipe is connected to the tank that can fill it in 4 hours. How much time will it take to fill the half-full tank?

Solution:
ingoing pipe in one hour can fill = 1/4
outgoing pipe in one hour can empty = 1/10

in one hour total fill up = 1/4 - 1/10 = 3/20

to fill half the tank it will take = 20/6 = 10/3 hours.

১,০৮১.

ব্যাখ্যা

Question:

Solution:

১,০৮২.

A can complete a work in 24 days and B in 16 days. They work together for 6 days. How many more days will A take alone to finish the remaining work?

ক
15 days
খ
9 days
গ
12 days
ঘ
10 days

ব্যাখ্যা

Question: A can complete a work in 24 days and B in 16 days. They work together for 6 days. How many more days will A take alone to finish the remaining work?

Solution:
A একা কাজটি করতে পারে = 24 দিনে
∴ A এর একদিনের কাজ = 1/24 অংশ
এবং,
B একা কাজটি করতে পারে = 16 দিনে
∴ B এর একদিনের কাজ = 1/16 অংশ

∴ A ও B একসাথে একদিনের কাজ = (1/24) + (1/16) = (2 + 3)/48 = 5/48 অংশ
তারা 6 দিনে একসাথে কাজ করে = 6 × (5/48) = 5/8 অংশ

বাকি কাজ = 1 - (5/8) = 3/8 অংশ

অতএব,
A, 1/24 অংশ কাজ করে 1 দিনে
∴ 3/8 অংশ কাজ করে = (24 × 3)/8 = 9 দিনে

অতএব, A একা বাকি কাজ শেষ করতে ৯ দিন লাগবে।

১,০৮৩.

If you divide 40 by half and add 5 with the resulting figure, then what is the final result?

ক
ক) 25
খ
খ) 30
গ
গ) 35
ঘ
ঘ) 85

ব্যাখ্যা

Question: If you divide 40 by half and add 5 with the resulting figure, then what is the final result?

Solution:
40/(1/2)
= 80

adding 5 = 80 + 5
= 85

১,০৮৪.

If the ratio of two numbers is 5 : 8, and their Least Common Multiple is 200, what are the two numbers?

ক
20, 32
খ
15, 24
গ
25, 40
ঘ
30, 48

ব্যাখ্যা

Question: If the ratio of two numbers is 5 : 8, and their Least Common Multiple is 200, what are the two numbers?

Solution:
ধরি,
সংখ্যা দুইটি যথাক্রমে 5x, 8x
5x, 8x এর লসাগু = 40x

প্রশ্নমতে,
40x = 200
⇒ x = 200/40
∴ x = 5

∴ সংখ্যা দুইটি যথাক্রমে = 5 × 5 = 25, 8 × 5 = 40

১,০৮৫.

A works twice as fast as B. If B can complete a work in 18 days independently, the number of days in which A and B can together finish the work is -

ক
ক) 4 days
খ
খ) 6 days
গ
গ) 8 days
ঘ
ঘ) 9 days

ব্যাখ্যা

Question: A works twice as fast as B. If B can complete a work in 18 days independently, the number of days in which A and B can together finish the work is -

Solution:
Ratio of rates of working of A and B = 2 : 1
So, the ratio of time taken = 1 : 2

Since, B takes 18 days, A takes 9 days

∴ (A + B)'s 1 day's work = (1/9) + (1/18)
= 3/18
= 1/6

∴ A + B can finish the work in (6/1) = 6 days

১,০৮৬.

Sum of three different positive integers is the same as their product. What is the largest of these 3 integers?

ক
ক) 1
খ
খ) - 1
গ
গ) 2
ঘ
ঘ) 3

ব্যাখ্যা

ধরি,
প্রথম সংখ্যা = 1 দ্বিতীয় সংখ্যা = 2 এবং তৃতীয় সংখ্যা = 3

সংখ্যা তিনটির যোগফল = 1 + 2 + 3 = 6
সংখ্যা তিনটির গুণফল = 1 × 2 × 3 = 6
সবচেয়ে বড় সংখ্যা = 3

১,০৮৭.

The average age of a group of 20 students is 16 years. When 5 more students join the group, the average age increase by 1 year. The average age of the new students is?

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

ব্যাখ্যা

Question: The average age of a group of 20 students is 16 years. When 5 more students join the group, the average age increase by 1 year. The average age of the new students is?

Solution:
Total age of 20 students = 20 × 16 = 320 years

Total age of 25 students = 25 × 17 = 425 years

Total age of 5 new students = 425 - 320 = 105 years

∴ Average age of 5 new students = 105/5 = 21 years

১,০৮৮.

The sum of four consecutive two-digit odd numbers, when divided by 10, become a perfect square. Which of the following can possibly be one of these four numbers?

ক
ক) 21
খ
খ) 41
গ
গ) 25
ঘ
ঘ) 67

ব্যাখ্যা

Suppose the numbers are x, x+2, x+4, x+6
Their sum will be 4x +12, which when divided by 10 gives us (4x+12)/10 becoming a perfect square.
The expression 4x+12 ends in 0 to be divisible by 10, which means 4x has to end at 8.
Any odd number multiplied by 4, giving us product ending at 8 means that odd number ends at 7.
Let the first number now be n7 then the 4 numbers would be n7, n9, (n+1)1 and (n+1)3.
The expression of sum of the 4 numbers divided by 10 can be expressed now as (10*n+7)+(10n+9)+{(n+1)*10+1}+{(n+1)*10+3}/10.
This can be simplified to (40n+40)/10 or can be expressed as 4n+4.
Value of 4n+4 for all values of n from 2 to 8 is 12(for n=2), 16(for n=3), 20(for n=4), 24(for n=5), 28(for n=6), 32(for n=7) and 36(for n=8).
Out of these, only 16 and 36 are perfect squares
Therefore the two possible sets of 4 such numbers will be 37, 39, 41 & 43 and 87, 89, 91 & 93.
---------------------------------------------
---------------------------------------------
বিকল্প - ১:
Using options,
We find that four consecutive odd numbers are 37, 39, 41 and 43
The sum of these 4 numbers is 160, when divided by 10 we get 16 which is a perfect square.
Thus, 41 is one of the odd numbers

---------------------------------------------
---------------------------------------------
বিকল্প - ২:
দুই অঙ্কবিশিষ্ট চারটি ক্রমিক বিজোড় সংখ্যার যোগফল ১০ দ্বারা বিভাজ্য হলে, পূর্ণ বর্গ সংখ্যা পাওয়া যাবে এমন সংখ্যা চারটির একটি হচ্ছে ৪১
দুই অঙ্কবিশিষ্ট চারটি ক্রমিক বিজোড় সংখ্যা ৩৭, ৩৯, ৪১ ও ৪৩
এদের যোগফল = ৩৭ + ৩৯ + ৪১ + ৪৩ = ১৬০
১৬০ কে ১০ দ্বারা ভাগ করলে ১৬ পাওয়া যায় যা পূর্ণ বর্গ সংখ্যা
অপশনে ৪১ থাকায় সঠিক উত্তর ৪১

১,০৮৯.

The line perpendicular to y = x - 2 is

ক
ক) y = 2x + 1
খ
খ) 2y = - 2x - 5
গ
গ) 2y = x + 7
ঘ
ঘ) y = 3x + 1

ব্যাখ্যা

একটি সরলরেখা অপর একটি সরলরেখার উপর তখনই লম্ব হবে যখন ঢালদ্বয়ের গুণফল -1 হবে।
y = x - 2 সমীকরণের ঢাল হলো 1 [ঢাল = x এর সহগ]
এখন দেখতে হবে কোন সমীকরণের ঢাল -1
অপশন b তে, 2y = - 2x - 5
⇒ y = - x - 5/2
অতএব, y = - x - 2 সরলরেখার উপর লম্ব রেখার সমীকরণ 2y = - 2x - 5

১,০৯০.

How many perfect squares lie from 100 to 300?

ক
ক) 6
খ
খ) 7
গ
গ) 8
ঘ
ঘ) 9

ব্যাখ্যা

Question: How many perfect squares lie from 100 to 300?

Solution:
(10)²= 100 And
(17)²= 289

So, the perfect squares between 100 and 300 are the squares of numbers from 10 to 17.
Clearly, there are 8 in number.

১,০৯১.

3/4 part of the tank is full of water. When 30 litres of water is taken out, the tank becomes empty. The capacity of the tank is -

ক
36 litres
খ
32 litres
গ
44 litres
ঘ
40 liters

ব্যাখ্যা

Question: 3/4 part of the tank is full of water. When 30 litres of water is taken out, the tank becomes empty. The capacity of the tank is -

Solution:
Let us consider,
The tank has 4x litres of total capacity and holds 3x litres of water.
And if 30 litres of water is taken out, then the tank becomes empty.

It means 3x litres of water is taken out.
∴ 3x = 30 litres
⇒ x = 10 litres

Capacity of tank
= 4x
= 4 × 10
= 40 liters.

১,০৯২.

Eight years back, Akib's age was 1/8^th of Mahin's age. Ten years from now, Mahin's age will be double of Akib's age. How many years old is Mahin now?

ক
28 years
খ
30 years
গ
32 years
ঘ
35 years
ঙ
None

ব্যাখ্যা

Question: Eight years back, Akib's age was 1/8^th of Mahin's age. Ten years from now, Mahin's age will be double of Akib's age. How many years old is Mahin now?

Solution:
ধরি,
মাহিনের বর্তমান বয়স ক বছর
৮ বছর পূর্বে মাহিনের বয়স ছিল = (ক - ৮) বছর
∴ ৮ বছর পূর্বে আকিবের বয়স ছিল = (ক - ৮)/৮ বছর

∴ বর্তমানে, আকিবের বয়স = {(ক - ৮)/৮} + ৮ বছর
= (ক - ৮ + ৬৪)/৮ বছর
= (ক + ৫৬)/৮ বছর

প্রশ্নমতে,
ক + ১০ = ২ × [{(ক + ৫৬)/৮} + ১০]
⇒ ক + ১০ = ২ × {(ক + ৫৬ + ৮০)/৮}
⇒ ক + ১০ = ২ × {(ক + ১৩৬)/৮}
⇒ ক + ১০ = (ক + ১৩৬)/৪
⇒ ৪(ক + ১০) = ক + ১৩৬
⇒ ৪ক + ৪০ = ক + ১৩৬
⇒ ৪ক - ক = ১৩৬ - ৪০
⇒ ৩ক = ৯৬
⇒ ক = ৯৬/৩
∴ ক = ৩২

∴ মাহিনের বর্তমান বয়স = ৩২ বছর

১,০৯৩.

A student multiplied 765 by a certain number and got 448835 as their answer. If in the answer both 8s are wrong, but the other digits are correct, then what will be the correct answer?

ক
446435
খ
445935
গ
444635
ঘ
442935
ঙ
442560

ব্যাখ্যা

Question: A student multiplied 765 by a certain number and got 448835 as their answer. If in the answer both 8s are wrong, but the other digits are correct, then what will be the correct answer?

Solution:
The correct answer must be divisible by 765.

By checking possible corrections:

446435 ÷ 765 → remainder ≠ 0
445935 ÷ 765 → remainder ≠ 0
444635 ÷ 765 → remainder ≠ 0
442935 ÷ 765 → remainder = 0 → quotient = 579

Hence, the correct answer = 442935

১,০৯৪.

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

ক
2
খ
4
গ
5
ঘ
9

ব্যাখ্যা

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

Solution:
- S = {2, 4, 5, 9}
- Number of elements 4,
- Total = (2+4+5+9)= 20.
∴ Average = 20/ 4= 5

Try removing each number and check if the new average is still 5,
After removing 2, we get S = {4,5,9}
- Summation = 18
- Number of elements = 3
∴ Average = 18/3 =6. Which is not equal to 5.

Again, removing 4 from set S, we get S = {2,5,9}
- Summation = 16
- Number of elements = 3
- Average = 16/3 = 5.33

Again, removing 5 from set S, we get S = {2,4,9}
- Summation = 15
- Number of elements = 3
- Average = 15/3 = 5.

- Final Answer: 5 can be removed without changing the average.

১,০৯৫.

Time required by two pipes A and B working separately to fill a tank is 36 seconds and 45 seconds respectively. Another pipe C can empty the tank in 30 seconds. Initially, A and B are opened and after 7 seconds, C is also opened. In how much more time the tank would be completely filled?

ক
47 seconds
খ
43 seconds
গ
39 seconds
ঘ
35 seconds

ব্যাখ্যা

Question: Time required by two pipes A and B working separately to fill a tank is 36 seconds and 45 seconds respectively. Another pipe C can empty the tank in 30 seconds. Initially, A and B are opened and after 7 seconds, C is also opened. In how much more time the tank would be completely filled?

Solution:
Let the capacity of the tank be LCM (36, 45, 30) = 180 units
∴ Efficiency of pipe A = 180/36 = 5 units/second
Efficiency of pipe B = 180/45 = 4 units/second
Efficiency of pipe C = - 180 / 30 = - 6 units/second

Now,
for the first 7 seconds, A and B were open.
Combined efficiency of A and B = 5 + 4 = 9 units/second
∴ Part of the tank filled in 7 seconds = 7 × 9 = 63 units

Part of tank empty = 180 - 63 = 117 units

Now, all pipes are opened.
Combined efficiency of all pipes = 5 + 4 - 6 = 3 units/second
Therefore, more time required = 117/3 = 39 seconds.

১,০৯৬.

A dice is thrown in the air. The probability of getting odd numbers is-

ক
1/2
খ
1/3
গ
2/3
ঘ
1/6

ব্যাখ্যা

Question: A dice is thrown in the air. The probability of getting odd numbers is-

Solution:
Numbers on dice are {1, 2, 3, 4, 5, 6}
Numbers on dice which is odd {1, 3, 5}
Number of favorable outcomes = 3
Total possible outcomes = 6

∴ The probability of getting odd numbers is = 3/6 = 1/2

১,০৯৭.

The average of three times a number and its square is nine times the number. The number is -

ক
ক) 9
খ
খ) 12
গ
গ) 15
ঘ
ঘ) 18

ব্যাখ্যা

Question: The average of three times a number and its square is nine times the number. The number is -

Solution:
Let, the number be x

Now,
(3x + x²)/2 = 9x
⇒ 3x + x² = 18x
⇒ x² - 15x = 0
⇒ x(x - 15) = 0
Now, x = 0 or 15

১,০৯৮.

Noman bought a ticket to a cricket match for Tk. 25 and later sold the ticket to Nazmul for Tk 75. What was Noman's percentage profit?

ক
50%
খ
100%
গ
200%
ঘ
300%

ব্যাখ্যা

Question: Noman bought a ticket to a cricket match for Tk. 25 and later sold the ticket to Nazmul for Tk 75. What was Noman's percentage profit?

Solution:
নোমান একটি টিকিট ২৫ টাকায় কিনে।
সে ৭৫ টাকায় টিকিটটি নাজমুলের কাছে বিক্রি করে।
∴ লাভ = (৭৫ -২৫) টাকা
= ৫০ টাকা

∴ শতকরা লাভ = (৫০ × ১০০)/২৫%
= ২০০%

১,০৯৯.

Jennifer flipped a coin three times and got heads each time. What is the probability that she gets heads on the next flip?

ক
ক) 1
খ
খ) 1/16
গ
গ) 1/2
ঘ
ঘ) 0

ব্যাখ্যা

Every time a coin flips, its independent outcome has a 50-50 chance. So, the probability is 1/2

১,১০০.

If (1/x) + (1/y) = 1/3, then xy/(x + y) = ?

ক
3
খ
1
গ
1/3
ঘ
1/5

ব্যাখ্যা

Question: If (1/x) + (1/y) = 1/3, then xy/(x + y) = ?

Solution:

(1/x) + (1/y) = 1/3

⇒ (y + x)/xy = 1/3

∴ xy/(x + y) = 3