Bank Math

PrepBank · পাতা ৭২ / ১৬১ · ৭,১০১–৭,২০০ / ১৬,১২৪

৭,১০১.

P and Q can do a job in 3 days. Q and R can do the same job in 4 days, while R and P can do it in 12 days. In how many days the job will be finished when P, Q and R working together.

ক
10 days
খ
3 days
গ
4 days
ঘ
12 days

সঠিক উত্তর: খ

3 days

উত্তর

সঠিক উত্তর: খ

3 days

খ

ব্যাখ্যা

Question: P and Q can do a job in 3 days. Q and R can do the same job in 4 days, while R and P can do it in 12 days. In how many days the job will be finished when P, Q and R working together.

Solution:
Given that,
(P + Q)'s 1 day's work = 1/3
(Q + R )'s 1 day's work= 1/4
and
(R + P)'s 1 day's work = 1/12

Now, (P + Q) + (Q + R) + (R + P) = 2(P + Q + R)'s 1 day's work = (1/3) + (1/4) + (1/12)
= (4 + 3 + 1)/12
= 8/12 = 2/3
∴ (P + Q + R)'s 1 day's work = 2/(3 × 2) = 1/3

So, P + Q + R can complete the job in = 3 days

৭,১০২.

If x = - 3 and y = - 2, what is xy³=?

ক
ক) - 12
খ
খ) - 24
গ
গ) 24
ঘ
ঘ) 12

সঠিক উত্তর: গ

গ) 24

উত্তর

সঠিক উত্তর: গ

গ) 24

গ

ব্যাখ্যা

দেয়া আছে,
x = - 3
y = - 2

xy³ = (- 3) × (- 2)³
= (- 3) × (- 8)
= 24

৭,১০৩.

What are the roots of the equation √(2x + 9) = 13 - x?

ক
8
খ
6
গ
12
ঘ
20

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: What are the roots of the equation √(2x + 9) = 13 - x?

Solution:
√(2x + 9) = 13 - x
Squaring both sides, we get
2x + 9 = (13 - x)²
⇒ 2x + 9 = 169 - 26x + x²
⇒ - x² + 28x - 160 = 0
⇒ x² - 28x + 160 = 0
⇒ (x - 8)(x - 20) = 0
⇒ x - 8 = 0 or x - 20 = 0
⇒ x = 8 or x = 20

But x = 20 does not satisfy the given equation, so it is rejected. Hence, the root of the given equation is 8.

৭,১০৪.

A rectangle has a perimeter of 64 cm, with the lengths of its adjacent sides in a 7 : 9 ratio. What are the measurements of these sides?

ক
10 cm, 8 cm
খ
14 cm, 18 cm
গ
12 cm, 14 cm
ঘ
20 cm, 15 cm

সঠিক উত্তর: খ

14 cm, 18 cm

উত্তর

সঠিক উত্তর: খ

14 cm, 18 cm

খ

ব্যাখ্যা

Question: A rectangle has a perimeter of 64 cm, with the lengths of its adjacent sides in a 7 : 9 ratio. What are the measurements of these sides?

Solution:
Perimeter of a rectangle = 2(Length + Breadth)
Also Length : Breadth = 9 : 7
Let, the actual values are 9p and 7p

Hence,
2(9p + 7p) = 64
⇒ 16p = 32
∴ p = 2

∴ The sides will be 14 cm and 18 cm.

৭,১০৫.

On a certain sum of money, the simple interest for 2 years is Tk. 200 at the rate of 5% per annum. If it was invested at compound interest at the same rate for the same duration as before, how much more interest would be earned?

ক
9 Tk
খ
3 Tk
গ
5 Tk
ঘ
10 Tk

সঠিক উত্তর: গ

5 Tk

উত্তর

সঠিক উত্তর: গ

5 Tk

গ

ব্যাখ্যা

Question: On a certain sum of money, the simple interest for 2 years is Tk. 200 at the rate of 5% per annum. If it was invested at compound interest at the same rate for the same duration as before, how much more interest would be earned?

Solution:
Simple Interest (I) = (P × r × n)/100
⇒ 200 = (P × 2 × 5)/100
∴ P = 2000

∴ Compound Interest = P(1 + r)ⁿ - P
= 2000(1 + 0.05)² - 2000
= 2000 × 1.1025 - 2000
= 2205 - 2000
= 205

∴ Difference = 205 - 200 = 5 Tk

৭,১০৬.

What will come at the place of the question mark?
7, 13, 25, 49, ?

ক
97
খ
98
গ
90
ঘ
99
ঙ
None

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: What will come at the place of the question mark?
7, 13, 25, 49, ?

Solution:
Here,
First term = 7

Second term = (7 × 2 - 1) = 13
Third term = (13 × 2 - 1) = 25
Fourth term = (25 × 2 - 1) = 49

So,
Fifth term = (49 × 2 - 1) = 97

৭,১০৭.

The sum of all the interior angles of a hexagon is:

ক
ক) 540°
খ
খ) 360°
গ
গ) 180°
ঘ
ঘ) 270°
ঙ
ঙ) 720°

সঠিক উত্তর: ঙ

ঙ) 720°

উত্তর

সঠিক উত্তর: ঙ

ঙ) 720°

ঙ

ব্যাখ্যা

Number of sides in hexagon, n = 6
Sum of interior angles = (n - 2) x 180°
= (6 – 2) x 180°
= 720°

৭,১০৮.

Find the perimeter (in cm) of a square having an area equal to the area of a rhombus, measures of whose diagonals are 8 cm and 16 cm.

ক
ক) 38 cm
খ
খ) 36 cm
গ
গ) 32 cm
ঘ
ঘ) 39 cm

সঠিক উত্তর: গ

গ) 32 cm

উত্তর

সঠিক উত্তর: গ

গ) 32 cm

গ

ব্যাখ্যা

প্রশ্ন : Find the perimeter (in cm) of a square having an area equal to the area of a rhombus, measures of whose diagonals are 8 cm and 16 cm.

সমাধান :

Measures of the diagonals of the rhombus are 8 cm and 16 cm.
Area of the square = Area of the rhombus

Concept used:

Area of a rhombus = Product of two diagonals ÷ 2
Area of a square = side²

Perimeter of a square = 4 × side

Area of the rhombus = (8 × 16)/2 = 64 cm²

According to the question,

Area of the square = 64 cm²

The measure of each side of the square = √64 = 8 cm
Now, the perimeter of the square = 4 × 8 = 32 cm

∴ The perimeter of the square is 32 cm.

৭,১০৯.

In a certain school, 20% of students are below 8 years of age. The number of students above 8 years of age is 2/3 of the number of students of 8 years of age which is 48. What is the total number of students in the school?

ক
72
খ
100
গ
124
ঘ
180

সঠিক উত্তর: খ

100

উত্তর

সঠিক উত্তর: খ

100

খ

ব্যাখ্যা

Question: In a certain school, 20% of students are below 8 years of age. The number of students above 8 years of age is 2/3 of the number of students of 8 years of age which is 48. What is the total number of students in the school?

Solution:
Given,
20% of students are below 8 years of age
Number of students of 8 years of age = 48
Number of students above 8 years = 2/3 of students of 8 years

∴ Students above 8 years = 2/3 × 48 = 32
∴ Students of 8 years or above = 48 + 32 = 80

Since 20% are below 8 years, then 80% are of 8 years or above.
So, 80% of total students = 80

Let, total number of students = x
80% of x = 80
⇒ (80/100) × x = 80
⇒ x = 80 × (100/80)
∴ x = 100

৭,১১০.

3 pumps, working 8 hours a day, can empty a tank in 2 days, How many hours a day must 4 pumps work to empty the tank in 1 day?

ক
18 hours
খ
15 hours
গ
9 hours
ঘ
12 hours

সঠিক উত্তর: ঘ

12 hours

উত্তর

সঠিক উত্তর: ঘ

12 hours

ঘ

ব্যাখ্যা

Question: 3 pumps, working 8 hours a day, can empty a tank in 2 days, How many hours a day must 4 pumps work to empty the tank in 1 day?

Solution:
3 pumps, working 8 hours a day, can empty a tank in 2 days
Formula used: M₁ × T₁ = M₂ × T₂
Where M₁ and M₂ is men and T₁ and T₂ is time

Calculation:
Let H hours be the number of hours required Applying the above formula
⇒ 3 × 8 × 2 = 4 × 1 × H
⇒ H = 48/4
⇒ H = 12 hours

∴ 4 pump need to work 12 hours to complete the work in 1 day.

৭,১১১.

Using all the letters of the word 'GIFT' how many distinct words can be formed?

ক
22 words
খ
24 words
গ
200 words
ঘ
256 words

সঠিক উত্তর: খ

24 words

উত্তর

সঠিক উত্তর: খ

24 words

খ

ব্যাখ্যা

Question: Using all the letters of the word 'GIFT' how many distinct words can be formed?

Solution:
The word 'GIFT' has 4 letters.
The number of distinct words using all the letters can be calculated as 4! = 4 × 3 × 2 × 1 = 24.

So, there are 24 distinct words that can be formed from the letters in the word 'GIFT'.

৭,১১২.

What percentage of numbers from 1 to 70 have 1 or 9 in the unit's digit?

ক
ক) 10%
খ
খ) 20%
গ
গ) 15%
ঘ
ঘ) 7%

সঠিক উত্তর: খ

খ) 20%

উত্তর

সঠিক উত্তর: খ

খ) 20%

খ

ব্যাখ্যা

Clearly, the numbers which have 1 or 9 in the unit's digit, have squares that end in the digit 1.
Such numbers from 1 to 70 are
1, 9, 11, 19, 21, 29, 31, 39, 41, 49, 51, 59, 61, 69.

Number of such number =14

Required percentage = {(14/70) × 100}%
= 20%

৭,১১৩.

The diagonal of a rectangle is √41 cm and its area is 20 sq. cm. The perimeter of the rectangle must be-

ক
ক) 9 cm
খ
খ) 12 cm
গ
গ) 30 cm
ঘ
ঘ) 18 cm

সঠিক উত্তর: ঘ

ঘ) 18 cm

উত্তর

সঠিক উত্তর: ঘ

ঘ) 18 cm

ঘ

ব্যাখ্যা

Question: The diagonal of a rectangle is √41 cm and its area is 20 sq.cm . The perimeter of the rectangle must be-

Solution:
l²+ b²= (√41)²= 41
lb = 20

We know that,
(l + b)²= l²+ b²+ 2lb
(l + b)² = 41 + 2 × 20
(l + b)² = 41 + 40
(l + b)² = 81
l + b = √81
= 9

Perimeter= 2(l + b) = 2 × 9 = 18 cm

৭,১১৪.

A boat travels at 12 km/h in still water, and the stream's speed is 3 km/h. If a boatman rows 90 km to a destination and comes back, how much time does he take in total?

ক
9 hours
খ
15 hours
গ
16 hours
ঘ
18 hours

সঠিক উত্তর: গ

16 hours

উত্তর

সঠিক উত্তর: গ

16 hours

গ

ব্যাখ্যা

Question: A boat travels at 12 km/h in still water, and the stream's speed is 3 km/h. If a boatman rows 90 km to a destination and comes back, how much time does he take in total?

Solution:
Given,
Speed of the boat in still water = 12 km/h
Speed of the current = 3 km/h

Speed of the boat downstream = (Speed in still water + Speed of current) = (12 + 3) km/h = 15 km/h
Speed of the boat upstream = (Speed in still water − Speed of current) = (12 − 3) km/h = 9 km/h

Time taken to go downstream = Distance / Speed = (90/15) hours = 6 hours
Time taken to go upstream = Distance / Speed = (90/9) hours = 10 hours

Total time for the round trip = (6 + 10) hours = 16 hours

৭,১১৫.

A sum of money amounts to Tk. 1800 in 5 years and Tk. 2400 in 8 years at simple interest. Find the annual rate of interest.

ক
10%
খ
15%
গ
20%
ঘ
25%

সঠিক উত্তর: ঘ

25%

উত্তর

সঠিক উত্তর: ঘ

25%

ঘ

ব্যাখ্যা

Question: A sum of money amounts to Tk. 1800 in 5 years and Tk. 2400 in 8 years at simple interest. Find the annual rate of interest.

Solution:
দেওয়া আছে,
5 বছরের সুদ-আসল = 1800 টাকা
8 বছরের সুদ-আসল = 2400 টাকা
^{——————————————————}∴ 3 বছরের সুদ = (2400 - 1800) = 600 টাকা
∴ 1 বছরের সুদ = 600/3 = 200 টাকা

এখন,
আসল = 5 বছরের সুদ-আসল - 5 বছরের সুদ
⇒ আসল = 1800 - (5 × 200)
⇒ আসল = 1800 - 1000
⇒ আসল = 800 টাকা

এখানে,
মোট সুদ (I) = 1000 টাকা
আসল (P) = 800 টাকা
সময় (n) = 5 বছর
আমরা জানি, সুদের হার, r = (I × 100)/(P × n)
⇒ r = (1000 × 100)/(800 × 5)
⇒ r = 100000/4000
⇒ r = 25
অতএব, বার্ষিক সুদের হার 25%।

৭,১১৬.

I we double a certain number and add 30 to it, we get the same value as four times that number. What is the value of thrice the number?

ক
ক) 12
খ
খ) 15
গ
গ) 30
ঘ
ঘ) 45
ঙ
ঙ) None

সঠিক উত্তর: ঘ

ঘ) 45

উত্তর

সঠিক উত্তর: ঘ

ঘ) 45

ঘ

ব্যাখ্যা

Question: I we double a certain number and add 30 to it, we get the same value as four times that number. What is the value of thrice the number?

Solution:
ধরি
সংখ্যাটি x

প্রশ্নমতে
2x + 30 = 4x
30 = 4x - 2x
2x = 30
x = 15

সংখ্যাটির তিনগুণ = 3 × 15 = 45

৭,১১৭.

A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions. The number of choices available to him is

ক
ক) 140
খ
খ) 196
গ
গ) 280
ঘ
ঘ) 346

সঠিক উত্তর: খ

খ) 196

উত্তর

সঠিক উত্তর: খ

খ) 196

খ

ব্যাখ্যা

If the student answers 4 questions out of the first five questions he can choose 6 questions from the remaining 8 questions.
Number of combinations will be = ⁵C₄ × ⁸C₆ = 140
If the student answers 5 questions from the first five questions he can choose 5 questions from the remaining 8 questions; Number of combinations will be = ⁵C₅ × ⁸C₅ = 56
So, total number of choices are = 140+56 =196

৭,১১৮.

What is the volume of a hemisphere having radius 3 cm?

ক
18π m³
খ
9π m³
গ
27π m³
ঘ
36π m³
ঙ
None of these

সঠিক উত্তর: ঙ

None of these

উত্তর

সঠিক উত্তর: ঙ

None of these

ঙ

ব্যাখ্যা

Question: What is the volume of a hemisphere having radius 3 cm?

Solution:
The radius of a hemisphere is 3cm
A hemisphere's volume is equal to (2/3)πr³

Volume = (2/3)π.3³)
⇒ 2π × 9
⇒ 18π

∴ The volume of a hemisphere having a radius 3 cm is 18π cm³

৭,১১৯.

If x² + 9/x² = 31, what is the value of x - 3/x?

ক
ক) 36
খ
খ) 25
গ
গ) 9
ঘ
ঘ) 5

সঠিক উত্তর: ঘ

ঘ) 5

উত্তর

সঠিক উত্তর: ঘ

ঘ) 5

ঘ

ব্যাখ্যা

Given, x² + 9/x² = 31
Or, x² + (3/x)² = 31
Or, (x - 3/x)² + 2.x.3/x = 31
Or, (x - 3/x)² = 31 - 6
Or, (x - 3/x)² = 25
So, x - 3/x = 5

৭,১২০.

Q. (56-61): Answer the questions based on the following information.
A farmer has three fields, 1, 2 and 3, and is deciding which crops to plant. The crops are F, G, H, I & J.
F will grow only in fields 1 and 3, but in order for F to grow it must be fertilized with X.
G will grow in fields 1, 2, and 3, but in order for G to grow, fertilizer X must not be used.
H will grow in fields 1, 2, and 3 but in order for H to grow in field 3. it must be fertilized with Y.
I will grow only in fields 2 and 3, but in order for I to grow in field 2 it must be sprayed with pesticide Z, and in order for I to grow in field 3, it must not be sprayed with Z.
J will grow only in field 2, but in order for J to grow. H must not be planted in the same field.
All crops are planted and harvested at the same time. More than one crop may be planted in a field.

It is possible to grow which of the following pairs of crops together in field 1?
I. F and G
II. G and H
III. F and H
IV. H and J

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

সঠিক উত্তর: গ

গ) II and III only

উত্তর

সঠিক উত্তর: গ

গ) II and III only

গ

ব্যাখ্যা

Question: It is possible to grow which of the following pairs of crops together in field 1?
I. F and G
II. G and H
III. F and H
IV. H and J

Solution:
ফিল্ড ১ এ F, G, H উৎপাদন করা যায়।
কিন্তু, F ও G এক সাথে উৎপাদন করা যায় না। কারণ, F উৎপাদন করতে X সার দিতে হবে অনুদিকে G উৎপাদন করতে X সার ব্যবহার করা যাবে না।

অতএব, FH ও GH একসাথে ফিল্ড ১ এ উৎপাদন করা যাবে।

৭,১২১.

A boat takes 6 hours to cover a distance while travelling upstream, whereas while travelling downstream it takes 4 hours. If the speed of the current is 6 kmph, what is the speed of the boat in still water?

ক
ক) 18 kmph
খ
খ) 24 kmph
গ
গ) 30 kmph
ঘ
ঘ) 38 kmph

সঠিক উত্তর: গ

গ) 30 kmph

উত্তর

সঠিক উত্তর: গ

গ) 30 kmph

গ

ব্যাখ্যা

Let the speed of the boat in still water be x kmph
Then Speed downstream = (x + 6) kmph
Speed up stream = (x - 6) kmph

Now
(x + 6) × 4 = (x - 6) × 6
4x + 24 = 6x - 36
6x - 4x = 36 + 24
2x = 60
x = 30 kmph

৭,১২২.

A boat goes 13 km upstream in 39 minutes. The speed of stream is 3 km/h. What is the speed of the boat in still water?

ক
6 km/h
খ
16 km/h
গ
23 km/h
ঘ
26 km/h

সঠিক উত্তর: গ

23 km/h

উত্তর

সঠিক উত্তর: গ

23 km/h

গ

ব্যাখ্যা

Question: A boat goes 13 km upstream in 39 minutes. The speed of stream is 3 km/h. What is the speed of the boat in still water?

Solution:
Against the stream,
Speed of the boat = Distance/Time
= 13/(39/60) km/h
= (13 × 60)/39 km/h
= 20 km/h

We know,
Speed of the boat upstream = Speed in still water − Speed of the stream
⇒ Speed in still water = Speed of the boat upstream + Speed of the stream
⇒ Speed in still water = (20 + 3) km/h = 23 km/h

∴ Speed of the boat in still water = 23 km/h.

৭,১২৩.

A milkman wants to gain 25% on selling the mixture at cost price, then in what ratio must he mix water with milk?

ক
ক) 1 : 4
খ
খ) 1 : 2
গ
গ) 1 : 3
ঘ
ঘ) 2 : 5

সঠিক উত্তর: ক

ক) 1 : 4

উত্তর

সঠিক উত্তর: ক

ক) 1 : 4

ক

ব্যাখ্যা

Question: A milkman wants to gain 25% on selling the mixture at cost price, then in what ratio must he mix water with milk?

Solution:
Let, the milkman has milk of 100 Tk.
after mixing he sold it in = 100 + 25 = 125 Tk.

In 125, milk is of 100 Tk. and water is of 25 Tk.

∴ The ratio of water and milk is 25 : 100 = 1 : 4

৭,১২৪.

The present age of a father is 5 times that of his son. After 8 years, the ratio will be 3 : 1. Find their age gap.

ক
8
খ
32
গ
40
ঘ
48

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: The present age of a father is 5 times that of his son. After 8 years, the ratio will be 3 : 1. Find their age gap.

Solution:
Let the present age of the son = x years.
Present age of the father = 5x years.

After 8 years:
Son = x + 8
Father = 5x + 8
Given ratio after 8 years = 3 : 1

Accordingly,
(5x + 8) / (x + 8) = 3
⇒ 5x + 8 = 3(x + 8)
⇒ 5x + 8 = 3x + 24
⇒ 5x - 3x = 24 - 8
⇒ 2x = 16
∴ x = 8

Present ages of
Son = x = 8 years
And Father = 5x = 40 years

Age gap = 40 - 8 = 32

So the age gap is 32 years.

৭,১২৫.

A rectangular carpet has an area of 120 sq. metres and a perimeter of 46 metres. What is the length of the diagonal?

ক
ক) 15
খ
খ) 16
গ
গ) 20
ঘ
ঘ) 17

সঠিক উত্তর: ঘ

ঘ) 17

উত্তর

সঠিক উত্তর: ঘ

ঘ) 17

ঘ

ব্যাখ্যা

Question: A rectangular carpet has an area of 120 sq. metres and a perimeter of 46 metres. What is the length of the diagonal?

Solution:
Let the length and breadth of the rectangle be x and y metres.

given 2(x + y) = 46
x + y = 23
x = 23 - y

and,
xy = 120
(23 - y) y = 120
23y - y² = 120
y² - 23y + 120 = 0
y² - 15y - 8y + 120 = 0
y(y - 15) - 8(y - 15) = 0
(y - 15) (y - 8) = 0
y = 15 or, 8

∴ length, x = 15 and breadth, y = 8

so, the diagonal is = √{(15)² + (8)²}
= √289
= 17

৭,১২৬.

ΔABC is a right triangle. In ΔABC, hypotenuse AC = 2, normal AB = 1; then which of the following is correct?

ক
∠ABC = 90°
খ
∠ACB = 30°
গ
tan(A) = √3
ঘ
A, B and C
ঙ
None of these

সঠিক উত্তর: ঘ

A, B and C

উত্তর

সঠিক উত্তর: ঘ

A, B and C

ঘ

ব্যাখ্যা

Question: ΔABC is a right triangle. In ΔABC, hypotenuse AC = 2, normal AB = 1; then which of the following is correct?

Solution:

Given,
In right triangle ABC, hypotenuse AC = 2, normal AB = 1
Let, base BC = a

2² = a² + 1²
⇒ a² = 4 - 1
⇒ a = √3

sin(C) = AB/AC = 1/2 = sin 30°
⇒ C = 30°
∴ ∠ACB = 30°

Again, for ∠BAC, hypotenuse AC = 2, base AB = 1 and normal BC = √3
tan(A) = BC/AB = √3/1 = √3
∴ tan(A) = √3

৭,১২৭.

MONSOON is related to SEASON in the same way APRIL is related to -

ক
ক) SUMMER
খ
খ) SPRING
গ
গ) WINTER
ঘ
ঘ) MONTH

সঠিক উত্তর: ঘ

ঘ) MONTH

উত্তর

সঠিক উত্তর: ঘ

ঘ) MONTH

ঘ

ব্যাখ্যা

Question: MONSOON is related to SEASON in the same way APRIL is related to -

সমাধান:
MONSOON (বর্ষাকাল) হচ্ছে ঋতুর (SEASON) নাম,
তেমনি APRIL (এপ্রিল) হচ্ছে মাসের (MONTH) নাম।

৭,১২৮.

In the figure below, AB is perpendicular to BC and DB = DC. If AD = √10 and AC = 4 cm, what is the value of BC?

ক
2√2 cm
খ
√2 cm
গ
3 cm
ঘ
2√3 cm
ঙ
5 cm

সঠিক উত্তর: ক

2√2 cm

উত্তর

সঠিক উত্তর: ক

2√2 cm

ক

ব্যাখ্যা

Question: In the figure below, AB is perpendicular to BC and DB = DC. If AD = √10 and AC = 4 cm, what is the value of BC?

Solution:
ΔABD-এ,
BD² + AB² = AD²
⇒ BD² + AB² = (√10)²
⇒ BD² + AB² = 10

আবার, ΔABC-এ,
BC² + AB² = AC²
⇒ (BD + DC)² + AB² = 4²
⇒ BD² + DC² + 2BD.DC + AB² = 16
⇒ DC² + 2BD.DC = 6 (যেহেতু BD² + AB² = 10)
⇒ DC² + 2DC² = 6 (যেহেতু BD = DC)
⇒ 3DC² = 6
⇒ DC² = 2
⇒ DC = √2

অতএব, BC = 2DC (যেহেতু BD = DC)
= 2√2 cm

৭,১২৯.

A pair of runners begin a race at the same time. One runs at 5 km/h and the other at 8 km/h. The faster runner finishes 1.5 hours sooner. Find the total distance.

ক
18 km
খ
20 km
গ
25 km
ঘ
30 km
ঙ
None of the above

সঠিক উত্তর: খ

20 km

উত্তর

সঠিক উত্তর: খ

20 km

খ

ব্যাখ্যা

Question: A pair of runners begin a race at the same time. One runs at 5 km/h and the other at 8 km/h. The faster runner finishes 1.5 hours sooner. Find the total distance.
(দুইজন দৌড়বিদ সমান সময়ে একটি দৌড় প্রতিযোগিতা শুরু করেন। প্রথম ব্যক্তি ঘণ্টায় ৫ কিমি এবং দ্বিতীয় ব্যক্তি ঘণ্টায় ৮ কিমি গতিতে এগোন। দ্রুততম ব্যক্তি ১.৫ ঘণ্টা আগে সমাপ্ত করেন। মোট দূরত্ব নির্ণয় করুন।)

Solution:
ধরা যাক, দূরত্ব = Z
প্রথম দৌড়বিদের সময় = Z/5 hour
দ্বিতীয় দৌড়বিদের সময় = Z/8

প্রশ্নমতে,
Z/5 - Z/8 = 3/2
⇒ 3Z/40 = 3/2
⇒ 3Z = 60
⇒ Z = 20 km

৭,১৩০.

The original retail price of an appliance was 60 percent more than its wholesale cost. If the appliance was actually sold for 20 percent less than the original retail price, then it was sold for what percent more than its wholesale cost?

ক
20%
খ
28%
গ
36%
ঘ
40%
ঙ
42%

সঠিক উত্তর: খ

28%

উত্তর

সঠিক উত্তর: খ

28%

খ

ব্যাখ্যা

Question: The original retail price of an appliance was 60 percent more than its wholesale cost. If the appliance was actually sold for 20 percent less than the original retail price, then it was sold for what percent more than its wholesale cost?

Solution:
Let Wholesale cost be = 100
Original retail price = 60% more than wholesale cost = 160% of 100 = (160/100) × 100 = 160

Selling price = 20% less than original retail price = 80% of 160 = (80/100) × 160 = 128

Difference between Wholesale Cost and Selling price = 128 - 100 = 28

∴ Percent change = (Difference in cost / initial cost) × 100 = (28/100) × 100 = 28%

৭,১৩১.

Two packs of cards are thoroughly mixed and shuffled and two cards are drawn at random, one after the other. What is the probability that both of them are jacks?

ক
1/169
খ
1/179
গ
7/1339
ঘ
8/1153

সঠিক উত্তর: গ

7/1339

উত্তর

সঠিক উত্তর: গ

7/1339

গ

ব্যাখ্যা

Question: Two packs of cards are thoroughly mixed and shuffled and two cards are drawn at random, one after the other. What is the probability that both of them are jacks?

Solution:
Total number of cards = 104 = 2 × 52
and total number of jacks = 8 = 2 × 4
∴ Probability for the jack in first draw = 8/104
and probability for the jack in second draw = 7/103

Since both the events are independent events.
Hence the probability that both of them are jacks = (8/104) × (7/103)
= 7/1339

৭,১৩২.

A fruit seller had some apples. He sold 70% apples and still has 510 apples left. Originally he had:

ক
1700
খ
1500
গ
1440
ঘ
1260

সঠিক উত্তর: ক

1700

উত্তর

সঠিক উত্তর: ক

1700

ক

ব্যাখ্যা

He sold 70% apples
He has left = 100 - 70 = 30%
30% of apples = 510
100% of apples
= 510 × 100/30
= 1700

৭,১৩৩.

In how many ways, a committee of 5 members be selected from 7 men and 5 ladies, consisting of 3 men and 2 ladies?

ক
ক) 250
খ
খ) 350
গ
গ) 450
ঘ
ঘ) 320

সঠিক উত্তর: খ

খ) 350

উত্তর

সঠিক উত্তর: খ

খ) 350

খ

ব্যাখ্যা

Question: In how many ways, a committee of 5 members be selected from 7 men and 5 ladies, consisting of 3 men and 2 ladies?

Solution:
there are total 7 men and 5 ladies

∴ number of ways a committee of 5 members can be slected = (⁷C₃) × (⁵C₂)
= 35 × 10
= 350 ways

৭,১৩৪.

In the figure below, what is the value of x?

ক
55°
খ
60°
গ
70°
ঘ
90°

সঠিক উত্তর: গ

70°

উত্তর

সঠিক উত্তর: গ

70°

গ

ব্যাখ্যা

Question: In the figure below, what is the value of x?

Solution:
AB = √(4² + 3²) = √(16 + 9) = √25 = 5
13² = AC² + 12²
⇒ AC2 = 13² - 12² = 25
AC = 5

∠B = ∠C = 55°

∠x = 180 - 55 - 55
= 180 - 110
= 70°

৭,১৩৫.

A does half as much work as B in one-sixth of the time. If together they take 15 days to complete a work, how much time shall B alone take to do it?

ক
ক) 30 days
খ
খ) 45 days
গ
গ) 60 days
ঘ
ঘ) 75 days

সঠিক উত্তর: গ

গ) 60 days

উত্তর

সঠিক উত্তর: গ

গ) 60 days

গ

ব্যাখ্যা

Question: A does half as much work as B in one-sixth of the time. If together they take 15 days to complete a work, how much time shall B alone take to do it?

Solution:
Let B takes x days to do the work.
A takes 1/6 of x time to do 1/2 of the work.
∴ the work will be dy by A in (x/6) × 2 days
= x/3 days

ATQ,
1/x + 3/x = 1/15
⇒ 4/x = 1/15
⇒ x = 60

৭,১৩৬.

The total monthly salary of 4 men and 2 women is Tk. 46000. If a woman earns Tk. 500 more than a man, what is the monthly salary of a woman?

ক
Tk. 5000
খ
Tk. 8000
গ
Tk. 10000
ঘ
Tk. 12000

সঠিক উত্তর: খ

Tk. 8000

উত্তর

সঠিক উত্তর: খ

Tk. 8000

খ

ব্যাখ্যা

Question: The total monthly salary of 4 men and 2 women is Tk. 46000. If a woman earns Tk. 500 more than a man, what is the monthly salary of a woman?

Solution:
Let
The monthly salary of a man be Tk. x
The monthly salary of a woman be Tk. x + 500

Now
4x + 2( x + 500) = 46000
4x + 2x + 1000 = 46000
6x + 1000 = 46000
6x = 46000 - 1000
6x = 45000
x = 45000/6
x = 7500

The monthly salary of a woman be Tk. (7500 + 500) = Tk. 8000

৭,১৩৭.

Arif, Kamal and Shajib invested Tk.8000, Tk.4000 and Tk.8000 respectively in a business. Arif left after six months. If after eight months, there was a gain of Tk.4005, then what will be the share of Kamal in this gain?

ক
ক) 890 Taka
খ
খ) 1335 Taka
গ
গ) 1602 Taka
ঘ
ঘ) None of these

সঠিক উত্তর: ক

ক) 890 Taka

উত্তর

সঠিক উত্তর: ক

ক) 890 Taka

ক

ব্যাখ্যা

প্রশ্ন: Arif, Kamal and Shajib invested Tk.8000, Tk.4000 and Tk.8000 respectively in a business. Arif left after six months. If after eight months, there was a gain of Tk.4005, then what will be the share of Kamal in this gain?

সমাধান:
Arif : Kamal : Shajib = (8000 x 6) : (4000 x 8) : (8000 x 8)
= 48 : 32 : 64
= 3 : 2 : 4

∴ Kamal's share = {4005 x (2/9)}
= 890 Taka

৭,১৩৮.

In a single throw of a die, what is the probability of getting a number greater than 4?

ক
ক) 1/3
খ
খ) 2/3
গ
গ) 1/6
ঘ
ঘ) 1/4

সঠিক উত্তর: ক

ক) 1/3

উত্তর

সঠিক উত্তর: ক

ক) 1/3

ক

ব্যাখ্যা

Question: In a single throw of a die, what is the probability of getting a number greater than 4?

Solution:
In a single throw of dice, total results are 1, 2, 3, 4, 5 and 6 = 6 results
Event of getting numbers greater than 4 ={5, 6} = 2 possible results
Required probability to get a number greater than 4 is 2/6 = 1/3

৭,১৩৯.

If log₅y - log₅√y = 2log_y5, then find the value of y.

ক
25
খ
35
গ
10
ঘ
15
ঙ
None of these

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: If log₅y - log₅√y = 2log_y5, then find the value of y.

Solution:
log₅y - log₅√y = 2log_y5

৭,১৪০.

Four friends planned to rent a car for a trip and spilt the cost equally. If at the last minute, two of the friends do not attend the trip, the remaining people will each have to pay 40 taka more to rent the car. How much did each person originally have to pay to rent the car?

ক
20
খ
40
গ
60
ঘ
120

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: Four friends planned to rent a car for a trip and spilt the cost equally. If at the last minute, two of the friends do not attend the trip, the remaining people will each have to pay 40 taka more to rent the car. How much did each person originally have to pay to rent the car?

Solution:
ধরি,
গাড়ি ভাড়া x টাকা
4 জনকে দেওয়া লাগত x/4 টাকা
2 জনকে দেওয়া লাগত x/2 টাকা

প্রশ্নমতে,
(x/2) - (x/4) = 40
বা, (2x - x)/4 = 40
বা, x/4 = 40
∴ x = 160

4 জনকে দেওয়া লাগত 160/4 টাকা
= 40 টাকা করে

৭,১৪১.

If 3 men or 9 boys can do a job in 60 days, how long will it take 11 men and 27 boys to finish the same job?

ক
6 days
খ
9 days
গ
10 days
ঘ
12 days

সঠিক উত্তর: খ

9 days

উত্তর

সঠিক উত্তর: খ

9 days

খ

ব্যাখ্যা

Question: If 3 men or 9 boys can do a job in 60 days, how long will it take 11 men and 27 boys to finish the same job?

Solution:
9 জন বালক = 3 জন পুরুষ

1 জন বালক = 3/9 = 1/3 জন পুরুষ
∴ 27 জন বালক = 27/3 = 9 জন পুরুষ

11 জন পুরুষ + 27 জন বালক = (11 + 9) = 20 জন পুরুষ

3 জন পুরুষ কাজটি করতে সময় নেয় = 60 দিন
∴ 1 জন পুরুষ কাজটি করতে সময় নেয় = (60 × 3) দিন
∴ 20 জন পুরুষ কাজটি করতে সময় নেয় = (60 × 3)/20 = 9 দিন

৭,১৪২.

The HCF and LCM of two numbers are 8 and 96 respectively. If one of the two numbers is 24, what is the other one?

ক
ক) 28
খ
খ) 30
গ
গ) 32
ঘ
ঘ) 34

সঠিক উত্তর: গ

গ) 32

উত্তর

সঠিক উত্তর: গ

গ) 32

গ

ব্যাখ্যা

Question: The HCF and LCM of two numbers are 8 and 96 respectively. If one of the two numbers is 24, what is the other one?

Solution:
Let the other number be x.

We know that,
H.C.F.× L.C.M.=Product of two numbers
⇒8 × 96 = 24 × x
⇒x = 32

৭,১৪৩.

If e^x = 7, then x =?

ক
0
খ
e
গ
7
ঘ
ln 7

সঠিক উত্তর: ঘ

ln 7

উত্তর

সঠিক উত্তর: ঘ

ln 7

ঘ

ব্যাখ্যা

Question: If e^x = 7, then x =?

Solution:
Given,
e^x = 7
⇒ ln(e^x) = ln 7
∴ x = ln 7; [Formula: ln(e^x) = x]

৭,১৪৪.

A rhombus has one diagonal of 16 centimeters and an area of 192 square centimeters. What is the length of the second diagonal?

ক
12 cm
খ
16√2 cm
গ
24 cm
ঘ
32 cm

সঠিক উত্তর: গ

24 cm

উত্তর

সঠিক উত্তর: গ

24 cm

গ

ব্যাখ্যা

Question: A rhombus has one diagonal of 16 centimeters and an area of 192 square centimeters. What is the length of the second diagonal?

solution:
দেওয়া আছে,
রম্বসের ক্ষেত্রফল = 192 বর্গ সে.মি.
একটি কর্ণের দৈর্ঘ্য, d₁ = 16 সে.মি.
ধরি, অপর কর্ণের দৈর্ঘ্য = d₂ সে.মি.

আমরা জানি,
রম্বসের ক্ষেত্রফল = (1/2) × (কর্ণদ্বয়ের গুণফল)
∴ 192 = 1/2 × d₁ × d₂
⇒ 192 = 1/2 × 16 × d₂
⇒ 192 = 8 × d₂
⇒ d₂ = 192/8
∴ d₂ = 24 সে.মি.

অতএব, অপর কর্ণের দৈর্ঘ্য = 24 সে.মি.

৭,১৪৫.

Find the equation of the line passing through (2, -3) and parallel to 5x - 2y + 6 = 0.

ক
5x - 2y - 16 = 0
খ
5x - 2y - 4 = 0
গ
2x + 5y - 11 = 0
ঘ
5x + 2y + 4 = 0

সঠিক উত্তর: ক

5x - 2y - 16 = 0

উত্তর

সঠিক উত্তর: ক

5x - 2y - 16 = 0

ক

ব্যাখ্যা

Question: Find the equation of the line passing through (2, -3) and parallel to 5x - 2y + 6 = 0.

Solution:
Slope of given line:
5x - 2y + 6 = 0 ---------(1)
⇒ y = (5/2)x + 3
∴ slope, m = 5/2

If a line has slope = m and passes through a point (x₁, y₁).
Then the equation of the line is -
∴ y – y₁ = m(x – x₁)

Now, parallel to equation (1), has the same slope and pass through (2, -3).
So, the required line,
y + 3 = (5/2)(x - 2)
⇒ 2y + 6 = 5x - 10
⇒ 5x - 2y - 16 = 0

∴ the required line 5x - 2y - 16 = 0.

৭,১৪৬.

If sec²θ + tan²θ = 5/12 then, sec⁴θ - tan⁴θ = ?

ক
7/12
খ
5/12
গ
5/7
ঘ
3/8

সঠিক উত্তর: খ

5/12

উত্তর

সঠিক উত্তর: খ

5/12

খ

ব্যাখ্যা

প্রশ্ন: If sec²θ + tan²θ = 5/12 then, sec⁴θ - tan⁴θ = ?

সমাধান:
sec⁴θ - tan⁴θ
= (sec²θ - tan²θ)(sec²θ + tan²θ)
= 1 × (sec²θ + tan²θ) [cause 1 + tan²θ = sec²θ]
= 1 × (7/12)
= 5/12

৭,১৪৭.

Tk. 850 was distributed among P, Q, R, S, T in ascending order forming an arithmetic progression. T received Tk. 60 more than P. How much did R receive?

ক
Tk. 140
খ
Tk. 150
গ
Tk. 170
ঘ
Tk. 180

সঠিক উত্তর: গ

Tk. 170

উত্তর

সঠিক উত্তর: গ

Tk. 170

গ

ব্যাখ্যা

Question: Tk. 850 was distributed among P, Q, R, S, T in ascending order forming an arithmetic progression. T received Tk. 60 more than P. How much did R receive?

Solution:
Given that, P + Q + R + S + T = Tk. 850
And, T - P = 60

Now, Arithmetic progression: a, a + d, a + 2d, a + 3d, a + 4d
∴ Amount of T is = (a + 4d) and Amount of P is = a

According to the question,
⇒ a + 4d - a = 60
⇒ 4d = 60
⇒ d = 60/4
⇒ d = 15

Also,
a + (a + d) + (a + 2d) + (a + 3d) + (a + 4d) = 850
⇒ 5a + 10d = 850
⇒ 5a + 10 × 15 = 850
⇒ 5a + 150 = 850
⇒ 5a = 700
⇒ a = 140

So, amount R = a + 2d
= 140 + 2 × 15
= 140 + 30
= Tk. 170

৭,১৪৮.

The angle of elevation of a ladder leaning against a wall is 60° and the foot of the ladder is 4.6 m away from the wall. The length of the ladder is:

ক
7.9 m
খ
8.4 m
গ
5.3 m
ঘ
8.2 m
ঙ
9.2 m

সঠিক উত্তর: ঙ

9.2 m

উত্তর

সঠিক উত্তর: ঙ

9.2 m

ঙ

ব্যাখ্যা

Let AB be the wall and BC be the ladder

Then, ACB = 60°
and AC = 4.6 m.
AC/BC = cos 60° = 1/2
BC = 2 × AC = (2 × 4.6) m = 9.2 m.

৭,১৪৯.

'A' can do a work in 12 days, and 'B' in 24 days. They work together for 6 days. How much of the work is left?

ক
1/2
খ
1/4
গ
2/3
ঘ
5/8

সঠিক উত্তর: খ

1/4

উত্তর

সঠিক উত্তর: খ

1/4

খ

ব্যাখ্যা

Question: 'A' can do a work in 12 days, and 'B' in 24 days. They work together for 6 days. How much of the work is left?

সমাধান:
মনে করি,
সম্পূর্ণ কাজ = 1 অংশ

∴ A একা একদিনে করে = 1/12 অংশ।
B একা একদিনে করে = 1/24 অংশ।

∴ A ও B একসাথে একদিনে করে = (1/12) + (1/24) অংশ
= (2 + 1)/24
= 3/24
= 1/8 অংশ

∴ A ও B একসাথে 6 দিনে করে = 6 × (1/8) অংশ
= 3/4 অংশ

∴ কাজ বাকি থাকে = 1 - (3/4) অংশ
= (4 - 3)/4
= 1/4 অংশ

৭,১৫০.

A dog takes 3 leaps for every 5 leaps of a hare. If one leap of the dog is equal to 3 leaps of the hare, the ratio of the speed of the dog to that of the hare is:

ক
8 : 5
খ
9 : 7
গ
9 : 5
ঘ
8 : 7

সঠিক উত্তর: গ

9 : 5

উত্তর

সঠিক উত্তর: গ

9 : 5

গ

ব্যাখ্যা

Question: A dog takes 3 leaps for every 5 leaps of a hare. If one leap of the dog is equal to 3 leaps of the hare, the ratio of the speed of the dog to that of the hare is:

Solution:
খরগোশের 3 লাফ = কুকুরের 1 লাফ
খরগোশের 1 লাফ = কুকুরের 1/3 লাফ
খরগোশের 5 লাফ = কুকুরের 5/3 লাফ

কুকুরের গতিবেগ : খরগোশের গতিবেগ = 3 : 5/3
= 9 : 5

৭,১৫১.

The GCD of 1.08, 0.36, 0.9 is-

ক
ক) 0.03
খ
খ) 0.18
গ
গ) 0.9
ঘ
ঘ) 0.108

সঠিক উত্তর: খ

খ) 0.18

উত্তর

সঠিক উত্তর: খ

খ) 0.18

খ

ব্যাখ্যা

প্রশ্ন: 1.08, 0.36, 0.9 এর গ.সা.গু কত?

সমাধান:
প্রদত্ত সংখ্যাগুলো হলো 1.08, 0.36, 0.9
108, 36, 90 এর গ.সা.গু = 18

∴ 1.08, 0.36, 0.9 এর গ.সা.গু = 0.18

৭,১৫২.

The breadth of a room is twice its height, one half of its length and the volume of the room is 512cu.m. The length of the room :

ক
ক) 4 m
খ
খ) 8 m
গ
গ) 12 m
ঘ
ঘ) 16 m

সঠিক উত্তর: ঘ

ঘ) 16 m

উত্তর

সঠিক উত্তর: ঘ

ঘ) 16 m

ঘ

ব্যাখ্যা

Let
breadth of a room (b) = x
Then height (h)= x/2
and length (l)= 2x

Now,
Volume of the room =2x × x × x/2 = x³

Volume of the room is given as 512 m³.
⇒ x³= 512
⇒ x³= 8³
⇒ x = 8
∴ Length = 2x = 2 × 8 = 16 m

৭,১৫৩.

A biker was riding at 60 km/h, but if he had gone at 80 km/h, he could’ve gone 100 km more in the same time. How far did he actually travel?

ক
300 km
খ
250 km
গ
380 km
ঘ
290 km

সঠিক উত্তর: ক

300 km

উত্তর

সঠিক উত্তর: ক

300 km

ক

ব্যাখ্যা

Question: A biker was riding at 60 km/h, but if he had gone at 80 km/h, he could’ve gone 100 km more in the same time. How far did he actually travel?

Solution:
Let, the actual distance travelled be a km.

Then,
a/60 = (a + 100)/80
⇒ a/6 = (a + 100)/8
⇒ 6(a + 100) = 8a
⇒ 6a + 600 = 8a
⇒ 8a - 6a = 600
⇒ 2a = 600
⇒ a = 600/2
⇒ a = 300 km

So the actual distance travelled = 300 km

৭,১৫৪.

In a box, there are 9 green balls, 5 blue balls, and 4 yellow balls. One ball is picked at random. What is the probability that it is neither green nor yellow?

ক
5/18
খ
2/9
গ
1/3
ঘ
4/18

সঠিক উত্তর: ক

5/18

উত্তর

সঠিক উত্তর: ক

5/18

ক

ব্যাখ্যা

Question: In a box, there are 9 green balls, 5 blue balls, and 4 yellow balls. One ball is picked at random. What is the probability that it is neither green nor yellow?

Solution:
Here,
Green balls = 9
Blue balls = 5
Yellow balls = 4
Total balls = 9 + 5 + 4 = 18

Let, E = event that the ball drawn is neither green nor yellow
= event that the ball drawn is blue

অর্থাৎ বলটি যদি সবুজ বা হলুদ না হয় তাহলে বলটি হবে নীল।

∴ n(E) = 5
∴ P(E) = 5/18

৭,১৫৫.

Two sides of a triangle are 7 and 16. Which of the following is not the length of the third side?

ক
ক) 22
খ
খ) 17
গ
গ) 12
ঘ
ঘ) 9

সঠিক উত্তর: ঘ

ঘ) 9

উত্তর

সঠিক উত্তর: ঘ

ঘ) 9

ঘ

ব্যাখ্যা

Question: Two sides of a triangle are 7 and 16. Which of the following is not the length of the third side?

Solution:
আমরা জানি,
একটি ত্রিভুজের যেকোন দুই বাহুর সমষ্টি তৃতীয় বাহু অপেক্ষা বৃহত্তর হবে।
16 - 7 < তৃতীয় বাহু < 16 + 7
9 < তৃতীয় বাহু < 23

9 কখনো তৃতীয় বাহুর দৈর্ঘ্য হতে পারে না।

৭,১৫৬.

What is the minimum number of chocolate that must be added to the existing batch of 260 chocolate, so that the total batch can equally be divided among 3, 4 or 6 person?

ক
ক) 4
খ
খ) 8
গ
গ) 12
ঘ
ঘ) 16

সঠিক উত্তর: ক

ক) 4

উত্তর

সঠিক উত্তর: ক

ক) 4

ক

ব্যাখ্যা

Question: What is the minimum number of chocolate that must be added to the existing batch of 260 chocolate, so that the total batch can equally be divided among 3, 4 or 6 persons?

Solution:
L.C.M of 260 is = 12

Now, 260 is divided by 12 and reminder is 8.
So, chocolate must be added = 12 - 8 = 4

৭,১৫৭.

A train speeds past a pole in 20 seconds and speeds past a platform 100 meters in length in 30 seconds. What is the length of the train?

ক
ক) 100 meters
খ
খ) 150 meters
গ
গ) 180 meters
ঘ
ঘ) 200 meters

সঠিক উত্তর: ঘ

ঘ) 200 meters

উত্তর

সঠিক উত্তর: ঘ

ঘ) 200 meters

ঘ

ব্যাখ্যা

Let the length of the train be x meters and its speed be y m/sec.
Then, xy = 20
⇒ y = x/20
∴ (x+100)/30 = x/20
⇒ 30x = 20x + 2000
⇒ 10x = 2000
⇒ x = 200 meters

৭,১৫৮.

What is the solution of the inequality, -10 < 3x - 4 ≤ 8 ?

ক
(- 4, 2]
খ
(- 1, 3)
গ
(- 2, 4]
ঘ
[- 3, 5)

সঠিক উত্তর: গ

(- 2, 4]

উত্তর

সঠিক উত্তর: গ

(- 2, 4]

গ

ব্যাখ্যা

Question: What is the solution of the inequality, -10 < 3x - 4 ≤ 8 ?

Solution:
-10 < 3x - 4 ≤ 8
⇒ -10 + 4 < 3x - 4 + 4 ≤ 8 + 4
⇒ - 6 < 3x ≤ 12
⇒ - 6/3 < 3x/3 ≤ 12/3
⇒ - 2 < x ≤ 4

∴ Solution of the inequality: (- 2, 4]

The parenthesis "(" means - 2 is not included (open interval).
The bracket "]" means 4 is included (closed interval).

৭,১৫৯.

A, B and C start together from the same place to walk round a circular path of length 12km. A walks at the rate of 6 km/h, B 3 km/h and C 3/2 km/h. They will meet together at the starting place at the end of-

ক
8 hours
খ
9 hours
গ
9.5 hours
ঘ
10 hours

সঠিক উত্তর: ক

8 hours

উত্তর

সঠিক উত্তর: ক

8 hours

ক

ব্যাখ্যা

Question: A, B and C start together from the same place to walk round a circular path of length 12km. A walks at the rate of 6 km/h, B 3 km/h and C 3/2 km/h. They will meet together at the starting place at the end of-

Solution:
We know,
Time = Distance/speed

Time taken to complete the revolution
A = 12/6 = 2 hours
B = 12/3 = 4 hours
C = 12 × (2/3) = 8 hours

Now, required time = LCM of 2, 4, and 8
= 8 hours

৭,১৬০.

An employee may claim Tk. 7 for each kilometer when he travels by taxi and Tk. 6 for each kilometer when he drives his own car. If in one week he claimed Tk. 900 for travelling 135 km, how many kilometers did he drive his own car?

ক
90 km
খ
75 km
গ
55 km
ঘ
45 km

সঠিক উত্তর: ঘ

45 km

উত্তর

সঠিক উত্তর: ঘ

45 km

ঘ

ব্যাখ্যা

Question: An employee may claim Tk. 7 for each kilometer when he travels by taxi and Tk. 6 for each kilometer when he drives his own car. If in one week he claimed Tk. 900 for travelling 135 km, how many kilometers did he drive his own car?

Solution:
Let,
The distance travelled by taxi x km
The distance travelled by own car 135 - x km

Now,
7x + 6(135 - x) = 900
⇒ 7x + 810 - 6x = 900
⇒ x + 810 = 900
⇒ x = 900 - 810
∴ x = 90

∴ The distance travelled by own car 135 - 90 = 45 km

৭,১৬১.

The ratio of the cost of two articles is 7 : 3 . The first one was sold at a loss of 20% and the second one was sold at a gain of 40% What is the overall percentage of gain/loss?

ক
ক) 2% loss
খ
খ) 2% gain
গ
গ) 4% loss
ঘ
ঘ) 4% gain

সঠিক উত্তর: ক

ক) 2% loss

উত্তর

সঠিক উত্তর: ক

ক) 2% loss

ক

ব্যাখ্যা

দেওয়া আছে,
১ম টির ক্রয়মূল্য : ২য় টির ক্রয়মূল্য = 7 : 3
মনেকরি, ১ম টির ক্রয়মূল্য = 70 টাকা
এবং ২য় টির ক্রয়মূল্য = 30 টাকা
∴ মোট ক্রয়মূল্য = 70 + 30 = 100 টাকা
এখন,
১ম টির বিক্রয়মূল্য = 80/100 × 70 = 56 টাকা
২য় টির বিক্রয়মূল্য = 140/100 × 30 = 42 টাকা
∴ মোট বিক্রয়মূল্য = 56 + 42 = 98 টাকা
∴ ক্ষতি = 100 - 98 = 2 টাকা

৭,১৬২.

The slope of the line perpendicular to the line y = - 5x + 9 is-

ক
ক) 5
খ
খ) - 5
গ
গ) 1/5
ঘ
ঘ) - 1/5

সঠিক উত্তর: গ

গ) 1/5

উত্তর

সঠিক উত্তর: গ

গ) 1/5

গ

ব্যাখ্যা

প্রশ্ন: The slope of the line perpendicular to the line y = - 5x + 9 is-

সমাধান:
y = -5x + 9
⇒ y + 5x = 9
সরল রেখার উপর লম্ব রেখার সমীকরণ হবে,
5y - x = k
⇒ 5y = x + k
∴ y = x/5 + k/5

লম্ব রেখাটির ঢাল = 1/5

৭,১৬৩.

A man can row upstream at 12km/hr and downstream at 18 km/hr. The man rowing speed in still water is?

ক
5 km/h
খ
10 km/h
গ
15 km/h
ঘ
20 km/h

সঠিক উত্তর: গ

15 km/h

উত্তর

সঠিক উত্তর: গ

15 km/h

গ

ব্যাখ্যা

Speed of boat in still water = (18 + 12)/2
= 15 km/h

৭,১৬৪.

Quantity in A : 1 - (1/27) and Quantity B : (8/9) + (1/81)

ক
ক) Quantity in A is greater
খ
খ) Quantity in B is greater
গ
গ) The two quantities are equal
ঘ
ঘ) the relationship indeterminate
ঙ
ঙ) None of these

সঠিক উত্তর: ক

ক) Quantity in A is greater

উত্তর

সঠিক উত্তর: ক

ক) Quantity in A is greater

ক

ব্যাখ্যা

Question: Quantity in A : 1 - (1/27) and Quantity B : (8/9) + (1/81)

Solution:
Quantity in A : 1 - 1/27
= (27 - 1)/27
= 26/27
= (26 × 3)/(27 × 3)
= 78/81

Quantity B : 8/9 + 1/81
= (72 + 1)/81
= 73/81

Quantity in A is greater.

৭,১৬৫.

In triangle ABC, AB = AC . All of the following statements are true except

ক
ক) AB < AC + BC
খ
খ) AC < AB + BC
গ
গ) BC + AC > AB + BC
ঘ
ঘ) AC + BC = AB + BC

সঠিক উত্তর: গ

গ) BC + AC > AB + BC

উত্তর

সঠিক উত্তর: গ

গ) BC + AC > AB + BC

গ

ব্যাখ্যা

BC + AC > AB + BC can’t be true because AB = AC,
So, BC + AC > AC + BC
or, BC > BC, which is impossible

৭,১৬৬.

Two pipes A and B can fill a tank in 10 and 15 minutes respectively. If both the pipes are used together, how long will it take to fill the tank?

ক
ক) 6 minutes
খ
খ) 8 minutes
গ
গ) 9 minutes
ঘ
ঘ) 10 minutes

সঠিক উত্তর: ক

ক) 6 minutes

উত্তর

সঠিক উত্তর: ক

ক) 6 minutes

ক

ব্যাখ্যা

Question: Two pipes A and B can fill a tank in 10 and 15 minutes respectively. If both the pipes are used together, how long will it take to fill the tank?

Solution:
Part filled by A in 1 min = 1/10
Part filled by B in 1 min = 1/15
Part filled by ( A + B ) in 1 min = ( 1/10 + 1/15 ) = 5/30 = 1/6
∴ Both the pipes can fill the tank in 6 minutes.

৭,১৬৭.

What is the perimeter of the rectangle shown at the right?

ক
ক) 16
খ
খ) 24
গ
গ) 30
ঘ
ঘ) 28

সঠিক উত্তর: ঘ

ঘ) 28

উত্তর

সঠিক উত্তর: ঘ

ঘ) 28

ঘ

ব্যাখ্যা

প্রশ্ন: What is the perimeter of the rectangle shown at the right?

সমাধান:
আয়তক্ষেত্রে,
কর্ণ^২ = দৈর্ঘ্য^২ + প্রস্থ^২
⇒ 10² = দৈর্ঘ্য২ + 6²
⇒ দৈর্ঘ্য^২ = 10² - 6²
⇒ দৈর্ঘ্য^২ = 100 - 36
⇒ দৈর্ঘ্য^২ = 64
⇒ দৈর্ঘ্য = √64
∴ দৈর্ঘ্য = 8 মিটার

পরিসীমা = ২ (দৈর্ঘ্য + প্রস্থ)
= 2(6 + 8) মিটার
= (2 × 14) মিটার
= 28 মিটার

৭,১৬৮.

A tank is 7 metre long and 4 meter wide wide. At what speed should water run through a pipe 5 cm broad and 4 cm deep so that in 6 hours and 18 minutes water level in the tank rises by 4.5 meter?

ক
ক) 10 km/hr
খ
খ) 12 km/hr
গ
গ) 9 km/hr
ঘ
ঘ) 8 km/hr

সঠিক উত্তর: ক

ক) 10 km/hr

উত্তর

সঠিক উত্তর: ক

ক) 10 km/hr

ক

ব্যাখ্যা

Rate of flow of water = x cm/minute
∴ The volume of water that flowed in the in 1 minute
= (5 × 4 × x) = 20x cu.cm.

∴ The volume of water that flowed in the tank in 6 hours 18 minutes.
i.e. (6 × 60) + 18 = 378 minutes
= 2x × 378 cu. cm.

According to question,
20x × 378 = 700 × 400 × 450
⇒ x = (700 × 400 × 450)/(20 × 378) cm/minutes
⇒ x = (700 × 400 × 450 × 60)/(20 × 378 × 100000) km/hours
⇒ x = 10 km/hrs.

৭,১৬৯.

The expression (11.98 × 11.98 + 11.98 × Q + 0.02 × 0.02) will be a perfect square for Q =

ক
0.02
খ
0.2
গ
0.04
ঘ
0.4
ঙ
None

সঠিক উত্তর: গ

0.04

উত্তর

সঠিক উত্তর: গ

0.04

গ

ব্যাখ্যা

Question: The expression (11.98 × 11.98 + 11.98 × Q + 0.02 × 0.02) will be a perfect square for Q =

Solution:
We know that,
(a + b)² = a² + 2ab + b² = a × a + 2ab + b × b

Now,
(11.98 × 11.98 + 2 × 11.98 × 0.02 + 0.02 × 0.02) = (11.98 + 0.02)²

We can say that, (11.98 × 11.98 + 11.98 × Q + 0.02 × 0.02) will be a perfect square if Q = 2 × 0.02 = 0.04

৭,১৭০.

The ratio between the perimeter and the length of a rectangle is 7 : 2. If the area of the rectangle is 0.12 sq. m, what is the breadth of the rectangle?

ক
30 cm
খ
10 cm
গ
12 cm
ঘ
15 cm
ঙ
18 cm

সঠিক উত্তর: ক

30 cm

উত্তর

সঠিক উত্তর: ক

30 cm

ক

ব্যাখ্যা

2 ( length + breadth ) / length = 7/2
or, ( length + breadth ) / length = 7/4
or, 4 × breadth + 4 × length = 7 × length
∴ length = 4 × breadth / 3

Area = 0.12 square meter = 0.12 × 100 × 100 square centimeter
∴ length × breadth = 1200 square centimeter
or, breadth × 4 × breadth / 3 = 1200
or, breadth ² = 900 = 30² square centimeter
∴ breadth = 30cm

৭,১৭১.

If (2x - 2y)/(x - 4y) = 4, then find the value of (x + y)/(x + 3y) = ?

ক
ক) 5/4
খ
খ) 4/5
গ
গ) 3/5
ঘ
ঘ) 5/6

সঠিক উত্তর: খ

খ) 4/5

উত্তর

সঠিক উত্তর: খ

খ) 4/5

খ

ব্যাখ্যা

Question: If (2x - 2y)/(x - 4y) = 4, then find the value of (x + y)/(x + 3y) = ?

Solution:
(2x - 2y)/(x - 4y) = 4
⇒ 2x - 2y = 4(x - 4y)
⇒ 2x - 2y = 4x - 16y
⇒ 2x - 4x = - 16y + 2y
⇒ - 2x = - 14y
∴ x = 7y

∴ (x + y)/(x + 3y) = (7y + y)/(7y + 3y)
= 8y/10y
= 4/5

৭,১৭২.

The annual incomes and expenditures of a man and his wife are in the ratio of 5 : 3 and 3 : 1 respectively. If they decide to save equally and find a balance of Tk. 4000 at the end of the year, what was the income of the husband?

ক
ক) 2000
খ
খ) 3000
গ
গ) 4000
ঘ
ঘ) 5000
ঙ
ঙ) 6000

সঠিক উত্তর: ঘ

ঘ) 5000

উত্তর

সঠিক উত্তর: ঘ

ঘ) 5000

ঘ

ব্যাখ্যা

Let, annual income of husband and wife respectively = 5x and 3X
Annual expenditure husband and wife respectively = 3y and y
Saving of wife = (3x - y)
Saving of husband = (5x - 3y)
ATQ,
3x - y = 2000
=> y = (3x - 2000)
And,
5x - 3y = 2000
=> 5x - 3(3x - 2000) = 2000
=> 5x - 9x + 6000 = 2000
=> x = 1000
So, Husband's Income = 5 × 1000 = 5000

৭,১৭৩.

= ?

ক
secA
খ
cosecA
গ
1/cosecA
ঘ
1/tanA

সঠিক উত্তর: খ

cosecA

উত্তর

সঠিক উত্তর: খ

cosecA

খ

ব্যাখ্যা

Question: = ?

Solution:
1/{tanA√(1 - sin²A)}
= 1/(tanA × √cos²A)
= 1/(tanA × cosA)
= 1/{(sinA/cosA) × cosA}
= 1/sinA
= cosecA

৭,১৭৪.

A school only 3 classes having 20, 30 and 40 students respectively. The percentages of students passed are 30%, 50% and 60% respectively. Find the percentage of passed students of the entire school.

ক
ক) 40
খ
খ) 50
গ
গ) 60
ঘ
ঘ) 30

সঠিক উত্তর: খ

খ) 50

উত্তর

সঠিক উত্তর: খ

খ) 50

খ

ব্যাখ্যা

১ম শ্রেণিতে পাস = 20 এর 30%
= 20 এর 30/100
= 6 জন

২য় শ্রেণিতে পাস = 30 এর 50%
= 30 এর 50/100
= 15 জন

৩য় শ্রেণিতে পাস = 40 এর 60%
= 40 এর 60/100
= 24 জন

মোট পাস করে = (6 + 15 + 24)= 45 জন

এখন, মোট শিক্ষার্থী = (20 + 30 + 40) = 90 জন

:: শতকরা পাস করে = {(45/90) × 100}%
=50%

৭,১৭৫.

A refrigerator is on sale for 20% off the original price. A store-wide sale results in an additional reduction of 25%. What is the total discount based on the original price?

ক
40%
খ
60%
গ
55%
ঘ
45%

সঠিক উত্তর: ক

40%

উত্তর

সঠিক উত্তর: ক

40%

ক

ব্যাখ্যা

Question: A refrigerator is on sale for 20% off the original price. A store-wide sale results in an additional reduction of 25%. What is the total discount based on the original price?

Solution:
Let
x = original price.

First discount:
x(1 - 0.20)
= 0.80x (sale price).

Second discount:
0.80x(1 - 0.25)
= 0.60x (final sale price)

Therefore, the final price is 60% of the original price.
∴ the total discount = 1 - 0.60 = 0.40
∴ % total discount = 40%

৭,১৭৬.

If x³ = 117 + y³, y = x - 3, then x + y =?

ক
5
খ
6
গ
7
ঘ
8

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: If x³ = 117 + y³, y = x - 3, then x + y =?

Solution:
y = x - 3
⇒ x - y = 3

x³ = 117 + y³
⇒ x³ - y³ = 117
⇒ (x - y) (x² + xy + y²) = 117
⇒ 3 {(x - y)² + 2xy + xy} = 117
⇒ 3 (3² + 3xy) = 117
⇒ 9 + 3xy = 39
⇒ 3xy = 39 - 9 = 30
⇒ xy = 30/3 = 10

(x + y)² = (x - y)² + 4xy
= 3² + 4 × 10
= 9 + 40
= 49

∴ (x + y) = √49 = 7

৭,১৭৭.

x^2/3/45 = 4/x^1/3 , what is the value of x?

ক
210
খ
180
গ
160
ঘ
40

সঠিক উত্তর: খ

180

উত্তর

সঠিক উত্তর: খ

180

খ

ব্যাখ্যা

প্রশ্ন: x^2/3/45 = 4/x^1/3 , what is the value of x?

Solution:
Given that,
⇒ x^2/3/45 = 4/x^1/3
⇒ x^2/3 × x^1/3 = 4 × 45
⇒ x^{2/3 + 1/3} = 180
⇒ x^{(2 + 1)/3} = 180
⇒ x^3/3 = 180
∴ x = 180

৭,১৭৮.

What is the value of sin45° ?

ক
1
খ
1/2
গ
1/√2
ঘ
1/√3

সঠিক উত্তর: গ

1/√2

উত্তর

সঠিক উত্তর: গ

1/√2

গ

ব্যাখ্যা

Question: What is the value of sin45°?

Solution:
sin45° = 1/√2
sin30° = 1/2
sin90° = 1
sin60° = √3/2

৭,১৭৯.

The ratio of milk and water in a pot is 5:2. If the quantity of milk is 6 litre more than quantity of water, then what is the quantity of milk in the pot in litre?

ক
4
খ
16
গ
20
ঘ
10

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: The ratio of milk and water in a pot is 5:2. If the quantity of milk is 6 litre more than quantity of water, then what is the quantity of milk in the pot in litre?

Solution:
দেওয়া আছে,
একটি পাত্রে দুধ ও পানির পরিমাণের অনুপাত = 5:2
এবং পাত্রে দুধের পরিমাণ পানির পরিমাণ অপেক্ষা 6 লিটার বেশি।

মনে করি,
পাত্রে দুধের পরিমাণ 5x লিটার
পাত্রে পানির পরিমাণ 2x লিটার

প্রশ্নমতে,
5x - 2x = 6
⇒ 3x = 6
⇒ x = 6/3
∴ x = 2

∴ পাত্রে দুধের পরিমাণ (5 × 2) বা 10 লিটার।

৭,১৮০.

A mobile and a monitor have the same price. If the price of the mobile goes up by 20% and that of the monitor goes down by 10%, how much more will it cost be buy 4 mobiles and 4 computers?

ক
4%
খ
5%
গ
6%
ঘ
10%

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: A mobile and a monitor have the same price. If the price of the mobile goes up by 20% and that of the monitor goes down by 10%, how much more will it cost be buy 4 mobiles and 4 computers?

Solution:
ধরি,
মোবাইল বা মনিটরের মূল্য = ক টাকা

২০% বৃদ্ধিতে,
মোবাইলের দাম = ক + ক এর ২০%
= ক + {ক × (২০/১০০)}
= (৫ক + ক)/৫
= ১.২ক

১০% ক্ষতিতে,
মনিটরের দাম = ক - ক এর ১০%
= ক - {ক × (১০/১০০)}
= (১০ক - ক)/১০
= ০.৯ক

পূর্বে ৪টি মোবাইল ও ৪টি মনিটরের দাম = ৮ক টাকা
বর্তমানে ৪টি মোবাইল ও ৪টি মনিটরের দাম = (৪ক × ১.২) + (৪ক × ০.৯) = ৮.৪ক টাকা

∴ শতকরা বৃদ্ধি = {(৮.৪ক - ৮ক)/৮} × ১০০
= (.৪/৮) × ১০০
= ৫ টাকা বা ৫%

৭,১৮১.

What will come at the place of question mark ?
7, 26, 63, 124, 215, 342, ?

ক
421
খ
511
গ
481
ঘ
391
ঙ
527

সঠিক উত্তর: খ

511

উত্তর

সঠিক উত্তর: খ

511

খ

ব্যাখ্যা

Question: What will come at the place of question mark ?
7, 26, 63, 124, 215, 342, ?

Solution:
The terms are given in a series
(2³ - 1) = 7
(3³ - 1) = 26
(4³ - 1) = 63
(5³ - 1) = 124
(6³ - 1) = 215
(7³ - 1) = 342
So, the missing term is,
(8³ - 1) = 511.

৭,১৮২.

Rakib completes two rounds of a circular field of radius 50 metres in 5 minutes. What is his speed in m/s?

ক
ক) 2.09 m/s
খ
খ) 3.5 m/s
গ
গ) 4 m/s
ঘ
ঘ) 2.5 m/s

সঠিক উত্তর: ক

ক) 2.09 m/s

উত্তর

সঠিক উত্তর: ক

ক) 2.09 m/s

ক

ব্যাখ্যা

Question: Rakib completes two rounds of a circular field of radius 50 metres in 5 minutes. What is his speed in m/s?

Solution:
বৃত্তাকার মাঠের ব্যাসার্ধ, r = 50 metres
সময়, t = 5 minutes = 300 seconds
দুইবার সম্পূর্ণ ঘুরলে মোট অতিক্রান্ত দূরত্ব = 2 × 2πr
= (4πr)
= 4 × 3.1416 × 50
= 628.32 metres

speed = মোট দূরত্ব / মোট সময়
= ৬২৮.৩২/৩০০
= ২.০৯ মি/সে

৭,১৮৩.

18 years ago, a man was three times as old as his son. Now, the man is twice as old as his son. The sum of the present ages of the man and his son is = ?

ক
ক) 54 years
খ
খ) 72 years
গ
গ) 105 years
ঘ
ঘ) 108 years
ঙ
ঙ) 111 years

সঠিক উত্তর: ঘ

ঘ) 108 years

উত্তর

সঠিক উত্তর: ঘ

ঘ) 108 years

ঘ

ব্যাখ্যা

Let the son's age 18 years ago be x years,
Then man's age 18 years ago = 3x years
(3x+18) = 2(x+18)
⇒ 3x+18 = 2x+36
⇒ x = 18
Sum of their present ages
⇒ (3x+18+x+18) years
⇒ (4x+36) years
⇒ (4×18+36) years
⇒ 108 years

৭,১৮৪.

If REAL = 185112 and MAN = 13114, then PEACH will be equal to-

ক
165138
খ
125166
গ
148133
ঘ
145035

সঠিক উত্তর: ক

165138

উত্তর

সঠিক উত্তর: ক

165138

ক

ব্যাখ্যা

Question: If REAL = 185112 and MAN = 13114, then PEACH will be equal to-

Solution:
REAL:
R = 18
E = 5
A = 1
L = 12
So, REAL = 185112

MAN:
M = 13
A = 1
N = 14
So, MAN = 13114

Apply the same logic to PEACH
P = 16
E = 5
A = 1
C = 3
H = 8
So, PEACH = 165138

৭,১৮৫.

A pipe can fill 1/6th of a tank in 30 minutes. How much time will it take to fill two tanks?

ক
9 hours
খ
8 hours
গ
4 hours
ঘ
6 hours

সঠিক উত্তর: ঘ

6 hours

উত্তর

সঠিক উত্তর: ঘ

6 hours

ঘ

ব্যাখ্যা

Question: A pipe can fill 1/6th of a tank in 30 minutes. How much time will it take to fill two tanks?

Solution:
full tank is filled in = (6 × 30) = 180 minutes = 3 hours.

two tanks is filled in = 6 hours

৭,১৮৬.

A student multiplied a number by 3/5 instead of 5/3. What is the percentage error in the calculation?

ক
ক) 74%
খ
খ) 64%
গ
গ) 84%
ঘ
ঘ) 54%

সঠিক উত্তর: খ

খ) 64%

উত্তর

সঠিক উত্তর: খ

খ) 64%

খ

ব্যাখ্যা

Let the number be x.
The number was asked to be multiplied by 5/3.
Hence Correct result would be = x×(5/3) = 5x/3.
By mistake the multiplication is = 3x/5
Then, error = (5x/3 - 3x/5) = 16x/15

∴ Error % = (error/True value) × 100%
= {(16x/15) / (5x/3)} × 100%
= 16/25 × 100%
= 64 %

৭,১৮৭.

If A travels to his school from his house at the speed of 3 km/h, then he reaches the school 7 minutes late. If he travels at the speed of 4 km/h, he reaches the school 8 minutes earlier than school time. The distance of his school from his house is:

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

সঠিক উত্তর: খ

খ) 3 km

উত্তর

সঠিক উত্তর: খ

খ) 3 km

খ

ব্যাখ্যা

Question: If A travels to his school from his house at the speed of 3 km/h, then he reaches the school 7 minutes late. If he travels at the speed of 4 km/h, he reaches the school 8 minutes earlier than school time. The distance of his school from his house is:

Solution:
Let the distance between school and home be x km.
The difference of time when A goes school to school with these two different speeds is = 8 - (- 7) = 15 min
=15/60 hour

Now
(x/3) - (x/4) = 15/60
(4x - 3x)/12 = 1/4
x/12 = 1/4
x = 12/4
x = 3

৭,১৮৮.

In a family of 8, the men eat on average 72 kg of food and women eat on an average 50 kg of food. The men and women are equal in number. A hungry woman named Akhi joined the family for dinner and the average consumption became 67. How much did Akhi eat (in kgs)?

ক
115
খ
80
গ
90
ঘ
85
ঙ
None of these

সঠিক উত্তর: ক

115

উত্তর

সঠিক উত্তর: ক

115

ক

ব্যাখ্যা

Question: In a family of 8, the men eat on average 72 kg of food and women eat on an average 50 kg of food. The men and women are equal in number. A hungry woman named Akhi joined the family for dinner and the average consumption became 67. How much did Akhi eat (in kgs)?

Solution:

As men and women are equal so , there are 4 women and 4 men so, total consumption will be 72 × 4 = 288(by men) and 50 × 4 = 200(by women)

Total consumption = 488.

But after including Akhi the average consumption for 9 people is given to be 67.
So the total consumption will be 67 × 9 = 603.
So, Neetu’s consumption will be = 603 - 488 = 115

৭,১৮৯.

Find the equation of the line with x-intercept = 6 and y-intercept = 2.

ক
x + 3y - 6 = 0
খ
3x + y - 6 = 0
গ
x + 3y - 8 = 0
ঘ
3x + 2y - 12 = 0

সঠিক উত্তর: ক

x + 3y - 6 = 0

উত্তর

সঠিক উত্তর: ক

x + 3y - 6 = 0

ক

ব্যাখ্যা

Question: Find the equation of the line with x-intercept = 6 and y-intercept = 2.

Solution:
x- intercept = 6, so, the line passes through (6, 0)
y- intercept = 2, so, the line passes through (0, 2)

We know, the intercept form of a line is:
(x/a) + (y/b) = 1, where a = x- intercept and b = y- intercept
⇒ (x/6) + (y/2) = 1
⇒ (x + 3y)/6 = 1
⇒ x + 3y = 6
∴ x + 3y - 6 = 0

so, the equation of the line is x + 3y - 6 = 0.

৭,১৯০.

In a room of 36 people, 20 players play chess while 28 players play poker. How many players play both?

ক
48
খ
20
গ
12
ঘ
28

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: In a room of 36 people, 20 players play chess while 28 players play poker. How many players play both?

Solution:
Given that,
Total number of people in the room = 36.
Number of people who play chess = 20.
Number of people who play poker = 28.
We need to find the number of people who play both chess and poker.

We know that,
∣C ∪ P∣ = ∣C∣ + ∣P∣ - ∣C ∩ P∣
⇒ 36 = 20 + 28 - ∣C ∩ P∣
⇒ 36 = 48 - ∣C ∩ P∣
⇒ ∣C ∩ P∣ = 48 - 36
∴ ∣C ∩ P∣ = 12

৭,১৯১.

A boat covers a certain distance downstream in 1 hour, while it comes back in 7/2 hours. If the speed of the stream be 5 kmph, what is the speed of the boat in still water?

ক
ক) 13 kmph
খ
খ) 11 kmph
গ
গ) 9 kmph
ঘ
ঘ) 7 kmph

সঠিক উত্তর: গ

গ) 9 kmph

উত্তর

সঠিক উত্তর: গ

গ) 9 kmph

গ

ব্যাখ্যা

Let the speed of the boat in still water be x kmph.
Then,
Speed downstream = (x + 5) kmph,
Speed upstream = (x - 5) kmph.

Now
(x + 5) x 1 =(x - 5)(7/2)
2(x + 5) = 7(x - 5)
2x + 10 = 7x - 35
7x - 2x = 35 +10
5x = 45
x = 9 kmph

৭,১৯২.

Three numbers A, B and C are in the ratio 1 : 2 : 3. Their average is 600. If A is increased by 10% and B is decreased by 20%, then to get the average increased by 5%, C will be increased by-

ক
180 Tk
খ
280 Tk
গ
380 Tk
ঘ
580 Tk

সঠিক উত্তর: ক

180 Tk

উত্তর

সঠিক উত্তর: ক

180 Tk

ক

ব্যাখ্যা

Question: Three numbers A, B and C are in the ratio 1 : 2 : 3. Their average is 600. If A is increased by 10% and B is decreased by 20%, then to get the average increased by 5%, C will be increased by-

Solution:
Let
A= x, B = 2x, C = 3x.
Then,
x + 2x + 3x = 600 × 3
→ 6x = 1800
→ x = 300

So, A =300, B = 600, C = 900.
New value of A = 110% of 300 = 330
New value of B = 80% of 600 = 480
New average = 105% of 600 = 630

∴ New value of C = (630 × 3) - (330 + 480)=1080
→ Increase in value of C = 1080 - 900 = 180 Tk.

৭,১৯৩.

The number of positive integers which can be formed by using any number of digits from 0, 1, 2, 3, 4, 5 without repetition.

ক
ক) 1630
খ
খ) 1820
গ
গ) 1350
ঘ
ঘ) 1160

সঠিক উত্তর: ক

ক) 1630

উত্তর

সঠিক উত্তর: ক

ক) 1630

ক

ব্যাখ্যা

One digit positive numbers = 5
Two digit positive numbers = 25
Three digit positive numbers = 100
4 digit positive numbers = 300
5 digit positive numbers = 600
Six digit positive numbers = 600
Total positive numbers,
= 5 + 25 + 100 + 300 + 600 + 600
= 1630
---------------------------------------
Alternative way:

৭,১৯৪.

A train, 800metre long is running with a speed of 78 km/hr. It crosses a tunnel in 1 minute. What is the length of the tunnel (in metres)?

ক
ক) 440 metre
খ
খ) 260 metre
গ
গ) 500 metre
ঘ
ঘ) 430 metre

সঠিক উত্তর: গ

গ) 500 metre

উত্তর

সঠিক উত্তর: গ

গ) 500 metre

গ

ব্যাখ্যা

Let the length of the tunnel = x metre
Then, distance = (800 + x) metre.
Time = 1 minute = 60 seconds.

∴ Speed = 78 km/hr
= 78 × (5/18)
= (65/3) m/s

According to the question,
800 + x = {60 × (65/3)}
⇒ 800 + x = 1300
⇒ x = 500 metre.
Hence, The length of the tunnel is 500 metre.

৭,১৯৫.

x + y = 5, x + 4y = 4 what is the Value 4x² + 20xy + 16y²?

ক
ক) 60
খ
খ) 40
গ
গ) 20
ঘ
ঘ) 80

সঠিক উত্তর: ঘ

ঘ) 80

উত্তর

সঠিক উত্তর: ঘ

ঘ) 80

ঘ

ব্যাখ্যা

4x² + 20xy + 16y²
= 4x² + 4xy + 16xy + 16y²
= 4x(x + y) + 16y(x + y)
= (x + y)(4x + 16y)
= 4(x + y)(x + 4y)
= 4.5.4
= 80

৭,১৯৬.

If n be any natural number then by which largest number (n³ - n) is always divisible?

ক
ক) 3
খ
খ) 6
গ
গ) 12
ঘ
ঘ) 18

সঠিক উত্তর: খ

খ) 6

উত্তর

সঠিক উত্তর: খ

খ) 6

খ

ব্যাখ্যা

⇒ Let n = 1 then, (n³ – n) = 1³ - 1= 0
⇒ n = 2 then, (n³ – n) = 2³ - 2= 6
⇒ n = 3 then, (n³– n) = 3³ - 3 = 24

∴ From above we can say that (n³ – n) is always divisible by 6 if n is integer.

৭,১৯৭.

Find the area of the trapezium if height is 5 cm and AB and CD are given as 10 and 6 cm respectively.

ক
40 cm²
খ
60 cm²
গ
90 cm²
ঘ
80 cm²

সঠিক উত্তর: ক

40 cm²

উত্তর

সঠিক উত্তর: ক

40 cm²

ক

ব্যাখ্যা

Question: Find the area of the trapezium if height is 5 cm and AB and CD are given as 10 and 6 cm respectively.

Solution:
Given,
AB = 10cm, CD = 6cm, height = 5cm

According to the formulae,
Area of Trapezium = (1/2) h (AB+CD)
⇒ Area of Trapezium = 1/2 × 5 × (10 + 6)
⇒ Area of Trapezium = 40 cm²

৭,১৯৮.

একটি ত্রিভুজাকৃতি কক্ষের বাহুগুলোর দৈর্ঘ্য যথাক্রমে ৫ মি. ১২ মি. ১৩ মি. হলে এর ক্ষেত্রফল কত?

ক
৫০ বর্গমিটার
খ
৪৫ বর্গমিটার
গ
৬০ বর্গমিটার
ঘ
৩০ বর্গমিটার
ঙ
৬৫ বর্গমিটার

সঠিক উত্তর: ঘ

৩০ বর্গমিটার

উত্তর

সঠিক উত্তর: ঘ

৩০ বর্গমিটার

ঘ

ব্যাখ্যা

প্রশ্ন: একটি ত্রিভুজাকৃতি কক্ষের বাহুগুলোর দৈর্ঘ্য যথাক্রমে ৫ মি. ১২ মি. ১৩ মি. হলে এর ক্ষেত্রফল কত?

সমাধান:
ধরি, বাহুগুলোর দৈর্ঘ্য a = ৫ মি. B = ১২ মি. C = ১৩ মি.
পরিসীমা, ২s = (৫ + ১২ + ১৩) মি.
বা, s = ৩০/২ মি. = ১৫ মি.

আমরা জানি,
ক্ষেত্রফল = √{s(s - a) (s - b) (s - c)}
= √{১৫ (১৫ - ৫) (১৫ - ১২) (১৫ - ১৩)} বর্গমিটার
= √{১৫ × ১০ × ৩ × ২} বর্গমিটার
= √{৯০০} বর্গমিটার
= ৩০ বর্গমিটার

৭,১৯৯.

If p + q = √7 and p - q = √5 the, 8pq(p² + q²) = ?

ক
16
খ
42
গ
48
ঘ
24

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: If p + q = √7 and p - q = √5 the, 8pq(p² + q²) = ?

Solution:
Given,
p + q = √7
and p - q = √5

Now,
8pq(p² + q²)
= 4pq × 2(p² + q²)
= {(p + q)² - (p - q)²}{(p + q)² + (p - q)²}
= {(√7)² - (√5)²}{(√7)² + (√5)²}
= (7 - 5)(7 + 5)
= 2 × 12
= 24

৭,২০০.

The product and LCM of two numbers are 54 and 18. Find their HCF-

ক
3
খ
2
গ
7
ঘ
6

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: The product and LCM of two numbers are 54 and 18. Find their HCF-

Solution:
আমরা জানি,
দুইটি সংখ্যার গুণফল = ল.সা.গু × গ.সা.গু।
গ.সা.গু = দুইটি সংখ্যার গুণফল/ল.সা.গু
গ.সা.গু = ৫৪/১৮
গ.সা.গু = ৩