Bank Math

PrepBank · পাতা ৪৯ / ১৬১ · ৪,৮০১–৪,৯০০ / ১৬,১২৪

৪,৮০১.

The average of 4 consecutive numbers is 5.5 . The largest of these numbers is:

ক
5
খ
6
গ
7
ঘ
9

ব্যাখ্যা

প্রশ্ন: The average of 4 consecutive numbers is 5.5 . The largest of these numbers is:

সমাধান:
৪ টি সংখ্যার গড় ৫.৫
৪ টি সংখ্যার সমষ্টি = ৫.৫ × ৪ = ২২

ধরি, সংখ্যাগুলি হল a, a + ১, a + ২, a + ৩
প্রশ্নমতে,
a + a + ১ + a + ২ + a + ৩ = ২২
⇒ ৪a + ৬ = ২২
⇒ ৪a = ১৬
⇒ a = ৪

∴ বড় সংখ্যাটি হল a + ৩
= ৪ + ৩
= ৭

৪,৮০২.

If 13:11 is the ratio of the present age of Jolly and Lopa respectively and 15:9 is the ratio between Jolly’s age 4 years hence and Lopa's age 4 years ago. Then what will be the ratio of Jolly’s age 4 years ago and Lopa's age 4 years hence?

ক
15:9
খ
9:15
গ
11:13
ঘ
13:11

ব্যাখ্যা

Let the present age of Jolly and lopa be 13X and 11X respectively.
Given, Jlly's age 4 years hence and lopa's age 4 years ago in the ratio 15:9.
That is,
(13X + 4)/(11X - 4) = 15/9
⇒ 9(13X + 4) = 15(11X - 4)
⇒ 117X + 36 = 165X - 60
⇒ 48X = 96
⇒ X = 2.
Now, required ratio is (13X-4)/(11X + 4)
= 13(2) - 4/11(2) + 4
= 22/26
= 11/13.
Hence the answer is 11:13.

৪,৮০৩.

In a class, 30 students study Mathematics, 20 students study Physics, and 8 students study both. 12 students study neither Mathematics nor Physics. What is the total number of students in the class?

ক
45
খ
50
গ
54
ঘ
60

ব্যাখ্যা

Question: In a class, 30 students study Mathematics, 20 students study Physics, and 8 students study both. 12 students study neither Mathematics nor Physics. What is the total number of students in the class?

Solution:
Number of students who study Mathematics, n(M) = 30
Number of students who study Physics, n(P) = 20
Number of students who study both Mathematics and Physics, n(M ∩ P) = 8
Number of students who study neither = 12

n(M ∪ P) = n(M) + n(P) - n(M ∩ P)
= 30 + 20 - 8 = 42

Total students in the class = students who study Mathematics or Physics + students who study neither
= 42 + 12 = 54

∴ There are 54 students in the class.

৪,৮০৪.

A cylinder and a cone have the same base radius and height. What is the ratio of their volumes?

ক
1 : 2
খ
2 : 3
গ
3 : 1
ঘ
3 : 2

ব্যাখ্যা

Question: A cylinder and a cone have the same base radius and height. What is the ratio of their volumes?

Solution:
The radius of the base of both the cylinder and the cone be r.
The height of both the cylinder and the cone be h.

The volume of a cylinder is = πr²h
The volume of a cone is = (1/3)πr²h

∴ Ratio = (πr²h)/{(1/3)πr²h} = 1/(1/3) = 3
The ratio of the volumes of the cylinder to the cone is 3 : 1

৪,৮০৫.

In a race of 1km, A can beat B by 100m. In a 400m, B beats C by 40m. In a race of 500m. A will beat C by-

ক
45 m
খ
75 m
গ
50 m
ঘ
95 m

ব্যাখ্যা

Question: In a race of 1km, A can beat B by 100m. In a 400m, B beats C by 40m. In a race of 500m. A will beat C by-

Solution:
We know,
1km = 1000m

∴ While A covers 1000 B covers 900
∴ while A covers 500 B covers 450m

∴ While B covers 400, C covers 360m
∴ While B covers 450, C covers (360 × 450)/400 = 405m

∴ in a 500m race A will beat C by = (500 - 405) = 95m

That means when A runs 500 meter then B can run 450m then C runs 405m.

৪,৮০৬.

If secA + tanA = 5/2, then what is the value of secA - tanA?

ক
√3/2
খ
1
গ
1/√2
ঘ
2/5

ব্যাখ্যা

Question: If secA + tanA = 5/2, then what is the value of secA - tanA?

Solution:
দেয়া আছে,
secA + tanA = 5/2

আমরা জানি,
sec²A - tan²A = 1
⇒ (secA + tanA) (secA - tanA ) =1
⇒ 5/2(secA - tanA) = 1
∴ (secA - tanA) = 2/5

৪,৮০৭.

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

ক
8
খ
15
গ
27
ঘ
35
ঙ
None of these

ব্যাখ্যা

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

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

Thus, students failed in both subjects = 100 - 92 = 8%.

৪,৮০৮.

Which of the following numbers is a prime number?

ক
167
খ
213
গ
350
ঘ
437

ব্যাখ্যা

Question: Which of the following numbers is a prime number?

Solution:
Step 1: Find a whole number 'X' for each number such that X² > the number;
13² > 167
15² > 213
19² > 350
21² > 437

Step 2: Get all the prime numbers which are less than 'X'.
Prime numbers less than 13 are 2, 3, 5, 7, and 11
Prime numbers less than 15 are 2, 3, 5, 7, 11, and 13
Prime numbers less than 19 are 2, 3, 5, 7, 11, 13, and 17
Prime numbers less than 21 are 2, 3, 5, 7, 11, 13, 17, and 19

Step 3: Check divisibility of each number with prime numbers which are less than 'X'.
167 is not divisible by any prime number
213 is divisible by 3
352 is divisible by 2 and 11
437 is divisible by 19

∴ 167 is the required prime number as it is not divisible by any prime number.

৪,৮০৯.

A train 125m long passes a man, running at 5 km/hr in the same direction in which the train is going, in 10 seconds. The speed of the train is-

ক
ক) 45 km/hr
খ
খ) 50 km/hr
গ
গ) 54 km/hr
ঘ
ঘ) 55 km/hr

ব্যাখ্যা

Speed of the train relative to man
= (125/10) m/sec
= (25/2) m/sec
= (25/2)×(18/5) km/hr
= 45 km/hr
Let the speed of the train be x km/hr.
Then, relative speed = (x−5) km/hr
∴ x−5 = 45
⇒ x = 50km/hr

৪,৮১০.

Tk. 6200 amounts to Tk. 9176 in 4 years at simple interest. If the interest rate is increased by 2% it would amount to how much?

ক
Tk. 9672
খ
Tk. 3362
গ
Tk. 3862
ঘ
Tk. 3472

ব্যাখ্যা

Question: Tk. 6200 amounts to Tk. 9176 in 4 years at simple interest. If the interest rate is increased by 2% it would amount to how much?

Solution:
Principle = Tk. 6200
Amount = Tk. 9176
Time = 4 years
Rate = ?
Interest = Tk. 9176 - Tk. 6200 = Tk. 2976

We know,
I = Pnr
Or, r = I/Pn
Or, r = (2976 x 100)/(6200 x 4)
= 297600/24800
= 12%

Now, if the rate is increased by 2% then final rate = 14%

Then, Interest = (6200 × 14 × 4)/100 = Tk. 3472

New amount =Tk.(6200 + 3472) = TK.9672

৪,৮১১.

A, B, C subscribe Tk. 50,000 for a business. A subscribes Tk. 4000 more than B and B Tk. 5000 more than C. Out of a total profit of Tk. 35,000, A receives-

ক
14500 taka
খ
14700 taka
গ
14800 taka
ঘ
15000 taka

ব্যাখ্যা

Question: A, B, C subscribe Tk. 50,000 for a business. A subscribes Tk. 4000 more than B and B Tk. 5000 more than C. Out of a total profit of Tk. 35,000, A receives-

Solution:
let, C subscribes x taka
B subscribes x + 5000 taka
A subscribes x + 5000 + 4000
= x + 9000 taka

x + x + 5000 + x + 9000 = 50000
⇒ 3x + 14000 = 50000
⇒ 3x = 36000
⇒ x = 12000 taka

A receives = {(x + 9000)/50000} × 35000
= (21000/50000) × 35000
= 14700 taka

৪,৮১২.

What must be added to x/y to make 2y/x?

ক
2x²/y
খ
(xy - y²)/2
গ
(2y² - x²)/xy
ঘ
x/y

ব্যাখ্যা

Question: What must be added to x/y to make 2y/x?

Solution:

৪,৮১৩.

ক
ক) 0
খ
খ) 1
গ
গ) 2
ঘ
ঘ) 3

ব্যাখ্যা

Question:

Solution:

৪,৮১৪.

At 3 : 40, the hour hand and the minute hand of a clock form an angle of-

ক
120°
খ
125°
গ
130°
ঘ
135°

ব্যাখ্যা

Question: At 3 : 40, the hour hand and the minute hand of a clock form an angle of-

Solution:
কোণ =│(১১×মিনিট - ৬০×ঘন্টা)/২│°
=│(১১ × ৪০ - ৬০ × ৩)/২│°
=│(৪৪০ - ১৮০)/২│°
=│২৬০/২│°
= ১৩০°

৪,৮১৫.

The price of sugar increases by 25%. How much sugar can now be bought with the same amount of money that was earlier sufficient to buy 25 kg of sugar?

ক
22kg
খ
20kg
গ
18kg
ঘ
15kg

ব্যাখ্যা

Question: The price of sugar increases by 25%. How much sugar can now be bought with the same amount of money that was earlier sufficient to buy 25 kg of sugar?

Solution:
Let
Tk. 100 be spent on sugar initially for 25 kg.

As the price increases by 25%, new price for 25 kg sugar = Tk. (100 + 25% of 100)
= Tk. (100 + 25)
= Tk. 125

New price of sugar = Tk. (125/25) = Tk. 5

Sugar can be bought now = 100/5 = 20 kg

৪,৮১৬.

The number of students in 3 classes is in the ratio 2 : 3 : 4. If 10 students are increased in each class this ratio changes to 8 : 11 : 14. The total number of students in the three classes in the beginning was-

ক
ক) 135
খ
খ) 150
গ
গ) 140
ঘ
ঘ) 125

ব্যাখ্যা

Question: The number of students in 3 classes is in the ratio 2 : 3 : 4. If 10 students are increased in each class this ratio changes to 8 : 11 : 14. The total number of students in the three classes in the beginning was-

Solution:
Let the number of students in the classes be 2x, 3x and 4x respectively;
Total students = 2x + 3x + 4x = 9x
According to the question,
(2x+10) : (3x+10) = 8 : 11
or, 11(2x+10) = 8(3x+10)
or, 22x + 110 = 24x + 80
or, 2x = 30
or, x = 15

Hence,
Original number of students,
9x = 9×15 = 135

৪,৮১৭.

If 5 - 3x ≤ 14, then what is the value of x?

ক
(- ∞, - 3]
খ
[3, ∞)
গ
(- ∞, 3)
ঘ
[- 3, ∞)

ব্যাখ্যা

প্রশ্ন: If 5 - 3x ≤ 14, then what is the value of x?

Solution:
5 - 3x ≤ 14
⇒ - 3x ≤ 14 - 5
⇒ - 3x ≤ 9
⇒ 3x ≥ -9 [উভয় পক্ষকে -1 দ্বারা গুণ করলে]
⇒ x ≥ - 9/3
∴ x ≥ - 3

সমাধানটিকে ব্যবধি (interval) আকারে প্রকাশ করলে হয়: [- 3, ∞)
এখানে তৃতীয় বন্ধনী [ দ্বারা বোঝায় যে - 3 সমাধান সেটের অন্তর্ভুক্ত, এবং ∞ এর পাশে প্রথম বন্ধনী ) বোঝায় যে এটি অসীম পর্যন্ত বিস্তৃত।

৪,৮১৮.

If the distribution of movie durations is symmetric around the mean of 1.5 hours following a bell curve and if the standard deviation is 20 minutes, approximately what percentage of all movies are longer than 110 minutes?

ক
12
খ
16
গ
20
ঘ
25
ঙ
50

ব্যাখ্যা

Question: If the distribution of movie durations is symmetric around the mean of 1.5 hours following a bell curve and if the standard deviation is 20 minutes, approximately what percentage of all movies are longer than 110 minutes?

Solution:
Mean movie duration, µ = 1.5 hours = 90 minutes
Standard deviation, σ = 20 minutes

We are to find the percentage of movies longer than 110 minutes.
So, X = 110
Here, z = (X - µ)/σ
= (110 - 90)/20
= 1

Using standard normal distribution,
Area to the left of z = 1 (i.e. movies ≤ 110 minutes) is approximately 84.13%.
So, the area to the right (i.e., movies > 110 minutes) is = 100% + (- 84.13)% = 15.87% approx 16%

৪,৮১৯.

Which of the following is irrational?

ক
3/5
খ
√2
গ
0.75
ঘ
1.2

ব্যাখ্যা

Question: Which of the following is irrational?

Solution:
অমূলদ সংখ্যা (irrational number):
- যে সংখ্যাকে p/q আকারে প্রকাশ করা যায় না, যেখানে p ও q পূর্ণসংখ্যা এবং q ≠ 0, সে সংখ্যাকে অমূলদ সংখ্যা বলা হয়।
- পূর্ণবর্গ নয় এরূপ যে কোনো স্বাভাবিক সংখ্যার বর্গমূল কিংবা তার ভগ্নাংশ একটি অমূলদ সংখ্যা। যেমন, √2 = 1.414213..., √6 = 2.229489... ইত্যাদি অমূলদ সংখ্যা।
- কোনো অমূলদ সংখ্যাকে দুইটিপূর্ণ সংখ্যার অনুপাত হিসেবে প্রকাশ করা যায় না।
-অমূলদ সংখ্যাকে একটি মূলদ সংখ্যা দ্বারা গুণ করলে অমূলদ সংখ্যা পাওয়া যায়।
অর্থাৎ, non zero rational number × irrational number = irrational number.

অপশন গুলো ব্যাখ্যা করে,
ক) 3/5 ; এটি একটি ভগ্নাংশ, তাই rational
খ) √2 ; এটি কোন পূর্ণসংখ্যার বর্গমূল নয়, তাই irrational
গ) 0.75 = 3/4 ; ভগ্নাংশের সমান, তাই rational
ঘ) 1.2 = 6/5 ; ভগ্নাংশের সমান, তাই rational

সুতরাং, √2 একটি অমূলদ সংখ্যা।

৪,৮২০.

A number when multiplied by 16 increases by 540. What is the number?

ক
30
খ
36
গ
42
ঘ
46

ব্যাখ্যা

Question: A number when multiplied by 16 increases by 540. What is the number?

Solution:
Let the number is x.

As per question;
16x - x = 540
⇒ 15x = 540
∴ x = 36

৪,৮২১.

Ahsan borrows 2,500 taka from a Leasing Company at 4.5% compound interest per year. Calculate the total must be paid after 36 months?

ক
ক) 2852.92
খ
খ) 2837.5
গ
গ) 2934.28
ঘ
ঘ) 2935.63

ব্যাখ্যা

দেওয়া আছে,
মূলধন P = 2500 টাকা
সময়, n = 36 মাস = 3 বছর
সুদের হার, r = 4.5% = 4.5/100

চক্রবৃদ্ধি মুনাফায় সবৃদ্ধিমূল,
C = P(1 + r)ⁿ
= 2500(1 + 4.5/100 )³
= 2500 × (104.5/100)³
= 2500 × (1.045)³
= 2500 × 1.141166125
= 2,852.915
= 2,852.92

৪,৮২২.

Half of 1 percent written as a decimal is ________

ক
ক) 0.2
খ
খ) 0.05
গ
গ) 0.02
ঘ
ঘ) 0.005

ব্যাখ্যা

Question: Half of 1 percent written as a decimal is ________

Solution:
As we know,
1% = 1/100

Hence,
(1/2)% = (1/2) × (1/100) = 1/200 = 0.005

৪,৮২৩.

Which of the following comes first in dictionary order?

ক
Mausoleum
খ
Mauve
গ
Mane
ঘ
Maundy

ব্যাখ্যা

Question: Which of the following comes first in dictionary order?

Solution:
প্রথম শব্দটি হবে Mane, কারণ এটি শুরুতেই আলাদা (Mane ≠ Mau...)

বাকি তিনটি শব্দ: Mausoleum, Mauve, Maundy – এদের শুরুতে রয়েছে "Mau"
এর পরের অক্ষরগুলো হলো: s (Mausoleum), v (Mauve), n (Maundy)

Dictionary order অনুযায়ী: n < s < v
⇒ Mane → ১ম
⇒ Maundy → ২য়
⇒ Mausoleum → ৩য়
⇒ Mauve → ৪র্থ

৪,৮২৪.

The average length of the sides of ∆ABC is 12. What is the perimeter at ∆ABC?

ক
24
খ
36
গ
12
ঘ
18

ব্যাখ্যা

Question: The average length of the sides of ∆ABC is 12. What is the perimeter at ∆ABC?

Solution:
তিনটি বাহুর গড় = 12
তিন বাহুর সমষ্টি = 12 × 3 = 36

∴ ∆ABC এর পরিসীমা = 36

৪,৮২৫.

The difference between the compound interest and simple interest on Tk. x at 8.5% per annum for 2 years is Tk. 28.90. The value of x is

ক
3300
খ
3500
গ
4000
ঘ
4200

ব্যাখ্যা

Question: The difference between the compound interest and simple interest on Tk. x at 8.5% per annum for 2 years is Tk. 28.90. The value of x is-

Solution :
Here,
Principal P = x Tk.
Rate of Interest per year r = 8.5%
= 8.5/100
Time n = 2 years

∴ Simple Interest =Prn
= x × 8.5/100 × 2
= 17x/100
= 0.17x

∴ Compound Interest = P (1 + r)ⁿ - P
= x × (1 + 8.5/100)² - x
= x × (1 + 0.085 )² - x
= x × (1.085)² - x
= 1.177225 x - x
= (1.177225 - 1)x
= 0.177225x

According to the question,
Compound Interest - Simple Interest = 28.90
⇒ 0.177225 x - 0.17x = 28.90
⇒ (0.177225 - 0.17) x = 28.90
⇒ 0.007225x = 28.90
⇒ x = 28.90/0.007225
∴ x = 4000

So the value of x is = 4000

৪,৮২৬.

A room has a length of 10 m, width of 6 m, and height of 4 m. What is the area of the four walls of the room?

ক
144 m²
খ
136 m²
গ
128 m²
ঘ
120 m²

ব্যাখ্যা

Question: A room has a length of 10 m, width of 6 m, and height of 4 m. What is the area of the four walls of the room?

Solution:
Given that,
Length of the room = 10 m
Width = 6 m
Height = 4 m

The area of the four walls = 2 × (length × height) + 2 × (width × height)
= 2 × (10 × 4) + 2 × (6 × 4)
= 2 × 40 + 2 × 24
= (80 + 48)
= 128 m²

So the area of the four walls is 128 m².

৪,৮২৭.

In how many ways can 5 examination papers be arranged so that the best and the worst papers never come together?

ক
ক) 120 ways
খ
খ) 72 ways
গ
গ) 48 ways
ঘ
ঘ) 20 ways

ব্যাখ্যা

Question: In how many ways can 5 examination papers be arranged so that the best and the worst papers never come together?

Solution:
Total ways = 5!
= 120 ways

if two papers come together, we can consider them one.
ways that they will come together = 4! × 2!
= 24 × 2
= 48 ways

∴ ways the best and the worst papers never come together = 120 - 48 ways
= 72 ways

৪,৮২৮.

Evaluate :

ক
ক) 20
খ
খ) 16
গ
গ) 24
ঘ
ঘ) 30

ব্যাখ্যা

Question: Evaluate :

Solution:
√{248 + √(52 + √144)}
= √{248 + √(52 + 12)}
= √(248 + √64)
= √(248 + 8)
= √256
= 16

৪,৮২৯.

In how many ways can a group of 5 men and 2 women be made out of a total of 7 men and 3 women?

ক
45
খ
63
গ
90
ঘ
126
ঙ
135

ব্যাখ্যা

Question: In how many ways can a group of 5 men and 2 women be made out of a total of 7 men and 3 women?

Solution:
Required number of ways = (⁷C₅ × ³C₂)
= 21 × 3
= 63

৪,৮৩০.

If a, b, and c are the lengths of the three sides of a triangle, then which of the following is true?

ক
a + b < c
খ
a - b < c
গ
a + b = c
ঘ
a + b ≥ c

ব্যাখ্যা

Question: If a, b, and c are the lengths of the three sides of a triangle, then which of the following is true?

Solution:
ত্রিভুজ হওয়ার শর্ত:
• ত্রিভুজের যেকোনো দুই বাহুর সমষ্টি তার তৃতীয় বাহু অপেক্ষা বৃহত্তর।
• যেকোনো দুই বাহুর অন্তর তার তৃতীয় বাহু অপেক্ষা ক্ষুদ্রতর।

দেওয়া আছে
ত্রিভুজের তিনটি বাহু যথাক্রমে a, b এবং c
সুতরাং, a - b < c অবশ্যই সঠিক।

৪,৮৩১.

A motor boat takes 12 hours to go downstream and it takes 24 hours to return the same distance. What is the time taken by boat in still water?

ক
15 h
খ
16 h
গ
8 h
ঘ
20 h
ঙ
None of the above

ব্যাখ্যা

If t1 and t2 are the upstream and down stream times.
Then time taken in still water is given by
(2×t1×t2)/(t1+t2)
= (2×12×24)/36
= 16h

৪,৮৩২.

A takes 30 minutes more than B to reach a destination. The speeds of A and B are in the ratio 4 : 6. Time in which A reaches the destination?

ক
1.5 hr
খ
2 hr
গ
2.5 hr
ঘ
1.25 hr

ব্যাখ্যা

Question: A takes 30 minutes more than B to reach a destination. The speeds of A and B are in the ratio 4 : 6. Time in which A reaches the destination?

Solution:
here, ratio of speed is 4 : 6
as the ratio of time is inversly proportional to speed,
ratio of time = 6 : 4

ATQ,
6x - 4x = 30 min
2x = 30 min
x = 15 min = 1/4 hour

A takes total time of = 6 × 1/4 = 1.5 hr

৪,৮৩৩.

If 2/3 of a number is 5 more than 1/4 of the number then 7/2 of the number is-

ক
41
খ
42
গ
43
ঘ
44

ব্যাখ্যা

Question: If 2/3 of a number is 5 more than 1/4 of the number then 7/2 of the number is-

Solution:
Let,
the number be x

According to the question,
⇒ (2x/3) - x/4 = 5
⇒ (8x - 3x)/12 = 5
⇒ 5x = 5 × 12
⇒ 5x = 60
⇒ x = 12

Then 7/2 of the number will be = x × 7/2
= (12 × 7)/2
= 42

৪,৮৩৪.

Find the missing letter marked (?) in the series
C, H, M, ?, W, B.

ক
ক) S
খ
খ) P
গ
গ) Q
ঘ
ঘ) R
ঙ
ঙ) Z

ব্যাখ্যা

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

Solution:

C, H, M, ?, W, B
3, 8, 13, ?, 23, 28 [ Here the position number of B shown 28 because of cyclic count]
Here,
8 - 3 = 5
13 - 8 = 5
and,
28 - 23 = 5
So, the position number of '?' marked place will be 13 + 5 = 18
So, the Letter will be 'R'.

৪,৮৩৫.

The owner of a bike buys petrol for 3 years continuously at Tk. 64, Tk. 80 and Tk. 320 per litre respectively. If he spends Tk. 32,000 every year buying petrol, what is the average price per litre of petrol?

ক
Tk. 100
খ
Tk. 92
গ
Tk. 90
ঘ
Tk. 106
ঙ
Tk. 96

ব্যাখ্যা

Question: The owner of a bike buys petrol for 3 years continuously at Tk. 64, Tk. 80 and Tk. 320 per litre respectively. If he spends Tk. 32,000 every year buying petrol, what is the average price per litre of petrol?

Solution:
Number of litres consume of first year = 32000/64 = 500 litres
Number of litres consume of second year = 32000/80 = 400 litres
Number of litres consume of third year = 32000/320 = 100 litres
Total consume petrol in three years = 500 + 400 + 100 = 1000 litres
Total expenditure on petrol in three years = 32000 × 3
Average cost per litres of petrol = (32000 × 3)/1000 = Tk. 96

৪,৮৩৬.

If 0 ≤ θ ≤ π/2 and cos θ + √3sinθ = 2, then what is the value of θ?

ক
π/2
খ
π/3
গ
π/4
ঘ
π/6

ব্যাখ্যা

Question: If 0 ≤ θ ≤ π/2 and cos θ + √3sinθ = 2, then what is the value of θ?

Solution:
cos θ + √3sinθ = 2
⇒ (1/2) cos + (√3/2) sinθ = 1
⇒ sin30° cosθ + cos30° sinθ = 1
⇒ sin(30° + θ) = sin90°
⇒ 30° + θ = 90°
∴ θ = 60° = π/3

৪,৮৩৭.

The current of a stream keeps running at 4 km in 60 minutes. A boat goes 6 km and back to the beginning stage in 2 hours. The rate of the boat in still water is:

ক
ক) 6 km/hr
খ
খ) 7.5 km/hr
গ
গ) 8 km/hr
ঘ
ঘ) 6.8 km/hr

ব্যাখ্যা

Let the speed in still water be x km/hr.
Then,
Speed downstream = (x+ 4) km/hr,
speed upstream = (x-4) km/hr.
6/(x+4) + 6 /(x-4) = 2
=> 1/(x+4) +1/(x-4)=2/6 = 1/3
=> (x+4)+(x-4)/x2-16= 1/3
=> x²-16= 6x
=> x² -6x-16= 0
=> (x-8) (x+2) = 0
=> x = 8.
∴ Speed of boat in still water = 8 km/hr.

৪,৮৩৮.

The length of the sides of a triangle are in the ratio of 3 to 5 to 6. If the perimeter of the triangle is 70. what is the length of the longest side?

ক
ক) 26
খ
খ) 35
গ
গ) 25
ঘ
ঘ) 30

ব্যাখ্যা

ত্রিভুজের তিন বাহুর দৈর্ঘ্যের অনুপাত = 3 : 5 : 6
ধরি ত্রিভুজের বাহু তিনটির দৈর্ঘ্য যথাক্রমে 3x, 5x, 6x
প্রশ্নমতে,
3x + 5x + 6x = 70
=> 14x = 70
=> x = 5
অতএব, বৃহত্তম বাহুর দৈর্ঘ্য = 6x = 6 × 5 = 30

৪,৮৩৯.

A room is 12.25m long and 7m wide. The maximum length of a square tile to fill the floor of the room with a whole number of tiles should be:

ক
110 cm
খ
175 cm
গ
157 cm
ঘ
150 cm
ঙ
170 cm

ব্যাখ্যা

Length of largest tile:

= H.C.F. of 12.25 m and 7 m

= H.C.F. of 1225 cm and 700 cm

= 175 cm

৪,৮৪০.

In a sugar-water solution, the ratio of water to sugar is 8 : 3. If you add 2 kgs of sugar, the ratio becomes 2 : 1. What is the amount of sugar in the original solution in kg?

ক
4 kg
খ
5 kg
গ
6 kg
ঘ
7 kg
ঙ
8 kg

ব্যাখ্যা

Question: In a sugar-water solution, the ratio of water to sugar is 8 : 3. If you add 2 kgs of sugar, the ratio becomes 2 : 1. What is the amount of sugar in the original solution in kg?

Solution:
Let's say the initial amount of sugar is x kg
Given the 8:3 ratio, the initial amount of water is (8x/3) kg

After adding 2 kg of sugar, new amount of sugar = x + 2 kg
Amount of water remains same = (8x/3) kg

ATQ,
⇒ (8x/3)/(x + 2) = 2/1
⇒ 8x/3 = 2(x + 2)
⇒ 8x/3 = 2x + 4
⇒ 8x = 6x + 12
⇒ 2x = 12
∴ x = 6
Therefore, the original amount of sugar was 6 kg.

৪,৮৪১.

Two boats are spotted on the two sides of a lighthouse. If the angle of depression made by both the boats from top of the lighthouse is 30° and 45° and the height of the light house is 125 m then find the distance between the two boats.

ক
1 m
খ
188.56 m
গ
250 m
ঘ
197.17 m

ব্যাখ্যা

Let AB be the lighthouse and the two boats be at C and D
AB = 125 m

tan30° = BC/AB
= x/125
= 1/v3
x = 72.17 m

tan45° = BD/AB
= y/125
= 1
y = 125 m
Therefore,
the distance between the two boats is = x + y
= 72.17 + 125
= 197.17 m

৪,৮৪২.

Three numbers are in the ratio 3 : 4 : 5, and the sum of their squares is 1250. Find the smallest number.

ক
15
খ
18
গ
21
ঘ
25

ব্যাখ্যা

Question: Three numbers are in the ratio 3 : 4 : 5, and the sum of their squares is 1250. Find the smallest number.

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

ATQ,
(3x)² + (4x)² + (5x)² = 1250
⇒ 9x² + 16x² + 25x² = 1250
⇒ 50x² = 1250
⇒ x² = 25
∴ x = 5

∴ Smallest number = 3x
= 3 × 5
= 15

৪,৮৪৩.

The area of a circle of radius √2 is approximately-

ক
ক) 1.4142
খ
খ) 6.2832
গ
গ) 9.4298
ঘ
ঘ) 3.1416

ব্যাখ্যা

Area of the circle = πr²
= π(√2)²
= 2π
= 6.2832

৪,৮৪৪.

What is the value of

ক
6
খ
4
গ
2
ঘ
1

ব্যাখ্যা

Question: What is the value of

Solution:

৪,৮৪৫.

The smallest number which is divided by 4, 6, 8, 12 and 16, leaving the remainder 2, is-

ক
ক) 50
খ
খ) 46
গ
গ) 48
ঘ
ঘ) 56

ব্যাখ্যা

প্রশ্ন: The smallest number which is divided by 4, 6, 8, 12 and 16, leaving the remainder 2, is

সমাধান:
4, 6, 8, 12, 16 এর ল.সা.গু = 48

∴ যেহেতু প্রতিক্ষেত্রে 2 ভাগশেষ থাকে তাই সংখ্যাটি = 48 + 2 = 50

৪,৮৪৬.

The ratio of the four angles of a quadrilateral is 3 : 4 : 5 : 6. What is the largest angle?

ক
ক) 100°
খ
খ) 105°
গ
গ) 120°
ঘ
ঘ) 135°

ব্যাখ্যা

Question: The ratio of the four angles of a quadrilateral is 3 : 4 : 5 : 6. What is the largest angle?

Solution:
Sum of the angles of a quadrilateral = 360°
Largest angle = 360° × (6/18) = 120°

৪,৮৪৭.

A boat can travel 48 km upstream in 6 hours. If the speed of the stream is 2 km/hr, how much time will the boat take to cover a distance of 120 km downstream?

ক
20 hours
খ
18 hours
গ
10 hours
ঘ
19 hours

ব্যাখ্যা

Question: A boat can travel 48 km upstream in 6 hours. If the speed of the stream is 2 km/hr, how much time will the boat take to cover a distance of 120 km downstream?

Solution:
Distance covered by a boat in 6 hours = 48 km
Rate upstream of boat = 48/6
= 8 km/hr

Now,
Speed of stream = 2 km/hr
∴ Speed of boat in still water = (8 + 2)
= 10 km/hr

∴ Rate downstream of boat = (10 + 2) km/hr
= 12 km/hr

∴ Time taken in covering 120 km distance = 120/12
= 10 hours

৪,৮৪৮.

If A and B can together do a work in 15 days and B alone can finish the job in 20 days. In how many days, A alone can finish the job?

ক
ক) 60
খ
খ) 45
গ
গ) 40
ঘ
ঘ) 30
ঙ
ঙ) None of above

ব্যাখ্যা

A and B complete a work in = 15 days
One day's work of (A + B) = 1/15
B complete the work in = 20 days;
One day's work of B = 1/20
Then, A's one day's work = 1/15 - 1/20
= (4 - 3)/60
= 1/60
Thus, A can complete the work in = 60 days.

৪,৮৪৯.

Tk. 7800 are distributed among A, B, and C. The share of "A" is the 3/4 of the share of B, and the share of B is the 2/3 of the share of C. Find the difference between the share of B and C.

ক
1200
খ
1300
গ
1500
ঘ
800

ব্যাখ্যা

Question: Tk. 7800 are distributed among A, B, and C. The share of "A" is the 3/4 of the share of B, and the share of B is the 2/3 of the share of C. Find the difference between the share of B and C.

Solution:
The share of A: B is 3: 4
The share of B: C is 2: 3

Note: Whenever such form is given, multiply a to b, then b to b, and then b to c.

i.e., A: B: C = 3×2: 4×2: 4×3
Or, A: B: C = 6: 8: 12
Or, A: B: C = 3: 4: 6
Sum of ratios = 13
Now, the share of B = [4/13] × 7800 = 2400
Share of C = [6/13]× 7800 = 3600

The difference between the share of B and C = 3600- 2400 = 1200

৪,৮৫০.

A sum of money lent at compound interest for 2 year at 20% per annum would fetch Tk.482 more, if the interest was payable half yearly than if it was payable annually. The sum is

ক
ক) 10000
খ
খ) 20000
গ
গ) 40000
ঘ
ঘ) 50000

ব্যাখ্যা

Let sum=Tk.x
C.I. when compounded half yearly = [x(1+10/100)⁴−x]=4641/10000
C.I. when compounded annually =[x(20/100)²−x]=11/25
4641/10000x−11/25x=482
=> x=20000

৪,৮৫১.

The price of diesel increases by 50%. Find by how much percent a truck owner must reduce his consumption in order to maintain the same budget?

ক
11.11%
খ
22.22%
গ
33.33%
ঘ
44.44%

ব্যাখ্যা

Question: The price of diesel increases by 50%. Find by how much percent a truck owner must reduce his consumption in order to maintain the same budget?

Solution:
Let,
the initial price of the diesel be Tk. x
Then, after 50% increase in price, it will become Tk. x + (x × 50)/100 = Tk. 1.5x
Now,
we have to reduce the consumption to keep expenditure Tk. x.

Increase in price =1.5x - x = 0.5x
We have to reduce the consumption by (0.5x × 100)/(1.5x) = 33.33%

৪,৮৫২.

Which number will continue the following series? 3, 8, 15, 24, 35, 48, 63, 80 ___

ক
ক) 92
খ
খ) 91
গ
গ) 95
ঘ
ঘ) 97
ঙ
ঙ) 99

ব্যাখ্যা

ধারাটিঃ
3 + 5 = 8
8 + 7 = 15
15 + 9 = 24
24 + 11 = 35
35 + 13 = 48
48 + 15 = 63
63 + 17 = 80
80 + 19 = 99

৪,৮৫৩.

Two pipes A and B can fill a cistern in 37.5 minutes and 45 minutes respectively. This cistern will be filled in half an hour if both the pipes are opened together initially and pipe B is then turned off after X minutes. What is X?

ক
8 minutes
খ
12 minutes
গ
9 minutes
ঘ
11 minutes

ব্যাখ্যা

Question: Two pipes A and B can fill a cistern in 37.5 minutes and 45 minutes respectively. This cistern will be filled in half an hour if both the pipes are opened together initially and pipe B is then turned off after X minutes. What is X?

Solution:
Let the volume of tank = V liters
Rate of pipe A = (V/37.5) L/min
Rate of pipe B = (V/45) L/min
In X minutes (both pipes open): Part filled = (V/37.5) + (V/45) × X
= V(45 + 37.5)/(37.5 × 45) × X
= V × 82.5/(1687.5) × X
= V × (X/20.45)

In remaining time (30 - X) minutes (only pipe A): Part filled = (V/37.5) × (30 - X)
Total part filled = 1 (complete tank)
X/20.45 + (30-X)/37.5 = 1
⇒ 37.5X + 20.45(30-X) = 37.5 × 20.45
⇒ 37.5X + 613.5 - 20.45X = 766.875
⇒ 17.05X = 153.375
∴ X = 9

৪,৮৫৪.

Prova travels a distance of 9 km from her house to the school by auto-rickshaw at 18 km/hr and returns on rickshaw at 15 km/hr. Find the average speed for the whole journey.

ক
ক) 12.2 km/hr
খ
খ) 16.3 km/hr
গ
গ) 19.8 km/hr
ঘ
ঘ) 26.6 km/hr

ব্যাখ্যা

Question: Prova travels a distance of 9 km from her house to the school by auto-rickshaw at 18 km/hr and returns on rickshaw at 15 km/hr. Find the average speed for the whole journey.

Solution:
total distance = 9 + 9 = 18 km

time taken by auto rickshaw = 9/18 hr
= 1/2 hr

time taken by auto rickshaw = 9/15 hr
= 3/5 hr

total time = 1/2 hr + 3/5 hr
= 11/10 hr

average speed = 18/11/10
= 180/11 km/hr
= 16.3 km/hr

৪,৮৫৫.

30 kgs of rice costing Tk. 12/kg is mixed with some kgs of rice costing Tk. 18 to get the mixture costing Tk. 13.5. Find the quantity of rice costing Tk. 18.

ক
10 kg
খ
12 kg
গ
18 kg
ঘ
20 kg

ব্যাখ্যা

Question: 30 kgs of rice costing Tk. 12/kg is mixed with some kgs of rice costing Tk. 18 to get the mixture costing Tk. 13.5. Find the quantity of rice costing Tk. 18.

Solution:
Let, the amount of rice costing Tk.18 is x kg

ATQ,
30 × 12 + 18x = 13.5 × (x + 30)
⇒ 360 + 18x = 13.5x + 405
⇒ 18x - 13.5x = 405 - 360 = 45
⇒ 4.5x = 45
∴ x = 45/4.5 = 10 kg

৪,৮৫৬.

The average age of A, B and C was 25 years and that of B and C was 25 years. A’s present age is-

ক
30 years
খ
25 years
গ
40 years
ঘ
42 years
ঙ
None of these

ব্যাখ্যা

Question: The average age of A, B and C was 25 years and that of B and C was 25 years. A’s present age is-

Solution:

Average of A,B,C is 25

So, sum of their ages =75

Now, the sum of B and C will be 50 (because their average is 25)

So age of A =75 - 50 = 25 years

৪,৮৫৭.

In order to obtain an income of Tk. 650 from 10% stock at Tk. 96, one must make an investment of -

ক
ক) Tk. 9600
খ
খ) Tk. 6500
গ
গ) Tk. 6240
ঘ
ঘ) Tk. 3100

ব্যাখ্যা

Market Value = Tk. 96.
Required Income = Tk. 650.
Here face value is not given. Take face value as Tk. 100 if it is not given in the question
To obtain Tk. 10 (ie,10% of the face value 100), investment = Tk. 96
To obtain Tk. 650, investment = {(96/10) × 650}
= Tk. 6240.

৪,৮৫৮.

Anita had to do a multiplication. instead of taking 35 as one of the multipliers, she took 53. As a result, the product went up by 540. What is the new product?

ক
ক) 1,050
খ
খ) 1,250
গ
গ) 1,440
ঘ
ঘ) 1,590

ব্যাখ্যা

Question: Anita had to do a multiplication. instead of taking 35 as one of the multipliers, she took 53. As a result, the product went up by 540. What is the new product?

Solution:
ধরি
সংখাটি = x
প্রশ্নমতে
53x - 35x = 540
⇒18x = 540
⇒x = 540/18
⇒ x =30

নতুন গুণফল
53 × 30 = 1590

৪,৮৫৯.

Junayed starts a business with TK 9000 and after one year, Rayhan joins with junayed by investing a certain amount. At the end of 2 years, If 3 : 2 is the proportion of the profit then Rayhan's contribution to the capital is-

ক
12000 TK
খ
15000 TK
গ
18000 TK
ঘ
16000 TK

ব্যাখ্যা

Question: Junayed starts a business with TK 9000 and after one year, Rayhan joins with junayed by investing a certain amount. At the end of 2 years, If 3 : 2 is the proportion of the profit then Rayhan's contribution to the capital is-

Solution:
Let,
Rayhan's capital be TK P
Junayed's investment = TK 9000 for 24 months
Rayhan's investment = TK p for 12 months
we know that, profit ratio = investing ratio

ATQ,
(9000 × 24)/(p × 12) = 3 : 2
or, (9000 × 24) : 12p = 3 : 2
or, (9000 × 24)/12p = 3/2
or, 36p = (2 × 24 × 9000)
or, p = (48 × 9000)/36
∴ p = 12000

The required answer is 12000 TK

৪,৮৬০.

If a cube has a length of 8 cm, what is its total surface area?

ক
ক) 64
খ
খ) 96
গ
গ) 384
ঘ
ঘ) 256

ব্যাখ্যা

প্রশ্ন: If a cube has a length of 8 cm, what is its total surface area?

সমাধান:
দেওয়া আছে,
ঘনকের এক বাহুর দৈর্ঘ্য, a = 8 cm

∴ ঘনকের সমগ্রতলের ক্ষেত্রফল = 6a²
= 6 × 8 × 8 cm²
= 384 cm²

৪,৮৬১.

The average of 6 numbers is 7. The average of three numbers of them is 5. What will be the average of the remaining numbers?

ক
ক) 15
খ
খ) 30
গ
গ) 42
ঘ
ঘ) 9

ব্যাখ্যা

Question: The average of 6 numbers is 7. The average of three numbers of them is 5. What will be the average of the remaining numbers?

Solution:
Average of 6 numbers = 7
Sum of 6 numbers = 6 × 7 = 42
Average of three numbers = 5
Sum of three numbers = 5 × 3 = 15

∴ Sum of the remaining three numbers = 42 - 15 = 27

∴ Required average = 27/3 = 9

৪,৮৬২.

A sum of Taka 100,000 is invested at 5% simple interest for the first 4 years and 10% compound interest for the next 2 years. What is the total amount after 6 years?

ক
Tk. 1,40,000
খ
Tk. 1,45,200
গ
Tk. 1,50,500
ঘ
Tk. 1,60,100

ব্যাখ্যা

Question: A sum of Taka 100,000 is invested at 5% simple interest for the first 4 years and 10% compound interest for the next 2 years. What is the total amount after 6 years?

সমাধান:
দেওয়া আছে,
মূলধন, P = 1,00,000 টাকা
প্রথম 4 বছরের জন্য বার্ষিক সরল সুদের হার, r = 5%
পরবর্তী 2 বছরের জন্য বার্ষিক চক্রবৃদ্ধি সুদের হার, r = 10%
মোট সময়কাল = 6 বছর

প্রথম 4 বছরের সরল সুদ (Simple Interest), I = (P × n × r)/100
⇒ I = (100000 × 4 × 5)/100
⇒ I = 20,000 টাকা

4 বছর পর আসল (Principal) হবে = 1,00,000 + 20,000 = 1,20,000 টাকা
পরবর্তী 2 বছরের জন্য, এই 1,20,000 টাকা হবে নতুন মূলধন (New Principal)।
সুতরাং, P = 1,20,000 টাকা
সময়, n = 2 বছর
সুদের হার, r = 10%

চক্রবৃদ্ধি মূল (Compound Amount), C = P(1 + r/100)ⁿ
⇒ C = 120000(1 + 10/100)²
⇒ C = 120000(110/100)²
⇒ C = 120000 × (110/100) × (110/100)
⇒ C = 1,45,200 টাকা

সুতরাং, 6 বছর পর মোট পরিমাণ হবে 1,45,200 টাকা।

৪,৮৬৩.

In a class, 1/4 of the male students is equal to 3/5 of the female students. What fraction of the students in the room is female?

ক
4/15
খ
7/20
গ
5/17
ঘ
8/21
ঙ
None

ব্যাখ্যা

Question: In a class, 1/4 of the male students is equal to 3/5 of the female students. What fraction of the students in the room is female?

Solution:
Let the number of male students be m
And the number of female students be f.

Given that,
(1/4) of the male students = (3/5) of the female students
⇒ (1/4)m = (3/5)f
∴ m = (12/5)f ; [Cross-multiply]

∴ Total students = m + f = (12/5)f + f
= (12f + 5f)/5
= 17f/5

∴ Fraction of students that are female = f/(m + f) = f/(17f/5) = 5/17

So the fraction of students that are female is 5/17.

৪,৮৬৪.

10 minutes after a plane leaves the airport, it is reported to be 40 miles away. What is the average speed in miles per hour of the plane?

ক
ক) 560
খ
খ) 400
গ
গ) 240
ঘ
ঘ) 200

ব্যাখ্যা

Question: 10 minutes after a plane leaves the airport, it is reported to be 40 miles away. What is the average speed in miles per hour of the plane?

Solution:
গড় বেগ = {40/(10/60)} miles/hour
= 40 × 6 miles/hour
= 240 miles/hour

৪,৮৬৫.

A circus artist is climbing a 28 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with the ground level is 60°.

ক
32√2 m
খ
24√3 m
গ
11√5 m
ঘ
14√3 m

ব্যাখ্যা

Question: A circus artist is climbing a 28m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with the ground level is 60°.

Solution:

By observing the figure, AB is the pole.
In triangle ABC,
⇒ AB/AC = sin60°
⇒ AB/28 = √3/2
⇒ AB = 14√3

Therefore, the height of the pole is 14√3 m

৪,৮৬৬.

Find the value of 2(x + 3) - 3(x - 1) + x.

ক
x - 9
খ
9
গ
x + 9
ঘ
- x

ব্যাখ্যা

Question: Find the value of 2(x + 3) - 3(x - 1) + x.

Solution:
2(x + 3) - 3(x - 1) + x
= 2x + 6 - 3x + 3 + x
= 3x - 3x + 9
= 9

৪,৮৬৭.

ক
0
খ
11
গ
9
ঘ
5

ব্যাখ্যা

Question:

Solution:

৪,৮৬৮.

A sum of money placed at compound interest doubles itself in 4 years. In how many years will it amount to 8 times?

ক
9 years
খ
8 years
গ
27 years
ঘ
12 years

ব্যাখ্যা

Question: A sum of money placed at compound interest doubles itself in 4 years. In how many years will it amount to 8 times?

Solution:
Let,
Principal = Tk. 100.
Amount = Tk. 200.
Rate = r%
Time = 4 years.
Now,
A = P(1+ r/100)ⁿ
200 = 100(1 + r/100)⁴
2 = (1 + r/100)⁴ ........... (1)
If sum become 8 times in the time n years,
then,
8 = (1 + r/100)ⁿ
2³ = (1 + r/100)ⁿ ............. (2)
Using eqn (1) in (2), we get;
{(1 + r/100)⁴)³ = (1 + r/100)ⁿ
(1 + r/100)¹² = (1 + r/100)ⁿ
∴ n = 12

৪,৮৬৯.

If a^x=b, b^y=c and xyz=1 then what is the value of c^z?

ক
ক) a
খ
খ) b
গ
গ) ab
ঘ
ঘ) a / b

ব্যাখ্যা

Given. xyz=1,a^x=b,b^y=c
Now, b=a^x
=> b^y=a^xy
=> b^yz=a^xyz
=> c^z=a

৪,৮৭০.

A sum of money at simple interest amounts to Tk. 3600 in 4 years and Tk. 4200 in 6 years. What is the rate of interest?

ক
12%
খ
10.5%
গ
8%
ঘ
12.5%

ব্যাখ্যা

Question: A sum of money at simple interest amounts to Tk. 3600 in 4 years and Tk. 4200 in 6 years. What is the rate of interest?

Solution:
Simple interest for 2 years = (4200 - 3600) Tk. = Tk. 600

∴ Simple Interest for 2 years = Tk. 600

∴ Simple Interest for 6 years = 600 × (6/2) = Tk. 1800

∴ Principal = 4200 - 1800 = Tk. 2400

We know,
I = Pnr
⇒ r = I/Pn
⇒ r = (1800 × 100)/(2400 × 6)
⇒ r = 300/24
∴ r = 12.5%

৪,৮৭১.

If ABC and PQR are similar triangles in which ∠A = 47° and ∠Q = 83°, then ∠C is-

ক
50°
খ
75°
গ
67°
ঘ
56°

ব্যাখ্যা

Question: If ABC and PQR are similar triangles in which ∠A = 47° and ∠Q = 83°, then ∠C is-

Solution:
Since, ΔABC and ΔPQR are similar triangles.
And ∠A = 47°
then,
∠B = ∠Q = 83°

Thus, in ΔABC,
∠A + ∠B + ∠C = 180°
∠C = 180° - (∠A + ∠ B)
⇒ ∠C = 180° - (47° + 83°)
∴ ∠C = 50°

৪,৮৭২.

If the mean of 5 observations x, x + 2, x + 4, x + 6, and x + 8 is 11, then the mean of the last three observations is -

ক
11
খ
13
গ
15
ঘ
17

ব্যাখ্যা

Question: If the mean of 5 observations x, x + 2, x + 4, x + 6, and x + 8 is 11, then the mean of the last three observations is -

Solution:
ATQ,
{x + (x + 2) + (x + 4) + (x + 6) + (x + 8)}/5 = 11
⇒ 5x + 20 = 55
⇒ 5x = 55 - 20
⇒ x = 35/5
∴ x = 7

So, the numbers are 7, 9, 11, 13, 15

Required mean = (11 + 13 + 15)/3
= 39/3
= 13

৪,৮৭৩.

If m² < 225 and n - m = - 10, what is the difference between the smallest possible integer value of 3m + 2n and the greatest possible integer value of 3m + 2n?

ক
- 190
খ
- 188
গ
- 150
ঘ
- 148
ঙ
- 40

ব্যাখ্যা

Question: If m² < 225 and n - m = - 10, what is the difference between the smallest possible integer value of 3m + 2n and the greatest possible integer value of 3m + 2n?

Solution:
From n - m = - 10 it follows that n = m - 10.
Thus, 3m + 2n = 3m + 2(m - 10) = 5m - 20.
So, we need to find the difference between the smallest possible integer value of 5m - 20 and the greatest possible integer value of 5m - 20.

Now,
lets' work on m² < 225
Take the square root from both sides: |m| < 15
Get rid of the modulus sign: - 15 < m < 15
Multiply all three parts by 5: - 75 < 5m < 75
Subtract 20 from all three parts: - 95 < 5m - 20< 55

From - 95 < 5m - 20 < 55 it follows that the smallest possible integer value of 5m - 20 is - 94 and the greatest possible integer value of 5m - 20 is 54.

Therefore, the difference is: - 94 - 54 = - 148.

৪,৮৭৪.

How many number of pairs in there, where the pair contains consecutive odd positive integers,both of which are smaller than 10, such that their sum is more than 11.

ক
1
খ
2
গ
3
ঘ
None of the above

ব্যাখ্যা

Question: How many number of pairs in there, where the pair contains consecutive odd positive integers,both of which are smaller than 10, such that their sum is more than 11.

Solution:
Let x the smaller of the two consecutive odd positive integers be = x
Then the other odd integer is x + 2.

It is given that both the integers are smaller than 10 and their sum is more than 11.

∴ x + 2 < 10 and, x + (x + 2) > 11
⇒ x < 10 - 2 and 2x + 2 > 11
⇒ x < 8 and 2x > 9
⇒ x < 8 and x > 9/2
⇒ 9/2 < x < 8
⇒ x = 5, 7

The required pairs of odd integers are (5, 7) and (7, 9).

৪,৮৭৫.

From a point at some distance from the base of a tree, the angle of elevation to the top of the tree is 30°. If the tree is 12 meters tall, how far is that point from the tree?

ক
12√2 meters
খ
12√3 meters
গ
8 meters
ঘ
8√2 meters

ব্যাখ্যা

Question: From a point at some distance from the base of a tree, the angle of elevation to the top of the tree is 30°. If the tree is 12 meters tall, how far is that point from the tree?

Solution:

Let's assume the tree is located a meters away.

Thus, tan 30° = AB/AC
⇒ 1/√3 = 12/a
∴ a = 12√3 meters

৪,৮৭৬.

A circular well with a diameter of 14 meters, is dug to a depth of 3 meters. What is the volume of the earth dug out?

ক
300 m³
খ
462 m³
গ
250 m³
ঘ
148 m³

ব্যাখ্যা

Question: A circular well with a diameter of 14 meters, is dug to a depth of 3 meters. What is the volume of the earth dug out?
(একটি বৃত্তাকার কুয়া যার ব্যাস ১৪ মিটার, সেটি ৩ মিটার গভীর পর্যন্ত খনন করা হয়েছে। খননের ফলে উত্তোলিত মাটির আয়তন কত?)

Solution:
কূপের ব্যাস = 14 m
ব্যাসার্ধ 14/2 = 7 m

আয়তন = πr²h
= (22/7) × 7 × 7 × 3
= 462 m³

৪,৮৭৭.

If a : b = 1 : 3, b : c = 5 : 7, then what is the value of a : b : c?

ক
ক) 3 : 5 : 7
খ
খ) 5 : 9 : 15
গ
গ) 5 : 15 : 21
ঘ
ঘ) 4 : 13 : 24

ব্যাখ্যা

দেয়া আছে ,
a : b = 1 : 3 = 5 : 15
b : c = 5 : 7 = 15 : 21

a : b : c = 5 : 15 : 21

৪,৮৭৮.

If m : n = 3 : 2, then find the ratio (4m + 5n) : (4m - 5n).

ক
ক) 1 : 10
খ
খ) 11 : 1
গ
গ) 10 : 1
ঘ
ঘ) 11 : 3

ব্যাখ্যা

Question: If m : n = 3 : 2, then find the ratio (4m + 5n) : (4m - 5n).

Solution:
here,
m : n = 3 : 2
2m = 3n
∴ 4m = 6n

hence,
(4m + 5n) : (4m - 5n) = (6n + 5n) : (6n - 5n)
= 11n : n
= 11 : 1

৪,৮৭৯.

If the base of a parallelogram is 8 and the height is 7 and perimeter is 24, what is the area of the parallelogram?

ক
ক) 28
খ
খ) 56
গ
গ) 65
ঘ
ঘ) 82

ব্যাখ্যা

Question: If the base of a parallelogram is 8 and the height is 7 and perimeter is 24, what is the area of the parallelogram?

Solution:
দেয়া আছে,
সামান্তরিকের ভূমি ৪ মিটার
সামান্তরিকের উচ্চতা 7 মিটার এবং
সামান্তরিকের পরিসীমা 24 মিটার

আমরা জানি,
সামান্তরিকের ক্ষেত্রফল = (ভূমি × উচ্চতা) বর্গমিটার
= 8 × 7 = 56 বর্গমিটার

৪,৮৮০.

P, Q and R can do a job in 20, 30 and 60 days respectively. In how many days can P do the job if he is assisted by Q and R every third day?

ক
12 days
খ
11 days
গ
15 days
ঘ
13 days
ঙ
None

ব্যাখ্যা

Question: P, Q and R can do a job in 20, 30 and 60 days respectively. In how many days can P do the job if he is assisted by Q and R every third day?

Solution:
P's 2 days' work = 2/20
= 1/10

∴ (P + Q + R)'s 1 day's work
= (1/20 + 1/30 + 1/60)
= 6/60
= 1/10

∴ Job done in 3 days [P alone 2 days + (P+Q+R) 1 day] = (1/10 + 1/10) = 1/5

Now, 1/5 jobs is done in 3 days

∴ The whole job will be done in (3 x 5) = 15 days.

৪,৮৮১.

Let x be a number. GCF of 2/5, 3/5 and 1/4 of that number is 5. find the number.

ক
100
খ
200
গ
120
ঘ
90

ব্যাখ্যা

Question: Let x be a number. GCF of 2/5, 3/5 and 1/4 of that number is 5. find the number.

Solution:
the numbers are 2x/5, 3x/5, x/4.
GCF of 2x, 3x, and x is x
and LCM of 5, 5 and 4 is 20
∴ the GCF of 2x/5, 3x/5, x/4 is x/20

ATQ,
x/20 = 5
x = 100

৪,৮৮২.

A man invested Tk. 14400 in Tk. 100 shares of a company at 20% premium. If the company declares 5% dividend at the end of the year, then how much does he get?

ক
ক) Tk. 550
খ
খ) Tk. 700
গ
গ) Tk. 650
ঘ
ঘ) Tk. 600

ব্যাখ্যা

Question: A man invested Tk. 14400 in Tk. 100 shares of a company at 20% premium. If the company declares 5% dividend at the end of the year, then how much does he get?

Solution:
Number of shares = 14400/120
= 120
Face value = Tk. (100 × 120)
= Tk. 12000

∴ Annual income = Tk. (5/100) × 12000
= Tk. 600

৪,৮৮৩.

5 years ago, sister’s age was 5 times the age of her brother and the sum of present ages of sister and brother is 34 years. What will be the age of her brother after 6 years?

ক
ক) 12 years
খ
খ) 15 years
গ
গ) 13.5 years
ঘ
ঘ) 20 years

ব্যাখ্যা

Question: 5 years ago, sister’s age was 5 times the age of her brother and the sum of present ages of sister and brother is 34 years. What will be the age of her brother after 6 years?

Solution:
Let,
The age of brother 5 years ago was x years
∴ The age of sister 5 years ago was 5x years

The present age of brother is x + 5 years
The present age of sisters is 5x + 5 years

ATQ,
x + 5 + 5x + 5 = 34
⇒ 6x + 10 = 34
⇒ 6x = 24
∴ x = 4

∴ the age of her brother after 6 years will be 4 + 5 + 6 = 15 years

৪,৮৮৪.

ক
8
খ
16
গ
36
ঘ
9
ঙ
None

ব্যাখ্যা

Question:

Solution:

৪,৮৮৫.

There are 5 doors to a cinema hall. In how many ways can a person enter the hall through a door and leave the hall by a different door?

ক
15
খ
18
গ
20
ঘ
25

ব্যাখ্যা

Question: There are 5 doors to a cinema hall. In how many ways can a person enter the hall through a door and leave the hall by a different door?

Solution:
প্রবেশের সময় যে কোন দরজা বাছাই করার উপায় = ⁵C₁
= 5

বের হবার সময় অন্য ৪ টি দরজা থেকে একটি বাছাই করার উপায় = ⁴C₁
= 4

∴ মোট উপায় = 5 × 4
= 20

৪,৮৮৬.

A company offers a bonus to employees who complete certain training modules. The probability that Emma will complete the "Leadership Skills" module is 0.8, and the probability that John will complete the "Time Management" module is 0.5. What is the probability that both Emma and John will complete their respective modules?

ক
0.4
খ
0.75
গ
0.3
ঘ
0.6

ব্যাখ্যা

Question: A company offers a bonus to employees who complete certain training modules. The probability that Emma will complete the "Leadership Skills" module is 0.8, and the probability that John will complete the "Time Management" module is 0.5. What is the probability that both Emma and John will complete their respective modules?

Solution:
Let,
Probability that Emma completes the "Leadership Skills" module P(E) = 0.8
P(J) = Probability that John completes the "Time Management" module P(J) = 0.5

Since the events are independent, the probability that both Emma and John will complete their respective modules,
P (E ∩ J) = P(E) × P (J)
= 0.8 × 0.5
= 0.4

৪,৮৮৭.

What percent is 3% of 5%?

ক
15%
খ
30%
গ
50%
ঘ
60%

ব্যাখ্যা

Question: What percent is 3% of 5%?

Solution:
3% = 3/100
5% = 5/100 = 1/20

percentage = {(3/100)/(1/20)} × 100%
= 60%

৪,৮৮৮.

cos(θ + 25°) = √3/2, then the value of θ?

ক
15°
খ
5°
গ
10°
ঘ
20°

ব্যাখ্যা

Question: cos(θ + 25°) = √3/2, then the value of θ?

Solution:
Given that,
cos(θ + 25°) = √3/2
⇒ cos(θ + 25°) = cos30°
⇒ θ + 25° = 30°
⇒ θ = 30° - 25°
∴ θ = 5°

৪,৮৮৯.

What is the length of a train that takes 12 seconds to pass a pole when it runs at a speed of 24 km/h?

ক
75 m
খ
80 m
গ
85 m
ঘ
95 m

ব্যাখ্যা

Question: What is the length of a train that takes 12 seconds to pass a pole when it runs at a speed of 24 km/h?

Solution:
speed = 24 km/h
= (24 × 5/18) m/s
= 20/3 m/s

∴ The length of the train = (12 × 20/3) m = 80 m

৪,৮৯০.

The number of students in each section of a school is 24, After admitting new students, three new sections were started. Now, the total number of sections is 16 and there are 21 students in each section. The number of new students admitted is-

ক
ক) 24
খ
খ) 14
গ
গ) 48
ঘ
ঘ) 114
ঙ
ঙ) None of these

ব্যাখ্যা

২৪ ছাত্র নিয়ে section ১৬ - ৩ = ১৩ টি।
তাহলে ১৩ টি সেকশনের ছাত্র ১৩×২৪ = ৩১২ জন।
নতুন ছাত্র ভর্তি হওয়ার পর ২১ জন করে ১৬ সেকশনে মোট ছাত্র ১৬×২১ = ৩৩৬ জন।
∴ নতুন ভর্তি হয়েছে ৩৩৬ - ৩১২ = ২৪ জন।

৪,৮৯১.

A fair coin is tossed 5 times. What is the probability of getting exactly 3 heads?

ক
5/16
খ
4/15
গ
3/8
ঘ
1/4

ব্যাখ্যা

Question: A fair coin is tossed 5 times. What is the probability of getting exactly 3 heads?

Solution:

For 5 tosses, the total number of possible outcomes is = 2⁵ = 32

The number of ways to choose 3 Heads out of 5 tosses is = ⁵C₃ = 10

∴ Probability of getting exactly 3 heads = 10/32 = 5/16

৪,৮৯২.

Every 3 minutes, 4 litres of water are poured into a 2,000 litre tank. After 2 hours, what percent of the tank is full?

ক
ক) 0.4%
খ
খ) 4%
গ
গ) 8%
ঘ
ঘ) 12%

ব্যাখ্যা

In 3 minutes, 4 liters is poured
In, 120 minutes = (120×4)/3 = 160 liters
So, percentage filled = (160×100)/2000 = 8%

৪,৮৯৩.

For all numbers x and y, x # y = xy + x. What is the value of 5 # 4?

ক
ক) 9
খ
খ) 24
গ
গ) 25
ঘ
ঘ) 36

ব্যাখ্যা

Question: For all numbers x and y, x # y = xy + x. What is the value of 5 # 4?

Solution:
Given,
x # y = xy + x.
∴ 5 # 4 = (5 × 4) + 5= 25

৪,৮৯৪.

If x^-7/2= 1/128 then the value of x is:

ক
ক) 1
খ
খ) 2
গ
গ) 3
ঘ
ঘ) 4

ব্যাখ্যা

Question: If x^-7/2= 1/128 then the value of x is:

Solution:
x-7/2 = 1/128
1/x^7/2 = 1/2⁷
x^7/2 = 2⁷
(x^7/2)^1/2 = (2⁷)^1/2(x^1/2)^7/2 =2^7/2
x^1/2 = 2
(x^1/2)² = 2²
x = 4

৪,৮৯৫.

M did a piece of work in 5 days. That piece of work was done by N in 9 days. If M and N worked together, they got total wages of Tk. 4200. Find the share of N.

ক
Tk. 1500
খ
Tk. 2000
গ
Tk. 1000
ঘ
Tk. 1200
ঙ
None of these

ব্যাখ্যা

Question: M did a piece of work in 5 days. That piece of work was done by N in 9 days. If M and N worked together, they got total wages of Tk. 4200. Find the share of N.

Solution:
M's 1 day's work 1/5
N's 1 day's work 1/9

M : N
Time = 5 : 9
Efficiency = 9 : 5

(Time and efficiency are inversely proportional)
N gets = 4200 × (5/14)
= 1500

Thus, N gets the wages of Tk. 1500.

৪,৮৯৬.

If an inspector rejects 0.08% of a product as defective, how many units of the product will he examine in order to reject 2?

ক
ক) 500
খ
খ) 1500
গ
গ) 2000
ঘ
ঘ) 2500

ব্যাখ্যা

Question: If an inspector rejects 0.08% of a product as defective, how many units of the product will he examine in order to reject 2?

Solution:
ধরি,
ক পরিমান পণ্য পরীক্ষা করতে হবে।

তাহলে,
ক এর ০.০৮% = ২
ক = (২ × ১০০)/০.০৮
= ১০০/০.০৪
= ১০০০০/৪
= ২৫০০

৪,৮৯৭.

ক
180
খ
294
গ
315
ঘ
322

ব্যাখ্যা

Question:

Solution:

৪,৮৯৮.

The angle of elevation of a ladder leaning against a wall is 60º and the foot of the ladder is 4.6 m away from the wall. The length of the ladder is:

ক
9.2 m
খ
9.3 m
গ
9.4 m
ঘ
9.5 m

ব্যাখ্যা

Let,
AB be the wall and BC be the ladder.
Then, ∠ACB = 60° = AC = 4.6m
AC/BC = cos60° = 1/2
⇒ BC = 2 × AC = 2 × 4.6 = 9.2m

৪,৮৯৯.

P, Q and R are three towns on a river which flows uniformly. Q is equidistant from P and R. I row from P to Q and back in 10 hours and I can row from P to R in 4 hours. Compare the speed of my boat in still water with that of the river.

ক
ক) 4 : 3
খ
খ) 5 : 3
গ
গ) 6 : 5
ঘ
ঘ) 7 : 3

ব্যাখ্যা

Let PQ = Qr = x km
Let speed downstream = a km/hr.
and speed upstream = b km/hr.

Then,
x/a + x/b = 10
x = 10ab/(a + b) .........(i)

And,
2x/a = 4
x = 4a/2
x = 2a .............(ii)

From (i) and (ii) we have:
2a = 10ab/(a + b)
5b = a + b
a = 4b

Required ratio = Speed in the water/Speed of river
= {1/2(a + b)}/{(1/2) (a - b)}
= (a + b)/(a - b)
= (4b + b)/(4b - b)
= 5b/3b
= 5/3

৪,৯০০.

Pipes A and B can fill a tank in 8 and 10 hours respectively. Pipe C can empty it in 20 hours. If all the three pipes are opened together, then the tank will be filled in -

ক
ক) 20/3 hours
খ
খ) 20/7 hours
গ
গ) 40/3 hours
ঘ
ঘ) 40/7 hours

ব্যাখ্যা

Question: Pipes A and B can fill a tank in 8 and 10 hours respectively. Pipe C can empty it in 20 hours. If all the three pipes are opened together, then the tank will be filled in -

Solution:
in one hour, A can fill = 1/8
B can fill = 1/10
C can reduce = 1/20

so, in one-hour, total fill up = 1/8 + 1/10 - 1/20
= 7/40

Hence, it will take 40/7 hours to fill the tank if all three pipes are opened together.