Bank Math

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

১.

The perimeter of a rectangle is 36 cm. If its length is 10 cm, what is its breadth in cm?

ক
6
খ
7
গ
9
ঘ
8

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: The perimeter of a rectangle is 36 cm. If its length is 10 cm, what is its breadth in cm?

Solution:
Given that,
Perimeter = 36 cm
Length = 10 cm
Let the breadth = b cm

Then we know,
Perimeter = 2 × (length + breadth)
⇒ 2 × (10 + b) = 36
⇒ 10 + b = 36/2 = 18
⇒ 10 + b = 18
⇒ b = 18 - 10
∴ b = 8

So the breadth is 8 cm.

২.

The average of 3, 8, 7, and x is 6 and the average of 19, 2, 7, x and y is 9. What is the value of y?

ক
12
খ
15
গ
11
ঘ
16

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: The average of 3, 8, 7, and x is 6 and the average of 19, 2, 7, x and y is 9. What is the value of y?

Solution:
Given that,
The average of 3, 8, 7, x is 6

Therefore,
6 = (3 + 8 + 7 + x)/4
⇒ 24 = 18 + x
⇒ x = 24 - 18
∴ x = 6

Therefore,
9 = (19 + 2 + 7 + x + y)/5
⇒ 45 = 28 + 6 + y
⇒ y = 45 - 34
∴ y = 11

৩.

A teacher says, 15% of the first number is equal to 21% of the second number. So, 18% of the first number will be what percent of the second number?

ক
ক) 18%
খ
খ) 20%
গ
গ) 24%
ঘ
ঘ) 25.2%

সঠিক উত্তর: ঘ

ঘ) 25.2%

উত্তর

সঠিক উত্তর: ঘ

ঘ) 25.2%

ঘ

ব্যাখ্যা

According to the question,
15% of 1st no. =21% of 2nd no.
1% of 1st no.= (21/15) % of 2nd no.
18% of 1st no.= {(21/15) × 18} % of 2nd no.
=25.2%

৪.

A train 125 m long passes a person, running at 8 km/hr in the same direction in which the train is going in 25 seconds. The speed of the train is :

ক
ক) 22
খ
খ) 36
গ
গ) 30
ঘ
ঘ) 26

সঠিক উত্তর: ঘ

ঘ) 26

উত্তর

সঠিক উত্তর: ঘ

ঘ) 26

ঘ

ব্যাখ্যা

Speed of the train relative to Person
= (125/25) m/s.
= 5 m/s.
= 5 × (18/5) km/hr
= 18 km/hr

Let the speed of the train be x km/hr.
then, relative speed = (x - 8) km/hr.

So, (x - 8) = 18
⇒ x = 26 km/hr.

৫.

A bank offers 12% compound interest calculated half-yearly. A customer deposits Tk. 5000 on 1st January and another Tk. 5000 on 1st July of the same year. How much interest will he earn at the end of the year?

ক
Tk. 900
খ
Tk. 918
গ
Tk. 956
ঘ
Tk. 1020

সঠিক উত্তর: খ

Tk. 918

উত্তর

সঠিক উত্তর: খ

Tk. 918

খ

ব্যাখ্যা

Question: A bank offers 12% compound interest calculated half-yearly. A customer deposits Tk. 5000 on 1st January and another Tk. 5000 on 1st July of the same year. How much interest will he earn at the end of the year?

Solution:
এখানে,
অর্ধ-বার্ষিক সুদের হার = 12% ÷ 2 = 6%
প্রথম জমা (1লা জানুয়ারী):
আসল, P₁ = 5000 টাকা
সময়, n₁ = 1 বছর = 2 অর্ধ-বছর
সুদের হার, r = 6%
চক্রবৃদ্ধি মূল (A₁) = P₁ (1 + r/100)^n₁
= 5000 × (1 + 6/100)²
= 5000 × (106/100)²
= 5000 × (106/100) × (106/100)
= 5618 টাকা

দ্বিতীয় জমা (1লা জুলাই):
আসল, P₂ = 5000 টাকা
সময়, n₂ = 6 মাস = 1 অর্ধ-বছর
সুদের হার, r = 6%
চক্রবৃদ্ধি মূল (A₂) = P₂ (1 + r/100)^n₂
= 5000 × (1 + 6/100)¹
= 5000 × (106/100)
= 5300 টাকা

মোট প্রাপ্ত মূল = A₁ + A₂ = 5618 + 5300 = 10918 টাকা
মোট জমা = 5000 + 5000 = 10000 টাকা

∴ মোট অর্জিত সুদ = মোট প্রাপ্ত মূল - মোট জমা
= 10918 - 10000 = 918 টাকা

৬.

Two students appeared at an examination. One of them secured 9 marks more than the other and his marks was 56% of the sum of their marks. The marks obtained by them are:

ক
42, 33
খ
47, 38
গ
68, 59
ঘ
75, 66

সঠিক উত্তর: ক

42, 33

উত্তর

সঠিক উত্তর: ক

42, 33

ক

ব্যাখ্যা

প্রশ্ন: Two students appeared at an examination. One of them secured 9 marks more than the other and his marks was 56% of the sum of their marks. The marks obtained by them are:

সমাধান:
ধরি,
একজন পায় ক নম্বর
অপরজন পায় ক + ৯ নম্বর

শর্তমতে,
ক + ৯ = (ক + ক + ৯) × (৫৬/১০০)
বা, ১০০(ক + ৯) = ৫৬(২ক +৯)
বা, ১০০ক + ৯০০ = ১১২ক + ৫০৪
বা, ১২ক = ৩৯৬
∴ ক = ৩৩

∴ একজন পায় ৩৩ নম্বর
অপরজন পায় ৩৩ + ৯ = ৪২ নম্বর

৭.

The difference between the two numbers is 16. If one-third of the smaller number is greater than one-seventh of the larger number by 4, then the two numbers are -

ক
ক) 9 and 25
খ
খ) 12 and 28
গ
গ) 33 and 49
ঘ
ঘ) 56 and 72

সঠিক উত্তর: গ

গ) 33 and 49

উত্তর

সঠিক উত্তর: গ

গ) 33 and 49

গ

ব্যাখ্যা

Let the numbers be x and (x + 16)
Then, x/3 - (x + 16)/7 = 4
⇔ 7x - 3(x +16) = 84
⇔ 4x = 84 + 48 = 132
⇔ x = 132/4 = 33
hence, the numbers are 33 and 49
Answer: 33 and 49

৮.

A table fan is quoted for Tk. 1500. Roni pays Tk. 1173 for it. If he gets a series of two discounts and the rate of the first discount is 15%, then the rate of the second discount is?

ক
4%
খ
6.69%
গ
10.02%
ঘ
8%

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: A table fan is quoted for Tk. 1500. Roni pays Tk. 1173 for it. If he gets a series of two discounts and the rate of the first discount is 15%, then the rate of the second discount is?

Solution:
After first discount = 1500 - 1500 × 15%
= 1500 - 1500 × 15/100
= 1500 -225
= 1275 taka

let second discount is x%

1275 - 1275 × x/100 = 1173
⇒ 1275x/100 = 1275 - 1173 = 102
⇒ x = 102 × 100/1275
= 8

৯.

In one hour, a boat goes 11 km along the stream and 5 km against the stream. The speed of the boat in still water(in km/hr) is

ক
ক) 5 km/hr
খ
খ) 6 km/hr
গ
গ) 7 km/hr
ঘ
ঘ) 8 km/hr

সঠিক উত্তর: ঘ

ঘ) 8 km/hr

উত্তর

সঠিক উত্তর: ঘ

ঘ) 8 km/hr

ঘ

ব্যাখ্যা

Let
speed of boat and stream be x km/hr and y km/hr.
⇒ Downstream speed = (x + y) km/hr
⇒ Upstream speed = (x - y) km/hr
According to the question
(x + y) = 11 km/hr ----(i)
(x - y) = 5 km/hr ----(ii)

Add equation (i) and equation (ii)
2x = 16
x = 8

∴ Speed of the boat is 8 km/hr in still water.

১০.

If a 64 meter ladder is placed against a 32√2 meter wall such that it just reaches the top of the wall, the angle of elevation of the wall is -

ক
ক) 30º
খ
খ) 45º
গ
গ) 60º
ঘ
ঘ) 90º

সঠিক উত্তর: খ

খ) 45º

উত্তর

সঠিক উত্তর: খ

খ) 45º

খ

ব্যাখ্যা

Question: If a 64 meter ladder is placed against a 32√2 metre wall such that it just reaches the top of the wall, the angle of elevation of the wall is

Solution:
Given that
Ladder's length = 64 m
Wall's height = 32√2 m

Perpendicular = Wall's height = 32√2 m
Hypotenuse = Ladder's length = 64 m

We know
Sinθ = Perpendicular/Hypotenuse
⇒ Sinθ = 32√2/64
⇒ Sinθ = 1/√2
⇒ Sinθ = sin 45º
∴ θ = 45º

১১.

Roni bought an old cycle for Tk. 1200 and spent Tk. 200 on its repair. He sold it for Tk. 1680. His profit percent is =?

ক
12%
খ
20%
গ
24%
ঘ
30%

সঠিক উত্তর: খ

20%

উত্তর

সঠিক উত্তর: খ

20%

খ

ব্যাখ্যা

Question: Roni bought an old cycle for Tk. 1200 and spent Tk. 200 on its repair. He sold it for Tk. 1680. His profit percent is =?

Solution:
Total cost = 1200 + 200 taka
= 1400 taka

profit = 1680 - 1400
= 280 taka

%profit = (280/1400) × 100%
= 20%

১২.

If a woman sells her table for Tk. 2,400, she loses 20%. To gain 20%, at what price should she sell it?

ক
Tk. 3000
খ
Tk. 3400
গ
Tk. 3600
ঘ
Tk. 4000

সঠিক উত্তর: গ

Tk. 3600

উত্তর

সঠিক উত্তর: গ

Tk. 3600

গ

ব্যাখ্যা

Question: If a woman sells her table for Tk. 2,400, she loses 20%. To gain 20%, at what price should she sell it?

Solution:
Let the Cost price of the Table is x.
Selling price = x - 20% of x
⇒ 2400 = x - (20x/100)
⇒ 2400 = 80x/100
⇒ 80x = 240000
∴ x = 3000

To gain 20% = 3000 + 20% of 3000
= 3000 + 600
= Tk. 3600

To gain 20%, she should sell the table for Tk. 3600.

১৩.

Four fifth of a number is 12 more than the two third of the number. The number is -

ক
55
খ
65
গ
80
ঘ
90

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: Four fifth of a number is 12 more than the two third of the number. The number is -

Solution:
Let,
The number is x.
ATQ
(4/5) × x = {(2/3) × x} + 12
⇒ (4x)/5 = (2x)/3 + 12
⇒ 3 × 4x = 5 × 2x + 180 [multiply with 15]
⇒ 12x = 10x + 180
⇒ 2x = 180
∴ x = 90

১৪.

A shopkeeper marks his goods 30% above the cost price and offers a discount of 10% on the marked price. What is his gain percent?

ক
25%
খ
20%
গ
17%
ঘ
15%

সঠিক উত্তর: গ

17%

উত্তর

সঠিক উত্তর: গ

17%

গ

ব্যাখ্যা

Question: A shopkeeper marks his goods 30% above the cost price and offers a discount of 10% on the marked price. What is his gain percent?

Solution:
Let,
the cost price (CP) be Tk. 100

Marked Price = 30% more than cost price
= 100 + 30
= 130 Tk.

Discount = 10% of 130
= (10/100) × 130
=13 Tk.

Selling Price (SP)= 130 - 13 = 117 Tk.

∴ Profit = SP - CP = 117 - 100 = 17 Tk.

∴ Gain percent = 17%

১৫.

9 : 10 : 12 : 14 : 15 are the ratio of the angles of a pentagon. What is the sum of measures of the smallest and largest angles?

ক
216°
খ
135°
গ
116°
ঘ
81°

সঠিক উত্তর: ক

216°

উত্তর

সঠিক উত্তর: ক

216°

ক

ব্যাখ্যা

Question: 9 : 10 : 12 : 14 : 15 are the ratio of the angles of a pentagon. What is the sum of measures of the smallest and largest angles?

Solution:
ধরি, পঞ্চভুজের কোণগুলো যথাক্রমে 9x, 10x, 12x, 14x, 15x

পঞ্চভুজের 5 কোণের সমষ্টি = 540°

প্রশ্নমতে,
9x + 10x + 12x + 14x + 15x = 540°
60x = 540°
∴ x = 9°

ক্ষুদ্রতম কোণের মান = 9 × 9° = 81°
বৃহত্তম কোণের মান = 15 × 9° = 135°
সমষ্টি = 81° + 135°
= 216°

১৬.

Over 27 innings, a cricket player averages 47 runs per innings. His maximum score is 157 runs more than his minimum score. Excluding these two innings, the average of the other 25 innings drops to 42 runs. Find the player’s highest score.

ক
198
খ
165
গ
190
ঘ
188
ঙ
176

সঠিক উত্তর: ঘ

188

উত্তর

সঠিক উত্তর: ঘ

188

ঘ

ব্যাখ্যা

Question: Over 27 innings, a cricket player averages 47 runs per innings. His maximum score is 157 runs more than his minimum score. Excluding these two innings, the average of the other 25 innings drops to 42 runs. Find the player’s highest score.

Solution:
Given that,
The batting average for 27 innings of a cricket player is 47 runs.
His highest score exceeds his lowest score by 157 runs.
If these two innings are excluded, the average of the remaining 25 innings is 42 runs.

Now,
Sum of runs for 27 innings of a cricket player = 47 × 27 = 1269
Sum of runs for 25 innings of a cricket player = 42 × 25 = 1050
∴ Sum of remaining 2 innings = 1269 - 1050 = 219

Let,
The minimum score be x and the maximum score be x + 157

According to the question,
x + x + 157 = 219
⇒ 2x = 219 - 157
⇒ 2x = 62
∴ x = 31

So, highest score = 157 + 31 = 188

১৭.

The distance between two places A and B is 370 km. the 1st car departs from place A to B, at a speed of 80 kmph at 10 am and 2nd car departs from place B to A at a speed of 50 kmph at 1 pm. At what time both cars meet each other?

ক
1:00 pm
খ
1:30 pm
গ
2:00 pm
ঘ
2:30 pm

সঠিক উত্তর: গ

2:00 pm

উত্তর

সঠিক উত্তর: গ

2:00 pm

গ

ব্যাখ্যা

Question: The distance between two places A and B is 370 km. the 1st car departs from place A to B, at a speed of 80 kmph at 10 am and 2nd car departs from place B to A at a speed of 50 kmph at 1 pm. At what time both cars meet each other?

Solution:
Total distance = 370 km
Now, the distance covered by first car in (10 am to 1 pm =) 3 hours
= 80 × 3
= 240 km

Remaining distance = 370 - 240 = 130 km
Relative speed = 80 + 50 = 130 kmph
Now, they cover 130 km in (130/130) = 1/1 hrs = 60 mins

So, they meet 60 mins after 1 pm.
So, requred answer = 2:00 pm.

১৮.

A man deposits Tk. 5000 in a Bank at 10% interest rate compounded annually. At the end of the three year, the total amount including interest will become?

ক
ক) 605
খ
খ) 6655
গ
গ) 7755
ঘ
ঘ) 5566

সঠিক উত্তর: খ

খ) 6655

উত্তর

সঠিক উত্তর: খ

খ) 6655

খ

ব্যাখ্যা

Question: A man deposits Tk. 5000 in a Bank at 10% interest rate compounded annually. At the end of the three year, the total amount including interest will become?

Solution:
Given,
Principal, P = 5000 Tk.
Rate of interest, r = 10% = 10/100 = 1/10
Time, n = 3 years.

We know,
Compound Amount = P (1 + r)ⁿ
= 5000 × (1 + 1/10)³
= 5000 × (11/10)³
= 5000 × (11/10) × (11/10) × (11/10)
= 6655

১৯.

The equation 12x² + 4kx + 3 = 0 has real and equal roots, if-

ক
k = ± 2
খ
k = ± 3
গ
k = ± 4
ঘ
k = ± 6

সঠিক উত্তর: খ

k = ± 3

উত্তর

সঠিক উত্তর: খ

k = ± 3

খ

ব্যাখ্যা

Question: The equation 12x² + 4kx + 3 = 0 has real and equal roots, if-

Solution:
Here a = 12, b = 4k, c = 3
Since the given equation has real and equal roots
∴ b² - 4ac = 0
⇒ (4k)² - 4 × 12 × 3 = 0
⇒ 16k² - 144 = 0
⇒ 16k² = 144
⇒ k² = 9
⇒ k = ± 3

২০.

After dividing a number successively by 3, 4, and 7, the remainder obtained is 2, 1, and 4 respectively. What will be the remainder if 84 is divided by the same number?

ক
49
খ
51
গ
53
ঘ
57

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: After dividing a number successively by 3, 4, and 7, the remainder obtained is 2, 1, and 4 respectively. What will be the remainder if 84 is divided by the same number?

Solution:
As the Number gives a remainder of 4 when it is divided by 7, then the number = (7x + 4)
The same gives remainder 1 when it is divided 4, then the number = {4 × (7x + 4) + 1}
Also, the number when divided by 3 gives the remainder 2, thus number = [3 × {4 × (7x + 4) + 1} + 2]
= 3 × (28x + 16 + 1) + 2
= 84x + 51 + 2
= 84x + 53

We get the final number 53 more than a multiple of 84 Hence, if the number is divided by 84,
The remainder will be 53

২১.

If the sum of a number and 10 exceeds 1/21 of 5/7 of the number by 294, then what will be the number?

ক
21
খ
127
গ
294
ঘ
81

সঠিক উত্তর: গ

294

উত্তর

সঠিক উত্তর: গ

294

গ

ব্যাখ্যা

Let the number be X.
Then the sum of the number and 10 = X + 10.
and 1/21 of 5/7 of the number = 1/21 × 5/7 × X
From given,
We have X + 10 - 294 = 1/21 × 5/7 × X
X + 10 - 294 = 1/21 × 5/7 × X
⇒ X - 284 = 5X/147
⇒ 5X/147 - X = -284
⇒ 5X - 147X/147 = -284
⇒ -142X = 147 × (-284)
⇒ X = 147 × 2
⇒ X = 294
Hence the required number is 294.

২২.

If 400 gm of sugar costs 60 taka, how many taka will 1.2 kg cost?

ক
180 taka
খ
250 taka
গ
320 taka
ঘ
290 taka

সঠিক উত্তর: ক

180 taka

উত্তর

সঠিক উত্তর: ক

180 taka

ক

ব্যাখ্যা

Question: If 400 gm of sugar costs 60 taka, how many taka will 1.2 kg cost?

Solution:
400 gm চিনির দাম = 60 taka
∴ 1 gm চিনির দাম = 60/400 taka
= 3/20 taka

এখন, 1.2 kg = 1200 gm

∴ 1200 gm চিনির দাম = 1200 × (3/20) taka
= (1200 × 3)/20 taka
= 3600/20 taka
= 180 taka

২৩.

Two numbers are respectively 20% and 50% more than a third number. The ratio of the two numbers is

ক
ক) 2:5
খ
খ) 3:5
গ
গ) 4:5
ঘ
ঘ) 6:5

সঠিক উত্তর: গ

গ) 4:5

উত্তর

সঠিক উত্তর: গ

গ) 4:5

গ

ব্যাখ্যা

Let the third number be x.
First Number (120/100) × x = 6x/5
Second Number (150/100) × x = 3x/2
Ratio = 6x/5 : 3x/2
=> 4:5

২৪.

The average of smallest and largest primes between 60 and 80 is -

ক
ক) 60
খ
খ) 70
গ
গ) 60
ঘ
ঘ) 77

সঠিক উত্তর: খ

খ) 70

উত্তর

সঠিক উত্তর: খ

খ) 70

খ

ব্যাখ্যা

60 ও 80 এর মধ্যে বৃহত্তম মৌলিক সংখ্যা 79
60 ও 80 এর মধ্যে ক্ষুদ্রতম মৌলিক সংখ্যা 61
সংখ্যা দুটির সমষ্টি = 140
∴ সংখ্যা দুটির গড় = 140/2 = 70

২৫.

31, 29, 24, 22, 17, . . . What number should come next?

ক
15
খ
12
গ
14
ঘ
13

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: 31, 29, 24, 22, 17, . . . What number should come next?

Solution:
In this series at first subtracts 2, then 5.

31 - 2 = 29
29 - 5 = 24
24 - 2 = 22
22 - 5 = 17
17 - 2 = 15

২৬.

If both the length and the breadth of a rectangle are increased by 20%, what is the percentage increase in its area?

ক
22%
খ
44%
গ
33%
ঘ
38%

সঠিক উত্তর: খ

44%

উত্তর

সঠিক উত্তর: খ

44%

খ

ব্যাখ্যা

Question: If both the length and the breadth of a rectangle are increased by 20%, what is the percentage increase in its area?

Solution:
মনে করি,
দৈর্ঘ্য = x একক এবং
প্রস্থ = y একক
∴ ক্ষেত্রফল = xy বর্গ একক

২০% বৃদ্ধিতে,
নতুন দৈর্ঘ্য = x + x এর ২০%
= ১২x /১০ একক

২০% বৃদ্ধিতে,
প্রস্থ = y + y এর ২০%
= ১২y /১০ একক

∴ নতুন ক্ষেত্রফল = (১২x/১০) ×( ১২y/১০) = ১৪৪xy/১০০ বর্গ একক

∴ ক্ষেত্রফল বৃদ্ধি =১৪৪xy/১০০ - xy
=(১৪৪xy - ১০০xy)/১০০
= ৪৪xy/১০০
∴ শতকরা ক্ষেত্রফল বৃদ্ধি = {(৪৪xy/১০০) × (১/xy) × ১০০}%
= ৪৪%

২৭.

A group of workers promises to complete a piece of work in 10 days. But five of them do not report for work. If it took the remaining workers 12 days to complete the work, then the number of workers originally hired was-

ক
15
খ
25
গ
30
ঘ
45

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: A group of workers promises to complete a piece of work in 10 days. But five of them do not report for work. If it took the remaining workers 12 days to complete the work, then the number of workers originally hired was-

Solution:
Let, the workers promised were = x
The workers worked were = x - 5

ATQ,
10x = 12(x - 5)
⇒ 10x = 12x - 60
⇒ 12x - 10x = 60
⇒ 2x = 60
∴ x = 30

২৮.

Abir and Asif started a business investing Tk. 27500 and Tk. 35000 respectively. Out of a total profit of Tk. 14200, Asif's share is?

ক
ক) Tk. 6248
খ
খ) Tk. 7950
গ
গ) Tk. 7952
ঘ
ঘ) Tk. 6250

সঠিক উত্তর: গ

গ) Tk. 7952

উত্তর

সঠিক উত্তর: গ

গ) Tk. 7952

গ

ব্যাখ্যা

Question: Abir and Asif started a business investing Tk. 27500 and Tk. 35000 respectively. Out of a total profit of Tk. 14200, Asif's share is?

Solution:
Given,
Investment ratio = 27500 : 35000
= 11 : 14
Sum of the ratio's = 11 + 14 = 25

∴ Asif's share = 14200 × (14/25)
= 7952

২৯.

Find the missing letter marked (?) in the series:
C, E, H, ?, Q, W.

ক
K
খ
L
গ
M
ঘ
O

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: Find the missing letter marked (?) in the series:
C, E, H, ?, Q, W.

Solution:

এখানে প্যাটার্নটি হলো +2, +3, +4, +5, ...
C (3) + 2 = E (5)
E (5) + 3 = H (8)
H (8) + 4 = 12তম অক্ষর, অর্থাৎ L
L (12) + 5 = Q (17)
Q (17) + 6 = W (23)

সুতরাং প্রশ্নবোধক চিহ্নিত স্থানে L বসবে।

৩০.

⁷P_r = 210 and ⁷C_r = 35 then what is the value of r?

ক
3
খ
4
গ
5
ঘ
6

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: ⁷P_r = 210 and ⁷C_r = 35 then what is the value of r?

Solution:
Given that,
⁷P_r = 210 and ⁷C_r = 35

We know that,
ⁿP_r = r! × ⁿC_r
⇒ 210 = r! × 35
⇒ r! = 210/35
⇒ r! = 6
⇒ r! = 3!
∴ r = 3

৩১.

In an election 2 candidates participate, 20% of the votes are declared invalid, and the winner gets 70% of the valid votes and wins the election by 3200 votes. Find the valid votes.

ক
10000
খ
8000
গ
5620
ঘ
12000

সঠিক উত্তর: খ

8000

উত্তর

সঠিক উত্তর: খ

8000

খ

ব্যাখ্যা

Question: In an election 2 candidates participate, 20% of the votes are declared invalid, and the winner gets 70% of the valid votes and wins the election by 3200 votes. Find the valid votes.

Solution:
Let total votes = 100
Then, 80 votes are valid only.
As per question, the winner gets = 80 × (70/100) = 56 (valid votes),
So, the loser's votes = 80 - 56 = 24
Difference = 56 - 24 = 32, but ATQ, it is 3200
i.e., 32 × 100 = 3200
Hence 80 × 100= 8000
So, we have to multiply 32 by 100 to make it equal to 3200.
So, 80 is also multiplied with 100.
Therefore, the total valid votes = 80 × 100= 8000

৩২.

If you count from 1 to 100, how often will you encounter the number 5?

ক
25
খ
19
গ
12
ঘ
20

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: If you count from 1 to 100, how often will you encounter the number 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

৩৩.

A man travels from his home to office at 4km/hr and reaches his office 20 min late. If the Speed had been 6 km/hr he would have reached 10 min early. Find the distance from his home to office?

ক
6 km
খ
8 km
গ
12 km
ঘ
10 km

সঠিক উত্তর: ক

6 km

উত্তর

সঠিক উত্তর: ক

6 km

ক

ব্যাখ্যা

Question: A man travels from his home to office at 4km/hr and reaches his office 20 min late. If the Speed had been 6 km/hr he would have reached 10 min early. Find the distance from his home to office?

Solution:
Let the distance from home to office = d km

Time taken at 4 km/h = d/4 hours
And time taken at 6 km/h = d/6 hours

The difference between these two times = late time + early time = 20 min + 10 min = 30 min = 1/2 hours

ATQ,
(d/4) - (d/6) = 1/2
⇒ (3d - 2d)/12 = 1/2
⇒ d = 12/2
∴ d = 6 km

So the distance from home to office is 6 km.

৩৪.

If ‘x’ is an integer then solve (log₂x)² - log₂x⁴ - 32 = 0.

ক
125
খ
256
গ
375
ঘ
373
ঙ
None of these

সঠিক উত্তর: খ

256

উত্তর

সঠিক উত্তর: খ

256

খ

ব্যাখ্যা

Question: If ‘x’ is an integer then solve (log₂x)² - log₂x⁴ - 32 = 0.

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

৩৫.

A and B are partners in a business. A contributes 1/4 of the capital for 15 months and B received 2/3 of the profit. For how long B's money was used?

ক
3 months
খ
6 months
গ
10 months
ঘ
12 months
ঙ
5 months

সঠিক উত্তর: গ

10 months

উত্তর

সঠিক উত্তর: গ

10 months

গ

ব্যাখ্যা

Question: A and B are partners in a business. A contributes 1/4 of the capital for 15 months and B received 2/3 of the profit. For how long B's money was used?

Solution:
Let the total profit be Tk. z.
Then,
B's share = Tk. 2z/3,
A's share = Tk. z - (2z/3 ) = Tk. z/3.

A : B = z/3 : 2z/3 = 1 : 2

Let the total capital be Tk. x and suppose B's money was used for y months. Then.
{(x/ 4) × 15}/{(3x/4) × y} = 1/2
⇒ y = (15 × 2)/3 = 10 .

Thus, B's money was used for 10 months.

৩৬.

The ratio of three numbers is 3 : 4 : 5 and the sum of their squares is 1250. The sum of the numbers is-

ক
30
খ
45
গ
60
ঘ
90

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: The ratio of three numbers is 3 : 4 : 5 and the sum of their squares is 1250. The sum of the numbers is-

Solution:
Let, the number be 3x, 4x, 5x

According to the question,
(3x)² + (4x)² + (5x)² = 1250
⇒ 9x² + 16x² + 25x² = 1250
⇒ 50x² = 1250
⇒ x² = 1250/50
⇒ x² = 25
∴ x = 5

∴ The sum of the numbers = 3x + 4x + 5x
= 12x
= 12 × 5
= 60

৩৭.

A person crosses a 450 meters long street in 2.5 minutes. What is his speed in kilometers per hour?

ক
12.2
খ
10.8
গ
11.4
ঘ
None of these

সঠিক উত্তর: খ

10.8

উত্তর

সঠিক উত্তর: খ

10.8

খ

ব্যাখ্যা

Speed = 450 meters / 2.5 minutes
= (450 × 60)/(2.5 × 1000) km/hr
= 10.8 km/hr

৩৮.

By selling 100 notebooks, a shopkeeper gains the selling price of 20 notebooks. What is his gain percentage?

ক
15%
খ
20%
গ
25%
ঘ
10%

সঠিক উত্তর: গ

25%

উত্তর

সঠিক উত্তর: গ

25%

গ

ব্যাখ্যা

Question: By selling 100 notebooks, a shopkeeper gains the selling price of 20 notebooks. What is his gain percentage?

Solution:

Let the SP of 1 notebook be Tk. 1

SP of 100 notebooks = Tk. 100

Now,
gain = selling price of 20 notebooks = Tk. 20

Then
CP = SP - Gain = 100 - 20 = 80

Gain% = (20/80) × 100% = 25%

৩৯.

The average weight of 8 person's increases by 2.5 kg when a new person comes in place of one of them weighing 65 kg. What might be the weight of the new person?

ক
76 kg
খ
76.5 kg
গ
85 kg
ঘ
Data inadequate

সঠিক উত্তর: গ

85 kg

উত্তর

সঠিক উত্তর: গ

85 kg

গ

ব্যাখ্যা

Question: The average weight of 8 person's increases by 2.5 kg when a new person comes in place of one of them weighing 65 kg. What might be the weight of the new person?

Solution:

Total weight increased = (8 × 2.5) kg = 20 kg.
Weight of new person = (65 + 20) kg = 85 kg.

৪০.

If 25% of a = b, then b% of 16 is equal to- ?

ক
2% of a
খ
8% of a
গ
6% of a
ঘ
4% of a

সঠিক উত্তর: ঘ

4% of a

উত্তর

সঠিক উত্তর: ঘ

4% of a

ঘ

ব্যাখ্যা

Question: If 25% of a = b, then b% of 16 is equal to- ?

Solution:
Given that,
⇒ 25% of a = b
⇒ (25/100) × a = b
⇒ a/4 = b
∴ b = a/4

Now,
b% of 16
= (b/100) × 16
= (a/4) × (16/100)
= 4a/100
= 4% of a

৪১.

If an investor gets Tk. 19,500 as interest after 5 years on his savings of Tk. 13,000. What is the rate of interest?

ক
ক) 8%
খ
খ) 5%
গ
গ) 10%
ঘ
ঘ) 30%

সঠিক উত্তর: ঘ

ঘ) 30%

উত্তর

সঠিক উত্তর: ঘ

ঘ) 30%

ঘ

ব্যাখ্যা

Question: If an investor gets Tk. 19,500 as interest after 5 years on his savings of Tk. 13,000. What is the rate of interest?

Solution:
এখানে
মুনাফা I = 19,500 টাকা
আসল P = 13,000 টাকা
সময় n = 5 বছর
মুনাফার হার r = ?

আমরা জানি,
I = Pnr
r = I/Pn
= {(19500 × 100)/(13000 × 5)}%
= 30%

৪২.

What is the largest possible value of cos θ?

ক
90
খ
0
গ
- 1
ঘ
1

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: What is the largest possible value of cos θ?

Solution:
cosθ এর সর্বনিম্ন মান - 1 এবং সর্বোচ্চ মান 1
sinθ এর সর্বনিম্ন মান - 1 এবং সর্বোচ্চ মান 1

৪৩.

A dishonest dealer sells the goods at 6(1/4)% loss on cost price but uses 12(1/2)% less weight. What is the percentage profit or loss?

ক
7(1/2)%
খ
7(5/7)%
গ
7(1/7)%
ঘ
7(2/5)%

সঠিক উত্তর: গ

7(1/7)%

উত্তর

সঠিক উত্তর: গ

7(1/7)%

গ

ব্যাখ্যা

Let, the C.P (cost price) of 1 kg goods be 1 tk
Then, S.P. of ⟮100 - (25/2%) of 1 kg} i.e., g 875 g goods = .9375 tk
S.P of 1 kg goods = (.9375/875 × 1000) tk = 1 (1/14) tk
& there4; Profit % = (1/14 × 100)% = 7(1/7)%
Answer : 7(1/7)%

৪৪.

A and B can do a work in 9 days, B and C can do it in 12 days and A and C can do it in 18 days. If all of them work together, in how many days they can finish the work?

ক
ক) 5 days
খ
খ) 8 days
গ
গ) 10 days
ঘ
ঘ) 15 days

সঠিক উত্তর: খ

খ) 8 days

উত্তর

সঠিক উত্তর: খ

খ) 8 days

খ

ব্যাখ্যা

Question: A and B can do a work in 9 days, B and C can do it in 12 days and A and C can do it in 18 days. If all of them work together, in how many days they can finish the work?

Solution:
A + B can do in 1 day = 1/9 part
B + C can do in 1 day = 1/12 part
A + C can do in 1 day = 1/18 part
2(A + B + C ) can do in 1 day = ((1/9) + (1/12) + (1/18)) = 1/4 part
∴ A + B + C can do in 1 day = 1/(4 × 2) part = 1/8 part

∴ Total days needed = 1/(1/8) = 8 days

৪৫.

Coffee worth Tk. 180 per kg and Tk. 210 per kg are mixed with a third variety in the ratio 2 : 1 : 3. If the mixture is worth Tk. 225 per kg, the price of the third variety per kg will be:

ক
220 Tk.
খ
200 Tk.
গ
260 Tk.
ঘ
280 Tk.

সঠিক উত্তর: গ

260 Tk.

উত্তর

সঠিক উত্তর: গ

260 Tk.

গ

ব্যাখ্যা

Question: Coffee worth Tk. 180 per kg and Tk. 210 per kg are mixed with a third variety in the ratio 2 : 1 : 3. If the mixture is worth Tk. 225 per kg, the price of the third variety per kg will be:

Solution:
Let price of third variety be x Tk. per kg
180 × 2y + 210y + x × 3y = 225(2y + y + 3y)
⇒ 360y + 210y + 3xy = 225 × 6y
⇒ 360 + 210 + 3x = 1350
⇒ 570 + 3x = 1350
⇒ 3x = 780
∴ x = 260 Tk.

৪৬.

Salman bought 1600 eggs at Tk. 3.75 a dozen. He sold 900 of them at 2 for Tk.1 and the remaining at 5 for Tk. 2. His percent gain or loss is-

ক
ক) 26% gain
খ
খ) 46% gain
গ
গ) 36% loss
ঘ
ঘ) 16% loss

সঠিক উত্তর: খ

খ) 46% gain

উত্তর

সঠিক উত্তর: খ

খ) 46% gain

খ

ব্যাখ্যা

C. P of 1600 eggs =Tk. (3.75/12) × 1600
= Tk. 500
S. P of 1600 eggs =Tk. {(1/2) × 900 + (2/5) × 700}
= Tk. (450 + 280)
= Tk. 730

Gain = Tk. (730 - 500)
= TK.230
Gain% = {(230/500) × 100}% = 46%

৪৭.

An amount of money is invested in a savings account for two years. It increased by Tk. 102.50 in two years after annual compounding at the rate of 5% per year. What is the amount invested initially?

ক
Tk. 800
খ
Tk. 1000
গ
Tk. 1250
ঘ
Tk. 1460

সঠিক উত্তর: খ

Tk. 1000

উত্তর

সঠিক উত্তর: খ

Tk. 1000

খ

ব্যাখ্যা

Question: An amount of money is invested in a savings account for two years. It increased by Tk. 102.50 in two years after annual compounding at the rate of 5% per year. What is the amount invested initially?

Solution:
ধরি, আসল = P টাকা
সময়, n = 2 বছর
মুনাফার হার, r = 5% = 5/100 = 1/20
চক্রবৃদ্ধি মুনাফা = 102.50 টাকা

আমরা জানি, চক্রবৃদ্ধি মুনাফা = P(1 + r)ⁿ - P
⇒ P(1 + 1/20)² - P = 102.50
⇒ P(21/20)² - P = 102.50
⇒ P(441/400) - P = 102.50
⇒ (441P - 400P)/400 = 102.50
⇒ 41P/400 = 102.50
⇒ 41P = 102.50 × 400
⇒ 41P = 41000
⇒ P = 41000/41
∴ P = 1000

সুতরাং, শুরুতে বিনিয়োগকৃত আসলের পরিমাণ ছিল 1000 টাকা।

৪৮.

If x² + 4x + 3 is odd, then which one of the following could be the value of x?

ক
ক) 3
খ
খ) 5
গ
গ) 9
ঘ
ঘ) 16

সঠিক উত্তর: ঘ

ঘ) 16

উত্তর

সঠিক উত্তর: ঘ

ঘ) 16

ঘ

ব্যাখ্যা

x = 3 হলে x² + 4x + 3 = 32 + 4 × 3 + 3 = 9 + 12 + 3 = 24
x = 5 হলে x² + 4x + 3 = 5²+ 4 × 5 + 3 = 25 + 20 + 3 = 48
x = 9 হলে x² + 4x + 3 = 9² + 4 × 9 + 3 = 81 + 36 + 3 = 120
x = 16 হলে x² + 4x + 3 = 16² + 4 × 16 + 3 = 256 + 64 + 3 = 323

৪৯.

The value of cos1° cos2° cos3° ............... cos88° cos89° cos90° is?

ক
0
খ
1/2
গ
1
ঘ
1/√2

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

প্রশ্ন: The value of cos1° cos2° cos3° ............... cos88° cos89° cos90° is?

সমাধান:
cos1° cos2° cos3° ............... cos88° cos89° cos90°
= cos90°
= 0 [0 will make whole series 0]
= 0

৫০.

The ratio, by weight, of the four ingredients A, B, C, and D of a certain mixture is 4 : 7 : 8 : 12. The mixture will be changed so that the ratio of A to C is quadrupled and the ratio of A to D is decreased. The ratio of A to B will be held constant. If B will constitute 20% of the weight of the new mixture, by approximately what percent will the ratio of A to D be decreased?

ক
45%
খ
40%
গ
50%
ঘ
None of the Above

সঠিক উত্তর: ক

45%

উত্তর

সঠিক উত্তর: ক

45%

ক

ব্যাখ্যা

Question: The ratio, by weight, of the four ingredients A, B, C, and D of a certain mixture is 4 : 7 : 8 : 12. The mixture will be changed so that the ratio of A to C is quadrupled and the ratio of A to D is decreased. The ratio of A to B will be held constant. If B will constitute 20% of the weight of the new mixture, by approximately what percent will the ratio of A to D be decreased?

Solution:
Lets say our thing has the weight x(4 + 7 + 8 + 12) = 31x.
When we change mixture our weight changes to x(4 + 7 + 2 + 12y)
=13x + 12xy

We also know that 7x = 0.2(13x + 12xy)
Reduce that last equation by x and lets just solve it for y : y = {7 - (0.2 × 13)}/2.4
= 11/6
So the original ratio was 4/12 = 1/3
The new ratio is (1 × 6)/(3 × 11) = 2/11
The resulting percentages are: {(1/3) - (2/11)}/(1/3) = 1 - (6/11)
= 5/11
= 0.4545
≈ 45%

৫১.

Two taps, P and Q, together can fill a tank in 12 minutes. If Tap Q takes 10 minutes more than Tap P to fill the tank separately, how much time will be taken by Tap P alone to fill the tank?

ক
15 minutes
খ
20 minutes
গ
24 minutes
ঘ
40 minutes

সঠিক উত্তর: খ

20 minutes

উত্তর

সঠিক উত্তর: খ

20 minutes

খ

ব্যাখ্যা

Question: Two taps, P and Q, together can fill a tank in 12 minutes. If Tap Q takes 10 minutes more than Tap P to fill the tank separately, how much time will be taken by Tap P alone to fill the tank?

সমাধান:
ধরি,
নল P একা ট্যাঙ্কটি পূর্ণ করতে সময় নেয় x মিনিট।
তাহলে, নল Q একা ট্যাঙ্কটি পূর্ণ করতে সময় নেবে (x + 10) মিনিট।

প্রশ্নমতে, তারা একসাথে 12 মিনিটে পূর্ণ করে। অর্থাৎ,
(1/x) + 1/(x + 10) = 1/12
⇒ (x + 10 + x)/{x(x + 10)} = 1/12
⇒ (2x + 10)/(x² + 10x) = 1/12
⇒ x² + 10x = 12(2x + 10)
⇒ x² + 10x - 24x - 120 = 0
⇒ x² - 14x - 120 = 0
⇒ x² - 20x + 6x - 120 = 0
⇒ x(x - 20) + 6(x - 20) = 0
⇒ (x - 20)(x + 6) = 0

যেহেতু সময় ঋণাত্মক হতে পারে না, তাই x = 20
∴ নল P একা ট্যাঙ্কটি পূর্ণ করতে 20 মিনিট সময় নেবে।

৫২.

Which of the following exactly denotes the average price of all the goods together if, Ramesh buys ‘a’ number of goods of type ‘A’ at price of Tk. ‘E’ each, ‘b’ number of goods of type ‘B’ at price of Tk. ‘F’ each and ‘c’ number of goods of type ‘C’ at price of Tk. ‘G’ each?

ক
(aE + bF + cG)/(a + b + c)
খ
(aA + bB + cC)/(a + b + c)
গ
(E + F + G)/(a + b + c)
ঘ
None of the above

সঠিক উত্তর: ক

(aE + bF + cG)/(a + b + c)

উত্তর

সঠিক উত্তর: ক

(aE + bF + cG)/(a + b + c)

ক

ব্যাখ্যা

Question: Which of the following exactly denotes the average price of all the goods together if, Ramesh buys ‘a’ number of goods of type ‘A’ at price of Tk. ‘E’ each, ‘b’ number of goods of type ‘B’ at price of Tk. ‘F’ each and ‘c’ number of goods of type ‘C’ at price of Tk. ‘G’ each?

Solution:
'a' items of type 'A' at Tk. 'E' each
'b' items of type 'B' at Tk. 'F' each
'c' items of type 'C' at Tk. 'G' each

To find the average price:
First we need total cost of all items:
Cost of A items = a × E
Cost of B items = b × F
Cost of C items = c × G
Total cost = aE + bF + cG

Then divide by total number of items (a + b + c)
Therefore, average price = (aE + bF + cG)/(a + b + c)

৫৩.

A canteen requires 672 bananas for a week. Totally, how many bananas will it require for the months of March, April and May?

ক
8632
খ
8836
গ
8328
ঘ
8832

সঠিক উত্তর: ঘ

8832

উত্তর

সঠিক উত্তর: ঘ

8832

ঘ

ব্যাখ্যা

Question: A canteen requires 672 bananas for a week. Totally, how many bananas will it require for the months of March, April and May?

Solution:
Total number of days = (31 + 30 + 31) = 92
Let the number of bananas be x

ATQ,
7/92 = 672/x
⇒ 7x = 92 × 672
⇒ x = (92 × 672)/7
∴ x = 8832

৫৪.

What is the difference between the compound interests on Tk. 5000 for 3/2 years at 4% per annum compounded yearly and half-yearly?

ক
ক) 5.78
খ
খ) 4.65
গ
গ) 3.24
ঘ
ঘ) 2.04

সঠিক উত্তর: ঘ

ঘ) 2.04

উত্তর

সঠিক উত্তর: ঘ

ঘ) 2.04

ঘ

ব্যাখ্যা

C.I. when interest compound yearly
= Tk. [5000 × (1 + 4/100) (1 + 1/2 × 4/100)]
= Tk. 5304.

C.I. when interest is compounded half-yearly
= Tk. 5000 (1 + 2/100)3
= Tk. 5306.04

Difference
= Tk. (5306.04 - 5304)
= Tk. 2.04

৫৫.

A vegetable cart sells a potato for $0.24 and a tomato for $0.76. Fred bought 12 vegetables in total, he only bought potatoes and tomatoes. If Fred paid $6.52 total, how many potatoes did he buy?

ক
ক) 4
খ
খ) 5
গ
গ) 8
ঘ
ঘ) 9

সঠিক উত্তর: খ

খ) 5

উত্তর

সঠিক উত্তর: খ

খ) 5

খ

ব্যাখ্যা

Let, Fred bought x potatoes and (12 - x) tomatoes

ATQ,
0.24x + 0.76(12 - x) = 6.52
Or, 0.24x + 9.12 - 0.76x = 6.52
Or, 0.52x = 2.6
Or, x = 5

৫৬.

Water flows through a cylindrical pipe of an internal diameter of 7cm at the rate of 5m/s. The time, in minutes, the pipe would take to fill an empty rectangular tank of 4m × 3m × 2.31m is -

ক
20 min
খ
18 min
গ
28 min
ঘ
24 min

সঠিক উত্তর: ঘ

24 min

উত্তর

সঠিক উত্তর: ঘ

24 min

ঘ

ব্যাখ্যা

Question: Water flows through a cylindrical pipe of an internal diameter of 7cm at the rate of 5m/s. The time, in minutes, the pipe would take to fill an empty rectangular tank of 4m × 3m × 2.31m is -

Solution:
the total volume of the tank is = 400 × 300 × 231 cc
total water flow per second through the pipe is = πr²h
= (22/7) × (7/2)² × 500

∴ total time = (400 × 300 × 231)/{(22/7) × (7/2)² × 500}
= (400 × 300 × 231 × 4 × 7)/(22 × 49 × 500)
= 1440 s
= 24 min

৫৭.

P does one-third as much work as Q in one-fourth of the time. If together they take 24 days to complete a work, how much time shall Q alone take to do it?

ক
42 days
খ
48 days
গ
56 days
ঘ
62 days

সঠিক উত্তর: গ

56 days

উত্তর

সঠিক উত্তর: গ

56 days

গ

ব্যাখ্যা

Question: P does one-third as much work as Q in one-fourth of the time. If together they take 24 days to complete a work, how much time shall Q alone take to do it?

Solution:
Let Q takes x days to do the work.
P takes 1/4 of x time to do 1/3 of the work.
∴ the work will be done by P in (x/4) × 3 days
= 3x/4

ATQ,
(1/x) + (4/3x) = 1/24
⇒ 7/3x = 1/24
⇒ x = 56
∴ Q alone will take 56 days

৫৮.

0.014 × 0.014 = ?

ক
0.000196
খ
0.00196
গ
19.6
ঘ
196

সঠিক উত্তর: ক

0.000196

উত্তর

সঠিক উত্তর: ক

0.000196

ক

ব্যাখ্যা

Question: 0.014 × 0.014 = ?

Solution:
0.014 × 0.014 = 0.000196

৫৯.

A vendor buys mangoes at a score for Tk. 300 and retails them at a dozen for Tk. 240. Did he gain or lose in the transaction, and what percentage was his gain or loss?

ক
27.67% loss
খ
27.67% gain
গ
33.33% loss
ঘ
33.33% gain

সঠিক উত্তর: ঘ

33.33% gain

উত্তর

সঠিক উত্তর: ঘ

33.33% gain

ঘ

ব্যাখ্যা

Question: A vendor buys mangoes at a score for Tk. 300 and retails them at a dozen for Tk. 240. Did he gain or lose in the transaction, and what percentage was his gain or loss?

Solution:
Cost Price (C.P) of 1 mango = Tk. 300/score
= 300/20 = 15 (Note: 1 score = 20)

Selling Price (S.P) of 1 mango = Tk. 240/dozen
= 240/12
= 20

∴ Profit on 1 mango = 20 - 15 = 5

∴ % Profit = (5/15) × 100 % = 33.33%

৬০.

If {1/|2q - 7|} > 1/5, then what is the value of q?

ক
- 1 < q < 6
খ
- 1 < q < 3
গ
1 < q < - 4
ঘ
1 < q < 6

সঠিক উত্তর: ঘ

1 < q < 6

উত্তর

সঠিক উত্তর: ঘ

1 < q < 6

ঘ

ব্যাখ্যা

Question: If {1/|2q - 7|} > 1/5, then what is the value of q?

Solution:
Given that,
{1/|2q - 7|} > 1/5
⇒ |2q - 7| < 5
⇒ - 5 < 2q - 7 < 5
⇒ - 5 + 7 < 2q - 7 + 7 < 5 + 7
⇒ 2 < 2q < 12
∴ 1 < q < 6

৬১.

A man invested Tk. 45000 when he bought Tk. 100 shares at Tk. 150. If 20% dividend is declared, find his annual income.

ক
Tk. 4500
খ
Tk. 3000
গ
Tk. 5000
ঘ
Tk. 6000

সঠিক উত্তর: ঘ

Tk. 6000

উত্তর

সঠিক উত্তর: ঘ

Tk. 6000

ঘ

ব্যাখ্যা

Question: A man invested Tk. 45000 when he bought Tk. 100 shares at Tk. 150. If 20% dividend is declared, find his annual income.

Solution:
For Tk. 150 he get Tk. 20
∴ For Tk. 1 He get Tk. (20/150)
∴ For Tk. 45000 He get Tk. (20 × 45000)/150
= Tk. 6000

৬২.

How many revolutions per minute does a 140 cm diameter scooter wheel need to maintain a speed of 132 km/h?

ক
500
খ
501
গ
850
ঘ
1000
ঙ
250

সঠিক উত্তর: ক

500

উত্তর

সঠিক উত্তর: ক

500

ক

ব্যাখ্যা

Question: How many revolutions per minute does a 140 cm diameter scooter wheel need to maintain a speed of 132 km/h?

Solution:
Distance travelled by wheel in one revolution = circumference of wheel
= (22/7) × 140 = 440 cm.

And
Speed of scooter = 132 km/hr = (132 × 1000 × 100)/60 cm/min = 220000 cm/min.

∴ Revolutions per minute = Distance covered per minute/Distance per revolution
= 220000/440 = 500

So the answer is indeed 500 revolutions per minute.

৬৩.

If a³ - 8b³ = - 2 and a = - 1, then b =?

ক
- 1/2
খ
2
গ
- 2/3
ঘ
1/2

সঠিক উত্তর: ঘ

1/2

উত্তর

সঠিক উত্তর: ঘ

1/2

ঘ

ব্যাখ্যা

Question: If a³ - 8b³ = - 2 and a = - 1, then b =?

Solution:
a = - 1

a³ - 8b³ = - 2
⇒ (- 1)³ - 8b³ = - 2
⇒ - 1 - 8b³ = - 2
⇒ - 8b³ = - 2 + 1
⇒ - 8b³ = - 1
⇒ 8b³ = 1
⇒ b³ = 1/8
⇒ b³ = (1/2)³
∴ b = 1/2

৬৪.

The difference of the areas of two squares drawn on two line segments of different lengths is 45 sq. cm. Find the length of the greater line segment if one is longer than the other by 3 cm.

ক
ক) 6 cm
খ
খ) 8 cm
গ
গ) 9 cm
ঘ
ঘ) 10 cm

সঠিক উত্তর: গ

গ) 9 cm

উত্তর

সঠিক উত্তর: গ

গ) 9 cm

গ

ব্যাখ্যা

Let
the length of the smaller line segment =x cm
length of greater line segment =(x + 3) cm
Given,
(x + 3)²- x²= 45
⇒x²+ 6x + 9 - x²= 45
⇒6x = 45 - 9
⇒ x = 36/6
⇒x = 6 cm.

∴ Length of the greater line segment = 9 cm.

৬৫.

A train passes a station platform in 36 seconds and a man standing on the platform in 52/3 seconds. If the speed of the train is 54 km/hr, what is the length of the platform?

ক
ক) 300 meters
খ
খ) 280 meters
গ
গ) 250 meters
ঘ
ঘ) 240 meters

সঠিক উত্তর: খ

খ) 280 meters

উত্তর

সঠিক উত্তর: খ

খ) 280 meters

খ

ব্যাখ্যা

Question: A train passes a station platform in 36 seconds and a man standing on the platform in 52/3 seconds. If the speed of the train is 54 km/hr, what is the length of the platform?

Solution:
Speed = 54 km/hr = (54 × 1000)/3600 m/s
= 15 m/s

Length of the train = 15 × (52/3) m = 260 m.

Let,
The length of the platform be x meters
∴ (260 + x)/36 = 15
⇒ 260 + x = 540
⇒ x = 540 - 260
∴ x = 280

∴ The length of the platform 280 meters

৬৬.

ক
ক) 7
খ
খ) 9
গ
গ) 21
ঘ
ঘ) 49

সঠিক উত্তর: ঘ

ঘ) 49

উত্তর

সঠিক উত্তর: ঘ

ঘ) 49

ঘ

ব্যাখ্যা

Question:

Solution:

৬৭.

If the ratio of two numbers is 3:4 and their L.C.M is 84, then find the greater number.

ক
ক) 28
খ
খ) 34
গ
গ) 82
ঘ
ঘ) 54

সঠিক উত্তর: ক

ক) 28

উত্তর

সঠিক উত্তর: ক

ক) 28

ক

ব্যাখ্যা

Let, these two numbers be 3x and 4x and their LCM = 12x
ATQ,
12x = 84
Or, x = 7
So, the greater number is = 4 × 7 = 28

৬৮.

An observer 1.6 m tall is 20√3 away from a tower. The angle of elevation from his eye to the top of the tower is 30º. The height of the tower is:

ক
19.8 m
খ
20.4 m
গ
21.6 m
ঘ
22.6 m
ঙ
23.9 m

সঠিক উত্তর: গ

21.6 m

উত্তর

সঠিক উত্তর: গ

21.6 m

গ

ব্যাখ্যা

Question: An observer 1.6 m tall is 20√3 away from a tower. The angle of elevation from his eye to the top of the tower is 30º. The height of the tower is:

Solution:

Let AB be the observer and CD tower
Draw BE perpendicular to CD

Then CE = AB = 1.6 m
And BE = AC = 20√3 m

Then right angle triangle DEB
∴ tan30° = DE/BE
⇒ 1/√3 = DE/20√3
⇒ DE = 20√3m

Then CD = CE + DE
= 1.6 + 20
= 21.6 m

৬৯.

50 cats caught 50 rats in 50 seconds. How many cats are required to catch 500 rats in 500 seconds?

ক
50
খ
500
গ
10
ঘ
None of these

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

50 cats caught 50 rats in 50 seconds.
50 cats caught 1 rat in 50/50 second or 1 second
50 cats caught 500 rats in 1 × 500 seconds or 500 seconds

৭০.

If 3^{x +1} = 243, what is the value of 2^{2x - 7}?

ক
ক) 1
খ
খ) 2
গ
গ) 5
ঘ
ঘ) 8

সঠিক উত্তর: খ

খ) 2

উত্তর

সঠিক উত্তর: খ

খ) 2

খ

ব্যাখ্যা

Question: If 3^{x +1} = 243, what is the value of 2^{2x - 7}?

Solution:
3^{x +1} = 243
⇒ 3^{x + 1} = 3⁵
⇒ x + 1 = 5
⇒ x = 5 - 1
⇒ x = 4

2^{2x - 7} = 2^{2 × 4 - 7} = 2^{8 - 7} = 2¹ = 2

৭১.

All possible three digit numbers are formed by 1, 3, 5. If one number is chosen randomly, the probability that it would be divisible by 5 is

ক
0
খ
2/9
গ
1/3
ঘ
1/4

সঠিক উত্তর: গ

1/3

উত্তর

সঠিক উত্তর: গ

1/3

গ

ব্যাখ্যা

Question: All possible three digit numbers are formed by 1, 3, 5. If one number is chosen randomly, the probability that it would be divisible by 5 is

Solution:
1, 3, 5 এই তিনটি অংক দ্বারা 3! = 6 উপায়ে সংখ্যা গঠন করা যায়
5 দ্বারা বিভাজ্য হতে হলে শেষের অংক 5 রেখে সংখ্যার প্রথম দুটি স্থান বাকি দুইটি অংক দিয়ে 2! = 2 উপায়ে গঠন করা যায়।
∴ সংখ্যাটি 5 দ্বারা বিভাজ্য হবার সম্ভাবনা = 2/6
= 1/3

৭২.

Which man is the tallest?
When
A is smaller than B
B is smaller than C
C is smaller than D
D is taller than E

ক
D
খ
B
গ
C
ঘ
A

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: Which man is the tallest?
When
A is smaller than B
B is smaller than C
C is smaller than D
D is taller than E

Solution:
প্রশ্নানুযায়ী,
B থেকে A ছোট
C থেকে B ছোট
D থেকে C ছোট
কিন্তু, E থেকে D লম্বা।

∴ A, B, C, D ও E এর মধ্যে D সবচেয়ে লম্বা।

৭৩.

The ratio between the speeds of two trains is 5 : 8. If the second train runs 600 km in 5 hours, then the speed of the first train is -

ক
100 kmph
খ
95 kmph
গ
80 kmph
ঘ
75 kmph

সঠিক উত্তর: ঘ

75 kmph

উত্তর

সঠিক উত্তর: ঘ

75 kmph

ঘ

ব্যাখ্যা

Question: The ratio between the speeds of two trains is 5 : 8. If the second train runs 600 km in 5 hours, then the speed of the first train is -

Solution:
Let the speed of the trains are 5x and 8x

the speed of the second train = 600/5 kmph = 120 kmph

∴ 8x = 120
x = 15

∴ speed of first train = 5x = 75 kmph

৭৪.

If a² - 8 = 2√15, Than what is the value of a.

ক
√3 + √2
খ
√15
গ
√3 - √2
ঘ
√5 + √3

সঠিক উত্তর: ঘ

√5 + √3

উত্তর

সঠিক উত্তর: ঘ

√5 + √3

ঘ

ব্যাখ্যা

Question: If a² - 8 = 2√15, Than what is the value of a.

Solution:
Given that,
a² - 8 = 2√15
⇒ a² = 8 + 2√15
⇒ a² = 5 + 2 × √5 × √3 + 3
⇒ a² = (√5)² + + 2 × √5 × √3 + (√3)²
⇒ a² = (√5 + √3)² ; [(a + b)² = a² + 2ab + b²]
∴ a = √5 + √3

৭৫.

A father said to his son, ''I was as old as you are at the present at the time of your birth''. If the father's age is 38 years now, the son's age five years back was:

ক
ক) 14 years
খ
খ) 19 years
গ
গ) 33 years
ঘ
ঘ) 38 years
ঙ
ঙ) 40 years

সঠিক উত্তর: ক

ক) 14 years

উত্তর

সঠিক উত্তর: ক

ক) 14 years

ক

ব্যাখ্যা

Let the son's present age be x years.
Then, (38 - x) = x
2x = 38.
x = 19.
Son's age 5 years back = (19 - 5)
= 14 years.

৭৬.

A certain elevator has a safe weight limit of 2,000 pounds. What is the greatest possible number of people who can safely ride on the elevator at one time with the average (arithmetic mean) weight of half the riders being 180 pounds and the average weight of the others being 215 pounds?

ক
10
খ
9
গ
8
ঘ
7

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: A certain elevator has a safe weight limit of 2,000 pounds. What is the greatest possible number of people who can safely ride on the elevator at one time with the average (arithmetic mean) weight of half the riders being 180 pounds and the average weight of the others being 215 pounds?

Solution:
Lets assume there are 2X people.
Half of them have average weight of 180 and other half has 215.
Maximum Weight is = 2000
∴ 180 × X + 215 × X = 2000
⇒ 395X = 2000
⇒ X is approximately equal to 5.
So total people is 2 × 5 = 10

We are not taking 11 as answer because say 11th person has minimum of 180 weight then
180 × 6 + 215 × 5 = 2155 (Which is more than 2000)

∴ Correct answer 10.

৭৭.

The range of f(x) = 1/(x + 1) is:

ক
R\{0}
খ
x > -1
গ
x < -1
ঘ
R\{- 1}

সঠিক উত্তর: ক

R\{0}

উত্তর

সঠিক উত্তর: ক

R\{0}

ক

ব্যাখ্যা

Question: The range of f(x) = 1/(x + 1) is:

Solution:
দেওয়া আছে,
f(x) = 1/(x + 1)
⇒ y = 1/(x + 1)
⇒1/y = x + 1
⇒ x = (1/y) - 1
⇒ x = (1 - y)/y

∴ f^-1(x) = y = (1 - x)/x
x এর মান 0 ব্যতীত যেকোনো বাস্তব সংখ্যা হবে। কারণ x এর মান 0 হলে ফাংশনটি অসঙ্গায়িত হবে।

অতএব, নির্ণেয় রেঞ্জ: R\{0}

৭৮.

The factors of the polynomial a³ - 6a² + 12a - 9

ক
(a - 1) (a² - 3a + 1)
খ
(a + 3) (a² + 3a + 3)
গ
(a - 3) (a² - 3a + 2)
ঘ
(a - 3) (a² - 3a + 3)

সঠিক উত্তর: ঘ

(a - 3) (a² - 3a + 3)

উত্তর

সঠিক উত্তর: ঘ

(a - 3) (a² - 3a + 3)

ঘ

ব্যাখ্যা

Question: The factors of the polynomial a³ - 6a² + 12a - 9 are:

Solution:
a³ - 6a² + 12a - 9
= a³ - 3 . a² . 2 + 3 . a . 2² - 2³ - 1
= (a - 2)³ - 1³
= (a - 2 - 1) {(a - 2)² + (a - 2) . 1 + 1²}
= (a - 3) (a² - 4a + 4 + a - 2 + 1)
= (a - 3) (a² - 3a + 3)

৭৯.

If Titu is three years older than double the age of Puja and Pavel is 7 years senior to Puja. Tithi is Pavel's wife and 3 years junior to Pavel. What is the age of Tithi if Titu is 57 years old?

ক
31
খ
32
গ
33
ঘ
29

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: If Titu is three years older than double the age of Puja and Pavel is 7 years senior to Puja. Tithi is Pavel's wife and 3 years junior to Pavel. What is the age of Tithi if Titu is 57 years old?

Solution:
As per the question,
Let
Puja's age be x
Titu's age = 2x + 3
Now
2x + 3 = 57
2x = 57 - 3
2x = 54
x = 27

Pavel = x + 7 = 27 + 7 = 34

Tithi is = 34 - 3 = 31

৮০.

Some articles were bought at 6 articles for TK. 5 and sold at 5 articles for Tk. 6. Gain percent is:

ক
ক) 20%
খ
খ) 30%
গ
গ) 33.33%
ঘ
ঘ) 35%
ঙ
ঙ) 44%

সঠিক উত্তর: ঙ

ঙ) 44%

উত্তর

সঠিক উত্তর: ঙ

ঙ) 44%

ঙ

ব্যাখ্যা

Suppose, number of articles bought = L.C.M. of 6 and 5 = 30.
C.P. of 30 articles = TK.(5/6)x 30 = Tk. 25.
S.P. of 30 articles = Tk. (6/5)x 30 = Tk. 36.
Gain % = (11/25)x 100 = 44%.

৮১.

If 2 kg of metal, of which 1/3 is zinc and the rest is copper, be mixed with 3 kg of metal, of which 1/4 is zinc and the rest is copper, then what will be the ratio of zinc to copper in the mixture?

ক
ক) 17 : 43
খ
খ) 13 : 42
গ
গ) 19 : 43
ঘ
ঘ) 15 : 42

সঠিক উত্তর: ক

ক) 17 : 43

উত্তর

সঠিক উত্তর: ক

ক) 17 : 43

ক

ব্যাখ্যা

Quantity of zinc in the mixture = 2(1/3) + 3(1/4) = 17/12
Quantity of copper in the metal = (3 + 2) - 17/12 = (60 - 17)/12 = 43/12
So, the ratio of Zinc and Copper = 17/12 : 43/12 = 17 : 43

৮২.

Out of 10 persons, 9 persons spent Tk. 20 each for their meals. The tenth one spent Tk. 9 more than the average expenditure of all the ten. The total money spent by all of them was-

ক
ক) 180 Tk.
খ
খ) 200 Tk.
গ
গ) 210 Tk.
ঘ
ঘ) 300 Tk.

সঠিক উত্তর: গ

গ) 210 Tk.

উত্তর

সঠিক উত্তর: গ

গ) 210 Tk.

গ

ব্যাখ্যা

Question: Out of 10 persons, 9 persons spent Tk. 20 each for their meals. The tenth one spent Tk. 9 more than the average expenditure of all the ten. The total money spent by all of them was-

Solution:
৯ জনের প্রতি জন খরচ করে ২০ টাকা
৯ জন মোট খরচ করে = (৯ × ২০) টাকা
= ১৮০ টাকা

ধরি, দশম জন খরচ করে x টাকা
দশজন মোট খরচ করে ১৮০ + x টাকা
গড় = (১৮০ + x)/১০ টাকা

প্রশ্নমতে,
{(১৮০ + x)/১০} + ৯ = x
⇒ {(১৮০ + x)/১০} = x - ৯
⇒ ১৮০ + x = ১০x - ৯০
⇒ ১০x - x = ১৮০ + ৯০
⇒ ১০x - x = ২৭০
⇒ ৯x = ২৭০
∴ x = ৩০

দশজন মোট খরচ করে = ১৮০ + ৩০ টাকা
= ২১০ টাকা

৮৩.

The average of ten number is 7. If each number is multiplied by 10, then the average of the new set of number is-

ক
ক) 42
খ
খ) 63
গ
গ) 70
ঘ
ঘ) 84

সঠিক উত্তর: গ

গ) 70

উত্তর

সঠিক উত্তর: গ

গ) 70

গ

ব্যাখ্যা

Question: The average of ten number is 7. If each number is multiplied by 10, then the average of the new set of number is-

Solution:
let, 7 numbers are a₁, a₂, a₃,.......,a₇
so, (a₁ + a₂ + a₃+.......+a₇)/10 = 7
a₁ + a₂ + a₃+.......+a₇ = 70

If each number is multiplied by 10, Then sum = (a₁ × 10) + (a₂× 10) + (a₃ × 10) + .... + (a₇ × 10)
= 10 (a₁ + a₂ + a₃+.......+a₇)
= 10 × 70
= 700

then average will be = 700/10
= 70

৮৪.

In a boat, there are 8 men whose average weight is increased by 1 kg when 1 man of 60 kg is replaced by a new man. What is the weight of the newcomer?

ক
66 kg
খ
68 kg
গ
70.5 kg
ঘ
71 kg

সঠিক উত্তর: খ

68 kg

উত্তর

সঠিক উত্তর: খ

68 kg

খ

ব্যাখ্যা

Question: In a boat, there are 8 men whose average weight is increased by 1 kg when 1 man of 60 kg is replaced by a new man. What is the weight of the newcomer?

সমাধান:
মোট লোকের সংখ্যা = 8 জন
গড় ওজন 1 কেজি বৃদ্ধি পেয়েছে।
∴ মোট ওজন বৃদ্ধি = 8 × 1 = 8 কেজি

যেহেতু নতুন ব্যক্তি আসার ফলে গড় ওজন বৃদ্ধি পেয়েছে, তাই নতুন ব্যক্তির ওজন অবশ্যই পূর্বের ব্যক্তির ওজনের চেয়ে বেশি হবে।

দেওয়া আছে, পূর্বের ব্যক্তির ওজন = 60 কেজি
∴ নতুন ব্যক্তির ওজন = 60 + 8 = 68 কেজি

৮৫.

A thief is noticed by a policeman from a distance of 200 m. The thief starts running and the policeman chases him. The thief and the policeman run at the rate of 10 km and 11 km per hour respectively. What is the distance between them after 6 minutes?

ক
100 m
খ
150 m
গ
190 m
ঘ
200 m
ঙ
None of these

সঠিক উত্তর: ক

100 m

উত্তর

সঠিক উত্তর: ক

100 m

ক

ব্যাখ্যা

Question: A thief is noticed by a policeman from a distance of 200 m. The thief starts running and the policeman chases him. The thief and the policeman run at the rate of 10 km and 11 km per hour respectively. What is the distance between them after 6 minutes?

Solution:
Relative speed of the thief and policeman = (11 - 10) km/hr = 1 km/hr
Distance covered in 6 minutes = (1/60) × 6 km = 1/10 km = 100 m
Therefore, Distance between the thief and policeman = (200 - 100) m = 100 m.

৮৬.

24 workers can complete a construction job in 15 days, working 6 hours a day. How many additional workers are needed to complete the same job in 10 days, working 8 hours a day?

ক
6
খ
3
গ
12
ঘ
15

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: 24 workers can complete a construction job in 15 days, working 6 hours a day. How many additional workers are needed to complete the same job in 10 days, working 8 hours a day?

Solution:
দৈনিক 6 ঘণ্টা করে কাজ করে 15 দিনে শেষ করতে লোক লাগে 24 জন
∴ দৈনিক 1 ঘণ্টা করে কাজ করে 1 দিনে শেষ করতে লোক লাগে (24 × 15 × 6) জন
∴ দৈনিক 8 ঘণ্টা করে কাজ করে 10 দিনে শেষ করতে লোক লাগে (24 × 15 × 6)/(10 × 8) জন
= 27 জন

∴ অতিরিক্ত লোক লাগবে = (27 - 24) = 3 জন

৮৭.

What is the greatest number of four digits which is divisible by 15, 25, 40 and 75?

ক
ক) 9200
খ
খ) 9400
গ
গ) 9600
ঘ
ঘ) 9800

সঠিক উত্তর: গ

গ) 9600

উত্তর

সঠিক উত্তর: গ

গ) 9600

গ

ব্যাখ্যা

Greatest number of four digits = 9999
LCM of 15, 25, 40 and 75 = 600
9999/600 = 16,
remainder = 399
Hence, greatest number of four digits which is divisible by 15, 25, 40 and 75
= 9999 - 399
= 9600

৮৮.

A does one-third as much work as B in one-fourth of the time. If together they take 12 days to complete a work, how much time shall B alone take to do it?

ক
16 days
খ
28 days
গ
24 days
ঘ
20 days

সঠিক উত্তর: খ

28 days

উত্তর

সঠিক উত্তর: খ

28 days

খ

ব্যাখ্যা

Question: A does one-third as much work as B in one-fourth of the time. If together they take 12 days to complete a work, how much time shall B alone take to do it?

Solution:
Let B takes x days to do the work.
A takes 1/4 of x time to do 1/3 of the work.
∴ the work will be done by A in (x/4) × 3 days
= 3x/4
ATQ,
1/x + 4/3x = 1/12
⇒ 7/3x = 1/12
⇒ x = 28
∴ B alone will take 28 days.

৮৯.

If - 3 is 6 more than x, what is the value of x/3?

ক
- 9
খ
- 6
গ
- 3
ঘ
1

সঠিক উত্তর: গ

- 3

উত্তর

সঠিক উত্তর: গ

- 3

গ

ব্যাখ্যা

Question: If - 3 is 6 more than x, what is the value of x/3?

Solution:
x + 6 = - 3
⇒ x = - 3 - 6
⇒ x = - 9
⇒ x/3 = - 9/3
= - 3

৯০.

If the points A(1, 5), B(k, 1), and C(4, 11) are collinear, then find the value of k.

ক
2
খ
- 1
গ
3
ঘ
- 2

সঠিক উত্তর: খ

- 1

উত্তর

সঠিক উত্তর: খ

- 1

খ

ব্যাখ্যা

Question: If the points A(1, 5), B(k, 1), and C(4, 11) are collinear, then find the value of k.

Solution:
আমরা জানি,
তিনটি বিন্দু A(1, 5), B(k, 1) এবং C(4, 11) সরলরেখায় অবস্থিত হলে, তাদের ঢাল (slope) সমান হবে।

আমরা জানি,
ঢাল = (y₂ - y₁)/(x₂ - x₁)

এখন,
A এবং B-এর মধ্যে ঢাল, m_AB = (1 - 5)/(k - 1)
= - 4/(k - 1)

B এবং C-এর মধ্যে ঢাল, m_BC = (11 - 1)/(4 - k)
= 10/(4 - k)

শর্তমতে,
m_AB= m_BC
⇒ - 4/(k - 1) = 10/(4 - k)
⇒ 10(k - 1) = - 4(4 - k)
⇒ 10k - 10 = - 16 + 4k
⇒ 10k - 4k = - 16 + 10
⇒ 6k = - 6
∴ k = - 1

৯১.

The average of x₁, x₂, x₃ and x₄ is 16. Half of the sum x₂, x₃ and x₄ is 23. What is the value of x₁?

ক
ক) 18
খ
খ) 20
গ
গ) 22
ঘ
ঘ) 24

সঠিক উত্তর: ক

ক) 18

উত্তর

সঠিক উত্তর: ক

ক) 18

ক

ব্যাখ্যা

x₁ + x₂ + x₃ + x₄ = 16 × 4 = 64
⇒ 1/2(x₂ + x₃ + x₄) = 23
⇒ x₂ + x₃ + x₄ = 46

∴ x₁ = 64 - 46
= 18

৯২.

A select group of 4 is to be formed from 8 men and 6 women in such a way that the group must have at least 1 women. In how many different ways can it be done?

ক
931
খ
360
গ
1050
ঘ
720

সঠিক উত্তর: ক

931

উত্তর

সঠিক উত্তর: ক

931

ক

ব্যাখ্যা

Question: A select group of 4 is to be formed from 8 men and 6 women in such a way that the group must have at least 1 women. In how many different ways can it be done?

Solution:
The required number of ways = ⁸C₃ × ⁶C₁ +⁸C₂ × ⁶C₂ + ⁸C₁ × ⁶C₃ + ⁶C₄
= 56 × 6 + 28 × 15 + 8 × 20 + 15
= 336 + 420 + 160 + 15
= 931

৯৩.

Everyone present at a party shakes hands with each other. If the total number of handshakes is 66, how many people were present at the party?

ক
10
খ
11
গ
12
ঘ
13

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: Everyone present at a party shakes hands with each other. If the total number of handshakes is 66, how many people were present at the party?

Solution:
ধরি, পার্টিতে মোট x জন লোক উপস্থিত ছিল।

যেহেতু করমর্দন করতে দুইজন লোকের প্রয়োজন হয়, তাই মোট করমর্দন সংখ্যা = ^xC₂
= x!/{2!(x - 2)!}
= {x(x - 1)(x - 2)!}/{2!(x - 2)!}
= x(x - 1)/2
= (x² - x)/2

প্রশ্নমতে,
(x² - x)/2 = 66
⇒ x² - x - 132 = 0
⇒ (x - 12)(x + 11) = 0
∴ x = 12 or x = - 11

কিন্তু, x = - 11 হতে পারে না।
∴ x = 12

৯৪.

A water tank can be filled in 15 minutes by two pipes A and B working together. Both pipes are opened simultaneously, but after 9 minutes, pipe A is closed. If pipe B takes an additional 10 minutes to fill the remaining part of the tank, how many minutes would pipe B alone take to fill the entire tank?

ক
15 minutes
খ
18 minutes
গ
25 minutes
ঘ
30 minutes

সঠিক উত্তর: গ

25 minutes

উত্তর

সঠিক উত্তর: গ

25 minutes

গ

ব্যাখ্যা

Question: A water tank can be filled in 15 minutes by two pipes A and B working together. Both pipes are opened simultaneously, but after 9 minutes, pipe A is closed. If pipe B takes an additional 10 minutes to fill the remaining part of the tank, how many minutes would pipe B alone take to fill the entire tank?

Solution:
পাইপ A ও B একত্রে 15 মিনিটে পূর্ণ করে 1 অংশ
∴ পাইপ A ও B একত্রে 1 মিনিটে পূর্ণ করে (1/15) অংশ
∴ পাইপ A ও B একত্রে 9 মিনিটে পূর্ণ করে (9/15) অংশ = 3/5 অংশ

ট্যাংকটির অবশিষ্ট অংশ = {1 - (3/5)} অংশ = 2/5 অংশ

প্রশ্নমতে, পাইপ A বন্ধ করার পর পাইপ B অবশিষ্ট 2/5 অংশ পূর্ণ করে 10 মিনিটে।
∴ পাইপ B দ্বারা 2/5 অংশ পূর্ণ হয় 10 মিনিটে
∴ পাইপ B দ্বারা 1 বা সম্পূর্ণ অংশ পূর্ণ হয় (10 × 5) / 2 মিনিটে
= 50 / 2 মিনিট
= 25 মিনিটে

সুতরাং, পাইপ B একাকী 25 মিনিটে সম্পূর্ণ ট্যাংকটি পূর্ণ করতে পারবে।

৯৫.

Solve |2x - 3| ≤ 1

ক
ক) - 1 ≤ x ≤ - 2
খ
খ) - 1 < x < 12
গ
গ) 1 ≤ x ≤ 2
ঘ
ঘ) 1 ≤ x ≤ - 2

সঠিক উত্তর: গ

গ) 1 ≤ x ≤ 2

উত্তর

সঠিক উত্তর: গ

গ) 1 ≤ x ≤ 2

গ

ব্যাখ্যা

Question: Solve |2x - 3| ≤ 1

Solution:
|2x - 3| ≤ 1
বা, - 1 ≤ 2x - 3 ≤ 1
বা, - 1 + 3 ≤ 2x - 3 + 3 ≤ 1 + 3
বা, 2 ≤ 2x ≤ 4
বা, 2/2 ≤ 2x/2 ≤ 4/2
∴ 1 ≤ x ≤ 2

৯৬.

In the adjoining figure ABCD is a rhombus. If ∠A = 70° then ∠CDB =?

ক
65°
খ
55°
গ
35°
ঘ
45°
ঙ
25°

সঠিক উত্তর: খ

55°

উত্তর

সঠিক উত্তর: খ

55°

খ

ব্যাখ্যা

Question: In the adjoining figure ABCD is a rhombus. If ∠A = 70° then ∠CDB =?

Solution:
Let ∠CDB= x°.
then , CD = CB ⇒ ∠CBD = ∠CDB = x°.
∠BCD = ∠BAD = 70° (opp. s of a rhombus)
∴ x + x + 70 = 180° (sum of the angles of a ∆ is 180°)
⇒ 2x = 110°
⇒ x = 55°
∴ ∠CDB = 55°.

৯৭.

An outgoing pipe pours water at half the amount of an ingoing pipe. After 4 hours of running both pipes, a tank was filled. If the outgoing pipe was closed, how much time would it take to fill the tank with the ingoing pipe?

ক
4 hours
খ
8 hours
গ
1 hours
ঘ
2 hours

সঠিক উত্তর: ঘ

2 hours

উত্তর

সঠিক উত্তর: ঘ

2 hours

ঘ

ব্যাখ্যা

Question: An outgoing pipe pours water at half the amount of an ingoing pipe. After 4 hours of running both pipes, a tank was filled. If the outgoing pipe was closed, how much time would it take to fill the tank with the ingoing pipe?

Solution:
Let,
ingoing pipe needs X hours,
The outgoing pipe needs 2X hours.

together in one hour, these pipes can fill = 1/X - 1/2X = 1/2X

ATQ,
2X = 4
X = 2
∴ Ingoing pipe will take 2 hours to fill the tank.

৯৮.

The sum of the L.C.M. and H.C.F. of two numbers is 1260, and the L.C.M. is 900 more than the H.C.F. What is the product of these two numbers?

ক
194800
খ
194000
গ
149400
ঘ
194400

সঠিক উত্তর: ঘ

194400

উত্তর

সঠিক উত্তর: ঘ

194400

ঘ

ব্যাখ্যা

Question: The sum of the L.C.M. and H.C.F. of two numbers is 1260, and the L.C.M. is 900 more than the H.C.F. What is the product of these two numbers?

Solution:
Let the HCF be x

LCM = HCF + 900
LCM = x + 900 ...............(1)

And, LCM + HCF = 1260
LCM + x = 1260 .................(2)

From (1) and (2) equation,
(x + 900) + x = 1260
⇒ 2x + 900 = 1260
⇒ 2x = 1260 - 900
⇒ 2x = 360
⇒ x = 360/2
∴ x = 180

∴ HCF = 180
And, from equation (1),
LCM = HCF + 900
LCM = 180 + 900
∴ LCM = 1080

By formula, the product of the numbers is equal to the product of their HCF and LCM.

Product of numbers = HCF × LCM = 180 × 1080
∴ Product = 194400

৯৯.

An amount of money is to be distributed among P, Q and R in the ratio of 2 : 7 : 9. The total of P’s and Q’s share is equal to R’s share. What is the difference between the shares of P and Q?

ক
5500
খ
8000
গ
9000
ঘ
Information inadequate

সঠিক উত্তর: ঘ

Information inadequate

উত্তর

সঠিক উত্তর: ঘ

Information inadequate

ঘ

ব্যাখ্যা

Questions: An amount of money is to be distributed among P, Q and R in the ratio of 2 : 7 : 9. The total of P’s and Q’s share is equal to R’s share. What is the difference between the shares of P and Q?

Solution:
Let the amount to be distributed be Tk x.
P : Q : R = 2 : 7 : 9
Sum of the ratios = 2 + 7 + 9 = 18

P = (2/18) × x = x/9
Q = 7x/18
R = 9x/18 = x/2

As given, (x/9) + (7x/18) = (x/2)
⇒ (2x + 3x)/18 = (x/2)
⇒ x/2 = x/2
Thus, we get no conclusion. Amount should necessarily be known.

১০০.

A ladder rests against a wall that is perpendicular to the ground. If the bottom of the ladder is 4 meter away from the bottom of the wall, while the top of the ladder is at a height of 3 meter. What is the length of the ladder?

ক
ক) 5 m
খ
খ) 6 m
গ
গ) 9 m
ঘ
ঘ) 12 m

সঠিক উত্তর: ক

ক) 5 m

উত্তর

সঠিক উত্তর: ক

ক) 5 m

ক

ব্যাখ্যা

Question: A ladder rests against a wall that is perpendicular to the ground. If the bottom of the ladder is 4 meter away from the bottom of the wall, while the top of the ladder is at a height of 3 meter. What is the length of the ladder?

Solution:

মইয়ের দৈর্ঘ্য, AC = x m
ভূমি হতে মইয়ের শীর্ষবিন্দুর উচ্চতা, AB = 3 m
মই হতে দেয়ালের দূরত্ব, BC = 4 m

পীথাগোরাসের উপপাদ্য হতে পাই,
AC² = AB² + BC²
⇒ x² = 3² + 4²
⇒ x² = 25
⇒ x = √25
∴ x = 5