Bank Math

PrepBank · পাতা ১০১ / ১৬১ · ১০,০০১–১০,১০০ / ১৬,১২৪

১০,০০১.

Munna covers one-third of a road at 3 km/h and the remaining distance at 6 km/h. His average speed is-

ক
ক) 4 km/hr
খ
খ) 4.5 km/hr
গ
গ) 5 km/hr
ঘ
ঘ) 5.5 km/hr

ব্যাখ্যা

Question: Munna covers one-third of a road at 3 km/h and the remaining distance at 6 km/h. His average speed is-

Solution:
ধরি, রাস্তাটির দূরত্ব x km
3 km/h বেগে অতিক্রম করে x/3 কিমি ও
6 km/h বেগে অতিক্রম করে = (x - x/3) কিমি
= 2x/3 কিমি

গড় বেগ = (2x)/{(x/3/3) + (2x/3/6)}
= x/(x/9 + x/9)
= 9/2
= 4.5 kmph

১০,০০২.

5 years back Himel got married at the age of 30. After 2 years he was blessed with a baby child. What will be the ratio of Himel and his children's age 5 years from now?

ক
ক) 4 : 1
খ
খ) 1 : 4
গ
গ) 5 : 1
ঘ
ঘ) 5 : 2

ব্যাখ্যা

Question: 5 years back Himel got married at the age of 30. After 2 years he was blessed with a baby child. What will be the ratio of Himel and his children's age 5 years from now?

Solution:
বর্তমানে হিমেলের বয়স = ৩০ + ৫ = ৩৫ বছর
তার সন্তানের বয়স = ৩ বছর

৫ বছর পর,
হিমেলের বয়স হবে = ৩৫ + ৫ = ৪০ বছর
তার সন্তানের বয়স = ৩ + ৫ = ৮ বছর

অনুপাত = ৪০ : ৮ = ৫ : ১

১০,০০৩.

In the figure given below, what is ∠BCD equal to?

ক
70°
খ
60°
গ
80°
ঘ
90°

ব্যাখ্যা

Question: In the figure given below, what is ∠BCD equal to?

Solution:
বৃত্তের একই চাপের উপর দন্ডায়মান বৃত্তস্থ কোণ পরস্পর সমান।
BC চাপের উপর বৃত্তস্থ কোণ ∠BAC ও ∠BDC
∠BAC = ∠BDC = 30°

আবার
ত্রিভুজের তিন কোণের সমষ্টি দুই সমকোণ বা 180°
ΔBCD এ
⇒ ∠CBD + ∠BDC + ∠BCD = 180°
⇒ 30° + 70° + ∠BCD = 180°
∴ ∠BCD = 180° - 100° = 80°

১০,০০৪.

If x + (2/x) = 4, what is the value of x³ + (8/x³)?

ক
30
খ
40
গ
45
ঘ
50

ব্যাখ্যা

Question: If x + (2/x) = 4, what is the value of x³ + (8/x³)?

Solution:
Here, x + (2/x) = 4

Now,
x³ + (8/x³)
= (x)³ + (2/x)³
= {(x + (2/x)}³ - 3 . x . 2/x {x + (2/x)}
= 4³ - 3 . 2 . 4
= 64 - 24
= 40

১০,০০৫.

What is the H.C.F. of the following fractions?
2/4, 4/6, 6/8.

ক
1/6
খ
1/12
গ
1/8
ঘ
1/4

ব্যাখ্যা

Question: What is the H.C.F. of the following fractions?
2/4, 4/6, 6/8.

Solution:
আমরা জানি, ভগ্নাংশের গসাগু = (লবের গসাগু)/(হরের লসাগু)

এখানে লব = 2, 4 এবং 6
2 = 2 × 1
4 = 2 × 2
6 = 2 × 3
∴ লবের গসাগু (H.C.F.) = 2

হর = 4, 6 এবং 8
4 = 2²
6 = 2 × 3
8 = 2³
∴ হরের লসাগু (L.C.M.) = 2³ × 3 = 8 × 3 = 24

ভগ্নাংশের গসাগু = লবের গসাগু/হরের লসাগু
= 2/24
= 1/12

১০,০০৬.

The perimeter of two equilateral triangles are 24 m and 36 m respectively. What is the difference between the area of the larger triangle and the smaller triangle?

ক
34.64 m²
খ
36.64 m²
গ
32.64 m²
ঘ
30.64 m²

ব্যাখ্যা

Let 'a' be the side of the 1st triangle.
Area = [√3/4] x a² and perimeter = 3a.
Perimeter of 1st triangle 3a = 24m and a = 8m
Area₁ = [√3/4] x a²
[√3/4] x 8²
= 1.732 x 16 m²
Perimeter of 2nd triangle 3A = 36m and A = 12m.
Area₂ = [√3/4] x 12²
= 1.732 x 36 m²
Required difference = 1.732 x 36 m² - 1.732 x 16 m²
= 1.732(36 - 16)
= 1.732 x 20 m²
= 34.64 m².

১০,০০৭.

If log_x(1/81^-1) = - 4, what is the value of x?

ক
1/2
খ
9
গ
1/3
ঘ
3

ব্যাখ্যা

Question: If log_x(1/81^-1) = - 4, what is the value of x?

Solution:
দেওয়া আছে,
log_x(1/81^-1) = - 4
⇒ log_x{1/(1/81)} = - 4
⇒ log_x(81) = - 4
⇒ x^{- 4} = 81 [log_ab = c ⇒ a^c = b]
⇒ x^{- 4} = 3⁴⇒ (1/x)⁴ = 3⁴⇒ 1/x = 3
∴ x = 1/3

১০,০০৮.

A, B and C enter into a partnership. A contributes one - third of the capital while B contributes as much as A and C together contribute. If the profit at the end of the year amounts to Tk. 900, What would C receive?

ক
ক) Tk. 200
খ
খ) Tk. 150
গ
গ) Tk. 100
ঘ
ঘ) Tk. 300

ব্যাখ্যা

ধরি, মোট বিনিয়োগ x টাকা
A বিনিয়োগ করে x/3 টাকা
B + C = (x - x/3) = 2x/3
কিন্তু, B এর বিনিয়োগ পরিমান A ও C এর মোট বিনিয়োগের সমান।
∴ x/3 + C + C = 2x/3
⇒ 2C = 2x/3 - x/3
⇒ 2C = x/3
⇒ C = x/6 এবং B = 2x/3 - x/6 = 3x/6 = x/2
A : B : C = x/3 : x/2 : x/6 = 2 : 3 : 1
∴ C এর লাভের পরিমাণ { 900 এর 1/(3 + 2 + 1)} = (900 এর 1/6) = 150 টাকা

১০,০০৯.

The sum of 5 consecutive integers is 35. Which of the following is the highest integer among them?

ক
5
খ
7
গ
9
ঘ
10

ব্যাখ্যা

Question: The sum of 5 consecutive integers is 35. Which of the following is the highest integer among them?

Solution:
Let,
The first integer is x
ATQ
x + x + 1 + x + 2 + x + 3 + x + 4 = 35
⇒ 5x + 10 = 35
⇒ 5x = 25
∴ x = 5
∴ Highest integer is = x + 4 = 5 + 4 = 9.

১০,০১০.

A man bought some toys at the rate of 10 for Tk. 40 and sold them at 8 for Tk. 35. Find his gain or loss percent.

ক
ক) 10%
খ
খ) 9.5%
গ
গ) (75/8)%
ঘ
ঘ) (71/5)%

ব্যাখ্যা

প্রশ্ন: A man bought some toys at the rate of 10 for Tk. 40 and sold them at 8 for Tk. 35. Find his gain or loss percent.

সমাধান:
Cost price of 10 toys = Tk. 40
∴ CP of 1 toy = Tk. 4

Selling price of 8 toys = Tk. 35
∴ SP of 1 toy = Tk. 35/8

Therefore, Gain = 35/8 – 4 = 3/8.

Gain percent = {(3/8)/4} × 100% =(75/8)%

১০,০১১.

If y{(4x - 2)/4} = y and y ≠ 0, then x = ?

ক
ক) 2/3
খ
খ) 2
গ
গ) 4
ঘ
ঘ) 4/3
ঙ
ঙ) 3/2

ব্যাখ্যা

Question: If y{(4x - 2)/4} = y and y ≠ 0, then x = ?

Solution:
y(4x - 2/4) = y
⇒ 4x - 2/4 = 1
⇒ 16x - 2 = 4
⇒ x = 6/16
∴ x = 3/8

১০,০১২.

A sum of money becomes 4 times itself in 20 years at simple interest. What is the annual interest rate?

ক
10%
খ
15%
গ
20%
ঘ
12.5%

ব্যাখ্যা

Question: A sum of money becomes 4 times itself in 20 years at simple interest. What is the annual interest rate?

Solution:
Let, Principal amount = P
Sum of amount = 4P
∴ Interest, I = 4P - P = 3P
Time, n = 20 years
Rate of interest = r

We know, I = Pnr/100
⇒ Pnr/100 = 3P
⇒ nr/100 = 3
⇒ 20 × r/100 = 3
⇒ 20r = 300
⇒ r = 300/20
⇒ r = 15

∴ The annual interest rate is 15%.

১০,০১৩.

A person can swim in water with a speed of 14 km/hr in still water. If the speed of the stream is 4 km/hr, what will be the time taken by the person to go 90 km downstream?

ক
5 hrs.
খ
7 hrs.
গ
4 hrs.
ঘ
6 hrs.
ঙ
None of these

ব্যাখ্যা

Question: A person can swim in water with a speed of 14 km/hr in still water. If the speed of the stream is 4 km/hr, what will be the time taken by the person to go 90 km downstream?

Solution:
Speed downstream = Speed of Boat in still water + Speed od the stream =
(14 + 4)km/hr.
= 18 km/hr.

∴ Time taken to travel 90 km downstream = Distance/Speed
= (90/18)hrs
= 5 hrs.

১০,০১৪.

If x and y are negative integers, which of the following must be true?
i. x - y > 0
ii. (x/y) > y
iii. x² > y

ক
i only
খ
ii only
গ
iii only
ঘ
i and iii
ঙ
ii and iii

ব্যাখ্যা

Question: If x and y are negative integers, which of the following must be true?
i. x - y > 0
ii. (x/y) > y
iii. x² > y

Solution:
X and Y are -ve integers

(i) x - y > 0
Case 1:
x = - 2 and y = - 3
- 2 + 3 = 1 > 0 (YES)

Case 2:
x = - 3 and y = - 2
- 3 + 2 = - 1 > 0 (NO)
Hence option 1 & option 4 can be ruled out

(ii) (x/y) > y
Case 1:
x = - 2 and y = - 3
(- 2/- 3) > - 3
= 0.66 > - 3 (YES)
LHS will be always positive since numerator and denominator are -ve and RHS is always -ve
Hence it must be true

(III) x² > y
Case 1:
x = - 2 and y = - 3
⇒ (- 2)² > - 3
∴ 4 > - 3
This case is always true because a -ve integer will turn +ve once squared and RHS will be always -ve
Hence it must be true

১০,০১৫.

The average of 2, 7, 6 and x is 5 and the average of 18, 1, 6, x and y is 10. What is the value of y?

ক
ক) 5
খ
খ) 20
গ
গ) 10
ঘ
ঘ) 30

ব্যাখ্যা

Question: The average of 2, 7, 6 and x is 5 and the average of 18, 1, 6, x and y is 10. What is the value of y?

Solution:
Given that
average of 2, 7, 6, x is 5

Therefore,
(2 + 7 + 6 + x)/4 = 5
⇒ 15 + x = 20
⇒ x = 20 - 15
∴ x = 5

Therefore,
(18 + 1 + 6 + x + y)/5 = 10
⇒ (18 + 1 + 6 + 5 + y)/5 = 10
⇒ 30 + y = 50
⇒ y = 50 - 30
∴ y = 20

১০,০১৬.

There were 35 students in a hostel. If the number of students increases by 7, the expenses of the mess increase by Tk. 42 per day while the average expenditure per head diminishes by Tk. 1. Find the original expenditure of the mess?

ক
ক) 400
খ
খ) 420
গ
গ) 440
ঘ
ঘ) 450

ব্যাখ্যা

Suppose the average expenditure was Tk. a.
Then total expenditure = 35a
When 7 more students join the mess, total expenditure = 35a + 42
Now, the average expenditure= (35a + 42)/(35 + 7)

Now, we have,
(35a + 42)/42 = (a - 1)
⇒ 35a + 42 = 42a – 42
⇒ 7a = 84
⇒ a = 12
Thus the original expenditure of the mess = 35 x 12 = 420.

১০,০১৭.

If the sum of 3 consecutive integers is 210, then the sum of the two smaller integer is-

ক
145
খ
154
গ
139
ঘ
213

ব্যাখ্যা

Question: If the sum of 3 consecutive integers is 210, then the sum of the two smaller integer is-

Solution:
Let,
Three consecutive integers is, x - 1 , x, x + 1

ATQ,
x - 1 + x + x + 1 = 210
⇒ 3x = 210
∴ x = 70

The sum of the two smaller integer is = x - 1 + x
= 70 - 1 + 70
= 140 - 1
= 139

১০,০১৮.

If a > 7 and b > - 3 then which of the following is true?

ক
ab > - 21
খ
ab < - 21
গ
- a > 2b
ঘ
None of these

ব্যাখ্যা

Question: If a > 7 and b > - 3 then which of the following is true?

Solution:
Given,
a > 7 and b > - 3
⇒ ab > 7 × (- 3)
⇒ ab > - 21

১০,০১৯.

Three unbiased coins are tossed. What is the probability of getting at least two heads?

ক
ক) 7/8
খ
খ) 3/8
গ
গ) 3/4
ঘ
ঘ) 1/2

ব্যাখ্যা

Question: Three unbiased coins are tossed. What is the probability of getting at least two heads?

Solution:
The events when three unbiased coins are tossed = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
Total number of events 8

The events of getting at least two heads {HHH, HHT, HTH, THH}

Number of expected events = 4

∴ The probability of getting at least two heads is = 4/8 = 1/2

১০,০২০.

If x² = 2, what is the value of (x + 1/x)(x - 1/x)?

ক
ক) 1
খ
খ) 2
গ
গ) 1.5
ঘ
ঘ) 2.5

ব্যাখ্যা

Question: If x² = 2, what is the value of (x + 1/x)(x - 1/x)?

Solution:
Given that,
x² = 2
∴ x = √2

1/x = 1/√2

Now,
(x + 1/x)(x - 1/x)
= x² - (1/x)²
= (√2)² - (1/√2)²
= 2 - (1/2)
= 3/2
= 1.5

১০,০২১.

The sum of two integers is 48 and product (multiplicative result) is 432. What is the smaller number?

ক
48
খ
36
গ
16
ঘ
12

ব্যাখ্যা

Question: The sum of two integers is 48 and their product is 432. What is the smaller number?

Solution:
Let the two integers be x and y.
Then we get,
x + y = 48
y = 48 - x .....(1)

And, xy = 432
⇒ x(48 - x) = 432 ; [From (1)]
⇒ 48x - x² = 432
⇒ x² - 48x + 432 = 0
⇒ x² - 36x - 12x + 432 = 0
⇒ x(x - 36) - 12(x - 36) = 0
⇒ (x - 36)(x - 12) = 0
Now, x - 36 = 0
∴ x = 36
Or, x - 12 = 0
∴ x = 12
Therefore the two numbers are 12 and 36.

∴ The smaller number is 12.

১০,০২২.

If 2 kg of almonds cost as much as 8 kg of walnuts, and the cost of 5 kg of almonds and 16 kg of walnuts is Tk. 1080, what is the cost of almonds per kg?

ক
Tk. 160
খ
Tk. 120
গ
Tk. 150
ঘ
None of these.

ব্যাখ্যা

Question: If 2 kg of almonds cost as much as 8 kg of walnuts, and the cost of 5 kg of almonds and 16 kg of walnuts is Tk. 1080, what is the cost of almonds per kg?

Solution:
Let the cost of almond per kg be Tk. x, and the cost of walnuts per kg be Tk. y.
Now, ATQ,
2x = 8y
or, x = 4y

Now,
5x + 16y = 1080
Or, 5x + 4x = 1080
Or, x = 120

Therefore, the cost price of almond per kg = Tk. 120

১০,০২৩.

A sum of TK. 24,500 amounts to TK. 34,300 in 5 years at a simple interest rate. What is the rate of interest?

ক
ক) 5 %
খ
খ) 5%
গ
গ) 6%
ঘ
ঘ) 8%

ব্যাখ্যা

Here, interest = 34300 - 24500 = 9800 taka
We know, I = pnr
Or, r = (I×100)/pn
= (9800×100) / (24500×5)
= 8%

১০,০২৪.

A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.

ক
35 km/hr
খ
40 km/hr
গ
42 km/hr
ঘ
45 km/hr

ব্যাখ্যা

Question: A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.

Solution:
Given distance=360 km.
Let the speed of the train be x km/hr.
Speed when increased by 5 km/hr = (x + 5) km/hr

(360/x) - {360/(x + 5)} = 1
⇒ [360x + 1800 - 360x]/x(x + 5) =1
⇒ 1800/(x² + 5x) = 1
⇒ x² + 5x = 1800
⇒ x² + 5x - 1800 = 0
⇒ x² + 45x - 40x - 1800 = 0
⇒ x(x + 45) - 40(x + 45) = 0
⇒ (x- 40)(x + 45) = 0
∴ x = 40, - 45

The speed of the train is 40 km/hr.

১০,০২৫.

If x⁴- 5x² + 1 = 0, what is the value of x² + 1/x²?

ক
ক) 5
খ
খ) 0
গ
গ) 2
ঘ
ঘ) 3

ব্যাখ্যা

Question: If x⁴- 5x² + 1 = 0, what is the value of x² + 1/x²?

Given that
x⁴ - 5x² + 1 = 0
x⁴ + 1 = 5x²
x⁴/x² + 1/x² = 5x²/x²
x² + 1/x² = 5

১০,০২৬.

If sinθ = 1/√2 then, tanθ =?

ক
0
খ
1
গ
1/√3
ঘ
√3

ব্যাখ্যা

প্রশ্ন: If sinθ = 1/√2 then, tanθ =?

সমাধান:
sinθ = 1/√2
⇒ sinθ = sin45°
∴ θ = 45°

∴ tanθ
= tan45°
= 1

১০,০২৭.

A man travels for 2 hours at 35 miles an hour and he covers 65 miles in the next 3 hours. What is the average speed per hour for the entire trip?

ক
ক) 22
খ
খ) 24
গ
গ) 26
ঘ
ঘ) 27

ব্যাখ্যা

লোকটি প্রথম 2 ঘণ্টায় অতিক্রম করে 35 × 2 মাইল
= 70 মাইল

লোকটি পরবর্তী 3 ঘণ্টায় অতিক্রম করে = 65 মাইল

গড় গতিবেগ = (70 + 65)/(2 + 3) মাইল
= 135/5
= 27 মাইল

১০,০২৮.

Three gentlemen and two ladies are candidates for two vacancies. A voter has to vote for two candidates. In how many ways can one cast his vote?

ক
ক) 5
খ
খ) 10
গ
গ) 12
ঘ
ঘ) 15

ব্যাখ্যা

Question: Three gentlemen and two ladies are candidates for two vacancies. A voter has to vote for two candidates. In how many ways can one cast his vote?

Solution:
ways one can cast his vote = ⁵C₂
= 5!/2! 3!
= 10

১০,০২৯.

Three numbers which are co-prime to each other are such that the product of the first two is 551 and that of the last two is 1073. The sum of the three numbers is-

ক
75
খ
81
গ
91
ঘ
85
ঙ
89

ব্যাখ্যা

Question: Three numbers which are co-prime to each other are such that the product of the first two is 551 and that of the last two is 1073. The sum of the three numbers is-

Solution:
Let the three numbers be a, b, and c.

Given that,
a × b = 551 = 19 × 29 ...… (1) ; [both 19 and 29 are prime numbers]
∴ a = 19, b = 29

And,
b × c = 1073 = 29 × 37 ...… (2) ; [both 29 and 37 are prime numbers]
∴ b = 29, c = 37

∴ a : b : c = 19 : 29 : 37
So the three numbers are 19, 29, 37

∴ sum = 19 + 29 + 37 = 85

১০,০৩০.

The father explained to his son, “When you were born, I had reached the age you are now.” If the father is 52, what was the son’s age five years ago?

ক
18 years
খ
21 years
গ
23 years
ঘ
19 years
ঙ
None

ব্যাখ্যা

Question: The father explained to his son, “When you were born, I had reached the age you are now.” If the father is 52, what was the son’s age five years ago?

Solution:
let, at present son is x years old
so, at time of his birth the father was x years old

∴ at present the age of father is = x + x year
= 2x years

2x = 52
⇒ x = 52/2
= 26 years

∴ the son's age five years back was = 26 - 5
= 21 years

১০,০৩১.

A does half as much as work as B in 1/6 of the time taken by B. Together they can complete the work in 10 days. B alone can complete the work in how many days?

ক
20 days
খ
30 days
গ
40 days
ঘ
50 days

ব্যাখ্যা

Let B does 1 work in x days
Then, A does [1/2] work in x/6 days
To do 1 unit work, A will take x/3 days

That means the time ratio of A: B = x: x/3
Or time ratio = 1: 1/3
Or time ratio = 3: 1

As we know that the efficiency ratio is inversely proportional to the time ratio

i.e., Efficiency ratio of A: B = 1: 3

ATQ,

(A + B) = 10 days
Total work = days × efficiency of (A + B)
Or, total work = 10 × 4 = 40 units
i.e., B can finish the work in = total work/ efficiency of B

So, B can finish the work in 40/1 = 40 days.

১০,০৩২.

Rifat ranks 13th in the class of 33 students. There are 5 students below Sifat rankwise. How many students are there between Rifat and Sifat?

ক
ক) 12
খ
খ) 13
গ
গ) 14
ঘ
ঘ) 15

ব্যাখ্যা

Question: Rifat ranks 13th in the class of 33 students. There are 5 students below Sifat rankwise. How many students are there between Rifat and Sifat?

Solution:
মোট শিক্ষার্থী = ৩৩ জন
রিফাত তাদের মধ্যে ১৩তম

সিফাতের পিছনে ৫ জন আছে।
অর্থাৎ সিফাত শেষ থেকে ৬ষ্ঠ।

তাই, রিফাত ও সিফাতের মধ্যে আছে = ৩৩ - (১৩ + ৬) = ১৪

১০,০৩৩.

A monkey climbs a 60 m high pole. In the first minute, he climbs 6 m and slips down 3 m in the next minute. How much time is required by it to reach the top?

ক
25 minutes
খ
33 minutes
গ
35 minutes
ঘ
37 minutes
ঙ
None of the above

ব্যাখ্যা

Question: A monkey climbs a 60 m high pole. In the first minute, he climbs 6 m and slips down 3 m in the next minute. How much time is required by it to reach the top?

Solution:
According to the question,
For 1^st min, he climbs 6 m and comes down 3 m in 2^nd min.
Effectively in 2 min, he climbs 3 m.

In 1 min, he climbs (3/2) m

So, in 36 mins he climbs = (3/2) × 36 = 54 m

So, in the 37^th minute, he climbs the next 6 m.

∴ The required value is 37 minutes.

১০,০৩৪.

x = - y - z হলে (1/3)(x³ + y³ + z³) এর মান কত?

ক
xyz/3
খ
3xyz
গ
xyz
ঘ
1
ঙ
কোনটিই নয়

ব্যাখ্যা

প্রশ্ন: x = - y - z হলে 1/3(x³ + y³ + z³) এর মান কত?

সমাধান:
দেওয়া আছে,
x = - y - z
x + y + z = 0

প্রদত্ত রাশি = 1/3(x³ + y³ + z³)
= 1/3(x³ + y³ + z³ - 3xyz + 3xyz)
= 1/3(x³ + y³ + z³ - 3xyz) + (1/3) . 3xyz
= 1/3 {(x + y + z) (x² + y² + z² - xy - yz - zx)} + xyz
= 1/3 {0 . (x² + y² + z² - xy - yz - zx)} + xyz
= xyz

১০,০৩৫.

If a sequence of 8 consecutive odd integers with increasing values has 9 as its 7th term, what is the sum of the terms of the sequence?

ক
22
খ
32
গ
36
ঘ
40
ঙ
44

ব্যাখ্যা

Question: If a sequence of 8 consecutive odd integers with increasing values has 9 as its 7th term, what is the sum of the terms of the sequence?

Solution:
Seventh term, a₇ = 9
Number of terms, n = 8
a_n = a₁ + (n - 1)d
a₇ = a₁ + 6d
⇒ 9 = a₁ + 12
⇒ a₁ = - 3

8th term , a₈ = 11

Sum of terms of the sequence = (n/2)[2a + (n - 1)d]
= (8/2)[2 × (- 3) + (8 - 1) × 2]
= 4 × [- 6 + 14]
= 4 × 8
= 32

১০,০৩৬.

A train passes a station platform in 42 seconds and a man standing on the platform in 20 seconds. If the speed of the train is 72 km/hr, what is the length of the platform?

ক
360 m
খ
420 m
গ
440 m
ঘ
380 m
ঙ
None

ব্যাখ্যা

Question: A train passes a station platform in 42 seconds and a man standing on the platform in 20 seconds. If the speed of the train is 72 km/hr, what is the length of the platform?

Solution:
ট্রেনটি 3600 সেকেন্ডে অতিক্রম করে 72000 মিটার
ট্রেনটি 1 সেকেন্ডে অতিক্রম করে 72000/3600 মিটার
ট্রেনটি 20 সেকেন্ডে অতিক্রম করে (72000 × 20)/3600 মিটার
= 400 মিটার

ট্রেনের দৈর্ঘ্য = 400 মিটার
আবার
ট্রেনটি 3600 সেকেন্ডে অতিক্রম করে 72000 মিটার
ট্রেনটি 1 সেকেন্ডে অতিক্রম করে 72000/3600 মিটার
ট্রেনটি 42 সেকেন্ডে অতিক্রম করে (72000 × 42)/3600 মিটার
= 840 মিটার

প্লাটফর্মের দৈর্ঘ্য = (840 - 400) = 440 মিটার

১০,০৩৭.

A student has bought x pencils at Tk. 5 each and (x + 4) notebooks at Tk. 8 each. If the total cost does not exceed Tk. 97, what is the maximum number of notebooks he has bought?

ক
10
খ
14
গ
9
ঘ
12

ব্যাখ্যা

Question: A student has bought x pencils at Tk. 5 each and (x + 4) notebooks at Tk. 8 each. If the total cost does not exceed Tk. 97, what is the maximum number of notebooks he has bought?

Solution:
A student has bought x pencils at Tk. 5 each and (x + 4) notebooks at Tk. 8 each.
Total cost = 5x + 8(x + 4)
= 5x + 8x + 32
= 13x + 32

So, 13x + 32 ≤ 97
⇒ 13x ≤ 97 - 32
⇒ 13x ≤ 65
⇒ x ≤ 5
⇒ x + 4 ≤ 5 + 4
∴ x + 4 ≤ 9

So, the maximum number of notebooks he has bought is 9

১০,০৩৮.

In a school having roll strength 286, the ratio of boys and girls is 8 : 5. If 22 more girls get admitted into the school, the ratio of boys and girls becomes

ক
12 : 7
খ
4 : 3
গ
10 : 7
ঘ
8 : 7

ব্যাখ্যা

Question: In a school having roll strength 286, the ratio of boys and girls is 8 : 5. If 22 more girls get admitted into the school, the ratio of boys and girls becomes

Solution:
Boys : girls = 8 : 5
Let, the boys = 8x, girl = 5x

According to the 1st condition,
8x + 5x = 286
⇒ 13x = 286
⇒ x = 286/13
∴ x = 22
Boys = 8 × 22 = 176 and girls = 5 × 2 = 110

22 more girls get admitted then the number of girls become
(5x + 22) = 110 + 22 = 132

Now, new ratio of boys and girls = 176 : 132 = 4 : 3

১০,০৩৯.

The price of a pen is 25% more than the price of a book. The price of a pen holder is 50% more than the price of the book. How much is the price of the pen holder more than the price of the pen?

ক
ক) 20%
খ
খ) 25%
গ
গ) 50%
ঘ
ঘ) 37.5%

ব্যাখ্যা

ধরি
বইয়ের দাম = ১০০ টাকা
কলমের দাম = ১০০ + ২৫ টাকা
= ১২৫ টাকা

কলমদানির দাম = ১২৫ + ২৫ = ১৫০ টাকা

কলম থেকে কলমদানির দাম = (১৫০ - ১২৫) টাকা
= ২৫ টাকা

কলম থেকে কলম দানি শতকরা বেশি =(২৫ × ১০০)/১২৫
= ২০ টাকা

১০,০৪০.

Fresh fruit contains 68% water and dry fruit contains 20% water. How much dry fruit can be obtained from 100 kg of fresh fruits?

ক
20 kg
খ
30 kg
গ
40 kg
ঘ
50 kg

ব্যাখ্যা

Question: Fresh fruit contains 68% water and dry fruit contains 20% water. How much dry fruit can be obtained from 100 kg of fresh fruits?

Solution:
The fruit content in both the fresh fruit and dry fruit is the same.

Given, fresh fruit has 68% water. So remaining 32% is fruit content. Weight of fresh fruits is 100kg

Dry fruit has 20% water. So remaining 80% is fruit content. Let weight of dry fruit be y kg.

Fruit % in freshfruit = Fruit% in dryfruit
(32/100) × 100 = (80/100 ) × y
⇒ 32 = (8/10) × y
⇒ y = (32 × 10)/8
∴ y = 40

We get, y = 40 kg

১০,০৪১.

If 0 ≤ x ≤ 4 and y < 12, which of the following can not be the value of xy?

ক
- 2
খ
0
গ
6
ঘ
24
ঙ
48

ব্যাখ্যা

Question: If 0 ≤ x ≤ 4 and y < 12, which of the following can not be the value of xy?

Solution:
If,
x = 1, y = - 2, then xy = - 2, this one is possible.
x = 0, y = 1, then xy = 0, this one is possible.
x = 1, y = 6, then xy = 6, this one is possible.
x = 4, y = 6, then xy = 24, this one is possible.

The maximum value of x is 4 and the maximum value of y is less than 12 hence xy cannot be 4 × 12=48, notice that (- 4) × (- 12) = 48 is also impossible as x cannot be less than 0.

১০,০৪২.

A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km downstream. Determine the speed of the stream and that of the boating till water.

ক
ক) 3 km/hr
খ
খ) 4 km/hr
গ
গ) 8 km/hr
ঘ
ঘ) 6 km/hr

ব্যাখ্যা

Let,
the speed of the stream be = u kmph
the speed of the boat be = v kmph
speed upstream will be = v - u kmph
speed downstream will be = v + u kmph

30 km upstream in time duration = 30 / (v - u) hrs
44 km downstream in time duration = 44 / (v + u) hrs
44/(v + u) + 30/(v - u)
= 10 hrs .........(i)

Similarly,
40 /(v - u) + 55 / (v + u) = 13 hrs Multiply with 3/4:
30/(v - u) + 165 / 4(v + u) = 39/4 .......(ii)

Now (i) - (ii)
⇒ [44 - 165/4] / (v + u) = 10 - 39/4 = 1/4
⇒ v + u = 11 ..........(iii)
Substitute this in (1) to get: 44/11 + 30/(v - u) = 10
⇒ v - u = 30/6 = 5.........(iv)

Solving (iii) and (iv) ,
we get :
v = 8 kmph and
u = 3 kmph

১০,০৪৩.

Which of the following numbers will replace the question mark (?) in the given series?
?, 11, 35, 107, 323

ক
3
খ
4
গ
5
ঘ
8

ব্যাখ্যা

Question: Which of the following numbers will replace the question mark (?) in the given series?
?, 11, 35, 107, 323

Solution:
?, 11, 35, 107, 323

(11 × 3) + 2 = 33 + 2 = 35
(35 × 3) + 2 = 105 + 2 = 107
(107 × 3) + 2 = 321 + 2 = 323

∴ (? × 3) + 2 = 11
⇒ ? × 3 = 9
∴ ? = 3

১০,০৪৪.

Find the value of x, if log(x + 1) + log(x - 1) = 3log2

ক
3
খ
2
গ
4
ঘ
1/2

ব্যাখ্যা

Question: Find the value of x, if log(x + 1) + log(x - 1) = 3log2

Solution:
Given,
⇒ log(x + 1) + log(x - 1) = 3log2
⇒ log{(x + 1)(x - 1)} = log2³
⇒ log(x² - 1) = log8
⇒ x² - 1 = 8
⇒ x² = 8 + 1
⇒ x² = 9
⇒ x = ± √9
⇒ x = ± 3
∴ x = 3 ; [Only positive value]

১০,০৪৫.

What is the average of the sum of the first 11 logical terms of the Fibonacci series if the series starts with zero?

ক
7
খ
9
গ
15
ঘ
17
ঙ
None of the above

ব্যাখ্যা

Question: What is the average of the sum of the first 11 logical terms of the Fibonacci series if the series starts with zero?

Solution:
The first 9 logical terms of the Fibonacci series if the series starts with zero = 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55
So, the average = (0 + 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55)/11
= 143/11
= 13

১০,০৪৬.

The sides of a triangle are consecutive integers. The perimeter of the triangle is 120 cm. Find the length of the greatest side:

ক
21 cm
খ
31 cm
গ
41 cm
ঘ
51 cm

ব্যাখ্যা

Question: The sides of a triangle are consecutive integers. The perimeter of the triangle is 120 cm. Find the length of the greatest side:

Solution:
Let the sides of the triangles be x cm, (x + 1) cm and (x + 2) cm respectively.
Then,
x + (x + 1) + (x + 2) = 120
⇒ 3x + 3 = 120
⇒ 3x = 117
⇒ x = 39

∴ Length of the greatest side
= (39 + 2) cm
= 41 cm

১০,০৪৭.

A man deposits Tk. 10000 in a Bank at 10% interest rate compounded annually. At the end of the second year, the total amount(nearest) including interest will become?

ক
Tk. 15000
খ
Tk. 11325
গ
Tk. 10100
ঘ
Tk. 18300
ঙ
Tk. 12100

ব্যাখ্যা

Question: A man deposits Tk. 10000 in a Bank at 10% interest rate compounded annually. At the end of the second year, the total amount(nearest) including interest will become?

Solution:
Given,
Principal, P = 10000 Tk.
Rate of interest, r = 10% = 10/100 = 0.1
Time, n = 2 years

We know,
Compound Amount = P(1 + r)ⁿ
= 10000 × (1 + 0.1)²
= 10000 × (1.1)²
= 10000 × 1.21
= 10000 × (19,683/15,625)
= 12100

১০,০৪৮.

The average of 10 numbers is 12, and the average of another 15 numbers is 18. What is the average of all 25 numbers combined?

ক
14
খ
15.6
গ
16
ঘ
18.5

ব্যাখ্যা

Question: The average of 10 numbers is 12, and the average of another 15 numbers is 18. What is the average of all 25 numbers combined?

Solution:
10 টি সংখ্যার গড় = 12
∴10 টি সংখ্যার সমষ্টি = 10 × 12 = 120

15 টি সংখ্যার গড় = 18
∴ 15 টি সংখ্যার সমষ্টি = 15 × 18 = 270

মোট সমষ্টি = 120 + 270 = 390
মোট সংখ্যা = 10 + 15 = 25
সুতরাং, সম্মিলিত গড় = 390/25 = 15.6

১০,০৪৯.

Five years ago, Rahul was twice as old as his sister. Now, Rahul is 6 years older than his sister. What is the present age of Rahul?

ক
22 years
খ
19 years
গ
21 years
ঘ
15 years
ঙ
17 years

ব্যাখ্যা

Question: Five years ago, Rahul was twice as old as his sister. Now, Rahul is 6 years older than his sister. What is the present age of Rahul?

Solution:
Let,
Present age of Rahul's sister = x years
Present age of Rahul = x + 6 years (since Rahul is 6 years older)

Five years ago,
Rahul’s age = (x + 6) - 5 = x + 1
And sister’s age = x - 5

According to the question,
x + 1 = 2(x - 5)
⇒ x + 1 = 2x - 10
⇒ 2x - x = 10 + 1
∴ x = 11

∴ Rahul's present age = x + 6 = 11 + 6 = 17 years.

১০,০৫০.

What is the solution of the inequality,
- 12 < 4x - 8 ≤ 20 ?

ক
[- 1, 7]
খ
[- 1, 8)
গ
(- 1, 7]
ঘ
[- 3, 8]

ব্যাখ্যা

Question: What is the solution of the inequality,
- 12 < 4x - 8 ≤ 20 ?

Solution:
- 12 < 4x - 8 ≤ 20
⇒ - 12 + 8 < 4x - 8 + 8 ≤ 20 + 8
⇒ - 4 < 4x ≤ 28
⇒ - 4/4 < 4x/4 ≤ 28/4
⇒ - 1 < x ≤ 7

∴ solution of the inequality: (- 1, 7]

১০,০৫১.

If the sum of the first n positive integers is S, what is the sum of the first n positive even integers, in terms of S?

ক
S/2
খ
S
গ
2S
ঘ
2S + 2
ঙ
4S

ব্যাখ্যা

Question: If the sum of the first n positive integers is S, what is the sum of the first n positive even integers, in terms of S?

Solution:
Sum of firs n natural numbers = n(n+1)/2 = S
Sum of first n even natural numbers = n(n+1) = 2S

1 + 2 + 3 + ... + n = S
⇒ 2 + 4 + 6 + ... + 2n = 2(1 + 2 + 3 + ... + n) = 2S

১০,০৫২.

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

ক
600
খ
500
গ
810
ঘ
300

ব্যাখ্যা

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

Solution:
Minimum pass percentage for Boys = 60%
Minimum pass percentage for girls = 45%

A boy gets 767 and failed by 313
∴ Passing marks for boys= 767 + 313 = 1080

Now
60% = 1080
⇒ 1% = 1080/60
⇒ 45% = (1080 × 45)/60
= 810

১০,০৫৩.

A jar contains white, red and green marbles in the ratio of 2 : 3 : 5. Six more green marbles are added to the jars, and then the ratio becomes 2 : 3 : 7. How many white marbles are there in the jar?

ক
ক) 2
খ
খ) 3
গ
গ) 4
ঘ
ঘ) 5
ঙ
ঙ) 6

ব্যাখ্যা

W : R : G = 2 : 3 : 5 [Initial]
W : R : G = 2 : 3 : 7 [Final]
Let, Initial Green = 5x Final Green = 7x
ATQ,
(7x - 5x) = 6
=> 2x = 6
=> x = 3
So, white marbles in the jar = (2 × 3) = 6

১০,০৫৪.

The marked price of a watch was Tk. 720. A man bought the same for Tk. 550.80 after getting two successive discounts. the first being 10%. then the second discount rate is-

ক
25%
খ
20%
গ
15%
ঘ
12%

ব্যাখ্যা

Question: The marked price of a watch was Tk. 720. A man bought the same for Tk. 550.80 after getting two successive discounts. the first being 10%. then the second discount rate is-

Solution:
Marked price = 720
Actual price = 550.80
First discount = 10%
Let the second discount be x%
Then, we can write
⇒ 720(1 - 0.1)(1 - x/100) = 550.80
⇒ 720 × 0.9(1 - 0.01x) = 550.8
⇒ 648(1 - x/100) = 550.8
⇒ 1 - x/100 = 550.8/648
⇒ x/100 = 1 - 0.85
⇒ x/100 = 0.15
⇒ x = 0.15 × 100
∴ x = 15

∴ Second discount = 15%

১০,০৫৫.

যদি m একটি বিজোড় স্বাভাবিক সংখ্যা হয়, নিচের কোনটি অবশ্যই একটি জোড় স্বাভাবিক সংখ্যা হবে?

ক
m + ১
খ
m²
গ
৩m
ঘ
m² + m + ১
ঙ
কোনোটিই নয়

ব্যাখ্যা

প্রশ্ন: যদি m একটি বিজোড় স্বাভাবিক সংখ্যা হয়, নিচের কোনটি অবশ্যই একটি জোড় স্বাভাবিক সংখ্যা হবে?

সমাধান:
ধরি, m = ৩ (বিজোড় সংখ্যা)

ক) m + ১ = ৩ + ১ = ৪ (জোড়)
খ) m² = ৩² = ৯ (বিজোড়)
গ) ৩m = ৩ × ৩ = ৯ (বিজোড়)
ঘ) m² + m + ১ = ৯ + ৩ + ১ = ১৩ (বিজোড়)

১০,০৫৬.

A shopkeeper marks the price of an article at TK. 120. What will be the selling price. if he allows two successive discounts of 5% each?

ক
ক) Tk. 104.30
খ
খ) Tk. 106.30
গ
গ) Tk. 108.30
ঘ
ঘ) Tk. 105.30

ব্যাখ্যা

প্রশ্ন: A shopkeeper marks the price of an article at TK. 120. What will be the selling price. if he allows two successive discounts of 5% each?

সমাধান:
At 5% discount for the first time = (120 × 95)/100 = Tk.114

At 5% discount for second time = (114 × 95)/100 = Tk. 108.30

১০,০৫৭.

ক
ক) 14
খ
খ) 16
গ
গ) 18
ঘ
ঘ) 19

১০,০৫৮.

A sum of Tk. 30,000 yields a compound interest of Tk. 4347 when invested at 7% per annum. What is the investment period in years?

ক
2 years
খ
4 years
গ
3 years
ঘ
5 years

ব্যাখ্যা

Question: A sum of Tk. 30,000 yields a compound interest of Tk. 4347 when invested at 7% per annum. What is the investment period in years?

Solution:
Given,
Principal, P = 30000
Rate, r = 7% per annum
Compound Interest, CI = 4347

We know,
Amount, A = P + CI = 30000 + 4347 = 34347

Using the compound amount formula:
A = P(1 + r/100)ⁿ
⇒ 34347 = 30000 × (1 + 7/100)ⁿ
⇒ 34347 = 30000 × (107/100)ⁿ
⇒ (107/100)ⁿ = 34347/30000
⇒ (1.07)ⁿ = 1.1449
⇒ (1.07)ⁿ = (1.07)²∴ n = 2 years

১০,০৫৯.

30% of a number when subtracted from 91, gives the number itself. Find the number.

ক
75
খ
70
গ
65
ঘ
60

ব্যাখ্যা

Question: 30% of a number when subtracted from 91, gives the number itself. Find the number.

Solution:
Let the number be x

According to the question,
91 - (30x/100) = x
⇒ 9100 - 30x = 100x
⇒ 9100 = 130x
⇒ x = 9100/130
∴ x = 70.

১০,০৬০.

A dealer marks the price of all the goods at 30% above the cost price and assumes that he will make a profit of 15% if he offers a discount of 15%. Find what will be his actual profit on sales?

ক
ক) 15%
খ
খ) 30%
গ
গ) 12.50%
ঘ
ঘ) 10.50%

ব্যাখ্যা

Let cost-price goods be Tk. 100
Marked price (Selling Price) marked by the shopkeeper on goods = Tk. 130
He sells the goods at a discount of 15%

Therefore,
Selling price = 85% of Tk. 130
= Tk. 110.50
Gain = S.P. – C.P.
= 110.5 – 100
= 10.50%

১০,০৬১.

The expense of 9 pens and 5 pencils is the same as the expense of 7 pens and 8 pencils. What is the ratio between the price of one pen and one pencil?

ক
5 : 2
খ
3 : 2
গ
1 : 2
ঘ
3 : 1

ব্যাখ্যা

Question: The expense of 9 pens and 5 pencils is the same as the expense of 7 pens and 8 pencils. What is the ratio between the price of one pen and one pencil?

Solution:
Let,
The price of one pen X Tk.
The price of one pencil Y Tk.

ATQ,
9X + 5Y = 7X + 8Y
⇒ 9X - 7X = 8Y - 5Y
⇒ 2X = 3Y
⇒ X/Y = 3/2
∴ X : Y = 3 : 2

১০,০৬২.

If England is written as 1234526 and France is written as 785291, then Greece is written as-

ক
352293
খ
293522
গ
381191
ঘ
130429

ব্যাখ্যা

Question: If England is written as 1234526 and France is written as 785291, then Greece is written as-

Solution:
England is written as 1234526
E = 1
n = 2
g = 3
l = 4
a = 5
n = 2
d = 6

France is written as 785291
F = 7
r = 8
a = 5
n = 2
c = 9
e = 1

So, GREECE will be = 381191

১০,০৬৩.

A map has a scale of 1 cm to 3 km, what length on actual ground does a 3 cm length on the map represent?

ক
5 km
খ
9 km
গ
12 km
ঘ
None of these

ব্যাখ্যা

Question: A map has a scale of 1 cm to 3 km, what length on actual ground does a 3 cm length on the map represent?

Solution:
Given that,
1 cm represents 3 km
∴ 3 cm represent (3 × 3) km
= 9 km

১০,০৬৪.

If 20 boys or 10 men can make 20 cricket bats in 10 days, then how many cricket bats will be made by 4 boys and 8 men in 20 days?

ক
20 cricket bats
খ
30 cricket bats
গ
40 cricket bats
ঘ
60 cricket bats

ব্যাখ্যা

Question: If 20 boys or 10 men can make 20 cricket bats in 10 days, then how many cricket bats will be made by 4 boys and 8 men in 20 days?

Solution:
Here,
10 men = 20 boys
8 men = (2 × 8) boys = 16 boys

If 20 boys can make 20 cricket bats in 10 days,
8 men and 4 boys or, (16 + 4) = 20 boys can make in 20 days = (20 × 2) cricket bats
= 40 cricket bats

১০,০৬৫.

Two brands of detergent are to be combined. Detergent A contains 30 percent bleach and 70 percent soap, while Detergent B contains 50 percent bleach and 50 percent soap. If the combined mixture is to be 40 percent bleach, what percent of the final mixture should be Detergent A?

ক
30%
খ
40%
গ
50%
ঘ
60%

ব্যাখ্যা

Question: Two brands of detergent are to be combined. Detergent A contains 30 percent bleach and 70 percent soap, while Detergent B contains 50 percent bleach and 50 percent soap. If the combined mixture is to be 40 percent bleach, what percent of the final mixture should be Detergent A?

Solution:
ধরি, মিশ্রণে Detergent A এর অংশ x%
∴ Detergent B এর অংশ হবে (100 - x%)

এখানে,
ডিটারজেন্ট A থেকে প্রাপ্ত ব্লিচের পরিমাণ হলো (x% এর 30%)।
ডিটারজেন্ট B থেকে প্রাপ্ত ব্লিচের পরিমাণ হলো (100 - x)% এর 50%।
চূড়ান্ত মিশ্রণে ব্লিচের মোট পরিমাণ হলো (100% এর 40%)।

প্রশ্নমতে,
0.30x + 0.50 × (100 - x) = 0.40 × 100
⇒ 0.30x + 50 - 0.50x = 40
⇒ - 0.20x = - 10
⇒ x = - 10/(- 0.20)
∴ x = 50

অর্থাৎ, মিশ্রণের 50% হবে Detergent A

১০,০৬৬.

A sum of Tk. 1360 has been divided among A, B and C such that A gets 2/3 of what B gets and B gets 1/4 of what C gets. B's share is?

ক
Tk. 420
খ
Tk. 280
গ
Tk. 240
ঘ
Tk. 350

ব্যাখ্যা

Question: A sum of Tk. 1360 has been divided among A, B and C such that A gets 2/3 of what B gets and B gets 1/4 of what C gets. B's share is?

Solution:
Given that,
Total amount = Tk. 1360
A gets 2/3 of what B gets ⇒ A = 2B/3
B gets 1/4 of what C gets ⇒ B = C/4 ⇒ C = 4B

ATQ,
A + B + C = 1360
⇒ (2B/3) + B + 4B = 1360
⇒ (2B + 3B + 12B)/3 = 1360
⇒ 17B/3 = 1360
⇒ B = (1360 × 3)/17
∴ B = 240

∴ B’s share = Tk. 240

১০,০৬৭.

If ratio of three numbers is 3 : 2 : 5 and their sum is 150, what is the largest number?

ক
ক) 30
খ
খ) 45
গ
গ) 60
ঘ
ঘ) 75

ব্যাখ্যা

Question: If ratio of three numbers is 3 : 2 : 5 and their sum is 150, what is the largest number?

Solution:
let the numbers be 3x, 2x, 5x

3x + 2x + 5x = 150
⇒ 10x = 150
∴ x = 15

largest number = 5 × 15
= 75

১০,০৬৮.

The difference between the greatest and least prime numbers which are less than 70 is-

ক
69
খ
67
গ
65
ঘ
64

ব্যাখ্যা

Question: The difference between the greatest and least prime numbers which are less than 70 is-

Solution:
Greatest prime number (less than 70) = 67
Least prime number = 2
So, their difference = 67 - 2 = 65

১০,০৬৯.

The average of five consecutive numbers is x. If the next two numbers are also included, how shall the average vary?

ক
ক) The average remains the same
খ
খ) The average increased by 1
গ
গ) The average decreased by 1
ঘ
ঘ) The average increased by 1.4

ব্যাখ্যা

Question: The average of five consecutive numbers is x. If the next two numbers are also included, how shall the average vary?

Solution:
Let the five consecutive numbers be 5, 6, 7, 8, and 9 respectively.

So, the average is = (5 + 6 + 7 + 8 + 9)/5 = 35/5 = 7
Suppose, the average is, x = 7

If the next two numbers are added, then average is = (5 + 6 + 7 + 8 + 9 + 10 + 11)/7 = 56/7 = 8
So, the new average is, x = 7 + 1

So, the average increased by 1

১০,০৭০.

A small company employs 3 men and 5 women. If a team of 4 employees is to be randomly selected to organize the company retreat, what is the probability that the team will have exactly 3 women?

ক
ক) 1/7
খ
খ) 3/7
গ
গ) 3/8
ঘ
ঘ) 5/7

ব্যাখ্যা

Question: A small company employs 3 men and 5 women. If a team of 4 employees is to be randomly selected to organize the company retreat, what is the probability that the team will have exactly 3 women?

Solution:
A small company employs 3 men and 5 women.
Total people = 8

ways of selecting 4 people from 8 = ⁸C₄
= 70

ways of selecting 3 women from 5 = ⁵C₃
ways of selecting 1 men from 3 = ³C₁

∴ probability = (⁵C₃ × ³C₁)/ 70
= (10 × 3)/70
= 3/7

১০,০৭১.

A can complete a certain work in 4 minutes, B in 5 minutes, C in 6 minutes, D in 10 minutes and E in 12 minutes. The average number of units of work completed by them per minute will be ______

ক
ক) 0.40
খ
খ) 0.16
গ
গ) 0.80
ঘ
ঘ) None of these

ব্যাখ্যা

Question: A can complete a certain work in 4 minutes, B in 5 minutes, C in 6 minutes, D in 10 minutes and E in 12 minutes. The average number of units of work completed by them per minute will be ______

Solution:
A 1 মিনিটে করতে পারে = 1/4 অংশ
B 1 মিনিটে করতে পারে = 1/5 অংশ
C 1 মিনিটে করতে পারে = 1/6 অংশ
D 1 মিনিটে করতে পারে = 1/10 অংশ
E 1 মিনিটে করতে পারে = 1/12 অংশ

প্রতি মিনিটে তারা গড়ে করে = (1/4 + 1/5 + 1/6 + 1/10 + 1/12)/5
= (48/60) × (1/5)
= 4/25
= 0.16 অংশ

১০,০৭২.

The average of five consecutive integers is P. If the next two numbers are added, how shall the average vary?

ক
increase by 3
খ
increase by 1
গ
increase by 2
ঘ
remain the same

ব্যাখ্যা

Question: The average of five consecutive integers is P. If the next two numbers are added, how shall the average vary?

Solution:
Let the five consecutive numbers be 5, 6, 7, 8, and 9 respectively.

So, the average is = (5 + 6 + 7 + 8 + 9)/5 = 35/5 = 7
Suppose, the average is, P = 7

If the next two numbers are added, then average is = (5 + 6 + 7 + 8 + 9 + 10 + 11)/7 = 56/7 = 8
So, the new average is, P = 7 + 1

So, the average increased by 1

১০,০৭৩.

For what value of x is 8^{2x - 4} = 16^x?

ক
5
খ
3
গ
4
ঘ
6

ব্যাখ্যা

Question: For what value of x is 8^{2x - 4} = 16^x?

Solution:
8^{2x - 4} = 16^x
⇒ (2³)^{2x - 4} = 2^4x
⇒ 2 ^{6x - 12} = 2^4x
⇒ 6x - 12 = 4x
⇒ 6x - 4x = 12
⇒ 2x = 12
⇒ x = 12/2
∴ x = 6

১০,০৭৪.

A man can do a work in 20 days and a woman in 15 days. If they work on it together for 5 days, then the fraction of the work that is left-

ক
8/15
খ
1/12
গ
5/12
ঘ
None of these

ব্যাখ্যা

Question: A man can do a work in 20 days and a woman in 15 days. If they work on it together for 5 days, then the fraction of the work that is left-

Solution:
Man's 1 day's work = 1/20
Woman's 1 day's work = 1/15

∴ (Man + woman)'s 1 day's work = (1/20) + (1/15) = (3 + 4)/60 = 7/60
∴ (Man + woman)'s 5 day's work = (7/60) × 5 = 7/12

Thus, Remaining work = 1 - (7/12) = (12 - 7)/12 = 5/12

∴ The fraction of the work that is left = 5/12

১০,০৭৫.

The average of first five multiples of 3 is -

ক
ক) 9
খ
খ) 12
গ
গ) 3
ঘ
ঘ) 15

ব্যাখ্যা

Question: The average of first five multiples of 3 is -

Solution:
First five multiples of 3 are = 3, 6, 9, 12, 15.

∴ Average = (3 + 6 + 9 + 12 + 15)/5
= 45/5
= 9

১০,০৭৬.

400 grams of sugar solution has 30% sugar in it. How much sugar should be added to make 50% in the solution?

ক
ক) 160 gm
খ
খ) 180 gm
গ
গ) 150 gm
ঘ
ঘ) 140 gm

ব্যাখ্যা

Question: 400 grams of sugar solution has 30% sugar in it. How much sugar should be added to make 50% in the solution?

Solution:
Sugar = 400 × 30% = 120 grams
Let,
We have to add x grams

ATQ,
120 + x = 50% (400 + x)
⇒ 120 + x = (50/100) × (400 + x)
⇒ 120 + x = (1/2) × (400 + x)
⇒ 240 + 2x = 400 + x
∴ x = 160

১০,০৭৭.

Ms. Rahman's monthly salary is Tk. 8000.00 and she gets 15% commission on her salary. If her salary increased by 20%, by what percent will her commission increase?

ক
10%
খ
20%
গ
5%
ঘ
15%

ব্যাখ্যা

Question: Ms. Rahman's monthly salary is Tk. 8000.00 and she gets 15% commission on her salary. If her salary increased by 20%, by what percent will her commission increase?

Solution:
পূর্বের কমিশন = 8000 এর 15% টাকা
= 8000 × 15/100
= 1200 টাকা

২০% বৃদ্ধিতে বর্তমান বেতন = 8000 + 8000 এর 20%
= 8000 + 8000 × 20/100
= 8000 + 1600
=9600 টাকা

নতুন কমিশন = 9600 এর 15%
= 9600 × 15/100 = 1440 টাকা

কমিশন বৃদ্ধি পায় = 1440 - 1200 = 240 টাকা

শতকরা কমিশন বৃদ্ধি পায় = (240/1200) × 100%
= 20%

১০,০৭৮.

If ⁶P_r = 360 and⁶C_r = 15, find the value of r.

ক
6
খ
5
গ
4
ঘ
3

ব্যাখ্যা

Question: If ⁶P_r = 360 and⁶C_r = 15, find the value of r.

Solution:
We know that,
ⁿP_r = ⁿC_r × r!
360 = 15 × r!
r! = 24
r! = 4!
r = 4

১০,০৭৯.

A solid cube of edge 6 is first painted red and then cut into smaller cubes of edge 2. How many of the smaller cubes have paint on exactly 2 edges?

ক
3
খ
6
গ
12
ঘ
15
ঙ
18

ব্যাখ্যা

Question: A solid cube of edge 6 is first painted red and then cut into smaller cubes of edge 2. How many of the smaller cubes have paint on exactly 2 edges?

Solution:
Number of small cubes along each edge, n = Edge of the larger cube/Edge of the smaller cube
= 6/2
= 3

Cubes with exactly 2 painted faces lie on the edges, but not at the corners of the cube.
Each edge contains (n − 2) such cubes. A cube has 12 edges

∴ Required number of smaller cubes = 12 × (n - 2)
= 12 × (3 - 2)
= 12

১০,০৮০.

Find the area of a triangle with side lengths of 7 meters, 8 meters, and 9 meters.

ক
8√5 m
খ
10√5 m
গ
12√5 m
ঘ
None of the above

ব্যাখ্যা

Question: Find the area of a triangle with side lengths of 7 meters, 8 meters, and 9 meters.
(একটি ত্রিভুজের তিন বাহুর দৈর্ঘ্য যথাক্রমে ৭, ৮ ,৯ মি. হলে এর ক্ষেত্রফল কত?)

Solution:

আমরা জানি,
ত্রিভুজের অর্ধপরিসীমা,
S = (a + b + c)/২
S = (৭ + ৮ + ৯)/২
= ২৪/২
∴ S = ১২

∴ ত্রিভুজের ক্ষেত্রফল
= √{s(s - a)(s - b)(s - c)}
= √{১২(১২ - ৭)(১২ - ৮) (১২ - ৯)}
= √(১২ × ৫ × ৪ × ৩)
= √(৭২০)
= ১২√৫
∴ ত্রিভুজের ক্ষেত্রফল = ১২√৫ বর্গ মি.

যেহেতু, ক্ষেত্রফলের একক m² বা, বর্গ মিটার; কিন্তু এখানে, অপশনগুলোয় কেবল m দেয়া। তাই "None of the above" - উত্তর হবে।

১০,০৮১.

A man took out a loan from a bank at the rate of 8%per annum simple interest. After 4 years he had to pay Tk. 6200 interest only for the period. What was the principal amount borrowed by him?

ক
ক) Tk. 20245
খ
খ) Tk. 18230
গ
গ) Tk. 17322
ঘ
ঘ) Tk. 19375

ব্যাখ্যা

P = (100 × S.I)/RT
= (100 × 6200)/(8 × 4)
= 620000/32
= 775 × 25
= Tk. 19375.

১০,০৮২.

If one man or two women or three boys can finish a work in 88 days, then how many days will one man, one woman and one boy together take to finish the same work?

ক
ক) 46 days
খ
খ) 54 days
গ
গ) 48 days
ঘ
ঘ) 44 days
ঙ
ঙ) 45 days

ব্যাখ্যা

1 man = 2 women = 3 boys
1 man + 1 woman + 1 boy
= (3 + 3/2 + 1) boys
= 11/2 boys
Here, M₁ = 3, M₂ = 11/2,
D₁ = 88 and D₂ = ?

According to the formula,

M₁D₁ =M₂D₂
D₂ = (M₁D₁)/M₂

Required number of days,
= (3 × 2 × 88)/11
= 3 × 2 × 8
= 48 days

১০,০৮৩.

If x ≥ 10 and y ≥ 12, then which of the following must be true?

ক
ক) (x + y) ≤ 22
খ
খ) (x - y) ≤ 22
গ
গ) (x + y) = 22
ঘ
ঘ) (x - y) ≤ 0
ঙ
ঙ) (x + y) ≥ 22

ব্যাখ্যা

Question: If x ≥ 10 and y ≥ 12, then which of the following must be true?

Solution:
Here,
x ≥ 10 ..............(1)
y ≥ 12 ..............(2)

From (1) + (2) we get,
x + y ≥ 10 + 12
∴ x + y ≥ 22

১০,০৮৪.

What is the probability of getting a sum 9 from two throws of a dice?

ক
ক) 1/6
খ
খ) 1/8
গ
গ) 1/9
ঘ
ঘ) 1/12

ব্যাখ্যা

In two throws of a dice, n(S) = (6 x 6) = 36
Let E = event of getting a sum
= {(3, 6), (4, 5), (5, 4), (6, 3)}
∴P(E)=n(E)/n(S)
=4/36
=1/9

১০,০৮৫.

If TOUR is written in a certain code as 1234, CLEAR as 56784 and SPARE as 90847, what will be the 5th digit for SCULPTURE in the same code?

ক
3
খ
4
গ
0
ঘ
6

ব্যাখ্যা

Question: If TOUR is written in a certain code as 1234, CLEAR as 56784 and SPARE as 90847, what will be the 5th digit for SCULPTURE in the same code?

Solution:
T - 1
O - 2
U - 3
R - 4
C - 5
L - 6
E - 7
A - 8
S - 9
P - 0

5th digit of SCULPTURE - 0

১০,০৮৬.

The cost price of an article is 80% of its marked price for sale. How much percent does the tradesman gain after allowing a discount of 12%?

ক
ক) 10%
খ
খ) 8%
গ
গ) 12%
ঘ
ঘ) 15%

ব্যাখ্যা

Question: The cost price of an article is 80% of its marked price for sale. How much percent does the tradesman gain after allowing a discount of 12%?

Solution:
Let
the marked price =Tk. 100
Then the cost price = 80% of 100 = Tk. 80
After allowing 12% discount on MP,
SP become = 100 - (100 × 12)/100
= Tk. 88.
% profit = (8 × 100)/80
= 10%

১০,০৮৭.

Five years ago, Rahul was twice as old as his sister. Now, Rahul is 6 years older than his sister. What is the present age of Rahul?

ক
24 years.
খ
17 years.
গ
12 years.
ঘ
22 years.
ঙ
None of these

ব্যাখ্যা

Question: Five years ago, Rahul was twice as old as his sister. Now, Rahul is 6 years older than his sister. What is the present age of Rahul?

Solution:
Let,
Present age of Rahul's sister = x years
Present age of Rahul = x + 6 years (since Rahul is 6 years older)

Five years ago,
Rahul’s age = (x + 6) - 5 = x + 1
And sister’s age = x - 5

According to the question,
x + 1 = 2(x - 5)
⇒ x + 1 = 2x - 10
⇒ 2x - x = 10 + 1
∴ x = 11

∴ Rahul's present age = x + 6 = 11 + 6 = 17 years.

১০,০৮৮.

A swimmer can swim at 5 km/hour in still water. When the river flows at 1 km/hour, then he takes 75 minutes to go to a certain point and come back to the starting point. How far is this place?

ক
5 km
খ
6 km
গ
4 km
ঘ
3 km
ঙ
None of these

ব্যাখ্যা

Question: A swimmer can swim at 5 km/hour in still water. When the river flows at 1 km/hour, then he takes 75 minutes to go to a certain point and come back to the starting point. How far is this place?

Solution:
Given that,
Total time = 5 hours and Speed of river = 1 km/hour
Total time to go to a certain point and come back = 75 min = 75/60 h = 5/4 h

Now, Let the distance to the point be x km.
When swimming downstream, the swimmer's speed = (5 + 1) = 6 km/h.
When swimming upstream, the swimmer's speed = (5 - 1) = 4 km/h

Now,
T₁ + T₂ = T
⇒ (x/6) + (x/4) = 5/4
⇒ (2x + 3x)/12 = 5/4
⇒ 5x = 15
⇒ x = 15/5
∴ x = 3 km.

∴ The place is 3 km away from the starting point.

১০,০৮৯.

If a shopkeeper sells a watch for Tk. 600, he incurs a loss of 20%. At what price should he sell it to make a profit of 20%?

ক
Tk. 850
খ
Tk. 900
গ
Tk. 800
ঘ
Tk. 950

ব্যাখ্যা

Question: If a shopkeeper sells a watch for Tk. 600, he incurs a loss of 20%. At what price should he sell it to make a profit of 20%?

Solution:
Let the Cost Price (CP) of the watch be x.
Selling price = x - 20% of x
⇒ 600 = x - (20x/100)
⇒ 600 = 80x/100
⇒ 80x = 60000
⇒ x = 750 (This is the CP)

To gain 20% = 750 + 20% of 750
= 750 + (20 × 750)/100
= 750 + 150
= Tk. 900

১০,০৯০.

Which number is the odd one in oval A and B respectively?

ক
42, 18
খ
42, 52
গ
36, 6
ঘ
48, 52

ব্যাখ্যা

Question: Which number is the odd one in oval A and B respectively?

Solution:
In figure A,
42 is odd because except 42 all the other numbers are divisible by 12

In figure B,
52 is odd because except 52 all the other numbers are divisible by 6

১০,০৯১.

A volunteer group consists of 4 male volunteers and 6 female volunteers. If a team of 4 volunteers is to be chosen at random to lead the event, what is the probability that the team will include at least 2 women?

ক
37/42
খ
41/53
গ
47/59
ঘ
61/72
ঙ
None of the above

ব্যাখ্যা

Question: A volunteer group consists of 4 male volunteers and 6 female volunteers. If a team of 4 volunteers is to be chosen at random to lead the event, what is the probability that the team will include at least 2 women?

Solution:
Given,
Total people = 10

∴ Ways of selecting 4 people from 10 = ¹⁰C₄
= 210

We want at least 2 women, so there are 3 possible combinations:
1st combination: 2 women and 2 men
= ⁶C₂ × ⁴C₂
= 15 × 6
= 90

2nd combination: 3 women and 1 man
= ⁶C₃ × ⁴C₁
= 20 × 4
= 80

3rd combination: 4 women and 0 man
= ⁶C₄
= 15

∴ Total outcomes = 90 + 80 + 15
= 185

∴ Probability = 185/210
= 37/42

১০,০৯২.

A shopkeeper marks his goods 30% above his cost price but allows a discount of 10% at the time of sale. His gain is = ?

ক
ক) 21%
খ
খ) 20%
গ
গ) 18%
ঘ
ঘ) 17%

ব্যাখ্যা

Let the cost price of goods = Tk. 100
Mark price of goods = Tk.130 (30% discount)
Selling price of goods
= 90% of 130 = Tk 117
Gain = 117 - 100 = Tk. 17
Gain% = 17/100×100 = 17%

১০,০৯৩.

If sinA = 4/5 find the value of tanA.secA = ?

ক
20/9
খ
5/3
গ
3/20
ঘ
4/3

ব্যাখ্যা

Question: If sinA = 4/5 find the value of tanA.secA = ?

Solution:
sinA = লম্ব/অতিভুজ = 4/5
ভূমি = √{(5)² - (4)²}
= 3

tanA.secA = (4/3)(5/3)
= 20/9

১০,০৯৪.

For what values of x is 8^{2x - 4} = 4^2x?

ক
2
খ
4
গ
6
ঘ
12

ব্যাখ্যা

8^{2x - 4} = 4^2x
⇒ (2³)^{2x - 4} = (2²)^2x
⇒ 2^{6x - 12} = 2^4x
⇒ 6x - 12 = 4x
⇒ 2x = 12
∴ x = 6

১০,০৯৫.

2 years ago, the average age of a family of 5 members was 16 years. After a baby is born, the average age of family is the same today. Find the present age of the baby.

ক
ক) 4
খ
খ) 6
গ
গ) 8
ঘ
ঘ) 8(1/2)

ব্যাখ্যা

(1) First find total age of 5 members 2 years ago
(2) Present age of 5 members
(3) Total age of 6 members
(4) Age of baby = Total age of 6 members - Present age of 5 members

We know that,
Average = Sum of Quantities/Number of Quantities

1) First calculate total age of 5 members 2 years ago = (Average age of 5 members x number of members)
First calculate total age of 5 members 2 years ago = (16 x 5) = 80 years

2) Calculate the present age of 5 members
2 years ago, their total age was 80 years. Present age can be calculated as follows:
Present age of 5 members = [80 + (2 x 5)] = 90 years

3) Calculate total age of 6 members considering baby = (16 x 6 ) = 96 years

4) Age of baby = (96 – 90)
= 6 years.

১০,০৯৬.

What is the reflex angle between the hands of a clock at 9.20?

ক
125°
খ
150°
গ
180°
ঘ
200°
ঙ
None of them

ব্যাখ্যা

Question: What is the reflex angle between the hands of a clock at 9.20?

Solution:

The angle between the hands of the clock is |11M - 60H|°/2
= |(11 × 20) - (60 × 9)|°/2
= |220 - 540|°/2
= |- 320|°/2
= 160°

∴ Reflex angle = 360° - 160°
= 200°

১০,০৯৭.

A family had provision of food for 15 days. After 5 days 8 guests came and the provision lasted 6 days. How many are the members of the family?

ক
10
খ
12
গ
11
ঘ
13

ব্যাখ্যা

Question: A family had provision of food for 15 days. After 5 days 8 guests came and the provision lasted 6 days. How many are the members of the family?

Solution:
5 দিন পর খাদ্য অবশিষ্ট থাকে = 1 - (1/3) অংশ
= 2/3 অংশ

ধরি লোকসংখ্যা ছিল x জন

প্রশ্নমতে,
(10x/6) - x = 8
⇒ (10x - 6x)/6 = 8
⇒ 10x - 6x = 48
⇒ 4x = 48
∴ x = 12

১০,০৯৮.

What will be the number in the question mark?
2, 5, 11, 20, 32, ?

ক
47
খ
50
গ
48
ঘ
24

ব্যাখ্যা

Question: What will be the number in the question mark?
2, 5, 11, 20, 32, ?

Solution:
প্রদত্ত ধারাটি হলো: 2, 5, 11, 20, 32, ?
ধারার সংখ্যাগুলোর মধ্যে পার্থক্য নির্ণয় করি:
5 - 2 = 3
11 - 5 = 6
20 - 11 = 9
32 - 20 = 12

এখানে, প্রতিবার পার্থক্য 3 করে বৃদ্ধি পাচ্ছে।
∴ পরবর্তী পার্থক্য হবে = 12 + 3 = 15
∴ পরবর্তী সংখ্যাটি হবে = 32 + 15 = 47
অতএব, প্রশ্নবোধক স্থানে 47 বসবে।

Shortcut: 2 (+3)→ 5 (+6)→ 11 (+9)→ 20 (+12)→ 32 (+15) → 47.

১০,০৯৯.

A is the widow of B. B & C were the only children of E. C is unmarried and is a doctor. D is the grand-daughter of E and studies science. How is A related to D?

ক
aunt
খ
daughter
গ
mother
ঘ
sister

ব্যাখ্যা

Question: A is the widow of B. B & C were the only children of E. C is unmarried and is a doctor. D is the grand-daughter of E and studies science. How is A related to D?

Solution:
A is the widow of B ⇒ A's husband was B.

B & C were the only children of E ⇒ A is the daughter in law of E.

D is the grand-daughter of E ⇒ D is daughter of A [D can not be daughter of C because C is unmarried]

So, A is the mother of D.

১০,১০০.

The average of six numbers is 14. The average of four of these numbers is 15. The average of the remaining two numbers is

ক
4
খ
8
গ
12
ঘ
16

ব্যাখ্যা

Question: The average of six numbers is 14. The average of four of these numbers is 15. The average of the remaining two numbers is-

Solution:
6 টি সংখ্যার গড় = 14
6 টি সংখ্যার সমষ্টি = 14 × 6
= 84

4 টি সংখ্যার গড় = 15
4 টি সংখ্যার সমষ্টি = 15 × 4
= 60

2 টি সংখ্যার সমষ্টি = 84 - 60
= 24

2 টি সংখ্যার গড় = 24/2 = 12