Bank Math

PrepBank · পাতা ৬২ / ১৬১ · ৬,১০১–৬,২০০ / ১৬,১২৪

৬,১০১.

Arafat has TK. 420. He purchased fifty mangoes and thirty oranges with the whole amount. He then chose to return six mangoes for nine oranges as both quantities are equally priced. What is the price of each Mango in Tk?

ক
ক) 4.00
খ
খ) 4.50
গ
গ) 5.00
ঘ
ঘ) 6.00
ঙ
ঙ) None

ব্যাখ্যা

Price of 6 mangoes = Price of 9 oranges
⇒ (Price of 1 mango) : (Price of 1 orange) = 9/6 = 3 : 2
Suppose, the price of each mango is Tk. 3x, and the price of each orange is Tk. 2x.
According to the question,
50×3x + 30×2x = 420
⇒ x = 2
Price of each mango = 3×2 = Tk. 6.

৬,১০২.

An amount of Tk. 12,000 yields a simple interest of Tk. 2,160 in 4 years. What is the annual rate of interest?

ক
4.5%
খ
5%
গ
6.4%
ঘ
10%

ব্যাখ্যা

Question: An amount of Tk. 12,000 yields a simple interest of Tk. 2,160 in 4 years. What is the annual rate of interest?

Solution:
Given, Principal, P = 12000
Simple Interest, SI = 2160
Time, n = 4 years
Rate of interest, r = ?

We know, I = Pnr/100
⇒ r = (I × 100)/(P × n)
⇒ r = (2160 × 100)/(12000 × 4)
⇒ r = 216000/48000
⇒ r = 4.5%

So, the annual rate of interest is 4.5%.

৬,১০৩.

A rectangular block 6 cm by 12 cm by 15 cm is cut up into an exact number of equal cubes. Find the least possible number of cubes.

ক
40
খ
30
গ
50
ঘ
45
ঙ
55

ব্যাখ্যা

Volume of the block = (6 × 12 × 15) cm3

= 1080 cm3

Side of the largest cube = H.C.F of 6 cm, 12 cm, 15 cm

= 3 cm.

Volume of this cube = (3 × 3 × 3) cm3

= 27 cm3

Number of cubes = 1080/27

= 40.

৬,১০৪.

A sum of money at simple interest amounts to Tk. 721 in 3 years and to Tk. 754 in 4 years. The sum is-

ক
Tk. 622
খ
Tk. 655
গ
Tk. 755
ঘ
Tk. 645

ব্যাখ্যা

Question: A sum of money at simple interest amounts to Tk. 721 in 3 years and to Tk. 754 in 4 years. The sum is-

Solution:
Simple interest for 1 year = Tk. (754 - 721)
= Tk. 33

∴ Simple interest for 3 years = Tk.(33 × 3)
= Tk. 99

∴ Sum = (721 - 99)
= Tk. 622

৬,১০৫.

In how many ways can six different rings be worn on four fingers of one hand?

ক
ক) 6⁴
খ
খ) 4⁶
গ
গ) 6
ঘ
ঘ) 216

ব্যাখ্যা

Question: In how many ways can six different rings be worn on four fingers of one hand?

Solution:
Number of fingers n = 4
Number of rings r = 6
∴ 6 rings may be worn in = 4⁶ ways.

৬,১০৬.

Which number will complete the series:
1, 3, 7, 15, 31, 63, 127, __?

ক
259
খ
245
গ
255
ঘ
252

ব্যাখ্যা

Question: Which number will complete the series:
1, 3, 7, 15, 31, 63, 127, __?

Solution:
3 - 1 = 2
7 - 3 = 4 = 2 × 2
15 - 7 = 8 = 4 × 2
31 - 15 = 16 = 8 × 2
63 - 31 = 32 = 16 × 2
127 - 63 = 64 = 32 × 2

∴ The next number of 127 will be 127 + 64 × 2
= 127 + 128
= 255

৬,১০৭.

ক
ক) y = 0
খ
খ) x = √y
গ
গ) x = y
ঘ
ঘ) x = y/2

ব্যাখ্যা

প্রশ্ন:

সমাধান:

৬,১০৮.

7589 - ? = 3434

ক
4242
খ
4155
গ
1123
ঘ
11023

ব্যাখ্যা

Question: 7589 - ? = 3434

Solution:
7589 - ? = 3434
⇒ 7589 - 3434 = ?
∴ ? = 4155

৬,১০৯.

In a class, there are 15 boys and 10 girls. Three students are selected at random. The probability that 1 girl and 2 boys are selected, is-

ক
21/46
খ
25/117
গ
1/50
ঘ
3/25
ঙ
None of these

ব্যাখ্যা

Question: In a class, there are 15 boys and 10 girls. Three students are selected at random. The probability that 1 girl and 2 boys are selected, is-

Solution:
Let S be the sample space and E be the event of selecting 1 girl and 2 boys.
Then,
n(S) = Number ways of selecting 3 students out of 25
= ²⁵C₃
= 2300

n(E) = (¹⁰C₁ × ¹⁵C₂) = 10 × 105
= 1050

∴ P(E) = n(E)/n(S) = 1050/2300 = 21/46

৬,১১০.

X’s age 3 years ago was three times the present age of Y. At present Z’s age is twice the age of Y and also Z is 12 years younger than X. What is the present age of Z?

ক
17 years
খ
20 years
গ
16 years
ঘ
18 years

ব্যাখ্যা

Question: X’s age 3 years ago was three times the present age of Y. At present Z’s age is twice the age of Y and also Z is 12 years younger than X. What is the present age of Z?

Solution:
Let,
Their present age is X, Y, Z respectively.

As X’s age 3 years ago was three times the present age of Y.
X - 3 = 3y
X - 3y = 3……….(i)

Z’s age is twice the age of Y
∴ Z = 2Y……….(ii)

Z is 12 years younger than X
∴ X - 12 = Z
Or, X - 12 = 2Y [ putting Z = 2Y from equation (ii) ]
Or, X - 2Y = 12……..(iii)

(iii) - (i),
X - 2Y - X + 3Y = 12 - 3
Or, Y = 9

So, the present age of Y is 9.
∴ the present age of Z is = 2Y = 2 x 9 = 18 years.

৬,১১১.

A student's marks were wrongly entered as 83 instead of 63. Due to the error the average marks for the class got increased by 0.5. Find the number of students in the class.

ক
10
খ
20
গ
40
ঘ
73

ব্যাখ্যা

Question: A student's marks were wrongly entered as 83 instead of 63. Due to the error the average marks for the class got increased by 0.5. Find the number of students in the class.

Solution:
Let the number of students in the class be x.
Total increase in marks = x × 1/2 = x/2

According to the question
⇒ x/2 = (83 - 63)
⇒ x/2 = 20
⇒ x = 40.

∴The total number of students in the class is 40.

৬,১১২.

If A591 is divisible by 11, find the value of the smallest natural number A?

ক
ক) 8
খ
খ) 9
গ
গ) 7
ঘ
ঘ) 5

ব্যাখ্যা

Question: If A591 is divisible by 11, find the value of the smallest natural number A?

Solution:
কোন সংখ্যা ১১ দ্বারা বিভাজ্য হবে যদি তার জোড় স্থানীয় অংক গুলোর সমষ্টি এবং বিজোড় স্থানীয় অংকগুলোর সমষ্টির পার্থক্য ০ হয় বা ১১ দ্বারা বিভাজ্য হয়।

অর্থাৎ,
(A + 9) - (5 + 1) = 0 বা, 11 দ্বারা বিভাজ্য হবে।
A এর মান 8 হলে, এটি 11 দ্বারা বিভাজ্য হবে।
A এর মান -3 হলে, সমষ্টি 0 আসবে তখনও 11 দ্বরা বিভাজ্য হবে।

কিন্তু -3 স্বাভাবিক সংখ্যা না তাই উত্তর 8 হবে।

∴ A এর মান 8 হলে সংখ্যটি 11 দ্বারা বিভাজ্য হবে।

৬,১১৩.

If y > 1 and y < 4, then which of the following expressions is positive?
I. (y - 1)(y - 4)
II. (1 - y)(y - 4)
III. (1 - y)(4 - y)

ক
I only
খ
II only
গ
III only
ঘ
II and III

ব্যাখ্যা

Question: If y > 1 and y < 4, then which of the following expressions is positive?
I. (y - 1)(y - 4)
II. (1 - y)(y - 4)
III. (1 - y)(4 - y)

Solution:
Given,
y > 1 and y < 4

For expression I: (y - 1)(y - 4)
Since y > 1, (y - 1) will be positive.
Since y < 4, (y - 4) will be negative.
∴ (y - 1)(y - 4) = positive × negative = negative

For expression II: (1 - y)(y - 4)
Since y > 1, (1 - y) will be negative.
Since y < 4, (y - 4) will be negative.
∴ (1 - y)(y - 4) = negative × negative = positive

For expression III: (1 - y)(4 - y)
Since y > 1, so (1 - y) will be negative.
Since y < 4, so (4 - y) will be positive.
∴ (1 - y)(4 - y) = negative × positive = negative

∴ Only expression II is positive.

৬,১১৪.

Select the odd one.

ক
ক)
খ
খ)
গ
গ)
ঘ
ঘ)

ব্যাখ্যা

Question: Select the odd one.

Solution:
১ম চিত্রে, বাইরের ক্ষেত্রটি সীমাবদ্ধ নয়, ভেতরের ক্ষেত্রটি সীমাবদ্ধ (চতুর্ভুজ)।
২য় চিত্রে, বাইরের ক্ষেত্রটি সীমাবদ্ধ (ত্রিভুজ), ভেতরের ক্ষেত্রটি সীমাবদ্ধ নয়।
৩য় চিত্রে, বাইরের ক্ষেত্রটি সীমাবদ্ধ নয়, ভেতরের ক্ষেত্রটি সীমাবদ্ধ (চতুর্ভুজ)।

অতএব, তিনটি চিত্রেই একটি সীমাবদ্ধ ক্ষেত্র ও আরেকটি সীমাবদ্ধ নয়।
৪র্থ চিত্রে, দুটো ক্ষেত্রই সীমাবদ্ধ। তাই এতি অন্যগুলো থেকে ভিন্ন।

৬,১১৫.

If one-seventh of a number exceeds its eleventh part by 100, what is the number?

ক
770
খ
1100
গ
1825
ঘ
1925

ব্যাখ্যা

Question: If one-seventh of a number exceeds its eleventh part by 100, what is the number?

Solution:
Let the number be x. Then,
According to the question
x/7 - x/11 = 100
⇒ 11x - 7x = 7700
⇒ 4x = 7700
∴ x = 1925

৬,১১৬.

If p and q are two positive integers and p + q = 5 and we need to find the probability that p equals 1.

ক
1/4
খ
2/3
গ
1/5
ঘ
3/2

ব্যাখ্যা

Question: If p and q are two positive integers and p + q = 5 and we need to find the probability that p equals 1.

Solution:
Let's start by writing all the possible outcomes for p + q = 5, given that both p and q are positive integers.
The possible outcomes are:

p = 1, q = 4
p = 2, q = 3
p = 3, q = 2
p = 4, q = 1

So, Total Number of Outcomes = 4
Number of Favorable outcomes in which p = 1 is 1 (p = 1, q = 4)

∴ The probability = 1/4

৬,১১৭.

Poltu divided one-third of his money in a 2 : 3 ratio between Ratul and Shovon, and the remainder in the same ratio between Nitul and Sanhar. If Sanhar received 360 Tk, what was Poltu's total amount?

ক
1800 Tk
খ
600 Tk
গ
900 Tk
ঘ
1500 Tk

ব্যাখ্যা

Question: Poltu divided one-third of his money in a 2 : 3 ratio between Ratul and Shovon, and the remainder in the same ratio between Nitul and Sanhar. If Sanhar received 360 Tk, what was Poltu's total amount?

Solution:
If Sanhar received 360 Tk,
Nitul received = (360/3) × 2 Tk
= 240 Tk

Total amount Nitul and Sanhar received = 360 + 240 Tk
= 600 Tk [which is {1 - (1/3)} or 2/3 of total amount]

∴ Total amount = (600/2) × 3 Tk
= 900 Tk

৬,১১৮.

A is four years older than B, who is thrice as old as C. If the total of the ages of A, B, and C is 46, how old is A?

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

ব্যাখ্যা

Question: A is four years older than B, who is thrice as old as C. If the total of the ages of A, B, and C is 46, how old is A?

Solution:
Let
C's age be = a years
Then, B's age = 3a years
A's age = (3a + 4) years

∴ (3a + 4) + 3a + a = 32
⇒ 7a + 4 = 46
⇒ 7a = 42
⇒ a = 6

Hence, A's age = (3 × 6) + 4 = 22 years.

৬,১১৯.

If 8 workers can assemble a car in 9 hours, how long would it take 12 workers to assemble the same car?

ক
3 hours
খ
6 hours
গ
9 hours
ঘ
12 hours

ব্যাখ্যা

Question: If 8 workers can assemble a car in 9 hours, how long would it take 12 workers to assemble the same car?

Solution:

Here, M₁ = 8, M₂ = 12, W₁ = W₂ = 1, T₁ = 9, T₂ = ?

(M₁ × T₁)/(M₂ × T₂) = W₁/W₂
⇒ (8 × 9)/ (12 × T₂) = 1
⇒ 12 × T₂ = 72
⇒ T₂ = 72/12
∴ T₂ = 6

৬,১২০.

The sum of 2 positive numbers is 151. The lesser number is 19 more than the square root of the greater number. What is the value of the greater number minus the lesser number?

ক
19
খ
85
গ
91
ঘ
66
ঙ
None

ব্যাখ্যা

Question: The sum of 2 positive numbers is 151. The lesser number is 19 more than the square root of the greater number. What is the value of the greater number minus the lesser number?

Solution:
Let the two positive numbers be x and y; x > y

ATQ,
x + y = 151 ---------- (1)
y = 19 + √x ---------- (2)

Putting the value y = 19 + √x in equ (1)
x + 19 + √x = 151
⇒ x + √x = 151 - 19
⇒ x + √x = 132
⇒ √x = 132 - x
⇒ (√x)² = (132 - x)²
⇒ x = (132)² - 2 . 132 . x + x²
⇒ x² - 265x + 17424 = 0
∴ x = 121, 144

The value of x can be 121 and 144.
The value of x = 144 cannot be possible as it doesn’t satisfies the two condition.
Therefore the value of x is 121.

From equ. (1) we get,
y = 151 - 121
y = 30

∴ The value of greatest number minus the lesser would be x-y, i.e, 121 - 30 = 91

৬,১২১.

Pipe A can fill the tank in 8 hours and pipe B can fill it in 12 hours. If pipe A is opened at 8:00 AM and pipe B is opened at 10:00 AM, then at what time will the tank be full ?

ক
11 : 48 PM
খ
1 : 36 P.M.
গ
2 : 20 P.M.
ঘ
12 : 48 PM

ব্যাখ্যা

Question: Pipe A can fill the tank in 8 hours and pipe B can fill it in 12 hours. If pipe A is opened at 8:00 AM and pipe B is opened at 10:00 AM, then at what time will the tank be full ?

Solution:
A opened 2 hours early to B
In 2 hours A can do 3 × 2 = 6 unit work
Remaining work = 24 - 6 = 18
A + B can do it in
= 18/5 hours
= 3 hours 36 minutes
∴ Tank will be full in 10 A.M. + 3 hours 36 minutes = 1 : 36 P.M.

৬,১২২.

In a bucket there are 5 purple, 15 grey and 25 green balls. If the ball is picked up randomly, find the probability that it is neither grey nor purple?

ক
51/43
খ
2/7
গ
5/9
ঘ
12/13

ব্যাখ্যা

Question: In a bucket there are 5 purple, 15 grey and 25 green balls. If the ball is picked up randomly, find the probability that it is neither grey nor purple?

Solution:
If the ball is neither grey nor purple then it must be green.
There are 45 balls in total of which 25 are green
so the probability of picking a green ball is 25/45 = 5/9

৬,১২৩.

What is the value of 5 + 4 × 5 + 4 × 5²+ 4 × 5³ + 4 ×5⁴ + 4 × 5⁵?

ক
5⁶
খ
5⁷
গ
5⁸
ঘ
5⁹

ব্যাখ্যা

Question: What is the value of 5 + 4 × 5 + 4 × 5²+ 4 × 5³ + 4 ×5⁴ + 4 × 5⁵?

Solution:
5 + 4 × 5 + 4 × 5²+ 4 × 5³ + 4 ×5⁴ + 4 × 5⁵
= 5 + (5 - 1) × 5 + (5 - 1) × 5²+ (5 - 1) × 5³ + (5 - 1) × 5⁴ + (5 - 1) × 5⁵
= 5 + 5² - 5 + 5³ - 5² + 5⁴ - 5³ + 5⁵ - 5⁴ + 5⁶ - 5⁵
= 5⁶

৬,১২৪.

A box contains 5 green, 3 yellow, and 4 black balls. If one ball is drawn at random, what is the probability that it will not be a green ball?

ক
2/3
খ
5/8
গ
5/12
ঘ
7/12

ব্যাখ্যা

Question: A box contains 5 green, 3 yellow, and 4 black balls. If one ball is drawn at random, what is the probability that it will not be a green ball?

Solution:
Given that,
Green balls = 5
Yellow balls = 3
Black balls = 4
∴ Total balls = 5 + 3 + 4 = 12

And, number of non-green balls = Yellow + Black = 3 + 4 = 7

We know,
P(not green) = favorable outcomes/total outcomes
= 7/12

∴ The probability of drawing a non-green ball is 7/12

৬,১২৫.

If the area of the trapezium, whose parallel sides are 6 cm and 10 cm is 32 sq. cm, what will be the distance between the parallel sides?

ক
2 cm
খ
4 cm
গ
5 cm
ঘ
6 cm
ঙ
8 cm

ব্যাখ্যা

Question: If the area of the trapezium, whose parallel sides are 6 cm and 10 cm is 32 sq. cm, what will be the distance between the parallel sides?

Solution:
Parallel sides of a trapezium = 6 cm, and 10 cm
Area of trapezium = (1/2)(sum of the parallel sides) × distance between the parallel sides
32 = (1/2)(6 + 10 ) × distance
⇒ 32 = 8 × distance
⇒ distance = 32/8 = 4 cm

So, the distance between the parallel lines of trapezium = 4 cm.

৬,১২৬.

The side BC of a triangle ABC proceeds to D. If ∠ACD = 112° and ∠B = (3/4) ∠A, then the measure of ∠B is:

ক
30°
খ
45°
গ
48°
ঘ
64°

ব্যাখ্যা

Question: The side BC of a triangle ABC proceeds to D. If ∠ACD = 112° and ∠B = (3/4) ∠A, then the measure of ∠B is:

Solution:

∠ACB = 180 - 112 = 68

∠B + (4/3) ∠B + 68 = 180
⇒ (7/3) ∠B = 112
⇒ ∠B = 48°

৬,১২৭.

If A is the center of the circle shown above and AB = BC = CD, What is the value of x? [Note: Figure not drawn to scale]

ক
15
খ
30
গ
45
ঘ
60

ব্যাখ্যা

Question:

If A is the center of the circle shown above and AB = BC = CD, What is the value of x? [Note: Figure not drawn to scale]

Solution:
Given that AB = BC = CD,
also since AB is the radius then AB = AC = AD = radius,
so we have that: AB = BC = CD = AC = AD,
so basically we have two equilateral triangles ABC and ACD with common base of AC (ABC and ACD are mirror images of each other). Line segment BD cuts the angle ABC in half and since all angles in equilateral triangle equal to 60 degrees then x=60/2=30 degrees.

৬,১২৮.

A man purchased a cow for Tk. 3000 and sold it the same day for Tk. 3600, allowing the buyer a credit of 2 years. If the rate of interest be 10% per annum, then the man has a gain of

ক
ক) 0%
খ
খ) 5%
গ
গ) 7.5%
ঘ
ঘ) 10%

ব্যাখ্যা

Cost price = Tk 3000
Selling price = [{3600 × 100}/{100 + (10 × 2)}]
= Tk. 3000
Gain = 0%.

৬,১২৯.

Find the product of two consecutive numbers if three times the first number is 5 more than twice the second number.

ক
20
খ
30
গ
42
ঘ
56

ব্যাখ্যা

Question: Find the product of two consecutive numbers if three times the first number is 5 more than twice the second number.

Solution:
Let the numbers be a and a + 1.

According to the question:
3 × (first number) = 2 × (second number) + 5
⇒ 3a = 2(a + 1) + 5
⇒ 3a = 2a + 2 + 5
⇒ 3a = 2a + 7
⇒ 3a - 2a = 7
⇒ a = 7

∴ The numbers are 7 and 8.
Product = 7 × 8 = 56

৬,১৩০.

A hat contains a total of 30 cards, of which 12 are marked with a star and the remaining 18 are unmarked. If a card is drawn at random from the hat, what is the probability that it will be a card marked with a star?

ক
1/3
খ
2/3
গ
3/5
ঘ
2/5

ব্যাখ্যা

Question: A hat contains a total of 30 cards, of which 12 are marked with a star and the remaining 18 are unmarked. If a card is drawn at random from the hat, what is the probability that it will be a card marked with a star?

Solution:

Total number of cards = 12 (marked) + 18 (unmarked) = 30

Number of favorable outcomes (marked with a star) = 12

Probability = (Number of favorable outcomes)/(Total number of outcomes)
= 12/30
= 2/5

৬,১৩১.

A fan is listed at Tk. 150 and a discount of 20% is given. Then the selling price is = ?

ক
ক) 180
খ
খ) 150
গ
গ) 120
ঘ
ঘ) 110

ব্যাখ্যা

Selling price = 150×80 / 100 = Tk. 120

৬,১৩২.

The volume of a cone is 300π cubic centimeters. If the radius of its base is 6 cm, what is the height of the cone?

ক
15 cm
খ
20 cm
গ
25 cm
ঘ
32 cm

ব্যাখ্যা

Question: The volume of a cone is 300π cubic centimeters. If the radius of its base is 6 cm, what is the height of the cone?

solution:
দেওয়া আছে,
কোণকের আয়তন, V = 300π ঘন সে.মি.
ভূমির ব্যাসার্ধ, r = 6 সে.মি.
ধরি, কোণকের উচ্চতা = h সে.মি.

আমরা জানি,
কোণকের আয়তন, V = 1/3 × π × r² × h
∴ 300π = 1/3 × π × 6² × h
⇒ 300 = 1/3 × 36 × h (π উভয় পক্ষ থেকে বাদ দিয়ে)
⇒ 300 = 12h
⇒ h = 300 / 12
∴ h = 25 সে.মি.

অতএব, কোণকটির উচ্চতা = 25 সে.মি.

৬,১৩৩.

The quadratic equation whose roots are 1 and (- 1/2):

ক
2x² + x - 1 = 0
খ
2x² - x - 1 = 0
গ
2x² + x + 1 = 0
ঘ
2x² - x + 1 = 0

ব্যাখ্যা

Question: The quadratic equation whose roots are 1 and (- 1/2):

Solution:
The quadratic equation whose roots are 1 and (- 1/2)

৬,১৩৪.

Tickets numbered 1 to 20 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 5?

ক
2/5
খ
1/2
গ
3/10
ঘ
9/20

ব্যাখ্যা

Question: Tickets numbered 1 to 20 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 5?

Solution:
Here, S = {1, 2, 3, 4, ...., 19, 20}
Let E = event of getting a multiple of 3 or 5 = {3, 6, 9, 12, 15, 18, 5, 10, 20}

∴ P(E) = n(E)/n(S)
= 9/20

৬,১৩৫.

Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed -

ক
ক) 210
খ
খ) 1050
গ
গ) 25200
ঘ
ঘ) 21400

ব্যাখ্যা

Number of ways of selecting (3 consonants out of 7) and (2 vowels out of 4)
= ⁷C₃× ⁴C₂
= {(7 × 6 × 5)/(3 × 2 × 1)} × {(4 × 3)/(2 × 1)}
= 210.

৬,১৩৬.

When bent in the form of a circle, a wire has a radius of 28 cm. If the same wire is bent into the shape of a square, what will be its area in cm²?

ক
1936 cm²
খ
1681 cm²
গ
2025 cm²
ঘ
1849 cm²

ব্যাখ্যা

Question: When bent in the form of a circle, a wire has a radius of 28 cm. If the same wire is bent into the shape of a square, what will be its area in cm²?

Solution:
প্রদত্ত, বৃত্তের ব্যাসার্ধ, r = 28 cm

অতএব, পরিধি = 2πr
= 2 × (22/7) × 28 = 176 cm

ধরি, বর্গের বাহু = a cm
তাহলে, বর্গের পরিসীমা = 4a

এখন,
বৃত্তের পরিধি = বর্গের পরিসীমা
⇒ 176 = 4a
⇒ a = 176/4
= 44 cm

∴ বর্গের ক্ষেত্রফল = a²
= 44²
= 1936 cm²

৬,১৩৭.

If (11x - 1)² = 441, then which one of the following could equal x?

ক
4
খ
3
গ
2
ঘ
1

ব্যাখ্যা

Question: If (11x - 1)² = 441, then which one of the following could equal x?

Solution:
(11x - 1)² = 441
⇒ 11x - 1 = √441
⇒ 11x - 1 = 21
⇒ 11x = 22
∴ x = 2

৬,১৩৮.

How many digits will be there to the right of the decimal point in the product of 95.75 and .02554?

ক
ক) 3
খ
খ) 4
গ
গ) 5
ঘ
ঘ) 6

ব্যাখ্যা

Sum of decimal places = 7.
Since the last digit to the extreme right will be zero (since 5 x 4 = 20),
so there will be 6 significant digits to the right of the decimal point.

৬,১৩৯.

If A = {p, q, r, s, t}, then how many proper subsets does A have?

ক
32
খ
16
গ
15
ঘ
31

ব্যাখ্যা

Question: If A = {p, q, r, s, t}, then how many proper subsets does A have?

Solution:
Given that,
A = {p, q, r, s, t}
The number of elements in set A is 5.

We know that,
Number of proper subsets = 2ⁿ - 1 ; [where n = number of elements in the set]
∴ Number of proper subsets of A = 2⁵ - 1
= 32 - 1
= 31

৬,১৪০.

A person rowing against the current can go 2 km per hour. If the speed of the current is 3 km per hour, how much time will he take to cover 32 km, rowing along the current?

ক
3 hr
খ
4 hr
গ
6 hr
ঘ
8 hr

ব্যাখ্যা

Question: A person rowing against the current can go 2 km per hour. If the speed of the current is 3 km per hour, how much time will he take to cover 32 km, rowing along the current?

Solution:
ধরি, ব্যক্তির বেগ x কিমি/ঘণ্টা
স্রোতের বেগ ৩ কিমি/ঘণ্টা

ব্যক্তি স্রোতের বিপরীতে ২ কিমি/ঘণ্টা বেগে যায়।

x - ৩ = ২
∴ x = ৫ কিমি/ঘণ্টা

স্রোতের অনুকূলে বেগ = ৩ + ৫ কিমি/ঘণ্টা
= ৮ কিমি/ঘণ্টা

স্রোতের অনুকূলে যেতে সময় লাগে = ৩২/৮ ঘণ্টা
= ৪ ঘন্টা

৬,১৪১.

sin(θ + 15°) = 3/√12 হলে cos²θ = কত?

ক
ক) 1/√2
খ
খ) 1/4
গ
গ) 1/2
ঘ
ঘ) 1

ব্যাখ্যা

Question: sin(θ + 15°) = 3/√12 হলে cos²θ = কত?

Solution:
sin(θ + 15°) = 3/√12
⇒ sin(θ + 15°) = 3/(2√3)
⇒ sin(θ + 15°) = √3/2
⇒ sin(θ + 15°) = sin60°
⇒ θ + 15° = 60°
⇒ θ = 45°

Now,
cos²θ = (cos 45°)²
= (1/√2)²
= 1/2

৬,১৪২.

What number should come next in the series:
3, 7, 15, 31, 63, .......?

ক
121
খ
95
গ
127
ঘ
135

ব্যাখ্যা

Question: What number should come next in the series:
3, 7, 15, 31, 63, .......?

Solution: দেওয়া আছে,
সিরিজটি হলো: 3, 7, 15, 31, 63, .......

প্রতিটি পার্থক্য আগের পার্থক্যের 2 গুণ।

3 থেকে 7 পর্যন্ত পার্থক্য: 4
7 থেকে 15 পর্যন্ত পার্থক্য: 8 (4 × 2)
15 থেকে 31 পর্যন্ত পার্থক্য: 16 (8 × 2)
31 থেকে 63 পর্যন্ত পার্থক্য: 32 (16 × 2)
সুতরাং, পরবর্তী পার্থক্যটি হবে: 32 × 2 = 64

পরবর্তী সংখ্যাটি হবে শেষ সংখ্যা এবং এই পার্থক্যের যোগফল: 63 + 64 = 127

অতএব, পরবর্তী সংখ্যাটি হলো 127

৬,১৪৩.

The mean weight of 100 students in a class is 46 kg. The mean weight of boys is 50 and of girls is 40 kg. Therefore, the number of boys is-

ক
64
খ
70
গ
55
ঘ
60

ব্যাখ্যা

Question: The mean weight of 100 students in a class is 46 kg. The mean weight of boys is 50 and of girls is 40 kg. Therefore, the number of boys is-

Solution:
Given that,
Total students = 100
Mean weight of all students = 46 kg
∴ Total weight of all students = 100 × 46 = 4600 kg.

Let,
The number of boys = x. Then, the number of girls = 100 - x
Mean weight of boys = 50 kg,
∴ total weight of boys = 50x
And,
Mean weight of girls = 40 kg,
∴ total weight of girls = 40(100 - x)

ATQ,
50x + 40 × (100 - x) = 4600
⇒ 50x + 4000 - 40x = 4600
⇒ 10x = 4600 - 4000
⇒ x = 600/10
∴ x = 60

So the number of boys is 60.

৬,১৪৪.

The ratio of red balls, to yellow balls, to green balls, to blue balls in a basket is 2 : 3 : 4 : 5. What is the probability that a ball chosen at random from the basket is a red ball?

ক
1/7
খ
2/7
গ
3/14
ঘ
3/7

ব্যাখ্যা

Question: The ratio of red balls, to yellow balls, to green balls, to blue balls in a basket is 2 : 3 : 4 : 5. What is the probability that a ball chosen at random from the basket is a red ball?

Solution:
The ratio of red balls, to yellow balls, to green balls in a basket is = 2 : 3 : 4 : 5
let, there are 2x red balls, 3x yellow balls, 4x green balls and 5x blue balls.

∴ Total balls = 2x + 3x + 4x + 5x
= 14x

∴ probability that a ball chosen at random from the basket is a red ball = 2x/14x
= 1/7

৬,১৪৫.

A and B can complete a work in 15 days and 10 days respectively. They started doing the work together but after 2 days, B had to leave and A alone completed the remaining work. The whole work was completed in:

ক
10 days
খ
12 days
গ
14 days
ঘ
16 days

ব্যাখ্যা

Question: A and B can complete a work in 15 days and 10 days respectively. They started doing the work together but after 2 days, B had to leave and A alone completed the remaining work. The whole work was completed in:

Solution:
Work done by (A + B) in 1 day
= (1/15) + (1/10)
= (2 + 3)/30
= 5/30
= 1/6

(A + B)’s 2 days’ work = 2/6
= 1/3

∴ Remaining work = 1 - (1/3)
= 2/3

This part is done by A alone.
Since, one work is done by A in 15 days.
2/3 work is done in = 15 × (2/3)
= 10 days

So, Total number of days = (10 + 2) 12 days
= 12 days

৬,১৪৬.

A city's population increases by 10% each year. If its present population is 50,000, what will be the population after 2 years?

ক
59400
খ
60500
গ
61600
ঘ
63600
ঙ
None

ব্যাখ্যা

Question: A city's population increases by 10% each year. If its present population is 50,000, what will be the population after 2 years?

Solution:
Here we can use the compound interest based formula,
Population after n years
= P × [1 + (r/100)]n
∴ Population after 2 years = 50000 × [1 + (10/100)]²
= 50000 × (110/100)²
= 50000 × 1.21
= 60,500

৬,১৪৭.

If M = 14, TANK = 62, then STARDOM =?

ক
79
খ
89
গ
99
ঘ
109

ব্যাখ্যা

Question: If M = 14, TANK = 62, then STARDOM =?

Solution:

M = 14
TANK = 7 + 26 + 13 + 16 = 62
STARDOM = 8 + 7 + 26 + 9 + 23 + 12 + 14 = 99

৬,১৪৮.

In a ΔABC, AB = BC, ∠B = x° and ∠A = (2x - 20)°. Then, ∠B = ?

ক
30°
খ
40°
গ
35°
ঘ
32°
ঙ
44°

ব্যাখ্যা

AB = BC
⇒ ∠C = ∠A = (2x - 20)°.
∠A+ ∠B + ∠C =180°
⇒ (2x - 20) + x + (2x - 20 ) = 180
⇒ 5x - 40 = 180
⇒ 5x = 220
⇒ x = 44.
∴ ∠B = 44°

৬,১৪৯.

Which number replaces the question mark?

ক
ক) 4
খ
খ) 7
গ
গ) 9
ঘ
ঘ) 10

ব্যাখ্যা

Question: Which number replaces the question mark?

Solution:
১ম চিত্রে,
(4 × 4) - (5 × 3)
= 16 - 15
= 1

২য় চিত্রে,
(6 × 8) - (6 × 7)
= 48 - 42
= 6

৩য় চিত্রে,
(5 × 8) - (5 × 6)
= 40 - 30
= 10

৬,১৫০.

A 240 metre long train crosses a platform thrice its length in 60 seconds. What is the speed of the train in km/hr?

ক
ক) 55.6 km/hr
খ
খ) 57.6 km/hr
গ
গ) 66.6 km/hr
ঘ
ঘ) 62.6 km/hr

ব্যাখ্যা

Length of train = 240 m
Length of platform = (3 × 240) m = 720m

∴Speed of train (240 + 720)/60 m/sec
= 960/60 m/sec
= 16 × (18/5) km/hr
= 57.6 km/hr

৬,১৫১.

Three pipes A, B, and C are connected to a tank. Out of the three, A and B are the inlet pipes and C is the outlet pipe. If opened separately, A fills the tank in 10 hours, B fills the tank in 12 hours and C empties the tank in 30 hours. If all three are opened simultaneously, how much time does it take to fill/empty the tank?

ক
13/3 hours to empty
খ
25/3 hours to fill
গ
19/3 hours to empty
ঘ
20/3 hours to fill
ঙ
17/3 hours to empty

ব্যাখ্যা

Question: Three pipes A, B, and C are connected to a tank. Out of the three, A and B are the inlet pipes and C is the outlet pipe. If opened separately, A fills the tank in 10 hours, B fills the tank in 12 hours and C empties the tank in 30 hours. If all three are opened simultaneously, how much time does it take to fill / empty the tank?

Solution:
Solution:
Part of tank filled by pipe A in one hour working alone = 1/10
Part of tank filled by pipe B in one hour working alone = 1/12
Part of tank emptied by pipe C in one hour working alone = 1/30

Part of tank filled by pipes A, B and C in one hour working together = (1/10) + (1/12) - (1/30)
= (6 + 5 - 2)/60
= 9/60
= 3/20

Therefore, time taken to completely fill the tank if A, B and C work together = 20/3 hours

৬,১৫২.

Find the value of n, if 27^n-(1/3) = 243.

ক
2
খ
3
গ
4
ঘ
5

ব্যাখ্যা

Question: Find the value of n, if 27^{n - (1/3)} = 243.

Solution:
27^{n - (1/3)} = 243
⇒ (3³)^{n - (1/3)} = 3⁵
⇒ 3^{3n - 1} = 3⁵
⇒ 3n - 1 = 5
⇒ 3n = 5 + 1
⇒ 3n = 6
∴ n = 2

৬,১৫৩.

A dealer buys dry fruits at Tk. 100, Tk. 80, and Tk. 60 per kilogram. He mixes them in the ratio 3 : 4 : 5 by weight and sells at a profit of 50%. At what price per kilogram does he sell the dry fruits?

ক
80
খ
100
গ
95
ঘ
115

ব্যাখ্যা

Question: A dealer buys dry fruits at Tk. 100, Tk. 80, and Tk. 60 per kilogram. He mixes them in the ratio 3 : 4 : 5 by weight and sells at a profit of 50%. At what price per kilogram does he sell the dry fruits?

Solution:
Let the dealer buy 3 kg, 4 kg and 5 kg.
∴ Price of total dry fruits = (3 × 100) + (4 × 80) + (5 × 60) = Tk. 920

At 50% Profit,
Selling Price, SP = 920 + 50% of 920
= 920 + (50/100) × 920
= 1380

Hence,
Price of dry fruits per kg = 1380/12 = 115 Tk.

৬,১৫৪.

If the average of four consecutive odd numbers is 42, find the largest numbers.

ক
41
খ
43
গ
45
ঘ
47

ব্যাখ্যা

Question: If the average of four consecutive odd numbers is 42, find the largest numbers.

Solution:
Let
the first number is x,
then the next three odd numbers would be (x + 2), (x + 4), (x + 6)

ATQ,
{x + (x + 2) + (x + 4) + (x + 6)}/4 = 42
⇒ (4x + 12)/4 = 42
⇒ 4x + 12 = 168
⇒ 4x = 156
∴ x = 39

Largest number would be = 39 + 6 = 45

৬,১৫৫.

A truck travels 150 km from A to B at 75 km/hr. and returns from B to A along the same route at 50 km/hr. The average speed, in kilometers per hour, for the round trip is:

ক
ক) 60 km/hr
খ
খ) 55 km/hr
গ
গ) 65 km/hr
ঘ
ঘ) 50 km/hr

ব্যাখ্যা

Question: A truck travels 150 km from A to B at 75 km/hr and returns from B to A along the same route at 50 km/hr. The average speed, in kilometers per hour, for the round trip is:

Solution:
Here round trip means total distance 150 + 150 = 300 km
And total hours = (150/75) + (150/50) = 2+3 = 5 hours

∴ Average Speed: 300/5 = 60 km/hr. [ মোট দূরত্বকে মোট সময় দ্বারা ভাগ করলে গড় গতিবেগ পাওয়া যায়]

৬,১৫৬.

If 18 is 15 percent of 30 percent of a certain number, what is the number?

ক
ক) 9
খ
খ) 26
গ
গ) 40
ঘ
ঘ) 81
ঙ
ঙ) 400

ব্যাখ্যা

Question: If 18 is 15 percent of 30 percent of a certain number, what is the number?

Solution:
let,the number is x

x × 30% × 15% = 18
⇒ x × 0.3 × 0.15 = 18
⇒ x × 0.045 = 18
⇒ x = 18/0.045
∴ x = 400

৬,১৫৭.

What is the angle formed between the hour hand and the minute hand of a clock at half past two in the afternoon?

ক
95°
খ
145°
গ
60°
ঘ
105°

ব্যাখ্যা

Question: What is the angle formed between the hour hand and the minute hand of a clock at half past two in the afternoon?

Solution:
দুপুর আড়াইটায় ঘড়ির ঘণ্টার কাঁটা ও মিনিটের কাঁটার ব্যবধান
= ।(১১M - ৬০H)/২।°
= ।{(১১ × ৩০) - (৬০ × ২)}/২।°
= ।(৩৩০ - ১২০)/২।°
= ১০৫°

৬,১৫৮.

A can complete a work in 20 days, B in 30 days, and C in 60 days. A stops working 4 days before the completion of the work, and B stops 6 days before completion. C continues working alone till the end. What was the total number of days taken to complete the entire work?

ক
10 days
খ
14 days
গ
18 days
ঘ
21 days

ব্যাখ্যা

Question: A can complete a work in 20 days, B in 30 days, and C in 60 days. A stops working 4 days before the completion of the work, and B stops 6 days before completion. C continues working alone till the end. What was the total number of days taken to complete the entire work?

Solution:
Let the total work be completed in y days.

∴ A worked for (y - 4) days
So his contribution = (y - 4)/20

B worked for (y - 6) days
So his contribution = (y - 6)/30

C worked full y days, so his contribution = y/60

Therefore,
(y - 4)/20 + (y - 6)/30 + y/60 = 1
⇒ 3(y - 4) + 2(y - 6) + y = 60
⇒ 3y - 12 + 2y - 12 + y = 60
⇒ 6y - 24 = 60
⇒ 6y = 84
⇒ y = 14

∴ The total work was completed in 14 days.

৬,১৫৯.

A short distance athlete has taken 60 seconds to cover 100 meters. If he makes 30 steps in 9 seconds, how many steps has he taken in that time?

ক
ক) 130
খ
খ) 170
গ
গ) 173
ঘ
ঘ) None of these

ব্যাখ্যা

Question: A short distance athlete has taken 60 seconds to cover 100 meters. If he takes 30 steps in 9 seconds, how many steps has he taken in that time?

Solution:
৯ সেকেন্ডে স্টেপ দেয় ৩০ টি
৬০ সেকেন্ডে স্টেপ দেয় = (৩০ × ৬০)/৯
= ২০০ টি

৬,১৬০.

A dice is thrown. What is the probability that the number shown on the dice is odd number ?

ক
ক) 1/6
খ
খ) 1/3
গ
গ) 1/2
ঘ
ঘ) 1/4

ব্যাখ্যা

When a dice is thrown once.
The total number of outcomes is 6 (1, 2, 3, 4, 5, and 6)
Odd numbers = 3 (1, 3, 5)

Probability = No of Favorable Outcomes/Total no of Outcomes
P(Odd numbers) = 3/6
= 1/2

∴ The required probability is 1/2

৬,১৬১.

You bought 11 pencils and erasers worth BDT. 80. If erasers cost half that of a pencil and you bought one extra eraser, how much is the eraser worth?

ক
5
খ
8
গ
10
ঘ
12

ব্যাখ্যা

Let,
Pencil's price x
And eraser's price = x/2
ATQ, 5x + 6x/2 = 80 [As there is an extra eraser]
Or, 16x = 160
Or, x = 10
∴ eraser costs = x/2 = 10/2 = 5

বিকল্প পদ্ধতি:
5 pencils + 6 erasers = ৳80
The five pencils have the same value as ten erasers, so if we include the extra (non-paired) eraser, we have a total value of 16 erasers:
80 ÷ 16 = 5

৬,১৬২.

If a number x is 10% less than another number y and y is 10% more than 125, then x is equal to:

ক
133
খ
123.75
গ
152
ঘ
140.55

ব্যাখ্যা

Question: If a number x is 10% less than another number y and y is 10% more than 125, then x is equal to:

Solution:
y is 10% more than 125
y = 125 × (110/100)
= 137.5

and x is 10% less than y
x = (90/100) × y
= (90/100) × 137.5
= 123.75

৬,১৬৩.

In a two-digit positive number, the digit in the unit’s place is equal to the square of the digit in ten’s place, and the difference between the number and the number obtained by interchanging the digits is 54. What is 40% of the original number?

ক
15.6
খ
20
গ
21.2
ঘ
30

ব্যাখ্যা

Question: In a two-digit positive number, the digit in the unit’s place is equal to the square of the digit in ten’s place, and the difference between the number and the number obtained by interchanging the digits is 54. What is 40% of the original number?

Solution:
Let, ten’s digit = x.
Then, unit’s digit = x².
∴ number = 10x + x².

Since x² > x,
so, the number formed by interchanging the digits is greater than the original number.

∴ (10x² + x) - (10x + x²) = 54
⇒ 9x² - 9x = 54
⇒ x² - x = 6
⇒ x² - x - 6 = 0
⇒ x² - 3x + 2x - 6 = 0
⇒ x(x - 3) + 2(x - 3) = 0
⇒ (x - 3) (x + 2) = 0
⇒ x = 3. [∵ it is positive number]

So, ten’s digit = 3, unit’s digit = 3² = 9.
∴ Original number = 39.
Required result = 40% of 39
= (40/100) × 39 = 15.6.

৬,১৬৪.

Find the root of the quadratic equation: 3x² - 2√6x + 2 = 0

ক
√3/2, - (√2/3)
খ
√(2/3), √(2/3)
গ
1/√3, √(2/3)
ঘ
√(2/5), √2/3

ব্যাখ্যা

Question: Find the root of the quadratic equation: 3x² - 2√6x + 2 = 0

Solution:
3x² - 2√6x + 2 = 0
⇒ 3x² - √6x - √6x + 2 = 0
⇒ √3x(√3x - √2) - √2(√3x - √2) = 0
⇒ (√3x - √2)(√3x - √2) = 0
∴ x = √2/√3, √2/√3

৬,১৬৫.

If 4x² - px + 16 is a square number, then p =?

ক
ক) 9
খ
খ) 16
গ
গ) 20
ঘ
ঘ) 25

ব্যাখ্যা

Question: If 4x² - px + 16 is a square number, then p =?

Solution:
4x² - px + 16
= (2x)² - 2. 2x. 4 + 4² - px + 16x
= (2x - 4)² - px + 16x

Now,
- px + 16x = 0 [ the expression is a square number]
⇒ px = 16
∴ p = 16

৬,১৬৬.

Weekly incomes of two persons are in the ratio of 7 : 3 and their weekly expenses are in the ratio of 5 : 2. If each of them saves Tk. 300 per week, find their weekly incomes.

ক
Tk. 6000 and Tk. 2500
খ
Tk. 6300 and Tk. 2700
গ
Tk. 6500 and Tk. 2800
ঘ
Tk. 6200 and Tk. 2600
ঙ
None of these

ব্যাখ্যা

Question: Weekly incomes of two persons are in the ratio of 7 : 3 and their weekly expenses are in the ratio of 5 : 2. If each of them saves Tk. 300 per week, find their weekly incomes.

Solution:
Let the incomes of the two persons be 7x and 3x
And their expenses be 5y and 2y respectively.

Then we get,
7x - 5y = 3x - 2y
⇒ 4x = 3y
∴ y = 4x/3

Now, 7x - 5y = 300
⇒ 7x - 5(4x/3) = 300
⇒ (21x - 20x)/3 = 300
∴ x = Tk. 900

∴ Weekly income of first person = (7 × 900) = Tk. 6300
∴ Weekly income of second person = (3 × 900) = Tk. 2700

So weekly incomes are Tk. 6300 and Tk. 2700

৬,১৬৭.

The hypotenuse of a right angled isosceles triangle is 6 cm then its area is-

ক
ক) 6 cm²
খ
খ) 9 cm²
গ
গ) 10 cm²
ঘ
ঘ) 12 cm²

ব্যাখ্যা

Question:The hypotenuse of a right angled isosceles triangle is 6 cm then its area is-
Solution:

Now,
x² + x² = 6²
2x² = 36
x² = 36/2
x =√(36/2)
x = 6/√2

Area =(1/2) × (6/√2) × (6/√2) = 9 cm²

৬,১৬৮.

A train running at the speed of 90 km/h crosses a pole in 10 seconds. What is the length of the train?

ক
150 meters
খ
250 meters
গ
100 meters
ঘ
160 meters

ব্যাখ্যা

Question: A train running at the speed of 90 km/h crosses a pole in 10 seconds. What is the length of the train?

Solution:
Speed = 90 km/h
= [90 × (5/18)] m/sec
= 25 m/sec

∴ Length of the train = (25 × 10) m
= 250 m

So, the length of the train is 250 meters.

৬,১৬৯.

A truck driver must complete a 180 mile trip in 4 hours. If his average speed is 50 miles per hour for the first 3 hours, then how fast must he travel for the final hour?

ক
ক) 25
খ
খ) 35
গ
গ) 40
ঘ
ঘ) 30

ব্যাখ্যা

Question: A truck driver must complete a 180-mile trip in 4 hours. If his average speed is 50 miles per hour for the first 3 hours, then how fast must he travel for the final hour?

Solution:
প্রথম ৩ ঘণ্টায় গড় গতিবেগ ৫০ মাইল/ঘণ্টা
৩ ঘণ্টায় অতিক্রান্ত দূরত্ব (৫০ × ৩) মাইল
= ১৫০ মাইল

অর্থাৎ, শেষ ঘণ্টায় ট্রাকচালককে (১৮০ - ১৫০) মাইল বা ৩০ মাইল যেতে হবে
ট্রাকচালকের গতিবেগ = ৩০ মাইল/ঘণ্টা

৬,১৭০.

A sum of money is distributed equally among 8 persons. If 4 more persons were included, each person would get Tk. 50 less. What was the total sum?

ক
Tk. 1000
খ
Tk. 1200
গ
Tk. 1300
ঘ
Tk. 1500
ঙ
None

ব্যাখ্যা

Question: A sum of money is distributed equally among 8 persons. If 4 more persons were included, each person would get Tk. 50 less. What was the total sum?

Solution:
Let, the sum be Tk. x
When the sum is distributed among 8 persons, each person gets, x/8
And,
If 4 more persons were included, making it 12 persons, each person would get, x/12

ATQ,
(x/8) - (x/12) = 50
⇒ (3x - 2x)/24 = 50
⇒ x = 50 × 24
∴ x = 1200

So, The total sum of money is Tk. 1200

৬,১৭১.

Select an appropriate term that completes the series K o E, M p F, O q G, Q r H, _____.

ক
S s I
খ
T t J
গ
R s I
ঘ
S s J

ব্যাখ্যা

Question: Select an appropriate term that completes the series K o E, M p F, O q G, Q r H, _____.

Solution:
প্রথম অক্ষর এর ক্ষেত্রে,
K এর পর L নেই,
M এর পর N নেই,
O এর পরে P নেই,
তাহলে অবশ্যই Q এর পরে R না হয়ে S হবে,

দ্বিতীয় অক্ষরের জন্য
o, p, q, r, তারপর অবশ্যই s হবে।

তৃতীয় অক্ষরের জন্য
E, F, G, H, এর পর অবশ্যই I হবে।

তাহলে খালি ঘরে অবশ্যই S s I

৬,১৭২.

A courtyard is 25 meter long and 16 meter board is to be paved with bricks of dimensions 20 cm by 10 cm. The total number of bricks required is :

ক
ক) 16000
খ
খ) 18000
গ
গ) 20000
ঘ
ঘ) 22000

ব্যাখ্যা

Number of bricks
=Courtyard area/1 brick area
=(2500×1600 / 20×10)=20000

৬,১৭৩.

Strawberries are bought at 5 for Tk 10 and sold at 6 for Tk 15. The gain percent is-

ক
20%
খ
22.5%
গ
25%
ঘ
27.5%

ব্যাখ্যা

Question: Strawberries are bought at 5 for Tk 10 and sold at 6 for Tk 15. The gain percent is-

Solution:
Cost per strawberry = 10/5 Tk
= 2 Tk

Sell per strawberry = 15/6 Tk
= 2.5 Tk

So, gain percentage = {(2.5 - 2)/2} × 100
= 25 Tk or 25%

৬,১৭৪.

The greatest value of sin⁴2θ + cos⁴2θ is?

ক
5/2
খ
2
গ
1
ঘ
√3/2

ব্যাখ্যা

Question: The greatest value of sin⁴2θ + cos⁴2θ is?

Solution:
sin²2θ + cos²2θ = 1

(sin²2θ + cos²2θ)² = 1²
⇒ sin⁴2θ + cos⁴2θ + 2 sin²2θ cos²2θ = 1
⇒ sin⁴2θ + cos⁴2θ = 1 − 2 sin²2θ cos²2θ [2 sin²2θ cos²2θ = 0 (when θ = 0° or 90°)]
∴ sin⁴2θ + cos⁴2θ = 1

∴ greatest value = 1

৬,১৭৫.

A train 360 m long passes a pole in 30 seconds. How long will it take to pass a platform 540 m long?

ক
45 seconds
খ
64 seconds
গ
75 seconds
ঘ
96 seconds

ব্যাখ্যা

Question: A train 360 m long passes a pole in 30 seconds. How long will it take to pass a platform 540 m long?

Solution:
সমাধান:
ট্রেনটির মোট দূরত্ব অতিক্রম করতে হবে = (360 + 540) মিটার = 900 মিটার

ট্রেনটি 360 মিটার অতিক্রম করতে সময় নেয় = 30 সেকেন্ড
ট্রেনটি1 মিটার অতিক্রম করতে সময় নেয় = 30/360 সেকেন্ড
ট্রেনটি 900 মিটার অতিক্রম করতে সময় নেয় = (30 × 900)/360 সেকেন্ড
= 75 সেকেন্ড

৬,১৭৬.

A takes twice as much time as B or thrice as much time as C to finish a piece of work. Working together, they can finish the work in 2 days. B can do the work alone in:

ক
ক) 6 days
খ
খ) 8 days
গ
গ) 12 days
ঘ
ঘ) 4 days

ব্যাখ্যা

Suppose,
A, B and C take x, x/2 and x/3 days respectively to finish the work.
Then,
(1/x) + (2/x) + (3/x) = 1/2
⇒ 6/x = 1/2
⇒ x = 12.
So, B takes 6 days to finish the work.

৬,১৭৭.

What is the solution of - 13 < 3x + 2 ≤ 11?

ক
(- 5, 3)
খ
(- 5, 3]
গ
[- 5, 3]
ঘ
(- 5, 5]

ব্যাখ্যা

Question: What is the solution of - 13 < 3x + 2 ≤ 11?

Solution:
- 13 < 3x + 2 ≤ 11
⇒ - 13 - 2 < 3x + 2 - 2 ≤ 11 - 2
⇒ - 15 < 3x ≤ 9
⇒ - 5 < x ≤ 3

∴ x ∈ (- 5, 3]

৬,১৭৮.

Gold is 17 times as heavy as water and copper is 8 times as heavy as water. In what ratio should these be mixed to get an alloy 13 times as heavy as water?

ক
ক) 3 : 2
খ
খ) 5 : 4
গ
গ) 6 : 7
ঘ
ঘ) 8 : 5

ব্যাখ্যা

Question: Gold is 17 times as heavy as water and copper is 8 times as heavy as water. In what ratio should these be mixed to get an alloy 13 times as heavy as water?

Solution:
Let, gold is 17x times as heavy as water and copper is 8y times as heavy as water.

ATQ,
17x + 8y = 13(x + y)
⇒ 17x + 8y = 13x + 13y
⇒ 17x - 13x = 13y - 8y
⇒ 4x = 5y
⇒ x/y = 5/4
∴ x : y = 5 : 4

৬,১৭৯.

rsinθ = 1, rcosθ = √3 then the value of (√3tanθ + 1) = ?

ক
2
খ
3
গ
4
ঘ
1

ব্যাখ্যা

Question: rsinθ = 1, rcosθ = √3 then the value of (√3tanθ + 1) = ?

Solution:
rsinθ = 1
rcosθ = √3

Now,
rsinθ/rcosθ = 1/√3
⇒ tanθ = 1/√3
⇒ √3tanθ = 1
⇒ √3tanθ + 1 = 1 + 1
∴ √3tanθ + 1 = 2

৬,১৮০.

What is the measure of the radius of the circle that circumscribes a triangle whose sides measure 9, 40, and 41?

ক
10.5 units
খ
16 units
গ
18.25 units
ঘ
20.5 units
ঙ
None

ব্যাখ্যা

Question: What is the measure of the radius of the circle that circumscribes a triangle whose sides measure 9, 40, and 41?

Solution:
The sides of the triangle are 9, 40, and 41.

Here,
9² + 40² = 81 + 1600 = 1681 = 41², this is a Pythagorean triplet,
∴ the triangle is right-angled, with 41 as the hypotenuse.

We know,
In a right-angled triangle, the radius of the circumscribed circle is half the hypotenuse.

∴ The radius of the circle that circumscribes the triangle = 41/2 = 20.5 units

৬,১৮১.

Two pipes can fill a tank in 20 and 24 minutes respectively and a waste pipe can empty 3 gallons per minute. All the three pipes working together can fill the tank in 15 minutes. The capacity of the tank is-

ক
60 gallons
খ
100 gallons
গ
120 gallons
ঘ
180 gallons

ব্যাখ্যা

Question: Two pipes can fill a tank in 20 and 24 minutes respectively and a waste pipe can empty 3 gallons per minute. All the three pipes working together can fill the tank in 15 minutes. The capacity of the tank is-

Solution:
Work done by the waste pipe in 1 minute = 1/15 - (1/20 + 1/24)
= 1/15 - 11/120
= - 1/40 [- ve sign means emptying]

Volume of 1/40 part = 3 gallons
∴ Volume of whole = (3 × 40) gallons
= 120 gallons.

৬,১৮২.

If a and b are positive integers and (a - b)/3.5 = 4/7, then

ক
ক) b < a
খ
খ) b > a
গ
গ) b = a
ঘ
ঘ) b > a
ঙ
ঙ) None of these

ব্যাখ্যা

Question: If a and b are positive integers and (a - b)/3.5 = 4/7, then

Solution:
(a - b)/3.5 = 4/7
⇒ a - b = (4 × 3.5)/7
⇒ a - b = 14/7
⇒ a - b = 2
∴ a = b + 2

So, we can say that, a > b ⇔ b < a

৬,১৮৩.

A mixture contains alcohol and water in the ratio 4 : 3. If 5 liters of water is added to the mixture, the ratio becomes 4 : 5. Find the quantity of alcohol in the given mixture.

ক
10 litres
খ
12 litres
গ
15 litres
ঘ
18 litres

ব্যাখ্যা

Question: A mixture contains alcohol and water in the ratio 4 : 3. If 5 liters of water is added to the mixture, the ratio becomes 4 : 5. Find the quantity of alcohol in the given mixture.

Solution:

Let the quantity of alcohol and water be 4x litres and 3x litres respectively

ATQ,
4x/(3x + 5) = 4/5

⇒ 20x = 4(3x + 5)
⇒ 20x = 12x + 20

⇒ 8x = 20

∴ x = 2.5

Quantity of alcohol = (4 × 2.5) litres = 10 litres.

৬,১৮৪.

Fifth part of a number increased by 4 is equal to its fourth part diminished by 10. What is the number?

ক
120
খ
240
গ
280
ঘ
360

ব্যাখ্যা

Let the number be x
x/5 + 4 = x/4 - 10
or, x/4 - x/5 = 10 + 4
or, (5x - 4x)/20 = 14
or, x/20 = 14
or, x = 280
Therefore, the number is 280

৬,১৮৫.

A hospital uses a mixture of salt and water at Tk. 7.62/litre. This mixture contains 5% salt. Another mixture containing 75% water costs Tk. 7.82/litre. How much does the patient pay if he buys 5 litres of mixture containing 18% salt?

ক
Tk. 83.75
খ
Tk. 73.85
গ
Tk. 37.85
ঘ
Tk. 38.75

ব্যাখ্যা

Question: A hospital uses a mixture of salt and water at Tk. 7.62/litre. This mixture contains 5% salt. Another mixture containing 75% water costs Tk. 7.82/litre. How much does the patient pay if he buys 5 litres of mixture containing 18% salt?

Solution:
1st mixture contains 5% salt.
2nd mixture contains 75% water i.e. 25 % salt.
Required % of salt = 18%.
∴ Required ratio = 25 - 18 : 18 - 5 = 7 : 13
and required price of the mixture = 7.82 - x : x - 7.62 = 7 : 13 [Let, patient pays Tk. x per kg]
⇒ 13(7.82 - x) = 7(x - 7.62)
⇒ 101.66 - 13x = 7x - 53.34
⇒ 20x = 155
⇒ x = 7.75
Hence price of 5 liters of this mixture = 7.75 × 5 = Tk. 38.75.

৬,১৮৬.

30 men working 8 hours per day can dig a pond in 16 days. By working how many hours per day can 32 men dig the same pond, in 20 days?

ক
5 hours/day
খ
6 hours/day
গ
7 hours/day
ঘ
8 hours/day
ঙ
None

ব্যাখ্যা

Question: 30 men working 8 hours per day can dig a pond in 16 days. By working how many hours per day can 32 men dig the same pond, in 20 days?

Solution:
30 men can dig a pond in 16 days by working 8 hours per day
∴ 1 men can dig a pond in 1 days by working (8 × 30 × 16) hours per day
∴ 32 men can dig a pond in 20 days by working (8 × 30 × 16)/(20 × 32) hours per day
= 6 hours per day

৬,১৮৭.

What is the unit digit in the product 84 × 59 × 13 × 76?

ক
ক) 2
খ
খ) 4
গ
গ) 6
ঘ
ঘ) 8

ব্যাখ্যা

84 × 59 × 13 × 76 = 4896528
Without the use of calculator, to count the unit digit = 4 × 9 × 3 × 6 = 36 × 18 = 648

So, 8 is the unit digit

৬,১৮৮.

12 spheres of the same size are made by melting a solid cylinder of 16 cm. diameter and 2 cm. height. The diameter of each sphere is -

ক
ক) 3
খ
খ) 2
গ
গ) 4
ঘ
ঘ) 6

ব্যাখ্যা

Question: 12 spheres of the same size are made by melting a solid cylinder of 16 cm. diameter and 2 cm. height. The diameter of each sphere is -
Solution:
Volume of sphere = 4/3 πr³
Volume of cylinder = πr²h

The volume of 12 spheres = Volume of cylinder
⇒ 12 × (4/3)πr³ = π × 8 × 8 × 2
⇒ 16r³ = 8 × 8 × 2
r³ = (8 × 8 × 2)/16
r³ = 8
r³ = 2³
r = 2
The diameter of each sphere is= 4 cm

৬,১৮৯.

একটি কোম্পানির ৪৬% কর্মচারি পুরুষ। যদি ৬০% কর্মচারি ইউনিয়ন করে এবং এর মধ্যে ৭০% কর্মচারি পুরুষ হয়, তাহলে শতকরা কতজন মহিলা কর্মচারি ইউনিয়ন করে না?

ক
ক) ৯০%
খ
খ) ৮৭.৫%
গ
গ) ৫০%
ঘ
ঘ) ৩৬%

ব্যাখ্যা

প্রশ্ন: একটি কোম্পানির ৪৬% কর্মচারি পুরুষ। যদি ৬০% কর্মচারি ইউনিয়ন করে এবং এর মধ্যে ৭০% কর্মচারি পুরুষ হয়, তাহলে শতকরা কতজন মহিলা কর্মচারি ইউনিয়ন করে না?

সমাধান:
ধরি,
কোম্পানির কর্মচারির সংখ্যা ১০০ জন।
∴ কোম্পানির পুরুষ কর্মচারি ৪৬ জন।
কোম্পানির মহিলা কর্মচারি (১০০ - ৪৬) জন = ৫৪ জন

ইউনিয়ন করে ৬০ জন।
ইউনিয়নে মহিলা কর্মচারির সংখ্যা ৬০ × (৩০/১০০) জন
= ১৮ জন

ইউনিয়ন না করা মহিলা কর্মচারির সংখ্যা (৫৪ - ১৮) জন
= ৩৬ জন

∴ শতকরা ৩৬ জন মহিলা কর্মচারি ইউনিয়ন করে না।

৬,১৯০.

A tap can fill a tank in 8 hrs. After half the tank is filled, three more similar taps are opened. What is the total time taken to fill the tank completely?

ক
4 hrs
খ
5 hrs
গ
3 hrs
ঘ
2 hrs

ব্যাখ্যা

Question: A tap can fill a tank in 8 hrs. After half the tank is filled, three more similar taps are opened. What is the total time taken to fill the tank completely?

Solution:
Time taken by one tap to fill half the tank = 4 hrs
Remaining part after 4 hrs = (1 - 1/2) = 1/2
Part filled by the four taps in 1 hours = 4 × (1/8) = 1/2

Total time = 4 + 1 = 5 hrs

৬,১৯১.

ক
ক) Tk. 75
খ
খ) Tk. 80
গ
গ) Tk. 125
ঘ
ঘ) Tk. 210

ব্যাখ্যা

(61 -63): Answer the questions on the basis of the information given below

61.Your income for a year is Tk. 26,000. You receive a raise so that next year's income will be Tk. 29,000. How much more will will be Tk. 29,000. you pay in taxes next year if the tax rate remains the same?

Solution:
২৬০০০ টাকা বেতন থাকাকালীন টেক্স দিতে হত = ১০৭০ + (২৫০০০ এর বেশি ইনকামের ৭%)
= ১০৭০ + (২৬০০০ - ২৫০০০) এর ৭%
= ১০৭০ + (১০০০ এর ৭%)
= ১০৭০ + ৭০
= ১১৪০ টাকা

নতুন বেতন ২৯০০০ এর জন্য টেক্স দিতে হবে = ১০৭০ + (২৯০০০ - ২৫০০০) এর ৭%
= ১০৭০ + (৪০০০ এর ৭%)
= ১০৭০ + ২৮০ টাকা
= ১৩৫০ টাকা

∴ অতিরিক্ত টেক্স দিতে হবে = (১৩৫০ - ১১৪০) = ২১০ টাকা

৬,১৯২.

33-56: Read the following questions carefully and choose the right answer.
৩৩) Two buses start at the same time from Delhi and Agra, which are 300km apart, towards each other. After what time will they cross each other if their speeds are 38km per hour and 37km per hour?

ক
ক) 4 hours
খ
খ) 3 hours
গ
গ) 5 hours
ঘ
ঘ) 6 hours

ব্যাখ্যা

Since we know that, Speed=Distance/Time
In order to find time, we use the following,
Time=Distance/Speed
So the distance between two buses is 300 km, let bus A speed be 38 km per hr and speed of bus B be 37 km per hr.
Therefore, Time=300/38+37
Thus on solving the above we get the answer as 4hrs.

৬,১৯৩.

Polash and Hasan started a business with the investment of Tk.6000 and Tk.5000 respectively and after 6 months, Mehedi joined with the investment of Tk.3000. At the end of the year, the total profit is Tk.5000. Find the Hasan’s profit share?

ক
Tk. 600
খ
Tk. 1200
গ
Tk. 2000
ঘ
Tk. 2400

ব্যাখ্যা

Question: Polash and Hasan started a business with the investment of Tk.6000 and Tk.5000 respectively and after 6 months, Mehedi joined with the investment of Tk.3000. At the end of the year, the total profit is Tk.5000. Find the Hasan’s profit share?

Solution:
Polash invest = 6000 × 12 =72,000
Hasan invest = 5000 × 12 = 60,000
Mehedi invest = 3000 × 6=18,000

The ratio of their investments is = 72,000 : 60,000 : 18,000
= 12 : 10 : 3

The total profit is Tk. 5000

Now,
Hasan’s share of the profit is = (10/25) × 5000 = 2000

∴ Hasan’s share of the profit is Tk. 2000

৬,১৯৪.

Three unbiased coins are tossed simultaneously. What is the probability of getting at most one tail?

ক
1/3
খ
1/4
গ
1/2
ঘ
1/5

ব্যাখ্যা

Question: Three unbiased coins are tossed simultaneously. What is the probability of getting at most one tail?

Solution: Total outcomes = 2³ = 8 (HHH, HHT, HTH, THH, HTT, THT, TTH, TTT)

Favorable outcomes = cases with at most one tail (zero tails or exactly one tail) = (HHH, HHT, HTH, THH)

At most one tail refers to zero or one tail only.

Therefore, Probability = Favorable outcomes / Total outcomes = 4/8 = 1/2

৬,১৯৫.

A bag contains red, blue, and green balls in the ratio 3 : 5 : 7. If there are 45 blue balls, how many total balls are there in the bag?

ক
90
খ
105
গ
120
ঘ
135

ব্যাখ্যা

Question: A bag contains red, blue, and green balls in the ratio 3 : 5 : 7. If there are 45 blue balls, how many total balls are there in the bag?

Solution: Given ratio = Red : Blue : Green = 3 : 5 : 7
Let the numbers be:
Red = 3x, Blue = 5x, Green = 7x

Given:
5x = 45
⇒ x = 45 ÷ 5 = 9

Now calculate total balls:
= 3x + 5x + 7x
= (3 + 5 + 7)x = 15x
= 15 × 9 = 135

৬,১৯৬.

If p : q = 7 : 3, then the value of 5p + 70 : 3q + 18 is

ক
30 : 7
খ
35 : 9
গ
25 : 9
ঘ
None of these

ব্যাখ্যা

p : q = 7 : 3
or, p/q = 7/3
or, 5p/3q = (5 × 7)/(3 × 3)
or, 5p/3q = 35/9
or, 5p/35 = 3q/9
or, (5p + 70)/35 = (3q + 18)/9
or, (5p + 70)/(3q + 18) = 35/9
or, (5p + 70) : (3q + 18) = 35 : 9
---------------------------------
Short-cut
p = 7, q = 3
5p + 70 : 3q + 18 = 5 × 7 + 70 : 3 × 3 + 18 = 105 : 27 = 35 : 9

৬,১৯৭.

The difference of two numbers is 1365. On dividing the larger number by the smaller, we get 6 as quotient and the 15 as remainder. What is the smaller number?

ক
ক) 270
খ
খ) 1270
গ
গ) 350
ঘ
ঘ) 720

ব্যাখ্যা

Let the bigger number be x and smaller number be y
ATQ, x - y = 1365 ...... (i)
and, 6y + 15 = x ....... (ii)
By solving the equation we get, y = 270
So, the smaller number is 270

৬,১৯৮.

A fruit seller had some oranges. He sells 40% oranges and still has 420 oranges. How many oranges did he have originally?

ক
500
খ
700
গ
850
ঘ
675
ঙ
600

ব্যাখ্যা

He sells 40% of oranges and still there are 420 oranges remaining.
=> 60% of oranges = 420
=> Total oranges × 60 /100 = 420
=> Total oranges = 420 × 100/ 60 = 700

৬,১৯৯.

আবির একটি নির্দিষ্ট কাজ সম্পন্ন করার জন্য বাবুলের দ্বিগুণ বা কবিরের তিনগুন সময় নেয়। তারা একত্রে ২ দিন কাজ করলে কাজটি শেষ করতে পারে। তাহলে বাবুল কতদিনে কাজটি করতে পারে?

ক
৪ দিন
খ
৬ দিন
গ
৮ দিন
ঘ
১০ দিন
ঙ
কোনটিই নয়

ব্যাখ্যা

প্রশ্ন: আবির একটি নির্দিষ্ট কাজ সম্পন্ন করার জন্য বাবুলের দ্বিগুণ বা কবিরের তিনগুন সময় নেয়। তারা একত্রে ২ দিন কাজ করলে কাজটি শেষ করতে পারে। তাহলে বাবুল কতদিনে কাজটি করতে পারে?

সমাধান:
ধরি,
আবির সময় নেয় = ক দিন
বাবলু সময় নেয় = ক/২ দিন
কবির সময় নেয় = ক/৩ দিন

প্রশ্নমতে,
(১/ক) + (২/ক) + (৩/ক) = ১/২
⇒ (১ + ২ + ৩)/ক = ১/২
⇒ ৬/ক = ১/২
⇒ ক = ১২

∴ কাজটি শেষ করতে বাবলু সময় নেয় = ১২/২ = ৬ দিন

৬,২০০.

The area of a circle whose radius is the diagonal of a square whose area is 9 sq. units is-

ক
ক) 8π
খ
খ) 10π
গ
গ) 18π
ঘ
ঘ) 24π

ব্যাখ্যা

Area of a square = 9sq. units
Side of square = 3
Diagonal of square = 3√2 units

Radius of the circle =3√2

Area of circle = πr²
= π(3√2)²
=18π