Bank Math

PrepBank · পাতা ২৬ / ১৬১ · ২,৫০১–২,৬০০ / ১৬,১২৪

২,৫০১.

In how many ways can 5 people be seated around a circular table?

ক
20
খ
24
গ
60
ঘ
120

ব্যাখ্যা

Question: In how many ways can 5 people be seated around a circular table?

Solution:
For n people around a circle, the number of distinct arrangements is = (n -1)!
So, for 5 people = (5 − 1)! = 4! = 24

২,৫০২.

Find the least number which will leaves remainder 5 when divided by 8, 12, 16 and 20.

ক
230
খ
245
গ
240
ঘ
235

ব্যাখ্যা

Question: Find the least number which will leaves remainder 5 when divided by 8, 12, 16 and 20.

Solution:
We have to find the Least number, therefore we find out the LCM of 8, 12, 16 and 20.
8 = 2 × 2 × 2;
12 = 2 × 2 × 3;
16 = 2 × 2 × 2 × 2;
20 = 2 × 2 × 5;
LCM = 2 × 2 × 2 × 2 × 3 × 5 = 240;
This is the least number which is exactly divisible by 8, 12, 16 and 20
Thus,
Required number which leaves remainder 5 is,
240 + 5 = 245

২,৫০৩.

Out of six men and seven women, a five-member committee is to be chosen. What’s the likelihood that it will have exactly two men and three women?

ক
75/429
খ
177/429
গ
49/327
ঘ
175/429
ঙ
None of the above

ব্যাখ্যা

Question: Out of six men and seven women, a five-member committee is to be chosen. What’s the likelihood that it will have exactly two men and three women?
(ছয়জন পুরুষ ও সাতজন নারী থেকে পাঁচ সদস্যের একটি কমিটি গঠন করতে হবে। এর মধ্যে ঠিক দুইজন পুরুষ ও তিনজন নারী থাকার সম্ভাবনা কত?)

Solution:
Total member = 6 + 7 = 13
2 men can be selected out of 6 men in 6C2 ways
3 women can be selected out of 7 women in 7C3 ways
Required number of ways = ⁶C₂ × ⁷C₃ = 15 × 35 = 525

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

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

২,৫০৪.

If (p/q) + (q/p) = 6 the value of (p³/q³) + (q³/p³) is:

ক
172
খ
166
গ
188
ঘ
198

ব্যাখ্যা

Question: If (p/q) + (q/p) = 6 the value of (p³/q³) + (q³/p³) is:

Solution:
Here, (p/q) + (q/p) = 6

Given that = (p³/q³) + (q³/p³)
= (p/q)³ + (q/p)³
= {(p/q) + (q/p)}³ - 3 . p/q . q/p {(p/q) + (q/p)}
= 6³ - 3 . 6
= 216 - 18
= 198

২,৫০৫.

To gain 10% on selling sample of milk at the cost price of pure milk, the quantity of water to be mixed with 100 kg of pure milk is -

ক
2.5 kg
খ
5 kg
গ
7.5 kg
ঘ
10 kg

ব্যাখ্যা

Question: To gain 10% on selling sample of milk at the cost price of pure milk, the quantity of water to be mixed with 100 kg of pure milk is -

Solution:
Let the quantity of water mixed be x kg.
Let,
CP of 1 kg of pure milk = Tk. 1
CP of 100 kg of pure milk = Tk. 100

Hence,
% gain = (x/100) × 100
⇒ 10 = x

২,৫০৬.

If the compound interest in the third year at 8% per annum on a certain sum is Tk 3600, then what is the difference between the compound interest in the 4th and 5th year?

ক
ক) 210.05
খ
খ) 315.07
গ
গ) 421.07
ঘ
ঘ) 375.07
ঙ
ঙ) 311.04

ব্যাখ্যা

Compound interest for the 4th year = 3600 + (3600 × 8 × 1)/100.
= 3600 + 288.
= Tk. 3888.

Compound interest for the 5th year = 3888 + (3888 × 8 × 1)/100.
= 3888 + 311.04.
= Tk. 4199.04.

Difference between the compound interest in the 4th and 5th year = 4199.04 - 3888.
= Tk. 311.04.

২,৫০৭.

A grocer buys some eggs at tk. 3 each. He finds 12 of them are broken, but he sells others at tk.4 each and makes a profit of tk. 96. How many eggs did he buy?

ক
ক) 140
খ
খ) 142
গ
গ) 144
ঘ
ঘ) 150

ব্যাখ্যা

ধরি, মুদি ব্যবসায়ী x টি ডিম কিনেছিল।
প্রশ্নমতে, 4 (x - 12) - 3x = 96
বা, 4x - 48 - 3x = 96
বা, x = 96 + 48
বা, x = 144

২,৫০৮.

A bag contains 3 red balls and 2 blue balls. If two balls are drawn without replacement, what is the probability that both are red?

ক
3/10
খ
3/5
গ
6/25
ঘ
9/25

ব্যাখ্যা

Question: A bag contains 3 red balls and 2 blue balls. If two balls are drawn without replacement, what is the probability that both are red?

Solution:
Total balls = 3 red + 2 blue = 5 balls.
Probability that the first ball is red = 3/5

After removing 1 red ball, we have:

Remaining red balls = 2

Total remaining balls = 4

So, the probability that the second ball is red = 2/4 = 1/2

∴ Total probability (both red) = 3/5 × 1/2
= 3/10

২,৫০৯.

The polynomial equation x(x + 3) + 8 = (x + 2)(x - 2) is-

ক
quadratic equation
খ
cubic equation
গ
linear equation
ঘ
bi-quadratic equation

ব্যাখ্যা

Question: The polynomial equation x(x + 3) + 8 = (x + 2)(x - 2) is-

Solution:
We have
x(x + 1) + 8 = (x + 2)(x - 2)
⇒ x² + 3x + 8 = x² - 4
⇒ x² + 3x + 8 - x² + 4 = 0
⇒ 3x + 12 = 0 which is a linear equation.

২,৫১০.

60% of a smaller number is 6 less than 50% of a larger number. The larger number is 60 greater than the smaller one. The sum of these two number is-

ক
520
খ
640
গ
500
ঘ
540

ব্যাখ্যা

Question: 60% of a smaller number is 6 less than 50% of a larger number. The larger number is 60 greater than the smaller one. The sum of these two number is-

Solution:
Let the smaller number be x and larger number be y
According to the question,
60% of x + 6 = 50% of y
⇒ (3x/5) + 6 = y/2
⇒ 6x + 60 = 5y
⇒ 6x - 5y = - 60 ................ (1)
and
y - x = 60 ----------- (2)

By using (1) and (2), we get
x = 240
y = 300

∴ The sum of these two number =(300 + 240) = 540

২,৫১১.

Rasel obtained an amount of Tk. 8376 as simple interest on a certain amount at 8 p.c.p.a. after 6 years. What is the amount invested by Rasel?

ক
16450 Tk.
খ
18470 Tk.
গ
17450 Tk.
ঘ
19472 Tk.
ঙ
None

ব্যাখ্যা

Question: Rasel obtained an amount of Tk. 8376 as simple interest on a certain amount at 8 p.c.p.a. after 6 years. What is the amount invested by Rasel?

Solution:
Here,
Simple interest, I = Tk. 8376
Rate of interest, r = 8%
Time, n = 6 years

We know,
I = Pn(r/100)
Or, P = 100I/nr
= (8376 × 100)/(6 × 8)
∴ P = 17450 Tk.

২,৫১২.

The slant height of a right circular cone is 10 m and its height is 8 m. Find the area of its curved surface.

ক
72π m²
খ
90π m²
গ
60π m²
ঘ
66π m²

ব্যাখ্যা

Question: The slant height of a right circular cone is 10 m and its height is 8 m. Find the area of its curved surface.

Solution:
Here, l = 10m
h = 8m

So, r = √(l² - h²)
= √(10² - 8²)
=√(100 - 64)
= √36
= 6

∴Curved surface area = πrl
= (π × 6 × 10) m² = 60π m²

২,৫১৩.

Rahim and Shafiq are standing at two ends of a room with a width of 30 m. They start walking towards each other along the width of the room with a speed of 2 m/s and 1 m/s respectively. Find the total distance travelled by Rahim when he meets Shafiq for the third time.

ক
ক) 110 m
খ
খ) 112 m
গ
গ) 120 m
ঘ
ঘ) 100 m

ব্যাখ্যা

When Rahim meets Shafiq for the third time,
they together would have covered a Distance of 5d, i.e 5 × 30m = 150 m.

The ratio of Speed of Rahim and Shafiq = 2 : 1,
so the total distance traveled by them will also be in the ratio 2 : 1
as the Time is taken is constant.

So the Distance traveled by Rahim will be (2/3) × 150= 100 m.

২,৫১৪.

Which of the following equations represents a conic?

ক
x - 5y = 8
খ
x² + 5x + 6 = 0
গ
x² + x= 0
ঘ
x² + y = 0

ব্যাখ্যা

x² + y = 0
or, x² = - y which is the shape of x² = 4ay
x² = - y is the equation of conic.

২,৫১৫.

Tickets numbered 1 to 24 are mixed up and then a ticket is drawn at random. What is the probability that the ticket drawn has a number which is a multiple of 3 or 4?

ক
11/24
খ
1/2
গ
5/12
ঘ
3/8

ব্যাখ্যা

Question: Tickets numbered 1 to 24 are mixed up and then a ticket is drawn at random. What is the probability that the ticket drawn has a number which is a multiple of 3 or 4?

Solution:
Here, S = {1, 2, 3, 4, ...., 23, 24}
Let E = event of getting a multiple of 3 or 4
= {3, 4, 6, 8, 9, 12, 15, 16, 18, 20, 21, 24}

∴P(E) = n(E)/n(S)
= 12/24
= 1/2

২,৫১৬.

Salim sells 900 shares via his broker who charges a flat rate of commission of Tk. 20 on all transmission of less than Tk. 1600. His bank account is credited with Tk. 340 from the share sale. What price were his shares sold at?

ক
ক) 12 cents
খ
খ) $7 cents
গ
গ) 60 cents
ঘ
ঘ) 40 cents

ব্যাখ্যা

সেলিমের একাউন্টে ৩৪০ টাকা জমা হয়েছে ২০ টাকা কর্তনের পরে৷
অন্যথায় তার একাউন্টে থাকতো ৩৬০ টাকা।
সে ৯০০ শেয়ার বিক্রি করে ৩৬০ টাকায়।
তাহলে, তার শেয়ার প্রতি বিক্রয়মূল্য ৩৬০/৯০০ = ০.৪ বা ৪০ সেন্টস

২,৫১৭.

The perimeter of a rectangle is 400 meters. The breadth is 3/7 part of the length. What is the length?

ক
148 m
খ
126 m
গ
160 m
ঘ
140 m
ঙ
None of the above

ব্যাখ্যা

Question: The perimeter of a rectangle is 400 meters. The breadth is 3/7 part of the length. What is the length?

Solution:
Let
Breadth of rectangle = 3p
Length of rectangle = 7p

Now,
2(3p + 7p) = 400
⇒ 2 × 10p = 400
⇒ 20p = 400
⇒ p = 20

∴ Length of rectangle = 7 × 20 = 140 m

২,৫১৮.

A man on tour travels first 60 km at 20 km/hr and the next 60 km at 30 km/hr. The average speed for the first 120 km of the tour is :

ক
20 km/hr
খ
24 km/hr
গ
25 km/hr
ঘ
26 km/hr

ব্যাখ্যা

Question: A man on tour travels first 60 km at 20 km/hr and the next 60 km at 30 km/hr. The average speed for the first 120 km of the tour is :

সমাধান:
প্রথম অংশের জন্য সময় = দূরত্ব/গতিবেগ
= 60 কিমি/20 কিমি/ঘন্টা
= 3 ঘন্টা

দ্বিতীয় অংশের জন্য সময় = দূরত্ব/গতিবেগ
= 60 কিমি/30 কিমি/ঘন্টা
= 2 ঘন্টা

মোট অতিক্রান্ত দূরত্ব = 60 কিমি + 60 কিমি = 120 কিমি
মোট সময় = 3 ঘন্টা + 2 ঘন্টা = 5 ঘন্টা

∴ গড় গতিবেগ = 120 কিমি/5 ঘন্টা
= 24 কিমি/ঘন্টা

২,৫১৯.

A sum of money is distributed equally among 10 persons, but if 2 more persons were included, each person would get Tk. 80 less. What was the total sum?

ক
Tk. 5850
খ
Tk. 3800
গ
Tk. 4750
ঘ
Tk. 6400
ঙ
Tk. 4800

ব্যাখ্যা

Question: A sum of money is distributed equally among 10 persons, but if 2 more persons were included, each person would get Tk. 80 less. What was the total sum?

Solution:
Let, the sum be Tk. x
When the sum is distributed among 10 persons, each person gets, x/10
And,
If 2 more persons were included, making it 12 persons, each person would get, x/12

ATQ,
⇒ (x/10) - (x/12) = 80
⇒ (6x - 5x)/60 = 80
⇒ x = 80 × 60
⇒ x = 4800

So,The total sum of money is Tk. 4800

২,৫২০.

A man is 24 years older than his son. In two years, his age will be twice the age of his son. The present age of his son is-

ক
ক) 20
খ
খ) 21
গ
গ) 22
ঘ
ঘ) 23
ঙ
ঙ) 25

ব্যাখ্যা

Let the son's present age be x years.
Then, man's present age = (x + 24) years
=> (x + 24) + 2 = 2(x + 2)
=> x + 26 = 2x + 4
So, x = 22

২,৫২১.

What is the sum of all 3-digit positive integers such that all the digits of each of the number is even?

ক
55,500
খ
44,400
গ
247,275
ঘ
54,400

ব্যাখ্যা

Question: What is the sum of all 3-digit positive integers such that all the digits of each of the number is even?

Solution:
Constraint: All the digits of the number are even. Therefore, the digits of each of the numbers are from the set {0, 2, 4, 6, 8}

Step 1: Compute the total possible numbers
Hundreds place: 4 possible values {2, 4, 6, 8}. Hundreds place cannot be zero.
Tens place and units place: All 5 values possible.
Therefore, total possible 3-digit positive integers such that all the digits are even = 4 × 5 × 5 = 100

Step 2: Compute the value of sum of units and tens digits
Each of the 5 digits is equally likely to appear in the units place.
Therefore 100/5 = 20 is the number of times each digit will appear in the units place.
Therefore, value of sum of digits in units place = 20 × (0 + 2 + 4 + 6 + 8 ) = 400.

For the same reason, sum of tens digits = 400
Hence, value of the sum of digits in tens place = 400 × 10 = 4000

Step 3: Compute the value of sum of hundreds digits
Each of the 4 digits is equally likely to appear in the hundreds place.
Therefore 1004
100/4 = 25 numbers will begin with each of {2, 4, 6, and 8}
Therefore sum of digits in hundreds place = 25(2 + 4 + 6 + 8) = 500
Value of the sum of digits in 100s place = 500 × 100 = 50,000

Step 4: Compute the sum of such 3-digit numbers
Required Sum = Sum of units place + Sum of tens place + Sum of hundreds place
= 400 + 4000 + 50000 = 54,400

২,৫২২.

একজন ব্যক্তি ৯০০০ টাকার উপরে ৩০% হারে খাজনা ১২টি সমান কিস্তিতে পরিশোধ করে। প্রতি কিস্তির পরিমাণ কত?

ক
ক) ২২৫ টাকা
খ
খ) ২.৫০ টাকা
গ
গ) ২.২৫ টাকা
ঘ
ঘ) ২৫০ টাকা

ব্যাখ্যা

প্রশ্ন: একজন ব্যক্তি ৯০০০ টাকার উপরে ৩০% হারে খাজনা ১২টি সমান কিস্তিতে পরিশোধ করে। প্রতি কিস্তির পরিমাণ কত?

সমাধান:
১০০ টাকার খাজনা ৩০ টাকা
∴ ৯০০০ টাকার খাজনা (৩০ × ৯০০০)/১০০ টাকা
= ২৭০০ টাকা

প্রতি কিস্তির পরিমাণ (২৭০০ ÷ ১২) টাকা
= ২২৫ টাকা

২,৫২৩.

Find the HCF of 1/3 , 8/7, 9/11.

ক
1/231
খ
231
গ
1/72
ঘ
72
ঙ
None of these

ব্যাখ্যা

Question: Find the HCF of 1/3 , 8/7, 9/11.

Solution:
HCF of given Numbers : HCF(1, 8, 9)/LCM(3, 7, 11) = 1/231.

২,৫২৪.

A man can reach a certain place in 30 hours .If the reduces his speed by 1/15th, he goes 10 km less in that time. Find his speed.

ক
ক) 10 km/hr.
খ
খ) 8 km/hr.
গ
গ) 6 km/hr.
ঘ
ঘ) 5 km/hr

ব্যাখ্যা

Let’s the speed of man = x km/hr.

According to the question
x × 30 - x × (14/15) × 30 = 10
⇒ 30x - 28x = 10
⇒ 2x = 10
⇒x = 10/2
∴ x = 5

২,৫২৫.

An aeroplane covers a certain distance at a speed of 240 kmph in 5 hours. To cover the same distance in 100 minutes, it must travel at a speed of:

ক
300 kmph
খ
360 kmph
গ
600 kmph
ঘ
720 kmph

ব্যাখ্যা

Question: An aeroplane covers a certain distance at a speed of 240 kmph in 5 hours. To cover the same distance in 100 minutes, it must travel at a speed of:

Solution:
Total Distance = (240 × 5) = 1200 km.
Time = 100 minutes = 100/60 hr
= 5/3 hr

We know that,
Speed = Distance/Time
∴ Required speed = 1200/(5/3) km/hr
= 720 km/hr.

২,৫২৬.

4 men and 6 women can complete a work in 8 days, while 3 men and 7 women can complete it in 10 Days. In how many days will 10 women complete it?

ক
ক) 10 days
খ
খ) 20 days
গ
গ) 30 days
ঘ
ঘ) 40 days
ঙ
ঙ) 50 days

ব্যাখ্যা

(4M + 6W) × 8 = (3M + 7W) × 10
=> M / W = 11 : 1
Now , (4 × 11 + 6 × 1) × 8 = 10 × 1 × T
=> T = 40 days

২,৫২৭.

A forester wants to plant 66 coconut trees, 88 date trees and 110 plum trees in equal sized rows (in terms of number of tress). Also he wants to make distinct rows of trees (i.e only one type of trees in one row). What is the minimum number of rows required?

ক
ক) 12
খ
খ) 22
গ
গ) 11
ঘ
ঘ) 24
ঙ
ঙ) None

ব্যাখ্যা

Question: A forester wants to plant 66 coconut trees, 88 date trees and 110 plum trees in equal sized rows (in terms of number of tress). Also he wants to make distinct rows of trees (i.e only one type of trees in one row). What is the minimum number of rows required?

Solution:
এখানে
66, 88 এবং 110 এর গ.সা.গু = 22

নারিকেল গাছের সারি লাগবে = 66/22 = 3টি
খেজুর গাছের সারি লাগবে = 88/22 = 4টি
পাম গাছের সারি লাগবে = 110/22 = 5টি

মোট সারি লাগবে = 3 + 4 + 5 = 12 টি

২,৫২৮.

A sales person earns a commission of 5% on all sales between Tk. 2000 and Tk. 6000 and 8% on all sales over Tk. 6000. What is his total commission in a week in which his sales total Tk. 10,000?

ক
500
খ
540
গ
620
ঘ
720
ঙ
800

ব্যাখ্যা

Commission for sale upto 6000 = 6000 × 5% = 300
Commission for sales over 6000 = 4000 × 8% = 320
Total commission = 620.

২,৫২৯.

A shopkeeper has sufficient money to buy 50 books. On reduction in the price of each book by Tk. 4, he could buy 10 books more. How much money does he have?

ক
Tk. 800
খ
Tk. 1000
গ
Tk. 1200
ঘ
Tk. 1500

ব্যাখ্যা

Question: A shopkeeper has sufficient money to buy 50 books. On reduction in the price of each book by Tk. 4, he could buy 10 books more. How much money does he have?

Solution:
১টি বইয়ে দাম কমে ৪ টাকা
∴ ৫০টি বইয়ে দাম কমে (৫০ × ৪) টাকা
= ২০০ টাকা

সে মোট বই কিনে (৫০ + ১০) টি
= ৬০টি

১০টি বইয়ের দাম ২০০ টাকা
∴ ৬০টি বইয়ের দাম (২০০ × ৬০)/১০ টাকা
= ১২০০ টাকা

∴ তার কাছে ১২০০ টাকা আছে।

২,৫৩০.

Two, trains, one from Sylhet to Dhaka and the other from Dhaka to Sylhet, start simultaneously. After they meet, the trains reach their destinations after 16 hours and 25 hours respectively. The ratio of their speeds is-

ক
4 : 5
খ
25 : 16
গ
5 : 4
ঘ
16 : 25

ব্যাখ্যা

Question: Two, trains, one from Sylhet to Dhaka and the other from Dhaka to Sylhet, start simultaneously. After they meet, the trains reach their destinations after 16 hours and 25 hours respectively. The ratio of their speeds is-

Solution:

let, they meet at X, and Train A and B have the speed of p and y respectively.

in the time of t,
A covers = pt distance
B covers = qt distance

after meeting at X point,
A covers the rest part in 16 hours and B in 25 hours.
so,
distance covered by A is = 16p
distance covered by B is = 25q

∴ pt = 25q......(i)
and, qt = 16p.......(ii)

from equations (i) and (ii) we we get,
p/q = 25q/16p
p²/q² = 25/16
p/q = 5/4
p : q = 5 : 4

২,৫৩১.

P can build a wall in 24 days and Q can do it in 20 days. With help of R, they completed the work in 6 days. Find in how many days R alone can do the work.

ক
56/3 days
খ
44/5 days
গ
40/3 days
ঘ
29/4 days

ব্যাখ্যা

Question: P can build a wall in 24 days and Q can do it in 20 days. With help of R, they completed the work in 6 days. Find in how many days R alone can do the work.

Solution:
P can do 1/24 part per day
Q can do 1/20 part per day
R can do per day = (1/6) - {(1/24) + (1/20)}
= (1/6) - (11/120)
= (20/120) - (11/120)
= 3/40
Time taken by R alone = 40/3 days

২,৫৩২.

The sum of squares of two numbers is 48 and the square of their difference is 16. The product of the two number is:

ক
ক) 16
খ
খ) 17
গ
গ) 18
ঘ
ঘ) 32

ব্যাখ্যা

ধরি,
সংখ্যা দুটি যথাক্রমে a ও b

a² + b²= 48............ (1)
(a - b)² = 16............ (2)
এখন
(a - b)² = a² + b² - 2ab
16 = 48 - 2ab
2ab = 48 - 16
2ab = 32
ab = 16

২,৫৩৩.

Three numbers are in ratio 1 : 2 : 3 and HCF is 12. The largest number is:

ক
12
খ
24
গ
36
ঘ
40

ব্যাখ্যা

Question: Three numbers are in ratio 1 : 2 : 3 and HCF is 12. The largest number is:

Solution:
let, the numbers are x, 2x, 3x
HCF is x
x = 12

largest number = 12 × 3
= 36

২,৫৩৪.

15 buckets of water fill a tank when the capacity of each bucket is 16 liters. How many buckets will be needed to fill the same tank, if the capacity of each bucket is 12 liters?

ক
20
খ
18
গ
25
ঘ
12

ব্যাখ্যা

Question: 15 buckets of water fill a tank when the capacity of each bucket is 16 liters. How many buckets will be needed to fill the same tank, if the capacity of each bucket is 12 liters?

Solution:
total capacity of the tank is = (15 × 16) = 240 liters.

total buckets of 12 liters = 240/12 = 20 buckets

২,৫৩৫.

Find the minimum number of straight lines required to make the given figure.

ক
15
খ
16
গ
17
ঘ
18
ঙ
None

ব্যাখ্যা

Question: Find the minimum number of straight lines required to make the given figure.

Solution:

The horizontal lines are = IK, AB, HG and DC= 4 in number
The vertical lines are = AD, EH, JM, FG and BC= 5 in number
The slanting lines are IE, JE, JF, KF, DE, DH, FC and GC = 8 in number.

Thus, there are 4 + 5 + 8 = 17 straight lines in the figure.

২,৫৩৬.

The greatest four digits number which is divisible by 15, 25, 40 and 75 is -

ক
ক) 9600
খ
খ) 9700
গ
গ) 9800
ঘ
ঘ) 9900

ব্যাখ্যা

Question: The greatest four digits number which is divisible by 15, 25, 40 and 75 is -

Solution:
৪ অংকের বৃহত্তম সংখ্যা = ৯৯৯৯
১৫ = ৩ × ৫
২৫ = ৫ × ৫
৪০ = ২ × ২ × ২ × ৫
৭৫ = ৩ × ৫ × ৫

১৫, ২৫, ৪০, ৭৫ এর ল.সা.গু = ২ × ২ × ২ × ৩ × ৫ × ৫
= ৬০০
৯৯৯৯ কে ৬০০ দিয়ে ভাগ করলে ভাগশেষ থাকে ৩৯৯।

∴ সংখ্যাটি = ৯৯৯৯ - ৩৯৯ = ৯৬০০

২,৫৩৭.

If p and q are the roots of the equation 2x² − 9x + 7 = 0, then what is the value of (1/p) + (1/q)?

ক
4
খ
1
গ
5/7
ঘ
9/7

ব্যাখ্যা

Question: If p and q are the roots of the equation 2x² − 9x + 7 = 0, then what is the value of (1/p) + (1/q)?

Solution:
Given equation:
2x²− 9x + 7 = 0
⇒ 2x² − 7x − 2x + 7 = 0
⇒ x(2x − 7) − 1(2x − 7) = 0
⇒ (x − 1)(2x − 7) = 0

So the roots are:
x = 1 = p
x = 7/2 = q

Now,
1/p + 1/q
= 1/1 + 1/(7/2)
= 1 + 2/7
= 9/7

২,৫৩৮.

In an election 4% of votes cast are invalid. A candidate gets 55% of casted votes and wins the election by 4800 votes. Find the total number of votes casted.

ক
50000
খ
45000
গ
42000
ঘ
60000

ব্যাখ্যা

Question: In an election 4% of votes cast are invalid. A candidate gets 55% of casted votes and wins the election by 4800 votes. Find the total number of votes casted.

Solution:
বিজয়ী প্রার্থী ভোট পায়= 55%
বৈধ ভোট = (100 - 4)% = 96%

বিজয়ী প্রার্থী বৈধ ভোটের শতকরা পায় = (55 × 96)/100
= 52.8%

পরাজিত প্রার্থী ভোট পায় = (96 - 52.8)%
= 43.2%

বিজয়ী প্রার্থী ও পরাজিত প্রার্থীর ভোটের পার্থক্য = (52.8 - 43.2)%
= 9.6%

প্রশ্নমতে
9.6% = 4800
1% = 4800/9.6
100% = (4800 × 100)/9.6
= 50000

২,৫৩৯.

A tap can fill a tank in 4 hours. After half the tank is filled, three more similar taps are opened. What is the total time taken to fill the tank completely?

ক
ক) 2 h 30 m
খ
খ) 3 h 20 m
গ
গ) 3 h 45 m
ঘ
ঘ) 4 h 30 m

ব্যাখ্যা

A tap can fill a tank in 4 hours.
Therefore the tap can fill half the tank in 2 hours.

Remaining = 1/2

After half the tank is filled, three more similar taps are opened.
Hence, the total number of taps becomes 4.

Part filled by one tap in 1 hour = 1/4
Part filled by four taps in 1 hour = 4 × (1/4) = 1
i.e., 4 taps can fill the remaining half in 30 minutes.

Total time taken
= 2 hour + 30 minute = 2 hour 30 minutes.

২,৫৪০.

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

ক
0.4%
খ
4%
গ
8%
ঘ
12%

ব্যাখ্যা

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

Solution:
In 3 minutes, 4 liters is poured
In, 120 minutes = (120 × 4)/3 = 160 liters

So, percentage filled = (160 × 100)/2000
= 8%

২,৫৪১.

A tank is 30% full with water. If 18 liters of water is added the tank become 3/4 full. What is the capacity of the tank?

ক
ক) 45 Liters
খ
খ) 40 Liters
গ
গ) 42 Liters
ঘ
ঘ) 35 Liters

ব্যাখ্যা

Question: A tank is 30% full with water. If 18 liters of water is added the tank become 3/4 full. What is the capacity of the tank?

Solution:
Let, Capacity of tank is x Liters.

ATQ,
30% of x + 18 = (3/4) × x
⇒ (30x/100) + 18 = 3x/4
⇒ (3x/10) + 18 = 3x/4
⇒ (3x/4) - (3x/10) = 18
⇒ (15x - 6x)/20 = 18
⇒ 9x = 18 × 20
⇒ 9x = 360
∴ x = 40

∴ Capacity of tank is 40 Liters.

২,৫৪২.

What is the smallest number of soldiers that can be arranged in groups of 12, 15, 18, and 20, and also arranged to form a perfect square?

ক
616
খ
900
গ
1000
ঘ
1296

ব্যাখ্যা

Question: What is the smallest number of soldiers that can be arranged in groups of 12, 15, 18, and 20, and also arranged to form a perfect square?

Solution:
LCM of 12, 15, 18, and 20 is-
12 = 2 × 2 × 3
15 = 3 × 5
18 = 2 × 3 × 3
20 = 2 × 2 × 5

∴ LCM = 2 × 2 × 3 × 5 × 3

Since the soldiers are in the form of a solid square.
Hence, LCM must be a perfect square. To make the LCM a perfect square, we have to multiply it by 5,

∴ The required number of soldiers = 2 × 2 × 3 × 3 × 5 × 5
= 900

২,৫৪৩.

If x and y are two different real Numbers and xz = yz then the value of z is?

ক
ক) x-y
খ
খ) y/x
গ
গ) x/y
ঘ
ঘ) 0

ব্যাখ্যা

Given, xz = yz
বা, xz-yz = 0
বা, z(x-y) = 0
বা, z = 0 [As, x≠y, and they are different real numbers so, (x-y)≠0]

২,৫৪৪.

If a + b = 7 and ab = 12, then what is 1/a + 1/b?

ক
19
খ
5
গ
5/12
ঘ
7/12

ব্যাখ্যা

Question: If a + b = 7 and ab = 12, then what is 1/a + 1/b?

Solution:
1/a + 1/b
= (b + a)/ab
= (a + b)/(ab)
= 7/12

২,৫৪৫.

A square and a circle have the same perimeter. The side of the length of square is 11 cm, what is the area of the circle?

ক
225 sq. cm
খ
154 sq. cm
গ
194 sq. cm
ঘ
144 sq. cm

ব্যাখ্যা

Question: A square and a circle have the same perimeter. The side of the length of square is 11 cm, what is the area of the circle?

Solution:
Perimeter of the square = 4 × 11
= 44 cm

∴ Circumference of circle = 44 cm
⇒ 2πr = 44
⇒ r = 44/2π
⇒ r = (44 × 7)/(2 × 22)
∴ r = 7 cm

∴ Area of circle = πr²
= (22/7) × (7)²
= 154 sq. cm

২,৫৪৬.

The combined ages of 5 children, with each child 4 years older than the next, equal 70 years. How old is the oldest?

ক
18 years
খ
20 years
গ
22 years
ঘ
24 years

ব্যাখ্যা

Question: The combined ages of 5 children, with each child 4 years older than the next, equal 70 years. How old is the oldest?

Solution:
Let the ages of children be x, (x + 4), (x + 8), (x + 12) and (x + 16) years.
Then,
x + (x + 4) + (x + 8) + (x + 12) + (x + 16) = 70
5x + 40 = 70
5x = 70 - 40
⇒ 5x = 30
⇒ x = 6
∴ Age of the eldest child = x + 16 = 6 + 16 = 22 years.

২,৫৪৭.

What is the third proportional to 9 and 75?

ক
ক) 675
খ
খ) 625
গ
গ) 695
ঘ
ঘ) 745

ব্যাখ্যা

If a , b, c are in proportion then
a/b = b/c

Let the third proportional be x
9/75 = 75/x
⇒ x = 75 × 75/9
⇒ x = 625

∴ The third proportional is 625.

২,৫৪৮.

If log_m 128 = 7, then find the value of m.

ক
2
খ
3
গ
5
ঘ
7

ব্যাখ্যা

Question: If log_m 128 = 7, then find the value of m.

Solution:
log_m 128 = 7
⇒ m⁷ = 128 [log_b A = C, then b^C = A]
⇒ m⁷ = 2⁷
∴ m = 2

২,৫৪৯.

What word can be formed by arranging the letters of 'AENIPNOMU'?

ক
A name of a disease
খ
A name of an animal
গ
A name of a game
ঘ
A name of a sea

ব্যাখ্যা

Question: What word can be formed by arranging the letters of 'AENIPNOMU'?

Solution:
Given letters: A, E, N, I, P, N, O, M, U
By rearranging the letters 'AENIPNOMU', the name of a disease can be formed:
AENIPNOMU ⇒ PNEUMONIA

• PNEUMONIA is a serious respiratory disease that causes inflammation of the lungs, typically caused by bacterial or viral infection.

∴ The correct answer is — A name of a disease.

২,৫৫০.

How many different ways can the letters in the word ATTEND be arranged?

ক
260
খ
180
গ
360
ঘ
420

ব্যাখ্যা

Question: How many different ways can the letters in the word ATTEND be arranged?

Solution:
There are 6 letter in the word 'ATTEND' whereas, T comes two times.
So, required number of ways = 6!/2!
= 360

২,৫৫১.

The ratio of three numbers is 3 : 4 : 5 and the sum of their squares is 5000. The sum of the numbers is:

ক
100
খ
60
গ
120
ঘ
50

ব্যাখ্যা

Question: The ratio of three numbers is 3 : 4 : 5 and the sum of their squares is 5000. The sum of the numbers is-

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

According to the question,
(3x)² + (4x)² + (5x)² = 5000
⇒ 9x² + 16x² + 25x² = 5000
⇒ 50x² = 5000
⇒ x² = 5000/50
⇒ x² = 100
∴ x = 10

∴ The sum of the numbers = 3x + 4x + 5x
= 30 + 40 + 50
= 120

২,৫৫২.

Solve the inequality {1/(x + 3)} ≤ {1/(2x + 5)}

ক
(- 3, 2) ∪ (2, ∞)
খ
(- 5/2, 3)
গ
(- ∞, - 5/2) ∪ (3, 7)
ঘ
(- ∞, - 3) ∪ (- 5/2, - 2)
ঙ
None of these

ব্যাখ্যা

Question: Solve the inequality {1/(x + 3)} ≤ {1/(2x + 5)}

Solution:

The critical points are x = - 3, - 5/2, - 2

The solution is (- ∞, - 3) ∪ (- 5/2, - 2)

২,৫৫৩.

What percentage of numbers from 1 to 30 has 1 or 9 in the unit's digit?

ক
22%
খ
20%
গ
15%
ঘ
12%

ব্যাখ্যা

Question: What percentage of numbers from 1 to 30 has 1 or 9 in the unit's digit?

Solution:
Such numbers from 1 to 30 are 1, 9, 11, 19, 21, 29
Number of such numbers =6
Required percentage is (6/30) × 100% = 20%

২,৫৫৪.

A bus covers a distance of 2500m in just 1/12 hour. What is the speed in km/h?

ক
ক) 25km/h
খ
খ) 30km/h
গ
গ) 35km/h
ঘ
ঘ) 40km/h

ব্যাখ্যা

Question: A bus covers a distance of 2500m in just 1/12 hour. What is the speed in km/h?

Solution:
here,
distance, D = 2500m = 2.5km
time, T = 1/12 hour

We know,
D = S × T
S = D/T
= 2.5km/(1/12)hour
= 2.5 × 12 km/h
= 30 km/h

২,৫৫৫.

What will be the next number of the series?
4, 7, 13, 25, 49, 97, ..........

ক
ক) 193
খ
খ) 192
গ
গ) 195
ঘ
ঘ) 191

ব্যাখ্যা

Question: What will be the next number of the series?
4, 7, 13, 25, 49, 97, ..........

Solution:
এখানে,
4 + 3 = 7
7 + 6 = 13
13 + 12 = 25
25 + 24 = 49
49 + 48 = 97
97 + 96 = 193

২,৫৫৬.

cos(24°) + cos(5°) + cos(175°) + cos(204°) + cos(300°) =

ক
1/√2
খ
1/2
গ
1/3
ঘ
- 1/2

ব্যাখ্যা

Question: cos(24°) + cos(5°) + cos(175°) + cos(204°) + cos(300°) =

Solution:
cos(175°) = cos(180° - 5°) = - cos(5°)
cos(204°) = cos(180° + 24°) = - cos(24°)
cos(300°) = cos(360° - 60°) = cos(60°)

∴ cos(24°) + cos(5°) + cos(175°) + cos(204°) + cos(300°)
= cos(24°) + cos(5°) - cos(5°) - cos(24°) + cos(60°)
= cos(60°)
= 1/2

২,৫৫৭.

Of three numbers, the average of the first and second numbers is 12 more than the average of the second and third numbers. What is the difference between the first and third numbers?

ক
24
খ
26
গ
30
ঘ
20

ব্যাখ্যা

Question: Of three numbers, the average of the first and second numbers is 12 more than the average of the second and third numbers. What is the difference between the first and third numbers?

Solution:
Let, these numbers are x, y and z respectively.

ATQ,
{(x + y)/2} - {(y + z)/2} = 12
⇒ {(x + y) - (y + z)}/2 = 12
⇒ (x + y - y - z)/2 = 12
∴ x - z = 24

∴ The difference between the first and the third number is = 24.

২,৫৫৮.

If a + b = √13 and a - b = √5, what is the value of 8ab(a² + b²)?

ক
96
খ
64
গ
104
ঘ
144

ব্যাখ্যা

Question: If a + b = √13 and a - b = √5, what is the value of 8ab(a² + b²)?

Solution:
দেওয়া আছে,
a + b = √13
a - b = √5

আমরা জানি,
2(a² + b²) = (a + b)² + (a - b)²
4ab = (a + b)² - (a - b)²

এখন,
8ab(a² + b²) = (4ab) × 2(a² + b²)
= [(a + b)² - (a - b)²][(a + b)² + (a - b)²]
= [(√13)² - (√5)²][(√13)² + (√5)²]
= (13 - 5)(13 + 5)
= (8)(18)
= 144

২,৫৫৯.

The average weight of P, Q and R is 45 kg. If the average weight of P and Q is 40 kg and that of Q and R is 43 kg, what is the weight of Q?

ক
31
খ
32
গ
65
ঘ
67

ব্যাখ্যা

Question: The average weight of P, Q and R is 45 kg. If the average weight of P and Q is 40 kg and that of Q and R is 43 kg, what is the weight of Q?

Solution:
Let P, Q, R represent their respective weights.
Then, we have:
P + Q + R = (45 × 3) = 135.... (i)
P + Q = (40 × 2) = 80.... (ii)
Q + R = (43 × 2) = 86.... (iii)

Adding (ii) and (iii), we get:
P + 2Q + R = 166.... (iv)
Subtracting (i) from (iv), we get: Q = 31.

২,৫৬০.

A businessman has 1000 Kg of rice, part of which he sells at 8% profit and the rest at 18% profit. He gains 14% on the whole. How much Kg of rice did he sell at 18% profit?

ক
600
খ
620
গ
640
ঘ
None

ব্যাখ্যা

Question: A businessman has 1000 Kg of rice, part of which he sells at 8% profit and the rest at 18% profit. He gains 14% on the whole. How much Kg of rice did he sell at 18% profit?

Solution:
Let the quantity sold at 8% profit = x
And the quantity sold at 18% profit = y.

8x +18y =14(x + y)
⇒ 4y = 6x
∴ x : y = 2 : 3

Quantity sold at 18% profit = (3/5) × 1000
= 600

২,৫৬১.

Some tanks of capacity 720 liters are meant to be filled by three pipes of capacity of loading water at 50 liters, 25 liters, and 15 liters in 10 minutes respectively. In one day they together can fill -

ক
18 tanks
খ
20 tanks
গ
9 tanks
ঘ
25 tanks

ব্যাখ্যা

Question: Some tanks of capacity 720 liters are meant to be filled by three pipes of capacity of loading water at 50 liters, 25 liters, and 15 liters in 10 minutes respectively. In one day they together can fill -

Solution:
in 10 minutes total fill-up = (50 + 25 + 15) = 90 liters

in 1 hour = 540 liters.
in 24 hours = (540 × 24) = 12960 liters.

total tank = 12960/720 = 18

২,৫৬২.

A monkey climbs a 12 meters-high slippery pillar. In his first minute, he climbs 2 meters, and in the next minute, he slip one meter down. In this way, how much time will he take to reach the top of the pillar?

ক
10 minutes
খ
12 minutes
গ
11 minutes
ঘ
21 minutes

ব্যাখ্যা

Question: A monkey climbs a 12 meters-high slippery pillar. In his first minute, he climbs 2 meters, and in the next minute, he slip one meter down. In this way, how much time will he take to reach the top of the pillar?

Solution:
On first minute monkey climb = 2 m
On the second minute it slips = 1 m
For every two minute, it climbs 1 m
So, average speed = 1 m/2 min For 10 m,
time is taken = 20 min

For the last 2 m jump add 1 min
So time taken = 20 + 1 = 21 min

∴ Monkey takes 21 minutes to reach the top of the pole.

২,৫৬৩.

A bucket contains 64 liters of petrol. 16 liters of petrol is removed and replaced with kerosene. 16 liters of this mixture is removed and replaced with kerosene. How much kerosene (in liters) is present now?

ক
28 liters
খ
36 liters
গ
16 liters
ঘ
64 liters

ব্যাখ্যা

Question: A bucket contains 64 liters of petrol. 16 liters of petrol is removed and replaced with kerosene. 16 liters of this mixture is removed and replaced with kerosene. How much kerosene (in liters) is present now?

Solution:
বালতিতে পেট্রোল ছিল ৬৪ লিটার
১৬ লিটার পেট্রোল সরিয়ে কেরোসিন দিলে মিশ্রণে পেট্রোল থাকে (৬৪ - ১৬) = ৪৮ লিটার
মিশ্রণে কেরোসিন হয় ১৬ লিটার

∴ ১৬ লিটার পেট্রোল সরিয়ে কেরোসিন দিলে মিশ্রণে পেট্রোল ও কেরোসিনের অনুপাত হয় = ৪৮ : ১৬
= ৩ : ১

এখন,
১৬ লিটার মিশ্রণে কেরোসিন থাকে = ১৬ × (১/৪) লিটার
= ৪ লিটার

সুতরাং ১৬ লিটার মিশ্রণ তোলে নেয়ার পর কেরোসিন থাকে = ১৬ - ৪ লিটার = ১২ লিটার

আবার মিশ্রণে ১৬ লিটার কেরোসিন দিলে মোট কেরোসিন হয় = (১২ + ১৬) লিটার
= ২৮ লিটার

২,৫৬৪.

A man travels equal distances of his journey at 15, 20 and 30 km/hour respectively. What is his average speed for whole journey?

ক
ক) 10
খ
খ) 12
গ
গ) 16
ঘ
ঘ) 20

ব্যাখ্যা

Required average speed
= (3 × 15 × 20 × 30)/(15 × 20 + 20 × 30 + 30 × 15)
= 20 km/hour
----------------------------------------------------------------
Alternative way:
Distance 30 km, Speed 30 km/h, Time 1 Hour
Distance 30 km, Speed 15 km/h, Time 2 Hours
Distance 30 km, Speed 20 km/h, Time 1 hour & 30 minutes.
Total Distance 90 km, Total Time 4.5 Hours,
Average Speed 90/4.5 km/h = 20 km/h
--------------------------------------------------------------
Alternative way:
Let, equal distances travelled by the man be ’s’.
Time taken to travel first distance, s at 15 kmph = s/15
Time taken to travel second distance, s at 20 kmph = s/20
Tine taken to travel last distance, s at 30 kmph = s/30
Therefore, total time taken by the man = s(1/15 +1/20 +1/30)
Total distance travelled by the man = 3s
Hence, average speed of the man
= 3s / [ s(1/15 +1/20 +1/30) ] =3/(1/15 +1/20 +1/30)
= 20 kmph
---------------------------------------------------------------
Alternative way:
Avg speed formula= 3 ÷ (1/x +1/y +1/z)
Now we have 1/30 +1/20 +1/15
L.C.M. of 20, 30 and15 is 60.
Then (3 + 4 + 8)/60=9/60 = 3/20
From above formula, average speed = 3 ÷ 3/20=20 km/h

২,৫৬৫.

A cistern has two pipes. Both working together can fill the cistern in 12 minutes. First pipe is 10 minutes faster than the second pipe. How much time would it take to fill the cistern if only second pipe is used?

ক
30 minutes
খ
20 minutes
গ
40 minutes
ঘ
60 minutes

ব্যাখ্যা

Question: A cistern has two pipes. Both working together can fill the cistern in 12 minutes. First pipe is 10 minutes faster than the second pipe. How much time would it take to fill the cistern if only second pipe is used?

Solution:
Let the time taken by first pipe working alone be ‘t’ minutes.
Time taken by second pipe working alone = t + 10 minutes.

Part of tank filled by pipe A in one hour working alone = 1/t
Part of tank filled by pipe B in one hour working alone = 1/(t + 10)

∴ Part of tank filled by pipe A and B in one hour working together = (1/t) + (1/t+10) = (2t + 10)/[t × (t + 10)]
But we are given that it takes 12 minutes to completely fill the cistern if both pipes are working together.
∴ (2t + 10)/[t × (t + 10)] = 1/12
⇒ t × (t + 10)/(2t + 10) = 12
⇒ t² + 10t = 24t + 120
⇒ t² - 14t - 120 = 0
⇒ (t - 20) (t + 6) = 0
∴ t = 20 minutes (Time cannot be negative)

Therefore, time taken by second pipe working alone = 20 + 10 = 30 minutes

২,৫৬৬.

Nazma's age is 1/5 th of her father's age. Nazma's father's age will be twice of Moni's age after 10 years. If Moni's 8th birthday was celebrated 2 years ago,then what is Nazma's present age?

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

ব্যাখ্যা

Question: Nazma's age is 1/5 th of her father's age. Nazma's father's age will be twice of Moni's age after 10 years. If Moni's 8th birthday was celebrated 2 years ago,then what is Nazma's present age?

Solution:
মণির বর্তমান বয়স = ৮ + ২ বছর
= ১০ বছর

১০ বছর পর, মণির বয়স = ১০ + ১০ বছর
= ২০ বছর

১০ বছর পর পিতার বয়স = ২০ × ২ বছর
= ৪০ বছর

বর্তমানে পিতার বয়স = ৪০ - ১০ বছর
= ৩০ বছর

নাজমার বর্তমান বয়স = ৩০/৫ বছর
= ৬ বছর

২,৫৬৭.

A stock increases in value by 20%. By what percent must the stock decrease to reach back to its former value?

ক
16.66%
খ
26%
গ
15%
ঘ
26.66%

ব্যাখ্যা

Question: A stock increases in value by 20%. By what percent must the stock decrease to reach back to its former value?

Solution:
এখানে,
20% বৃদ্ধিতে মূল্য = (100 + 20) = 120 টাকা

120 টাকায় মূল্য কমাতে হবে 20 টাকা
∴ 1 টাকায় মূল্য কমাতে হবে 20/120
∴ 100 টাকায় মূল্য কমাতে হবে (20 × 100)/120
= 16.66 টাকা

২,৫৬৮.

ক
ক) X, Z, A and B
খ
খ) X, Z, D and B
গ
গ) X, Z, C and B
ঘ
ঘ) None of these

ব্যাখ্যা

(64-66): Answer the questions on the busis of the information given below:
City High School must put together a debating team consisting of four debaters. There are candidates of equal ability: X, Y and Z who are all seniors; and A, B, C and D who are all juniors. The school requires that there should be two seniors and two juniors the team. It is also necessary that all of the debaters be able to work with one another.
i) Debaters Y and A cannot work together.
ii) Debaters Z and C cannot work together.
iii) Debaters A and B cannot work together.
64.If debater B is selected and debater Y is rejected, the team will consist of which four?

Solution:
X, Y, Z = সিনিয়র এবং সমান পারদর্শী।
A, B, C, D = জুনিয়র

৪ জনের টিমে, দুই জন সিনিয়র এবং ২ জন জুনিয়র থাকবে এবং
১) Y এবং A এক সাথে থাকতে পারবে না।
২) Z এবং C এক সাথে থাকতে পারবে না।
৩) A এবং B এক সাথে থাকতে পারবে না।

তাহলে যদি B কে সিলেক্ট করে Y কে রিজেক্ট করা হয় তাহলে ৪ জনের টিমে থাকবে =
সিনিয়রদের মধ্যে থাকবে = X, Z
জুনিয়রদের মধ্যে থাকবে = B, D

২,৫৬৯.

In a examination, a student attempted 15 questions correctly and secured 40 marks. If there were two types of questions ( 2 marks and 4 marks), how many questions of 2 marks did he attempt correctly ?

ক
ক) 5
খ
খ) 8
গ
গ) 10
ঘ
ঘ) 12

ব্যাখ্যা

Let the 2 marks questions attempted correctly =x
The 4 marks questions attempted correctly =y

Now,
x + y = 15...................(I)
2x + 4y = 40 .....................(II)

(II) - (I) × 4 ⇒
2x + 4y - 4x - 4y = 40 - 60
-2x = - 20
x = 10

২,৫৭০.

In a tourist group of 100 people, 55 speak French, 40 speak Spanish, and 20 speak none of the languages. How many of them speak just one language?

ক
15
খ
25
গ
40
ঘ
65

ব্যাখ্যা

Question: In a tourist group of 100 people, 55 speak French, 40 speak Spanish, and 20 speak none of the languages. How many of them speak just one language?

Solution:

ধরি,
উভয় ভাষায় কথা বলতে পারে = x জন
∴ শুধু ফ্রেঞ্চ বলতে পারে = (55 - x) জন
∴ শুধু স্প্যানিশ বলতে পারে = (40 - x) জন

দেওয়া আছে,
কোনো ভাষায় কথা বলে না = 20 জন

প্রশ্নমতে,
(55 - x) + x + (40 - x) = 100 - 20
95 - x = 80
x = 95 - 80
x = 15

শুধু ফ্রেঞ্চ বলতে পারে = (55 - 15) জন = 40 জন
শুধু স্প্যানিশ বলতে পারে = (40 - 15) জন = 25 জন

∴ শুধুমাত্র একটি ভাষায় (ফ্রেঞ্চ বা স্প্যানিশ) কথা বলতে পারে = (40 + 25) জন = 65 জন

২,৫৭১.

What is the unit digit in the product of 853 × 1346 × 452 × 226?

ক
4
খ
5
গ
6
ঘ
8

ব্যাখ্যা

Question: What is the unit digit in the product of 853 × 1346 × 452 × 226?

Solution:
Pick up the unit digit of each number and multiply them;

3 in 853
6 in 1346
2 in 452
6 in 226

∴ 3 × 6 × 2 × 6 = 216 (consider the unit digit in the product)

So, the unit digit in the product of 853 × 1346 × 452 × 226 is 6.

২,৫৭২.

The average age of 12 children is 15 years. If another child comes the average age comes to 13. What is the age of the new child?

ক
11 years
খ
7 years
গ
9 years
ঘ
5 years

ব্যাখ্যা

Question: The average age of 12 children is 15 years. If another child comes the average age comes to 13. What is the age of the new child?

Solution:
Sum of 12 children ages = 12 × 15 = 180
Sum of the 13 children ages = 13 × 13 = 169
So age of new children = 180 - 169
= 11

[বাস্তবিকভাবে নতুন একজনের বয়স যুক্ত হওয়ার পর বয়সের গড় কম-বেশি হতে পারে তবে ১২ জনের মোট বয়স (১৮০ বছর) অপেক্ষা ১৩ জনের মোট বয়স (১৬৯ বছর) কম হতে পারে না। এই প্রশ্নটি যেহেতু জব সল্যুশনের প্রশ্ন তাই গাণিতিক নিয়ম অনুযায়ী ১১ বছর উত্তর রাখা হয়েছে]

২,৫৭৩.

Tahir gets paid travelling expenses according to the distance he drives in his car plus a weekly sum of Tk. 21. He claims for 420 miles travelled and receives an expenses payment of Tk. 105. What is the payment rate per mile?

ক
ক) 20 cents
খ
খ) 32 cents
গ
গ) 48 cents
ঘ
ঘ) 51 cents

ব্যাখ্যা

যেহেতু তাহিরকে ২১ টাকা প্রতি সপ্তাহে বাধ্যতামূলক দিতে হয়,
সেহেতু তাহিরের পারিশ্রমিক নেয় ১০৫-২১ = ৮৪ টাকা৷
অর্থাৎ, সে ৪২০ কি.মি. এর জন্য পায় ৮৪ টাকা।
সুতরাং, প্রতি কি.মি. এর জন্য পাবে ৪২০/৮৪ = ০.২ = ২০ সেন্ট।

২,৫৭৪.

If Nita is 15 ahead in rank of Mita who ranks 13th from the last, then how many students are there in the class if Nita ranks 5th in order of merit?

ক
34
খ
32
গ
33
ঘ
31

ব্যাখ্যা

Question: If Nita is 15 ahead in rank of Mita who ranks 13th from the last, then how many students are there in the class if Nita ranks 5th in order of merit?

Solution:
Let the total number of students = n

Now,
Mita’s position from the last = 13th
∴ Mita’s position from the front = n - 13 + 1 = n - 12

Again,
Nita’s position from the front = 5th.

And, Nita is 15 ranks ahead of Mita.
∴ Mita’s position = Nita’s position + 15.

According to the problem,
n - 12 = 5 + 15
⇒ n - 12 = 20
⇒ n = 20 + 12
∴ n = 32

Therefore, the total number of students in the class = 32.

২,৫৭৫.

If θ is a positive acute angle and 4cos²θ - 1 = 0, then the value of tan(θ - 15°) is equal to?

ক
ক) 0
খ
খ) 1
গ
গ) 1/√3
ঘ
ঘ) √3

ব্যাখ্যা

Question: If θ is a positive acute angle and 4cos²θ - 1 = 0, then the value of tan(θ - 15°) is equal to?

Solution:
Given,
4cos²θ - 1 = 0
⇒ 4cos²θ = 1
⇒ cos²θ = 1/4
⇒ cosθ = 1/2
⇒ cosθ = cos60°
∴ θ = 60°

Now,
tan(θ - 15°) = tan(60° - 15°)
= tan 45°
= 1

২,৫৭৬.

A metal sphere weighing 84 kilograms is melted and recast into 4,000 identical nails. Calculate the weight of each nail in grams.

ক
0.21 grams
খ
2.4 grams
গ
21 grams
ঘ
24 grams

ব্যাখ্যা

Question: A metal sphere weighing 84 kilograms is melted and recast into 4,000 identical nails. Calculate the weight of each nail in grams.

Solution:
দেওয়া আছে,
ধাতুর বলের ওজন = 84 কেজি = 84 × 1000 = 84000 গ্রাম
পেরেকের সংখ্যা = 4000 টি

এখন,
4000 পেরেকের ওজন = 84000 গ্রাম
∴ 1 টি পেরেকের ওজন = (84000/4000) গ্রাম = 21 গ্রাম

২,৫৭৭.

A room 6.2m × 8m is to be carpeted leaving a margin of 10 cm from each wall. If the cost of the carpet is Tk. 15 per sq. meter, the cost of carpeting the room will be:

ক
Tk. 695
খ
Tk. 702
গ
Tk. 712
ঘ
Tk. 725

ব্যাখ্যা

Question: A room 6.2m × 8m is to be carpeted leaving a margin of 10 cm from each wall. If the cost of the carpet is Tk. 15 per sq. meter, the cost of carpeting the room will be:

Solution:
Area of the carpet :
= [(6.20 - 0.20) × (8 - 0.20)] m²
= (6 × 7.8) m²
= 46.8 m²

∴ Cost of carpeting :
= Tk. (46.8 × 15)
= Tk. 702

২,৫৭৮.

If PALE is coded at 2134, EARTH is coded as 41590, then how can PEARL be coded in that language?

ক
24152
খ
24123
গ
24153
ঘ
10153

ব্যাখ্যা

Question: If PALE is coded at 2134, EARTH is coded as 41590, then how can PEARL be coded in that language?

Solution:
P → 2
A → 1
L → 3
E → 4

এবং
E → 4
A → 1
R → 5
T → 9
H → 0

PEARL → 24153

২,৫৭৯.

Which one word cannot be formed from the letters of the word ''CHOREOGRAPHY"?

ক
ক) GEOGRAPHY
খ
খ) GRAPH
গ
গ) OGRE
ঘ
ঘ) PHOTOGRAPHY

ব্যাখ্যা

Question: Which one word cannot be formed from the letters of the word ''CHOREOGRAPHY"?

Solution:
CHOREOGRAPHY ওয়ার্ড টি তে T নেই তাই PHOTOGRAPHY ওয়ার্ড গঠন করা যবে না।

২,৫৮০.

The ratio of the present ages of Rahim and Karim is 9 : 10 respectively. Twenty years ago, the ratio of their ages was 4 : 5 respectively. What is the present age of Rahim?

ক
34 years
খ
36 years
গ
42 years
ঘ
24 years

ব্যাখ্যা

Question: The ratio of the present ages of Rahim and Karim is 9 : 10 respectively. Twenty years ago, the ratio of their ages was 4 : 5 respectively. What is the present age of Rahim?

Solution:
The ratio of the present ages of Rahim and Karim = 9 : 10
Let's consider the present ages of Rahim and Karim as 9x and 10x

According to the question,
(9x - 20) : (10x - 20) = 4 : 5
⇒ (9x - 20)/(10x - 20) = 4/5
⇒ 5(9x - 20) = 4(10x - 20)
⇒ 45x - 100 = 40x - 80
⇒ 45x - 40x = 100 - 80
⇒ 5x = 20
∴ x = 4

The present age of Rahim = (9 × 4) years.
= 36 years.

২,৫৮১.

A sum of money amounts to Tk. 920 in 3 years and to Tk. 1000 in five years. Find the rate percent per amount -

ক
ক) 2%
খ
খ) 11%
গ
গ) 4%
ঘ
ঘ) 5%

ব্যাখ্যা

After 5 years the sum = Tk. 1000
After 3 years the sum = Tk. 920
The interest of 2 years = Tk. (1000 - 920) = Tk. 80
The interest of 3 years, I = Tk. 80 x (3/2) = Tk. 120
The principle, P = Tk. (920 - 120) = Tk. 800
So, the rate = I/Pn = 120/(800 x 3) × 100% = 5%

২,৫৮২.

A basket contains three blue and four red balls. If three balls are drawn at random from the basket, what is the probability that all the three are either blue or red?

ক
2/5
খ
1/7
গ
3/8
ঘ
None of these

ব্যাখ্যা

Question: A basket contains three blue and four red balls. If three balls are drawn at random from the basket, what is the probability that all the three are either blue or red?

Solution:
Probability to be a Blue = ³C₃/⁷C₃ = 1/35
Probability to be a Red = ⁴C₃/⁷C₃ = 4/35

∴ Required probability = (1/35) + (4/35)
= (1 + 4)/35
= 5/35
= 1/7

২,৫৮৩.

If a^1/x = b^1/y = c^1/z and abc = 1, then find the value of x + y + z.

ক
- 1
খ
1
গ
1/2
ঘ
0

ব্যাখ্যা

Question: If a^1/x = b^1/y = c^1/z and abc = 1, then find the value of x + y + z.

Solution:
Let, a^1/x = b^1/y = c^1/z = k
a = k^x, b = k^y and c = k^z
abc = k^x × k^y × k^z = k^{(x + y + z)}Given, abc = 1
k^{(x + y + z)} = k⁰
x + y + z = 0

২,৫৮৪.

If 60% of A's income is equal to 75% of B's income, then B's income is equal to x% of A's income. The value of x is :

ক
110
খ
80
গ
60
ঘ
90

ব্যাখ্যা

Question: If 60% of A's income is equal to 75% of B's income, then B's income is equal to x% of A's income. The value of x is :

Solution:
ATQ,
60 × (A/100) = 75 × (B/100)
⇒ 60A = 75B
⇒ 4A = 5B
⇒ B = 4A/5

Again,
B = A × (x/100)
⇒ 4A/5 = A × (x/100)
⇒ 4/5 = x/100
⇒ 5x = 400
⇒ x = 400/5
∴ x = 80

২,৫৮৫.

The difference of two numbers is 20% of the larger number. If the smaller number is 20,what is the larger number?

ক
25
খ
65
গ
40
ঘ
60

ব্যাখ্যা

Let x be the larger number and y be the smaller number
Therefore x - y = 20%of x
Now put the value of the smaller number in the equation
x - 20 = (20/100) × x
⇒ x - 20 = (1/5) × x

Here 5 is in the denominator.
As we bring 5 on the left side of the equation it will be multiplied by x - 20
Now the equation will be 5 × (x - 20) = x
⇒ 5x - 100 = x

Bring 100 on the right side of = and x on the left side of =
So it will become 5x - x=100
⇒ 4x = 100
⇒ x = 100/4
⇒ x = 25
Therefore the larger number is 25.

২,৫৮৬.

If the summation of roots of two integers is √(11+8√2). What is the summation of the squares of these two integers?

ক
49
খ
55
গ
57
ঘ
63
ঙ
78

ব্যাখ্যা

Question: If the summation of roots of two integers is √(11+8√2). What is the summation of the squares of these two integers?

Solution:
Let,
The two integers are x and y.
ATQ,
√x + √y = √(11 + 8√2)
⇒ (√x + √y)² = {√(11 + 8√2)}²
⇒ x + 2√x . √y + y = 11 + 8√2
⇒ x + y + 2√(xy) = 11 + 8√2

So, x + y = 11

And,
2√(xy) = 8√2
⇒ √(xy) = 4√2
⇒ xy = 32

Now,
(x + y)² = 11²
⇒ x² + y² + 2xy = 121
⇒ x² + y² = 121 - 2 . 32
⇒ x² + y² = 57

২,৫৮৭.

Solution set of the inequality, 4x - 7 ≤ 2x + 5 is-

ক
[- ∞, 4]
খ
(- ∞, - 6)
গ
(- ∞, 6]
ঘ
None of these

ব্যাখ্যা

Question: Solution set of the inequality, 4x - 7 ≤ 2x + 5 is-

Solution:
Given that,
4x - 7 ≤ 2x + 5
⇒ 4x - 2x ≤ 7 + 5
⇒ 2x ≤ 12
⇒ x ≤ 6
∴ x ≤ 6

∴ Solution set of the inequality is (- ∞, 6]

২,৫৮৮.

A man travelled a distance of 61 km in 9 hours. He travelled partly on foot at 4 km/hr and partly on bicycle at 9 km/hr. What is the distance travelled on foot?

ক
14 km
খ
16 km
গ
18 km
ঘ
12 km
ঙ
None of these

ব্যাখ্যা

Question: A man travelled a distance of 61 km in 9 hours. He travelled partly on foot at 4 km/hr and partly on bicycle at 9 km/hr. What is the distance travelled on foot?

Solution:
Let the time in which he travelled on foot = x hr
Then the time in which he travelled on bicycle =(9 - x) hr
distance = speed × time
⇒ 4x + 9(9 - x) = 61
⇒ 4x + 81 - 9x = 61
⇒ 5x = 20
⇒ x = 4

∴ The distance travelled on foot = 4 × 4 = 16 km

২,৫৮৯.

A field has a length of 60 meters and a width of 50 meters. Two roads, each 4 meters wide, cross each other perpendicularly through the middle of the field. What is the total area of the two roads?

ক
320 sq meters
খ
424 sq meters
গ
496 sq meters
ঘ
520 sq meters
ঙ
None

ব্যাখ্যা

Question: A field has a length of 60 meters and a width of 50 meters. Two roads, each 4 meters wide, cross each other perpendicularly through the middle of the field. What is the total area of the two roads?

Solution:
দৈর্ঘ্য বরাবর রাস্তার ক্ষেত্রফল = (60 × 4) বর্গমিটার
= 240 বর্গমিটার

প্রস্থ বরাবর রাস্তার ক্ষেত্রফল = (50 - 4) × 4 বর্গমিটার
= 46 × 4 বর্গমিটার
= 184 বর্গমিটার

রাস্তা দুইটির মোট ক্ষেত্রফল = (240 + 184) বর্গমিটার
= 424 বর্গমিটার

২,৫৯০.

A 12% stock yielding 10% is quoted at:

ক
ক) 105
খ
খ) 95
গ
গ) 110
ঘ
ঘ) 120
ঙ
ঙ) 115

ব্যাখ্যা

To earn Tk. 10, money invested = Tk. 100.
To earn Tk. 12, money invested = Tk (100/10 × 12)
= Tk. 120.
Market value of Tk.100 stock = Tk. 120.

২,৫৯১.

How much time will it take for an amount of Tk. 450 to yield Tk. 90 as interest at 2% per annum of simple interest?

ক
ক) 5 years
খ
খ) 8 years
গ
গ) 10 years
ঘ
ঘ) 12 years

ব্যাখ্যা

Question: How much time will it take for an amount of Tk. 450 to yield Tk. 90 as interest at 2% per annum of simple interest?

Solution:
Given that
Principal = Tk. 450,
Rate of interest = 2% and
simple interest = Tk. 90

Time = (100 × 90)/(450 × 2)
= 10 years.

২,৫৯২.

If a³ + b³ = a² - ab + b² and a = 13/17, find the vale of b.

ক
17/13
খ
4/13
গ
1
ঘ
4/17

ব্যাখ্যা

Question: If a³ + b³ = a² - ab + b² and a = 13/17, find the vale of b.

Solution:
Given,
a³ + b³ = a² - ab + b²
⇒ (a + b)(a² - ab + b²) = (a² - ab + b²) [ a³ + b³ = (a + b)(a² - ab + b²) ]
⇒ (a + b) = 1
⇒ b = 1 - a
⇒ b = 1 - (13/17)
⇒ b = 4/17

২,৫৯৩.

The area of a circle is increased by 22 square cm if its radius is increased by 1 cm. The original radius of the circle is -

ক
6 cm
খ
5 cm
গ
4 cm
ঘ
3 cm

ব্যাখ্যা

Question: The area of a circle is increased by 22 square cm if its radius is increased by 1 cm. The original radius of the circle is -

Solution:
Let the original radius of the circle be r cm.

ATQ,
π(r + 1)² - πr² = 22
⇒ π{(r + 1)² - r²} = 22
⇒ π(r² + 2r + 1 -r²) = 22
⇒ 2r + 1 = 22/π
⇒ 2r + 1 = (22 × 7)/22
⇒ 2r + 1 = 7
⇒ 2r = 6
⇒ r = 3 cm

২,৫৯৪.

The product of the ages of Minhaz and Rahul is 180. If twice the age of Rahul is more than Minhaz's age by 2 years, then find Rahul's age.

ক
ক) 10 years
খ
খ) 18 years
গ
গ) 20 years
ঘ
ঘ) 24 years

ব্যাখ্যা

Question: The product of the ages of Minhaz and Rahul is 180. If twice the age of Rahul is more than Minhaz's age by 2 years, then find Rahul's age.

Solution:
Let Minhaz's age be x years
Then, Rahul's age = 180/x years

ATQ,
2 × (180/x = x + 2
⇒ (360/x) - x = 2
⇒ 360 - x² = 2x
⇒ x² + 2x - 360 = 0
⇒ x² + 20x - 18x - 360 = 0
⇒ x(x + 20) - 18(x + 20) = 0
⇒ (x - 18) (x + 20) = 0
∴ x = 18

So, Rahul's age is = 180/18 = 10 years

২,৫৯৫.

If measures of the angles in a triangle are in the ratio of 1 : 3 : 5, then the degrees in the largest angle:

ক
60°
খ
90°
গ
120°
ঘ
100°

ব্যাখ্যা

Question: If measures of the angles in a triangle are in the ratio of 1 : 3 : 5, then the degrees in the largest angle:
(Officer Cash 2022 অনুযায়ী)

Solution:
Given that,
The angles of a triangle are in the ratio 1 : 3 : 5
Let,
x, 3x, 5x

We know that,
Sum of angles in a triangle = 180°

Now
x + 3x +5x = 180°
9x = 180°
x = 180°/9 = 20°
∴ x = 20°

∴ Largest angle = 5x = 5 × 20 = 100°

২,৫৯৬.

করিমের বার্ষিক আয় রহিমের চেয়ে ১০% বেশি, অন্যদিকে রহিমের বার্ষিক আয় খালেকের আয়ের চেয়ে ২০% বেশি । খালেকের মাসিক আয় ২০০০ টাকা হলে ৩ জনের মোট মাসিক আয় কত?

ক
ক) ৬৮৭২ টাকা
খ
খ) ৭০৪৬ টাকা
গ
গ) ৭৭৭২ টাকা
ঘ
ঘ) ৭০৪০ টাকা

ব্যাখ্যা

প্রশ্ন: করিমের বার্ষিক আয় রহিমের চেয়ে ১০% বেশি, অন্যদিকে রহিমের বার্ষিক আয় খালেকের আয়ের চেয়ে ২০% বেশি । খালেকের মাসিক আয় ২০০০ টাকা হলে ৩ জনের মোট মাসিক আয় কত?

সমাধান:
খালেকের মাসিক আয় ২০০০ টাকা

রহিমের মাসিক আয় = ২০০০ + (২০০০ × (২০/১০০) টাকা
= ২০০০ + ৪০০ টাকা
= ২৪০০ টাকা

করিমের মাসিক আয় = ২৪০০ + (২৪০০ × (১০/১০০) টাকা
= ২৪০০ + ২৪০ টাকা
= ২৬৪০ টাকা

৩ জনের মোট মাসিক আয় (২০০০ + ২৪০০ + ২৬৪০) টাকা
= ৭০৪০ টাকা

২,৫৯৭.

The speed of a car increases by 2 km after every one hour. If the distance travelled in the first one hour was 30 km, what was the total distance travelled in 10 hours?

ক
390 km
খ
400 km
গ
405 km
ঘ
415 km

ব্যাখ্যা

Question: The speed of a car increases by 2 km after every one hour. If the distance travelled in the first one hour was 30 km, what was the total distance travelled in 10 hours?

Solution:
Total distance travelled in 10 hours =( 30 + 32 + 34 +...... upto 10 terms)
This is an A.P with first term, a = 30 ,
number of terms, n = 10
d = 2

Required distance = (10/2)[(2 × 30) + {(10 - 1) × 2}] km
= 5 (60 + 18) km
= (5 × 78) km
= 390 km

২,৫৯৮.

Two discount of 8% and 12% are equal to a single discount of:

ক
20%
খ
19.04%
গ
18.96%
ঘ
18%

ব্যাখ্যা

Question: Two discount of 8% and 12% are equal to a single discount of:

Solution:
Let,
The initial price = Tk. 100
After first 8% discount,
100 - 8% of 100 = 100 - 8 = 92

After second 12% discount,
92 - 12% of 92 = 92 - 11.04 = 80.96

∴ Single discount = 100 - 80.96 = 19.04%

২,৫৯৯.

A and B together can do a piece of work in 20 days. A having worked for 10 days, B finishes the remaining work alone in 24 days. In how many days shall B finish the whole work alone?

ক
30 days
খ
36 days
গ
42 days
ঘ
48 days

ব্যাখ্যা

Question: A and B together can do a piece of work in 20 days. A and B having worked for 10 days, B finishes the remaining work alone in 24 days. In how many days shall B finish the whole work alone?

Solution:
A and B together in 20 days can do a piece of work
A and B together in 1 days can do 1/20 part
A and B together in 10 days can do (1 × 10)/20 part
= 1/2 part

So remaining work = (1 - 1/2) part
= 1/2 part

ATQ,
B finishes 1/2 part in 24 days.
∴ B finishes 1 part in (24 × 2) days
= 48 days

২,৬০০.

Which of the following numbers cannot be the last digit of a squared number?

ক
2
খ
0
গ
4
ঘ
9

ব্যাখ্যা

Question: Which of the following numbers cannot be the last digit of a squared number?

Solution:
বর্গ সংখ্যার শেষের অংক 0, 1, 4, 5, 6 এবং 9 হতে পারে।
2 বর্গ সংখ্যার শেষের অংক হতে পারে না।

1 এর বর্গ = 1² = 1
2 এর বর্গ = 2² = 4
3 এর বর্গ = 3² = 9
4 এর বর্গ = 4² = 16
5 এর বর্গ = 5² = 25
6 এর বর্গ = 6² = 36
10 এর বর্গ= 10² = 100