Bank Math

PrepBank · পাতা ১০২ / ১৬১ · ১০,১০১–১০,২০০ / ১৬,১২৪

১০,১০১.

A student got twice as many sums wrong as he got right. If he attempted 42 sums in all, how many did he solve correctly?

ক
15
খ
20
গ
28
ঘ
40
ঙ
14

ব্যাখ্যা

Question: A student got twice as many sums wrong as he got right. If he attempted 42 sums in all, how many did he solve correctly?
Solution:
Let he has solved correctly X no. of sums.
Therefore incorrect no. of sums = 2X
Now,
X + 2X = 42
⇒ 3X = 42
⇒ X = 14

∴ 14 sums he has done correctly.

১০,১০২.

What is the value of (√3 + 1)(3 - cot 30°) = ?

ক
3√2
খ
0
গ
2√3
ঘ
√3

ব্যাখ্যা

Question: What is the value of (√3 + 1)(3 - cot 30°) = ?

Solution:
Given that,
= (√3 + 1)(3 - cot 30°)
= (√3 + 1)(3 - √3)
= (√3 + 1)(√3.√3 - √3)
= √3(√3 + 1)(√3 - 1)
= √3{(√3)² - 1²} [(a + b)(a - b) = a² - b²]
= √3(3 - 1)
= 2√3

১০,১০৩.

A sells an article to B at a profit of 10% B sells the article back to A at a loss of 10%. In this transaction:

ক
A makes a profit of 20%
খ
B loses 20%
গ
A makes a profit of 11%
ঘ
A neither losses nor gains

ব্যাখ্যা

It could be easily shown by net percentage change graphic.
100(A) = 10%(Profit) ⇒110(B) = 10%(Loss) ⇒ 99(A)
In this transaction
A makes a profit of (110 - 99 = 11%) 11%
[10% on selling to B and 1% profit on buying back from B]

১০,১০৪.

If x, y, and z are said to be the real numbers, then what is the value of (x - y)³ + (y - z)³ + (z - x)³?

ক
0
খ
(x - y)
গ
(x - y)(y - z)(z - x)
ঘ
3(x - y)(y - z)(z - x)

ব্যাখ্যা

Question: If x, y, and z are said to be the real numbers, then what is the value of (x - y)³ + (y - z)³ + (z - x)³?

Solution:
Suppose a = (x - y), b = (y - z), and c = (z - x)
On adding a, b, and c we will get
a + b + c = x - y + y - z + z - x
∴ a + b + c =0

So, a³ + b³ + c³ = 3abc [because if a + b + c = 0, then a³ + b³ + c³ = 3abc]

We can say that (x - y)³ + (y - z)³ + (z - x)³ = 3(x - y)(y - z)(z - x)

১০,১০৫.

0.01 × (0.01)²× (10)⁶ = ?

ক
ক) 10
খ
খ) 1
গ
গ) 0.1
ঘ
ঘ) 0.01

ব্যাখ্যা

0.01 × (0.01)²× (10)⁶
= 0.01 × 0.0001 × 1000000
= (1/100) × (1/10000) × 1000000
= 1

১০,১০৬.

As the price of mango has reduced 20%, it is now possible to buy 2 more mangoes at Tk.12. What is the current price of 100 mangoes?

ক
ক) Tk.120
খ
খ) Tk.50
গ
গ) Tk.230
ঘ
ঘ) Tk.140

ব্যাখ্যা

Question: As the price of mango has reduced 20%, it is now possible to buy 2 more mangoes at Tk.12. What is the current price of 100 mangoes?

Solution:
ধরি,
1টি আমের পূর্বমূল্য = x টাকা
20% কমে
1টি আমের বর্তমানমূল্য = x - x এর 20%
= x - 20x /100
= 80x/100
= 0.8x

প্রশ্নমতে,
(12/0.8x) - (12/x) = 2
(15/x) - (12/x) = 2
(15 - 12)/x = 2
3/x = 2
2x = 3
x = 3/2
x = 1.5

1টি আমের বর্তমানমূল্য = 1.5 × 0.8 = 1.2 টাকা
100 টি আমের বর্তমানমূল্য = 1.2 × 100 টাকা
= 120 টাকা

১০,১০৭.

ক
6.91
খ
7
গ
5
ঘ
2
ঙ
3.84

১০,১০৮.

If the area of a right triangle is 6 square meters and the hypotenuse is 5 meters, what is the perimeter of the triangle?

ক
10
খ
12
গ
15
ঘ
18

ব্যাখ্যা

Question: If the area of a right triangle is 6 square meters and the hypotenuse is 5 meters, what is the perimeter of the triangle?

Solution:
যেহেতু ত্রিভুজটির অতিভুজ = 5 মিটার
∴ ত্রিভুজটি সমকোণী ত্রিভুজ।

ধরি, ত্রিভুজের দুই বাহু যথাক্রমে x ও y
∴ x² + y² = 5² = 25

আবার, ক্ষেত্রফল, (1/2)xy = 6
⇒ xy = 12

আমরা জানি, (x + y)² = x² + y² + 2xy
= 25 + (2 × 12)
= 25 + 24
= 49
x + y = 7

পরিসীমা = (x + y + 5)
= 7 + 5
= 12 মিটার

১০,১০৯.

How many degrees does a minute hand move in 20 minutes?

ক
ক) 180°
খ
খ) 20°
গ
গ) 80°
ঘ
ঘ) 120°

ব্যাখ্যা

Hand of minute moves 360° in 60 minutes
In 20 minutes it will move = (360° × 20) / 60 = 120°

১০,১১০.

Suppose x > y and xy < 0, which of the following must be negative?

ক
y
খ
x
গ
x - y
ঘ
None

ব্যাখ্যা

Question: Suppose x > y and xy < 0, which of the following must be negative?

Solution:
দেওয়া আছে,
x > y
xy < 0

x > y হলে, xy < 0 হবে যখন y < 0 হবে।

১০,১১১.

A rectangular field is to be fenced on three sides leaving a side of 20 feet uncovered. If the area of the field is 680 sq. feet, how many feet of fencing will be required?

ক
ক) 34
খ
খ) 40
গ
গ) 68
ঘ
ঘ) 88

ব্যাখ্যা

We have: l = 20 ft and lb = 680 sq. ft.
So, b = 34 ft.
Length of fencing = (l + 2b) = (20 + 68) ft = 88 ft.

১০,১১২.

Jishan took a loan of Tk. 1500 with simple interest for as many years as the rate of interest. If he paid Tk. 540 as interest at the end of loan period, what was the rate of interest?

ক
2%
খ
3%
গ
4%
ঘ
6%

ব্যাখ্যা

Simple interest is the same as the rate of interest.
Hence,
Rate of interest = R% and Time = R years

S.I. = (P × R × R)/100
⇒ 60 = (1500 × R²)/100
⇒ 15R²= 540
⇒ R² = 36
⇒ R=6 %

Rate of Interest = 6%.

১০,১১৩.

Two-fifth of one-fourth of three seventh of a number is 30. What is the half of the number?

ক
ক) 700
খ
খ) 600
গ
গ) 350
ঘ
ঘ) 300

ব্যাখ্যা

Question: Two-fifth of one-fourth of three seventh of a number is 30. What is the half of the number?

Solution:
let the number be x

(2/5) × (1/4) × (3/7)x = 30
⇒ x = (30 × 5 × 4 × 7)/(2 × 3)
⇒ x = 700
∴ x/2 = 350

১০,১১৪.

A merchant has 1000kg of sugar, part of which he sells at 8% profit and the rest at 18% profit. He gains 14% overall. The quantity sold at 18% profit is-

ক
400 kg
খ
600 kg
গ
500 kg
ঘ
630 kg

ব্যাখ্যা

Question: A merchant has 1000kg of sugar, part of which he sells at 8% profit and the rest at 18% profit. He gains 14% overall. The quantity sold at 18% profit is-

Solution:
Let the sugar sell at 18% profit is = x
So, the sugar sell at of 8% profit = 1000-x

According to question,
18% of x + 8% of (1000 -x) = 14% of 1000
→ 18x/100 + (8000 - 8x)/100 = 14000/100
→ 18x + 8000 - 8x = 14000 (multiplying both sides by 100)
→ 10x = 6000
→ x = 600 kg

১০,১১৫.

If a regular polygon has each of its interior angle equal to 175°, then find the number of sides of the polygon.

ক
72 sides
খ
80 sides
গ
180 sides
ঘ
90 sides

ব্যাখ্যা

Question: If a regular polygon has each of its interior angle equal to 175°, then find the number of sides of the polygon.

Solution:
Given that,
Each interior angle = 175°

We know,
Sum of interior angles = (n - 2) × 180
Each interior angle = Sum of interior angles ÷ n

Now,
175 = [(n - 2) × 180] ÷ n
⇒ 175 × n = (n - 2) × 180
⇒ 175n = 180n - 360
⇒ 180n - 175n = 360
⇒ 5n = 360
⇒ n = 360/5
∴ n = 72

So, the polygon has 72 sides.

১০,১১৬.

In the figure above, the two square regions have areas 16 and 25, respectively. What is the area of the shaded triangular region?

ক
6
খ
8
গ
9
ঘ
12
ঙ
15

ব্যাখ্যা

Question:

In the figure above, the two square regions have areas 16 and 25, respectively. What is the area of the shaded triangular region?

Solution:
Area of the first square = 16.
∴ Side of the first square = 4.

Area of the tilted square = 25.
∴ Side of the tilted square = 5.

The side of the tilted square becomes the hypotenuse of the triangle = 5.
The side of the first square becomes the height of the triangle. = 4.
Then by Pythagoras theorem,
base² + height² = hypotenuse²
⇒ base² + 4² = 5²
⇒ base² + 16 = 25
⇒ base² = 9
∴ base = 3

Area of the triangle = (1/2) × 3 × 4 = 6.

১০,১১৭.

Which of the following numbers will completely divide 7¹⁰ + 7¹¹ + 7¹² + 7¹³?

ক
15
খ
13
গ
11
ঘ
14

ব্যাখ্যা

Question: Which of the following numbers will completely divide 7¹⁰ + 7¹¹ + 7¹² + 7¹³?

Solution:
Factors of a number refers to those values that can exactly divide the original number without leaving a remainder.

7¹⁰ + 7¹¹ + 7¹² + 7¹³
= (1 + 7 + 7² + 7³) 7¹⁰
= (1 + 7 + 49 + 343) 7¹⁰
= 400 × 7¹⁰
= 2⁴ × 5² × 7¹⁰
So, the factors are 2, 4, 5, 7, 8, 10 etc.
So, out of given options, required factor = 2 × 7 = 14
∴ 14 will completely divide 7¹⁰ + 7¹¹ + 7¹² + 7¹³.

১০,১১৮.

A letter is taken out at random from the word "ENGINEERING", and another is taken out from the word "GREENHOUSE". What is the probability that both selected letters are the same

ক
3/22
খ
3/10
গ
4/11
ঘ
7/10
ঙ
None

ব্যাখ্যা

Question: A letter is taken out at random from the word "ENGINEERING", and another is taken out from the word "GREENHOUSE". What is the probability that both selected letters are the same

Solution:
For E: (3/11) × (3/10) = 9/110
For N: (3/11) × (1/10) = 3/110
For G: (2/11) × (1/10) = 2/110
For R: (1/11) × (1/10) = 1/110

Total probability = (9/110) + (3/110) + (2/110) + (1/110)
= 15/110
= 3/22

১০,১১৯.

A total of 300 coins of 25 paise and 50 paise make the sum of Tk 120. The number of 50 paise coins is-

ক
ক) 120
খ
খ) 150
গ
গ) 180
ঘ
ঘ) 200

ব্যাখ্যা

Question: A total of 300 coins of 25 paise and 50 paise make the sum of Tk 120. The number of 50 paise coins is-

Solution:
Let the number of 50 paise coins be = x
So, the number of 25 paise coins is = 300 - x

ATQ,
50x + {25 × (300 - x)} = 120 × 100
⇒ 50x + 7500 - 25x = 12000
⇒ 25x = 4500
⇒ x = 180

১০,১২০.

The square of a positive number is 21 more than 4 times the number. Find the number.

ক
ক) 14
খ
খ) 7
গ
গ) 11
ঘ
ঘ) 9

ব্যাখ্যা

Question: The square of a positive number is 21 more than 4 times the number. Find the number.

Solution:
প্রথমে প্রশ্নের স্টেটমেন্টকে সমীকরণে রূপান্তর করে পাই:
x² = 21+ 4x
বা, x²- 4x -21= 0
বা, (x-7)(x+3)= 0
Set each factor equal to zero:
x- 7 = 0 and x + 3 = 0
Solve each equation:
x = 7 and x = - 3

যেহেতু প্রশ্নে ধনাত্মক সংখ্যার কথা বলা হয়েছে, তাই সঠক উত্তর হবে 7.

১০,১২১.

If x ≥ 10 and y ≤ 7 which of the following must be true?

ক
ক) x - y ≤ 10
খ
খ) x + y ≤ 7
গ
গ) x - y ≥ 3
ঘ
ঘ) x + y ≥ 3

ব্যাখ্যা

Question: If x ≥ 10 and y ≤ 7 which of the following must be true?

Solution:
দেওয়া আছে,
x ≥ 10
এবং
y ≤ 7
বা, - y ≥ - 7 [- 1 দ্বারা গুণ করে পাই]

এখন,
x - y ≥ 10 - 7 [অসমতাদ্বয় যোগ করে পাই]
∴ x - y ≥ 3

১০,১২২.

The average age of 80 boys in a class is 15. The average age of a group of 20 boys in the class is 16 and the average age of another 25 boys in the class is 14. What is the average age of the remaining boys in the class?

ক
ক) 15.14 yrs.
খ
খ) 16.25 yrs.
গ
গ) 17.15 yrs.
ঘ
ঘ) 18.10 yrs.

ব্যাখ্যা

Total ages of 80 boys = 15 × 80 = 1200 yrs.
Total age of20 boys = 16 × 20 = 320 yrs
Total age of 25 boys = 14 × 25 = 350 yrs.

Average age of remaining boys = {1200 - (320 + 350)}/ {80 - (25 + 20)}
= 530/35
= 15.14 yrs.

১০,১২৩.

After decreasing 24% in the price of an article the discounted price becomes $912. Find the actual price of the article?

ক
ক) 1400
খ
খ) 1300
গ
গ) 1200
ঘ
ঘ) 1100

ব্যাখ্যা

Let the products actual price be x
ATQ, 76% of x = 912
Or, 76x/100 = 912
Or, x = (912 × 100) / 76
= 1200

১০,১২৪.

A is two years older than B, who is twice as old as C. If the total of the ages of A, B, and C is 32, how old is A?

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

ব্যাখ্যা

Question: A is two years older than B, who is twice as old as C. If the total of the ages of A, B, and C is 32, how old is A?

Solution:
Let,
C's age be = a years
Then, B's age = 2a years
A's age = (2a + 2) years

∴ (2a + 2) + 2a + a = 32
⇒ 5a = 30
⇒ a = 6
Hence, A's age = (2 × 6) + 2 = 14 years.

১০,১২৫.

In covering a certain distance, The speeds of A and B are in the ratio of 5 : 6. A takes 30 minutes more than B to reach the destination. The time taken by A to reach the destination is-

ক
ক) 2 hours
খ
খ) 3 hours
গ
গ) 4 hours
ঘ
ঘ) 5 hours

ব্যাখ্যা

Ratio of speeds =5 : 6
⇒ Ratio of times taken =6 : 5
Let A and B take 6x and 5x hrs. to reach the destination.
Then
6x - 5x=30/60
x =1/2

∴ Time taken by A =(6 × 1)/2 hours
= 3 hours

১০,১২৬.

Kamal started a business investing Tk. 9000. After five months, Sameer joined with a capital of Tk. 8000. If at the end of the year, they earn a profit of Tk. 6970, then what will be the share of Sameer in the profit ?

ক
Tk. 2380
খ
Tk. 2210
গ
Tk. 2720
ঘ
Tk. 2320

ব্যাখ্যা

Kamal invested for 12 months and Sameer invested for 7 months.
So, Kamal : Sameer = (9000 × 12) : (8000 × 7)
= 108 : 56
= 27 : 14
Sum of ratio = 27 + 14 = 41
Sameer Ratio in profit will be Tk. 6970 × 14/41 or Tk. 2380

১০,১২৭.

A sum of money is sufficient to pay A's wages for 21 days or B's wages for 28 days. The same money is sufficient to pay the wages of both for?

ক
24 days
খ
18 days
গ
12 days
ঘ
8 days

ব্যাখ্যা

Question: A sum of money is sufficient to pay A's wages for 21 days or B's wages for 28 days. The same money is sufficient to pay the wages of both for?

Solution:
Let
total money be Tk. x
A's 1 day's wages = Tk. x/21
B's 1 day's wages = Tk. x/28

∴ (A + B)'s 1 day's wages = Tk. (x/21 + x/28)
= Tk. (4x + 3x)/84
= Tk. 7x/84
= Tk. x/12

∴ Money is sufficient to pay the wages of both for 12 days.

১০,১২৮.

Two pipes X and Y can fill a cistern in 10 and 15 hours respectively. Both pipes are opened together. After how many hours should pipe X be turned off so that the cistern is filled in 9 hours?

ক
3 hours
খ
4 hours
গ
6 hours
ঘ
7.5 hours

ব্যাখ্যা

Question: Two pipes X and Y can fill a cistern in 10 and 15 hours respectively. Both pipes are opened together. After how many hours should pipe X be turned off so that the cistern is filled in 9 hours?

সমাধান:
ধরি, মোট সময় 9 ঘন্টা পর চৌবাচ্চাটি পূর্ণ হয়। এই সম্পূর্ণ সময়ে কেবল নল Y খোলা ছিল।

নল Y, 15 ঘন্টায় চৌবাচ্চাটি পূর্ণ করতে পারে।
∴ 1 ঘন্টায় Y পূর্ণ করে 1/15 অংশ।
∴ 9 ঘন্টায় Y পূর্ণ করে = 9/15 অংশ
= 3/5 অংশ।

অবশিষ্ট অংশ যা X পূর্ণ করেছিল = 1 - 3/5 অংশ
= 2/5 অংশ।

নল X, 10 ঘন্টায় পূর্ণ করে 1 অংশ।
∴ 1 অংশ পূর্ণ করে 10 ঘন্টায়।
∴ 2/5 অংশ পূর্ণ করে = (10 × 2/5) ঘন্টা
= 4 ঘন্টা।

∴ নল X কে 4 ঘন্টা পর বন্ধ করতে হবে।

১০,১২৯.

A student loses 1 mark for every wrong answer and scores 2 marks for every correct answer. If he answers all the 60 questions in an exam and scores 39 marks, how many of them were correct?

ক
27
খ
31
গ
33
ঘ
37

ব্যাখ্যা

Question: A student loses 1 mark for every wrong answer and scores 2 marks for every correct answer. If he answers all the 60 questions in an exam and scores 39 marks, how many of them were correct?

Solution:
ধরি,
মোট ভুল উত্তর = ক টি
প্রতিটি সঠিক উত্তরের জন্য প্রকৃতপক্ষে কাঁটা যায় = (২ + ১) = ৩ নম্বর।

প্রশ্নমতে,
(২ × ৬০) - ৩ক = ৩৯
৩ক = ১২০ - ৩৯
৩ক = ৮১
ক = ২৭

∴ সঠিক উত্তর = (৬০ - ২৭) = ৩৩ টি

১০,১৩০.

Jashim buys 10 CDs for BDT. 200. If DVDs cost BDT 20 more, how many DVDs can he buy for the same amount?

ক
ক) 4
খ
খ) 5
গ
গ) 6
ঘ
ঘ) 10

ব্যাখ্যা

Cost of a CD = 200/10 = 20
And, cost of a DVD = 20 + 20 = 40
∴ Number of DVD = 200/40 = 5

১০,১৩১.

x - y = 3, 2x = 2y + 6
The system of equations above has how many solutions?

ক
Exactly one
খ
Exactly two
গ
Infinitely many
ঘ
None

ব্যাখ্যা

Question: x - y = 3, 2x = 2y + 6
The system of equations above has how many solutions?

Solution:
Say, the given equations are:
ax + by = c
dx + ey = f

If a/d ≠ b/e, then the system of equations has a unique solution.

If a/d = b/e ≠ c/f, then the system of equations has no solution.

If a/d = b/e = c/f, then the system of equations has infinitely many solutions.

Here,
x - y = 3
2x = 2y + 6
⇒ 2x - 2y = 6

1/2 = (- 1)/(- 2) = 3/6
so there are infinitely many solutions.

১০,১৩২.

The average of the first five multiples of 5 is:

ক
20
খ
15
গ
12
ঘ
10

ব্যাখ্যা

Question: The average of the first five multiples of 5 is:

Solution:
We know,
The first five multiples of 5 = 5, 10, 15, 20, 25.

∴ Average = (5 + 10 + 15 + 20 + 25)/5
= 75/5
= 15

১০,১৩৩.

Quantity A = 0.012/3 and Quantity B = 4/100.

ক
Quantity A greater
খ
Quantity A equals Quantity B
গ
Quantity B is greater
ঘ
Relationship indeterminate

ব্যাখ্যা

Question: Quantity A = 0.012/3 and Quantity B = 4/100

Solution:
Quantity A = 0.012/3
= 12/(3 × 1000)
= 4/1000
= 0.004

Quantity B = 4/100
= 0.04

∴ Quantity B > Quantity A

১০,১৩৪.

A committee of 3 members is to be selected out of 3 men and 2 women. What is the probability that the committee has at least one woman?

ক
ক) 1/10
খ
খ) 9/20
গ
গ) 9/10
ঘ
ঘ) 1/20

ব্যাখ্যা

Question: A committee of 3 members is to be selected out of 3 men and 2 women. What is the probability that the committee has at least one woman?

Solution:
Total member = 3 + 2 = 5

Committee can be form with at least 1 one woman:
1 woman, 2 men : ²C₁× ³C₂ = 2 × 3 = 6
2 women, 1 man: ²C₂ × ³C₁ = 1 × 3 = 3
∴ The total number of ways to make committe with at least 1 one woman: 6 + 3 = 9

The total number of ways to make committe with all members = ⁵C₃ = 10

∴ The probability that the committee has at least woman = 9/10

১০,১৩৫.

A vendor bought toffees at 6 for a Taka. How many for a Taka must he sell to gain 20%?

ক
ক) 3
খ
খ) 4
গ
গ) 5
ঘ
ঘ) 6

ব্যাখ্যা

প্রশ্ন: A vendor bought toffees at 6 for a Taka. How many for a Taka must he sell to gain 20%?

সমাধান:
৬টি টফির ক্রয় মূল্য ১ টাকা

২০% লাভে,
ক্রয়মূল্য ১০০ টাকা হলে বিক্রয়মূল্য ১২০ টাকা
∴ ক্রয়মূল্য ১ টাকা হলে বিক্রয়মূল্য ১২০/১০০ টাকা
= ১.২ টাকা

১.২ টাকায় বিক্রয় হয় ৬টি
∴ ১ টাকায় বিক্রয় হয় ৬/১.২টি
= ৫টি

১০,১৩৬.

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 4 or 5?

ক
ক) 9/20
খ
খ) 2/5
গ
গ) 1/2
ঘ
ঘ) 3/10

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 4 or 5?

Solution:
Here, S = {1, 2, 3, 4, ...., 19, 20}. {4, 8, 12, 16, 20}{5,10,15,20}
Let E = event of getting a multiple of 3 or 5 = {4, 5, 8 , 10, 12, 15, 16, 20}

P(E) = n(E)/n(S) = 8/20 = 2/5

১০,১৩৭.

Which is the correct factor analysis of x² - 2xy - z² + 2yz:

ক
(x - z)(x - 2y + z)
খ
(x - y)(x - y - 2z)
গ
(x + z)(x - y + z)
ঘ
(x - y)(2x - y + z)

ব্যাখ্যা

Question: Which is the correct factor analysis of x² - 2xy - z² + 2yz:

Solution:
x² - 2xy - z² + 2yz
= x² - z² - 2xy + 2yz
= (x + z)(x - z) - 2y(x - z)
= (x - z)(x - 2y + z)

১০,১৩৮.

Students of a class stand in a queue. If Parimal is 21th in order from both ends, how many students are there in the queue?

ক
42
খ
41
গ
40
ঘ
43

ব্যাখ্যা

Question: Students of a class stand in a queue. If Parimal is 21th in order from both ends, how many students are there in the queue?

Solution:
এক দিক থেকে পরিমলের অবস্থান ২১ তম অর্থাৎ সে সহ ২১ জন। অপর দিক থেকে তার অবস্থান ২১ তম অর্থাৎ সে বাদে আরও ২০ জন আছে।
∴ ঐ সারিতে মোট ছাত্র আছে (২১ + ২০) জন = ৪১ জন

১০,১৩৯.

a is greater than b by 2 and b is greater than c by 10. If a + b + c = 130, then (b + c) - a =?

ক
ক) 42
খ
খ) 38
গ
গ) 34
ঘ
ঘ) 44

ব্যাখ্যা

Question: a is greater than b by 2 and b is greater than c by 10. If a + b + c = 130, then (b + c) - a =?

Solution:
According to the question,
b = c + 10
a = b + 2
Or, a = c + 10 + 2
Or, a = c + 12
Now, a + b + c = 130
Or, c + 12 + c + 10 + c = 130
Or, 3c + 22 = 130
Or, 3c = 108
∴ c = 36

Now, (b + c) - a = (c + 10 + c) - (c + 12)
= 2c + 10 - c - 12
= c - 2
= 36 - 2
= 34

১০,১৪০.

A man completes (1/3) of a job in 13 days. At this rate, how many more days will it take him to finish the job?

ক
20 days
খ
16 days
গ
36 days
ঘ
26 days

ব্যাখ্যা

Question: A man completes (1/3) of a job in 13 days. At this rate, how many more days will it take him to finish the job?

Solution:

Work done = 1/3

Balance work =(1−1/3) = 2/3

Less work, Less days ( Direct proportion)
Let the required number of days be x
Then,
⇔ (1/3) : (2/3) : : 13 : X
⇔ (1/3)/(2/3) = 13/X
⇔ 1/2 = 13/X
⇔ X = 13 × 2 = 26 days

১০,১৪১.

At what rate of compound interest per annum will a sum of Tk. 4000 becomes Tk. 4840 in 2 years?

ক
10%
খ
15%
গ
12%
ঘ
20%

ব্যাখ্যা

Question: At what rate of compound interest per annum will a sum of Tk. 4000 becomes Tk. 4840 in 2 years?

Solution:
Principal, P = Tk. 4000
Compound Amount, C = Tk. 4840
Time, n = 2 years
Rate, r = ?

We know,
C = P × (1 + r/100)ⁿ
⇒ 4840 = 4000 × (1 + r/100)²
⇒ (1 + r/100)² = 4840/4000
⇒ (1 + r/100)² = 484/400
⇒ 1 + r/100 = 22/20 [উভয়পাশে বর্গমূল করে]
⇒ r/100 = (11/10) - 1
⇒ r/100 = (11 - 10)/10
⇒ r/100 = 1/10
⇒ r = (1 × 100)/10
∴ r = 10

∴ Interest Rate = 10%

১০,১৪২.

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

ক
1/2
খ
3/4
গ
3/8
ঘ
5/16

ব্যাখ্যা

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

Then, E = {(1, 2), (1, 4), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 2), (3, 4), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 2), (5, 4), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}

∴ n(E) = 27
∴ P(E) = n(E)/n(S) = 27/36 = 3/4.

১০,১৪৩.

The average of 4 positive integers is 59. The highest integer is 83 and the lowest integer is 29. The difference between the remaining two integers is 28. Which of the following integers is higher of the remaining two integers?

ক
39
খ
48
গ
76
ঘ
Cannot be determined

ব্যাখ্যা

Question: The average of 4 positive integers is 59. The highest integer is 83 and the lowest integer is 29. The difference between the remaining two integers is 28. Which of the following integers is higher of the remaining two integers?

Solution:
Sum of four integers = 59 × 4 = 236
Let the required integers be x and x - 28
Then, x + (x - 28) = 236 - (83 + 29)
⇒ 2x - 28 = 124
⇒ 2x = 152
⇒ x = 76
Hence, required integer = 76

১০,১৪৪.

Thermometer : Temperature

ক
ক) Millimeter : Scale
খ
খ) Length : Breadth
গ
গ) Solar Energy : Sun
ঘ
ঘ) Cardiograph : Heart rate

ব্যাখ্যা

As temperature is measured from a thermometer in the same way heart rate is measured with cardiograph.

১০,১৪৫.

The ratio of length and breadth of a rectangular park is 7 : 5. A man runs along its boundary at 8 km/hr and takes 9 minutes for one round. Find its area in sq. meters.

ক
76800 sq. m.
খ
84320 sq. m.
গ
87500 sq. m.
ঘ
90400 sq. m.

ব্যাখ্যা

Question: The ratio of length and breadth of a rectangular park is 7 : 5. A man runs along its boundary at 8 km/hr and takes 9 minutes for one round. Find its area in sq. meters.

Solution:
One round of the park is equal to the perimeter of the park.
So, by completing one round, the man covers a distance equal to the perimeter of the park.
Now,
Distance or perimeter = speed × time
= 8 × (9/60)
= 1.2 km
= 1200 meters

Let,
Length = 7x and breadth = 5x
So, Perimeter,
2(7x + 5x) = 1200
⇒ 24x = 1200
∴ x =1200/24 = 50 meters

So, Length = 7 × 50 = 350 meters
And, Breadth = 5 × 50 = 250 meters

Area = Length × Breadth
= 350 × 250
= 87500 sq. m.

১০,১৪৬.

A cyclist rides 24 km at 16 kmph and further 36 km at 20 kmph. Find his average speed for the journey.

ক
ক) 18.18 km/hr.
খ
খ) 27.18 km/hr.
গ
গ) 15.18 km/hr.
ঘ
ঘ) 22.18 km/hr.

ব্যাখ্যা

Total distance travelled :
= (24 + 36) km
= 60 km

Total time taken :
(24/16) + (36/20) hours
= (3/2) + (9/5) hours
= (15 + 18)/10 hours
= 33/10 hours

∴ Average speed : 60 × (10/33) km/hr.
= 600/33 km/hr.
= 18.18 km/hr.

১০,১৪৭.

5 workers can complete work in 20 days. In how many days, 10 workers can complete the work?

ক
ক) 4 days
খ
খ) 6 days
গ
গ) 8 days
ঘ
ঘ) 10 days

ব্যাখ্যা

Question: 5 workers can complete work in 20 days. In how many days, 10 workers can complete the work?

Solution:
5 workers can complete work in 20 days
1 workers can complete work in 5 × 20 days
= 100 days
10 workers can complete work in 100/10 days
= 10 days

১০,১৪৮.

The average of 3, 5, 7, and x is 6, and the average of 6, 2, x, and y is 7. What are the values of x and y?

ক
0 and 1
খ
5 and 7
গ
9 and 11
ঘ
15 and 17
ঙ
2 and 3

ব্যাখ্যা

Question: The average of 3, 5, 7, and x is 6, and the average of 6, 2, x, and y is 7. What are the values of x and y?

Solution:
Given that,
The average of 3, 5, 7, and x is 6

Therefore,
6 = (3 + 5 + 7 + x)/4
⇒ 24 = 15 + x
⇒ x = 24 - 15
∴ x = 9

Therefore,
7 = (6 + 2 + x + y)/4
⇒ 7 = (6 + 2 + 9 + y)/4
⇒ 28 = 17 + y
⇒ y = 28 - 17
∴ y = 11

১০,১৪৯.

Sifat and Rifat can complete a task individually in 18 and 27 days, respectively. After working together for 9 days, Rifat leaves. How many days does Sifat need to complete the rest?

ক
1 days
খ
2 days
গ
3 days
ঘ
6 days

ব্যাখ্যা

Question: Sifat and Rifat can complete a task individually in 18 and 27 days, respectively. After working together for 9 days, Rifat leaves. How many days does Sifat need to complete the rest?

Solution:

১০,১৫০.

A, B, and C enter into a partnership. A contributes one-third of the capital while C contributes as much as A and B together contribute. If the profit at the end of the year amounts to Tk.19800, what would B receive?

ক
3300 Tk.
খ
4300 Tk.
গ
3800 Tk.
ঘ
5700 Tk.

ব্যাখ্যা

Question: A, B, and C enter into a partnership. A contributes one-third of the capital while C contributes as much as A and B together contribute. If the profit at the end of the year amounts to Tk.19800, what would B receive?

Solution:
As C contributes as much as A and B,
∴ C's share is = 1/2 of the partnership

given,
A's share = 1/3
∴ B's share = ( 1 - 1/3 - 1/2) = 1/6

∴ B will receive = (19800)/6 = 3300 Tk.

১০,১৫১.

If the cost price of an article is 80% of its selling price. Then find the profit%?

ক
ক) 20%
খ
খ) 25%
গ
গ) 30%
ঘ
ঘ) 50%

ব্যাখ্যা

Question: If the cost price of an article is 80% of its selling price. Then find the profit%?

Solution:
ধরি,
বিক্রয়মূল্য = ১০০ টাকা
ক্রয়মূল্য = ১০০ এর ৮০%
= ৮০ টাকা
লাভ = ১০০ - ৮০ = ২০ টাকা

শতকরা লাভ = {(২০/৮০) × ১০০}% = ২৫%

১০,১৫২.

The sum of all prime numbers from 1 to 20 is -

ক
ক) 75
খ
খ) 76
গ
গ) 77
ঘ
ঘ) 78

ব্যাখ্যা

The sum of all prime numbers from 1 to 20
= (2 + 3 + 5 + 7 + 11 + 13 + 17 + 19)
= 77

১০,১৫৩.

If xtan60° + cos45° = sec45°, then the value of x² + 1 is?

ক
5/3
খ
7/6
গ
9/4
ঘ
8/5

ব্যাখ্যা

Question: If xtan60° + cos45° = sec45°, then the value of x² + 1 is?

Solution:

১০,১৫৪.

Find the larger of the two positive numbers, such that sum of the numbers is 18 and difference of their squares is 9 times the larger number.

ক
20
খ
18
গ
16
ঘ
12
ঙ
None of these

ব্যাখ্যা

Question: Find the larger of the two positive numbers, such that sum of the numbers is 18 and difference of their squares is 9 times the larger number.

Solution:
Let
the numbers be x, y where x > y
∴ x + y = 18
⇒ y = 18 - x

ATQ,
x² - y² = 9x
⇒ x² - (18 - x)² = 9x
⇒ x² - 324 + 36x - x² = 9x
⇒ 36x - 9x = 324
⇒ 27x = 324
⇒ x = 324/27
∴ x = 12

১০,১৫৫.

Calculate the H.C.F of 2x² - 8 and x² + 4x + 4

ক
ক) (x + 2) (x - 2)
খ
খ) 2(x + 2)
গ
গ) x + 2
ঘ
ঘ) x - 2

ব্যাখ্যা

Question: Calculate the H.C.F of 2x² - 8 and x² + 4x + 4

Solution:
2x² - 8
= 2(x2 - 4)
= 2(x² - 2²)
= 2(x + 2)(x - 2)

And x² + 4x + 4
= (x² + 2 . x . 2 + 2²)
= (x + 2)²
= (x + 2)(x + 2)

∴ Required H.C.F = x + 2

১০,১৫৬.

If the one third of one fourth of a number is 15, then what is the 3/10 of the number?

ক
ক) 45
খ
খ) 54
গ
গ) 36
ঘ
ঘ) 56

ব্যাখ্যা

Question: If the one third of one fourth of a number is 15, then what is the 3/10 of the number?

Solution:
Let the number be x

ATQ,
(1/3) × (1/4) × x = 15
⇒ 1/12 × x = 15
x = 180

Now,
3x/10 = (3/10) × 180
∴ 3x/10 = 54

১০,১৫৭.

If AB and CD are two diameters of a circle of radius and they are mutually perpendicular then what is the ratio of the area of the circle to the area of the Δ ACD?

ক
ক) π/2
খ
খ) π/4
গ
গ) 2π
ঘ
ঘ) π

ব্যাখ্যা

ধরি, O কেন্দ্র বিশিষ্ট বৃত্তে ADC একটি ত্রিভুজ। বৃত্তটির ব্যাসার্ধ r
এখানে, ব্যাসার্ধ OA = OD = OC= r
ব্যাস, AB = DC = 2r
বৃত্তের ক্ষেত্রফল = πr² বর্গএকক

Δ ACD এর ক্ষেত্রফল = (1/2) × 2r × r
= r²

πr² : r²= π : 1 = π/1 = π

১০,১৫৮.

If 12 men or 18 boys can make 90 chairs in 9 days, then how many chairs will be made by 6 men and 9 boys in 10 days?

ক
50 chairs
খ
70 chairs
গ
80 chairs
ঘ
100 chairs

ব্যাখ্যা

Question: If 12 men or 18 boys can make 90 chairs in 9 days, then how many chairs will be made by 6 men and 9 boys in 10 days?

Solution:
Here,
12 men = 18 boys
∴ 6 men = (18 ÷ 2) boys
= 9 boys

∴ 6 men and 9 boys = (9 + 9) = 18 boys

Now,
18 boys can make 90 chairs in 9 days
In 1 day, 18 boys can make (90/9) =10 chairs
∴ In 10 day, 18 boys can make = (10 × 10) chairs
= 100 chairs

১০,১৫৯.

A, B, and C are partners in a business. Their shares are in the proposition of (1/3) : (1/4) : (1/5). A withdraws half of his capital after 15 months and after another 15 months, a profit of Tk. 4340 is divided. The share of C is-

ক
Tk. 1520
খ
Tk. 1435
গ
Tk. 1355
ঘ
Tk. 1240
ঙ
None

ব্যাখ্যা

Question: A, B, and C are partners in a business. Their shares are in the proposition of (1/3) : (1/4) : (1/5). A withdraws half of his capital after 15 months and after another 15 months the profit of Tk. 4340 is divided. The share of C is-

Solution:
Ratio of initial investments = 1/3 : 1/4 : 1/5
= 20 : 15 : 12

Let
their initial investments will be 20x, 15x and 12x respectively.

A : B : C = {(20x × 15) + (10x × 15)} : (15x × 30) : (12x × 30) [A's portion was calculated as half after the first 15 months.]
= 450x : 450x : 360x
= 5 : 5 : 4

Sum of the ratio = 5 + 5 + 4 = 14.
∴ C's share = 4340 × (4/14)
= Tk. 1240

১০,১৬০.

The cost of Type 1 rice is Tk. 15 per kg and Type 2 rice is Tk. 20 per kg. If both Type 1 and Type 2 are mixed in the ratio of 2 : 3, then the price per kg of the mixed variety of rice is-

ক
Tk. 19.50
খ
Tk. 19
গ
Tk. 18
ঘ
Tk. 18.50

ব্যাখ্যা

Question: The cost of Type 1 rice is Tk. 15 per kg and Type 2 rice is Tk. 20 per kg. If both Type 1 and Type 2 are mixed in the ratio of 2 : 3, then the price per kg of the mixed variety of rice is-

Solution:
Let,
Quantity of type 1 rice is 2x kg.
Quantity of type 2 rice is 3x kg.
The price per kg of the mixed variety of rice is y taka

∴ Total price of type 1 rice is 15 × 2x = 30x Taka
∴ Total price of type 2 rice is 20 × 3x = 60x Taka

ATQ,
30x + 60x = y(2x + 3x)
⇒ 90x = y × 5x
⇒ y = (90x)/(5x)
∴ y = 18

১০,১৬১.

Find the greatest number, which on dividing 1657 and 2037 leaves remainders 6 and 5 respectively.

ক
135
খ
127
গ
147
ঘ
119
ঙ
None of the above

ব্যাখ্যা

Question: Find the greatest number, which on dividing 1657 and 2037 leaves remainders 6 and 5 respectively.

Solution:
The number on dividing 1657 and 2037 leaves remainders 6 and 5 respectively.
Hence, make the dividend completely divisible by the divisor. This is possible, if we subtract remainders from the dividend.
Therefore,
1657 - 6 = 1651
2037 - 5 = 2032

H.C.F. of 1651 and 2032 is 127. 127 is the common factor.
127 × 13 = 1651
Thus by adding 6, we get 1651 + 6 = 1657

Hence, the required answer is 127

১০,১৬২.

The combined average age of a husband, wife, and child was 27 years three years back, and the average age of the wife and child was 20 years five years ago. Find their(wife and child) current total age.

ক
30 years
খ
35 years
গ
40 years
ঘ
50 years

ব্যাখ্যা

Question: The combined average age of a husband, wife, and child was 27 years three years back, and the average age of the wife and child was 20 years five years ago. Find their(wife and child) current total age.

Solution:
Sum of the present ages of husband, wife and child =(27 × 3 + 3 × 3) years
= (81 + 9) years
= 90 years

Sum of the present ages of wife and child =(20 × 2 + 5 × 2) years
= (40 + 10) years
= 50 years

১০,১৬৩.

Hemal bought a chair for Tk. 1540 and sold it to Tofail. If Hemal earned a profit of 25%, find the selling price of chair.

ক
Tk. 1875
খ
Tk. 1900
গ
Tk. 1925
ঘ
Tk. 1950

ব্যাখ্যা

Question: Hemal bought a chair for Tk. 1540 and sold it to Tofail. If Hemal earned a profit of 25%, find the selling price of chair.

Solution:
Cost Price of the chair = Tk. 1540
S.P. of the chair =?
Profit earned = 25%

∴ Selling Price = 1540 + 25% of 1540
= 1540 + (25 × 1540)/100
= 1540 + 385
= 1925

১০,১৬৪.

Joy can knit a pair of socks in 3 days. Belal can knit the same pair of socks in 9 days. If they are knitting together, then in how many days will they knit two pairs of socks?

ক
9/2 days
খ
9/4 days
গ
9/8 days
ঘ
None of these

ব্যাখ্যা

Question: Joy can knit a pair of socks in 3 days. Belal can knit the same pair of socks in 9 days. If they are knitting together, then in how many days will they knit two pairs of socks?

Solution:
They knit in one day = (1/3) + (1/9)
= 4/9 days

They knit a pair of socks in 9/4 days
they knit two pairs of socks in (9 × 2)/4 days
= 9/2 days

১০,১৬৫.

The greatest number that exactly divides 105, 1001 and 2436 is -

ক
3
খ
7
গ
11
ঘ
21

ব্যাখ্যা

Question: The greatest number that exactly divides 105, 1001 and 2436 is -

Solution:
The greatest number be GCD of 105, 1001 and 2436.
GCD of 105, 1001 and 2436 is 7
∴ The number is 7.

১০,১৬৬.

What is the greatest possible area of a triangle with one side of length 7 and another side of length 10?

ক
ক) 17
খ
খ) 30
গ
গ) 35
ঘ
ঘ) 70

ব্যাখ্যা

দেয়া আছে,
ত্রিভুজের উচ্চতা = 7
ভূমি = 10

ত্রিভুজের ক্ষেত্রফল = (1/2) × ভূমি × উচ্চতা
= (1/2) × 7 × 10
= 35

১০,১৬৭.

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

ক
16 km/hr
খ
18 km/hr
গ
20 km/hr
ঘ
22 km/hr

ব্যাখ্যা

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

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

ATQ,
Total time = {a/(3 × 10)} + {a/(3 × 20)} + {a/(3 × 60)}
= (a/30) + (a/60) + (a/180)
= (6a + 3a + a)/180
= 10a/180
= a/18

∴ Average speed = (a × 18)/a km/hr
= 18 km/hr

১০,১৬৮.

The perimeter of a rectangular field is 110 meters. If the length of the field is 5 meters less than three times the width, what is the area of that field in square meters?

ক
550 sq. m.
খ
600 sq. m.
গ
625 sq. m.
ঘ
575 sq. m.

ব্যাখ্যা

Question: The perimeter of a rectangular field is 110 meters. If the length of the field is 5 meters less than three times the width, what is the area of that field in square meters?

Solution:
ধরি, আয়তাকার ক্ষেত্রটির প্রস্থ = x মিটার
সুতরাং, ক্ষেত্রটির দৈর্ঘ্য = 3x - 5 মিটার

আয়তক্ষেত্রের পরিসীমা = 2(দৈর্ঘ্য + প্রস্থ)
প্রশ্নমতে,
2((3x - 5) + x) = 110
⇒ 2(4x - 5) = 110
⇒ 4x - 5 = 55
⇒ 4x = 60
⇒ x = 15 মিটার

সুতরাং, প্রস্থ = 15 মিটার।
দৈর্ঘ্য = 3x - 5 = (3 × 15) - 5
= 45 - 5 = 40 মিটার।

আয়তক্ষেত্রের ক্ষেত্রফল = দৈর্ঘ্য × প্রস্থ
ক্ষেত্রফল = 40 × 15 = 600 বর্গ মিটার।

সুতরাং, ক্ষেত্রটির ক্ষেত্রফল হলো 600 বর্গ মিটার।

১০,১৬৯.

An electric pole casts a √3 m long shadow on the ground at an elevation 60°, the height of the pole is -

ক
ক) 3 m
খ
খ) 3√3 m
গ
গ) 3√2 m
ঘ
ঘ) 2√3 m

ব্যাখ্যা

মনে করি, AB = h
খুঁটিটি ভূমির সাথে 60° কোণ তৈরি করে BC = √3 মিটার ছায়া তৈরি করে
তাহলে খুঁটির উচ্চতা h = ?
প্রশ্নমতে, tan60° = AB / BC
⇒ √3 = h/√3
∴ h = √3.√3 = 3 মিটার

১০,১৭০.

On February 12, 2002, it was Tuesday. The day of the week on February 12, 2001, was-

ক
Wednesday
খ
Saturday
গ
Sunday
ঘ
Monday

ব্যাখ্যা

Question: On February 12, 2002, it was Tuesday. The day of the week on February 12, 2001, was-

Solution:
2001 was an ordinary year, it had 1 odd day.

So, the day on February 12, 2002, would be one day beyond the day on February 12, 2001.

∴ Tuesday on February 12, 2002, would be one day beyond the day on February 12, 2001.

So the day on February 12, 2001, was Monday.

১০,১৭১.

The difference between the numerator and the denominator of a fraction is 5. If 5 is added to the denominator the fraction is decreased by 5/4 then the value of the fraction will be equal to:

ক
ক) 1/6
খ
খ) 13/4
গ
গ) 9/4
ঘ
ঘ) 5

ব্যাখ্যা

Question: The difference between the numerator and the denominator of a fraction is 5. If 5 is added to the denominator the fraction is decreased by 5/4 then the value of the fraction will be equal to:

Solution:
ধরি,
ভগ্নাংশটির হর = x
ভগ্নাংশটির লব = x + 5

∴ ভগ্নাংশটি = (x + 5)/x

প্রশ্নমতে,
{(x + 5)/x} - {(x + 5)/(x + 5)} = 5/4
⇒ {(x + 5)/x} - 1 = 5/4
⇒ {(x + 5)/x} = (5/4) + 1
∴ {(x + 5)/x} = 9/4

∴ ভগ্নাংশটি = 9/4

১০,১৭২.

A hall is 15 m long and 12 m broad. If the sum of the areas of the floor and the ceiling is equal to the sum of the areas of four walls, the volume of the hall is-

ক
720
খ
900
গ
1200
ঘ
1800

ব্যাখ্যা

Question: A hall is 15 m long and 12 m broad. If the sum of the areas of the floor and the ceiling is equal to the sum of the areas of four walls, the volume of the hall is-

Solution:
A hall is 15 m long and 12 m broad.
∴ The area of floor is = 15 × 12 = The area of ceiling

Let the height of the hall = h
∴ total area of four walls = 15 × h + 12 × h + 15 × h + 12 × h = 2(15h + 12h)

ATQ,
2(15h + 12h) = 2(15 × 12)
⇒ 27h = 180
⇒ h = 180/27
∴ h = 20/3

∴ Volume = 15 × 12 × (20/3) m³= 1200 m³

১০,১৭৩.

Which trigonometric ratio is undefined at 90°?

ক
tan90°
খ
sin90°
গ
cos90°
ঘ
cot90°

ব্যাখ্যা

Question: Which trigonometric ratio is undefined at 90°?

Solution:
sin90° = 1
cos90° = 0
cot90°= 0

∴ tan90° = sin90°/cos90°
= 1 / 0
= ∞ (Undefined)

১০,১৭৪.

A leak in the bottom of a tank can empty the whole tank in 6 hours. An inlet pipe fills water are the rate of 5 liters a minute. When the tank is full, the inlet is opened and due to the leak, the tank is empty in 10 hours. How many liters does the tank hold?

ক
ক) 4250 liters
খ
খ) 4500 liters
গ
গ) 4900 liters
ঘ
ঘ) 5250 liters

ব্যাখ্যা

Question: A leak in the bottom of a tank can empty the whole tank in 6 hours. An inlet pipe fills water are the rate of 5 liters a minute. When the tank is full, the inlet is opened and due to the leak, the tank is empty in 10 hours. How many liters does the tank hold?

Solution:
Work done by the inlet pipe in 1 hour = ( 1/6 - 1/10 ) = 2/30 = 1/15
Work done by the inlet pipe in 1 minute = ( 1/15 × 1/60 ) = 1/900
Volume of 1/900 part = 5 liters
Volume of the whole tank = ( 900 × 5 ) = 4500 liters

১০,১৭৫.

Which fraction has the smallest value?

ক
8/(3⁴ × 7³)
খ
27/(3⁵ × 7³)
গ
12/(3³ × 7³)
ঘ
2/(3³ × 7²)

ব্যাখ্যা

Question: Which fraction has the smallest value?

Solution:

ক) 8/(3⁴ × 7³) = 8/27,783 = 8/27,783
খ) 27/(3⁵ × 7³) = 1/3,087 = 8/24,696
গ) 12/(3³ × 7³) = 4/3,087 = 8/6,174
ঘ) 2/(3³ × 7²) = 2/1,323 = 8/5,292

লব একই হলে যে ভগ্নাংশের হর বড় সে ভগ্নাংশটি ছোট
সঠিক উত্তর: অপশন ক

১০,১৭৬.

An aeroplane covers a certain distance at a speed of 240 kmph in 5 hours. To cover the same distance in 5/3 hours, it must travel at a speed of-

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

ব্যাখ্যা

Question: An aeroplane covers a certain distance at a speed of 240 kmph in 5 hours. To cover the same distance in 5/3 hours, it must travel at a speed of-

Solution:
Given that,
Speed = 240 kmph
Time = 5 hours

We know,
Distance = Speed × Time
= 240 × 5
∴ Distance = 1200 km

Again,
Given:
Distance = 1200 km
New Time = 5/3 hours

New Speed = Distance/New Time
= 1200/(5/3)
= (1200 × 3)/5
= 720 kmph

∴ The aeroplane must travel at a speed of 720 kmph.

১০,১৭৭.

A can do a piece of work in 8 days working 12 hours per day. If B is two-thirds as efficient as A, then in how many days B alone do the same piece of work, working 4 hours per day?

ক
ক) 28 days
খ
খ) 31 days
গ
গ) 36 days
ঘ
ঘ) 43 days

ব্যাখ্যা

Question: A can do a piece of work in 8 days working 12 hours per day. If B is two-thirds as efficient as A, then in how many days B alone do the same piece of work, working 4 hours per day?

Solution:
Time taken by A alone to do the work = 8×12 = 96 hrs.
Since B is 2/3 efficient as A, so time taken by B is 3/2 times of A = ( 96×3/2 ) hrs = 144 hrs.
∴ Required days = ( 144/4 ) = 36 days.

১০,১৭৮.

If x - 1/x = 4, then x³ - 1/x³ =?

ক
76
খ
56
গ
64
ঘ
60

ব্যাখ্যা

Question: If x - 1/x = 4, then x³ - 1/x³ =?

Solution:
x³ - 1/x³
= (x - 1/x)³ + 3x.(1/x)(x - 1/x)
= 4³ + 3 × 4
= 64 + 12
= 76

১০,১৭৯.

A man invested Tk. 36,000 when he bought Tk. 100 shares at Tk. 144. If 16% dividend is declared, find his annual income.

ক
Tk. 3200
খ
Tk. 2620
গ
Tk. 4000
ঘ
Tk. 3600

ব্যাখ্যা

Question: A man invested Tk. 36,000 when he bought Tk. 100 shares at Tk. 144. If 16% dividend is declared, find his annual income.

Solution:
For Tk. 144 he gets Tk. 16
∴ For Tk. 1 he gets Tk. 16/144
∴ For Tk. 36,000 he gets Tk. (16 × 36,000)/144
= Tk. (1/9) × 36,000
= Tk. 4,000

∴ The man's annual income is Tk. 4000.

১০,১৮০.

A pupil's marks were wrongly entered as 83 instead of 63. Due to that the average marks for the class got increased by half (1/2). The number of pupils in the class is:

ক
ক) 10
খ
খ) 20
গ
গ) 40
ঘ
ঘ) 73
ঙ
ঙ) 30

ব্যাখ্যা

Let there be x pupils in the class.
Total increase in marks = x . 1/2
= x/2
∴ x/2 = 83 - 63
=> x/2 = 20
=> x = 40

১০,১৮১.

Ifthen what is the value of (5 - 3x) + (5 - 3x)²?

ক
4
খ
2
গ
1
ঘ
0

ব্যাখ্যা

Question: Ifthen what is the value of (5 - 3x) + (5 - 3x)²?

Solution:
Given,
√(5 - 3x) = 1
⇒ {√(5 - 3x)}² = 1²
⇒ (5 - 3x) = 1

∴ (5 - 3x) + (5 - 3x)² = 1 + 1² = 2

১০,১৮২.

If log_x16 = 4/3, what is the value of x?

ক
2
খ
4
গ
8
ঘ
16

ব্যাখ্যা

Question: If log_x16 = 4/3, what is the value of x?

Solution:
log_x16 = 4/3
⇒ x^4/3 = 16 [log_ba = c ⇒ b^c = a]
⇒ (x^4/3)^3/4 = 16^3/4
⇒ x = 16^3/4
⇒ x = (2⁴)^3/4
⇒ x = 2³
∴ x = 8

১০,১৮৩.

While calculating the average of a batsman as 36 in 100 matches that he played, one of the scores 90 was incorrectly noted as 40. The percentage error is -

ক
ক) 0.5%
খ
খ) 1.21%
গ
গ) 1.34%
ঘ
ঘ) 1.36%

ব্যাখ্যা

Correct sum = 36 × 100 + 90 - 40
= 3650
Correct average = 3650/100 = 36.5
Error = (36.5 - 36) = 0.5
∴ Error% = {(0.5/36.5) × 100}% = (100/73)%
= 1.36%

১০,১৮৪.

The value of (32)^0.08 × (32)^0.12 is -

ক
4
খ
3
গ
6
ঘ
2

ব্যাখ্যা

Question: The value of (32)^0.08 × (32)^0.12 is -

solution:
Given that,
(32)^0.08 × (32)^0.12
= (32)^{0.08 + 0.12}
= (32)^0.2
= (2⁵)^2/10
= 2^10/10
= 2¹
= 2
∴ the value is 2

১০,১৮৫.

Find the equation of the line with x-intercept = 5 and y-intercept = 2.

ক
2x + 5y - 10 = 0
খ
5x + 2y - 10 = 0
গ
2x - 5y - 10 = 0
ঘ
5x + 2y - 1 = 0

ব্যাখ্যা

Question: Find the equation of the line with x-intercept = 5 and y-intercept = 2.

Solution:
Given,
x-intercept = 5, the line passes through (5, 0).
y-intercept = 2, the line passes through (0, 2).

We know,
The intercept form of a line is:
(x/a) + (y/b) = 1, where a = x-intercept, b = y-intercept.
⇒ (x/5) + (y/2) = 1
⇒ (2x + 5y)/10 = 1
⇒ 2x + 5y = 10
⇒ 2x + 5y - 10 = 0

∴ The equation of the line is 2x + 5y - 10 = 0

১০,১৮৬.

A train 175 m long crosses a bridge which is 125 m long in 1 min 40 seconds. What is the speed of the train?

ক
ক) 2 m/sec
খ
খ) 3 m/sec
গ
গ) 4 m/sec
ঘ
ঘ) 5 m/sec

ব্যাখ্যা

Question: A train 175 m long crosses a bridge which is 125 m long in 1 min 40 seconds. What is the speed of the train?

Solution:
মোট অতিক্রান্ত দূরত্ব = ১৭৫ + ১২৫ মিটার
= ৩০০ মিটার

সময় = ১ মিনিট ৪০ সেকেন্ড
= ৬০ + ৪০ সেকেন্ড
= ১০০ সেকেন্ড

বেগ = ৩০০/১০০ মিটার/সেকেন্ড
= ৩ মিটার/সেকেন্ড

১০,১৮৭.

Which number is 40% less than 90?

ক
ক) 60
খ
খ) 54
গ
গ) 52
ঘ
ঘ) 36

ব্যাখ্যা

প্রশ্ন: Which number is 40% less than 90?

সমাধান:
৯০ এর ৪০% কম যে সংখ্যাটি তা ৯০ এর ৬০% হবে।

∴ সংখ্যাটি = ৯০ এর ৬০/১০০
= ৫৪

১০,১৮৮.

If an article is sold at 200 percent profit, then the ratio of its cost price to its selling price will be -

ক
ক) 1 : 3
খ
খ) 2 : 3
গ
গ) 1 : 2
ঘ
ঘ) 2 : 1

ব্যাখ্যা

Let
Cost price = Tk. 100
Sell price = 100 + 200% of 100
= 100 + (100 × 200)/100
= 100 + 200
= Tk. 300

Required ratio = 100 : 300 = 1 : 3

১০,১৮৯.

10 years ago, P was half of Q in age. If the ratio of their present age is 3 : 4. What will be the average present age of them?

ক
ক) 35 years
খ
খ) 30 years
গ
গ) 17.5 years
ঘ
ঘ) 22.5 years

ব্যাখ্যা

Let, the age of P be x years 10 years ago and so the age of Q is 2x years.
present age of P and Q are x + 10 and 2x + 10
(x + 10)/(2x + 10) = 3/4
⇒ 6x + 30 = 4x + 40
⇒ 2x = 10
⇒ x = 5
the average present age of them
= (x + 10 + 2x + 10)/2
= (5 + 10 + 2 × 5 + 10)/2
= 17.5 years

১০,১৯০.

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

ক
60 meters
খ
140 meters
গ
100 meters
ঘ
120 meters

ব্যাখ্যা

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

Solution:
Let,
Breadth of rectangle = 3x meters
∴ Length of rectangle = (3x) ×(7/3) = 7x meters

ATQ,
2(3x + 7x) = 400
⇒ 2 × 10x = 400
⇒ 20x = 400
∴ x = 20

∴ Length of rectangle = 7 × 20 = 140 meters.

১০,১৯১.

Average of five numbers is 27. If one number is excluded, the average becomes 25. What is the excluded number?

ক
32.5
খ
30
গ
35
ঘ
40

ব্যাখ্যা

Some of five numbers = 5 × 27
After excluding one number, the sum of the remaining four numbers = 4 × 25
Excluded number = (5 × 27) - (4 × 25)
= 135 - 100
= 35.

১০,১৯২.

What is the highest power of 5 in the prime factorization of 625?

ক
4
খ
5
গ
6
ঘ
7

ব্যাখ্যা

Question: What is the highest power of 5 in the prime factorization of 625?

Solution:
Given that, Highest power of 5 in the prime factorization of 625
Now, The prime factorization of 625 = 5 × 5 × 5 × 5 = 5⁴
Highest power of 5 = 4

∴ The highest power of 5 is 4.

১০,১৯৩.

The ratio of the angles of a triangle is 5 : 15 : 16. What is the largest angle in degrees?

ক
100°
খ
90°
গ
80°
ঘ
75°
ঙ
None

ব্যাখ্যা

Question: The ratio of the angles of a triangle is 5 : 15 : 16. What is the largest angle in degrees?

Solution:
Given
The ratio of the angles of a triangle = 5 : 15 : 16

Let,
the angles = 5x , 15x 16x

ATQ,
5x + 15x + 16x = 180°
⇒ 36x = 180°
⇒ x = 180°/36
∴ x = 5°

∴ the largest angle = 16 × 5° = 80°

১০,১৯৪.

If (2x² + 3x - 5)(x + 2) = ax³ + bx² + cx + d, then ac - bd = ?

ক
ক) 70
খ
খ) 71
গ
গ) 72
ঘ
ঘ) 73

ব্যাখ্যা

এখানে,
(2x²+ 3x - 5)(x + 2) = 2x³+ 4x²+ 3x²+ 6x - 5x - 10 = ax³+bx²+cx+d
⇒ 2x³+ 7x²+ x - 10 = ax³+ bx²+ cx + d
উভয় দিকে তুলনা করে পাই
a = 2, b = 7, c = 1, d = -10
অতএব, ac - bd = 2×1 -7(-10)
= 2 + 70
= 72

১০,১৯৫.

One fourth of the boys and three eight of the girls in a school participated in the annual sports. What proportional part of the total student population of the school participated in the annual sports?

ক
4/12
খ
5/8
গ
8/12
ঘ
6/12
ঙ
None of above

ব্যাখ্যা

Let total boys is 4 and participated in sports is 1
Total girls is 8 and participated in sports 3
So, total student 4 + 8 = 12,
and participant = 1 + 3 = 4
Therefore, the proportion = 4/12

১০,১৯৬.

Solve it. 4376 + 3209 - 1784 + 97 = 3125 + ?

ক
2237
খ
2663
গ
2769
ঘ
2773

ব্যাখ্যা

Question: Solve it. 4376 + 3209 - 1784 + 97 = 3125 + ?

Solution:
4376 + 3209 - 1784 + 97 = 3125 + x
⇒ 7682 - 1784 = 3125 + x
⇒ x = 7682 - 1784 - 3125
⇒ x = 2773

Hence,
The number is 2773

১০,১৯৭.

84 is divided into two parts so that 4 times one part and 12 times the another part are together equal to 544. The parts are?

ক
58 and 26
খ
62 and 22
গ
68 and 16
ঘ
54 and 30
ঙ
None of these

ব্যাখ্যা

Question: 84 is divided into two parts so that 4 times one part and 12 times the another part are together equal to 544. The parts are?

Solution:
let,
The two parts be x and (84 - x)

ATQ,
⇒ 4x + 12(84 - x) = 544
⇒ 4x + 1008 - 12x = 544
⇒ - 8x = 544 - 1008
⇒ - 7x = - 464
⇒ x = 464/8
∴ x = 58

So one part is 58 and other part is = 84 - 58 = 26

∴ The two parts are 58 and 26

১০,১৯৮.

HCF and LCM of two numbers are 6 and 180 respectively. If the numbers are between 30 and 80, what is the sum of the numbers?

ক
60
খ
66
গ
72
ঘ
76

ব্যাখ্যা

Question: HCF and LCM of two numbers are 6 and 180 respectively. If the numbers are between 30 and 80, what is the sum of the numbers?

Solution:
Given that,
HCF and LCM of two numbers are 6 and 180.

Let the numbers be 6x and 6y (where x and y are co-prime)
∴ LCM = 6xy
∴ 6xy = 180
⇒ xy = 180/6 = 30

Possible co-prime factor pairs of 30: (1, 30), (2, 15), (3, 10), (5, 6)

The numbers must lie between 30 and 80:
6 × 5 = 30 and 6 × 6 = 36, both between 30 and 80

Other pairs give at least one number ≤ 18 or ≥ 90, not valid
∴ Required numbers are 30 and 36
∴ Sum of the numbers = 30 + 36 = 66

So, the sum of the numbers is 66.

১০,১৯৯.

A chemist has two solutions, one containing 30% acid and the other containing 70% acid. How many liters of each solution should be mixed to get 10 liters of a solution containing 50% acid?

ক
ক) 2 liters
খ
খ) 3 liters
গ
গ) 4 liters
ঘ
ঘ) 5 liters

ব্যাখ্যা

Question: A chemist has two solutions, one containing 30% acid and the other containing 70% acid. How many liters of each solution should be mixed to get 10 liters of a solution containing 50% acid?

Solution:
Let x be the liters of the 30% acid solution.
Then, (10 - x) would be the liters of the 70% acid solution.

Amount of acid from 30% solution = 30% of x = 0.3x
Amount of acid from 70% solution = 70% of (10 - x) = 0.7(10 - x)

Total amount of acid in the mixture = Amount of acid from 30% solution + Amount of acid from 70% solution
0.3x + 0.7(10 - x) = 0.5(10)

Solve for x:
0.3x + 7 - 0.7x = 5
0.3x - 0.7x = 5 - 7
-0.4x = -2
x = -2 / -0.4
x = 5

So, 5 liters of the 30% acid solution should be mixed with (10 - 5) = 5 liters of the 70% acid solution to obtain 10 liters of the 50% acid solution.

১০,২০০.

A 42 liter mixture contains milk and water in the ratio 3 : 4. How many liters of milk must be added to the mixture so that the ratio of milk to water becomes 1 : 1?

ক
3 liters
খ
6 liters
গ
5 liters
ঘ
7 liters

ব্যাখ্যা

Question: A 42 liter mixture contains milk and water in the ratio 3 : 4. How many liters of milk must be added to the mixture so that the ratio of milk to water becomes 1 : 1?

Solution:
The ratio of milk to water is 3 : 4
Total portion = 3 + 4 = 7

Quantity of milk = 42 × (3/7) = 18 liters.
Quantity of water = 42 × (4/7) = 24 liters.

Let,
Quantity of milk to be added = x liters

According to the question,
(18 + x) : 24 = 1 : 1
⇒ (18 + x)/24 = 1/1
⇒ 18 + x = 24
⇒ x = 24 - 18
⇒ x = 6

∴ Quantity of milk to be added = 6 liters