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Question: Solve: (81.84 + 118.16) ÷ 5³ = 1.2 × 2 + ?

Solution:
Given that,
(81.84 + 118.16) ÷ 5³ = 1.2 × 2 + ?
⇒ 200 ÷ 5³ = 1.2 × 2 + ?
⇒ 200 ÷ 125 = 1.2 × 2 +?
⇒ 1.6 = 2.4 + ?
∴ ? = - 0.8

Question: A train 150 meters long passes a signal post in 15 seconds. How long will it take to pass a bridge that is 450 meters long?

Solution:
Train's speed = Distance/Time
= 150/15 = 10 m/s

Total distance to pass the bridge,
= Length of train + Length of bridge
= 150 m + 450 m
= 600 m

∴ Required time = Distance/Speed
= 600/10
= 60 seconds
​= 1 minute

∴ The train will take 60 seconds or 1 minute to pass platform.

Question: Find the value of x, if 3^{(2x + 1)} = 243.

Solution:
3^{(2x + 1)} = 243
⇒ 3^{(2x + 1)} = 3⁵ (since 243 = 3⁵)
⇒ 2x + 1 = 5
⇒ 2x = 5 - 1
⇒ 2x = 4
⇒ x = 4/2
∴ x = 2

দ্রব্যের ক্রয়মূল্য 150 টাকা
অতিরিক্ত ব্যয়ভার (150 এর 12%) = (150 × 12/100) = 18 টাকা
∴ মোট খরচ (150 + 18)  = 168 টাকা
10% লাভে বিক্রয়মূল্য (168 + 168 এর 10%) = 168 + 16.8 = 184.80 টাকা

Every polynomial of the form ax³ + bx + c with a, b > 0 has exactly one real roots.
Hence, 2x³ + 8x - 7 or 2x³ + 8x + (-7) has one real root.

Question: Six bells commence tolling together and toll at intervals of 3, 5, 6, 9, 10, and 15 seconds respectively. In 45 minutes, how many times do they toll together?

Solution:
3 = 3¹
5 = 5¹
6 = 2 × 3
9 = 3²
10 = 2 × 5
15 = 3 × 5

∴ ল.সা.গু. = 2¹ × 3² × 5¹
= 2 × 9 × 5
= 90

সুতরাং, ঘণ্টাগুলো প্রতি 90 সেকেন্ড পর পর একসাথে বাজবে।

এখন,
45 মিনিট = 45 × 60 = 2700 সেকেন্ড

মোট 2700 সেকেন্ডে ঘণ্টাগুলো যতবার একসাথে বাজবে = 2700/90 = 30 বার

যেহেতু ঘণ্টাগুলো প্রথমে একবার একসাথে বাজা শুরু করেছিল, তাই মোট সংখ্যাটি হবে 30 এর সাথে সেই প্রথমবারটি যোগ করে।
∴ মোট সংখ্যা = 30 + 1 = 31 বার।

সুতরাং, 45 মিনিটে ঘণ্টাগুলো মোট 31 বার একসাথে বাজবে।

Question: In a survey of 1,000 consumers it is found that 720 consumers liked product A and 450 liked product B. What is the least number that must have liked both the products?

Solution:
Given that,
Total consumers = 1000
Consumers who like product A = 720
Consumers who like product B = 450

We know, 
n(A U B) = n(A) + n(B) - n(A ∩ B)
⇒ 1000 = 720 + 450 - n(A ∩ B)  
⇒ 1000 = 1170 - n(A ∩ B)
⇒ n(A ∩ B) = 1170 - 1000
∴ n(A ∩ B) = 170

So 170 consumers like both the products A and B.

Question: From the figure, which of the following must be true?
(I) x + y = 90
(II) x is 35 units greater than y
(III) x is 35 units less than y


Solution:
চিত্রে কোণ x হলো ত্রিভুজটির একটি বহিঃস্থ কোণ। সুতরাং, এর মান বিপরীত অন্তঃস্থ কোণ দুটি, 35 এবং y-এর সমষ্টির সমান।

অর্থাৎ, x = y + 35

 এই সমীকরণ থেকে বোঝা যায় যে x এর মান y এর চেয়ে 35 একক বেশি। তাই, (II) বিবৃতিটি সত্য এবং (III) মিথ্যা।

এখন, যদি x একটি স্থূলকোণ (x > 90) হয়, তাহলে x + y এর মান 90 এর চেয়ে বেশি হবে। সুতরাং, x + y যে অবশ্যই 90 এর সমান হবে, এমন কোনো কথা নেই। তাই, (I) বিবৃতিটি অনিবার্যভাবে সত্য নয়।

অতএব, শুধুমাত্র (II) অবশ্যই সঠিক।

Question: Which of the following digits will replace the M marks in the following equation ?
9M + M8 + M6 = 251

9M+ M8 + M6 = 251
⇒ {(9 × 10) + M} + (10M + 8) + (10M + 6) = 251
⇒ 21M + 104 = 251
⇒ 21M = 251 - 104
⇒ 21M = 147
⇒ M = 7

Given that, a^2x+2=1
=> a^2x+2=a⁰
=> 2x+2=0
=> x=−2/ 2 =−1

Question: A square and a circle have the same perimeter. The length of the side of the square is 22 cm. What is the area of the circle?

Solution:
বর্গের পরিসীমা = 4 × বাহুর দৈর্ঘ্য
= 4 × 22 সেমি
= 88 সেমি

প্রশ্নমতে, বর্গ এবং বৃত্তের পরিসীমা সমান।
সুতরাং, বৃত্তের পরিধি = 88 সেমি

আমরা জানি,
বৃত্তের পরিধি = 2πr
⇒ 2πr = 88
⇒ 2 × (22/7) × r = 88
⇒ (44/7) × r = 88
⇒ r = 88 × (7/44)
∴ r = 14 সেমি

এখন, বৃত্তের ক্ষেত্রফল = πr²
= (22/7) × (14)²
= (22/7) × 196
= 22 × 28
= 616 বর্গ সেমি

Question: Find the missing number:

Solution: 
We can see that,
The elements of the 3rd row are 8, 6, and a missing element.
8 is equal to the summation of 2 and 6.
6 is equal to the summation of 1 and 5.

∴ So the rest of the missing element is the summation of 3 and 4, which is 7.

Question: Find the odd number out
6, 9, 15, 21, 24, 28, 30

Solution: 
6, 9, 15, 21, 24, 30 সবগুলো সংখ্যাই 3 দ্বারা বিভাজ্য। 
28, 3 দ্বার বিভাজ্য নয়। 

Question: 5.4 is 45 percent of 20 percent of a certain number. What is the number?

Solution:
ধরি, সংখ্যাটি = x

প্রশ্নমতে,
45% of (20% of x) = 5.4
⇒ (45/100) × {(20/100) × x} = 5.4 
⇒ (9/20) × (1/5) × (x )= 5.4
⇒ 9x/100 = 5.4
⇒ 9x = 5.4 × 100
⇒ 9x = 540
⇒ x = 540/9
∴ x = 60

Question: What sum of money will amount to Tk. 2,500 in 4 years and to Tk. 2,900 in 6 years at simple interest?

Solution:
Increase in 2 years = 2,900 - 2,500
= Tk. 400

∴ Simple interest for 2 years = Tk. 400
∴ Simple interest for 4 years = Tk. (400 × 4/2)
= Tk. 800

Principal, P = 2,500 - 800
= Tk. 1,700

Question: A vessel contains milk and water in the ratio 7 : 4. If 15 liters of milk are added to it, the ratio of milk to water becomes 10 : 4. Find the final amount of milk in the new mixture.

Solution:
Let the initial amount of milk be 7x liters
and the amount of water 4x liters.

According to the question, 
(7x + 15)/4x = 10/4
⇒ 4(7x + 15) = 10 × 4x
⇒ 28x + 60 = 40x
⇒ 60 = 12x
⇒ x = 60/12
⇒ x = 5

∴ Final amount of milk in mixture = 7x + 15
= 7 × 5 + 15
= 35 + 15
= 50 liters.

Question: In a geometric sequence, the third term is 16 and the sixth term is 128. What is the first term?

Solution:
Let the first term of the geometric sequence be a
and the common ratio be r.
Third term = 16
∴ ar² = 16 ....... (1)
Again,
Sixth term = 128
∴ ar⁵ = 128 ....... (2)

Now, divide equation (2) by equation (1),
ar⁵/ar² = 128/16
⇒ r³ = 8
⇒ r³ = 2³
∴ r = 2
Substitute the value of r into equation (1).
a(2)² = 16
⇒ 4a = 16
∴ a = 4

Therefore, the first term of the geometric sequence is 4.

Question: 45 toymakers can prepare 30 toys per day. Rifat wants 360 toys. How many toymakers should he employ to get the job done in 12 days?

Solution:
Let, the required number of toymakers x
45 toymakers make 30 toys per day
So, 1 toymaker makes = 30/45 = 2/3 toys per day
Each toymaker in 12 days makes = (2/3) × 12 = 8 toys
So, x toymakers will make = 8x toys

ATQ,
8x = 360
⇒ x = 360 × (1/8)
∴ x = 45

Let
True Discount be Tk.1
Then, Banker's Discount = Tk. 11/10 = Tk. 1.10
∴ Sum = Tk {(1.10 × 1)/(1.10 - 1)}
= Tk (110/10)
= Tk. 11
Simple interest on Tk. 11 for 2 years is Tk 1.10
∴ Rate = {(100 × 1.10)/(11 × 2)%
= 5%

Question: If 7 - 2x ≤ 15, then what is the value of x?

Solution:
Given inequality:
7 - 2x ≤ 15

Subtract 7 from both sides:
-2x ≤ 8

Divide both sides by -2 (and reverse the inequality sign):
x ≥ -4

So, the solution set is x ∈ [-4, ∞) 

ATQ, x×10% = y×25%
⇒ x = (25×16)/10 
∴ x = 40

Question: A cow is tethered in the middle of a field with a 14 feet long rope. If the cow grazes 100 sq.ft per day, they approximately what time will be taken by the cow to graze the whole field ?

Solution: 
Area of the field = π 14² sq.ft 
= 616  sq.ft 

Time required =616/100 
= 6.16 days 

Given,
x - 1/x = √3
⇒ (x - 1/x)² = (√3)²
⇒ (x + 1/x)² - 4.x.(1/x) = 3
⇒ (x + 1/x)² = 3 + 4 = 7
∴ x + 1/x = √7

The general equation of a sphere is: (x - a)² + (y - b)² + (z - c)² = r², where (a, b, c) represents the center of the sphere
As, here x² + y² + z² = 4
or, (x - 0)² + (y - 0)² + (z - 0)² = 2²,
So, it's an equation of a sphere where (0, 0, 0) represents the center of the sphere and '2' is it's radius

মোট বলের সংখ্যা = (5 + 3) = 8 টি
কালো বলের সংখ্যা = 3 টি
একটি কালো বল উঠার সম্ভাব্যতা = 3/8 টি

Let the Father be x years when he died.
Average Age 10 years ago be A.
Total Age 10 years ago = 6 × A
Total Age after 10 years(Just before father's Death) = 6A + 6 × 10 = 6A + 60
Father Died and Baby was born => the Total number of people in the family is Same (6)
Baby born today so age of baby = 0
(6A + 60 - x)/6 = 6A/6
=> A + 10 -(x/6) = A
=> x/6 = 10
=> x = 60
Therefore we can conclude that the father was 60 years old when he died.

Question: The area of the triangle whose vertices are given by the coordinates (1, 2), (- 4, -3) and (4, 1) is:

Solution:
Given that, 
Vertices of triangle = (1, 2), (- 4, -3), (4, 1)

We know,
Area of triangle = (1/2) × |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)| [whose vertices are (x₁, y₁), (x₂, y₂) and (x₃, y₃)]
= (1/2) × |1(- 3 - 1) + (- 4) (1 - 2) + 4{2 - (- 3)}|
= (1/2) × |(- 4) + 4 + 20|
= 20/2
= 10 sq. units

So the area of the triangle is 10 sq. units.

Let AB = 8 cm. And, be the chord of the circle with radius AO = 5 cm

Draw OP⊥AB, join OA
according to the theorem
AP = 1/2 AB = 1/2 × 8 = 4 cm
In △APO, ∠A = 90°
∴ (AO)² = (AP)² + (OP)²
OP = (AO)² − (AP)²
= √(5² − 4²)
= √9
∴ OP = 3cm

Question: In how many ways can the letters of the word "AUTHOR" be arranged such that the vowels are only in the odd positions?

Solution:
এখানে
মোট বর্ণ আছে 6টি
স্বরবর্ণ অর্থাৎ Vowel আছে (A, O, U) 3টি
ব্যঞ্জনবর্ণ অর্থাৎ Consonant আছে (T, H, R) 3টি

স্বরবর্ণ 3টি বিজোড় স্থানে রেখে বিন্যাস সংখ্যা = 3! = 6
বাকি 3টি ব্যঞ্জনবর্ণ 3টি জোড় স্থানে রেখে বিন্যাস সংখ্যা = 3! = 6

∴ স্বরবর্ণগুলোকে কেবল বিজোড় স্থানে রেখে মোট বিন্যাস সংখ্যা = 6 × 6
= 36

অতএব, AUTHOR শব্দটিকে স্বরবর্ণগুলোকে কেবল বিজোড় স্থানে রেখে মোট 36 উপায়ে সাজানো যাবে।

S.P of Tk. 5000 stock = {(156/100) × 5000}
= Tk. 7800

Income from this stock = tk. {(12/100) × 5000}
= Tk. 600

Let investment in 8 % stock be x and that in 9 % stock = (7800 - x).

Therefore,
x × (8/90) + (7800 - x) × 9/108 = 600 + 70
⇒ 4x/45 + {(7800 - x)/12} = 670
⇒ 16x + 117000 - 15x = 670 × 180
⇒ x = 3600

Money invested in 8% stock at 90 = Tk. 3600
Money invested in 9% at 108 = Tk. (7800 - 3600)
= Tk. 4200.

Question: Robin can paint 60 walls in 20 minutes. Tara can paint 8 walls in 16 minutes. Working together, how many walls can they paint in 40 minutes?

Solution:
Robin can paint in 1 minute = 60/20 = 3 walls
Tara can paint in 1 minute = 8/16 = 0.5 wall

∴ Working together, they can paint in 1 minute = 3 + 0.5 = 3.5 walls

∴ They can paint in 40 minutes = 3.5 × 40 = 140 walls

∴ Working together, Robin and Tara can paint 140 walls in 40 minutes.

By hit and trial,
we put x = 5 and y = 1
so that,
(3x + 7y) = (3 x 5 + 7 x 1) = 22, which is divisible by 11.
(4x + 6y) = ( 4 x 5 + 6 x 1) = 26, which is not divisible by 11;
(x + y + 4 ) = (5 + 1 + 4) = 10, which is not divisible by 11;
(9x + 4y) = (9 x 5 + 4 x 1) = 49, which is not divisible by 11;
(4x - 9y) = (4 x 5 - 9 x 1) = 11, which is divisible by 11.