Bank Math

PrepBank · পাতা ৯৫ / ১৬১ · ৯,৪০১–৯,৫০০ / ১৬,১২৪

৯,৪০১.

A beg contains an equal number of one rupee, 50 paisa and 25 paisa coins. If the total value of Tk. 35, how many coins of each type are there?

ক
20
খ
15
গ
18
ঘ
22

ব্যাখ্যা

Question: A beg contains an equal number of one rupee, 50 paisa and 25 paisa coins. If the total value of Tk. 35, how many coins of each type are there?

Solution:
Let X coins of each type of there
Total Value = Tk. 35

Now,
⇒ X + X/2 + X/4 = 35
⇒ 4X + 2X + X = 140
⇒ 7X = 140
⇒ X = 20

৯,৪০২.

The compound interest on a sum of Tk. 625 in 2 years is Tk. 51. Find the rate of interest.

ক
3%
খ
4%
গ
5%
ঘ
6%

ব্যাখ্যা

Question: The compound interest on a sum of Tk. 625 in 2 years is Tk. 51. Find the rate of interest.

Solution:
We know that A = CI + P
A = 51 + 625 = 676

Now going by the formula:
A = P[1+(R/100)]ⁿ
⇒ 676 = 625[1 + (R/100)]²
⇒ 676/625 = [1 + (R/100)]²
We can see that 676 is the square of 26 and 625 is the square of 25
Therefore,
⇒ (26/25)² = [1 + (R/100)]²
⇒ 26/25 = [1 + (R/100)]
⇒ 26/25 - 1 = R/100
⇒ 1/25 = R/100
⇒ R = 100/25
∴ R = 4

৯,৪০৩.

If the value of XYZ Company stock drops from Tk. 25 per share to Tk. 21 per share, what is the percent of the decrease?

ক
4%
খ
8%
গ
12%
ঘ
16%
ঙ
None of these

ব্যাখ্যা

Question: If the value of XYZ Company stock drops from Tk. 25 per share to Tk. 21 per share, what is the percent of the decrease?

Solution:
Decrease = 25 - 21 = 4

Percentage of decrease = (4/25) × 100% = 16%

৯,৪০৪.

If cosecθ - cotθ = 1/x, then (cosecθ + cotθ) = ?

ক
1
খ
cotx
গ
x²
ঘ
x

ব্যাখ্যা

Question: If cosecθ - cotθ = 1/x, then (cosecθ + cotθ) = ?

Solution:
Given that,
cosecθ - cotθ = 1/x

We know,
cosec² θ – cot²θ = 1
⇒ (cosecθ + cotθ) (cosecθ - cotθ) = 1
⇒ (cosecθ + cotθ) × (1/x) = 1
⇒ cosecθ + cotθ = x

∴ cosecθ + cotθ = x

৯,৪০৫.

A student was asked to find the arithmetic mean of the numbers 3, 11, 7, 9, 15, 13, 8, 19, 17, 21, 14, and x. He found the mean to be 12. What should be the number in place of x?

ক
ক) 7
খ
খ) 17
গ
গ) 31
ঘ
ঘ) 3

ব্যাখ্যা

(3 + 11 + 7 + 9 + 15 + 13 + 8 + 19 + 17 + 21 + 14 + x)/12 = 12
⇒ (137 + x)/12 = 12
⇒ x = 144 - 137
∴ x = 7

৯,৪০৬.

A committee of 5 members is to be formed by selecting out of 6 men and 7 women. What is the probability that the committee has exactly 2 men and 3 women?

ক
105/354
খ
96/363
গ
102/455
ঘ
175/429

ব্যাখ্যা

Question: A committee of 5 members is to be formed by selecting out of 6 men and 7 women. What is the probability that the committee has exactly 2 men and 3 women?

Solution:
Total member = 3 + 2 = 5

2 men can be selected out of 6 men = ⁶C₂ ways
3 women can be selected out of 7 women in = ⁷C₃ ways
Required number of ways = ⁶C₂ × ⁷C₃ = 15 × 35 = 525

The total number of ways to make committee with all members = ¹³C₅ = 1287

∴ The probability that the committee has exactly 2 men and 3 women = 525/1287
= 175/429

৯,৪০৭.

If P denotes +, Q denotes -, R denotes ÷ and S denotes ×, then what is the value of 18S36R12Q6P7?

ক
115
খ
55
গ
25
ঘ
none

ব্যাখ্যা

Question: If P denotes +, Q denotes -, R denotes ÷ and S denotes ×, then what is the value of 18S36R12Q6P7?

Solution:
18S36R12Q6P7
=18 × 36 ÷ 12 - 6 + 7
= 18 × 3 - 6 + 7
= 54 - 6 + 7
= 61 - 6
= 55

৯,৪০৮.

A man wishes to cross a river perpendicularly. In still water, it takes 4 minutes to cross the river, but in the flowing river, it takes 5 minutes. If the river is 100 meters wide, the velocity of the flowing water of the river is -

ক
ক) 10 m/min
খ
খ) 15 m/min
গ
গ) 20 m/min
ঘ
ঘ) 20 m/min

ব্যাখ্যা

Velocity of the river = [100 √{(1/4²) - (1/5²)}] m/min
= [100 √{(1/16) - (1/25)}] m/min
= 100{√(9/400)} m/min
= 100 × (3/20) m/min
= 15 m/min.

৯,৪০৯.

Zayed sold an article at 6% loss. Had he sold it for Tk. 32 more, he would have made a profit of 10%. Then the cost of the article is =?

ক
Tk. 380
খ
Tk. 220
গ
Tk. 400
ঘ
Tk. 200

ব্যাখ্যা

Question: Zayed sold an article at 6% loss. Had he sold it for Tk. 32 more, he would have made a profit of 10%. Then the cost of the article is =?

Solution:
Let, the cost of the article is x taka

Selling price = 0.94x

According to the question,
0.94x + 64 = 1.1x
⇒ 1.1x - 0.94x = 32
⇒ 0.16x = 32
⇒ x = 32/0.16
∴ x = 200 taka

৯,৪১০.

If A = {1, 3, 4, 5, 6} then the number of proper subsets of A is -

ক
ক) 30
খ
খ) 31
গ
গ) 32
ঘ
ঘ) 29

ব্যাখ্যা

Question: If A = {1, 3, 4, 5, 6} then the number of proper subsets of A is -

Solution:
কোনো সেটে সদস্য সংখ্যা n হলে ঐ সেটের প্রকৃত উপসেটের সংখ্যা = 2ⁿ - 1
এখানে,
A = {1, 3, 4, 5, 6}
যার সদস্য সংখ্যা, n = 5

∴ প্রকৃত উপসেটের সংখ্যা = 2⁵ - 1
= 32 - 1
= 31

৯,৪১১.

A and B are two positive integers such that AB = 60. Which of the following cannot be the value of A + B?

ক
18
খ
23
গ
32
ঘ
61

ব্যাখ্যা

Question: A and B are two positive integers such that AB = 60. Which of the following cannot be the value of A + B?

Solution:
Factor pairs of 60:
(1, 60) → A + B = 61
(2, 30) → A + B = 32
(3, 20) → A + B = 23
(4, 15) → A + B = 19
(5, 12) → A + B = 17
(6, 10) → A + B = 16

So, possible values of A + B are: 61, 32, 23, 19, 17, 16.

Among the options, 18 is not possible.

৯,৪১২.

The length of the longest rod that can be placed in a room of dimensions 10m × 10m × 5m is-

ক
16.8m
খ
13.5m
গ
15m
ঘ
17.5m

ব্যাখ্যা

Question: The length of the longest rod that can be placed in a room of dimensions 10m × 10m × 5m is-

Solution:
সর্বোচ্চ রড রাখতে হলে, রুমের কর্ণ বরাবর রাখতে হবে।

রুমের কর্ণের দৈর্ঘ্য = √{(১০)^২ + (১০)^২ + (৫)^২}
= √২২৫
= ১৫ মিটার

৯,৪১৩.

6 pipes are required to fill a tank in 1 hour 20 minutes. If we use 5 such types of pipes, how much time will it take to fill the tank?

ক
120 minutes
খ
96 minutes
গ
80 minutes
ঘ
85 minutes
ঙ
75 minutes

ব্যাখ্যা

For 6 pipes, it takes 1 hour 20 minutes
1 hour 20 minutes = 60 + 20 = 80 minutes

For 5 pipes, let the time taken be x.

This is inverse proportion case:
80 × 6 = x × 5
x = 480/5
= 96

৯,৪১৪.

The sum of three numbers is 144. The ratio of the first number to the second number is 2 : 3 and between the second and third numbers, this ratio is 3 : 4, ﬁnd the second number.

ক
ক) 16
খ
খ) 32
গ
গ) 48
ঘ
ঘ) 64

ব্যাখ্যা

Question: The sum of three numbers is 144. The ratio of the first number to the second number is 2 : 3 and between the second and third numbers, this ratio is 3 : 4, ﬁnd the second number.

Solution:
Given that,
The sum of three numbers is = 144
The ratio of first number to second number is = 2 : 3
The ratio of second number of third number is = 3 : 4

The ratio of first, second and third number = 2 : 3 : 4
Let the numbers are 2x, 3x , 4x

According to the question
2x + 3x + 4x = 144
⇒ 9x = 144
⇒ x = 144/9
⇒ x = 16

∴ The second number is
= 3 × 16
= 48

৯,৪১৫.

Two trains of equal length are running on parallel lines in the same direction at 48km/hr and 38 km/hr. The faster train passes the slower train in 54 seconds. The length of each train is

ক
ক) 50m
খ
খ) 75m
গ
গ) 65m
ঘ
ঘ) 60m

ব্যাখ্যা

Let,
the length of each train be x metres.
Then, distance covered = 2x metres.
Relative speed = (48 - 38) km/hr
= (10 x 5/18) m/sec [km/hr কে m/sec পরিণত করতে 5/18 গুণ করতে হয় ]
= (25 /9) m/sec

∴ 2x/ 54 = 25 /9
⇒ 2x = 150
x = 75

৯,৪১৬.

Ιx - 2Ι < 3

ক
1 < x < 5
খ
- 1 < x < 1
গ
- 1 < x < 2
ঘ
- 1 < x < 5

ব্যাখ্যা

Question: Solve Ιx - 2Ι < 3

Solution:
Given
Ιx - 2Ι < 3
⇒ - 3 < x - 2 < 3
⇒ - 3 + 2 < x - 2 + 2 < 3 + 2
⇒ - 1 < x < 5

৯,৪১৭.

log₅(√5 × 25) =?

ক
1/2
খ
1/4
গ
2/3
ঘ
5/2

ব্যাখ্যা

প্রশ্ন: log₅(√5 × 25) =?

সমাধান:
= log₅(√5 × 25)
= log₅(5^1/2 × 5²)
= log₅5^{{(1/2) + 2}}
= log₅5^5/2
= 5/2 log₅5
= 5/2

৯,৪১৮.

The slope of the line 3x - 6y = 12 is not the same as the slope of which one of the following lines?

ক
x - 2y = 4
খ
2x - 4y = 16
গ
y = 2x - 1
ঘ
y = x/2 - 3

ব্যাখ্যা

Question: The slope of the line 3x - 6y = 12 is not the same as the slope of which one of the following lines?

Solution:
প্রথমে, প্রদত্ত রেখাটির ঢাল নির্ণয় করতে হবে।
রেখাটির সমীকরণকে y = mx + c আকারে রূপান্তর করতে হবে।
এখানে 'm' হলো ঢাল (Slope)।

প্রদত্ত রেখার সমীকরণ: 3x - 6y=12
⇒ - 6y = - 3x + 12
⇒ y = (- 3/- 6)x + (12/ - 6)
⇒ y = (1/2)x - 2
∴ এই রেখাটির ঢাল (m) হলো 1/2.

এবার, প্রদত্ত অপশনগুলোর প্রত্যেকটির ঢাল নির্ণয় করি:

ক) x - 2y = 4
⇒ - 2y = - x + 4
⇒ y = (- x/- 2) + (4/- 2)
⇒ y = (1/2)x - 2
∴ ঢাল, m = 1/2

খ) 2x - 4y = 16
⇒ - 4y = - 2x + 16
⇒ y = (- 2x/- 4) + (16/- 4)
⇒ y = (1/2)x - 4
∴ ঢাল, m = 1/2

গ) y = 2x - 1
∴ ঢাল, m = 2

ঘ) y = x/2 - 3
⇒ y = (1/2)x - 3
∴ ঢাল, m = 1/2

সুতরাং, দেখা যাচ্ছে যে শুধু মাত্র অপশন (গ) এর রেখার ঢাল মূল রেখার ঢাল থেকে ভিন্ন।

৯,৪১৯.

Four pipes can fill a reservoir in 15, 20, 30 and 60 hours respectively. The first pipe was opened at 7 am, second 8 am, third at 9 am and fourth at 10 am. When will the reservoir be full?

ক
ক) 1 p.m.
খ
খ) 2 p.m.
গ
গ) 3 p.m.
ঘ
ঘ) 4 p.m.

ব্যাখ্যা

Question: Four pipes can fill a reservoir in 15, 20, 30 and 60 hours respectively. The first pipe was opened at 7 am, second 8 am, third at 9 am and fourth at 10 am. When will the reservoir be full?

Solution:
Let,
the time be x hours after 7 am.
Then, the first pipe worked for x hours
Second pipe for (x - 1) hours;
Third pipe for (x - 2) hours;
Fourth pipe for (x - 3) hours.

ATQ,
(x/15) + {(x - 1)/20} + {(x - 2)/30} + {(x - 3)/60} = 1
⇒ (4x + 3x - 3 + 2x - 4 + x - 3)/60 = 1
⇒ 10x - 10 = 60
⇒ 10x = 70
∴ x = 7

So, the reservoir will be full 7 hours after 7 am = 7 + 7 = 14 = 2 p.m.

৯,৪২০.

Which trigonometric ratio is undefined in value?

ক
cos 90°
খ
sec 0°
গ
sin 0°
ঘ
tan90°

ব্যাখ্যা

Question: Which trigonometric ratio is undefined in value?

Solution:
cos90° = 0
sec0° = 1
sin0° = 0
tan90° = ∞(Undefined)

৯,৪২১.

Tanvir purchased an item for Tk. 8200 and sold it at a gain of 25%. From that amount, he purchased another item and sold it at a loss of 20%. What is his overall gain or loss?

ক
ক) Loss of Tk. 120
খ
খ) Loss of Tk. 180
গ
গ) Gain of Tk. 80
ঘ
ঘ) Neither gain nor loss.

ব্যাখ্যা

Question: Tanvir purchased an item for Tk. 8200 and sold it at a gain of 25%. From that amount, he purchased another item and sold it at a loss of 20%. What is his overall gain or loss?

Solution:
Initial investment=Tk.8200

S.P.of 1st term=Tk. {(125/100) × 8200} = Tk. 10250

C.P. of 2nd term = Tk. 10250
loss = 20%

S.P.of 2nd term = Tk. {(80/100) × 10250}
=Tk. 8200

Since initial investment = final receipt,
so, there was neither gain nor loss.

৯,৪২২.

A student scores 55% marks in 8 papers of 100 marks each. He scores 15% of his total marks in English. How much does he score in English?

ক
ক) 44
খ
খ) 45
গ
গ) 66
ঘ
ঘ) 77

ব্যাখ্যা

Total marks obtained by the student = 55% of 800
= {(55/100) × 800}
= 440
∴ Marks scored in English
= 15% of 440
= {(15/100) × 440}
= 66

৯,৪২৩.

sin(A + 14°) = 1/2, find the value of A?

ক
46°
খ
16°
গ
76°
ঘ
36°

ব্যাখ্যা

Question: sin(A + 14°) = 1/2, find the value of A?

Solution:
sin(A + 14°) = 1/2
⇒ sin(A + 14°) = sin30°
⇒ A + 14° = 30°
⇒ A = 30° - 14°
∴ A = 16°

৯,৪২৪.

A can do a piece of work in 9 days, B can do it in 12 days and C can do it in 30 days. The efficiency of D is given to be twice the efficiency of C. If A worked for 1 day, B worked for 2 days and C worked for 3 days. Find the number of days D will take to complete the remaining work alone?

ক
20/3 days
খ
25/3 days
গ
23/3 days
ঘ
28/3 days

ব্যাখ্যা

Question: A can do a piece of work in 9 days, B can do it in 12 days and C can do it in 30 days. The efficiency of D is given to be twice the efficiency of C. If A worked for 1 day, B worked for 2 days and C worked for 3 days. Find the number of days D will take to complete the remaining work alone?

Solution:
A can do a piece of work in = 9 days
Work is done by A in one day = 1/9

B can do a piece of work in = 12 days
Work is done by B in one day = 1/12

C can do a piece of work in = 30 days
Work is done by C in one day = 1/30

The efficiency of D = 2 x Efficiency of C
∴ Work done by D in one day = 1/15

A worked for = 1 day
B worked for = 2 days
C worked for = 3 days

∴ Remaining work done = 1 - (1/9) - (2/12) - (3/30)
= 1- [(1/9) + (1/6) + (1/10)]
= 1 - [(10 + 15 + 9)/90]
= 1 - [34/90]
= 56/90
= 28/45

Let the time taken by D to complete the remaining work alone be x days
∴ x/15 = 28/45
∴ x = 28/3 days

D alone will take 28/3 days to finish the work.

৯,৪২৫.

The minimum value of 2sin²θ + 3cos²θ is?

ক
1
খ
2
গ
3
ঘ
0

ব্যাখ্যা

Question: The minimum value of 2sin²θ + 3cos²θ is?

Solution:
Let,
p = 2sin²θ + 3cos²θ
⇒ p = 2sin²θ + 2cos²θ + cos²θ
⇒ p = 2(sin²θ + cos²θ) + cos²θ
⇒ p = 2 + cos²θ [since sin²θ + cos²θ = 1]

Therefore p will be the minimum when cosθ = 0.
So, the minimum value of p will 2.

৯,৪২৬.

A school has a total of 90 students. There are 32 students taking Physics, 26 taking English, and 13 taking both. What percentage of the students is taking either Physics or English?

ক
32%
খ
48%
গ
50%
ঘ
51%

ব্যাখ্যা

Question: A school has a total of 90 students. There are 32 students taking Physics, 26 taking English, and 13 taking both. What percentage of the students is taking either Physics or English?

Solution:
Students taking physics n(A) = 32 (these 32 include those 13 that take both)
Students taking english n(B) = 26 (these 26 also include those 13)
Students taking both n(A ∩ B) = 13
Students taking either Physics or English n(A ∪ B) = ?

We know,
n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
= 32 + 26 - 13 = 45

Required percentage = (45/90) × 100
= 50%

৯,৪২৭.

ক
1/4
খ
1/2
গ
3/4
ঘ
1/8

ব্যাখ্যা

Question:

Solution:

৯,৪২৮.

The ratio of investments of two partners A and B is 11 : 12, and the ratio of their profits is 2 : 3. If A invested the money for 8 months, then for how much time B invested his money?

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

ব্যাখ্যা

Question: The ratio of investments of two partners A and B is 11 : 12, and the ratio of their profits is 2 : 3. If A invested the money for 8 months, then for how much time B invested his money?

Solution:
let, A invested Tk 11x for 8 months
and B invested Tk 12x for y months

now,
(11x × 8) : (12x × y) = 2 : 3
⇒ (11x × 8) / (12x × y) = 2/3
⇒ 88x/12xy = 2/3
⇒ 24y = 264
⇒ y =11

৯,৪২৯.

The wheel of an engine of 300 cm in circumference makes 10 revolutions in 6 seconds. What is the speed of the wheel (in km/h)?

ক
12 kmh^-1
খ
15 kmh^-1
গ
17 kmh^-1
ঘ
18 kmh^-1

ব্যাখ্যা

Question: The wheel of an engine of 300 cm in circumference makes 10 revolutions in 6 seconds. What is the speed of the wheel (in km/h)?

Solution:
total distance = 10 × 300 cm
= 3000 cm
= 30 m

speed = 30/6 ms^-1
= 5 ms^-1
= (5 × 3600)/(1000) kmh^-1
= 18 kmh^-1

৯,৪৩০.

If n and p are both odd numbers, which of the following is an even number?

ক
ক) np
খ
খ) n + p
গ
গ) np + 2
ঘ
ঘ) n + p + 1

ব্যাখ্যা

Option 1: When n and p is odd number then product of n and p may be odd.
Option 2: When n and p is odd number then n + p is always even number

Option 3: When n and p is odd number then product of n and p may be odd.
⇒ np is odd
⇒ np + 2is odd

Option 4:
When n and p is odd number then n + p is always even number
⇒ n + p is even ⇒ n + p + 1 is become odd

∴ n + p will give even number

৯,৪৩১.

The one-third of the Supplementary angle to 30° is–

ক
ক) 150
খ
খ) 100
গ
গ) 50
ঘ
ঘ) 10

ব্যাখ্যা

Supplementary angle to 30 is 180 - 30 = 150°
It's one third is = 150 × 1/3 = 50°

৯,৪৩২.

The radius of the wheel of a vehicle is 70 cm. The wheel makes 10 revolutions in 4 seconds. The speed of the vehicle is:

ক
45.6 km/hr
খ
38.6 km/hr
গ
39.6 km/hr
ঘ
52.6 km/hr
ঙ
63.6 km/hr

ব্যাখ্যা

Question: The radius of the wheel of a vehicle is 70 cm. The wheel makes 10 revolutions in 4 seconds. The speed of the vehicle is :

Solution:
Distance covered in 4 sec :
= {2 × (22/7) × 70 × 10} cm
= 4400 cm
= 44 m

Distance covered in 1 sec :
= (44/4) m
= 11 m

∴ speed = 11 m/sec.
= {11 × (18/5) } km/hr
= 39.6 km/hr

৯,৪৩৩.

X, Y and Z share Tk. 1425 in such a way that X has 2.5 times as much as Y, and Y has 4 times as much as Z. How much money does Y receive?

ক
380 Tk
খ
460 Tk
গ
550 Tk
ঘ
620 Tk
ঙ
None

ব্যাখ্যা

Question: X, Y and Z share Tk. 1425 in such a way that X has 2.5 times as much as Y, and Y has 4 times as much as Z. How much money does Y receive?

Solution:
Given,
X = 2.5Y
Y = 4Z

∴ X = 2.5 × 4Z = 10Z

So, the ratio of X, Y, Z = 10 : 4 : 1

∴ Y receive = (1425 × 4/15) Tk
= 380 Tk

৯,৪৩৪.

If y = 5, then what is the value of 20y√(y³ - y²)?

ক
1500
খ
2000
গ
1000
ঘ
4000

ব্যাখ্যা

Question: If y = 5, then what is the value of 20y√(y³ - y²)?

Solution:
Given,
y = 5

∴ 20y√(y³ - y²)
= 20. 5. √(53 - 52)
= 100√(125 - 25)
= 100√(100)
= 100 × 10
= 1000

৯,৪৩৫.

If x : y = 5 : 3, then (3x - 2y) : (3x + 2y) =?

ক
ক) 5 : 7
খ
খ) 3 : 1
গ
গ) 3 : 7
ঘ
ঘ) 4 : 7

ব্যাখ্যা

Question: If x : y = 5 : 3, then (3x - 2y) : (3x + 2y) =?

Solution:
x : y = 5 : 3
⇒ x/y = 5/3
⇒ 3x/2y = (5 × 3)/(3 × 2) [ 3/2 দ্বারা গুণ করে ]
⇒ 3x/2y = 15/6
⇒ (3x - 2y)/(3x + 2y) = (15 - 6)/(15 + 6)
⇒ (3x - 2y)/(3x + 2y) = 9/21 [বিয়োজন - যোজন করে]
∴ (3x - 2y):(3x + 2y) = 3/7
= 3 : 7

৯,৪৩৬.

Alif can fill 60 envelopes per minute, and Tonoy can fill 40 envelopes per minute. Working together, how long will they take to fill 500 envelopes?

ক
5 minutes
খ
4 minutes
গ
6 minutes
ঘ
8 minutes
ঙ
10 minutes

ব্যাখ্যা

Question: Alif can fill 60 envelopes per minute, and Tonoy can fill 40 envelopes per minute. Working together, how long will they take to fill 500 envelopes?

Solution:
Given,
Alif can fill 60 envelopes in 1 minute
Tonoy can fill 40 envelopes in 1 minute

So together, they can fill in 1 minute = 60 + 40 = 100 envelopes

∴ 500 envelopes can be filled in 500 ÷ 100 = 5 minutes

৯,৪৩৭.

= ?

ক
1.25
খ
1
গ
12.5
ঘ
25

ব্যাখ্যা

Question: = ?

Solution:

৯,৪৩৮.

A juice vendor has two cans of juice. The first contains 30% water and the rest juice. The second contains 60% water. How much juice should he mix from each of the containers so as to get 18 litres of juice such that the ratio of water to juice is 1:2?

ক
4 litre, 12 litre
খ
6 litre, 20 litre
গ
2 litre, 16 litre
ঘ
None of the above

ব্যাখ্যা

Question: A juice vendor has two cans of juice. The first contains 30% water and the rest juice. The second contains 60% water. How much juice should he mix from each of the containers so as to get 18 litres of juice such that the ratio of water to juice is 1:2?

Solution:
Let the cost of 1 litre of pure juice be Tk. 1.

Juice in 1 litre mix from 1st can = 7/10 litre; C.P. of 1 litre mix. in 1st can Tk. 7/10
Juice in 1 litre mix from 2nd can = 4/10 litre; C.P. of 1 litre mix. in 2nd can Tk. 4/10
Juice in 1 litre of the final mix = 2/3 litre; C.P. of 1 litre of the final mix = Tk. 2/3

By the rule of alligation, we have:
Quantity of Cheaper/ Qunatity of Dearer = (CP of Dearer - Mean Price)/ (Mean Price - CP of Cheaper)
⇒ Quantity of 2nd can : Quantity of 1st can = {(7/10) - (2/3)} : {(2/3) - (4/10)}
= (21 - 20)/30 : (20 - 12)/30
= (1/30) : (8/30)
= 1 : 8

Thus, the quantity from the first can is = (1/9) × 18 = 2
The quantity from the second can is= (8/9) × 18 = 16

To get 18 litres of juice with a water-to-juice ratio of 1:2, the vendor should mix 2 litres from the first can and 16 litres from the second can.

৯,৪৩৯.

If (0.0015×10^m)/(0.03×10^k) = 5×10⁷, m - k = ?

ক
ক) 9
খ
খ) 8
গ
গ) 1
ঘ
ঘ) 6

ব্যাখ্যা

(0.0015×10^m)/(0.03×10^k) = 5×10⁷
⇒ 0.15×10^m-k / 3 = 5×10⁷
⇒ 15×10^m-k / 3×100 = 5×10⁷
⇒ 10^m-k/10² = 10⁷
⇒ 10^m-k =10⁷× 10² = 10⁹
∴ m - k = 9

৯,৪৪০.

Two trains P and Q are moving in opposite direction at a rate of 36km/hr and 45km/hr respectively. A passenger is sitting in train P, finds that his train passes train Q in 8 seconds. What is the length of train Q?

ক
ক) 195m
খ
খ) 175m
গ
গ) 190m
ঘ
ঘ) 180m

ব্যাখ্যা

Question: Two trains P and Q are moving in opposite direction at a rate of 36km/hr and 45km/hr respectively. A passenger is sitting in train P, finds that his train passes train Q in 8 seconds. What is the length of train Q?

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

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

৯,৪৪১.

To construct a rectangle, we need to know:

ক
ক) All the interior angles
খ
খ) All the Sides
গ
গ) Only Length and breadth
ঘ
ঘ) Only one angle measure
ঙ
ঙ) None of these

ব্যাখ্যা

A rectangle has its parallel sides equal and all the interior angles measure 90 degrees. Hence, if the length and breadth of the rectangle is known, then we can construct it easily.

৯,৪৪২.

The sum of three consecutive odd integers is 40 more than the first of the numbers. What is the middle number?

ক
19
খ
21
গ
23
ঘ
None

ব্যাখ্যা

Question: The sum of three consecutive odd integers is 40 more than the first of the numbers. What is the middle number?

Solution:
Let's denote the three consecutive odd integers as x, x + 2, and x + 4

x + (x + 2) + (x + 4) = x + 40
⇒ 3x + 6 = x + 40
⇒ 2x + 6 = 40
⇒ 2x = 34
∴ x = 17

The three consecutive odd integers are 17, 19, and 21
∴ The middle number is 19

৯,৪৪৩.

The perimeter of a rectangle is 72 cm. If the ratio of the lengths of two adjacent sides is 7 : 5, find the lengths of these sides.

ক
28 cm and 20 cm
খ
21 cm and 15 cm
গ
24 cm and 18 cm
ঘ
30 cm and 12 cm

ব্যাখ্যা

Question: The perimeter of a rectangle is 72 cm. If the ratio of the lengths of two adjacent sides is 7 : 5, find the lengths of these sides.

Solution:
Perimeter of a rectangle = 2(Length + Breadth)
Also Length : Breadth = 7 : 5
Let actual values are 7x and 5x.

Hence,
2(7x + 5x) = 72
⇒ 12x = 36
∴ x = 3

Now,
Length = 7x = 7 × 3 = 21 cm
Breadth = 5x = 5 × 3 = 15 cm

∴ sides will be of 21 cm and 15 cm.

৯,৪৪৪.

tanA + cotA =?

ক
secA.cosecA
খ
secA.sinA
গ
secA.tanA
ঘ
None of these

ব্যাখ্যা

প্রশ্ন: tanA + cotA =?

সমাধান:
tanA + cotA
= (sinA/cosA) + (cosA/sinA)
= (sin²A + cos²A)/cosA.sinA
= 1/cosA.sinA
= (1/cosA)(1/sinA)
= secA.cosecA

৯,৪৪৫.

By investing in (50/3)% stock at Tk. 64, one earns Tk. 1500. The investment made is:

ক
ক) Tk. 9600
খ
খ) Tk. 7560
গ
গ) Tk. 5760
ঘ
ঘ) Tk. 5640

ব্যাখ্যা

Question: By investing in (50/3)% stock at Tk. 64, one earns Tk. 1500. The investment made is:

Solution:
To earn Tk. 50/3, investment = Tk. 64
To earn Tk. 1, investment = Tk. 64 × (3/50)
To earn Tk. 1500, investment = Tk. {64 × (3/50) × 1500}
= Tk. 5760

৯,৪৪৬.

How many numbers are there between 100 and 1000 inclusive, having at least one of their digits 7?

ক
250
খ
252
গ
255
ঘ
260

ব্যাখ্যা

Question: How many numbers are there between 100 and 1000 inclusive, having at least one of their digits 7?

Solution:
total numbers = 901

3 digit number without 7 = (8 × 9 × 9)
= 648
so, numbers are there between 100 and 1000 inclusive without 7 is = 648 + 1 = 649

∴ numbers are there between 100 and 1000 inclusive, having at least one of their digits 7 = 901 - 649
= 252

৯,৪৪৭.

9 years ago a brother was 7 years older than his sister. At present their total age is 53. What is the present age of the sister?

ক
23 years
খ
22 years
গ
24 years
ঘ
26 years

ব্যাখ্যা

Question: 9 years ago a brother was 7 years older than his sister. At present their total age is 53. What is the present age of the sister?

Solution:
Let,
sister was x years old 9 years ago.
∴ brother was (x + 7) years old

ATQ,
x + x + 7 + 9 + 9 = 53
or, 2x = 53 - 25
or, x = 14

present age of the sister is = 14 + 9 = 23 years

৯,৪৪৮.

A ladder leans against a vertical wall, making an angle of 30° with the ground. if the foot of the ladder is 15√3 meters away from the wall, what is the height on the wall reached by the ladder?

ক
12 meters
খ
15 meters
গ
10√3 meters
ঘ
20 meters

ব্যাখ্যা

Question: A ladder leans against a vertical wall, making an angle of 30° with the ground. if the foot of the ladder is 15√3 meters away from the wall, what is the height on the wall reached by the ladder?

Solution:

ধরি, মইটি দেয়ালে যে উচ্চতায় পৌঁছায় = h মিটার
দেয়াল থেকে মইয়ের পাদদেশের দূরত্ব, BC = 15√3
ভূমির সাথে যে কোণ তৈরি করে, ∠ACB = 30°

আমরা জানি,
tanθ = লম্ব/ভূমি
∴ tan 30° = AB/BC
⇒ 1/√3 = h/15√3
⇒ h√3 = 15√3
⇒ h = 15√3/√3
∴ h = 15 m

অতএব, মইটি দেয়ালের 15 m উচ্চতায় পৌঁছায়।

৯,৪৪৯.

একজন মোটর গাড়ি চালক ১৩০ কি.মি. দূরত্বে অবস্থিত দুটি শহর ২ ঘণ্টা ১০ মিনিটে ভ্রমণ করতে পারে। তাহলে প্রতি মিনিটে তার গতিবেগ কত?

ক
ক) ১২০০ মিটার/মিনিট
খ
খ) ৬০০ মিটার/মিনিট
গ
গ) ৫০০ মিটার/মিনিট
ঘ
ঘ) ১০০০ মিটার/মিনিট

ব্যাখ্যা

দূরত্ব = ১৩০ কি.মি. = (১৩০ × ১০০০)মিটার = ১৩০০০০ মিটার
সময় = ২ ঘণ্টা ১০ মিনিট
= (২ × ৬০) মিনিট + ১০ মিনিট
= ১২০ মিনিট + ১০ মিনিট
= ১৩০ মিনিট

গতিবেগ = ১৩০০০০/ ১৩০
= ১০০০ মিটার/মিনিট

৯,৪৫০.

If 3x + 2y = 13, 3x - y = 7, what is the value of (x, y)?

ক
ক) (3, 4)
খ
খ) (2, 2)
গ
গ) (3, 2)
ঘ
ঘ) (3, 3)

ব্যাখ্যা

Given that
3x + 2y = 13.............. (1)
3x - y = 7...........(2)

(1) - (2)⇒
3x + 2y - (3x - y) = 13 - 7
3x + 2y - 3x + y = 6
3y = 6
y = 2

Substituting the value of y into equation (2), we get
3x - y = 7
3x - 2 = 7
3x = 7 + 2
3x = 9
x = 3

∴ Determined solution (x , y) = (3, 2)

৯,৪৫১.

Car A travels at the speed of 65 km/hr and reaches its destination in 8 hours. Car B travels at a speed of 70 km/hr and reaches its destination in 4 hours. What is the ratio of the distance covered by car A and car B respectively?

ক
ক) 7:11
খ
খ) 13:7
গ
গ) 7:13
ঘ
ঘ) 11:7

ব্যাখ্যা

Required ratio:
= (65×8):(70×4)
= 520:280
= 13:7

৯,৪৫২.

The present ages of three cousins are in the ratio of 5 : 6 : 7. Three years ago, their total age was 45 years. In two years, what will be the age of the youngest cousin?

ক
15 years
খ
17 years
গ
19 years
ঘ
21 years

ব্যাখ্যা

Question: The present ages of three cousins are in the ratio of 5 : 6 : 7. three years ago, their total age was 45 years. In two years, what will be the age of the youngest cousin?

Solution:
Present age ratio of three cousins is 5 : 6 : 7
Let their ages be 5x, 6x, and 7x, respectively

ATQ,
5x - 3 + 6x - 3 + 7x - 3 = 45
⇒ 18x - 9 = 45
⇒ 18x = 54
⇒ x = 3

∴ The present age of the youngest cousin is = 5x = 3 × 5 = 15 years.
In two years, his age will be = (15 + 2) = 17 years

৯,৪৫৩.

A rectangular plot measuring 90 metres by 50 metre needs to be enclosed by wire fencing such that poles of the fence will be kept 5 metres apart. How many poles will be needed?

ক
ক) 30
খ
খ) 60
গ
গ) 44
ঘ
ঘ) 56

ব্যাখ্যা

Length of the wire fencing
= perimeter
= 2(90+50)
= 280
Two poles are kept 5 metres apart.
Note that the poles are placed along the perimeter of the rectangular plot, not in a single straight line.
Hence, the number of poles required
= 280/5
= 56.

৯,৪৫৪.

If p × q = p + q + (p/q) then according to this the value of 8 × 2 is?

ক
6
খ
10
গ
14
ঘ
16

ব্যাখ্যা

Question: If p × q = p + q + (p/q) then according to this the value of 8 × 2 is?

Solution:
Given that,
p × q = p + q + (p/q)

Now, Substitute p = 8 and q = 2 into the formula,
∴ 8 × 2 = 8 + 2 + (8/2)
= 10 + 4
= 14

৯,৪৫৫.

Volleyball, Hockey, Football

ক
ক) Aquatics
খ
খ) Baseball
গ
গ) Athletes
ঘ
ঘ) Sports

ব্যাখ্যা

Baseball is like volleyball, Hockey and Football.

৯,৪৫৬.

If a + b + c = 6, a² + b² + c² = 14, the value of ab + bc + ca is:

ক
17
খ
11
গ
21
ঘ
23

ব্যাখ্যা

Question: If a + b + c = 6, a² + b² + c² = 14, the value of ab + bc + ca is:

Solution:
দেয়া আছে,
a + b + c = 6
a² + b² + c² = 14

আমরা জানি
(a + b + c)² = a² + b²+ c² + 2(ab + bc + ca)
⇒ 2(ab + bc + ca) = (a + b + c)² - (a²+ b² + c²)
⇒ 2(ab + bc + ca) = 6² - 14
⇒ 2(ab + bc + ca) = 22
∴ (ab + bc + ca) = 11

৯,৪৫৭.

In first 1000 natural numbers, how many integers exist such that they leave a remainder 4 when divided by 7 and a remainder 9 when divided by 11?

ক
ক) 11
খ
খ) 13
গ
গ) 15
ঘ
ঘ) 17

ব্যাখ্যা

When Divided by 7,
A = 7x + 4
So, numbers can be: 4, 11, 18, 25, 32, 39, 46, 53…….

Again,
when divided by 11,
A = 11y + 9
So, numbers can be: 9, 20, 31, 42, 53…….

Here, 53 is common.

Now, LCM of 7 & 11 is 77.
So, the numbers pattern is : 77x + 53

Then,
77x ≤ 1000
Or, x ≤ 12.2
x can be 0, 1, 2 ...... 12 = total 13 integers
It can't be x = 13, then the number will be bigger than 1000

৯,৪৫৮.

The ratio between the two numbers is 3 : 4. If each number is increased by 12, the ratio becomes 5 : 6. The difference between the numbers will be:

ক
ক) 5
খ
খ) 6
গ
গ) 7
ঘ
ঘ) 11

ব্যাখ্যা

The ratio between the two numbers = 3 : 4
Each number is increased by 12

Let the numbers be 3x and 4x

According to the question:
(3x + 12)/(4x + 12) = 5/6
⇒ 6 × (3x + 12) = 5 × (4x + 12)
⇒ 18x + 72 = 20x + 60
⇒ 20x - 18x = 72 - 60
⇒ 2x = 12
⇒ x = 6

Required numbers = 3x = 3 × 6 = 18
4x = 4 × 6 = 24

The difference between the numbers = 24 - 18 = 6
∴ The difference between the numbers will be 6.

৯,৪৫৯.

The simple interest at m% for m years will be Tk. m on a sum of?

ক
Tk. 100/m
খ
Tk. m/100
গ
Tk. 100m
ঘ
Tk. m

ব্যাখ্যা

Question: The simple interest at m% for m years will be Tk. m on a sum of?

Solution:
Given,
Rate of interest, r = m%
Interest, I = m Tk.
Time, n = m yeras.

We know,
I = Pnr
P = I/nr
= (m × 100)/(m × m)
= 100/m Tk.

৯,৪৬০.

Kowshik earns taka 11 for each ticket that he sells and a bonus of taka 2 per ticket for each ticket he sells over 100. If Kowshik was paid taka 3700, how many tickets did he sell?

ক
ক) 120
খ
খ) 250
গ
গ) 280
ঘ
ঘ) 300

ব্যাখ্যা

Suppose, Kowshik sells x ticket
price of all ticket without bonus = Tk. 11x
Number of bonus ticket = (x - 100)
Therefore, 11x + 2(x - 100) = 3700
⇒ 11x + 2x - 200 = 3700
⇒ 13x = 3900
⇒ x = 300

৯,৪৬১.

A train 300 m long passes a pole in 0.4 minutes. How long will it take to cross a bridge 450 m long (in minutes)?

ক
ক) 1.8 min
খ
খ) 1 min
গ
গ) 1.2 min
ঘ
ঘ) 1.5 min

ব্যাখ্যা

প্রশ্ন : A train 300 m long passes a pole in 0.4 minutes. How long will it take to cross a bridge 450 m long (in minutes)?

সমাধান :

Length of train = 300m
Train cross a pole in = 0.4 min
Length of bridge = 450

We Know,
Speed = distance/time
Speed of train = 300/0.4
Speed of train = 750 m/min

∴ Time taken by train to cross bridge = (450 + 300)/750 = 750/750 = 1 min

৯,৪৬২.

ক
(42 - 5√11)/19
খ
(42 + 5√11)/19
গ
(49 - 5√11)/19
ঘ
None of the these

ব্যাখ্যা

Question:

Solution:

৯,৪৬৩.

What is the angle between any two sides of an equilateral triangle?

ক
30°
খ
60°
গ
90°
ঘ
120°

ব্যাখ্যা

Question: What is the angle between any two sides of an equilateral triangle?

Solution:
- যে ত্রিভুজের তিন কোণ সমান, তার বাহুগুলোও সমান হয় বলে তাকে সমবাহু ত্রিভুজ বলে।
- সমবাহু ত্রিভুজের প্রতিটি কোণের মান ৬০° ।
- সমবাহু ত্রিভুজ-কে সুক্ষ্মকোণী ত্রিভুজও বলে।
- সমবাহু ত্রিভুজের মধ্যমাত্রয় পরস্পর সমান।

৯,৪৬৪.

Out of 8 people in a lift, a person weighing 65 kg gets off, and a new person enters. As a result, the average weight of the 8 people increases by 2.5 kg. What is the weight of the new person?

ক
82 kg
খ
84 kg
গ
85 kg
ঘ
70 kg

ব্যাখ্যা

Question: Out of 8 people in a lift, a person weighing 65 kg gets off, and a new person enters. As a result, the average weight of the 8 people increases by 2.5 kg. What is the weight of the new person?

Solution:
ধরি,
লিফটে থাকা 8 জন ব্যাক্তির গড় ওজন = x kg
তাহলে, মোট ওজন = 8x
65 কেজি ওজনের ব্যক্তি বের হয়ে যাওয়ার পর মোট ওজন = 8x − 65

আবার, মনে করি, নতুন ব্যাক্তির ওজন= y kg
তাহলে নতুন ব্যক্তি প্রবেশ করার পর নতুন মোট ওজন হবে = (8x − 65 + y) kg
এবং নতুন গড় হবে = x + 2.5

প্রশ্নমতে,
(8x − 65 + y) /8 = x + 2.5
⇒ 8x − 65 + y = 8x + 20
⇒ 8x + y - 8x = 20 + 65
⇒ y = 85

সুতরাং, নতুন ব্যাক্তির ওজন 85 kg ।

৯,৪৬৫.

The perimeter of a rectangle is 64 cm. If the ratio of the lengths of two adjacent sides is 7 : 9, find the lengths of these sides.

ক
24 cm, 28 cm
খ
14 cm, 18 cm
গ
7 cm, 9 cm
ঘ
None of these

ব্যাখ্যা

Question: The perimeter of a rectangle is 64 cm. If the ratio of the lengths of two adjacent sides is 7 : 9, find the lengths of these sides.

Solution:
Perimeter of a rectangle = 2(L + B) [where L = length, B = Breadth]
Also B : L = 7 : 9
Let actual values are 7x and 9x.
Hence
2(9x + 7x) = 64
⇒ 16x = 32
∴ x = 2

∴ sides will be of 7 × 2 = 14 cm and 9 × 2 = 18 cm.

৯,৪৬৬.

X and Y share profits in the ratio 3 : 2. If 10% of the total profit is donated to a fund and X's share is Tk. 5400, find the total profit.

ক
Tk. 6000
খ
Tk. 9200
গ
Tk. 10000
ঘ
Tk. 14000

ব্যাখ্যা

Question: X and Y share profits in the ratio 3 : 2. If 10% of the total profit is donated to a fund and X's share is Tk. 5400, find the total profit.

Solution:
মনে করি, মোট লাভ = Tk. x
10% ফান্ডে দেওয়ার পর বাকি থাকে = 100% - 10% = 90% of x
= 90x/100

X এবং Y এর লাভের অনুপাত 3 : 2
অর্থাৎ, X এর অংশ = 90x/100 এর 3/(3 + 2)
= 90x/100 এর 3/5

প্রশ্নমতে,
90x/100 × (3/5) = 5400
⇒ 9x/10 × (3/5) = 5400
⇒ 27x/50 = 5400
⇒ 27x = 5400 × 50
⇒ x = 270000/27
∴ x = 10000

সুতরাং, মোট লাভ হলো Tk. 10000

৯,৪৬৭.

The sum of the digits of two - digit number is 10, while when the digits are reversed, the number decrease by 54. Find the changed number.

ক
19
খ
46
গ
37
ঘ
28

ব্যাখ্যা

Question: The sum of the digits of two - digit number is 10, while when the digits are reversed, the number decrease by 54. Find the changed number.

Solution:
Let number be (10x + y)

ATQ
(10x + y) - (10y + x) = 54
⇒ 10x - 10y + y - x = 54
⇒ 9x - 9y = 54
⇒ x - y = 6 ................ (1)

Sum of digits,
x + y = 10 .................(2)

(1) + (2)
x - y + x + y = 6 + 10
⇒ 2x = 16
∴ x = 8
Put the value of x in (2)
We get,
x + y = 10
⇒ y = 10 - 8
∴ y = 2

The required number is = (10x + y)
= (10 × 8) + 2
= 82

Changed number = 28

৯,৪৬৮.

The sum of two numbers is 528 and their H.C.F. is 33. The number of pairs of numbers satisfying the above conditions is:

ক
ক) 4
খ
খ) 6
গ
গ) 8
ঘ
ঘ) 12

ব্যাখ্যা

Let the required number numbers be 33a and 33b
Then, 33a + 33b = 528
⇒a + b = 16
Now, co - primes with sum 16 are (1,15), (3,13), (5,11) and (7,9)
∴ Required numbers are (33 × 1, 33 × 15), (33 × 3, 33 × 13), ( 33 × 5, 33 × 11), (33 × 7, 33 ×9)
The numbers of such pairs are 4

৯,৪৬৯.

Hanif started a 6-mile hike with a full 10-cup canteen of water and finished the hike in 2 hours with 1 cup of water remaining in the canteen. If the canteen leaked at the rate of 1 cup per hour and Hanif drank 3 cups of water during the last mile, how many cups did he drink per mile during the first 5 miles of the hike?

ক
4/5
খ
5/6
গ
6/5
ঘ
5/4
ঙ
None of these

ব্যাখ্যা

Question: Hanif started a 6-mile hike with a full 10-cup canteen of water and finished the hike in 2 hours with 1 cup of water remaining in the canteen. If the canteen leaked at the rate of 1 cup per hour and Hanif drank 3 cups of water during the last mile, how many cups did he drink per mile during the first 5 miles of the hike?

Solution:
No of cups leaked during the trip = 2 cups.
No of cups Harry drank = 7 cups.
No of cups harry drank during the first 5 miles = 7 - 3 = 4 cups
∴ drink/mile = 4/5

৯,৪৭০.

On a river, C is the mid-point between two points A and B on the same bank of the river. A boat can go from A to C and back in 14 hours and from A to B in 20 hours 20 min. How long it would take to go from B to A?

ক
7 h 40 m
খ
3 h 50 m
গ
10 h 10 m
ঘ
None of these

ব্যাখ্যা

Question: On a river, C is the mid-point between two points A and B on the same bank of the river. A boat can go from A to C and back in 14 hours and from A to B in 20 hours 20 min. How long it would take to go from B to A?

Solution:
Time required to travel from A to B = 20 hour 20 min
Time required to travel from A to C = 1/2 (20 h 20 m)
= 10 h 10 m

Given total time from A to C and C to A = 14 h
∴ 10 h 10 m + C to A = 14 h
∴ C to A = 3 h 50 m

Time taken from B to A is twice of C to A
then, time taken from B to A =2 × (3 h 50 m) =7 h 40 m

৯,৪৭১.

Find the least number which when divided by 4, 5, 6, and 7 leaves remainder 2 in each case.

ক
ক) 210
খ
খ) 418
গ
গ) 420
ঘ
ঘ) 422

ব্যাখ্যা

Question: Find the least number which when divided by 4, 5, 6, and 7 leaves remainder 2 in each case.

Solution:
LCM of 4, 5, 6, and 7 = 420
So, the number is = 420 + 2 = 422

৯,৪৭২.

Which of the following equation of the line passing through the points (7, 4) and (5, 1)?

ক
3y = 2x - 17
খ
2y = 3x - 13
গ
3y = 2x + 17
ঘ
5y = 3x - 13

ব্যাখ্যা

Question: Which of the following equation of the line passing through the points (7, 4) and (5, 1)?

Solution:
Given that,
Two points are, (7, 4) and (5, 1)

We know,
Slope, m = (y₂ - y₁)/(x₂ - x₁)
= (4 - 1)/(7 - 5)
∴ m = 3/2

So the slope is 3/2.

Now check which option has slope 3/2 and passes through one of the points (we’ll use point (5, 1)),
খ) 2y = 3x - 13
⇒ y = (3/2)x - (13/2)
slope, m = 3/2

Similar answer for point (7, 4)

∴ Correct Answer: খ) 2y = 3x - 13

৯,৪৭৩.

The cost price of 20 articles is the same as the selling price of x articles. If the profit is 25%, then the value of x is -

ক
ক) Tk. 18
খ
খ) Tk. 16
গ
গ) Tk. 15
ঘ
ঘ) Tk. 14

ব্যাখ্যা

Question: The cost price of 20 articles is the same as the selling price of x articles. If the profit is 25%, then the value of x is -

Solution:
Let,
Cost price of each article be Tk. 1
Cost price of x articles = Tk. x
Selling price of x articles = Tk. 20

∴ Profit = Tk. (20 - x)

ATQ,
(20 - x)/x = 25%
Or, 100(20 - x)/x = 25
Or, 2000 - 100x = 25x
Or, 125x = 2000
∴ x = 16

৯,৪৭৪.

Maruf is travelling on his cycle and he calculated to reach point A at 3 p.m. if he travels at 10 kmph, he will reach there at 1 p.m. if he travels at 15 kmph. At what speed must he travel to reach A at 2 p.m.

ক
15 kmph.
খ
19 kmph.
গ
17 kmph.
ঘ
12 kmph.

ব্যাখ্যা

Question: Maruf is travelling on his cycle and he calculated to reach point A at 3 p.m. if he travels at 10 kmph, he will reach there at 1 p.m. if he travels at 15 kmph. At what speed must he travel to reach A at 2 p.m.

Solution:
Let the distance travelled by x km.

ATQ,
x/10 - x/15 = 2
⇒ 3x - 2x = 60
∴ x = 60

Time taken to travel 60 km at 10km/h = 60/10 hours
= 6 hours

So, Maruf started 6 hours before 3 P.M. i.e., at 9 Α.Μ.

∴ Required speed = 60/5 kmph.
= 12 kmph.

৯,৪৭৫.

If 3^{m + n} = 81, 81^{m - n} = 3, then what is the value of n?

ক
ক) 13/8
খ
খ) 15
গ
গ) 15/8
ঘ
ঘ) 17

ব্যাখ্যা

Question: If 3^{m+ n} = 81, 81^{m - n} = 3, then what is the value of n?

Solution:
3^{m + n} = 81
⇒ 3^{m + n} = 3⁴
⇒ m + n = 4

81^{m - n} = 3
⇒ 3^{4(m - n)} = 3
⇒ 4(m - n) = 1
⇒ m - n = 1/4

m + n - m + n = 4 - (1/4)
⇒ 2n = 15/4
∴ n = 15/8

৯,৪৭৬.

The product of two consecutive even numbers is 168. What is the smaller number?

ক
18
খ
12
গ
8
ঘ
14
ঙ
None of these

ব্যাখ্যা

Question: The product of two consecutive even numbers is 168. What is the smaller number?

Solution:
Let the smaller number be x,
so the next even number is x + 2

ATQ,
⇒ x(x + 2) = 168
⇒ x² + 2x = 168
⇒ x² + 2x - 168 = 0
⇒ x + 14x - 12x - 168 = 0
⇒ x(x + 14) - 12(x + 14) = 0
⇒ (x + 14) (x - 12) = 0
∴ x= -14 and x = 12

So, the smaller number is 12.

৯,৪৭৭.

The ratio of Solution “A” and Solution “B” in the container is 3 : 2 when 10 liters of the mixture is taken out and is replaced by the Solution “B”, the ratio become 2 : 3. The total quantity of the mixture in the container is:

ক
ক) 20
খ
খ) 25
গ
গ) 30
ঘ
ঘ) 35
ঙ
ঙ) 40

ব্যাখ্যা

Ration of A : B = 3 : 2
10 L taken out and replaced by B
So, A remain = 3x - 3×10/5 = 3x - 6 And
B remain = 2x - 2×10/5 +10
= 2x + 6
ATQ,
(3x - 6)/(2x + 6) = 2/3
Or, 9x - 18 = 4x + 12
Or, 5x = 30
or, x = 6
The total quantity of mixture = (3x + 2x) = 5x = 5 × 6 = 30 L [Answer.]

৯,৪৭৮.

What is the compound interest on Tk. 2500 for 2 years at rate of interest 4% per annum?

ক
Tk. 180
খ
Tk. 220
গ
Tk. 210
ঘ
Tk. 204

ব্যাখ্যা

Question: What is the compound interest on Tk. 2500 for 2 years at rate of interest 4% per annum?

Solution:
Here,
C = 2500(1 + 4/100)²
= 2500 × 1.04 × 1.04
= 2704

∴ compound interest = 2704 - 2500 = 204

৯,৪৭৯.

If a pole 6 m high casts a shadow 2√3 m long on the ground, then the elevation of the sun is -

ক
ক) 60°
খ
খ) 30°
গ
গ) 45°
ঘ
ঘ) 90°

ব্যাখ্যা

Question: If a pole 6 m high casts a shadow 2√3 m long on the ground, then the elevation of the sun is -

Solution:

ধরি,
AB = 6, BC = 2√3

ABC সমকোণী ত্রিভুজ হতে পাই,
tanθ = AB/BC
⇒ tanθ = 6/2√3
⇒ tanθ = 3/√3
⇒ tanθ = √3
⇒ tanθ = tan60°
∴ θ = 60°

৯,৪৮০.

5/9 of a number equals twenty-five percent of the second number. Second number equals 1/4 of third number. The value of third number is 2960. What is 30% of first number?

ক
37
খ
88.8
গ
99.9
ঘ
None of these

ব্যাখ্যা

Question: 5/9 of a number equals twenty-five percent of the second number. Second number equals 1/4 of third number. The value of third number is 2960. What is 30% of first number?

Solution:
second number = 1/4 of third number = 2960/4
= 740

(5/9)first number = 25% of second number
first number = (1/4) × 740 × (9/5)
= 333

30% of first number = (3/10) × 333
= 99.9

৯,৪৮১.

The average of 20, 70 and x is 40. If the average of 20, 70, x and y is 50, then y = ?

ক
100
খ
80
গ
70
ঘ
60

ব্যাখ্যা

Question: The average of 20, 70 and x is 40. If the average of 20, 70, x and y is 50, then y = ?

Solution:
The average of 20, 70 and x is 40

(x + 20 + 70)/3 = 40
⇒ x + 20 + 70 = (3 × 40) = 120
⇒ x + 90 = 120
⇒ x = 120 - 90 = 30

the average of 20, 70, x and y is 50

(20 + 70 + x + y)/4 = 50
⇒ 90 + x + y = (4 × 50)
⇒ 90 + 30 + y = 200
⇒ 120 + y = 200
⇒ y = 200 - 120
∴ y = 80

৯,৪৮২.

Of the students in a school, 20 percent are in the science club and 30 percent are in the band. If 25 percent of the students in the school are in the band but are not in the science club, what percent of the students who are in the science club are not in the band?

ক
ক) 5%
খ
খ) 20%
গ
গ) 75%
ঘ
ঘ) 60%

ব্যাখ্যা

প্রশ্ন : Of the students in a school, 20 percent are in the science club and 30 percent are in the band. If 25 percent of the students in the school are in the band but are not in the science club, what percent of the students who are in the science club are not in the band?

সমাধান :

ধরি, মোট ছাত্রসংখ্যা ১০০ জন

৩০ জন ছাত্র ব্যান্ড এর সদস্য, এর মধ্যে ২৫ জন ছাত্র বিজ্ঞানের নয়।

সুতরাং, বিজ্ঞান ও ব্যান্ড উভয়ের সদস্য = ৩০ - ২৫ = ৫ জন।

অপরদিকে, বিজ্ঞানের ছাত্র ২০ জন, এর মধ্যে ৫ জন উভয়ের মধ্যে অন্তর্ভুক্ত।

আমরা বলতে পারি, বিজ্ঞানে রয়েছে কিন্তু ব্যান্ড দলে নয় এরূপ ছাত্র সংখ্যা = ২০ - ৫ = ১৫ জন
বিজ্ঞানে রয়েছে কিন্তু ব্যান্ড দলে নয় এরূপ ছাত্রসংখ্যার হার = 15 / 20 = 75 / 100 = 75%

৯,৪৮৩.

Anis, who owns (200/3)% of a factory, sells half of his share for Tk. 33,333. The value of the entire factory is-

ক
ক) Tk. 9,999
খ
খ) Tk. 66,666
গ
গ) Tk. 99,999
ঘ
ঘ) Tk. 133,332

ব্যাখ্যা

Question: Anis, who owns (200/3)% of a factory, sells half of his share for Tk. 33,333. The value of the entire factory is-

Solution:
(200/3)% = 200/(3 × 100)
= 2/3 of factory

1/2 of 2/3 or 1/3 of factory = 33,333 tk
∴ value of entire factory = 33,333 × 3 tk
= 99,999 tk.

৯,৪৮৪.

If you write down all the numbers from 1 to 100, then how many times do you write 5?

ক
12
খ
19
গ
20
ঘ
25

ব্যাখ্যা

Question: If you write down all the numbers from 1 to 100, then how many times do you write 5?

Solution:
১ - ১০ পর্যন্ত ৫ আছে = ১ টি
১১ - ২০ পর্যন্ত ৫ আছে = ১ টি
২১ - ৩০ পর্যন্ত ৫ আছে = ১ টি
৩১ - ৪০ পর্যন্ত ৫ আছে = ১ টি
৪১ - ৫০ পর্যন্ত ৫ আছে = ২ টি
৫১ - ৬০ পর্যন্ত ৫ আছে = ১০ টি
৬১ - ৭০ পর্যন্ত ৫ আছে = ১ টি
৭১ - ৮০ পর্যন্ত ৫ আছে = ১ টি
৮১ - ৯০ পর্যন্ত ৫ আছে = ১ টি
৯১ - ১০০ পর্যন্ত ৫ আছে = ১ টি

∴ মোট ৫ রয়েছে = ২০ টি

৯,৪৮৫.

In the coordinate plane, line m passes through the origin and has a slope of 3. If points (6, y) and (x, 12) are on line m, then y - x = ?

ক
26
খ
24
গ
18
ঘ
14

ব্যাখ্যা

Question: In the coordinate plane, line m passes through the origin and has a slope of 3. If points (6, y) and (x, 12) are on line m, then y - x = ?

Solution:
আমরা জানি,
মূলবিন্দুগামী রেখার সমীকরণ y = mx
এখানে, ঢাল m = 3
y = 3x ------- (1)

(6, y) বিন্দুর জন্য (1) নং হতে পাই,
y = 3x
⇒ y = 3 . 6
∴ y = 18

আবার, (x, 12) বিন্দুর জন্য (1) নং হতে পাই,
y = 3x
⇒ 12 = 3x
∴ x = 4

∴ y - x = 18 - 4 = 14

৯,৪৮৬.

If a² − b² = 20; a + b = 5, What is the Value of a − b?

ক
ক) 3
খ
খ) 15
গ
গ) 5
ঘ
ঘ) 4

ব্যাখ্যা

a² − b² = 20
Or, (a + b) (a - b) = 20
Or, a - b = 20/(a + b) = 20/5 = 4

৯,৪৮৭.

The annual income of Mim, Zim and Jorna taken together is Tk. 46000. Mim spends 70 % of income, Zim spends 80 % of her income and Jorna spends 92 % of her income. If their annual savings are 15 : 11 : 10, find the annual saving of Mim?

ক
ক) Tk. 10000
খ
খ) Tk. 3000
গ
গ) Tk. 2200
ঘ
ঘ) Tk. 7000

ব্যাখ্যা

Question: The annual income of Mim, Zim and Jorna taken together is Tk. 46000. Mim spends 70 % of income, Zim spends 80 % of her income and Jorna spends 92 % of her income. If their annual savings are 15 : 11 : 10, find the annual saving of Mim?

Solution:
Let,
Income of Mim, Zim and Jorna are A, B and C.
Annual income given is Tk. 46000

If 70 % income is spent by Mim, then that means she saves 30% = 0.3A
Similarly, Zim saves 20% = 0.2B and Jorna saves 8% = 0.08C

Given ratio of their annual savings are 15 : 11 : 10
∴ (0.3A)/15 = (0.2B)/11 = (0.08C)/10
=A/50 = B/55 = C/125
= A/10 = B/11 = C/25 = (A + B + C)/(10 + 11 + 25) = 46000/46 = 1000

∴ A = 1000 × 10 = 10000
∴ B = 1000 x 11 = 11000
∴ C = 1000 x 25 = 25000

∴ The annual saving of Mim = 0.3 × 10000
= 3000

৯,৪৮৮.

(0.75 × 4.4 × 2.4)/0.6 =?

ক
ক) 4.752
খ
খ) 12
গ
গ) 16.944
ঘ
ঘ) 13.2

ব্যাখ্যা

Question: (0.75 × 4.4 × 2.4)/0.6 =?

Solution:
(0.75 × 4.4 × 2.4)/0.6
= 7.92/0.6
= 13.2

৯,৪৮৯.

If Tk. 3000 is loaned for 4 months at a 4.5% annual rate, how much interest is earned?

ক
Tk. 45
খ
Tk. 60
গ
Tk. 120
ঘ
Tk. 135

ব্যাখ্যা

Question: If Tk. 3000 is loaned for 4 months at a 4.5% annual rate, how much interest is earned?

Solution:
P = 3000
n = 4 months = 4/12 years = 1/3 year
r = 4.5%

I = Pnr
= {3000 × (1/3) × (4.5)}/100
= 45

৯,৪৯০.

Find the value of 1 + {tan²θ/(1 + secθ)}

ক
secθ
খ
1/secθ
গ
cosθ
ঘ
tanθ

ব্যাখ্যা

Question: Find the value of 1 + {tan²θ/(1 + secθ)}.

Solution:
1 + {tan²θ / (1 + secθ)}
= 1 + {(sec²θ − 1)/(1 + secθ)}
= (1 + secθ + sec²θ − 1)/(1 + secθ)
= (secθ + sec²θ)/(1 + secθ)
= secθ

৯,৪৯১.

A car covers the first 39 kms of its journey in 45 minutes and covers the remaining 25 km in 35 minutes. What is the average speed of the car?

ক
40 km/hr
খ
48 km/hr
গ
49 km/hr
ঘ
64 km/hr

ব্যাখ্যা

Total distance travelled = (39 + 25)
= 64 km
Total time taken = (45 + 35)
= 80 min.
= (80/60) hr.
= (4/3) hr.
∴ Average speed = {64 × (3/4)} km/hr
= 48 km/hr.
Hence, the average speed of the car is 48 km/hr.

৯,৪৯২.

Find out the wrong number in the following series:
5, 10, 17, 26, 36, 50, 65

ক
17
খ
50
গ
26
ঘ
36

ব্যাখ্যা

Question: Find out the wrong number in the following series:
5, 10, 17, 26, 36, 50, 65

Solution:
Given,
The following series: 5, 10, 17, 26, 36, 50, 65

The difference between consecutive numbers of the given series are respectively = 5, 7, 9, 11, 13 and 15.
Therefore, 26 + 11 = 37
But in this question it is given 36.

So 36 is wrong number.

৯,৪৯৩.

Quantity A =Time to travel 95 miles at 50 miles per hour; and Quantity B = Time to travel 125 miles at 60 miles per hour.

ক
ক) Quantity A is greater
খ
খ) Quantity A equals Quantity B
গ
গ) Quantity B is greater
ঘ
ঘ) Relationship indeterminate
ঙ
ঙ) None of these

ব্যাখ্যা

Question: Quantity A = Time to travel 95 miles at 50 miles per hour; and Quantity B = Time to travel 125 miles at 60 miles per hour.

Solution:
Quantity A = 95/50 = 19/10 = 1.9 hour

Quantity B = 125/60 = 25/12 = 2.083 hour

∴ Quantity B is greater.

৯,৪৯৪.

The ratio of milk to water in three containers of equal capacity is 1 : 2, 7 : 8 and 11 : 4, respectively. The three containers are mixed together. What is the ratio of milk to water after mixing?

ক
ক) 27 : 13
খ
খ) 29 : 12
গ
গ) 21 : 27
ঘ
ঘ) 23 : 22

ব্যাখ্যা

ধরি,
প্রত্যেকটি পাত্রের ধারণ ক্ষমতা x

১ম পাত্রে
দুধের পরিমাণ = x/3
পানির পরিমাণ = 2x/3

২য় পাত্রে
দুধের পরিমাণ = 7x/15
পানির পরিমাণ = 8x/15

৩য় পাত্রে
দুধের পরিমাণ =11x/15
পানির পরিমাণ = 4x/15

নতুন মিশ্রণে দুধের পরিমাণ = (x/3) + (7x/15) + (11x/15)
= (5x + 7x + 11x)/15
= 23x/15

নতুন মিশ্রণে পানির পরিমাণ = (2x/3) + (8x/15) + (4x/15)
= (10x + 8x + 4x)/15
= 22x/15

নতুন মিশ্রণে দুধ ও পানির অনুপাত = (23x/15) : (22x/15)
= 23 : 22

৯,৪৯৫.

A committee of 3 members is to be formed by selecting out of 5 men and 4 women. In how many different ways the committee can be formed if it should have 1 man and 2 women?

ক
ক) 10
খ
খ) 11
গ
গ) 25
ঘ
ঘ) 30

ব্যাখ্যা

Question: A committee of 3 members is to be formed by selecting out of 5 men and 4 women. In how many different ways the committee can be formed if it should have 1 man and 2 women?

Solution:
We can select 1 men from 5 men in = ⁵C₁ ways = 5
2 women from 4 women in = ⁴C₂ = 6
Committee can be formed = 5 × 6 = 30

৯,৪৯৬.

ক
1
খ
2
গ
2^1/2
ঘ
2³
ঙ
None

ব্যাখ্যা

Question:

Solution:

৯,৪৯৭.

Length of train A is 50 % more than the train B coming from its opposite direction. Speed of train A is 40 km/hour and that of for train B is 32 km/hour and they took 15 sec time to cross each other. Find the length of train A.

ক
120 m
খ
135 m
গ
150 m
ঘ
180 m

ব্যাখ্যা

Question: Length of train A is 50 % more than the train B coming from its opposite direction. Speed of train A is 40 km/hour and that of for train B is 32 km/hour and they took 15 sec time to cross each other. Find the length of train A.

Solution:
let, length of train B is x km
length of train A is x + .5x = 1.5x

(x + 1.5x) / (40 + 32) = 15/3600
⇒ 2.5x = (15 × 72) / 3600
⇒ x = (15 × 72) / (3600 × 2.5) = 0.12 km

length of train A = 1.5 × 0.12
= 0.18 km
= 180 m

৯,৪৯৮.

Which one of the following is not a prime number?

ক
89
খ
79
গ
69
ঘ
29

ব্যাখ্যা

Question: Which one of the following is not a prime number?

Solution:
69 = 3 × 23 which is not a prime number.
29, 79, 89 are prime numbers.

৯,৪৯৯.

A, B and C can do a piece of work in 20, 30 and 60 days respectively. In how many days can A do the work if he is assisted by B and C on every third day?

ক
18 days
খ
20 days
গ
15 days
ঘ
21 days

ব্যাখ্যা

Question: A, B and C can do a piece of work in 20, 30 and 60 days respectively. In how many days can A do the work if he is assisted by B and C on every third day?

Solution:
A's 2 day's work = (1/20) × 2 = 1/10
(A + B + C)'s 1 day's work = 1/20 + 1/30 + 1/60
= 6/60
= 1/10

Work done in 3 days = 1/10 + 1/10
= 2/10 part
= 1/ 5 part

Now,
1/5 work is done in 3 days.
∴ Whole work will be done in (3 × 5) = 15 days

৯,৫০০.

A train 240 m long passed a pole in 24 sec. How long will it take to pass the 650 m long platform?

ক
ক) 65 sec
খ
খ) 89sec
গ
গ) 100sec
ঘ
ঘ) 130sec

ব্যাখ্যা

Train’s speed = 240/24 = 10 m/s
The train has to cover = (240 + 650) = 890 m.
∴ Required time = 890/10 = 89 seconds