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Bank Math

PrepBank · পাতা ৭৭ / ১৬১ · ৭,৬০১–৭,৭০০ / ১৬,১২৪

৭,৬০১.
A shopkeeper incurs a loss by selling an article for Tk 800. If he had sold it for Tk 1200, he would have made a profit which is three times the initial loss. At what price should he sell the article to make 25% profit?
  1. ক
    Tk. 1280
  2. খ
    Tk. 960
  3. গ
    Tk. 1020
  4. ঘ
    Tk. 1125
ব্যাখ্যা

Question: A shopkeeper incurs a loss by selling an article for Tk 800. If he had sold it for Tk 1200, he would have made a profit which is three times the initial loss. At what price should he sell the article to make 25% profit?

Solution:
ধরি, পণ্যের ক্রয়মূল্য = x টাকা
800 টাকায় বিক্রি করলে ক্ষতি = x - 800 টাকা
1200 টাকায় বিক্রি করলে লাভ = 1200 - x টাকা

প্রশ্নমতে,
1200 - x = 3(x - 800)
⇒ 1200 - x = 3x - 2400
⇒ 1200 + 2400 = 3x + x
⇒ 3600 = 4x
⇒ x = 3600/4
∴ x = 900

এখন,
25% লাভে,
ক্রয়মূল্য 100 টাকা হলে বিক্রয়মূল্য 125 টাকা
ক্রয়মূল্য 1 টাকা হলে বিক্রয়মূল্য (125/100) টাকা
∴ ক্রয়মূল্য ৯০০ টাকা হলে বিক্রয়মূল্য (125 × 900)/100 টাকা
= 1125 টাকা

৭,৬০২.
If θ is an acute angle and 7sin2θ + 3cos2θ = 4, what is the value of tanθ?
  1. ক
    1/√3
  2. খ
    √3
  3. গ
    0
  4. ঘ
    1
ব্যাখ্যা
Question: If θ is an acute angle and 7sin2θ + 3cos2θ = 4, what is the value of tanθ?

Solution:
7sin2θ + 3cos2θ = 4
⇒ 7sin2θ + 3(1 − sin2θ) = 4
⇒ 7sin2θ + 3 − 3sin2θ = 4
⇒ 4sin2θ = 1
⇒ sin2θ = 1/4
⇒ sinθ = 1/2
⇒ sinθ = sin30°
⇒ θ = 30°

∴ tanθ = tan30° = 1/√3
৭,৬০৩.
log49/log7 = ?
  1. ক
    ক) 1/2
  2. খ
    খ) 2
  3. গ
    গ) 7
  4. ঘ
    ঘ) 6
ব্যাখ্যা
প্রশ্ন : log49/log7 = ?
সমাধান :
log49/log7
= log 72/log7
= 2.log7/log7
= 2
 
৭,৬০৪.
Which of the following numbers can be the last digit of a squared number? 
  1. ক
    ক) 2
  2. খ
    খ) 3
  3. গ
    গ) 5
  4. ঘ
    ঘ) 7
ব্যাখ্যা
বর্গ সংখ্যার শেষের অংক  0, 1, 4, 5, 6 এবং 9 হতে পারে। 

1 এর বর্গ = 12 = 1
2 এর বর্গ = 22 = 4
3 এর বর্গ = 32 = 9 
4 এর বর্গ = 42 = 16 
5 এর বর্গ = 52 = 25 
6 এর বর্গ = 62 = 36 
10 এর বর্গ= 102 = 100
৭,৬০৫.
The sum of the first 12 terms of the series 5, 9, 13, 17,......?
  1. ক
    124
  2. খ
    204
  3. গ
    270
  4. ঘ
    324
ব্যাখ্যা
Question: The sum of the first 12 terms of the series 5, 9, 13, 17,......?

Solution:
এখানে,
প্রথম পদ, a = 5
সাধারন অন্তর, d = 9 - 5 = 4
পদসংখ্যা, n = 12
∴ 12টি পদের সমষ্টি = (12/2) [2 × 5 + (12 - 1) × 4]
= 6(10 + 44)
= 6 × 54
= 324
৭,৬০৬.
The equation 4x2 + 12px + 9 = 0 has real and equal roots, if-
  1. ক
    p = ± 1
  2. খ
    p = 3
  3. গ
    p = ± 5
  4. ঘ
    p = ± 8
ব্যাখ্যা

Question: The equation 4x2 + 12px + 9 = 0 has real and equal roots, if-

Solution:
Given, 4x2 + 12px + 9 = 0

Here a = 4, b = 12p, c = 9

যেহেতু প্রদত্ত সমীকরণটির বাস্তব ও সমান মূল আছে,
∴ b2 - 4ac = 0
⇒ (12p)2 - 4 × 4 × 9 = 0
⇒ 144p2 - 144 = 0
⇒ 144p2 = 144
⇒ p2 = 1
⇒ p = ± 1

সুতরাং, p = 1 অথবা p = -1

• দ্বিঘাত সমীকরণের মূলের প্রকৃতি:
1. যদি b2 - 4ac = 0 হয় তবে দ্বিঘাত সমীকরণের মূলদ্বয় বাস্তব ও সমান হবে।
2. যদি b2 - 4ac > 0 হয় তবে দ্বিঘাত সমীকরণের মূলদ্বয় বাস্তব ও অসমান হবে।
3. যদি b2 - 4ac < 0 হয় তবে দ্বিঘাত সমীকরণের মূলদ্বয় অবাস্তব ও অসমান হবে।
4. যদি b2 - 4ac পূর্ণবর্গ সংখ্যা হয় তবে দ্বিঘাত সমীকরণের মূলদ্বয় মূলদ ও অসমান হবে।

৭,৬০৭.
A man borrowed some money for six months. He paid Tk. 500 at an interest rate of 10% per annum. What was the amount he borrowed?
  1. ক
    Tk. 8500
  2. খ
    Tk. 10000
  3. গ
    Tk. 12800
  4. ঘ
    Tk. 15000
ব্যাখ্যা

Question: A man borrowed some money for six months. He paid Tk. 500 at an interest rate of 10% per annum. What was the amount he borrowed?

Solution:
এখানে,
সরল সুদ (SI) = Tk. 500
সুদের হার, r = 10%
সময়, n = 6 মাস = 6/12 = 1/2 বছর
আসল (Principal), P = ?

আমরা জানি,
I = Pnr/100
⇒ 500 = (P × 1/2 ×10)/100
⇒ 500 = 5P/100
⇒ 500 = P/20
⇒ P = 20 × 500
∴ P = 10000

অতএব, তিনি মোট Tk. 10,000 ধার নেন।

৭,৬০৮.
The speed of the boat in still water is 5 times that of current, it takes 1.1 hour to row to point B from point A downstream. The distance between point A and point B is 13.2 km. How much distance (in km) will it cover in 312 minutes upstream?
  1. ক
    ক) 43.2 km
  2. খ
    খ) 48 km
  3. গ
    গ) 41.6 km
  4. ঘ
    ঘ) 44.8 km
ব্যাখ্যা

Let the speed of the current be x kmph
Then speed of the boat in still water = 5x

∴ Downstream speed
= (5x + x) = 6x km/hr.
Upstream speed
= (5x - x)
= 4x km/hr

Now,
According to the question,
1.1 × 6x = 13.2 km
6.6x = 13.2
x = 13.2/6.6
x = 2 km/hr.

Upstream speed
= 4x = 4 × 2 = 8 km/hr

∴ 312 minutes
= 312/60 hours
= 5(1/5) hours

∴ Required Distance travelled upstream
= Speed × Time
= 8 × (26/5)
= 41.6 km

৭,৬০৯.

Square RSTU shown above is rotated in a plane about its center in a clockwise direction the minimum number of degrees necessary for T to be in the position where S is now shown. The number of degrees through which RSTU is rotated is-
  1. ক
    135 degree
  2. খ
    270 degree
  3. গ
    225 degree
  4. ঘ
    180 degree
ব্যাখ্যা
Question:

Square RSTU shown above is rotated in a plane about its center in a clockwise direction the minimum number of degrees necessary for T to be in the position where S is now shown. The number of degrees through which RSTU is rotated is-

Solution:
T থেকে U পর্যন্ত ঘুরলে ৯০° অতিক্রম করা হবে।
U থেকে R পর্যন্ত ঘুরলে ৯০° অতিক্রম করা হবে।
R থেকে S পর্যন্ত ঘুরলে ৯০° অতিক্রম করা হবে।
∴ T যদি S এর অবস্থানে যেতে চায় তাহলে (৯০° + ৯০° + ৯০°) = ২৭০° ঘুরতে হবে।
৭,৬১০.
The surface area of a sphere is the same as the curved surface area of a right circular cylinder whose height and diameter are 12 cm each. The radius of the sphere is:
  1. ক
    4 cm
  2. খ
    5 cm
  3. গ
    6 cm
  4. ঘ
    8 cm
ব্যাখ্যা
Question: The surface area of a sphere is the same as the curved surface area of a right circular cylinder whose height and diameter are 12 cm each. The radius of the sphere is:

Solution:
The surface area of sphere = 4πr12
The curved surface area of cylinder =2πr2h
diameter = 12 cm
radius r2 = 6 cm

⇒ 4πr12 = 2πrh
⇒ r12 = (6 × 12)/2
⇒ r12 = 36
⇒ r1 = 6

radius of the sphere 6 cm
৭,৬১১.
At an animal shelter, the ratio of tigers to Zebras is 4 to 7. If there are 27 more zebras than tigers, how many tigers there in the shelter?
  1. ক
    36
  2. খ
    45
  3. গ
    52
  4. ঘ
    75
  5. ঙ
    .
ব্যাখ্যা
Question: At an animal shelter, the ratio of tigers to Zebras is 4 to 7. If there are 27 more zebras than tigers, how many tigers there in the shelter?

Solution:
ধরি
বাঘের সংখ্যা = 4x টি
জেব্রার সংখ্যা = 7x টি

প্রশ্নমতে
7x - 4x = 27
3x = 27
x = 9

বাঘের সংখ্যা = 4 × 9 টি
= 36টি
৭,৬১২.
In a quiz, a student gets 5 marks for each correct answer and loses 3 marks for each wrong answer. He attempts 50 questions and scores 170 marks. How many questions did he answer correctly?
  1. ক
    45
  2. খ
    30
  3. গ
    35
  4. ঘ
    40
ব্যাখ্যা
Question: In a quiz, a student gets 5 marks for each correct answer and loses 3 marks for each wrong answer. He attempts 50 questions and scores 170 marks. How many questions did he answer correctly?

Solution:
Let,
the number of correct answers be x,
and the number of wrong answers is (50 - x).

ATQ,
5x - 3(50 - x) = 170
⇒ 5x - 150 + 3x = 170
⇒ 8x - 150 = 170
⇒ 8x = 170 + 150
⇒ x = 320/8
∴ x = 40

∴ He attempted 40 questions correctly.
৭,৬১৩.
What is the HCF of 408 and 1032?
  1. ক
    8
  2. খ
    12
  3. গ
    18
  4. ঘ
    24
ব্যাখ্যা
Question: What is the HCF of 408 and 1032?

Solution:
The prime factorisation of 408 is 2 × 2 × 2 × 3 × 17

The prime factorisation of 1032 is 2 × 2 × 2 × 3 × 43

Thus, the product of the prime factors of 2 × 2 × 2 × 3 is 24

Hence, the HCF of 408 and 1032 is 24.
৭,৬১৪.
Find the rate of interest if the amount after 2 years of simple interest on a capital of Tk. 1200 is Tk. 1440.
  1. ক
    ক) 20%
  2. খ
    খ) 15%
  3. গ
    গ) 12%
  4. ঘ
    ঘ) 10%
ব্যাখ্যা
Question: Find the rate of interest if the amount after 2 years of simple interest on a capital of Tk. 1200 is Tk. 1440.

Solution:
Given,
Amount, A = Tk.1440
Principal, P = Tk.1200
Interest, I = Tk.(1440 - 1200) = Tk. 240
Time, n = 2 years.

We know,
I = Pnr
⇒ r = I/Pn
⇒ r = (240 × 100)/(1200 × 2) 
∴ r = 10%
৭,৬১৫.
A man goes to his office from his house at a speed of 3 km/hr and returns at a speed of 2 km/hr. If he takes 5 hours in going and coming, what is the distance between his house and office?
  1. ক
    3 km
  2. খ
    5 km
  3. গ
    6 km
  4. ঘ
    4 km
ব্যাখ্যা

Average speed = (2 × 3 × 2)/(2 + 3)
= 12/5 km/h
Total time taken = 5 hours
Distance travelled = (12/5) × 5
= 12 km
Therefore, distance between his house and office
= 12/2
= 6 km.

৭,৬১৬.
A machine is purchased for Tk. 625000. Its value depreciates at the rate of 8% per annum. What will be its value after 2 years?
  1. ক
    Tk. 525000
  2. খ
    Tk. 600000
  3. গ
    Tk. 529000
  4. ঘ
    Tk. 590000
  5. ঙ
    None of these
ব্যাখ্যা
Question: A machine is purchased for Tk. 625000. Its value depreciates at the rate of 8% per annum. What will be its value after 2 years?

Solution:
To find the depreciated value of the machine after 2 years, we can use the formula for depreciation:
A = P(1 - r/100)t
P is the initial value of the machine = 625,000 Tk,
r is the rate of depreciation per annum = 8%,
t is the time in years = 2.

∴ A = 625000(1 - 8/100)2
= 625000 × 0.92 × 0.92
= 529000
৭,৬১৭.
তিন অঙ্ক বিশিষ্ট একটি সংখ্যার একক স্থানীয় অঙ্কটি দশক স্থানীয় অঙ্কের ৭৫%। দশক স্থানের অঙ্কটি শতক স্থানের অঙ্কের চেয়ে ১ বেশি। দশক ও শতক স্থানের অঙ্কদ্বয়ের সমষ্টি ১৫ হলে, সংখ্যাটি কত?
  1. ক
    ৭৯৫
  2. খ
    ৬৮৭
  3. গ
    ৭৮৬
  4. ঘ
    ৫৮৭
  5. ঙ
    কোনটিই নয়
ব্যাখ্যা
প্রশ্ন: তিন অঙ্ক বিশিষ্ট একটি সংখ্যার একক স্থানীয় অঙ্কটি দশক স্থানীয় অঙ্কের ৭৫%। দশক স্থানের অঙ্কটি শতক স্থানের অঙ্কের চেয়ে ১ বেশি। দশক ও শতক স্থানের অঙ্কদ্বয়ের সমষ্টি ১৫ হলে, সংখ্যাটি কত?

সমাধান:
ধরি,
শতক স্থানীয় অঙ্ক = ক
দশক স্থানীয় অঙ্ক = ক + ১
একক স্থানীয় অঙ্ক = (৭৫/১০০) × (ক + ১)
= ৩ক + ৩/৪

প্রশ্নমতে,
ক + ১ + ক = ১৫
⇒ ২ক = ১৪
⇒ ক = ৭

তাহলে,
শতক স্থানীয় অঙ্ক = ৭
দশক স্থানীয় অঙ্ক = ৭ + ১ = ৮
একক স্থানীয় অঙ্ক = {(৩ × ৭) + ৩}/৪ = ৬
∴ সংখ্যাটি = ৭৮৬
৭,৬১৮.
A rectangular field will be fenced on three side leaving one side of 20 feet uncovered. If the area of the field 680 square feet. How many feet of fencing is required?
  1. ক
    ক) 88
  2. খ
    খ) 34
  3. গ
    গ) 40
  4. ঘ
    ঘ) 68
ব্যাখ্যা
আয়তাকার মাঠের এক পাশের দৈর্ঘ্য = 20 ফুট 
আয়তাকার মাঠের  ক্ষেত্রফল = 680 বর্গ ফুট 

আয়তাকার মাঠের অন্য পাশের দৈর্ঘ্য = 680/20 ফুট = 34 ফুট

বেড়ার দৈর্ঘ্য = (34 + 20 + 34) = 88 ফুট
৭,৬১৯.
A tank has two leakages. The first leakage alone can empty the tank in 9 minutes and the second alone would have done it in 6 minutes. If water leaks out at a constant rate, how long does it take both the leakage together to empty the tank?
  1. ক
    ক) 3.6 min
  2. খ
    খ) 3.1 min
  3. গ
    গ) 3.5 min
  4. ঘ
    ঘ) 4.0 min
ব্যাখ্যা

In the 1st case,
1/9 part of the tank is emptied in 1 min
In the 2nd case,
1/6 part of the tank is emptied in 1 min
So, together in 1 min, they empties = (1/9 + 1/6) = 5/18 part

Now, 5/18 part is emptied in 1 minute
∴ 1 or, total part is emptied in = 18/5 = 3.6 minutes

৭,৬২০.
A is five times as old as B. 5 years ago, A was ten times as old as B. What will be the sum of their ages after 8 years?
  1. ক
    65 years
  2. খ
    70 years
  3. গ
    93 years
  4. ঘ
    84 years
ব্যাখ্যা

Question: A is five times as old as B. 5 years ago, A was ten times as old as B. What will be the sum of their ages after 8 years?

সমাধান:
ধরি,
B-এর বর্তমান বয়স = x বছর
A-এর বর্তমান বয়স = 5x বছর

5 বছর আগে,
B-এর বয়স ছিল = (x - 5) বছর
A-এর বয়স ছিল = (5x - 5) বছর

প্রশ্নমতে,
5x - 5 = 10(x - 5)
⇒ 5x - 5 = 10x - 50
⇒ 50 - 5 = 10x - 5x
⇒ 45 = 5x
⇒ x = 45/5
∴ x = 9

সুতরাং, B-এর বর্তমান বয়স 9 বছর এবং A-এর বর্তমান বয়স 5 × 9 = 45 বছর।

8 বছর পর,
B-এর বয়স হবে = 9 + 8 = 17 বছর
A-এর বয়স হবে = 45 + 8 = 53 বছর

∴ 8 বছর পর তাদের বয়সের সমষ্টি হবে = 17 + 53 = 70 বছর।

৭,৬২১.
4 men can do one work in 8 days.  How many men can do the half work in 2 days?
  1. ক
    ক) 6 men
  2. খ
    খ) 8 men
  3. গ
    গ) 10 men
  4. ঘ
    ঘ) 5 men
ব্যাখ্যা
Question: 4 men can do one work in 8 days.  How many men can do the half work in 2 days?

Solution: 
From the first part of the question, we get - 4 men can do one work in 8 days.
here, 
men, M1 = 4,
work, W1 = 1,
day, D1 = 8.

From the second part of the question, we get - How many men can do the half work in 2 days?
here, 
men, M2 =?
work, W2 = 1/2,
day, D = 2.

we know,
M1 × W2 × D1 = M2 × W1 × D2
M2 = (M1 × W2 × D1) / (W1 × D2)
= {4 × (1/2) × 8}/ (1 × 2)
= 16/2
= 8 men
৭,৬২২.
A man riding his bicycle covers 150 meters in 25 seconds. What is his speed in km per hour?
  1. ক
    ক) 20
  2. খ
    খ) 23
  3. গ
    গ) 21.6
  4. ঘ
    ঘ) 25
ব্যাখ্যা
Question: A man riding his bicycle covers 150 meters in 25 seconds. What is his speed in km per hour?

Solution:
Given,
Distance = 150 meters.
TIme = 25 seconds

∴ Speed = Distance/Time
= 150/25 m/s
= (150 × 3600)/(25 × 1000) km/hr.
= 21.6 km/hr.
৭,৬২৩.
Tea worth tk.126 per kg and Tk.135 per kg are mixed with a third variety in the ratio 1:1:2. If the mixture is worth Tk.153 per kg, The price of third variety per kg will be:
  1. ক
    ক) Tk. 169.50
  2. খ
    খ) Tk. 170
  3. গ
    গ) Tk. 175.50
  4. ঘ
    ঘ) Tk. 180
ব্যাখ্যা

Assume that the Third Variety of Tea quantity is X.
Then (126 X 1 + 135 X 1 + 2 X x)/(1+1+2) = 153
On solving,
261 + 2x = 4 X 153
2x = 612-261
x = 351/2
∴ x = 175.50

৭,৬২৪.
A dishonest trade mixes 2kg of vegetable ghee costing Tk. 45 a kg with 3kg of standard ghee costing TK. 70 per kg. He sells the mixed ghee at TK. 65 per kg. What is his percentage of profit?
  1. ক
    8.33%
  2. খ
    16.67%
  3. গ
    25%
  4. ঘ
    6%
ব্যাখ্যা
Question: A dishonest trade mixes 2kg of vegetable ghee costing Tk. 45 a kg with 3kg of standard ghee costing TK. 70 per kg. He sells the mixed ghee at TK. 65 per kg. What is his percentage of profit?

Solution:
Costing for ghee is (2 × 45 + 3 × 70) Taka
= 90 + 210 Taka
= 300 Taka

Selling Price = (2 + 3) × 65 taka 
= 5 × 65 Taka
= 325 Taka

∴ Profit 325 - 300 Taka 
= 25 Taka

∴ Percentage of profit = (25 × 100)/300 %
= 8.33%
৭,৬২৫.
If A : B = 3 : 4, C : B = 5 : 4, C : D = 10 : 9 then A : B : C : D is -
  1. ক
    ক) 8 : 6 : 9 : 10
  2. খ
    খ) 8 : 6 : 10 : 9
  3. গ
    গ) 6 : 8 : 10 : 9
  4. ঘ
    ঘ) 6 : 8 : 9 : 10
ব্যাখ্যা

A: B = 3 : 4 = (3 × 4) : (4 × 4) = 12 : 16
C : B = 5 : 4 = (5 × 4) : (4 × 4) = 20 : 16
C : D = 10 : 9 = (10 × 2) : (9 × 2) = 20 : 18
∴ A : B : C : D = 12 : 16 : 20 : 18 = 6 : 8 :10 : 9

৭,৬২৬.
Find the largest fraction from the following:
  1. ক
    ক) - 5/11
  2. খ
    খ) - 8/13
  3. গ
    গ) - 7/19
  4. ঘ
    ঘ) - 15/97
ব্যাখ্যা
প্রশ্ন: Find the largest fraction from the following:

সমাধান: 
এখানে,
-৫/১১ এবং -৮/১৩ থেকে আলাদাভাবে হর ও লব গুণ করে পাই, 
-৬৫ > -৮৮
সুতরাং, -৫/১১ > -৮/১৩ 

আবার,
-৫/১১ এবং -৭/১৯ থেকে আলাদাভাবে হর ও লব গুণ করে পাই, 
-৯৫ < -৭৭
সুতরাং, -৫/১১ < -৭/১৯ 

আবার,
-৭/১৯ এবং -১৫/৯৭ থেকে আলাদাভাবে হর ও লব গুণ করে পাই, 
-৬৭৯ < -২৮৫
সুতরাং, -৭/১৯ < -১৫/৯৭

সবগুলো সম্পর্ক থেকে দেখা যায়,
বৃহত্তম ভগ্নাংশ -১৫/৯৭
৭,৬২৭.
There are two numbers. HCF of both the numbers is 11, and their LCM is 693. If the first number is 77, find the second number?
  1. ক
    89
  2. খ
    56
  3. গ
    78
  4. ঘ
    99
ব্যাখ্যা
Question: There are two numbers. HCF of both the numbers is 11, and their LCM is 693. If the first number is 77, find the second number?

Solution:
The product of two numbers = HCF × LCM

Let the required number is = x
So, 77 × x = 11 × 693
⇒ x = (11 × 693)/77
∴ x = 99
৭,৬২৮.
Excluding stoppages, the speed of a bus is 80 kmph and including stoppages, it is 60 kmph. For how many minutes does the bus stop per hour?
  1. ক
    10 minutes
  2. খ
    12 minutes
  3. গ
    15 minutes
  4. ঘ
    18 minutes
ব্যাখ্যা

Question: Excluding stoppages, the speed of a bus is 80 kmph and including stoppages, it is 60 kmph. For how many minutes does the bus stop per hour?

Solution:
স্টপেজ ছাড়া বাসের গতিবেগ = 80 কিমি/ঘন্টা
স্টপেজ সহ বাসের গতিবেগ = 60 কিমি/ঘন্টা

প্রতি ঘন্টায় বাসটি যে দূরত্ব কম অতিক্রম করে = (80 - 60) কিমি = 20 কিমি।

এই 20 কিমি দূরত্ব অতিক্রম করতে বাসটির যে সময় লাগতো, সেই সময়টুকুই বাসটি থেমে থাকে।

গতিবেগ 80 কিমি/ঘন্টা হলে 20 কিমি যেতে সময় লাগে:
সময় = দূরত্ব/গতিবেগ
= 20 কিমি/80 কিমি/ঘন্টা
= 1/4 ঘন্টা
= (1/4 × 60) মিনিট
= 15 মিনিট

সুতরাং, প্রতি ঘন্টায় বাসটি 15 মিনিট থামে।

৭,৬২৯.
An employer pays 3 workers X, Y and Z a total of Tk. 36600 a week. X is paid 125% of the amount Y is paid and 80% of the amount Z is paid. How much does X make a week?
  1. ক
    ক) 9000
  2. খ
    খ) 12000
  3. গ
    গ) 10800
  4. ঘ
    ঘ) 11700
ব্যাখ্যা

x = 125% of y = 125y/100 = 5y/4
⇒ y = 4x/5
Again,
x = 80% of z = 80z/100 = 4z/5
⇒ z = 5x/4
x : y : z = x : 4x/5 : 5x/4 = 20 : 16 : 25
Sum of ratio = 20 + 16 + 25 = 61
So, x makes a week = 20/61 × 36600 = 20×600 = 12000 tk.

৭,৬৩০.
Machine A produces bolts at a uniform rate of 120 every 40 seconds and Machine B produce bolts at a uniform rate of 100 every 20 seconds. If the two machines run simultaneously, how many seconds will it take for them to produce a total of 200 bolts?
  1. ক
    ক) 22
  2. খ
    খ) 25
  3. গ
    গ) 28
  4. ঘ
    ঘ) 32
  5. ঙ
    ঙ) 56
ব্যাখ্যা
Question: Machine A produces bolts at a uniform rate of 120 every 40 seconds and Machine B produce bolts at a uniform rate of 100 every 20 seconds. If the two machines run simultaneously, how many seconds will it take for them to produce a total of 200 bolts?

Solution:
Given
Machine a produces bolts at a uniform rate of 120 every 40 second
∴ machine a produce 120/40 = 3 bolts per second

Machine b produces bolts at a uniform rate of 100 every 20 second
∴ machine b produces 100/20 = 5 bolts per second

Both machine produce 3 + 5 = 8 bolts running simultaneously per second.

Machine produces 8 bolts for 1 second
∴ For 200 bolts the time taken is 200/8 seconds
= 25 seconds

The machine takes 25 seconds to produce 200 bolts.
৭,৬৩১.
Which number will complete the series: 1, 3, 7, 15, 31, 63,__?
  1. ক
    ক) 123
  2. খ
    খ) 125
  3. গ
    গ) 127
  4. ঘ
    ঘ) 129
ব্যাখ্যা
Question: Which number will complete the series: 1, 3, 7, 15, 31, 63, __?

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

∴ The next number of 63 will be 63 + 32 × 2
= 63 + 64
= 127
৭,৬৩২.
If set A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}. Then write the universal set for all three sets.
  1. ক
    ক) {0, 1, 2, 3, 4, 5, 6, 8}
  2. খ
    খ) {0, 1, 2, 3, 4, 5, 6, 7, 8}
  3. গ
    গ) {1, 2, 3, 4, 5, 6, 7, 8}
  4. ঘ
    ঘ) {1, 2, 3, 4, 5, 6, 7}
ব্যাখ্যা
Question: If set A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}. Then write the universal set for all three sets.

Solution:
Let U is the universal set for sets A, B and C, 
Here,
U = A ∪ B ∪ C
U = {1, 3, 5} ∪ {2, 4, 6} ∪ {0, 2, 4, 6, 8}
U = {0, 1, 2, 3, 4, 5, 6, 8}
৭,৬৩৩.
A and B invest in a business in the ratio 3 : 2. If 5% of the total profit goes to charity and A's share is Tk. 1425, the total profit is:
  1. ক
    Tk. 2500
  2. খ
    Tk. 3200
  3. গ
    Tk. 2000
  4. ঘ
    Tk. 1500
ব্যাখ্যা
Question: A and B invest in a business in the ratio 3 : 2. If 5% of the total profit goes to charity and A's share is Tk. 1425, the total profit is:

Solution:
Let,
the total profit be = Tk.100
After paying to charity, A's share = Tk. {95 × (3/5)}
= Tk. 57

If A's share is = Tk. 57
∴ Total profit = Tk. 100

If A's share = Tk. 1425
∴ Total profit = {(100/57) × 1425}
= 2500
৭,৬৩৪.
A shopkeeper buys 40 kg sugar at taka 45 per kg. and 60 kg rice at taka 50 per kg. At what price approximately in taka per kg should he sell to make profit 15% of cost?
  1. ক
    Tk. 55.20
  2. খ
    Tk. 65.50
  3. গ
    Tk. 58
  4. ঘ
    Tk. 50.20
  5. ঙ
    Tk. 55
ব্যাখ্যা
Question: A shopkeeper buys 40 kg sugar at taka 45 per kg. and 60 kg rice at taka 50 per kg. At what price approximately in taka per kg should he sell to make profit 15% of cost?

Solution:
Cost of 40 kg sugar is = 40 × 45 = Tk. 1800
Cost of 60 kg sugar is = 60 × 50 = Tk. 3000

∴ Total cost = 1800 + 3000 = Tk. 4800
And total weight = 40 + 60 = 100 kg

Now, Total selling price = 4800 + 4800 of 15% = 4800 + 720 = Tk. 5520 

∴ Selling price per kg = 5520 ÷ 100 = Tk. 55.20 per kg
৭,৬৩৫.
What will be the profit percentage, if bananas purchased 10 for Tk. 100 and sold 8 for Tk. 100?
  1. ক
    ক) 20%
  2. খ
    খ) 30%
  3. গ
    গ) 15%
  4. ঘ
    ঘ) 25%
ব্যাখ্যা
Question: What will be the profit percentage, if bananas purchased 10 for Tk. 100 and sold 8 for Tk. 100?
Solution:

১০ টি কলার ক্রয়মূল্য ১০০ টাকা
∴ ১ টি কলার ক্রয়মূল্য ১০ টাকা

৮ টি কলার বিক্রয়মূল্য ১০০ টাকা
∴ ১ টি কলার বিক্রয়মূল্য ২৫/২ টাকা

∴ লাভ = (২৫/২ - ১০) = ৫/২ টাকা
 
১০ টাকায় লাভ হয় ৫/২ টাকা
∴ ১০০ টাকায় লাভ হয় (৫ × ১০০)/(১০ × ২) = ২৫ টাকা।
৭,৬৩৬.
A, B, C subscribe Tk. 50,000 for a business. A subscribes Tk. 4000 more than B and B tk. 5000 more than C. Out of a total profit of tk. 35,000, A receives
  1. ক
    ক) 8,400
  2. খ
    খ) 11,900
  3. গ
    গ) 13,600
  4. ঘ
    ঘ) 14,700
ব্যাখ্যা

Let C = x.
Then, B = x + 5000 and A = x + 5000 + 4000 = x + 9000
So, x + x + 5000 + x + 9000 = 50000
⇒ 3x = 36000
⇒ x = 12000
A : B : C = 21000 : 17000 : 12000 = 21 : 17 : 12
So A's Share 
= Tk 35000× (21/50)
= Tk 14700

৭,৬৩৭.
Of 132 examinees of a certain school, the ratio of successful to unsuccessful candidates is 9 : 2. If 4 more students passed, what would have been the ratio of successful to unsuccessful students?
  1. ক
    21 : 5
  2. খ
    18 : 5
  3. গ
    23 : 5
  4. ঘ
    28 : 5
ব্যাখ্যা
Question: Of 132 examinees of a certain school, the ratio of successful to unsuccessful candidates is 9 : 2. If 4 more students passed, what would have been the ratio of successful to unsuccessful students?

Solution: 
successful candidates = (9/11) × 132
= 108 

unsuccessful candidates = (2/11) × 132
= 24 

new successful candidates = 108 + 4 
= 112 
new unsuccessful candidates = 24 - 4 = 20

new ratio = 112 : 20 
= 28 : 5 
৭,৬৩৮.
If the difference between the circumference and diameter of a circle is 240 cm, then the diameter of the circle is:
  1. ক
    164 cm
  2. খ
    112 cm
  3. গ
    144 cm
  4. ঘ
    None of the above
ব্যাখ্যা

Question: If the difference between the circumference and diameter of a circle is 240 cm, then the diameter of the circle is:

Solution:
ধরি,
বৃত্তের ব্যাসার্ধ = r
বৃত্তের ব্যাস = 2r
বৃত্তের পরিধি = 2πr

প্রশ্নমতে,
2πr - 2r = 240
⇒ 2r(π - 1) = 240
⇒ r = (240/2){(22/7) - 1}
⇒ r = 120/(22 - 7)/7
⇒ r = (120 × 7)/15
∴ r = 56

∴ বৃত্তের ব্যাস = 2r = 2 × 56 = 112 সে.মি.

৭,৬৩৯.
If a + (1/a) = √7, what is the value of a3 + (1/a3)?
  1. ক
    3√7
  2. খ
    5√5
  3. গ
    6√7
  4. ঘ
    4√7
ব্যাখ্যা

Question: If a + (1/a) = √7, what is the value of a3 + (1/a3)?

Solution:
দেওয়া আছে, a + (1/a) = √7

এখন,
a3 + (1/a3)
= (a + 1/a)3 - 3 . a . 1/a(a + 1/a)
= (√7)3 - 3 × (√7)
= 7√7 − 3√7
= 4√7

৭,৬৪০.
A committee of 5 members is to be formed by selecting out of 8 men and 6 women. In how many different ways the committee can be formed if it should have at least 4 men?
  1. ক
    ক) 376
  2. খ
    খ) 476
  3. গ
    গ) 746
  4. ঘ
    ঘ) 647
ব্যাখ্যা
Question: A committee of 5 members is to be formed by selecting out of 8 men and 6 women. In how many different ways the committee can be formed if it should have at least 4 men? 

Solution:
      Men (8)       Women (6)
1)    4                     1
2)    5                     0

From (1) Number of ways = 8C4 × 6C1 = 70 × 6 = 420
From (2) Number of ways = 8C5 × 6C0 = 56 × 1 = 56

Total number of ways = 420 + 56 = 476
৭,৬৪১.
An investment doubles in 8 years at simple interest. In how many years will it become four times? 
  1. ক
    8
  2. খ
    16
  3. গ
    24
  4. ঘ
    32
ব্যাখ্যা

Question: An investment doubles in 8 years at simple interest. In how many years will it become four times?

Solution:
Given:
In 8 years, interest earned = P (because, P + interest = 2P)

To become 4 times (Total amount = 4P):
Interest needed = 4P - P = 3P

Since interest is earned linearly with time at simple interest:

P interest in = 8 years
3P interest in = 8 × 3 = 24 years

∴ 24 years

৭,৬৪২.
In an examination, a student must score 35% of the total marks to pass. A student scores 189 marks and is declared failed by 7% marks. What is the maximum total marks in the exam?
  1. ক
    650
  2. খ
    675
  3. গ
    710
  4. ঘ
    750
ব্যাখ্যা
Question: In an examination, a student must score 35% of the total marks to pass. A student scores 189 marks and is declared failed by 7% marks. What is the maximum total marks in the exam?

Solution:
Let
the maximum marks be x.

Then,
(35 - 7)% of x = 189
⇒ 28% of x = 189
⇒ 28x/100 = 189
⇒ 28x = (189 × 100)
⇒ x = (189 × 100)/28
∴ x = 675
৭,৬৪৩.
The average weight of A, B, and C is 60 kg, and that of B and C is 62 kg. A’s present weight is-
  1. ক
    48 kg
  2. খ
    50 kg
  3. গ
    56 kg
  4. ঘ
    52 kg
ব্যাখ্যা
Question: The average weight of A, B, and C is 60 kg, and that of B and C is 62 kg. A’s present weight is-

Solution:
The average of A, B and C is 60

So, the sum of their weights = 180
The sum of the weights of B and C is = 62 × 2 = 124

So, A’s present weight is = 180 - 124 = 56 kg
৭,৬৪৪.
A sum of Tk. 8000 will amount to Tk. 8820 in 2 years if the interest is calculated every year. The rate of compound interest is?
  1. ক
    ক) 8%
  2. খ
    খ) 6%
  3. গ
    গ) 5%
  4. ঘ
    ঘ) 4%
ব্যাখ্যা
Question: A sum of Tk. 8000 will amount to Tk. 8820 in 2 years if the interest is calculated every year. The rate of compound interest is?

Solution:
Principal = Tk. 8000
Amount = Tk. 8820
Let Rate = R
Time = 2 years

By using formula, 
8000{1 + (R/100)}2 = 8820
⇒ {1 + (R/100)}2 = 8820/8000
⇒ {1 + (R/100)}2 = 441/200
⇒ {1 + (R/100)}2 = (21/20)2
⇒ 1 + (R/100) = 21/20
⇒ R/100 = (21/20) - 1
⇒ R/100 = 1/20
∴ R = 5%
৭,৬৪৫.
What will be the least number which when doubled will be exactly divisible by 18, 24, 28, and 36?
  1. ক
    ক) 1008
  2. খ
    খ) 504
  3. গ
    গ) 360
  4. ঘ
    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
৭,৬৪৬.
Find the product of two consecutive numbers where four times the first number is 10 more than thrice the second number.
  1. ক
    ক) 182
  2. খ
    খ) 128
  3. গ
    গ) 821
  4. ঘ
    ঘ) 218
ব্যাখ্যা
Question: Find the product of two consecutive numbers where four times the first number is 10 more than thrice the second number.
Solution:
Let the numbers are ‘a' and ‘a + 1’.
According to the question :
4a = 3 × (a + 1) + 10
4a = 3a + 3 + 10
4a - 3a = 13
⇒ a = 13
Hence, the numbers are 13 and 14.

∴ Product = 13 × 14 = 182
৭,৬৪৭.
Rafi got twice as many sums wrong as he got right. If he attempted 48 sums in all, how many did he solve correctly?
  1. ক
    24
  2. খ
    18
  3. গ
    16
  4. ঘ
    10
ব্যাখ্যা
Question: Rafi got twice as many sums wrong as he got right. If he attempted 48 sums in all, how many did he solve correctly?

Solution: 
let, right sum = x 
wrong sum = 2x 

2x + x = 48 
⇒ 3x = 48
∴ x = 16 
৭,৬৪৮.
A square park is surrounded by a path of uniform width 2 meters all around it. The area of the path is 288 sq. meters. Find the perimeter of the park with path.
  1. ক
    136 m
  2. খ
    1444 m
  3. গ
    152 m
  4. ঘ
    1156 m
ব্যাখ্যা
Question: A square park is surrounded by a path of uniform width 2 meters all around it. The area of the path is 288 sq. meters. Find the perimeter of the park with path.

Solution:
Let,
One side of the park is x meter.
So, one side of the park with path = x + (2 + 2) meter
= x + 4

We know,
Area of the park = x2
Area of the path, (x + 4)2 - x2 = 288
⇒ x2 + 8x + 16 - x2 = 288 
⇒ 8x + 16 = 288
⇒ 8x = 288 - 16
⇒ 8x = 272
⇒ x = 272/8
∴ x = 34

One side of the square park = 34 m.
One side of the square park with path = 34 + 4 = 38 m.

So, perimeter of the square park with path = 4 × 38
= 152m
৭,৬৪৯.
A watch priced at Tk. 1200 discount two times and bought at Tk. 600. The first discount was 20%. What is the second discount? 
  1. ক
    32%
  2. খ
    30.25%
  3. গ
    25%
  4. ঘ
    37.5%
ব্যাখ্যা

Question: A watch priced at Tk. 1200 discount two times and bought at Tk. 600. The first discount was 20%. What is the second discount?

 Solution:
Marked Price = 1200 taka
First discount = 20%
Second discounted price = 600

Suppose,
Second discount % = x

First discounted price = 1200 - (20% of 1200)
= 1200 - 240
= 960

Second discounted price = 960 - (x% of 960)
600 = 960 - (x% of 960) 
960(1 - x / 100) = 600

৭,৬৫০.
At Company K, 15 percent of the employees are secretaries and 60 percent are salespeople. If there are 45 other employees of Company K, how many employees does Company K have?
  1. ক
    160
  2. খ
    180
  3. গ
    190
  4. ঘ
    200
  5. ঙ
    400
ব্যাখ্যা
Question: At Company K, 15 percent of the employees are secretaries and 60 percent are salespeople. If there are 45 other employees of Company K, how many employees does Company K have?

Solution:
Let the total number of employees in the company be x
% of secretaries = 15%
% of salespeople = 60%
% of of employees other than secretaries and salespeople = 100 - 75 = 25%
But this number is given as 45
so 25% of x = 45
⇒ (25/100) × x = 45
⇒ (1/4)x = 45
∴x = 180

Therefore there a total of 180 employees in the company K
৭,৬৫১.
A shopkeeper sold a product at loss of 10%. He had sold it for Tk. 108 more, he would have earned a profit of 10%. Find the cost of the product.
  1. ক
    ক) Tk. 540
  2. খ
    খ) Tk. 580
  3. গ
    গ) Tk. 600
  4. ঘ
    ঘ) Tk. 650
ব্যাখ্যা
At loss of 10%, if cost price is Tk. 100, selling price is Tk. (100 - 10) = Tk. 90
At profit of 10%, if cost price is Tk. 100, selling price is Tk. (100 + 10) = Tk. 110
More selling price = Tk. (110 - 90) = Tk. 20
if more selling price is Tk. 20, the cost price is Tk. 100
if more selling price is Tk. 108, the cost price is Tk. 100 × 108/20 = Tk. 540
-------------------------------------------------------------------------------------
Shortcut way:
more selling price = 10 - ( - 10) = 20 
Therefore, 20% of cost price = More selling price
20% of cost price = 108
cost price = Tk. 100 × 108/20 = Tk. 540
৭,৬৫২.
The area of a square inscribed in a circle is 140 cm2. What is the area of the semi-circle?
  1. ক
    ক) 220 cm2
  2. খ
    খ) 200 cm2
  3. গ
    গ) 150 cm2
  4. ঘ
    ঘ) 110 cm2
ব্যাখ্যা
Question: The area of a square inscribed in a circle is 140 cm2. What is the area of the semi-circle?

Solution:
The area of a square inscribed in a circle is 140 cm2
side of square = √140 cm
= 2√35 cm

diagonal of the square = √2 × 2√35
= 2√70 cm

diameter of circle = 2√70 cm
radius of the circle = √70 cm
∴ area of the circle = π (√70)2 cm2
= (22/7) × 70 cm2
= 220 cm2

area of semi-circle = 220/2 
= 110 cm2
৭,৬৫৩.
A wheel that has 6 cogs is meshed with a larger wheel of 12 cogs. If the smaller wheel has made 22 revolutions, then find the number of revolutions made by the larger wheel.
  1. ক
    11
  2. খ
    13
  3. গ
    15
  4. ঘ
    17
  5. ঙ
    None of these
ব্যাখ্যা
Question: A wheel that has 6 cogs is meshed with a larger wheel of 12 cogs. If the smaller wheel has made 22 revolutions, then find the number of revolutions made by the larger wheel.

Solution:
As number of cogs increase, the revolutions made decrease. Hence, this is a problem related to indirect proportion.
Let the number of wheels be x.
More cogs (↑),Less revolutions (↓)

12 : 6 : : 22 : x
⇒ 12 × x = 6 × 22
⇒ x = (6 × 22)/12
∴ x = 11
৭,৬৫৪.
A square is inscribed in a circle of diameter 2a and another square is circumscribing circle. The difference between the areas of outer and inner squares is:
  1. ক
    ক) a2
  2. খ
    খ) 2a2
  3. গ
    গ) 3a2
  4. ঘ
    ঘ) 4a2
ব্যাখ্যা
Question: A square is inscribed in a circle of diameter 2a and another square is circumscribing circle. The difference between the areas of outer and inner squares is:

Solution:
বৃত্তটির ব্যাস হলো অন্তবর্গের কর্ণ 

অন্তবর্গের এক বাহুর দৈর্ঘ্য x একক হলে 
x√2 = 2a
বা, x = 2a/√2
∴ x = √2a
অন্তবর্গের ক্ষেত্রফল = (√2a)2 = 2a2

বহিবর্গের এক বাহুর দৈর্ঘ্য = 2a
বহিবর্গের ক্ষেত্রফল = (2a)2
= 4a2

∴ বহিবর্গ ও অন্তবর্গের ক্ষেত্রফলের পার্থক্য = 4a2 - 2a2 = 2a2
৭,৬৫৫.
The sum of ages of 5 children born at the intervals of 4 years each is 60 years. What is the age of the third oldest among them?
  1. ক
    3 years
  2. খ
    8 years
  3. গ
    10 years
  4. ঘ
    12 years
ব্যাখ্যা
Question: The sum of ages of 5 children born at the intervals of 4 years each is 60 years. What is the age of the third oldest among them?

Solution:
ধরি,
ছোট সন্তানের বয়স = ক বছর

প্রশ্নমতে,
ক + (ক + 4) + (ক + 8) + (ক + 12) + (ক + 16) = 60
⇒ 5ক + 40 = 60
⇒ 5ক = 60 - 40
⇒ 5ক = 20
⇒ ক = 20/5
⇒ ক = 4

অর্থাৎ ছোট সন্তানের বয়স = 4 বছর 
∴ তৃতীয় সন্তানের বয়স = (4 + 8) বছর = 12 বছর
৭,৬৫৬.
Calculate cube root:
  1. ক
    2
  2. খ
    4
  3. গ
    8
  4. ঘ
    None of these
ব্যাখ্যা
Question: Calculate cube root:

Solution:
৭,৬৫৭.
The mean weight of a group of seven boys is 56 kg. The individual weights (in kg) of six of them are 52, 57, 55, 60, 59 and 55. Find the weight of the seventh boy.
  1. ক
    50 kg
  2. খ
    52 kg
  3. গ
    54 kg
  4. ঘ
    55 kg
ব্যাখ্যা
Question: The mean weight of a group of seven boys is 56 kg. The individual weights (in kg) of six of them are 52, 57, 55, 60, 59 and 55. Find the weight of the seventh boy.

Solution:
Mean weight of 7 boys = 56 kg.
Total weight of 7 boys = (56 × 7) kg = 392 kg.

Total weight of 6 boys = (52 + 57 + 55 + 60 + 59 + 55) kg
= 338 kg.

Weight of the 7th boy = (total weight of 7 boys) - (total weight of 6 boys)
= (392 - 338) kg
= 54 kg.

Hence, the weight of the seventh boy is 54 kg.
৭,৬৫৮.
The number of prime factors in the expression 610 × 717 × 1127 is equal to:
  1. ক
    ক) 82
  2. খ
    খ) 52
  3. গ
    গ) 64
  4. ঘ
    ঘ) 72
ব্যাখ্যা
Here
 610 × 717 × 1127
= (2 × 3)10 × 717 × 1127
= 210 × 310 × 717 × 1127
Number of prime factors in the given expression
= (10 + 10 + 17 + 27)
= 64
৭,৬৫৯.
The percentage increase in the area of a rectangle, if each of its sides is increased by 20% is -
  1. ক
    ক) 40%
  2. খ
    খ) 42%
  3. গ
    গ) 44%
  4. ঘ
    ঘ) 48%
ব্যাখ্যা
Question: The percentage increase in the area of a rectangle, if each of its sides is increased by 20% is - 

Solution:
ধরি,
চতুর্ভুজের দৈর্ঘ্য ও প্রস্থ যথাক্রমে ক ও খ
ক্ষেত্রফল = কখ

দৈর্ঘ্য ২০% বাড়লে নতুন দৈর্ঘ্য = ক + ক এর ২০%
= ক + ২০ক/১০০
= ১২/১০ক

প্রস্থ ২০% বাড়লে নতুন প্রস্থ = খ + খ এর ২০%
= খ + ২০খ/১০০
= ১২/১০খ

নতুন ক্ষেত্রফল = (১২ক/১০) × (১২খ/১০)
= ১৪৪কখ /১০০
ক্ষেত্রফল বৃদ্ধি = ১৪৪কখ/১০০ - কখ
= ৪৪কখ/১০০

শতকরা বৃদ্ধির হার = {(৪৪কখ/১০০) / কখ }১০০%
= ৪৪%
৭,৬৬০.
If the radius and height of a right circular cylinder are 14 cm and 21 cm respectively, then the total surface area of the cylinder is: 
  1. ক
    ক) 890π cm2
  2. খ
    খ) 809π cm2
  3. গ
    গ) 908π cm2
  4. ঘ
    ঘ) 980π cm2
ব্যাখ্যা
Here, r = 14 cm and h= 21 cm.
Total surface area of cylinder = 2πr(h + r)
                                                = 2π × 14 × (14 + 21)
                                                = 2 × π × 14 × 35
                                                = 980π cm2
৭,৬৬১.
Two trains of equal length are running on parallel lines in the same direction at 46 km/hr and 36 km/hr. The faster train passes the slower train in 36 seconds. The length of each train is-
  1. ক
    50 m
  2. খ
    72 m
  3. গ
    80 m
  4. ঘ
    82 m
  5. ঙ
    None of these
ব্যাখ্যা
Question: Two trains of equal length are running on parallel lines in the same direction at 46 km/hr and 36 km/hr. The faster train passes the slower train in 36 seconds. The length of each train is-

Solution:
Let the length of each train be x metres.
Then, distance covered = 2x metres.
Relative speed = (46 - 36) km/hr
= 10(5/18) m/sec
= 25/9  m/sec

ATQ,
2x/36 = 25/9
⇒ 2x =100
∴ x = 50 m
৭,৬৬২.
A barrel contains a mixture of wine and water in the ratio 3 : 1. How much fraction of the mixture must be drawn off and substituted by water so that the ratio of wine and water in the resultant mixture becomes 1 : 1 = ?
  1. ক
    3/4
  2. খ
    1/5
  3. গ
    1/2
  4. ঘ
    1/3
ব্যাখ্যা
Question: A barrel contains a mixture of wine and water in the ratio 3 : 1. How much fraction of the mixture must be drawn off and substituted by water so that the ratio of wine and water in the resultant mixture becomes 1 : 1 = ?

Solution:
Original ratio of the mixture 3 : 1

Taking out the mixture actually means just taking out the wine because the water anyways is going to be added back.

If we remove 1 part of wine, it makes the ratio 2 : 2 (1 part water is added to keep the volume constant )
so, what we have actually done is remove 1 part wine from 3 part Wine. i.e. 1/3.

So, 1/3 mixture drawn.
৭,৬৬৩.
A man can row 6 km/hr in still water. If the speed of the current is 2 km/hr, it takes 3 hrs more in upstream than in the downstream for the same distance. The distance is-
  1. ক
    ক) 30km
  2. খ
    খ) 24km
  3. গ
    গ) 20km
  4. ঘ
    ঘ) 32km
ব্যাখ্যা
Question: A man can row 6 km/hr in still water. If the speed of the current is 2 km/hr, it takes 3 hrs more in upstream than in the downstream for the same distance. The distance is-

Solution: 
Let distance = x
Speed of man in still water, = 6 km/h
Speed of current, = 2 km/h

Now
{x/(6 - 2)} - {x/(6 + 2)} = 3
⇒ (x/4) - (x/8) = 3
⇒ (2x - x)/8 = 3
⇒ x/8 = 3
x = 24 

Distance = 24 km
৭,৬৬৪.
The difference between two numbers is 5 and the difference between their squares is 65. What is the larger number?
  1. ক
    ক) 13
  2. খ
    খ) 11
  3. গ
    গ) 8
  4. ঘ
    ঘ) 9
ব্যাখ্যা

ধরি,
বৃহত্তর সংখ্যাটি x এবং ক্ষুদ্রতর সংখ্যাটি y
প্রশ্নমতে,
x - y = 5 ............(i)
এবং x2 - y2 = 65...........(ii)
(ii) নং সমীকরণ হতে পাই,
x2 - y2 = 65
⇒ (x + y)(x - y) = 65
⇒ 5(x + y) = 65
⇒ x + y = 13 ................(iii)
এখন (i) ও (iii) নং সমীকরণ যোগ করে পাই,
(x - y) + (x + y) = 18
⇒ 2x = 18
⇒ x = 9
বৃহত্তর সংখ্যাটি 9.

৭,৬৬৫.
A and B started a business investing in the ratio 3 : 4. After 5 years, A withdrew his capital. B continued alone for 5 more years. If the total profit after 10 years is Tk. 88,000, what is A's share of the profit?
  1. ক
    Tk. 14000
  2. খ
    Tk. 18000
  3. গ
    Tk. 24000
  4. ঘ
    Tk. 30000
ব্যাখ্যা

Question: A and B started a business investing in the ratio 3 : 4. After 5 years, A withdrew his capital. B continued alone for 5 more years. If the total profit after 10 years is Tk. 88,000, what is A's share of the profit?

Solution:
Let A's capital = 3x
B's capital = 4x

A's investment for 5 years = 3x × 5 = 15x
B's investment for 10 years = 4x × 10 = 40x

Ratio of profits = 15x : 40x
= 15 : 40
= 3 : 8

Total contribution units = 3 + 8 = 11

Total profit = Tk. 88,000
∴ A's share = (3/11) × 88,000
= 3 × 8,000
= 24,000

∴ A's share = Tk. 24,000

৭,৬৬৬.
A boat can travel 20 km downstream in 24 min. The ratio of the speed of the boat in still water to the speed of the stream is 4 : 1. How much time will the boat take to cover 15 km upstream?
  1. ক
    25 min
  2. খ
    30 min
  3. গ
    45 min
  4. ঘ
    50 min
ব্যাখ্যা
Question: A boat can travel 20 km downstream in 24 min. The ratio of the speed of the boat in still water to the speed of the stream is 4 : 1. How much time will the boat take to cover 15 km upstream?

Solution:
Downstream speed = (20/24) × 60 km/hr
= 50 km/hr.

Let, speed of water and stream is 4x and x
Downstream speed = 4x + x = 5x and upstream speed = 4x - x = 3x

ATQ, 
5x = 50
∴ x = 10

So, upstream speed = 3 × 10 = 30 km/hr.
Time taken to go 15 km upstream = (15 × 60)/30 = 30 min.
৭,৬৬৭.
Mr. Power attends to, on the average, 12 TV service calls per day. Mr. Fixit attends to, on the average, 16 service calls per day. Mr. Power's average charge is 3/2 as much as that Mr. Fixit, who earns Tk. 7200 per month. The monthly earnings of Mr. Power, in Taka, is-
  1. ক
    3600
  2. খ
    7200
  3. গ
    8100
  4. ঘ
    None
ব্যাখ্যা
Question: Mr. Power attends to, on the average, 12 TV service calls per day. Mr. Fixit attends to, on the average, 16 service calls per day. Mr. Power's average charge is 3/2 as much as that Mr. Fixit, who earns Tk. 7200 per month. The monthly earnings of Mr. Power, in Taka, is-

Solution:
Mr. Power's average charge is 3/2 as much as that Mr. Fixit

Let,
Mr. Power got 3x Taka per call
Mr. Fixit got 2x Taka per call

ATQ,
For Mr. Fixit
16 × 30 × 2x = 7200
⇒ x = 7200/960
∴ x = 7.5

∴ Mr. Power erned = 12 × 30 × 3x = 360 × 3 × 7.5 = 8100 taka
৭,৬৬৮.
A sprinter runs 100 meters in 9.58 seconds. What decimal of a kilometer per second is this speed (approximate)?
  1. ক
    0.01044 km/s
  2. খ
    0.02044 km/s
  3. গ
    0.00144 km/s
  4. ঘ
    None of the above
ব্যাখ্যা
Question: A sprinter runs 100 meters in 9.58 seconds. What decimal of a kilometer per second is this speed?

Solution:
Distance = 100 meters
Time = 9.58 seconds

Converting meters to kilometers
100/1000 = 0.1 kilometers

​Speed = Distance/Time
= 0.1 km​/9.58 s
≈ 0.01044 km/s
​
The sprinter's speed is approximately 0.01044 kilometers per second (decimal fraction).
৭,৬৬৯.
Mr. Anis is a trader. He mixes 26 kg of rice at Tk. 20 per kg with 30 kg of rice of other variety at Tk. 36 per kg and sells the mixture at Tk. 30 per kg. His profit percent is:
  1. ক
    5% profit
  2. খ
    1% profit
  3. গ
    4% profit
  4. ঘ
    no profit
ব্যাখ্যা
Question: Mr. Anis is a trader. He mixes 26 kg of rice at Tk. 20 per kg with 30 kg of rice of other variety at Tk. 36 per kg and sells the mixture at Tk. 30 per kg. His profit percent is:

Solution: 
C.P. of 56 kg rice = (26 × 20 + 30 × 36)
= (520 + 1080)
= 1600 taka

S.P. of 56 kg rice = (56 × 30)
= 1680 taka

∴ Profit = 1680 - 1600
= 80 taka

∴ Profit percentage = (80/1600) × 100%
= 5%
৭,৬৭০.
No. of books bound by a child is half the number of books a man can bind. 20 men and 16 children take 10 days to bind all the books. If there are 8 men and 12 children working on the assignment, how many days will they take to bind all the books?
  1. ক
    ক) 20 days
  2. খ
    খ) 24 days
  3. গ
    গ) 25 days
  4. ঘ
    ঘ) 40 days
ব্যাখ্যা

We know,
M1D1T1W2 = M2D2T2W1 [Men = M; Days = D; Time/Hours = T; Work = W]

1 child binds half the number of books bound by 1 adult
∴ 1 adult = 2 children
Take work done = 1
∴ (20 adults + 16 children) x 10 days x 1 = (8 adults + 12 children) x ? days x 1
∴ 56 children x 10 days x 1 = 28 children x ? days x 1 [Convert either all adults to children or all children to adults.]
∴ ? = 20 days = they will need these many days.

৭,৬৭১.
The sum of present ages of a father and his son is 8 years more than the present age of the mother. The mother is 22 years older than the son. What will be the age of the father after 4 years?
  1. ক
    34 years
  2. খ
    36 years
  3. গ
    40 years
  4. ঘ
    38 years
ব্যাখ্যা
Question: The sum of present ages of a father and his son is 8 years more than the present age of the mother. The mother is 22 years older than the son. What will be the age of the father after 4 years?

Solution:
Let present age of father, mother and son be x, y and z respectively
Sum of present ages of father and son = (Mother's present age + 8 years)
x + z = y + 8 years.................(i)

Mother's present age = Son's present age + 22 years
⇒ y = z + 22 years..............(ii)

Put the value of y in equation (i) we get
x + z = z + 22 + 8
x + z = z + 30
x = 30 years

∴ Father's present age = 30 years
Age of father after four years = 30 + 4 = 34 years
∴ Required age of father = 34 years
৭,৬৭২.
If k is a positive integer, what is the smallest possible value of k such that 1512 × k is the square of an integer?
  1. ক
    15
  2. খ
    8
  3. গ
    35
  4. ঘ
    42
ব্যাখ্যা

Question: If k is a positive integer, what is the smallest possible value of k such that 1512 × k is the square of an integer?

Solution:
আমরা জানি, একটি সংখ্যা পূর্ণবর্গ হতে হলে এর মৌলিক গুণনীয়কের ঘাতসমূহ জোড় সংখ্যা হতে হবে।

1512 = 2 × 2 × 2 × 3 × 3 × 3 × 7
= 23 × 33 × 7

1512k = 23 × 33 × 7 × k
এখন k এর মান 2 × 3 × 7 = 42  হলে, 1512k একটি পূর্ণবর্গ সংখ্যা হবে।

1512 × 42 = (23 × 33 × 71) × (2 × 3 × 7)
=24 × 34 × 72 
যেহেতু এই গুণফলের সব মৌলিক উৎপাদকের ঘাত জোড়, তাই এটি একটি পূর্ণবর্গ সংখ্যা।

সুতরাং, k = 42, হলে 1512 × k পূর্ণবর্গ সংখ্যা হয়।

৭,৬৭৩.
A man invested Tk. 14400 in Tk. 100 shares of a company at 20% premium. If the company declares 5% dividend at the end of the year, then how much does he get? 
  1. ক
    ক) Tk. 600  
  2. খ
    খ) Tk. 450  
  3. গ
    গ) Tk. 630  
  4. ঘ
    ঘ) Tk. 500  
ব্যাখ্যা
Number of shares = 14400/120 = 120 
Face value = Tk. (120 × 100) = Tk. 12000 
 
Annual income = (5/100) × 12000 = Tk . 600  
৭,৬৭৪.
If θ is an acute angle, and cosθ =15/17, then find the value of cot(90 - θ)?
  1. ক
    ক) 3/4
  2. খ
    খ) 7/19
  3. গ
    গ) 8/15
  4. ঘ
    ঘ) 11/17
  5. ঙ
    ঙ) 2/5
ব্যাখ্যা

Given that,
cos θ = 15/17
⇒ secθ = 17/15
⇒ sec2θ = 289/225
⇒ 1 + tan2θ = 289/225
⇒ tan2θ = 289/225 - 1
⇒ tan2θ = 64/225
⇒ tanθ = 8/15
⇒ cot(90 - θ) = 8/15

৭,৬৭৫.
Find the average of all the number between 6 and 34 which are divisible by 5.
  1. ক
    18
  2. খ
    20
  3. গ
    24
  4. ঘ
    30
ব্যাখ্যা
Question: Find the average of all the number between 6  and 34  which are divisible by  5.

Solution:
Numbers between 6 and 34 divisible by 5 are: 10, 15, 20, 25, 30
Sum of these numbers = 10 + 15 + 20 + 25 + 30 = 100
Count of numbers = 5
Average = 100/5 = 20

So the average is 20.
৭,৬৭৬.
The ratio of two numbers is 3 : 4 and their H.C.F is 5. Their L.C.M is -
  1. ক
    20
  2. খ
    45
  3. গ
    60
  4. ঘ
    90
ব্যাখ্যা

Question: The ratio of two numbers is 3 : 4 and their H.C.F is 5. Their L.C.M is -

Solution:
ধরি, সংখ্যা দুটি হলো 3x এবং 4x
∴ গসাগু (H.C.F) = x = 5

∴ সংখ্যা দুটি হলো: 3 × 5 = 15 এবং 4 × 5 = 20
∴ সংখ্যাদ্বয়ের গুণফল = 15 × 20 = 300 এবং H.C.F = 5

আমরা জানি, L.C.M = (Product of two numbers)/H.C.F
= 300/5 = 60

∴ সংখ্যা দুটির লসাগু (L.C.M) = 60

৭,৬৭৭.
A wire can be bent in the form of a circle of radius 7cm. If it is bent in the form of a square, then what will be its area?
  1. ক
    100 cm2
  2. খ
    121 cm2
  3. গ
    130 cm2
  4. ঘ
    144 cm2
ব্যাখ্যা
প্রশ্ন: A wire can be bent in the form of a circle of radius 7cm. If it is bent in the form of a square, then what will be its area?

সমাধান: 
দেওয়া আছে,
বৃত্তের ব্যাসার্ধ r = 7 cm 
বৃত্তের পরিধি = 2πr 
= 2 × (22/7) × 7 
= 2 × 22 × 1
= 44 cm 

বর্গের এক বাহুর দৈর্ঘ্য = 44/4 cm 
= 11 cm 

∴ বর্গের ক্ষেত্রফল = (11)2 cm2 
= 121 cm2
৭,৬৭৮.
What percentage of the whole week does Nazma spend in the office, if her office hours are 9 am to 5 pm from Monday to Friday?
  1. ক
    23.51%
  2. খ
    23.61%
  3. গ
    23.71%
  4. ঘ
    23.81%
  5. ঙ
    None
ব্যাখ্যা
Question: What percentage of the whole week does Nazma spend in the office, if her office hours are 9 am to 5 pm from Monday to Friday?

Solution:
Time spent by Nazma in a day = 5 pm - 9 am = 8 hrs

Time spend by Nazma in a week = 8 × 5 = 40 hrs

Percentage time spent in a week
= [40/(24 × 7)] × 100
= (40/168) × 100
= 23.81%
৭,৬৭৯.
  1. ক
    0.03
  2. খ
    0.08
  3. গ
    0.3
  4. ঘ
    0.8
ব্যাখ্যা
Question:

৭,৬৮০.
If (1/y) = (8/3), then {1/(1+y)}2 = ?
  1. ক
    64/121
  2. খ
    64/9
  3. গ
    6/64
  4. ঘ
    81/121
  5. ঙ
    None of these
ব্যাখ্যা
Question: If (1/y) = (8/3), then {1/(1+y)}2 = ?

Solution:
(1/y) = (8/3)
⇒ y = 3/8
⇒ 1 + y = 1 + 3/8
⇒ 1 + y = 11/8
⇒ 1/(1 + y) = 8/11
⇒ {1/(1+y)}2 = 64/121
৭,৬৮১.
How many 4-digit numbers can be formed from the digits 2, 4, 5, 7, 8, which are divisible by 2 and have no digit repeated?
  1. ক
    92 ways
  2. খ
    72 ways
  3. গ
    36 ways
  4. ঘ
    18 ways
  5. ঙ
    None
ব্যাখ্যা
Question: How many 4-digit numbers can be formed from the digits 2, 4, 5, 7, 8, which are divisible by 2 and have no digit repeated?

Solution:
We know,
A number is divisible by 2 if its last digit is even.
The available digits are: 2, 4, 5, 7, 8
The even digits here are: 2, 4, 8
So, the last digit must be one of these 3 digits.
So, Last digit can be chosen in 3C1 ways 
= 3 ways

As the digit is not repeated
First digit (thousands place) can be chosen in = 4 ways

As the digit is not repeated
Second digit (hundreds place) can be chosen in = 3 ways

As the digit is not repeated
Third digit (tens place) can be chosen in = 2 ways

∴ Total ways = 3 × 4 × 3 × 2 ways
= 72 ways
৭,৬৮২.
If x = b + c - 2a, y = c + a - 2b, z = a + b - 2c, then the value of x2 + y2 - z2 + 2xy is?
  1. ক
    a + b - c
  2. খ
    1
  3. গ
    a - b + c
  4. ঘ
    0
ব্যাখ্যা
Question: If x = b + c - 2a, y = c + a - 2b, z = a + b - 2c, then the value of x2 + y2 - z2 + 2xy is?

Solution:
Given,
x = b + c - 2a,
y = c + a - 2b,
z = a + b - 2c
∴x + y + z = (b + c - 2a) + (c + a - 2b) + (a + b - 2c)
= 0

 Now,
⇒ x2 + y2 + 2xy - z2
= (x + y)2 - z2 
= (x + y - z) (x + y + z)  [(a2 - b2) = (a + b) (a - b)]
= (x + y - z) × 0
= 0
৭,৬৮৩.
The wheel of scooter has diameter 140 cm. How many revolutions per minute must the wheel make so that the speed of the scooter is kept at 132 km per hour?
  1. ক
    1100
  2. খ
    500
  3. গ
    250
  4. ঘ
    1000
ব্যাখ্যা
Question: The wheel of scooter has diameter 140 cm. How many revolutions per minute must the wheel make so that the speed of the scooter is kept at 132 km per hour?

Solution:
Distance travelled by wheel in one revolution = circumference of wheel = (22/7) × 140 = 440 cm.

Speed of scooter = 132 km/hr
= (132 × 1000 × 100)/60 cm/min = 220,000 cm/min.

The wheel has therefore got to travel 220,000 cm in 1 min 
It has to perform 220,000/440 = 500 revolutions in 1 min
৭,৬৮৪.
A bag contains 50p, 25p, and 10p coins in the ratio 2 : 5 : 3, amounting to Tk. 510. Find the number of 50p coins.
  1. ক
    400
  2. খ
    500
  3. গ
    600
  4. ঘ
    1000
  5. ঙ
    480
ব্যাখ্যা
Question: A bag contains 50p, 25p, and 10p coins in the ratio 2 : 5 : 3, amounting to Tk. 510. Find the number of 50p coins.

Solution:
Let the common ratio be 100k.
Number of 50p coins = 200k
Number of 25p coins = 500k
Number of 10p coins = 300k

Value of 50p coins = 0.5 × 200k = 100k
Value of 25p coins = 0.25 × 500k = 125k
Value of 10p coins = 0.1 × 300k = 30k

∴ Total value of all coins = 100k + 125k + 30k = 255k = 510 (given)
∴ k = 2

Therefore, Number of 50p coins = 200k = 200 × 2 = 400
৭,৬৮৫.
Which of the following is not a prime number?
  1. ক
    ক) 253
  2. খ
    খ) 263
  3. গ
    গ) 241
  4. ঘ
    ঘ) 233
ব্যাখ্যা
Question: Which of the following is not a prime number?
Solution: 
যে সকল সংখ্যার গুণনীয়ক কেবল ১ এবং ঐ সংখ্যা তাদের কে মৌলিক সংখ্যা বলে।
এখানে ২৬৩, ২৪১, ২৩৩ এর গুণনীয়ক কেবল ১ এবং ঐ সংখ্যা।
কিন্তু ২৫৩/১১ = ২৩ পাওয়া যায় তাই ২৫৩ এর গুণনীয়ক কেবল ১ এবং ঐ সংখ্যা নয় তাই ২৫৩ সংখ্যাটি মৌলিক নয়।
৭,৬৮৬.
For sets A = {x | x is an integer, 1 ≤ x ≤ 6} and B = {x | x is an even integer, 2 ≤ x ≤ 8}, find the set A - B.
  1. ক
    {1, 2, 3, 4, 5, 6, 8}
  2. খ
    {2, 4, 6}
  3. গ
    {1, 3, 5}
  4. ঘ
    {1, 3, 5, 8}
  5. ঙ
    None of these
ব্যাখ্যা
Question: For sets A = {x | x is an integer, 1 ≤ x ≤ 6} and B = {x | x is an even integer, 2 ≤ x ≤ 8}, find the set A - B.

Solution:
A = {x | x is an integer, 1 ≤ x ≤ 6} 
A = {1, 2, 3, 4, 5, 6},

B = {x | x is an even integer, 2 ≤ x ≤ 8},
B = {2, 4, 6, 8}.

A - B (difference) is the set of elements in A that are not in B.
A - B = {1, 3, 5}.
৭,৬৮৭.
The dimensions of a rectangular floor are 16 feet by 20 feet. When a rectangular rug is placed on the floor, a strip of floor 3 feet wide is exposed on all sides. What are the dimensions of the rug, in feet?
  1. ক
    10 by 14
  2. খ
    10 by 17
  3. গ
    13 by 14
  4. ঘ
    13 by 17
  5. ঙ
    14 by 16
ব্যাখ্যা
Question: The dimensions of a rectangular floor are 16 feet by 20 feet. When a rectangular rug is placed on the floor, a strip of floor 3 feet wide is exposed on all sides. What are the dimensions of the rug, in feet?

Solution:

Given 3 feet wide is exposed on all sides. Hence all 4 sides will have 3 feet gap.

Length of floor = 20
Length of rug = 20 - 3 -3 =14

Width of floor = 16
Width of rug =16 - 3 - 3 =10

Hence dimensions of rug = 10 by 14
৭,৬৮৮.
The volume V of a right circular cylinder is V = πr2h where r is the radius of the base and h is the height of the cylinder. If the volume of a right circular cylinder is 81π and its height is 9, what is the circumference of its base?
  1. ক
    3√2π
  2. খ
    6π
  3. গ
    9π
  4. ঘ
    2√3π
ব্যাখ্যা

Question: The volume V of a right circular cylinder is V = πr2h where r is the radius of the base and h is the height of the cylinder. If the volume of a right circular cylinder is 81π and its height is 9, what is the circumference of its base?

Solution: 
একটি সিলিন্ডারের উচ্চতা h একক ও ব্যাসার্ধ r একক হলে,
উক্ত সিলিন্ডারের আয়তন = πr2h ঘন একক
 
প্রশ্নমতে,
πr2 × h = 81π
⇒ πr2  × 9 = 81π
⇒ r2 = 9
∴r = 3
 
সুতরাং বৃত্তের পরিধি = 2πr = 2π × 3 = 6π

৭,৬৮৯.
Tk. 6100 was partly invested in Scheme A at 10% p.a. compound interest for 2 years and partly in Scheme B at 10% p.a. simple interest for 4 years. Both the schemes pay equal interests. How much was invested in Scheme A?
  1. ক
    Tk. 4500
  2. খ
    Tk. 2000
  3. গ
    Tk. 5000
  4. ঘ
    Tk. 4000
ব্যাখ্যা
Question: Tk. 6100 was partly invested in Scheme A at 10% p.a. compound interest for 2 years and partly in Scheme B at 10% p.a. simple interest for 4 years. Both the schemes pay equal interests. How much was invested in Scheme A?

Solution:
Total sum = Tk. 6100
In Scheme A, Compound interest
Principal = P
R = 10% per annum for 2 years

In Scheme B, Simple interest
Principal = 6100 - P
R = 10% per annum for 4 years

Here,
Compound interest = Simple interest
⇒ P[{1 + (R/100)}n - 1] = (PRN)/100
⇒ P[{1 + 10/100}2 - 1] = [(6100 - P) × 4 × 10]/100
⇒ P× (21/100) = [(6100 - P) × 4 × 10]/100
⇒ 21P = 6100 × 40 - 40P
⇒ 61P = 6100 × 40
∴  P = 4000

∴ The amount invested in Scheme A is Tk. 4000
৭,৬৯০.
If A = 3, B = 6, C = 9, and so on, what is the meaning of the following numbers 36, 27, 45, 42?
  1. ক
    TEAR
  2. খ
    LION
  3. গ
    LEAP
  4. ঘ
    MUTE
ব্যাখ্যা

Question: If A = 3, B = 6, C = 9, and so on, what is the meaning of the following numbers 36, 27, 45, 42?

Solution:
Given,
A = 3, B = 6, C = 9, ......

∴ প্রতিটি কোড = অক্ষরের অবস্থান × 3

So,
36 ÷ 3 = 12 → L
27 ÷ 3 = 9 → I
45 ÷ 3 = 15 → O
42 ÷ 3 = 14 → N

∴ The meaning of the following number = LION

৭,৬৯১.
The average age of parents is 38 years. The average age of parents and a child is 30 years. What is the age is the child?
  1. ক
    ক) 10 years
  2. খ
    খ) 12 years
  3. গ
    গ) 14 years
  4. ঘ
    ঘ) 15 years
ব্যাখ্যা
Question: The average age of parents is 38 years. The average age of parents and a child is 30 years. What is the age is the child?

Solution:
Total age of parents = 38 × 2 = 76 years
Total age of parents and the child = 30 × 3 = 90 years

So, Child's age = 90 - 76 = 14 years
৭,৬৯২.
A train is driven at the speed of 100 kmph and stops for 10 minutes at the end of every 150 km. To cover a distance of 1000 km, it will be-
  1. ক
    ক) 9 hr
  2. খ
    খ) 10 hr
  3. গ
    গ) 11 hr
  4. ঘ
    ঘ) 12 hr
ব্যাখ্যা
Question: A train is driven at the speed of 100 kmph and stops for 10 minutes at the end of every 150 km. To cover a distance of 1000 km, it will be-

Solution:
Here, time taken to 150 km at 100 kmph  = 1 hour 30 min + 10 min
= 1 hr 40 min
= 1 hr + 2/3 hr
= 5/3 hr

time taken to cover (150 × 6) = 900 km in (5/3 × 900) = 10 hrs
Remaining 100 km is covered in 1 hr

So, total time = 10 + 1 = 11 hr
৭,৬৯৩.
Four metal rods of lengths 78 cm, 104 cm, 117 cm, and 169 cm are to be cut into parts of equal length. Each part must be as long as possible. What is the maximum number of pieces that can be cut?
  1. ক
    27
  2. খ
    36
  3. গ
    40
  4. ঘ
    48
ব্যাখ্যা
Question: Four metal rods of lengths 78 cm, 104 cm, 117 cm, and 169 cm are to be cut into parts of equal length. Each part must be as long as possible. What is the maximum number of pieces that can be cut?

Solution:
Maximum length of each part
= H.C.F. of 78 cm, 104 cm, 117 cm, and 169 cm
= 13 cm.

That means, if we take the H.C.F the length we get can be used to cut all 4 rods equally, so that each piece will have the same length.

∴ Number of pieces = (78 + 104 + 117 + 169)/13
= 468/13 
= 36
৭,৬৯৪.
Two ships are sailing in the sea on the two sides of the lighthouse. The angles of elevation of the top of the lighthouse observed from the ships are 30° and 60°, respectively. If the lighthouse is 60 m high, what is the distance between the two ships?
  1. ক
    60√3 m
  2. খ
    120 m
  3. গ
    50√3 m
  4. ঘ
    80√3 m
ব্যাখ্যা

Question: Two ships are sailing in the sea on the two sides of the lighthouse. The angles of elevation of the top of the lighthouse observed from the ships are 30° and 60°, respectively. If the lighthouse is 60 m high, what is the distance between the two ships?

Solution:

দেওয়া আছে, বাতিঘরের উচ্চতা AD = 60 m।
জাহাজ দুটির অবস্থান B এবং C; মোট দূরত্ব BC = BD + DC
উন্নতি কোণ: ∠ABD = 30° এবং ∠ACD = 60°

প্রথমে, ΔADC থেকে DC নির্ণয় করি:
tan 60° = AD/DC
 ⇒ √3 = 60/DC
⇒ DC = 60/√3
⇒ DC = (60√3)/3 
⇒ DC = 20√3 m

এরপর, ΔADB থেকে BD নির্ণয় করি:
tan 30° = AD/BD
⇒ 1/√3 = 60/BD
⇒ BD = 60√3 m

তাহলে, BC = BD + DC = 60√3 + 20√3 = 80√3 m

অতএব, জাহাজ দুটির মাঝের দূরত্ব = 80√3 m

৭,৬৯৫.
A man's regular pay is Taka 30 per hour up to 40 hours. Overtime is twice the payment for regular time. If he was paid Taka 1680, how many hours of overtime did he work?
  1. ক
    ক) 8
  2. খ
    খ) 16
  3. গ
    গ) 28
  4. ঘ
    ঘ) 48
ব্যাখ্যা

40 ঘন্টার জন্য regular pay = (30 × 40) = 1200 টাকা।
Overtime এর টাকার পরিমান = (1680 - 1200) টাকা = 480 টাকা
যেহেতু, Overtime এর প্রতিদিনের টাকার পরিমান Regular Payment এর দ্বিগুন,
সেহেতু মোট overtime কাজ করার সময় = (480 ÷ (30×2) ঘন্টা = 8 ঘন্টা

৭,৬৯৬.
Which number will complete the series
3, 10, 31, 94, ____, 850
  1. ক
    197
  2. খ
    283
  3. গ
    353
  4. ঘ
    451
ব্যাখ্যা
Question: Which number will complete the series
3, 10, 31, 94, ____, 850

Solution:
Here,
3
(3 × 3) + 1 = 10
(10 × 3) + 1 = 31
(31 × 3) + 1 = 94
(94 × 3) + 1 = 283
(283 × 3) + 1 = 850
৭,৬৯৭.
If 6 students have an average (arithmetic mean) score of 88 on an exam, and one of those students scored a 93 on the exam, what is the average score on this exam for the other 5 students ?
  1. ক
    ক) 84
  2. খ
    খ) 85
  3. গ
    গ) 86
  4. ঘ
    ঘ) 87
ব্যাখ্যা
Question: If 6 students have an average (arithmetic mean) score of 88 on an exam, and one of those students scored a 93 on the exam, what is the average score on this exam for the other 5 students ?

Solution: 
6 জন শিক্ষার্থীর গড় নম্বর = 88
6 জন শিক্ষার্থীর মোট নম্বর = 88 × 6 = 528

5 জন শিক্ষার্থীর মোট নম্বর = 528 - 93 = 435

5 জন শিক্ষার্থীর গড় নম্বর = 435/5= 87
৭,৬৯৮.
A man earns the same amount every day, but on Sunday he earns three times as much as on the other days. What fraction of his total weekly earnings does he earn on Sunday?
  1. ক
    1/2
  2. খ
    1/6
  3. গ
    1/4
  4. ঘ
    1/3
ব্যাখ্যা

Question: A man earns the same amount every day, but on Sunday he earns three times as much as on the other days. What fraction of his total weekly earnings does he earn on Sunday?

Solution:
Let us assume that on each day except Sunday, he earns x taka.

Then,
Earnings on the other 6 days = 6x taka
Earnings on Sunday = 3x taka (since it is three times the usual daily amount)

∴ Total earnings in a week = 6x + 3x = 9x taka

Therefore, the fraction of the total weekly earnings that he earns on Sunday is,
= Sunday’s earnings/Total weekly earnings
= 3x/9x
= 1/3

So he earns 1/3 (one-third) of his total weekly earnings on Sunday.

৭,৬৯৯.
The numerical value of 1 + (1/cot263°) - sec227° + (1/sin263°) - cosec227° is?
  1. ক
    0
  2. খ
    - 1
  3. গ
    1
  4. ঘ
    None of these
ব্যাখ্যা
Question: The numerical value of 1 + (1/cot263°) - sec227° + (1/sin263°) - cosec227° is?

Solution:
৭,৭০০.
If 4x - 3y = 24.5 and 5x + 2y = -1, then x = ?
  1. ক
    ক) -3
  2. খ
    খ) -2
  3. গ
    গ) 2
  4. ঘ
    ঘ) 3
ব্যাখ্যা

Given, 4x - 3y = 24.5 ..... (i) and 5x + 2y = -1 ...... (ii)
Lets, (ii)×3 + (i)×2
⇒ 15x + 6y + 8x - 6y = - 3 + 49
⇒ 23x = 46
⇒ x = 2