Bank Math

PrepBank · পাতা ১৩২ / ১৬১ · ১৩,১০১–১৩,২০০ / ১৬,১২৪

১৩,১০১.

If 4x - 3y = 13 and 5x + 2y = - 1, then x = ?

ক
ক) - 3
খ
খ) -1
গ
গ) 1
ঘ
ঘ) 3

ব্যাখ্যা

4x - 3y =13 ................(1)
5x + 2y = -1 ................(2)

(1) × 2 + (2) × 3 ⇒
8x - 6y + 15x + 6y = 26 - 3
23x = 23
x = 1

১৩,১০২.

A grocer mixes two types of pulses, one costing 15 taka per kg and the other 20 taka per kg. If the mixture is sold at 16.50 Taka per kg, what is the ratio of the two types of pulses in the mixture?

ক
3 : 7
খ
5 : 3
গ
7 : 3
ঘ
7 : 9

ব্যাখ্যা

Question: A grocer mixes two types of pulses, one costing 15 taka per kg and the other 20 taka per kg. If the mixture is sold at 16.50 Taka per kg, what is the ratio of the two types of pulses in the mixture?

Solution:
ধরি,
প্রথম প্রকার ডালের পরিমাণ = x কেজি
দ্বিতীয় প্রকার ডালের পরিমাণ = y

প্রথম প্রকার ডালের x কেজির মূল্য = 15x টাকা
দ্বিতীয় প্রকার ডালের y কেজির মূল্য = 20y টাকা

প্রশ্নমতে,
15x + 20y = 16.50(x + y)
⇒ 15x + 20y = 16.50x + 16.50y
⇒ 20y - 16.50y = 16.50x - 15x
⇒ 3.50y = 1.50x
⇒ x/y = 3.50/1.50
⇒ x/y = 7 : 3

শর্টকাট:

∴ অনুপাত = 3.50 : 1.50
= 7 : 3

১৩,১০৩.

If 2x - 1 ≥ - 3, then-

ক
x ≥ - 1
খ
x ≤ - 1
গ
x ≥ 1
ঘ
x ≥ 0

ব্যাখ্যা

Question: If 2x - 1 ≥ - 3, then-

Solution:
2x - 1 ≥ - 3
⇒ 2x ≥ - 3 + 1
⇒ 2x ≥ - 2
⇒ x ≥ - 1

১৩,১০৪.

Find the value of k if (x + 2) is a factor of 3x{x + (k/3x) + (2/3)}

ক
- 5
খ
7
গ
6
ঘ
- 8

ব্যাখ্যা

Question: Find the value of k if (x + 2) is a factor of 3x{x + (k/3x) + (2/3)}

Solution:
Here,
3x{x + (k/3x) + (2/3)}
= 3x² + k + 2x
= 3x² + 2x + k

Given,
(x + 2) is a factor,
So, x + 2 = 0
∴ x = - 2

ATQ,
3 × (- 2)² + 2(- 2) + k = 0
⇒ 12 - 4 + k = 0
⇒ 8 + k = 0
∴ k = - 8

১৩,১০৫.

Two dice are thrown simultaneously. What is the probability of getting two numbers whose product is odd?

ক
1/3
খ
1/4
গ
5/6
ঘ
5/9
ঙ
None of these

ব্যাখ্যা

Question: Two dice are thrown simultaneously. What is the probability of getting two numbers whose product is odd?

Solution:
In a simultaneous throw of two dice,
we have n(S) = (6 × 6) = 36

Now,
E= {(1, 1), (1, 3), (1, 5), (3, 1), (3, 3), (3, 5), (5, 1), (5, 3), (5 ,5)}

∴n(E) = 9

∴ P(E) = n(E)/n(S)
= 9/36
= 1/4

১৩,১০৬.

Shishir bought a book and sold it at a loss of 10%. If he had bought it for 20% less and sold it for tk 55 more, he would have made a profit of 40%, What was the cost of the book?

ক
200 tk
খ
250 tk
গ
300 tk
ঘ
350 tk
ঙ
400 tk

ব্যাখ্যা

Question: Shishir bought a book and sold it at a loss of 10%. If he had bought it for 20% less and sold it for tk 55 more, he would have made a profit of 40%, What was the cost of the book?

Solution:
Let the cost price of the book be x tk.
The selling price of the book when sold at a 10% loss is 0.92 tk.

If the book had been bought for 20% less, the cost price would be 0.8x tk.
If the selling price had been 55 tk more, the selling price would be 0.9x + 55 tk.
According to the problem, in this case, Shishir would have made a profit of 40%, so the selling price would be 1.4 × 0.8x = 1.12x tk.

ATQ,
0.9x + 55 = 1.12x
⇒ 1.12x - 0.9x = 55
⇒ 0.22x = 55
⇒ x = 55/0.22
⇒ x = 5500/22
∴ x = 250 tk

১৩,১০৭.

A person bought an article for Tk. 1200 and sold it at a profit of 15%. The buyer sold it to a third person at a loss of 5%. Calculate the last selling price?

ক
ক) Tk. 1310
খ
খ) Tk. 1311
গ
গ) Tk. 1380
ঘ
ঘ) Tk. 1280

ব্যাখ্যা

Question: A person bought an article for Tk. 1200 and sold it at a profit of 15%. The buyer sold it to a third person at a loss of 5%. Calculate the last selling price?

Solution:
১৫% লাভে,
১০০ টাকার দ্রব্যের বিক্রয়মূল্য = ১১৫ টাকা
১ টাকার দ্রব্যের বিক্রয়মূল্য = ১১৫/১০০ টাকা
∴ ১২০০ টাকার দ্রব্যের বিক্রয়মূল্য = (১১৫ × ১২০০)/১০০ টাকা
= ১৩৮০ টাকা

৫% ক্ষতিতে,
১০০ টাকার দ্রব্যের বিক্রয়মূল্য = ৯৫ টাকা
১ টাকার দ্রব্যের বিক্রয়মূল্য = ৯৫/১০০ টাকা
∴ ১৩৮০ টাকার দ্রব্যের বিক্রয়মূল্য = (৯৫ × ১৩৮০)/১০০ টাকা
= ১৩১১ টাকা

১৩,১০৮.

A boy rides his bicycle 10 km at an average speed of 12 km/hr. and again travels 12km at an average speed of 10km/hr. His average speed of for the entire trip is approximately -

ক
11 km/hour
খ
10.8 km/hour
গ
11.2 km/hour
ঘ
10.5 km/hour

ব্যাখ্যা

Question: A boy rides his bicycle 10 km at an average speed of 12 km/hr. and again travels 12km at an average speed of 10km/hr. His average speed of for the entire trip is approximately -

Solution:
Total Distance = 10 + 12 = 22 km
Total time = (10/12) + (12/10)
= (5/6) + (6/5)
= (25 + 36)/30
= 61/30 hours

Average speed = 22/(61/30) = 660/61 = 10.8 km/hour

১৩,১০৯.

Three partners A, B, C start a business. B's Capital is four times C's capital and twice A's capital is equal to thrice B's capital. If the total profit is Tk. 28600 at the end of a year, Find out B's share in it.

ক
Tk. 8400
খ
Tk. 9400
গ
Tk. 10400
ঘ
Tk. 11400

ব্যাখ্যা

Question: Three partners A, B, C start a business. B's Capital is four times C's capital and twice A's capital is equal to thrice B's capital. If the total profit is Tk. 28600 at the end of a year, Find out B's share in it.

Solution:
Let
C's capital = x then
B's capital = 4x
A's capital = 6x
A : B : C = 6x : 4x : x
= 6 : 4 : 1

B's share = 28600 × 4/11
= Tk. 10400

১৩,১১০.

A “Buy 3, Get 1 Free” deal is available at a store. How much discount does this equal in percentage terms?

ক
16%
খ
25%
গ
30%
ঘ
20%

ব্যাখ্যা

Question: A “Buy 3, Get 1 Free” deal is available at a store. How much discount does this equal in percentage terms?

Solution:
Assume the price of each item = 1 Taka.
Then,
Cost of 3 items = 3 Taka
Since 1 item is received for free, total items = 3 + 1 = 4

After the discount,
Cost of 4 items = 3 Taka
∴ Cost per item = 3/4 Taka

Actual price per item = 1 Taka
∴ Discount per item = 1 − ( 3/4 ) = 1/4 Taka

So, discount on 1 Taka = 1/4 Taka
∴ Discount on 100 Taka = 100 × ( 1/4 ) = 25 Taka

Therefore, in a “Buy 3, Get 1 Free” offer, the equivalent discount is 25%.

১৩,১১১.

The incomes of A and B are in the ratio 3 : 2 and their expenditure are in the ratio 5 : 3. If each saves Tk 1000, then, A's income can be-

ক
ক) 4000 Tk
খ
খ) 6000 Tk
গ
গ) 3000 Tk
ঘ
ঘ) 8000 Tk

ব্যাখ্যা

Question: The incomes of A and B are in the ratio 3 : 2 and their expenditure are in the ratio 5 : 3. If each saves Tk 1000, then, A's income can be-

Solution:
Let the income of A and B be 3x and 2x respectively.
Also, their expenditure is 5y and 3y.

According to the question,
3x - 5y = 1000 ------- (i)
2x - 3y = 1000 ---------- (ii)

Now, {(i) × 3} - {(ii) × 5}
⇒ 9x - 15y - 10x + 15y = 3000 - 5000
⇒ - x = -2000
⇒ x = 2000

Then, income of A = 3x = 3 × 2000 = Tk 6000

১৩,১১২.

A certain hotel has 1,400 single rooms and 420 double rooms. Each room is cleaned by one person. If one person can clean a single room every 15 minutes and a double room every 20 minutes, how many cleaning persons are needed to clean all the rooms if each person works for exactly 7 hours?

ক
68
খ
70
গ
76
ঘ
88
ঙ
94

ব্যাখ্যা

Question: A certain hotel has 1,400 single rooms and 420 double rooms. Each room is cleaned by one person. If one person can clean a single room every 15 minutes and a double room every 20 minutes, how many cleaning persons are needed to clean all the rooms if each person works for exactly 7 hours?

Solution:
Number of single rooms = 1400
Number of double rooms = 420
Time taken by a person to clean a single room = 15 mins = 15/60 hour = 1/4 hour
Time taken by a person to clean a double room = 20 mins = 20/60 hour = 1/3 hour

∴ Total time to clean 1400 single and 420 double rooms
= 1400 × (1/4) + 420 × (1/3)
= 350 + 140 hours
= 490 hours

Each person works for = 7 hours

∴ Number of persons required = 490/7 = 70

১৩,১১৩.

A car travels the first one-third of a certain distance with a speed of 10 km/hr, the next one-third distance with a speed of 20 km/hr and the last one-third distance with a speed of 60 km/hr. The average speed of the car for the whole journey is :

ক
12 km/hr
খ
18 km/hr
গ
20 km/hr
ঘ
24 km/hr

ব্যাখ্যা

Question: A car travels the first one-third of a certain distance with a speed of 10 km/hr, the next one-third distance with a speed of 20 km/hr and the last one-third distance with a speed of 60 km/hr. The average speed of the car for the whole journey is :

Solution:
Let, the distance travelled by a car be x km
First x/3 km distance cover at a speed of 10 km/hr
Second x/3 km distance cover at speed = 20 km/hr
Third x/3 km distance cover at speed = 60 km/hr

According to the question,
Total time = x/(3 × 10) + x/(3 × 20) + x/(3 × 60)
= x/30 + x/60 + x/180
= (6x + 3x + x)/180
= 10x/180
= x/18

Average speed = (x × 18)/x km/hr = 18 km/hr

১৩,১১৪.

In how many ways can the letters of the word 'MISSISSIPPI' be arranged such that the first letter is always 'M'?

ক
3,150
খ
3200
গ
3250
ঘ
3600
ঙ
None of the above

ব্যাখ্যা

Question: In how many ways can the letters of the word 'MISSISSIPPI' be arranged such that the first letter is always 'M'?

Solution:
The word 'MISSISSIPPI' contains 11 letters.

Condition: The first letter is fixed as 'M'.

So, the remaining 11 - 1 = 10 positions are to be filled by the remaining letters.

Among the remaining 10 letters, the repeated letters are:
I (4 times), S (4 times), P (2 times).

∴ Number of arrangements of the remaining 10 letters
= 10!/(4! × 4! × 2!)
= 3,628,800/(24 × 24 × 2)
= 3,628,800/1,152
= 3,150

১৩,১১৫.

If two angles are said to be complementary angles and one angle is 52° then the other angle is of:

ক
100°
খ
68°
গ
128°
ঘ
38°

ব্যাখ্যা

Question: If two angles are said to be complementary angles and one angle is 52° then the other angle is of:

Solution:
দুটি কোণের সমষ্টি ৯০° হলে, একটিকে অপরটির পূরক কোণ বলে।

অপর কোণ = 90° - 52°
= 38°

১৩,১১৬.

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

ক
12 days.
খ
25 days.
গ
35 days.
ঘ
15 days

ব্যাখ্যা

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

Solution:
P's 2 day's work = (1/20) × 2
= 1/10
(P + Q + R)'s 1 day's work = (1/20) + (1/30) + (1/60)
= 6/60
= 1/10
(P + Q + R) work in 3 days = (1/10) + (1/10)
= 1/5.

Now, 1/5 work is done in 3 days.
∴ The whole work is done in (3 × 5)
= 15 days.

১৩,১১৭.

A bag contains 6 red, 5 blue, and 4 green balls. Two balls are drawn one after another without replacement. What is the probability that both balls are red?

ক
3/4
খ
1/7
গ
1/2
ঘ
3/7

ব্যাখ্যা

Question: A bag contains 6 red, 5 blue, and 4 green balls. Two balls are drawn one after another without replacement. What is the probability that both balls are red?

Solution:
Total balls = 6 + 5 + 4 = 15
Probability of first red = 6/15
Probability of second red = 5/14

∴ Probability (both red) = (6/15) × (5/14)
= 30/210
= 1/7

Therefore, the probability that both balls are red is 1/7.

১৩,১১৮.

A train covers a certain distance at a speed of 240 kmph in 6 hours. To cover the same distance in 4 hours, it must travel at a speed of-

ক
260 km/h
খ
360 km/h
গ
160 km/h
ঘ
220 km/h

ব্যাখ্যা

Question: A train covers a certain distance at a speed of 240 kmph in 6 hours. To cover the same distance in 4 hours, it must travel at a speed of-

Solution:
Given,
Distance = (240 × 6)
= 1440 km

We know,
Speed = (Distance ÷ Time)

∴ Required speed = (1440 ÷ 4) km/h
= 360 km/h

∴ The required speed is 360 km/h.

১৩,১১৯.

A rectangular floor is covered by a rug except fo a strip p meters along each of the four edges. If the floor is m meters by n meteres, What is the area of the rug in square meters?

ক
mn - p(m + n)
খ
mn - 2p(m + n)
গ
(m - p)(n - p)
ঘ
(m - 2p)(n - 2p)

ব্যাখ্যা

Question: A rectangular floor is covered by a rug except fo a strip p meters along each of the four edges. If the floor is m meters by n meteres, What is the area of the rug in square meters?

Solution:

The length of the rug = m - 2 × p = m - 2p
The width of the rug = n - 2 × p = n - 2p
∴ The area of the rectangular rug equals = (m - 2p)(n - 2p)

১৩,১২০.

Amin took a loan of Tk.1400 with simple interest for as many years as the rate of interest. If he paid Tk.686 as interest at the end of the loan period, what was the rate of interest?

ক
ক) 4%
খ
খ) 5%
গ
গ) 6%
ঘ
ঘ) 7%

ব্যাখ্যা

Question: Amin took a ^loan of Tk.1400 with simple interest for as many years as the rate of interest. If he paid Tk.686 as interest at the end of the loan period, what was the rate of interest?

Solution:
Given that n = r

S.I = (P × n × r)/100
⇒ 686 = (1400 × r × r)/100
⇒ 686 = 14r²
⇒ r² = 49
⇒ r = 7%

১৩,১২১.

A train 150 meters long takes 30 seconds to cross a 300-meter-long bridge. How much time will it take to cross a 300-meter-long platform?

ক
18 seconds
খ
24 seconds
গ
30 seconds
ঘ
36 seconds

ব্যাখ্যা

Question: A train 150 meters long takes 30 seconds to cross a 300-meter-long bridge. How much time will it take to cross a 300-meter-long platform?

Solution:
Length of the train = 150 meters
Length of the bridge = 300 meters
Total distance = 150 + 300 = 450 meters

∴ Speed of the train = 450/30 = 15 m/s

New total distance (train + platform) = 150 + 300 = 450 meters

Time taken to cross the platform = 450/15 = 30 seconds

১৩,১২২.

Three unbiased coins are tossed. What is the probability of getting at most two tails ?

ক
ক) 1/8
খ
খ) 3/8
গ
গ) 7/8
ঘ
ঘ) 5/8

ব্যাখ্যা

Here S = {TTT, TTH, THT, HTT, THH, HTH, HHT, HHH}

Let E = event of getting at most two tails.
Then E = {TTH, THT, HTT, THH, HTH, HHT, HHH}

P(E) = n(E)/n(S)
= 7/8

১৩,১২৩.

Mofiz got married 10 years ago. His present age is 6/5 times his age at the time of his marriage. What is the present age of Mofiz?

ক
60 years
খ
40 years
গ
45 years
ঘ
50 years

ব্যাখ্যা

Question: Mofiz got married 10 years ago. His present age is 6/5 times his age at the time of his marriage. What is the present age of Mofiz?

Solution:
Let, Mofiz's age 10 years ago be x years
His present age is = x + 10 years

ATQ,
x + 10 = 6x/5
⇒ 5x + 50 = 6x
⇒ x = 50

His present age is = 50 + 10 = 60 years

১৩,১২৪.

If y = 5x² - 2x, and x = 3, then y =?

ক
24
খ
27
গ
39
ঘ
51

ব্যাখ্যা

Question: If y = 5x² - 2x, and x = 3, then y =?

Solution:

Put x = 3 in the equation y = 5x² - 2x

⇒ y = 5 × (3)² - (2 × 3)

⇒ y = 45 - 6

∴ y = 39

১৩,১২৫.

Q. (36-55) Choose the correct answer.
৩৬) A train which is 1 km long traveling at a speed of 60 km/hour enters a tunnel 1km of length. What time does the train take to come fully out of the tunnel?

ক
ক) 1 min
খ
খ) 2 min
গ
গ) 30 min
ঘ
ঘ) 60 min

ব্যাখ্যা

Question: A train which is 1 km long traveling at a speed of 60 km/hour enters a tunnel 1km of length. What time does the train take to come fully out of the tunnel?

Solution:
টানেল থেকে পুরোপুরি বের হলে, অতিক্রান্ত দূরত্ব ১ + ১ কি.মি.
= ২ কি.মি.

৬০ কি.মি. যেতে সময় লাগে ১ ঘণ্টা বা ৬০ মিনিট
১ কি.মি. যেতে সময় লাগে ৬০/৬০ = ১ মিনিট
∴ ২ কি.মি. যেতে সময় লাগে = ২ × ১ মিনিট
= ২ মিনিট

১৩,১২৬.

If a number is in the form of 8⁵ × 9⁷ × 7⁶, find the total number of prime factors of the given number.

ক
ক) 30
খ
খ) 32
গ
গ) 35
ঘ
ঘ) 52

ব্যাখ্যা

The number 8⁵× 9⁷ × 7⁶ can be written as (2³)⁵ × (3²)⁷ × 7⁶The number can be written as 2¹⁵× 3¹⁴× 7⁶
Total number of prime factors = 15 + 14 + 6
∴ The total number of prime factors are 35

১৩,১২৭.

What will be the least number which when doubled will be exactly divisible by 18, 24, 28, and 36?

ক
1008
খ
504
গ
360
ঘ
252

ব্যাখ্যা

Question: What will be the least number which when doubled will be exactly divisible by 18, 24, 28, and 36?

Solution:
LCM of 18, 24, 28, and 36 is = 504
So, the number will be half of 504 = 504/2 = 252

১৩,১২৮.

The population of a town 2 years ago was 62,500. Due to migration to big cities, it decreases every year at the rate of 4%. The present population of the town is -

ক
ক) 56,700
খ
খ) 57,600
গ
গ) 58,800
ঘ
ঘ) 60,000

ব্যাখ্যা

Present population
=62500 × {1 - (4/100)}²
= 62500 × (24/25) × (24/25)
= 57600.

১৩,১২৯.

P, Q and R start a business. P invests 3 times as much as Q invests 2/3rd as much as R invests. Find the ratio of capitals of P, Q and R ?

ক
ক) 3 : 2 : 6
খ
খ) 6 : 2 : 3
গ
গ) 2 : 6 : 3
ঘ
ঘ) 5 : 2 : 3

ব্যাখ্যা

ধরি
R বিনিয়োগ করে = x টাকা
Q বিনিয়োগ করে = 2x/3 টাকা
P বিনিয়োগ করে = 3× (2x/3) টাকা
= 2x টাকা
P, Q এবং R এর বিনিয়োগের অনুপাত = 2x : (2x/3) : x
= 2 : (2/3) : 1
= 6 : 2 : 3

১৩,১৩০.

If the sum of the squares three consecutive natural numbers is 110, then the smallest of these natural numbers is = ?

ক
ক) 5
খ
খ) 6
গ
গ) 7
ঘ
ঘ) 8

ব্যাখ্যা

Let the three consecutive numbers are x, x + 1, x + 2
According to question
x² + (x + 1)² + (x + 2)² = 110
⇒ x² + x² + 1 + 2x + x² + 4 + 4x = 110
⇒ 3x² + 6x + 5 = 110
⇒ 3x² + 6x - 105 = 0
⇒ x² + 2x - 35 = 0
⇒ x^₂ + 7x - 5x - 35 = 0
⇒ x(x + 7) -5(x + 7) = 0
⇒ (x + 7)(x - 5) = 0
⇒ x = 5 & -7
[∴ (-) value can not considered]
∴ Smallest number is = 5

১৩,১৩১.

A computer programmer needs to print 144 documents. The documents have an average (arithmetic mean) length of 10 pages and the printer takes 15 seconds to print each page. Approximately how many hours will it take to print all the documents if they are printed without interruptions?

ক
0.5 hr
খ
2 hr
গ
2.5 hr
ঘ
6 hr
ঙ
24 hr

ব্যাখ্যা

Question: A computer programmer needs to print 144 documents. The documents have an average (arithmetic mean) length of 10 pages and the printer takes 15 seconds to print each page. Approximately how many hours will it take to print all the documents if they are printed without interruptions?

Solution:
Total number of documents = 144
Average length of a document = 144 × 10 = 1440

In 15 sec Pages printed 1
In 1 min or 60 sec Pages printed 60/15 = 4

Pages printed in an hour = 4 × 60 = 240

Time taken = 1440/ 240= 6 hours

১৩,১৩২.

A person takes 8 hours 30 minutes in cycling a distance and walking back to the starting place. He could cycle both ways in 10 hours 40 minutes. What is the time taken by him to walk both ways?

ক
4 hours 45 minutes
খ
6 hours 20 minutes
গ
5 hours 40 minutes
ঘ
7 hours 30 minutes

ব্যাখ্যা

Question: A person takes 8 hours 30 minutes in cycling a distance and walking back to the starting place. He could cycle both ways in 10 hours 40 minutes. What is the time taken by him to walk both ways?

Solution:
Time taken in cycling both ways = 10 hours 40 minutes ........... (1)
Time taken in cycling one way and walking back = 8 hours 30 minutes ........... (2)

Using the equation {(2) × 2} - (1), we have:
Time taken by the person to walk both ways = 17 hours - 10 hours 40 minutes
= 6 hours 20 minutes

১৩,১৩৩.

ক
2, 5
খ
1, 1/2
গ
5, 3/8
ঘ
2, 1/3

ব্যাখ্যা

Question:

Solution:

১৩,১৩৪.

Which of the following is a perfect square?

ক
45
খ
72
গ
49
ঘ
81.5

ব্যাখ্যা

Question: Which of the following is a perfect square?

Solution:
বর্গসংখ্যা (perfect square): সাধারণভাবে একটি স্বাভাবিক সংখ্যা m কে যদি অন্য একটি স্বাভাবিক সংখ্যা n এর বর্গ (n²) আকারে প্রকাশ করা যায় তবে m বর্গসংখ্যা।
m সংখ্যাগুলোকে পূর্ণবর্গসংখ্যা বলা হয়।

পূর্ণবর্গ সংখ্যার ধর্ম:
• যে সংখ্যার সর্ব ডানদিকের অঙ্ক অর্থাৎ একক স্থানীয় অঙ্ক ২ বা ৩ বা ৭ বা ৮ তা পূর্ণবর্গ নয় ।
• যে সংখ্যার শেষে বিজোড় সংখ্যক শূন্য থাকে, ঐ সংখ্যা পূর্ণবর্গ নয়।
• একক স্থানীয় অঙ্ক ১ বা ৪ বা ৫ বা ৬ বা ৯ হলে, ঐ সংখ্যা পূর্ণবর্গ হতে পারে। যেমন : ৮১, ৬৪, ২৫, ৩৬, ৪৯ ইত্যাদি বর্গসংখ্যা ।
• আবার সংখ্যার ডানদিকে জোড়সংখ্যক শূন্য থাকলে ঐ সংখ্যা পূর্ণবর্গ হতে পারে। যেমন : ১০০, ৪৯০০ ইত্যাদি বর্গসংখ্যা ।

প্রদত্ত অপশগুলোর মধ্যে
গ) 49 = 7² [যা পূর্ণবর্গসংখ্যা]

১৩,১৩৫.

Ratio of milk and water in 630 litre of mixture is 4 : 3. 140 litre mixture taken out. Find the quantity of milk now?

ক
190 liters
খ
400 liters
গ
220 liters
ঘ
320 liters
ঙ
280 liters

ব্যাখ্যা

Question: Ratio of milk and water in 630 litre of mixture is 4 : 3. 140 litre mixture taken out. Find the quantity of milk now?

Solution:
Given that,
The ratio of milk to water in 630 liters of mixture is 4 : 3
140 liters of the mixture is taken out.

Now,
The total parts of the mixture = 4 + 3 = 7 parts.
Quantity of milk in the mixture = (4/7) × 630 = 360 liters of milk.
Quantity of water in the mixture = (3/7) × 630 = 270 liters of water.

When 140 liters of the mixture is taken out, the ratio of milk and water in the 140 liters will also be 4 : 3.
Milk taken out = (4/7) × 140 = 80 liters.

So, the remaining milk = 360 - 80 = 280 liters.

∴ The quantity of milk now is 280 liters.

১৩,১৩৬.

The distance between two places A and B is 570 km. A train starts from A at 50 km/h at 6 am and another starts from B at 80 km/h at 7 am towards each other. At what time will they meet?

ক
8 am
খ
9 am
গ
10 am
ঘ
11 am
ঙ
12 am

ব্যাখ্যা

Question: The distance between two places A and B is 570 km. A train starts from A at 50 km/h at 6 am and another starts from B at 80 km/h at 7 am towards each other. At what time will they meet?

Solution:
Let,
the two trains meet at a distance d km from place A.
Time required by the train starting from A to cover p = p/50 hr
Time taken by the other train starting from B to cover (570 - p) km = (570 - p)/80

But the first train has started 1 hr early. So, it has traveled 50 km in this 1 hr.
Therefore,
(p/50) - 1 = (570 - p)/80
⇒ (p - 50)/50 = (570 - p)/80
⇒ 28500 - 50p = 80p - 4000
⇒ 130p = 32500
∴ p = 250

So, they will meet at a distance of 250 km from Place A.
So the time at which they will meet will be = (250/50) = 5 hrs [after 6 am]
Hence, they will meet at 11 am

১৩,১৩৭.

If n(A) = 16, n(B) = 18 and n(A ∩ B) = 7, then what is the value of n(A ∪ B)

ক
9
খ
15
গ
27
ঘ
41

ব্যাখ্যা

Question: If n(A) = 16, n(B) = 18 and n(A ∩ B) = 7, then what is the value of n(A ∪ B)?

Solution:
We know that,
n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
= 16 + 18 - 7
= 27

১৩,১৩৮.

Tk. 2100 is divided among A, B, C so that A receives half as much as B and B half as much as C. Then C's share is -

ক
ক) 300
খ
খ) 600
গ
গ) 1200
ঘ
ঘ) 1800

ব্যাখ্যা

Question: Tk. 2100 is divided among A, B, C so that A receives half as much as B and B half as much as C. Then C's share is -

Solution:
ধরি, A পাবে ক টাকা
B পাবে ২ক টাকা
C পাবে ৪ক টাকা

প্রশ্নমতে,
ক + ২ক + ৪ক = ২১০০
৭ক = ২১০০
ক = ৩০০

∴ C পাবে = ৪ × ৩০০ = ১২০০

১৩,১৩৯.

What is the value of the greater root of the equation x² - 5x + 4 = 0?

ক
1
খ
4
গ
3
ঘ
2

ব্যাখ্যা

Question: What is the value of the greater root of the equation x² - 5x + 4 = 0?

Solution:
x² - 5x + 4 = 0
⇒ x² - 4x - x + 4 = 0
⇒ x(x - 4) - (x - 4) = 0
⇒ (x - 1)(x - 4) = 0
so the roots of the equation are x₁ = 1 and x₂ = 4. The greater one is obviously 4.

১৩,১৪০.

If tanθ = 9/40, then secθ = ?

ক
23/40
খ
41/10
গ
21/40
ঘ
41/40

ব্যাখ্যা

Question: If tanθ = 9/40, then secθ = ?

Solution:
এখানে,
tanθ = 9/40 = লম্ব/ভূমি

∴ লম্ব = 9, ভূমি = 40
∴ অতিভুজ = √(9² + 40²)
= √(81 + 1600)
= √1681 = 41

∴ secθ = 1/cosθ = অতিভুজ/ভূমি
= 41/40

১৩,১৪১.

If 4^{x + y} = 1 and 4^{x - y} = 4, then the values of x and y respectively are -

ক
ক) 3/2 and 1/2
খ
খ) - 1/4 and 1/4
গ
গ) - 1/4 and - 1/2
ঘ
ঘ) 1/2 and - 1/2

ব্যাখ্যা

Question: If 4^{x + y} = 1 and 4^{x - y} = 4, then the values of x and y respectively are -

Solution:
দেওয়া আছে,
4^{x + y} = 1
বা, 4^{x + y} = 4⁰
বা, x + y = 0 .................... (1)
এবং
4^{x - y}= 4
বা, 4^{x - y} = 4¹
বা, x - y = 1 ................. (2)

(1) + (2) হতে পাই,
x + y = 0
x - y = 1

2x = 1
∴ x = 1/2

x এর মান (1) নং বসিয়ে পাই,
x + y = 0
⇒ (1/2) + y = 0
∴ y = - 1/2

∴ নির্ণেয় সমাধান (x, y) = (1/2, - 1/2)

১৩,১৪২.

Calculate the area of a rhombus if the length of its side is 2 cm and one of its angles A is 30 degrees.

ক
5 cm²
খ
4 cm²
গ
3 cm²
ঘ
2 cm²

ব্যাখ্যা

Question: Calculate the area of a rhombus if the length of its side is 2 cm and one of its angles A is 30 degrees.

Solution:
Given,
Side = s = 2 cm
Angle A = 30 degrees
Square of side = 2 × 2 = 4

Area, A = s² × sin (30°)
⇒ A = 4 × (1/2)
∴ A = 2 cm²

১৩,১৪৩.

Age of a father 10 years ago was 3 times the age of her son. After 10 years, the father's age will be twice that of his son. Find the ratio of their present ages?

ক
11 : 7
খ
9 : 5
গ
7 : 4
ঘ
7 : 3

ব্যাখ্যা

We are given that,
age of father 10 years ago was 3 times the age of her son
So,
let the age of the son be x and as the father's age is 3 times the age of her son, let it be 3x, three years ago.
At present,
Father's age will be (3x + 10) and son's age will be (x + 10)
After 10 years,
Father's age will be (3x + 10) +10 and son’s age will be (x + 10) + 10

Father's age is twice that of son
(3x + 10) +10 = 2 [(x + 10) + 10]
(3x + 20) = 2[x + 20]
Solving the equation, we get x = 20
We are asked to find the present ratio.
(3x + 10) : (x + 10) = 70 : 30
(3x + 10) : (x + 10) = 7 : 3.

১৩,১৪৪.

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

ক
10
খ
15
গ
30
ঘ
120

ব্যাখ্যা

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

Solution:
Here 1 man can be selected from 5 men in ⁵C₁ = 5 ways
2 women can be selected from 4 women in ⁴C₂ = 6 ways

∴ The total number of ways the committee can be formed = 5 × 6 ways
= 30 ways.

১৩,১৪৫.

Two baskets together have 640 oranges. If one-fifth of the oranges in the first basket be taken to the second basket then, numbers of oranges in both baskets become equal. The number of oranges in the first basket is-

ক
ক) 800
খ
খ) 600
গ
গ) 400
ঘ
ঘ) 300

ব্যাখ্যা

Let the number of oranges in first basket be x,
Number of oranges in second basket = 640 - x
ATQ, x - x/5 = 640 - x + x/5
⇒ 4x/5 = 640 - 4x/5
⇒ 4x/5 + 4x/5 = 640
⇒ 8x/5 = 640
⇒ x = 640 × (5/8)
⇒ x = 400
∴ Number of oranges in first basket = 400.

১৩,১৪৬.

A trader sells his goods at a discount of 20%. He still makes a profit of 25%. If he sells the goods at the marked price only, his profit will be:

ক
ক) 56.25%
খ
খ) 57%
গ
গ) 52.50%
ঘ
ঘ) None of these

ব্যাখ্যা

Question: A trader sells his goods at a discount of 20%. He still makes a profit of 25%. If he sells the goods at the marked price only, his profit will be:

Solution:
ধরি,
দ্রব্যটির বাজার মূল্য = 100 টাকা
∴ 20% ছাড়ে বিক্রয়মূল্য = (100 - 20) = 80 টাকা।

আবার, ধরি,
দ্রব্যটির উৎপাদন খরচ = x টাকা।

প্রশ্নমতে,
∴ x + x এর 25/100 = 80
বা, x + (x/4) = 80
বা, 5x/4 = 80
∴ x = 64

∴ বাজার মূল্যে পন্যটি বিক্রি করলে,
লাভ =(100 - 64) = 36 টাকা।

∴ 64 টাকায় লাভ হয় = 36 টাকা
∴ 100 টাকায় লাভ হয় = (36 × 100)/64
= 56.25%

১৩,১৪৭.

Average mark in Math in a class of 40 students is 45. Average mark of all the 30 boys is 50. Then the average mark obtained by the girls is:

ক
ক) 30
খ
খ) 35
গ
গ) 25
ঘ
ঘ) 40
ঙ
ঙ) 33

ব্যাখ্যা

Question: Average mark in Math in a class of 40 students is 45. Average mark of all the 30 boys is 50. Then the average mark obtained by the girls is:

Solution:
Average mark of 40 students is 45
Total mark of 40 students is (45 × 40)
= 1800

Average mark of all the 30 boys is 50
Total mark of all the 30 boys is (50 × 30)
= 1500

∴ Total marks of all the 10 girls is (1800 - 1500) = 300
The average mark of all the 10 girls is 300/10 = 30

১৩,১৪৮.

The area of a triangle with sides 3 cm, 5 cm and 6 cm is -

ক
ক) 2√3 cm²
খ
খ) 2√14 cm²
গ
গ) 5√12 cm²
ঘ
ঘ) 4√14 cm²

ব্যাখ্যা

অর্ধপরিসীমা, s = (3 + 5 + 6)/2 = 7 সে.মি
∴ ক্ষেত্রফল = √{s(s - a)(s - b)(s - c)} বর্গএকক
= √ {7 (7 - 3) (7 - 5) (7 - 6)} বর্গসে.মি
= √ (7 × 4 × 2 × 1)
= 2√14 বর্গসে.মি

১৩,১৪৯.

The average of several exam scores is 80. One make-up exam was given. Included with the other scores, the new average was 84. If the score on the make-up exam was 92, how many total exams were given?

ক
5
খ
4
গ
3
ঘ
2

ব্যাখ্যা

Question: The average of several exam scores is 80. One make-up exam was given. Included with the other scores, the new average was 84. If the score on the make-up exam was 92, how many total exams were given?

Solution:
let, old number of exams be n

ATQ,
(80n + 92)/(n + 1) = 84
⇒ 80n + 92 = 84n + 84
⇒ 4n = 8
⇒ n = 2

total exam = 2 + 1 = 3

১৩,১৫০.

Out of three numbers, the first is twice the second and is half of the third. If the average of the three numbers is 56, then the average of the first and the third number is -

ক
ক) 48
খ
খ) 72
গ
গ) 80
ঘ
ঘ) 96

ব্যাখ্যা

Question: Out of three numbers, the first is twice the second and is half of the third. If the average of the three numbers is 56, then the average of the first and the third number is -

Solution:
Let, the second number be = x
So, the first number is = 2x
and the third number is = 2x × 2 = 4x

According to the question,
2x + x + 4x = 56 × 3
⇒ 7x = 168
⇒ x = 24

So, the first number is = 2 × 24 = 48
and the third number is = 4 × 24 = 96

Then, the average of the first and the third number is = (48 + 96)/2 = 72

১৩,১৫১.

The difference between a number and its three-fifths is 240, What is the number?

ক
ক) 125
খ
খ) 300
গ
গ) 420
ঘ
ঘ) 600

ব্যাখ্যা

Question: The difference between a number and its three-fifths is 240, What is the number?

Solutionh:
let the number x

x - 3x/5 = 240
⇒ (5x - 3x)/5 = 240
⇒ 2x = 1200
∴ x = 600

১৩,১৫২.

A canteen requires 553 liters of water for a week. Totally, how many liters will it require for the months of May, June, July?

ক
8563 liter
খ
7268 liter
গ
7347 liter
ঘ
8278 liter

ব্যাখ্যা

Question: A canteen requires 553 liters of water for a week. Totally, how many liters will it require for the months of May, June, July?

Solution:
একদিনে পানি লাগে = 553/7 = 79 লিটার
May, June, July মাসে মোট দিন = (31 + 30 + 31) = 92 দিন।

∴ মোট পানি লাগবে = (92 × 79) = 7268 লিটার

১৩,১৫৩.

A batsman makes a score of 82 runs in the 15th innings and thus increases his average by 2. Find his average after the 15th innings.

ক
ক) 42
খ
খ) 48
গ
গ) 52
ঘ
ঘ) 54

ব্যাখ্যা

Question: A batsman makes a score of 82 runs in the 15th innings and thus increases his average by 2. Find his average after the 15th innings.

Solution:
Let, the average after the 15th innings be = x
Then, average after 14th innings = (x - 2)

ATQ,
14 × (x - 2) + 82 = 15x
⇒ 14x - 28 + 82 = 15x
⇒ x = 54

১৩,১৫৪.

In an exam 60% passed both math and bangla. 20% failed both the subjects. If 70% passed in bangla, how many passed math?

ক
60%
খ
70%
গ
75%
ঘ
80%

ব্যাখ্যা

Question: In an exam 60% passed both math and bangla. 20% failed both the subjects. If 70% passed in bangla, how many passed math?

Solution:
Total passed = 60%
total failed = 40%

fails in bangla = ( 100 - 70 ) = 30%
Let fails in math = X%
fail in both = 20%

∴ total fail = ( math fail + bangla fail - both fail )
40 = X + 30 - 20
X = 30

passed in math = ( 100 -30 ) = 70%

১৩,১৫৫.

Simple interest on a certain sum of money for 3 years at 8% per annum is half the compound interest on Tk. 4000 for 2 years at 10% per annum. The sum placed on simple interest is-

ক
Tk. 1455
খ
Tk. 1575
গ
Tk. 1750
ঘ
Tk. 1845

ব্যাখ্যা

Question: Simple interest on a certain sum of money for 3 years at 8% per annum is half the compound interest on Tk. 4000 for 2 years at 10% per annum. The sum placed on simple interest is

Solution:
Given,
principal P = Tk. 4000
Compound rate, r = 10% per annum
Time n = 2 years

Amount,
C = P(1 + r)ⁿ
= 4,000(1 + 10/100)²
= 4,000(11/10)²
= (4,000 × 121)/100
= 4840
∴ Compound interest =Tk. (4840 - 4000)
= Tk. 840

∴ Simple interest I = (840 ÷ 2) = Tk. 420
Rate r = 8%
Time n = 3 years

We know,
P = I/rn
= 420/{(8/100) × 3}
= (420 × 100)/(8 × 3)
= 1750

১৩,১৫৬.

What should be the value of "P" so that the expression (9 - 12x + Px²) becomes a perfect square?

ক
4
খ
6
গ
9
ঘ
12

ব্যাখ্যা

Question: What should be the value of "P" so that the expression (9 - 12x + Px²) becomes a perfect square?

Solution:
(9 - 12x + Px²)
= (3)² - 2 × 3 × 2x + (2x)² + Px² - (2x)² = (3 - 2x)² + Px² - 4x²

The expression becomes a perfect square if:
Px² - 4x² = 0
⇒ Px² = 4x²
∴ P = 4

১৩,১৫৭.

A metallic sheet is of rectangular shape with dimensions 48 m x 36 m. From each of its corners, a square is cut off so as to make an open box. If the length of the square is 8 m, the volume of the box (in m³) is:

ক
ক) 4830
খ
খ) 5120
গ
গ) 6420
ঘ
ঘ) 8960

ব্যাখ্যা

Clearly, l = (48 - 16)m = 32 m,
b = (36 -16)m = 20 m,
h = 8 m.
Volume of the box = (32 x 20 x 8) m³ = 5120 m³.

১৩,১৫৮.

If A is a zero matrix, then A + B = ?

ক
Matrix A
খ
Matrix B
গ
Zero matrix
ঘ
Identity matrix

ব্যাখ্যা

Question: If A is a zero matrix, then A + B = ?
(Senior Officer 2022 অনুযায়ী)

Solution:
যদি A একটি শূন্য ম্যাট্রিক্স (zero matrix) হয়, তবে এর সব উপাদান শূন্য।
ম্যাট্রিক্স যোগ করার সময় প্রতিটি অবস্থানের উপাদানগুলি যথাক্রমে যোগ করা হয়।

তাই A + B মানে প্রতিটি অবস্থানে A-এর উপাদান এবং B-এর উপাদান যোগ করা।
যেহেতু A-এর সব উপাদান শূন্য, প্রতিটি অবস্থানে যোগফল শুধু B-এর উপাদানই থাকবে। তাই A + B = B.

এটি একটি মৌলিক বৈশিষ্ট্য যা শূন্য ম্যাট্রিক্সের সাথে যে কোনো ম্যাট্রিক্স যোগ করলে মূল ম্যাট্রিক্স অপরিবর্তিত থাকে।
সুতরাং সঠিক উত্তর হলো Matrix B।

- উত্তর: খ) Matrix B

১৩,১৫৯.

A tiger is 50 of its own leaps behind a deer. The tiger takes 5 leaps per minute to the deer's 4. If the tiger and the deer cover 8 m and 5 m per leap respectively, what distance will the tiger have to run before it caches the deer?

ক
800 m
খ
600 m
গ
1200 m
ঘ
1600 m
ঙ
None of the above

ব্যাখ্যা

Question: A tiger is 50 of its own leaps behind a deer. The tiger takes 5 leaps per minute to the deer's 4. If the tiger and the deer cover 8 m and 5 m per leap respectively, what distance will the tiger have to run before it caches the deer?

Solution:
Speed of tiger = 40 m/min
Speed of deer = 20 m/min.

Relative speed = 40 - 20 = 20 m/min.
Initial difference in distance = 50 × 8 = 400 m

Time taken to catch = 400/20
= 20 min.

Distance travelled in 20 min,
= 20 × 40
= 800 m

১৩,১৬০.

210 men working 12 hours a day can finish a job in 18 days. How many men are required to finish the job in 20 days working 14 hours a day?

ক
160
খ
162
গ
164
ঘ
166

ব্যাখ্যা

Question: 210 men working 12 hours a day can finish a job in 18 days. How many men are required to finish the job in 20 days working 14 hours a day?

Solution:
210 men need (18 × 12) hours
= 216 hours

to finish the job in 20 days working 14 hours a day men needed = (210 × 216)/(14 × 20)
= 162

১৩,১৬১.

If (4/5)³ × (4/5)^{- 6}= (4/5)^{2x - 1}, the value of x is-

ক
- 1
খ
- 2
গ
1
ঘ
2

ব্যাখ্যা

Question: If (4/5)³ × (4/5)^{- 6}= (4/5)^{2x - 1}, the value of x is-

Solution:
(4/5)³ × (4/5)^{- 6}= (4/5)^{2x - 1}
⇒ (4/5)^{3 - 6} = (4/5)^{2x - 1}
⇒ (4/5)^{- 3} = (4/5)^{2x - 1}

⇒ 2x - 1 = - 3

⇒ 2x = - 2

⇒ x = - 1

১৩,১৬২.

In a survey of 400 students, 65% play basketball, 45% play football, and 15% play neither sport. What percent of students play both sports?

ক
20%
খ
25%
গ
30%
ঘ
15%

ব্যাখ্যা

Question: In a survey of 400 students, 65% play basketball, 45% play football, and 15% play neither sport. What percent of students play both sports?

Solution:
Let,
x% students play both sports
∴ (65 - x)% + (45 - x)% + x% + 15% = 100%
⇒ 65% + 45% + 15% - x% = 100%
⇒ 125% - x% = 100%
∴ x% = 125% - 100% = 25%

১৩,১৬৩.

Sharif ran a 2 mile race at an average speed of 8 miles per hour. If Arif ran the same race an average speed of 6 mile per hour, how many minutes longer than Sharif did Arif take to complete the race?

ক
15
খ
10
গ
8
ঘ
5
ঙ
None of these

ব্যাখ্যা

D = T × S
Sharif takes, 2 = T × 8
or, T = 1/4 hours
or T = 15 minutes
Arif Takes, 2 = T × 3
or, T = 1/3
or, T = 15 minutes
Difference = (20 - 15) = 5 minutes.

১৩,১৬৪.

Rasel started a project where the work ratio of a day is twice the previous day. If the work is fully finished in 20 days, how many days it took to finish 1/4th of the work?

ক
ক) 5 days
খ
খ) 10 days
গ
গ) 18 days
ঘ
ঘ) 15 days

ব্যাখ্যা

Question: Rasel started a project where the work ratio of a day is twice the previous day. If the work is fully finished in 20 days, how many days it took to finish 1/4th of the work?

Solution:
প্রতিদিনের কাজ পূর্বের দিনের দ্বিগুণ হয়।

২০ দিনে যদি সম্পূর্ণ শেষ হয় তাহলে,
১৯ দিনে শেষ হবে ১/২ কাজ
১৮ দিনে শেষ হবে (১/২)/২ বা, ১/৪ কাজ

shortcut:
১ অংশ কাজ হয় ২০ দিনে,
১/৪ অংশ বা ১/২^২অংশ কাজ হয় = (২০ - ২) দিনে = ১৮ দিনে।
যত অংশ দেয়া থাকবে তার হরকে ২ এর ভিত্তি আকারে প্রকাশ করে ২ এর সূচককে মোট দিন থেকে বাদ দিয়ে দিন বের করতে হবে।

একই ভাবে যদি ১/১৬ অংশ বা ১/২^৪ অংশ কাজ হবে = (২০ -৪) = ১৬ দিনে।

১৩,১৬৫.

7^3.04 : 7^2.03 is equal to-

ক
ক) 7 : 17
খ
খ) 7 : 5
গ
গ) 3 : 1
ঘ
ঘ) 7 : 1

ব্যাখ্যা

Question: 7^3.04 : 7^2.03 is equal to-

Solution:
7^3.04 : 7^2.03
= 7^2.03 × 7 : 7^2.03
= 7 : 1

১৩,১৬৬.

The perimeter of a square is equal to the perimeter of a rectangle. The length of the rectangle is three times its width, and the area of the rectangle is 1,728 square metres. Find the perimeter of the square.

ক
192 m
খ
168 m
গ
144 m
ঘ
216 m

ব্যাখ্যা

Question: The perimeter of a square is equal to the perimeter of a rectangle. The length of the rectangle is three times its width, and the area of the rectangle is 1,728 square metres. Find the perimeter of the square.

Solution:
Let the width of the rectangle = x metres
Then the length of the rectangle = 3x metres

We know,
Area of the rectangle = length × width
⇒ 3x × x = 1728
⇒ 3x² = 1728
⇒ x² = 1728/3
⇒ x² = 576
⇒ x = √576
⇒ x = 24

Thus,
Width of rectangle = 24 m
Length of rectangle = 3 × 24 = 72 m

∴ Perimeter of the rectangle = 2(length + width)
= 2(72 + 24)
= 2 × 96
= 192 metres

Since the perimeter of the square = perimeter of the rectangle,
∴ Perimeter of the square = 192 metres.

১৩,১৬৭.

A can do a piece of work in 10 days working, 8 hours per day. If B is two - thirds as efficient as A, then in how many days can B alone do the same piece of work, working 5 hours per day?

ক
ক) 15
খ
খ) 18
গ
গ) 20
ঘ
ঘ) 24

ব্যাখ্যা

Time taken by A alone to do the work = (10 × 8)
= 80 hrs.
Since B is two -thirds as efficient as A,
So the time taken by B to do the work
= 80 × (3/2)hrs
= 120 hrs.
∴ Required time = (120/5)
= 24 days.

১৩,১৬৮.

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

ক
65
খ
71
গ
82
ঘ
68

ব্যাখ্যা

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

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

Given,
1st condition: x + y = 8
⇒ x = 8 - y .......(1)

2nd condition:
(10x + y) - (10y + x) = 54
⇒ 9x - 9y = 54
⇒ 9(8 - y) - 9y = 54
⇒ 72 - 9y - 9y = 54
⇒ 72 - 18y = 54
⇒ - 18y = 54 - 72
⇒ - 18y = - 18
⇒ y = 1

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

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

১৩,১৬৯.

There are six numbers 30, 72, 53, 68, x and 87 out of which x is unknown. The average value of the numbers is 60. What is the value of x?

ক
ক) 48
খ
খ) 50
গ
গ) 51
ঘ
ঘ) 55

ব্যাখ্যা

According to the question,
(30 + 72 + 53 + 68 + x + 87)/6 = 60
310 + x = 360
x = 360 - 310
x = 50

১৩,১৭০.

Surface area of hollow cylinder with radius 'r' and height 'h' is measured by

ক
ক) 2πr - h
খ
খ) 2πrh
গ
গ) 2πh
ঘ
ঘ) πr²

ব্যাখ্যা

Question: Surface area of hollow cylinder with radius 'r' and height 'h' is measured by

Solution:
ফাঁপা সিলিন্ডারের বক্রতলের ক্ষেত্রফল = ভূমির পরিধি × উচ্চতা
=2πr × h
=2πrh

১৩,১৭১.

What is the slope of the line perpendicular to the line given by the equation y = 3/4x - 2?

ক
3/4
খ
- 3/4
গ
4/3
ঘ
- 4/3

ব্যাখ্যা

Question: What is the slope of the line perpendicular to the line given by the equation y = 3/4x - 2?

Solution:
The equation of the line is y = 3/4x - 2

This is in the slope-intercept form y = mx + c

So, Slope(m) = 3/4

For two lines to be perpendicular, the product of their slopes must equal -1.
∴ m₁ . m₂ = - 1

Here, m₁ = 3/4
∴ m₂ = -1/(3/4)
= - 4/3

১৩,১৭২.

In the beginning, Ram works at a rate such that he can finish a piece of work in 24 hrs, but he only works at this rate for 16 hrs. After that, he works at a rate such that he can do the whole work in 18 hrs. If Ram is to finish this work at a stretch, how many hours will he take to finish this work?

ক
20 hrs 30 minute
খ
21 hrs
গ
21 hrs 20 minute
ঘ
22 hrs
ঙ
None of the above

ব্যাখ্যা

Question: In the beginning, Ram works at a rate such that he can finish a piece of work in 24 hrs, but he only works at this rate for 16 hrs. After that, he works at a rate such that he can do the whole work in 18 hrs. If Ram is to finish this work at a stretch, how many hours will he take to finish this work?

Solution:
Ram’s 16 hr work = 16/24 = 2/3.
Remaining work = 1 - (2/3) = 1/3.

Using work and time formula, this will be completed in (1/3) × 18 i.e. 6 hrs.
So, total time to complete work = 16 + 6 = 22 hrs.

১৩,১৭৩.

The length of a rope, to which a cow is tied, is increased from 19 m to 30 m. How much additional ground will it be able to graze? [Assume that the cow is able to move on all sides with equal ease.]

ক
1696 sq. m
খ
1694 sq. m
গ
1594 sq. m
ঘ
1756 sq. m

ব্যাখ্যা

Question: The length of a rope, to which a cow is tied, is increased from 19 m to 30 m. How much additional ground will it be able to graze? [Assume that the cow is able to move on all sides with equal ease.]

Solution:
The cow can graze the area covered by the circle of radius 19 m initially, because the length of the rope is 19 m.
Area of a circle = π × (radius)²
Therefore, the initial area that the cow can graze = (22/7) × 19² sq. m.
When the length of the rope is increased to 30 m, grazing area becomes = (22/7) × 30² sq. m.
The additional area it could graze when length is increased from 19 m to 30 m = (22/7) × (30² - 19²) sq. m.
= (22/7) × (30 + 19)(30 - 19) = (22/7) × 49 × 11 = 1694 sq. m.

১৩,১৭৪.

A water tank is two-fifth full. Pipe A can fill a tank in 10 minutes and pipe B can empty it in 6 minutes. If both the pipes are open, how long will it take to empty or fill the tank completely?

ক
6 min.to empty
খ
6 min.to fill
গ
9 min.to empty
ঘ
9 min.to fill

ব্যাখ্যা

Question: A water tank is two-fifth full. Pipe A can fill a tank in 10 minutes and pipe B can empty it in 6 minutes. If both the pipes are open, how long will it take to empty or fill the tank completely?

Solution:
Clearly, pipe B is faster than pipe A and so,the tank will be emptied.

part to be emptied = 2/5

part emptied by (A + B) in 1 minute = (1/6 - 1/10) = 1/15

1/15 part emptied in 1 minute
∴ Full part emptied in 15 minutes
∴ 2/5 part emptied in (15 × 2)/5 minutes
= 6 minutes

so, the tank will be emptied in 6 min.

১৩,১৭৫.

If the radius of a cylinder is decreased by 40% and the height is increased by 50% to form a new cylinder, the volume will be decreased by -

ক
ক) 46%
খ
খ) 54%
গ
গ) 62%
ঘ
ঘ) 48%

ব্যাখ্যা

Question: If the radius of a cylinder is decreased by 40% and the height is increased by 50% to form a new cylinder, the volume will be decreased by -

Solution:
মনে করি,
ব্যাসার্ধ, r = 10 একক
এবং উচ্চতা, h = 10 একক

আয়তন = πr²h
= π × (10)² × 10
= 1000π একক³

40% হ্রাসে ব্যাসার্ধ = 10 - 4 = 6 একক
50% বৃদ্ধিতে উচ্চতা = 10 + 5 = 15 একক

নতুন আয়তন = π × (6)² × 15
= 540 একক³

আয়তন হ্রাস পেয়েছে = 1000 - 540 = 460 একক³
আয়তন শতকরা হ্রাস পেয়েছে = (460 × 100)/(1000 = 46%

১৩,১৭৬.

If n is an integer between 20 and 80, then any of the following could be n + 7 except-

ক
88
খ
80
গ
47
ঘ
56

ব্যাখ্যা

Question: If n is an integer between 20 and 80, then any of the following could be n + 7 except-

Solution:
maximum value of n + 7 = 80 + 7 = 87
88 > 87
So, the correct answer is A

১৩,১৭৭.

The water of a pond is increasing doubled the amount daily. If it becomes full in 24 days, when was it half-filled?

ক
12th day
খ
18th day
গ
22th day
ঘ
20th day
ঙ
23rd day

ব্যাখ্যা

Question: The water of a pond is increasing doubled the amount daily. If it becomes full in 24 days, when was it half-filled?

Solution:
As it is an exponential growth,
we should start from the end.

full on the 24th day
half on the 23rd day
1/4 th on the 22nd day
1/8 th on the 21st day

১৩,১৭৮.

If x = 2y = 4z and xyz = 64, find the value of x.

ক
11
খ
8
গ
12
ঘ
6

ব্যাখ্যা

Question: If x = 2y = 4z and xyz = 64, find the value of x.

Solution:
Given,
x = 2y = 4z
So, y = x / 2 and z = x / 4

Now,
xyz = 64
⇒ x × (x/2) × (x/4) = 64
⇒ x³/8 = 64
⇒ x³ = 64 × 8
⇒ x³ = 512
⇒ x³ = 8³
∴ x = 8

১৩,১৭৯.

Three years ago, the average age of Anik, Pritom, and Varsha was 27 years. If five years ago, the average age of Pritam and Varsha was 20 years, find the present age of Anik.

ক
30
খ
40
গ
60
ঘ
25

ব্যাখ্যা

Question: Three years ago, the average age of Anik, Pritom, and Varsha was 27 years. If five years ago, the average age of Pritam and Varsha was 20 years, find the present age of Anik.

Solution:
Sum of the present ages of Anik, Pritam and Varsha = (27 × 3 + 3 × 3) years = 90 years.
Sum of the present ages of Pritam and Varsha = (20 × 2 + 5 × 2) years = 50 years.
Anik's present age = (90 - 50) years = 40 years.

১৩,১৮০.

A, B, C have the total money of Tk. 1400. B have half of A and C have half of B. the amount of C is

ক
ক) 200
খ
খ) 400
গ
গ) 800
ঘ
ঘ) 300

ব্যাখ্যা

question: A, B, C have the total money of Tk. 1400. B have half of A and C have half of B. The amount of C is

Solution:
Let A have X

then,
B have X/2 and
C have X/4

so,
X + (X/2) + (X/4) = 1400
7X/4 = 1400
X = 800

hence,
the amount of C is (800/4) or, 200

১৩,১৮১.

The product of the roots of the equation 2a² - 5a + p = 10 is - 6. Find the value of p.

ক
ক) - 2
খ
খ) - 4
গ
গ) - 6
ঘ
ঘ) - 8

ব্যাখ্যা

Question: The product of the roots of the equation 2a² - 5a + p = 10 is - 6. Find the value of p.

Solution:
Rearranging the given equation we have 2a² - 5a + (p - 10) = 0

We know that,
if ax² + bx + c = 0 is a quadratic equation, then the product of their roots = c/a

Given the product of the roots = - 6
⇒ (p - 10)/2 = - 6
⇒ (p - 10) = - 12
⇒ p = - 12 + 10
p = - 2

১৩,১৮২.

If 3x - 4y > 2x + 3y, then which of the following must be true?

ক
ক) x > y
খ
খ) y > x
গ
গ) x > 0
ঘ
ঘ) y > 0

ব্যাখ্যা

Question: If 3x - 4y > 2x + 3y, then which of the following must be true?

Solution:
3x - 4y > 2x + 3y
⇒ 3x - 2x > 3y + 4y
⇒ x > 7y
From that we surely say that x > y.

১৩,১৮৩.

If a sum doubles in 12 years, how much will it be in 8 years ?

ক
3/2
খ
5/3
গ
2/5
ঘ
1/2

ব্যাখ্যা

Question: If a sum doubles in 12 years, how much will it be in 8 years ?

Solution:
let the rate is r
we know,
I = Pnr/100
r = (I × 100)/Pn
= (P × 100)/(P × 12)
= 100/12 %

in 8 years,
I = Pnr/100
= (P × 8 × 100/12)/100
= 2P/3
= 2/3 of P

It will be (1 + 2/3) = 5/3P

১৩,১৮৪.

A project team of 3 people is to be formed from 4 seniors and 5 juniors, with exactly 1 senior included. In how many ways can this be done?

ক
40
খ
120
গ
80
ঘ
60
ঙ
None of these

ব্যাখ্যা

Question: A project team of 3 people is to be formed from 4 seniors and 5 juniors, with exactly 1 senior included. In how many ways can this be done?

Solution:
Step₁:
Select 1 senior from 4,
⁴C₁ = 4!/1!(4 - 1)!
= (4 × 3!)/3!
= 4

Step₂:
Select 2 juniors from 5 (since total team size is 3),
⁵C₂ = 5!/2!(5 - 2)!
= (5 × 4 × 3!)/(2 × 3!)
= 10

∴ Total number of ways = 4 × 10 = 40

So the number of ways to form a team of 3 people with exactly 1 senior is 40.

১৩,১৮৫.

Express C = {x : x positive integer and x² < 18} by using roster method.

ক
{1, 2, 3, 4, 5}
খ
{1, 2, 3, 4}
গ
{1, 3, 5}
ঘ
{ }

ব্যাখ্যা

Question: Express C = {x : x positive integer and x² < 18} by using roster method.

Solution:
1, 2, 3, 4, 5,... are positive integers.
Here,
if x = 1 then x² = 1² = 1
if x = 2 then x² = 2² = 4
If x = 3 then x² = 3² = 9
if x = 4 then x² = 4² = 16
If x = 5 then x² = 5² = 25 ; which is greater than 18.

∴ as per conditions the acceptable positive integers are 1, 2, 3 and 4.
∴ the given set is C = {1, 2, 3, 4}

১৩,১৮৬.

If 3x + 2y = 10 and 2x - 2y = 5, then find the value of (6 + 4x).

ক
15
খ
12
গ
18
ঘ
24

ব্যাখ্যা

Question: If 3x + 2y = 10 and 2x - 2y = 5, then find the value of (6 + 4x).

Solution:
Given that,
3x + 2y = 10 ........... (1)
and 2x - 2y = 5 ...........(2)
Now,
(1) + (2) ⇒ 3x + 2y + 2x - 2y = 10 + 5
⇒5x = 15
⇒x = 15/5
∴ x = 3

∴ 6 + 4x = 6 + (4 × 3) = 6 + 12 = 18

১৩,১৮৭.

66 cubic centimeters of silver is drawn into a wire 1 mm in diameter. The length of the wire in meters will be:

ক
82 m
খ
84 m
গ
68 m
ঘ
64 m

ব্যাখ্যা

Question: 66 cubic centimeters of silver is drawn into a wire 1 mm in diameter. The length of the wire in meters will be:

Solution:
Let, the length of the wire be h
Radius = 1/2 mm = 1/20 cm

ATQ,
πr²h = 66
⇒ (22/7) × (1/20)² × h = 66
⇒ h = (66 × 7 × 20 × 20)/22
⇒ h = 8400 cm
⇒ h = 8400/100 m
∴ h = 84 m

১৩,১৮৮.

How many days are there in y weeks y days?

ক
ক) 8y
খ
খ) 14y
গ
গ) 7y + 1
ঘ
ঘ) 7y²

ব্যাখ্যা

প্রশ্ন : How many days are there in y weeks y days?

সমাধান : y weeks y days = (7y + y) days = 8y days

১৩,১৮৯.

39 persons can repair a road in 12 days, working 5 hours a day. In how many days will 30 persons, working 6 hours a day, complete the work?

ক
10 days
খ
13 days
গ
14 days
ঘ
15 days

ব্যাখ্যা

Question: 39 persons can repair a road in 12 days, working 5 hours a day. In how many days will 30 persons, working 6 hours a day, complete the work?

Solution:
৩৯ জন ৫ ঘণ্টা কাজ করে রাস্তা মেরামত করে ১২ দিনে
৩৯ জন ১ ঘণ্টা কাজ করে রাস্তা মেরামত করে ১২ × ৫ দিনে
১ জন ১ ঘণ্টা কাজ করে রাস্তা মেরামত করে ১২ × ৫ × ৩৯ দিনে
৩০ জন ১ ঘণ্টা কাজ করে রাস্তা মেরামত করে (১২ × ৫ × ৩৯)/৩০ দিনে
৩০ জন ৬ ঘণ্টা কাজ করে রাস্তা মেরামত করে (১২ × ৫ × ৩৯)/(৩০ × ৬) দিনে
= ১৩ দিনে

১৩,১৯০.

A boy traveled from the home to the college at the rate or 25 km/hr and walked back at the rate of 4 km/ hr. If the whole journey took 5 hours 48 minutes, find the distance of the college from the home.

ক
ক) 5 km
খ
খ) 10 km
গ
গ) 15 km
ঘ
ঘ) 20 km

ব্যাখ্যা

Let the distance is D
While going with the speed of 25kmph
Formula
Speed=Distance/Time
Speed=25kmph
Distance=D
So,Time=D/25
Similarly, for return journey
The distance will be same as he returns from post office to village
Time=D/4
Now,Total time is given and we have time of onward and return Journey
Add this two time and equate with given time
D/25 + D/4=5 hrs 48 Min………….(1)
5hrs 48min can be written as
48min=48/60=4/5hrs
5hrs adding in this makes 5 + 4/5 = 29/5…….(2)
From (1) and (2)
D/25+D/4=29/5
29D/100=29/5
D=(29/5)*(100/29)
D=20
Answer is 20Km

১৩,১৯১.

If the income of A is 10% more than that of B and the income of B is 20% less than that of C, then the income of A, B and C respectively are in the ratio

ক
ক) 22 : 18 : 25
খ
খ) 22 : 20 : 25
গ
গ) 10 : 9 : 7
ঘ
ঘ) 11 : 10 : 8

ব্যাখ্যা

ধরি, C এর ইনকাম 100 টাকা
তাহলে B এর ইনকাম (100 - 100 এর 20%) = 80 টাকা
এবং A এর ইনকাম (80 + 80 এর 10 %) = 88 টাকা।
∴ A : B : C = 88 : 80 : 100 = 22 : 20 : 25

১৩,১৯২.

A starts business with tk. 3500 and after 5 months, B joins with A as his partner. After a year, the profit is divided in a ratio 2 : 3. What is B's contribution in the capital?

ক
7500
খ
8000
গ
8500
ঘ
9000

ব্যাখ্যা

Question: A starts business with tk. 3500 and after 5 months, B joins with A as his partner. After a year, the profit is divided in the ratio 2 : 3. What is B's contribution in the capital?

Solution:
Let B's capital be tk. x.
Then, (3500 x 12)/7x = 2/3
⇒14x = 126000
⇒ x = 9000 tk

১৩,১৯৩.

If the price of onion goes up by 25%, by how much should usage be reduced in order to keep the total expense for onion as before?

ক
ক) 33(1/3)%
খ
খ) 25%
গ
গ) 20%
ঘ
ঘ) 24%

ব্যাখ্যা

পিয়াজের দাম বেড়ে 100 থেকে 125 হলো
∴ খরচ সমান রাখতে হলে, এখন 125 টাকায় কিনা যায় 1 অংশ
∴ 100 টাকায় কিনা যায় (1/125) × 100 অংশ
= 4/5 অংশ
∴ ব্যবহার কমাতে হবে = 1 - 4/5
= 1/5 অংশ।
অতএব, 1/5 × 100%
= 20% কমাতে হবে।

১৩,১৯৪.

Some months have 30 days and some have 31. How many months have at least 28 days?

ক
12
খ
1
গ
10
ঘ
3
ঙ
5

ব্যাখ্যা

Question: Some months have 30 days and some have 31. How many months have at least 28 days?

Solution:
Every month has at least 28 days. [February has 28 days in a common year and 29 in a leap year].

We know,
January, March, May, July, August, October, December is 31 days
April, June, September, November is 30 days
And February is 28 or 29 days

So all 12 months have at least 28 days.

১৩,১৯৫.

A train can travel 50% faster than a car. Both start from point A at the same time and reach point B 75 kms away from A at the same time. On the way, however, the train lost about 12.5 minutes while stopping at the stations. The speed of the car is:

ক
ক) 80 kmph
খ
খ) 90 kmph
গ
গ) 120 kmph
ঘ
ঘ) 140 kmph

ব্যাখ্যা

Question: A train can travel 50% faster than a car. Both start from point A at the same time and reach point B 75 kms away from A at the same time. On the way, however, the train lost about 12.5 minutes while stopping at the stations. The speed of the car is:

Solution:
Let speed of the car be x kmph.
Then, speed of the train = 150x/100 = 3x/2

Now
(75/x) - {75/(3x/2)} = 125/(10 × 60)
(75/x) - (50/x) = 5/24
(75 - 50)/x = 5/24
25/x = 5/24
5x = 24 × 25
x = ( 24 × 25)/5
x = 120 kmph

১৩,১৯৬.

If simple interest on 600 taka for 4 years and on 600 taka for 2 years combined together is180 taka, find the rate of interest.

ক
ক) 4%
খ
খ) 5%
গ
গ) 7%
ঘ
ঘ) 9%

ব্যাখ্যা

If the rate is r%,
600 × r × 4/100 + 600 × r × 2/100 = 180
or, r = 5%

১৩,১৯৭.

A firm pays fixed costs of Tk. 1500 plus another Tk. 60 for each unit produced. How much can it produce for a budget of Tk. 4,800?

ক
ক) 42
খ
খ) 55
গ
গ) 67
ঘ
ঘ) 71

ব্যাখ্যা

1500 টাকা হলো fixed cost এবং প্রতি ইউনিট উৎপাদনের জন্য লাগে, ৬০ টাকা।
বাজেট ৪৮০০ টাকা হলে,
প্রশ্নমতে, 1500+60x = 4800
বা, x = 3300/60 = 55.

১৩,১৯৮.

A point on the edge of a fan blade that is rotating in a plane is 10 centimeters from the center of the fan. WHat is the distance traveled, in centimeters, by this point in 15 seconds when the fan runs at the rate of 300 revolutions per minutes?

ক
750π
খ
1500π
গ
1875π
ঘ
3000π
ঙ
7500π

ব্যাখ্যা

Question: A point on the edge of a fan blade that is rotating in a plane is 10 centimeters from the center of the fan. WHat is the distance traveled, in centimeters, by this point in 15 seconds when the fan runs at the rate of 300 revolutions per minutes?

Solution:
Revolutions/sec = 300/60 = 5
In 15 sec, revolutions = 15 × 5 = 75

Distance traveled in one revolution is 2πr = 2 × π × 10 = 20π
Distance traveled in 75 revolution is = 75 × 20π = 1500π

১৩,১৯৯.

If x + 1/x = 2 then, find the value of x⁷ - 1/x⁷

ক
0
খ
1
গ
2
ঘ
14

ব্যাখ্যা

Question: If x + 1/x = 2 then, find the value of x⁷ - 1/x⁷

Solution:
Given,
x + 1/x = 2
⇒ x² + 1 = 2x
⇒ x² - 2x + 1 = 0
⇒ x² - x - x + 1 = 0
⇒ x(x - 1)- 1(x - 1) = 0
⇒ (x - 1)(x - 1) = 0
∴ x - 1 = 0
⇒ x = 1

∴ x⁷ - 1/x⁷ = 1⁷ - 1/1⁷
= 1 - 1
= 0

১৩,২০০.

In a test, minimum passing percentage for girls and boys are 45% and 60% respectively. A boy scored 767 marks and failed by 313 marks. What are the minimum passing marks for girls?

ক
920
খ
910
গ
810
ঘ
850

ব্যাখ্যা

Question: In a test, the minimum passing percentage for girls and boys is 45% and 60% respectively. A boy scored 767 marks and failed by 313 marks. What are the minimum passing marks for girls?

Solution:
pass mark = 767 + 313
= 1080

0.6 × total marks = 1080
⇒ total marks = 1080/0.6
= 1800

the minimum passing marks for girls is = 1800 × 0.45
= 810