Bank Math

PrepBank · পাতা ১১১ / ১৬১ · ১১,০০১–১১,১০০ / ১৬,১২৪

১১,০০১.

Jalal bought a mobile phone for Taka. 8000 and then sold it at a loss of Taka. 1200. What was the selling price of the mobile phone?

ক
Taka. 5800
খ
Taka. 6800
গ
Taka. 7000
ঘ
Taka. 5600

সঠিক উত্তর: খ

Taka. 6800

উত্তর

সঠিক উত্তর: খ

Taka. 6800

খ

ব্যাখ্যা

Question: Jalal bought a mobile phone for Taka. 8000 and then sold it at a loss of Taka. 1200. What was the selling price of the mobile phone?

Answer:
Given,
Cost Price (CP) = 8000 Taka
Loss = 1200 Taka
Using Formula
Selling Price (SP) = Cost Price - Loss
⇒ Selling Price (SP) = Taka. 8000 - Taka. 1200
⇒ Selling Price (SP) = Taka. 6800
Therefore, selling price of the mobile phone is Taka. 6800.

১১,০০২.

Which of the following fractions is the largest?

ক
3/2
খ
7/4
গ
5/3
ঘ
6/5

সঠিক উত্তর: খ

7/4

উত্তর

সঠিক উত্তর: খ

7/4

খ

ব্যাখ্যা

Question: Which of the following fractions is the largest?

Solution:
3/2 = 1.5
7/4 = 1.75
5/3 = 1.66
6/5 = 1.2
Hence the largest fraction is 7/4

১১,০০৩.

tanA√(1 - sin²A) = ?

ক
1
খ
sinA
গ
cosA
ঘ
0

সঠিক উত্তর: খ

sinA

উত্তর

সঠিক উত্তর: খ

sinA

খ

ব্যাখ্যা

Question: tanA√(1 - sin²A) = ?

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

১১,০০৪.

A man spent 1/2 of his money and then lost 1/4 of the reminder. He was left with Tk. 3,600. How much did he start with?

ক
ক) 7,200
খ
খ) 8,800
গ
গ) 9,600
ঘ
ঘ) 10,400

সঠিক উত্তর: গ

গ) 9,600

উত্তর

সঠিক উত্তর: গ

গ) 9,600

গ

ব্যাখ্যা

Question: A man spent 1/2 of his money and then lost 1/4 of the reminder. He was left with Tk. 3,600. How much did he start with?

Solution:
ধরি
মোট ছিল = x টাকা

সে খরচ করলো = x/2 টাকা
অবশিষ্ট থাকে = x - (x/2)
= (2x - x)/2
= x/2

হারিয়ে ফেলে = x/2 এর 1/4 = x/8

অবশিষ্ট থাকে = (x/2) - (x/8)
= (4x - x)/8
= 3x/8

প্রশ্নমতে,
3x/8 = 3600
3x = 3600 × 8
x = (3600 × 8)/3
x = 9600 টাকা

১১,০০৫.

If (2 + √x) > 2√x,
Which of these statements cannot be false?

ক
x < 4
খ
x < 1
গ
x < 3
ঘ
x < 2
ঙ
x < 5

সঠিক উত্তর: ক

x < 4

উত্তর

সঠিক উত্তর: ক

x < 4

ক

ব্যাখ্যা

Question: If (2 + √x) > 2√x,
Which of these statements cannot be false?

Solution:
2 + √x > 2√x
⇒ 2 > 2√x - √x
⇒ 2 > √x
⇒ 4 > x
∴ x < 4

১১,০০৬.

A motorist travels to a place 150 km away at in average speed of 50 km and returns at 30 km per hour. His average speed for the whole journey in km per hour is:

ক
35
খ
37
গ
37.5
ঘ
40

সঠিক উত্তর: গ

37.5

উত্তর

সঠিক উত্তর: গ

37.5

গ

ব্যাখ্যা

Question: A motorist travels to a place 150 km away at in average speed of 50 km and returns at 30 km per hour. His average speed for the whole journey in km per hour is:

Solution:
50 কি.মি. যেতে সময় লাগে = 1 ঘণ্টা
150 কি.মি. যেতে সময় লাগে = 150 /50 = 3 ঘণ্টা

30 কি.মি. ফিরে আসতে সময় লাগে = 1 ঘণ্টা
150 কি.মি. ফিরে আসতে সময় লাগে = 150/30 = 5 ঘণ্টা

গড় বেগ = (150 + 150)/(3 + 5) কি.মি./ঘণ্টা
= 300/8 কি.মি./ঘণ্টা
= 37.5 কি.মি./ঘণ্টা

১১,০০৭.

Given that (1² + 2² + 3² + .......... + 10²) = 385, then the value of (2² + 4² + 6² + .......... + 20²) is equal to = ?

ক
1480
খ
1520
গ
1540
ঘ
None of these

সঠিক উত্তর: গ

1540

উত্তর

সঠিক উত্তর: গ

1540

গ

ব্যাখ্যা

Question: Given that (1² + 2² + 3² + .......... + 10²) = 385, then the value of (2² + 4² + 6² + .......... + 20²) is equal to = ?

Solution:
(2² + 4² + 6² + .......... + 20²)
= (1 × 2)² + (2 × 2)² + (3 × 2)² + ............. + (2 × 10)²
= (1² × 2²) + (2² × 2²) + (3² × 2²) ............ + (2² × 10²)
= 2²(1² + 2² + 3² + .......... + 10²)
= 4 × 385 [Putting the value (1² + 2² + 3² + .......... + 10²) = 385]
= 1540

১১,০০৮.

A solution contains 200 liters with 25% salt. How many liters of water must be evaporated so that the concentration of salt in the solution increases to 50%?

ক
100 liters
খ
120 liters
গ
115 liters
ঘ
200 liters
ঙ
None

সঠিক উত্তর: ক

100 liters

উত্তর

সঠিক উত্তর: ক

100 liters

ক

ব্যাখ্যা

Question: A solution contains 200 liters with 25% salt. How many liters of water must be evaporated so that the concentration of salt in the solution increases to 50%?

Solution:
200 লিটারের 25% লবণ রয়েছে, অর্থাৎ,
(25/100) × 200 = 50 লিটার লবণ

∴ পানির পরিমাণ = 200 - 50 = 150 লিটার

ধরি,
x লিটার পানি বাষ্পীভূত করতে হবে।
তাহলে নতুন দ্রবণের পরিমাণ হবে, 200 - x লিটার

প্রশ্নমতে,
50/(200 - x) = 50/100
⇒ 100 × 50 = 50(200 - x)
⇒ 5000 = 10000 - 50x
⇒ 50x = 10000 - 5000
⇒ 50x = 5000
⇒ x = 5000/50
⇒ x = 100

∴ 100 লিটার পানি বাষ্পীভূত করতে হবে যাতে লবণের পরিমাণ ৫০% হয়।

১১,০০৯.

A and B undertake to do a piece of work for Tk. 450. A can do it in 20 days and B can do it in 40 days. With the help of C, they finish it in 8 days. How much should C be paid for his contribution?

ক
Tk. 180
খ
Tk. 40
গ
Tk. 120
ঘ
Tk. 60
ঙ
Tk. 50

সঠিক উত্তর: ক

Tk. 180

উত্তর

সঠিক উত্তর: ক

Tk. 180

ক

ব্যাখ্যা

Question: A and B undertake to do a piece of work for Tk. 450. A can do it in 20 days and B can do it in 40 days. With the help of C, they finish it in 8 days. How much should C be paid for his contribution?

Solution:
A & B would have done 8/20 & 8/40 of the work respectively in 8 days.
Together they have done 3/5th of the work.
This implies that C has done 2/5th of the work.
Thus, C should be paid 2/5th of the amount i.e. 450 × (2/5) = Tk. 180.

১১,০১০.

x% of y is y% of what-

ক
x
খ
10x
গ
100x
ঘ
x/100

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: x% of y is y% of what-

Solution:
let, the number p

x% of y = y% of p
⇒ xy/100 = py/100
∴ p = x

১১,০১১.

ক
21√5
খ
23√5
গ
34√5
ঘ
40√5

সঠিক উত্তর: ঘ

40√5

উত্তর

সঠিক উত্তর: ঘ

40√5

ঘ

ব্যাখ্যা

Question:

Solution:

১১,০১২.

Tk. 9800 are invested partly in 9% stock at 75 and 10% stock at 80 to have an equal amount of income. The investment in 9% stock is

ক
ক) Tk. 4800
খ
খ) Tk. 5000
গ
গ) Tk. 5200
ঘ
ঘ) Tk. 5400

সঠিক উত্তর: খ

খ) Tk. 5000

উত্তর

সঠিক উত্তর: খ

খ) Tk. 5000

খ

ব্যাখ্যা

Question: Tk. 9800 are invested partly in 9% stock at 75 and 10% stock at 80 to have an equal amount of income. The investment in 9% stock is:

Solution:
Let,
the investment in 9% stock be tk. x.
Then, investment in 10% stock = tk.(9800 - x)

According to the question,
(9/75) × x = (10/80) × (9800 - x)
⇒ 3x/25 = (9800 - x)/8
⇒ 24x = (9800 × 25) - 25x
⇒ 24x + 25x = 245000
⇒ 49x = 245000
⇒ x = 245000/49
∴ x = 5000

১১,০১৩.

If the cost price of 120 mangoes is equal to the selling price of 100 mangoes, what is the profit percent in this transaction?

ক
21%
খ
20%
গ
17%
ঘ
15%

সঠিক উত্তর: খ

20%

উত্তর

সঠিক উত্তর: খ

20%

খ

ব্যাখ্যা

Question: If the cost price of 120 mangoes is equal to the selling price of 100 mangoes, what is the profit percent in this transaction?

Solution:
Let the C.P. of 120 mangoes be Tk. 120
As per question,
S.P. of 100 mangoes = Tk. 120

C.P. of 100 mangoes would be = Tk. 100
∴ Profit = (S.P. - C.P.) = 120 - 100 = 20

Profit percent = (20/100) × 100%
= 20%

১১,০১৪.

A number x is 32% of a number y. If y is 20% of z, what is z in terms of x?

ক
0.064x
খ
0.64x
গ
6.4x
ঘ
15.625x

সঠিক উত্তর: ঘ

15.625x

উত্তর

সঠিক উত্তর: ঘ

15.625x

ঘ

ব্যাখ্যা

Question: A number x is 32% of a number y. If y is 20% of z, what is z in terms of x?

Solution:
x = (32/100) × y
⇒ x = (32y)/100
∴ 100x = 32y ..............(1)

y = (20/100) × z
⇒ y = (20z)/100
∴ y = z/5

Put the value of y in equation (1),
100x = 32 × (z/5)
⇒ 100x = (32z)/5
⇒ 500x = 32z
⇒ z = 500x/32
⇒ z = 15.625x

১১,০১৫.

A teacher accidentally entered a student's marks as 82 instead of 67. Due to this error, the class average increased by 0.3 marks. Find the number of students in the class.

ক
50
খ
55
গ
57
ঘ
60
ঙ
62

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: A teacher accidentally entered a student's marks as 82 instead of 67. Due to this error, the class average increased by 0.3 marks. Find the number of students in the class.

Solution:
Let the number of students in the class be x.
Total increase in marks = x × 0.3
According to the question
⇒ x × 0.3 = (82 - 67)
⇒ x × 0.3 = 15
⇒ x = 15/0.3
⇒ x = 50

১১,০১৬.

It takes two pipes X and Y, running together, to fill a tank in 6 minutes. It takes A 5 minutes less than Y to fill the tank, then what will be the time taken by Y alone to fill the tank?

ক
15 minutes
খ
20 minutes
গ
18 minutes
ঘ
25 minutes

সঠিক উত্তর: ক

15 minutes

উত্তর

সঠিক উত্তর: ক

15 minutes

ক

ব্যাখ্যা

Question: It takes two pipes X and Y, running together, to fill a tank in 6 minutes. It takes A 5 minutes less than Y to fill the tank, then what will be the time taken by Y alone to fill the tank?

Solution:
Let the time taken by pipe X to fill the tank be a minutes
Time is taken by pipe Y to fill the tank = a + 5 minutes

So,
⇒ (1/a) + {1/(a + 5)} = 1/6
⇒ (2a + 5)/a(a + 5) = 1/6
⇒ a² + 5a - 12a - 30 = 0
⇒ a² - 7a - 30 = 0
⇒ (a - 10)(a + 3) = 0
⇒ a = 10, - 3
∴ a = 10 ; [neglecting the negative value of a]

Thus, time taken by Y alone to fill the tank is 10 + 5 = 15 minutes

১১,০১৭.

If the sum of x and its multiplicative inverse is 3, then x³ + 1/x³ = ?

ক
ক) 36
খ
খ) 0
গ
গ) 1
ঘ
ঘ) 18

সঠিক উত্তর: ঘ

ঘ) 18

উত্তর

সঠিক উত্তর: ঘ

ঘ) 18

ঘ

ব্যাখ্যা

Given, x + 1/x = 3
x³ + 1/x³ = (x + 1/x)³ - 3.x.1/x(x + 1/x)
= (3)³ - 3(3)
= 18

১১,০১৮.

A book is listed at Tk. 920. A customer pays Tk. 742.90 for it after receiving two successive discounts. If the first discount is 15%, what is the rate of the second discount?

ক
5%
খ
7%
গ
8%
ঘ
3%
ঙ
None

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: A book is listed at Tk. 920. A customer pays Tk. 742.90 for it after receiving two successive discounts. If the first discount is 15%, what is the rate of the second discount?

Solution:
MP = 920
After first discount Marked Price (MP) become,
= 920 - 15% of 920 = 782

The Selling Price (SP) = 742.90

Let,
second discount was x% on 782
⇒ 782 - x% of 782 = 742.90
⇒ 782x/100 = 39.1
⇒ 782x = 3910
⇒ x = 5%

∴ Second Discount = 5%

১১,০১৯.

A sum of Tk. 4000 will amount to TK. 4410 in 2 years if the interest is calculated every year. What is the rate of compound interest?

ক
4%
খ
5%
গ
6%
ঘ
8%
ঙ
10%

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: A sum of Tk. 4000 will amount to TK. 4410 in 2 years if the interest is calculated every year. What is the rate of compound interest?

Solution:
Here,
Principal, P = 4000 Tk.
Amount, A = 4410 Tk.
Time, n = 2 years
Let, Rate = r

By using formula,
A = P[1 + (r/100)]ⁿ
∴ 4410 = 4000 [1 + (r/100)]²
⇒ 441/400 = [1 + (r/100)]²
⇒ 21/20 = 1 + (r/100) [Taking square root of both sides]
⇒ 21/20 = (100 + r)/100
⇒ 20r + 2000 = 2100
⇒ 20r = 2100 - 2000
⇒ 20r = 100
⇒ r = 100/20
⇒ r = 5

∴ Rate of compound interest is 5%

১১,০২০.

In the morning, Mr. Parimal was standing facing a pole. The shadow of the pole fell on his right side. What direction was Mr. Parimal facing?

ক
East
খ
West
গ
North
ঘ
South

সঠিক উত্তর: ঘ

South

উত্তর

সঠিক উত্তর: ঘ

South

ঘ

ব্যাখ্যা

Question: In the morning, Mr. Parimal was standing facing a pole. The shadow of the pole fell on his right side. What direction was Mr. Parimal facing?

Solution:
In morning, the sun rises in east. And the shadow of the pole fell on his right side. So the right side of Mr. Parimal is west.
As his left side is east and right side is west so he is facing toward south.

১১,০২১.

Two dice are tossed. The probability that the total score is a prime number is:

ক
7/12
খ
6/13
গ
5/12
ঘ
7/13

সঠিক উত্তর: গ

5/12

উত্তর

সঠিক উত্তর: গ

5/12

গ

ব্যাখ্যা

Question: Two dice are tossed. The probability that the total score is a prime number is:

Solution:
n(S) = (6 × 6) = 36.
Let E = Event that the sum is a prime number.
Then E= {(1, 1), (1, 2), (1, 4), (1, 6), (2, 1), (2, 3), (2, 5), (3, 2), (3, 4), (4, 1), (4,3),(5, 2), (5, 6), (6, 1), (6, 5)}
n(E) = 15.

∴ P(E) = n(E)/n(S)
= 15/36
= 5/12

১১,০২২.

Two pipes can fill a tank in 20 and 24 minutes respectively and a waste pipe can empty 3 gallons per minute. All three pipes working together can fill the tank in 15 minutes. The capacity of the tank is:

ক
120 gallons
খ
110 gallons
গ
100 gallons
ঘ
80 gallons
ঙ
None of the above

সঠিক উত্তর: ক

120 gallons

উত্তর

সঠিক উত্তর: ক

120 gallons

ক

ব্যাখ্যা

Question: Two pipes can fill a tank in 20 and 24 minutes respectively and a waste pipe can empty 3 gallons per minute. All three pipes working together can fill the tank in 15 minutes. The capacity of the tank is:

Solution:
Let the waste pipe empty the tank in x minutes.

According to the question,
(1/20 + 1/24) - 1/x = 1/15
⇒ 1/x = (1/20 + 1/24) - 1/15
⇒ 1/x = 1/40
∴ x = 40 minutes

A waste pipe can empty 3 gallons per minute
In 40 minutes it can empty = 3 × 40 = 120 gallons.

∴ Capacity of the tank = 120 gallons.

১১,০২৩.

একটি ট্রেনের গতিবেগ ৬০কি.মি./ঘণ্টা। ট্রেনটির দৈর্ঘ্য ১০০ মিটার। ১৪০ মিটার দৈর্ঘ্যের একটি ব্রিজ অতিক্রম করতে ট্রেনটির কত সময় লাগবে?

ক
ক) ১০.৪ সেকেন্ড
খ
খ) ১২.৪ সেকেন্ড
গ
গ) ১৬.৪ সেকেন্ড
ঘ
ঘ) ১৪.৪ সেকেন্ড

সঠিক উত্তর: ঘ

ঘ) ১৪.৪ সেকেন্ড

উত্তর

সঠিক উত্তর: ঘ

ঘ) ১৪.৪ সেকেন্ড

ঘ

ব্যাখ্যা

ট্রেনটির মোট অতিক্রম করতে হবে = (১০০ + ১৪০) মিটার = ২৪০ মিটার

ট্রেনের গতিবেগ = ৬০কি.মি./ঘণ্টা
= (৬০ × ১০০০)/৩৬০০ মিটার/সেকেন্ড
= ৫০/৩ মিটার/সেকেন্ড
মোট সময় লাগবে = ২৪০/(৫০/৩)
= ৭২/৫
= ১৪.৪ সেকেন্ড

১১,০২৪.

The average of 20 numbers is zero. Of them, at the most, how many may be greater than zero?

ক
1
খ
10
গ
11
ঘ
19
ঙ
None

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: The average of 20 numbers is zero. Of them, at the most, how many may be greater than zero?

Solution:
Mean of numbers = 0
∴ sum of 20 numbers = 0 × 20 =0
It is possible that 19 of these numbers may be positive and their sum is a , the 20th number is (-a).

Means of 20 numbers would be = (n₁ + n₂ + ... + n₂₀)/20 =0
Now we take at most case -
let us assume n₁, n₂, ...n₁₉ are greater then 0
then
n₂₀ = - (n₁ + n₂ +...+ n₁₉)
Hence, in at most case there are 19 elements which is greater than 0. So, there are 19 numbers which are greater than zero

১১,০২৫.

The interest on a certain deposit at 4.5% p.a. is TK. 202.50 in one year. How much will the additional interest in one year be on the same deposit at 5% p.a. ?

ক
ক) Tk. 20.50
খ
খ) Tk. 24.50
গ
গ) Tk. 25.50
ঘ
ঘ) Tk. 22.50

সঠিক উত্তর: ঘ

ঘ) Tk. 22.50

উত্তর

সঠিক উত্তর: ঘ

ঘ) Tk. 22.50

ঘ

ব্যাখ্যা

S.I = TK. 202.50
r = 4.5%
n = 1

Now
I = pnr/100
P = I × 100/nr = ( 202.50 × 100)/( 1× 4.5) = 4500 Tk.

Again
I = pnr
= 4500 × 1 × 5/100 = 225

The total additional interest = ( 225 - 202.50) Tk. = Tk. 22.50

১১,০২৬.

Even after reducing the marked price of a T.V by Tk. 320, a shopkeeper makes a profit of 15%. If the cost price be Tk. 3200, what percentage of profit would he have made if he had sold the T.V at the marked price?

ক
10%
খ
20%
গ
25%
ঘ
16(2/3) %
ঙ
50%

সঠিক উত্তর: গ

25%

উত্তর

সঠিক উত্তর: গ

25%

গ

ব্যাখ্যা

15% of 3200 = 480
MP = 3200 + 480 + 320 = 4000

P = 4000 - 3200 = 800
P% = 8003200 × 100

১১,০২৭.

The average of a class of 39 students is 15 years. If the age of the teacher is included, then the average increase's by 3 months. Find the age of the teacher.

ক
ক) 25 years
খ
খ) 30 years
গ
গ) 32 years
ঘ
ঘ) 40 years

সঠিক উত্তর: ক

ক) 25 years

উত্তর

সঠিক উত্তর: ক

ক) 25 years

ক

ব্যাখ্যা

Question: The average of a class of 39 students is 15 years. If the age of the teacher is included, then the average increase's by 3 months. Find the age of the teacher.

Solution:
Total age of 39 persons = 39 × 15 = 585 years
The average age of 40 persons = 15 years + 3 months = 15 + (3/12) = 61/4 years

Total age of 40 person = (61/4) × 40 = 610 years

So, the age of the teacher = 610 - 585 = 25 years

১১,০২৮.

The average of four consecutive even numbers is 27. Find the smallest of these numbers.

ক
24
খ
30
গ
20
ঘ
40

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Consider the consecutive even numbers as : x, (x + 2), (x + 4) and (x+ 6)
Average = Sum of Quantities/Number of Quantities
{x + (x + 2) + (x + 4) + (x + 6)}/4 = 27
⇒ (4x + 12)/4 = 27
⇒ x + 3 = 27
⇒ x = 27 - 3
⇒ x = 24.

Therefore,
Largest number = (x + 6) = (24 + 6) = 30
Smallest number = 24.
Hence, the answer is 24.

১১,০২৯.

√(0.01 + √0.0064)=?

ক
ক) 0.03
খ
খ) 0.42
গ
গ) 0.3
ঘ
ঘ) None of these

সঠিক উত্তর: গ

গ) 0.3

উত্তর

সঠিক উত্তর: গ

গ) 0.3

গ

ব্যাখ্যা

√(0.01 + √0.0064)
= √(0.01 + 0.08)
= √(0.09)
= 0.3

১১,০৩০.

A shopkeeper sells one shirt for Tk.840 at a gain of 20% and another for Tk.960 at a loss of 4%. His total gain or loss percent is-

ক
ক) 80/17%
খ
খ) 50/17%
গ
গ) 100/17%
ঘ
ঘ) 200/17%

সঠিক উত্তর: গ

গ) 100/17%

উত্তর

সঠিক উত্তর: গ

গ) 100/17%

গ

ব্যাখ্যা

Question: A shopkeeper sells one shirt for Tk.840 at a gain of 20% and another for Tk.960 at a loss of 4%. His total gain or loss percent is-

Solution:
C.P. of 1^stshirt=Tk.(100 × 840)/120=Tk.700.
C.P. of 2^nd shirt=Tk.(100 × 960)/96=Tk.1000

So, total C.P.=Tk.(700 + 1000)=Tk.1700.
Total S.P.=Tk.(840 + 960)=Tk.1800.

∴Gain% =((100 × 100)/1700}% =100/17%

১১,০৩১.

If 7sin²θ + 3cos²θ = 4, then tan30° = ?

ক
1/√3
খ
1/3
গ
1/√2
ঘ
2/3

সঠিক উত্তর: ক

1/√3

উত্তর

সঠিক উত্তর: ক

1/√3

ক

ব্যাখ্যা

Question: If 7sin²θ + 3cos²θ = 4, then tan30° = ?

Solution:
7sin²θ + 3cos²θ = 4
⇒ 7sin²θ + 3(1 - sin²θ) = 4
⇒ 7sin²θ + 3 - 3sin²θ = 4
⇒ 4sin²θ = 1
⇒ sin²θ = 1/4
⇒ sinθ = 1/2
⇒ sinθ = sin30°
⇒ θ = 30°
⇒ tanθ = tan30°
∴ tanθ = 1/√3

১১,০৩২.

In how many ways can all the letters of the word 'LEADER' be arranged?

ক
70
খ
180
গ
240
ঘ
360

সঠিক উত্তর: ঘ

360

উত্তর

সঠিক উত্তর: ঘ

360

ঘ

ব্যাখ্যা

Question: In how many ways can all the letters of the word 'LEADER' be arranged?

Solution:
এখানে, মোট বর্ণ সংখ্যা 6 টি যার মধ্যে 2 টি E এবং বাকি L, A, D, R প্রত্যেকটি 1 টি করে আছে।
∴ বিন্যাস সংখ্যা = 6!/2!
= (6 × 5 × 4 × 3 × 2!)/2!
= 6 × 5 × 4 × 3
= 360

১১,০৩৩.

A shopkeeper mixed low-quality vegetable oil costing Tk. 40 per litre with sunflower refined oil costing Tk. 80 per litre in a ratio of 2 ∶ 3 respectively. If he sold the mixture at Tk. 100 per litre, find his profit percentage.

ক
42.75%
খ
47.5%
গ
51.5%
ঘ
56.25%

সঠিক উত্তর: ঘ

56.25%

উত্তর

সঠিক উত্তর: ঘ

56.25%

ঘ

ব্যাখ্যা

Solution: A shopkeeper mixed low-quality vegetable oil costing Tk. 40 per litre with sunflower refined oil costing Tk. 80 per litre in a ratio of 2 ∶ 3 respectively. If he sold the mixture at Tk. 100 per litre, find his profit percentage.

Solution:
Let,
Quantity of low-quality vegetable oil in mixure 2x litre.
∴ The costing price of low-quality vegetable oil Tk. (40 × 2x) = Tk. 80x

Quantity of sunflower refined oil in mixure 3x litre.
∴ The costing price of sunflower refined oil Tk. (80 × 3x) = Tk. 240x

∴ Total costing price = Tk. (80x + 240x) = Tk. 320x

Total selling price = Tk. (100 × 5x) = Tk. 500x

∴ Profit = Tk. (500 - 320) = Tk. 180

∴ Profit percentage = (180/320) × 100%
= 56.25%

১১,০৩৪.

A sum of money amounts of Tk. 460 in 3 years and to Tk. 500 in five years. Find the rate percent per annum

ক
ক) 5 Tk
খ
খ) 10 Tk
গ
গ) 15 Tk
ঘ
ঘ) 25 Tk

সঠিক উত্তর: ক

ক) 5 Tk

উত্তর

সঠিক উত্তর: ক

ক) 5 Tk

ক

ব্যাখ্যা

৫ বছরের সুদ + আসল = ৫০০ টাকা
৩ বছরের সুদ + আসল = ৪৬০ টাকা

২ বছরের সুদ = ৪০ টাকা
৩ বছরের সুদ = (৪০ ×৩)/২ টাকা
= ৬০ টাকা
আসল = (৪৬০ - ৬০) টাকা = ৪০০ টাকা

৪০০ টাকায় ৩ বছরের সুদ ৬০ টাকা
১ টাকায় ১ বছরের সুদ ৬০/(৪০০ × ৩) টাকা
১০০ টাকায় ১ বছরের সুদ (৬০ × ১০০)/(৪০০ × ৩) টাকা
= ৫ টাকা

১১,০৩৫.

Find the odd man out.
396, 462, 572, 427, 671, 264

ক
396
খ
427
গ
671
ঘ
264

সঠিক উত্তর: খ

427

উত্তর

সঠিক উত্তর: খ

427

খ

ব্যাখ্যা

Question: Find the odd man out.
396, 462, 572, 427, 671, 264

Solution:
In each number except 427, the middle digit is the sum of other two.

১১,০৩৬.

The value of
is = ?

ক
1
খ
0
গ
3
ঘ
- 1

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: The value of
is = ?

Solution:

১১,০৩৭.

Solve the inequality |x - 2| < 5

ক
- 3 < x < 7
খ
3 < x < 7
গ
- 3 < x < - 7
ঘ
3 < x < - 7
ঙ
none

সঠিক উত্তর: ক

- 3 < x < 7

উত্তর

সঠিক উত্তর: ক

- 3 < x < 7

ক

ব্যাখ্যা

Question: Solve the inequality |x - 2| < 5

Solution:
|x - 2| < 5
⇒ - 5 < x - 2 < 5
⇒ - 5 + 2 < x - 2 + 2 < 5 + 2
⇒ - 3 < x < 7

১১,০৩৮.

X can do a piece of work in 40 days. He works at it for 8 days and then Y finishes it in 16 days. How long will they together take to complete the work?

ক
ক) 13(1/3) days
খ
খ) 15 days
গ
গ) 20 days
ঘ
ঘ) 26 days
ঙ
ঙ) None of Above

সঠিক উত্তর: ক

ক) 13(1/3) days

উত্তর

সঠিক উত্তর: ক

ক) 13(1/3) days

ক

ব্যাখ্যা

Work done by X in 8 days = 1/40 x 8 = 1/5
Remaining work = 1 - 1/5 = 4/5

Now, 4/5 work is done by Y in 16 days.
Whole work will be done by Y in = 16 x 5/4 = 20 days.
X's 1 day's work = 1/40
Y's 1 day's work = 1/20
(X + Y)'s 1 day's work = 1/40 + 1/20 = 3/40

Hence, X and Y will together complete the work in = 40/3 = 13(1/3) days.

১১,০৩৯.

(√6 + √6)² = ?

ক
24
খ
12
গ
36
ঘ
48

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: (√6 + √6)² = ?

Solution:
Given that,
(√6 + √6)²
= (2√6)²
= 2² × (√6)²
= 4 × 6
= 24

১১,০৪০.

In a departmental store, an Tk. 400 dress is marked with Save 25%. What will be the sale price of this dress?

ক
Tk. 300
খ
Tk. 325
গ
Tk. 275
ঘ
Tk. 500

সঠিক উত্তর: ক

Tk. 300

উত্তর

সঠিক উত্তর: ক

Tk. 300

ক

ব্যাখ্যা

Question: In a departmental store, an Tk. 400 dress is marked with Save 25%. What will be the sale price of this dress?

Solution:
The phrase, Save 25% (i.e. 0.25) , means discount rate.

The original price of the dress is given as Tk. 400.
The discount amount will be 0.25 × 400 = Tk.100

Sale price = 400 - 100 = Tk. 300

১১,০৪১.

If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, what is A ∩ B?

ক
{1, 2, 3, 4}
খ
{1, 2}
গ
{5, 6}
ঘ
{3, 4}
ঙ
{ }

সঠিক উত্তর: ঘ

{3, 4}

উত্তর

সঠিক উত্তর: ঘ

{3, 4}

ঘ

ব্যাখ্যা

Question: If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, what is A ∩ B?

Solution:
A ∩ B = {1, 2, 3, 4} ∩ {3, 4, 5, 6}
= {3, 4}

১১,০৪২.

A tap can fill an empty tank in 12 h and a leakage can empty the tank in 20 h. If tap and leakage both work together, then how long will it take to fill the tank?

ক
ক) 25 h
খ
খ) 40 h
গ
গ) 30 h
ঘ
ঘ) 35 h
ঙ
ঙ) 37 h

সঠিক উত্তর: গ

গ) 30 h

উত্তর

সঠিক উত্তর: গ

গ) 30 h

গ

ব্যাখ্যা

Part filled by tap in 1 h = 1/12
Part emptied by leak in 1 h = 1/20

Net part filled in 1h when both (tap and leakage) work.
= 1/12 - 1/20
= (5 - 3)/60
= 2/60
= 1/30
Therefore, Required time to fill the tank = 30 h

১১,০৪৩.

The average of a set of 12 numbers, which includes 34 is A. If 34 is removed from the set and 38 is included to the set. What is the average of new set of numbers in terms of A?

ক
A + (2/3)
খ
A + (1/5)
গ
A + (1/3)
ঘ
None of the above

সঠিক উত্তর: গ

A + (1/3)

উত্তর

সঠিক উত্তর: গ

A + (1/3)

গ

ব্যাখ্যা

Question: The average of a set of 12 numbers, which includes 34 is A. If 34 is removed from the set and 38 is included to the set. What is the average of new set of numbers in terms of A?
(১২টি সংখ্যার একটি সেটের গড় A, যেখানে ৩৪ রয়েছে। যদি ৩৪ বাদ দিয়ে ৩৮ যুক্ত করা হয়, তাহলে নতুন গড় কত হবে A-এর হিসেবে?)

Solution:
12টি সংখ্যার সমষ্টি = 12A - 34 + 38
= 12A + 4

12টি সংখ্যার গড় = (12A + 4)/12
= (12A/12) + (4/12)
= A + (1/3)

১১,০৪৪.

If 5x - (5/x) = 15, then what is the value of x³ - (1/x)³?

ক
24
খ
36
গ
45
ঘ
54

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: If 5x - (5/x) = 15, then what is the value of x³ - (1/x)³?

Solution:
দেওয়া আছে,
5x - 5/x = 15
⇒ (5x - 5/x) / 5 = 15 / 5
∴ x - 1/x = 3

এখন,
x³ - (1/x)³
= (x - 1/x)³ + 3 × x × (1/x)(x - 1/x)
= (x - 1/x)³ + 3(x - 1/x)
= 3³ + 3 × 3
= 27 + 9
= 36

১১,০৪৫.

What is the probability of getting a sum of six if two dice are thrown at once?

ক
3/5
খ
5/36
গ
11/36
ঘ
7/36

সঠিক উত্তর: খ

5/36

উত্তর

সঠিক উত্তর: খ

5/36

খ

ব্যাখ্যা

We know, the probability of an event = Favorable outcomes / Total outcomes.
In a case of two dices, Total outcomes = 6 × 6 = 36
A.T.Q favorable outcome is a sum of 6,
Ways of obtaining a sum of 6 : (1,5), (2,4), (3,3), (4,2) and (5,1).
Thus, there are 5 ways in which 6 can be obtained using two dices.
Therefore, required probability, P = 5/36

১১,০৪৬.

In how many ways can be a group of 5 men and 2 women be made out of a total of 9 men and 3 women?

ক
120 ways
খ
378 ways
গ
720 ways
ঘ
1040 ways

সঠিক উত্তর: খ

378 ways

উত্তর

সঠিক উত্তর: খ

378 ways

খ

ব্যাখ্যা

Question: In how many ways can be a group of 5 men and 2 women be made out of a total of 9 men and 3 women?

Solution:
There are 9 men and 3 women.
We have to select 5 men out of 9 and 2 women out of 3.

∴ The number of ways of making the selection = ⁹C₅ × ³C₂
= 126 × 3
= 378 ways.

১১,০৪৭.

A sells a scooter priced Tk. 36,000. He gives a discount of 8% on the first Tk. 20,000 and 5% on the next Tk. 10,000. How much discount can he offered on the remaining Tk. 6,000 if he is to get as much as when 7% discount is allowed on the total ?

ক
4%
খ
5%
গ
6%
ঘ
7%

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: A sells a scooter priced Tk. 36000. He gives a discount of 8% on the first Tk. 20000 and 5% on the next Tk. 10000. How much discount can he offered on the remaining Tk. 6000 if he is to get as much as when 7% discount is allowed on the total?

Solution:
Discount on Tk. 36000 = (36000 × 7)/100 = Tk. 2520
Discount on first Tk. 20000 = (20000 × 8)/100 = Tk. 1600
Discount on next Tk 10000 = (10000 × 5)/100 = Tk. 50

∴ Discount on remaining Tk. 6000 = {2520 - (1600 + 500)} = Tk. 420
So, Required percent = (420 × 100)/6000 = 7%

১১,০৪৮.

If, x - y = 5, xy = 6, then x + y =?

ক
ক) 7
খ
খ) ± 7
গ
গ) 1
ঘ
ঘ) None

সঠিক উত্তর: খ

খ) ± 7

উত্তর

সঠিক উত্তর: খ

খ) ± 7

খ

ব্যাখ্যা

Question: If, x - y = 5, xy = 6, then x + y =?

Solution:
Given that
x - y = 5
xy = 6

Now
(x + y)² = (x - y)2 + 4xy
(x + y)² = 5²+ 4 × 6
(x + y)² = 25 + 24
(x + y)² = 49
(x + y) =±√49
x + y = ±7

১১,০৪৯.

The lengths of the two sides of a right-angled triangle, adjacent to right angle are 8 cm and 15 cm respectively. Find the area of the triangle.

ক
60 cm²
খ
48 cm²
গ
36 cm²
ঘ
30 cm²

সঠিক উত্তর: ক

60 cm²

উত্তর

সঠিক উত্তর: ক

60 cm²

ক

ব্যাখ্যা

Question: The lengths of the two sides of a right-angled triangle, adjacent to right angle are 8 cm and 15 cm respectively. Find the area of the triangle.

Solution:

Let,
the sides adjacent to right angle are a = 15 cm and b = 8 cm

We know,
The area = (1/2) × ab
= (1/2) × 15 × 8
= 60 cm²

১১,০৫০.

How many 3-digit numbers can be formed from the digits 2, 3, 5, 6, 7 and 9, which are divisible by 5 and none of the digits is repeated?

ক
5
খ
10
গ
15
ঘ
20
ঙ
None of these

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: How many 3-digit numbers can be formed from the digits 2, 3, 5, 6, 7 and 9, which are divisible by 5 and none of the digits is repeated?

Solution:
Since each desired number is divisible by 5, so we must have 5 at the unit place. So, there is 1 way of doing it.

The tens place can now be filled by any of the remaining 5 digits (2, 3, 6, 7, 9). So, there are 5 ways of filling the tens place.

The hundreds place can now be filled by any of the remaining 4 digits. So, there are 4 ways of filling it.

∴ Required number of numbers = (1 × 5 × 4) = 20.

১১,০৫১.

If x is an odd integer, which of the following is true?

ক
5x - 2 is even
খ
5x² + 2 is odd
গ
5x³ + 3 is odd
ঘ
None of these

সঠিক উত্তর: খ

5x² + 2 is odd

উত্তর

সঠিক উত্তর: খ

5x² + 2 is odd

খ

ব্যাখ্যা

Question: If x is an odd integer, which of the following is true?

Solution:
let, x = 1
putting x to each option we get,
5x - 2 = (5 × 1) - 2 = 3; here 3 is an odd integer. so, option 1 is not true.
5x² + 2 = (5 × 1) + 2 = 7; here 7 is an odd integer. so, option 2 is true.
5x² + 3 = (5 × 1) + 3 = 8; here 8 is an even integer. so, option 3 is not true.

so only option 2 is true.

১১,০৫২.

A train travelling at a speed of 75 mph enters a tunnel 3.5 miles long. The train is 0.25 mile long. How long does it take for the train to pass through the tunnel from the moment the front enters to the moment the rear emerges?

ক
ক) 3 min
খ
খ) 3.5 min
গ
গ) 3.2 min
ঘ
ঘ) 2.5 min

সঠিক উত্তর: ক

ক) 3 min

উত্তর

সঠিক উত্তর: ক

ক) 3 min

ক

ব্যাখ্যা

Question: A train travelling at a speed of 75 mph enters a tunnel 3.5 miles long. The train is 0.25 mile long. How long does it take for the train to pass through the tunnel from the moment the front enters to the moment the rear emerges?

Solution:
Total distance covered = 3.5 + 0.25 mile = 3.75 miles
Train traveles 75 miles in 60 min
∴ Train traveles 3.75 mile in (60 × 3.75)/75 min
= 3 min

১১,০৫৩.

The cost of 9 mangoes and 5 apples is equal to the cost of 7 mangoes and 8 apples. Find the ratio between the cost of 1 mango and the cost of one apple.

ক
5 : 8
খ
5 : 4
গ
7 : 5
ঘ
3 : 2

সঠিক উত্তর: ঘ

3 : 2

উত্তর

সঠিক উত্তর: ঘ

3 : 2

ঘ

ব্যাখ্যা

Question: The cost of 9 mangoes and 5 apples is equal to the cost of 7 mangoes and 8 apples. Find the ratio between the cost of 1 mango and the cost of one apple.

Solution:
ধরি,
১টি আমের দাম ক টাকা
১টি আপেলের দাম খ টাকা

শর্তমতে,
৯ক + ৫খ = ৭ক + ৮খ
⇒ ৯ক - ৭ক = ৮খ - ৫খ
⇒ ২ক = ৩খ
⇒ ক/খ = ৩/২
∴ ক : খ = ৩ : ২

১১,০৫৪.

If (5^{3x - 5}b^{2x - 6})/5^{x + 1} = a^{2x - 6} where a>0, b>0 and 5b ≠ a then what is the value of x?

ক
2
খ
3
গ
4
ঘ
6

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: If (5^{3x - 5}b^{2x - 6})/5^{x + 1} = a^{2x - 6} where a>0, b>0 and 5b ≠ a then what is the value of x?

Solution:
(5^{3x - 5}b^{2x - 6})/5^{x + 1} = a^{2x - 6}
⇒ 5^{3x - 5}/5^{x + 1} = a^{2x - 6}/b^{2x - 6}
⇒ 5^{3x - 5 - x - 1}= (a/b)^{2x - 6}
⇒ 5^{2x - 6} = (a/b)^{2x - 6}
⇒ 5^{2x - 6}/(a/b)^{2x - 6} = 1
⇒ {5/(a/b)}^{2x - 6} = {5/(a/b)}⁰[∵ {5/(a/b)}⁰ = 1]
⇒ 2x - 6 = 0
⇒ 2x = 6
∴ x = 3

১১,০৫৫.

A man traveled a distance of 61 km in 9 hours. He traveled partly on foot at 4 km/hr and partly on bicycle at 9 km/hr. What is the distance (in km) traveled on foot?

ক
10
খ
12
গ
14
ঘ
16

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: A man traveled a distance of 61 km in 9 hours. He traveled partly on foot at 4 km/hr and partly on bicycle at 9 km/hr. What is the distance (in km) traveled on foot?

Solution:
Let the time in which he travelled on foot = x hr.
Then the time in which he travelled on bicycle = (9 - x) hr.

Now
⇒ 4x + 9(9 - x) = 61
⇒ 4x + 81 - 9x = 61
⇒ 5x = 20
⇒ x = 4

∴ Distance travelled on foot = 4x
⇒ 4 × 4
= 16 km

∴ The distance travelled on foot is 16 Km.

১১,০৫৬.

20 women take 20 minutes to roll 20 breads. How many breads can 400 women roll in 400 minutes?

ক
ক) 20
খ
খ) 2000
গ
গ) 4000
ঘ
ঘ) 8000

সঠিক উত্তর: ঘ

ঘ) 8000

উত্তর

সঠিক উত্তর: ঘ

ঘ) 8000

ঘ

ব্যাখ্যা

Men = M; Days = D; Time/Hours = T; Work = W
M₁D₁T₁W₂= M₂D₂T₂W₁
Note that W₂ is on left side and W₁ is on right side

∴ 20 women x 20 minutes x ? = 400 women x 400 minutes x 20 chapattis
∴ ? = 8000 breads = Are rolled by 400 women in 400 minutes.

১১,০৫৭.

If the sum of two numbers is 30 and the sum of their squares is 692, then the product of two numbers is -

ক
ক) 81
খ
খ) 144
গ
গ) 104
ঘ
ঘ) 125

সঠিক উত্তর: গ

গ) 104

উত্তর

সঠিক উত্তর: গ

গ) 104

গ

ব্যাখ্যা

Question: If the sum of two numbers is 30 and the sum of their squares is 692, then the product of two numbers is -

Solutuion:
ধরি,
একটি সংখ্যা = x
অপরটি = y

প্রশ্নমতে,
x + y = 30 ........... (i)
x² + y² = 692 ........... (ii)

(ii) নং হতে পাই,
x²+ y²= 692
⇒ (x + y)² - 2xy = 692
⇒ (30)² - 2xy = 692
⇒ 900 - 2xy = 692
⇒ 2xy = 208
∴ xy = 104

১১,০৫৮.

A water tank is 30 m long, 20 m wide and 12 m deep. It is made of iron sheet which is 3 m wide. The tank is open at the top. If the cost of iron sheet is TK. 10 per meter, what is the total cost of iron sheet required to build the tank?

ক
Tk. 6000
খ
Tk. 8000
গ
Tk. 9000
ঘ
Tk. 10000
ঙ
None of these

সঠিক উত্তর: ক

Tk. 6000

উত্তর

সঠিক উত্তর: ক

Tk. 6000

ক

ব্যাখ্যা

Question: A water tank is 30 m long, 20 m wide and 12 m deep. It is made of iron sheet which is 3 m wide. The tank is open at the top. If the cost of iron sheet is TK. 10 per meter, what is the total cost of iron sheet required to build the tank?

Solution:
Length of water tank = 30 m
Width of water tank = 20 m
Depth of water tank = 12 m

Area of water tank = 2(lb + bh +hl) - lb
= 2 (30 × 20 + 20 × 12 + 12 × 30) - 30 × 20
= 2 (600 + 240 + 360) - 600
= 2 (1200) - 600
= 2400 - 600
= 1800 m²

Area of iron sheet (L × B) = area of tank = 1800 m²
⇒ L × B = 1800
⇒ L × 3 = 1800
∴ L = 1800/3 = 600 m

Given that, the cost of iron sheet is Tk. 10 per meter.
So, the total cost of iron sheet to build the tank = 10 × 600 = Tk. 6000

১১,০৫৯.

The average age of a man and his son is 34 years. If the respective ratio of their ages after four years 14 : 5. What is the present age of the man?

ক
ক) 48 years
খ
খ) 40 years
গ
গ) 52 years
ঘ
ঘ) 60 years

সঠিক উত্তর: গ

গ) 52 years

উত্তর

সঠিক উত্তর: গ

গ) 52 years

গ

ব্যাখ্যা

Question: The average age of a man and his son is 34 years. If the respective ratio of their ages after four years 14 : 5. What is the present age of the son?

Solution:
Total age of the man and his son = 34 × 2 = 68 years

Let the man's age be x years
So, the son's age is = 68 - x years

Now,
(x + 4) / (68 - x + 4) = 14/5
⇒ 5x + 20 = 1008 - 14x
⇒ 19x = 988
⇒ x = 52

১১,০৬০.

A metallic sheet is of rectangular shape with dimensions 48 m x 36 m. From each of its corners, a square is cut off so as to make an open box. If the length of the square is 8 m, the volume of the box (in m cube) is:

ক
ক) 4120 m cube
খ
খ) 4140 m cube
গ
গ) 5140 m cube
ঘ
ঘ) 5120 m cube

সঠিক উত্তর: ঘ

ঘ) 5120 m cube

উত্তর

সঠিক উত্তর: ঘ

ঘ) 5120 m cube

ঘ

ব্যাখ্যা

l = (48 - 16)m = 32 m, [because 8+8 = 16]
b = (36 -16)m = 20 m,
h = 8 m.
Volume of the box = (32 x 20 x 8) m cube
= 5120 m cube.

১১,০৬১.

If p is a positive integer, what is the smallest possible value of p such that 1470 × p is a perfect square?

ক
4
খ
12
গ
15
ঘ
30

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: If p is a positive integer, what is the smallest possible value of p such that 1470 × p is a perfect square?

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

1470 = 2 × 3 × 5 × 7 × 7
= 2¹ × 3¹ × 5¹ × 7²

এখন, 1470 × p = 2¹ × 3¹ × 5¹ × 7² × p
এখানে, 2-এর ঘাত = 1 (বিজোড়), 3-এর ঘাত = 1 (বিজোড়), 5-এর ঘাত = 1 (বিজোড়), 7-এর ঘাত = 2 (জোড়)
পূর্ণবর্গ করতে হলে সব ঘাত জোড় হতে হবে। তাই p = 2 × 3 × 5 = 30 হলে,
1470 × 30 = 2¹ × 3¹ × 5¹ × 7² × (2 × 3 × 5)
= 2² × 3² × 5² × 7²

যেহেতু সব মৌলিক উৎপাদকের ঘাত জোড়, তাই এটি একটি পূর্ণবর্গ সংখ্যা।

সুতরাং, p = 30 হলে, 1470p পূর্ণবর্গ সংখ্যা হয়।

১১,০৬২.

What is the greatest number which divides 639, 1065 and 1491 exactly?

ক
193
খ
183
গ
223
ঘ
213
ঙ
233

সঠিক উত্তর: ঘ

213

উত্তর

সঠিক উত্তর: ঘ

213

ঘ

ব্যাখ্যা

Question: What is the greatest number which divides 639, 1065 and 1491 exactly?

Solution:
The greatest number will be H.C.F. of 639, 1065 and 1491

H.C.F. of 639 and 1065 is 213.
H.C.F. of 213 and 1491 is 213.

১১,০৬৩.

In what ratio must a grocer mix two varieties of tea worth Tk. 60 a kg and Tk. 65 a kg so that by selling the mixture at Tk. 68.20 a kg he may gain 10%?

ক
3 : 1
খ
1 : 3
গ
2 : 3
ঘ
3 : 2
ঙ
None of the above

সঠিক উত্তর: ঘ

3 : 2

উত্তর

সঠিক উত্তর: ঘ

3 : 2

ঘ

ব্যাখ্যা

Question: In what ratio must a grocer mix two varieties of tea worth Tk. 60 a kg and Tk. 65 a kg so that by selling the mixture at Tk. 68.20 a kg he may gain 10%?

Solution:
Quantity of Tk. 60 tea is x kg.
Quantity of Tk. 65 tea is y kg

S.P. of 1 kg of the mixture = Tk. 68.20,
Gain = 10%.
C.P of 1 kg of the mixture = Tk. (100/110 × 68.20) = Tk. 62

ATQ,
60x + 65y = (x + y)62
⇒ 60x + 65y = 62x + 62y
⇒ 62x - 60x = 65y - 62y
⇒ 2x = 3y

∴ x/y = 3/2

১১,০৬৪.

The average of 10 numbers is 12. If 5 is subtracted from each of 6 of these numbers, what is the new average?

ক
8.5
খ
9
গ
11.25
ঘ
7

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: The average of 10 numbers is 12. If 5 is subtracted from each of 6 of these numbers, what is the new average?

Solution:
দেওয়া আছে,
10 টি নম্বরের গড় = 12
∴ 10 টি নম্বরের সমষ্টি = (10 × 12) = 120

10 টি নম্বরের মধ্যে 6 টি নম্বরের প্রতিটি থেকে 5 বিয়োগ করা হলে নতুন সমষ্টি,
= 120 - (6 × 5)
= 120 - 30
= 90

সুতরাং 10 টি সংখ্যার নতুন গড় = 90/10 = 9

১১,০৬৫.

Labonno walks 5 miles from point A to point B in one hour, then bicycles back to point A along the same route at 15 miles per hour. Benu makes the same round trip but does so at half of Labonno's average speed. How many minutes does Benu spend on his round trip?

ক
80 minutes
খ
100 minutes
গ
120 minutes
ঘ
160 minutes

সঠিক উত্তর: ঘ

160 minutes

উত্তর

সঠিক উত্তর: ঘ

160 minutes

ঘ

ব্যাখ্যা

Question: Labonno walks 5 miles from point A to point B in one hour, then bicycles back to point A along the same route at 15 miles per hour. Benu makes the same round trip but does so at half of Labonno's average speed. How many minutes does Benu spend on his round trip?

Solution:
বাইসাইকেলে লাবণ্যের ফিরতে সময় লাগে = 5/15 = 1/3 hr = 20 min

লাবণ্যের গড় বেগ = (5 + 5)/(60 + 20) = 10/80 miles/min
= 1/8 miles/min

বেণুর গড় বেগ = 1/16 miles/min

10 মাইল যেতে বেণুর সময় লাগবে = 10 × 16 min
= 160 min

১১,০৬৬.

A retailer purchases a TV at a discount of 25% from the wholesaler. He marks it up by 50% on the discounted price and then offers a discount of 20% to the customer. What is his percentage profit?

ক
22%
খ
20%
গ
32%
ঘ
27%
ঙ
30%

সঠিক উত্তর: খ

20%

উত্তর

সঠিক উত্তর: খ

20%

খ

ব্যাখ্যা

Question: A retailer purchases a TV at a discount of 25% from the wholesaler. He marks it up by 50% on the discounted price and then offers a discount of 20% to the customer. What is his percentage profit?

Solution:
Let the original price of the TV be 1000 Taka.

Wholesaler’s discount = 25%
Retailer’s purchase price:
1000 − 25% of 1000 = 1000−250 = 750 Taka

Retailer’s markup = 50% on Taka 750
New marked price:
750 + 50% of 750 = 750 + 375 = 1125 Taka

Customer’s discount = 20% on Taka 1125
Selling price:
1125 − 20% of 1125 = 1125 − 225 = 900 Taka

So, Profit = {(900 - 750)/750} × 100%
= 20%

১১,০৬৭.

Present ages of A,B and C are in proportions 4 : 7 : 9. Eight years ago, the sum of their ages was 56. What are their present ages (in years)?

ক
ক) 16, 30, 40
খ
খ) 16, 28, 40
গ
গ) 16, 26, 38
ঘ
ঘ) 16, 28, 36

সঠিক উত্তর: ঘ

ঘ) 16, 28, 36

উত্তর

সঠিক উত্তর: ঘ

ঘ) 16, 28, 36

ঘ

ব্যাখ্যা

Let present ages of A, B, and C be 4x, 7x, and 9x respectively.
(4x − 8) + (7x − 8) + (9x 8) = 56
20x = 80
x = 4

Hence present ages of A, B, and C are (4 × 4), (7 × 4), and (9 × 4)
That is 16, 28, and 36.

১১,০৬৮.

A Man travelled a distance of 61 km in 9 hours. He travelled partly on foot at 4 km/hr and partly on bicycle at 9 km/hr. What is the distance travelled on foot?

ক
ক) 10 km
খ
খ) 16 km
গ
গ) 21 km
ঘ
ঘ) 23 km

সঠিক উত্তর: খ

খ) 16 km

উত্তর

সঠিক উত্তর: খ

খ) 16 km

খ

ব্যাখ্যা

Let the time in which he traveled on foot = x hour
Time for travelling on bicycle = (9-x) hr
Distance = Speed×Time, and Total distance = 61 km
So,
4x + 9(9-x) = 61
=> 5x = 20
=> x = 4
So distance traveled on foot = 4x4 = 16 km

১১,০৬৯.

A pen marked at Tk. 85 is sold for Tk. 68. The rate of discount is-

ক
ক) 10%
খ
খ) 15%
গ
গ) 18%
ঘ
ঘ) 20%

সঠিক উত্তর: ঘ

ঘ) 20%

উত্তর

সঠিক উত্তর: ঘ

ঘ) 20%

ঘ

ব্যাখ্যা

Discount = 85 - 68
= 17
Discount Rate = (17/85) × 100
= 20%

১১,০৭০.

A certain basketball team that has played 2/3 of its games has a record of 17 wins and 3 losses. What is the greatest number of the remaining games that the team can lose and still win at least 3/4 of all of its games?

ক
6
খ
5
গ
4
ঘ
3

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: A certain basketball team that has played 2/3 of its games has a record of 17 wins and 3 losses. What is the greatest number of the remaining games that the team can lose and still win at least 3/4 of all of its games?

Solution:
Total match x
∴ (2/3) × x = 20
∴ x = 30

Total match = 30
Played = 20
remaining = 10

total win = 3/4 of 30 = 22.5 ≅ 23
total losses = 30 - 23 = 7 games
so max losses from the remaining 10 games = 7 - 3 = 4

১১,০৭১.

If 2^log⁡₂(x) = 16, then x =?

ক
8
খ
16
গ
32
ঘ
64

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: If 2^log⁡₂(x)= 16, then x =?

Solution:
2^log⁡₂(x)= 16
⇒ 2^log⁡₂(x) = 2⁴
⇒ log⁡₂(x) = 4
⇒ 2⁴ = x
∴ x = 16

১১,০৭২.

Find the value of 25% of 10% of Tk. 800.

ক
25
খ
20
গ
30
ঘ
35

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: Find the value of 25% of 10% of Tk. 800.

Solution:
25% of 10% of 800
= (25/100) × (10/100) × 800
= 20

১১,০৭৩.

Today is Friday. After 58 days, it will be:

ক
Sunday
খ
Tuesday
গ
Wednesday
ঘ
Saturday

সঠিক উত্তর: ক

Sunday

উত্তর

সঠিক উত্তর: ক

Sunday

ক

ব্যাখ্যা

Question: Today is Friday. After 58 days, it will be:

Solution:
আমরা জানি যে সপ্তাহের প্রতিটি দিন 7 দিন পর পুনরাবৃত্তি হয়।
58 ÷ 7 = 8 (ভাগশেষ 2)
অর্থাৎ, (7 × 8) = 56 দিন পর আবার শুক্রবার হবে।
∴ 58 দিন পর হবে (শুক্রবার + 2 দিন) = রবিবার।

১১,০৭৪.

Two trains of equal length are running on parallel lines in the same direction at 12.78 m/s and 10 m/s. The faster train passes the slower train in 72 seconds. The length of each train is:

ক
200 meters
খ
150 meters
গ
120 meters
ঘ
100 meters

সঠিক উত্তর: ঘ

100 meters

উত্তর

সঠিক উত্তর: ঘ

100 meters

ঘ

ব্যাখ্যা

Question: Two trains of equal length are running on parallel lines in the same direction at 12.78 m/s and 10 m/s. The faster train passes the slower train in 72 seconds. The length of each train is:

Solution:
Let,
The length of each train be x metres.
Then, distance covered = 2x metres

Relative speed = (12.78 - 10) m/s
= 2.78 m/s

Now
2x/72 = 2.78
⇒ 2x = 72 × 2.78
⇒ 2x = 200.16
⇒ x = 200.16/2
∴ x = 100.08 ≅ 100

∴ The length of each train is 100 meters.

১১,০৭৫.

If the average of 5 positive integers is 40 and the difference between the largest and the smallest of these 5 numbers is 10, what is the maximum value possible for the largest of these 5 integers?

ক
44
খ
48
গ
50
ঘ
52
ঙ
None

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: If the average of 5 positive integers is 40 and the difference between the largest and the smallest of these 5 numbers is 10, what is the maximum value possible for the largest of these 5 integers?

Solution:
Since average is 40, Sum of all 5 numbers = 40 × 5 = 200
Let's call the largest number L and smallest number S
We know, L - S = 10
So, S = L - 10

The other 3 numbers must be Greater than or equal to S (L - 10); Less than or equal to L and Must be integers.
To maximize L, the other 3 numbers should be as small as possible while staying within constraints
They cannot be smaller than S (L - 10); So all 3 middle numbers must be (L - 10)

ATQ,
L + (L - 10) + 3(L - 10) = 200
⇒ L + (L - 10) + 3L - 30 = 200
⇒ 5L - 40 = 200
⇒ 5L = 240
∴ L = 48

১১,০৭৬.

If θ be an acute angle and 7sin²θ + 3cos²θ = 4, then the value of tanθ is?

ক
0
খ
1/√3
গ
1/√2
ঘ
1

সঠিক উত্তর: খ

1/√3

উত্তর

সঠিক উত্তর: খ

1/√3

খ

ব্যাখ্যা

Question: If θ be an acute angle and 7sin²θ + 3cos²θ = 4, then the value of tanθ is?

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

∴ tanθ = tan30° = 1/√3

১১,০৭৭.

In a box, there are 7 yellow, 8 black, and 5 white balls. One ball is picked randomly. What is the probability that it is neither yellow nor white?

ক
1/2
খ
3/10
গ
1/5
ঘ
2/5

সঠিক উত্তর: ঘ

2/5

উত্তর

সঠিক উত্তর: ঘ

2/5

ঘ

ব্যাখ্যা

Question: In a box, there are 7 yellow, 8 black, and 5 white balls. One ball is picked randomly. What is the probability that it is neither yellow nor white?

Solution:
মোট বলের সংখ্যা = 7 + 8 + 5 = 20 টি।

ধরি, E হলো এমন ঘটনা যেখানে বলটি হলুদ বা সাদা কোনোটিই নয়, অর্থাৎ বলটি কালো।
∴ অনুকূল ফলাফলের সংখ্যা, n(E) = 8

সম্ভাব্যতা = (অনুকূল ফলাফলের সংখ্যা)/(মোট ফলাফলের সংখ্যা)
= 8/20
= 2/5

অতএব, বলটি হলুদ বা সাদা না হওয়ার সম্ভাব্যতা হলো 2/5।

১১,০৭৮.

The sum of the circumference of a circle and the perimeter of a rectangle is 130 cm. The area of the rectangle is 104 cm² and the length of the rectangle is 13 cm. What is the area of the circle?

ক
ক) 516 cm²
খ
খ) 616 cm²
গ
গ) 816 cm²
ঘ
ঘ) 216 cm²

সঠিক উত্তর: খ

খ) 616 cm²

উত্তর

সঠিক উত্তর: খ

খ) 616 cm²

খ

ব্যাখ্যা

Given, Length of rectangle, L = 13 cm
breadth, B = 104/13 = 8 cm
So, its perimeter = 2(13+8) = 42
ATQ, 2πr = 130 - 42
Or, 2πr = 88
Or, r = 14
∴ Area of the circle = πr² = 22/7 × 14² = 616 cm²

১১,০৭৯.

At a certain diner, a hamburger and coleslaw cost $3.95, and a hamburger and french fries cost $4.40. If french fries cost twice as much as coleslaw, how much do french fries cost?

ক
$0.45
খ
$0.60
গ
$0.75
ঘ
$0.90

সঠিক উত্তর: ঘ

$0.90

উত্তর

সঠিক উত্তর: ঘ

$0.90

ঘ

ব্যাখ্যা

Question: At a certain diner, a hamburger and coleslaw cost $3.95, and a hamburger and french fries cost $4.40. If french fries cost twice as much as coleslaw, how much do french fries cost?

Solution:

Let,
Cost of hamburger = $h

Cost of coleslaw = $c

Cost of fries = $f

h + c = $3.95 ...............(1)

h + f = $4.40 ................(2)

french fries cost twice as much as coleslaw
∴ f = 2c

Putting the Value of f in equation (2)
h + 2c = $4.40 ..............(3)

From (3) - (1) we get,

c = $0.45

∴ f = $0.45 × 2 = $0.90

১১,০৮০.

A petrol tank that is 1/2 full has 8 gallons petrol removed. The tank is then 1/10 full. What is the capacity, in gallons of the tank?

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

সঠিক উত্তর: গ

20 gallons

উত্তর

সঠিক উত্তর: গ

20 gallons

গ

ব্যাখ্যা

Question: A petrol tank that is 1/2 full has 8 gallons petrol removed. The tank is then 1/10 full. What is the capacity, in gallons of the tank?

Solution:
Let the capacity of the tank be x gallons.

According to the question,
(x/2) - 8 = x/10
⇒ (x - 16)/2 = x/10
⇒ 10(x - 16) = 2x
⇒ 10x - 160 = 2x
⇒ 10x - 2x = 160
⇒ 8x = 160
⇒ x = 160/8 = 20 gallons

১১,০৮১.

The market value of a 10.5% stock, in which an income of Tk. 756 is derived by investing Tk. 9000, brokerage being (1/4)% is -

ক
ক) Tk. 108.25
খ
খ) Tk. 112.20
গ
গ) Tk. 124.75
ঘ
ঘ) Tk. 125.25

সঠিক উত্তর: গ

গ) Tk. 124.75

উত্তর

সঠিক উত্তর: গ

গ) Tk. 124.75

গ

ব্যাখ্যা

For an income of Tk. 756, investment
= Tk. 9000

For an income of Tk. (21/2), investment
= Tk. {(9000/756) × (21/2)}
= Tk. 125

∴ For a Tk. 100 stock, investment = Tk. 125

The market value of Tk. 100 stock
= Tk. {125 - (1/4)}
= Tk. 124.75

১১,০৮২.

A and B started a partnership business investing some amount in the ratio of 3:5. C joined then after six months with an amount equal to that of B. In what proportion should the profit at the end of one year be distributed among A, B and C?

ক
ক) 3 : 5 : 2
খ
খ) 3 : 5 : 5
গ
গ) 6 : 10 : 5
ঘ
ঘ) Data inadequate

সঠিক উত্তর: গ

গ) 6 : 10 : 5

উত্তর

সঠিক উত্তর: গ

গ) 6 : 10 : 5

গ

ব্যাখ্যা

Let initial investment of A is 3x and B is 5x, then C investment is also 5x, but most important to note in this question is the time duration of investment
Like, A invested for 12 months, B invested for 12 months and C invested for 6 months.
A : B : C = (3x x 12) : (5x x 12) : (5x x 6)
= 36 : 60 : 30
= 6 : 10 : 5.

১১,০৮৩.

A man goes downstream with a boat to some destination and returns upstream to his original place in 5 hours. If the speed of the boat in still water and the stream are 10 kmph and 4 kmph respectively, the distance of the destination from the starting place is:

ক
14 km
খ
21 km
গ
24 km
ঘ
28 km

সঠিক উত্তর: খ

21 km

উত্তর

সঠিক উত্তর: খ

21 km

খ

ব্যাখ্যা

Question: A man goes downstream with a boat to some destination and returns upstream to his original place in 5 hours. If the speed of the boat in still water and the stream are 10 kmph and 4 kmph respectively, the distance of the destination from the starting place is:

Let,
the distance of the destination from the starting point = x km.
Speed downstream = (10 + 4) = 14 kmph
Speed upstream = (10 - 4) = 6 kmph

∴ Total time taken = 5 hours

ATQ,
(x/14) + (x/6) = 5
⇒ (3x + 7x)/42 = 5
⇒ 10x = 42 × 5
⇒ x = (42 × 5)/10
∴ x = 21 km

১১,০৮৪.

If x = 1 + √2 + √3 and y = 1 + √2 - √3 then the value of x² + 4xy + y²/(x + y)?

ক
ক) 2√2
খ
খ) 2(2 + √2)
গ
গ) 1
ঘ
ঘ) 6

সঠিক উত্তর: ঘ

ঘ) 6

উত্তর

সঠিক উত্তর: ঘ

ঘ) 6

ঘ

ব্যাখ্যা

x = 1 + √2 + √3 ...........(i)
y = 1 + √2 - √3 .............(ii)

x² + 4xy + y²/(x + y)
= {(x + y)² + 2xy}/(x + y)

From (i) + (ii)
x + y = 2 + 2√2
xy = (1 + √2)² - (√3)²
= 3 + 2√2 - 3
= 2√2

{(x + y)² + 2xy}/(x + y)
= {(2 + 2√2)² + (2 × 2√2)}/(2 + 2√2)
= (12 + 12√2)/(2 + 2√2)
= 12(1 + √2)/2((1 + √2)
= 12/2
= 6.

১১,০৮৫.

A train 200 m long passes a man, running at 5 km/hr in the same direction in which the train is going, in 16 seconds. The speed of the train is:

ক
48 km/hr
খ
50 km/hr
গ
55 km/hr
ঘ
60 km/hr

সঠিক উত্তর: খ

50 km/hr

উত্তর

সঠিক উত্তর: খ

50 km/hr

খ

ব্যাখ্যা

Question: A train 200 m long passes a man, running at 5 km/hr in the same direction in which the train is going, in 16 seconds. The speed of the train is:

Solution:
Speed of the train relative to man = 200/16 m/sec
= 12.5 m/sec
= (12.5 × 3600)/1000 km/hr
= 45 km/hr

Let,
The speed of the train be x km/hr.
∴ Relative speed = (x - 5) km/hr.
⇒ x - 5 = 45
∴ x = 50 km/hr.

১১,০৮৬.

Alam starts working on a job and works on it for 12 days and completes 40% of the work. Then Babu joins Alam and together they complete the rest of the work in 12 days. How long (in days) will it take Babu to complete the job if he works alone?

ক
30
খ
45
গ
60
ঘ
None

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: Alam starts working on a job and works on it for 12 days and completes 40% of the work. Then Babu joins Alam and together they complete the rest of the work in 12 days. How long (in days) will it take Babu to complete the job if he works alone?

Solution:
Let
the total work is 100x
Alam starts working on a job and works on it for 12 days and completes 40% of the work.

Remaining work 100x - 40x = 60x

Babu done 20x in 12 days
∴ Babu done 100x in (12 × 100x)/20x days
= 60 days

১১,০৮৭.

Solve for x, log₂(x + 3) = 4

ক
x = 1
খ
x = 19
গ
x = 13
ঘ
x = 8

সঠিক উত্তর: গ

x = 13

উত্তর

সঠিক উত্তর: গ

x = 13

গ

ব্যাখ্যা

Question: Solve for x, log₂(x + 3) = 4

Solution:
Given that,
⇒ log₂(x + 3) = 4
⇒ x + 3 = 2⁴
⇒ x + 3 = 16
⇒ x = 16 - 3
x = 13

১১,০৮৮.

If x + (1/x) = 3, then x - (1/x) =?

ক
- 3
খ
√13
গ
√7
ঘ
√5
ঙ
None of these

সঠিক উত্তর: ঘ

√5

উত্তর

সঠিক উত্তর: ঘ

√5

ঘ

ব্যাখ্যা

Question: If x + (1/x) = 3, then x - (1/x) =?

Solution:
We know that,
(x - 1/x)² = (x + 1/x)² - 4.x.(1/x)
⇒ (x - 1/x)² = 3² - 4
⇒ (x - 1/x)² = 9 - 4
∴ (x - 1/x) = √5

১১,০৮৯.

Coffee A normally costs 100 taka per pound. It is mixed with coffee B, which normally costs 70 taka per pound, to form a mixture which costs 88 taka per pound. If there are 10 pounds of the mix, how many pounds of coffee A are used in the mix?

ক
4
খ
5
গ
6
ঘ
7
ঙ
None of these

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Amount of A in the mixture = 10 × (3/5)
= 6

১১,০৯০.

A square and a circle have the same perimeter. The side of the length of square is 44 cm, what is the area of the circle?

ক
1456 sq. cm.
খ
375 sq. cm.
গ
2464 sq. cm.
ঘ
1864 sq. cm.

সঠিক উত্তর: গ

2464 sq. cm.

উত্তর

সঠিক উত্তর: গ

2464 sq. cm.

গ

ব্যাখ্যা

Question: A square and a circle have the same perimeter. The side of the length of square is 44 cm, what is the area of the circle?

Solution:
Perimeter of the square = 4 × side length
= 4 × 44 cm
= 176 cm

As per the question, the square and circle have the same perimeter.
∴ Circumference of the circle = 176 cm
We know that, Circumference of the circle = 2πr
∴ 2πr = 176
⇒ r = 176 / (2π)
⇒ r = 88 / π
⇒ r = 88 / (22/7)
⇒ r = 88 × 7 / 22
⇒ r = 4 × 7
⇒ r = 28 cm

Area of the circle = πr²
= (22/7) × 28²
= (22/7) × (28 × 28)
= 22 × 4 × 28
= 2464 sq. cm

∴ The area of the circle is 2464 sq. cm.

১১,০৯১.

What is the odd man out?
835, 734, 642, 751, 853, 981, 532

ক
835
খ
734
গ
642
ঘ
751
ঙ
853

সঠিক উত্তর: ঘ

751

উত্তর

সঠিক উত্তর: ঘ

751

ঘ

ব্যাখ্যা

In each number except 751, the difference of third and first digit is the middle one.

১১,০৯২.

A right circular cylinder just encloses a sphere. If p is the surface area of the sphere and q is the curved surface area of the cylinder, then which one of the following is correct?

ক
P = q
খ
2p = 3
গ
2p = q
ঘ
p = 2q

সঠিক উত্তর: ক

P = q

উত্তর

সঠিক উত্তর: ক

P = q

ক

ব্যাখ্যা

Question: A right circular cylinder just encloses a sphere. If p is the surface area of the sphere and q is the curved surface area of the cylinder, then which one of the following is correct?

Solution:
When a right circular cylinder just encloses a sphere as shown below, the radius of the sphere and cylinder are equal and the height of cylinder is equal to the diameter of the sphereLet, radius of sphere = radius of cylinder = r
Here, height of cylinder = 2r
Now, Curved surface area of sphere = 4π × (radius)²
= p = 4πr² Curved surface area of cylinder = 2 × radius × height
⇒ q = 2πr × 2r = 4πr²
∴ p = q

১১,০৯৩.

When the diameter of a circle is tripled, the area of the circle will be increased by how many time?

ক
ক) 3
খ
খ) 6
গ
গ) 9
ঘ
ঘ) 12

সঠিক উত্তর: গ

গ) 9

উত্তর

সঠিক উত্তর: গ

গ) 9

গ

ব্যাখ্যা

Question: When the diameter of a circle is tripled, the area of the circle will be increased by how many time?

Solution:
সমাধান :
ধরি,
বৃত্তের ব্যাসার্ধ r
বৃত্তের ব্যাস = 2r
∴বৃত্তের ক্ষেত্রফল = πr²

ব্যাস তিনগুণ বৃদ্ধি পেলে হবে 6r
∴ব্যাসার্ধ =6r/2 = 3r
∴ঐ বৃত্তের ক্ষেত্রফল হবে π(3r)²= 9πr²

বৃত্তের ক্ষেত্রফল ৯ গুণ পাবে।

১১,০৯৪.

If (2x + 3y) : (3x + 5y) = 18 : 29, then what is the value of x : y?

ক
ক) 2 : 3
খ
খ) 3 : 4
গ
গ) 4 : 7
ঘ
ঘ) 9 : 13

সঠিক উত্তর: খ

খ) 3 : 4

উত্তর

সঠিক উত্তর: খ

খ) 3 : 4

খ

ব্যাখ্যা

Question: If (2x + 3y) : (3x + 5y) = 18 : 29, then what is the value of x : y?

Solution:
(2x + 3y) : (3x + 5y) = 18 : 29
⇒ (2x + 3y)/(3x + 5y) = 18/29
⇒ 58x + 87y = 54x +90y
⇒ 4x = 3y
⇒ x/y = 3/4
⇒ x : y = 3 : 4

১১,০৯৫.

Maruf, Salman, and Mir-Jafor have taught history for a combined total of 96 years. If Maruf has taught for 9 more years than Salman and for 9 fewer years than Mir-Jafor, for how many years has Mir-Jafor taught?

ক
23
খ
32
গ
35
ঘ
41
ঙ
44

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: Maruf, Salman, and Mir-Jafor have taught history for a combined total of 96 years. If Maruf has taught for 9 more years than Salman and for 9 fewer years than Mir-Jafor, for how many years has Mir-Jafor taught?

Solution:
Let number of years taught by Maruf = M
number of years taught by Salman = S
number of years taught by Mir-Jafor = J

M + S + J = 96 .......(1)

M = S + 9
⇒ S = M - 9 .......(2)

M = J - 9

from (2) ⇒
S = (J - 9) - 9 = J - 18

From(1) ⇒
J - 9 + J - 18 + J = 96
⇒ 3J = 96 + 27 = 123
∴ J = 41

১১,০৯৬.

When the positive integers x and y are divided by the positive integer z, they yield remainders 12 and 22, respectively. If (x + y) is divided by z, the remainder is 6. What is the value of z?

ক
12
খ
15
গ
24
ঘ
28
ঙ
None

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: When the positive integers x and y are divided by the positive integer z, they yield remainders 12 and 22, respectively. If (x + y) is divided by z, the remainder is 6. What is the value of z?

Solution:
ধরি, x কে z দিয়ে ভাগ করলে ভাগশেষ 12
অর্থাৎ, x হলো z এর কোনো গুণিতক থেকে 12 বেশি।

একইভাবে,
y কে z দিয়ে ভাগ করলে ভাগশেষ 22
অর্থাৎ, y হলো z এর কোনো গুণিতক থেকে 22 বেশি।
তাই (x + y) এর ভাগশেষ আগে হবে = 12 + 22 = 34

কিন্তু দেওয়া আছে, (x + y) কে z দিয়ে ভাগ করলে ভাগশে 6।
অর্থাৎ,
⇒ 34 - z = 6
⇒ z = 34 - 6
∴ z = 28

১১,০৯৭.

A man travels 6 miles towards west, 3 miles towards south, then again 4 miles towards west. What is the direct distance of destination from the starting point?

ক
√104 miles
খ
√109 miles
গ
√113 miles
ঘ
√117 miles

সঠিক উত্তর: খ

√109 miles

উত্তর

সঠিক উত্তর: খ

√109 miles

খ

ব্যাখ্যা

Question: A man travels 6 miles towards west, 3 miles towards south, then again 4 miles towards west. What is the direct distance of destination from the starting point?

সমাধান:

ধরি, যাত্রা শুরু করার স্থান A এবং গন্তব্যের স্থান B।
সরাসরি দূরত্ব নির্ণয় করতে, আমরা পিথাগোরাসের উপপাদ্য ব্যবহার করব।
এখানে, অতিক্রান্ত মোট উল্লম্ব দূরত্ব হলো 3 মাইল
এবং অতিক্রান্ত মোট অনুভূমিক দূরত্ব হলো (6 + 4) = 10 মাইল।
সুতরাং,
AB² = (3)² + (6 + 4)²
⇒ AB² = 3² + 10²
⇒ AB² = 9 + 100
⇒ AB² = 109
⇒ AB = √109 miles
∴ সরাসরি দূরত্ব √109 মাইল।

১১,০৯৮.

A hat seller had some hats. He sells 55% hats and still has 90 hats. Originally, he had:

ক
175
খ
180
গ
200
ঘ
225

সঠিক উত্তর: গ

200

উত্তর

সঠিক উত্তর: গ

200

গ

ব্যাখ্যা

Question: A hat seller had some hats. He sells 55% hats and still has 90 hats. Originally, he had:

Solution:
Let,
The hat seller had = x hats.

ATQ,
(100 - 55)% of x = 90
⇒ 45% of x = 90
⇒ 45x/100 = 90
⇒ 9x/20 = 90
⇒ x = (90 × 20)/9
∴ x = 200

∴ Originally, he had 200 hats

১১,০৯৯.

A train of length 180 meters crosses a man running at 10 km/hr in the same direction in 10 seconds. What is the speed of the train?

ক
68.8 km/hr
খ
70.4 km/hr
গ
72.8 km/hr
ঘ
74.8 km/hr

সঠিক উত্তর: ঘ

74.8 km/hr

উত্তর

সঠিক উত্তর: ঘ

74.8 km/hr

ঘ

ব্যাখ্যা

Question: A train of length 180 meters crosses a man running at 10 km/hr in the same direction in 10 seconds. What is the speed of the train?

Solution:
When the train and man are moving in same direction then relative speed will be the difference between their individual speeds. In this problem the other way to find the relative speed is to divide the distance covered (length of train) by the time taken by the train to cross the man.

Relative Speed = 180/10 m/s

We will convert it into Km/hr
(180/10) × (18/5) = 64.8 km/hr

Now, let the speed of the train is X km/hr.
So, the relative speed, 64.8 km/hr = X km/hr - 10 km/hr
⇒ X - 10 = 64.8
⇒ X = 64.8 + 10
∴ X = 74.8 km/hr

১১,১০০.

The compound interest on 15000 at 10% per annum is 3150 Tk. What is the time period?

ক
1.5 years
খ
2 years
গ
2.5 years
ঘ
3 years

সঠিক উত্তর: খ

2 years

উত্তর

সঠিক উত্তর: খ

2 years

খ

ব্যাখ্যা

Question: The compound interest on 15000 at 10% per annum is 3150 Tk. What is the time period?

Solution:
Let the time is = n
C = 15000 + 3150
= 18150 Tk.
r = 10%

we know,
C = P(1 + r)ⁿ
⇒18150 = 15000(1 + 1/10)ⁿ
⇒ 18150/15000 = (11/10)ⁿ
⇒ 121/100 = (11/10)ⁿ
⇒ (11/10)² = (11/10)ⁿ
⇒ n = 2