Bank Math

PrepBank · পাতা ৪ / ১৬১ · ৩০১–৪০০ / ১৬,১২৪

৩০১.

A box contains 10 red marbles and 15 blue marbles. If 5 marbles are drawn without replacement, what is the probability of getting exactly 2 red marbles?

ক
0.42
খ
0.45
গ
0.35
ঘ
0.39

সঠিক উত্তর: ঘ

0.39

উত্তর

সঠিক উত্তর: ঘ

0.39

ঘ

ব্যাখ্যা

Question: A box contains 10 red marbles and 15 blue marbles. If 5 marbles are drawn without replacement, what is the probability of getting exactly 2 red marbles?

Solution:

Total marbles = 10 (red) + 15 (blue) = 25.

Number of draws = 5.

Desired number of red marbles = 2.

Number of ways to choose 2 red marbles = ¹⁰C₂= 45
Number of ways to choose 3 blue marbles = ¹⁵C₃ = 455

Favorable outcomes = 45 × 455 = 20475
Total possible outcomes = ²⁵C₅ = 53130

Probability = 20475/53130 = 0.3853 = 0.39

৩০২.

ক
ক) 1
খ
খ) 2
গ
গ) 4
ঘ
ঘ) 643

সঠিক উত্তর: গ

গ) 4

উত্তর

সঠিক উত্তর: গ

গ) 4

গ

ব্যাখ্যা

প্রশ্ন:

সমাধান:

৩০৩.

A book was sold for Tk. 144. If the percentage of profit was numerically equal to the cost price, the cost of the clock was -

ক
Tk. 60
খ
Tk. 70
গ
Tk. 80
ঘ
Tk. 100

সঠিক উত্তর: গ

Tk. 80

উত্তর

সঠিক উত্তর: গ

Tk. 80

গ

ব্যাখ্যা

Question: A book was sold for Tk. 144. If the percentage of profit was numerically equal to the cost price, the cost of the clock was -

Solution:
Let, the percentage of profit be x%
cost of the clock was x taka.

x + x × x/100 = 144
⇒ x + (x²/100) = 144
⇒ (x² + 100x) = 14400
⇒ x²+ 100x - 14400 = 0
⇒ x²+ 180x - 80x - 14400 = 0
⇒ x(x + 180) - 80 (x + 180) = 0
⇒ (x + 180) (x - 80) = 0
∴ x = 80

৩০৪.

The area of an equilateral triangle whose side is 8 cm is-

ক
ক) 8√3cm²
খ
খ) 32√3cm²
গ
গ) 16√3cm²
ঘ
ঘ) 64√3cm²

সঠিক উত্তর: গ

গ) 16√3cm²

উত্তর

সঠিক উত্তর: গ

গ) 16√3cm²

গ

ব্যাখ্যা

Given, side of an equilateral triangle = 8 cm

Area of an equilateral triangle with length of side a= (√3/4)a²

∴ Area of this equilateral triangle = (√3/4) × 8²
= (√3/4) × 64
= 16√3 cm²

৩০৫.

A, B and C enter into partnership by making investments in the ratio of 3 : 5 : 7. After a year, C invests another Tk. 337600 while A withdraws Tk.45600. The ratio of investments then changes to 24 : 59 : 67. How much did A invest initially?

ক
ক) 10000
খ
খ) 12000
গ
গ) 141600
ঘ
ঘ) 14900
ঙ
ঙ) None Of the Above

সঠিক উত্তর: গ

গ) 141600

উত্তর

সঠিক উত্তর: গ

গ) 141600

গ

ব্যাখ্যা

Let, A : B : C = 3x : 5x : 7x
After 1 year C investment = 7x + 337600
After 1 year A investment = 3x - 45,600
ATQ,
(3x − 45600)/5x = 24/59
=> 59 × (3x - 45,600) = 120x
=> 177x - 120x = 45600 × 59
=> 57x = 45600 × 59
=> x = 800 × 59 = 47,200
A's initial investment was = 47,200 × 3 = 141600 [Answer.]

৩০৬.

The average of three numbers is P. If two numbers of them are R and S. What is the 3^rd number?

ক
ক) P - R - S
খ
খ) 3R - P - R
গ
গ) - S + 3P + S
ঘ
ঘ) - R + 3P - S
ঙ
ঙ) None

সঠিক উত্তর: ঘ

ঘ) - R + 3P - S

উত্তর

সঠিক উত্তর: ঘ

ঘ) - R + 3P - S

ঘ

ব্যাখ্যা

Question: The average of three numbers is P. If two numbers of them are R and S. What is the 3^rd number?

Solution:
3টি সংখ্যার সমষ্টি = 3P
তৃতীয় সংখ্যাটি = 3P - (R + S)
= 3P - R - S
= - R + 3P - S

৩০৭.

If a = m - 1, then which of the following is true when m = 1/2

ক
a⁰ > a² > a³ > a
খ
a⁰ > a² > a > a³
গ
a⁰ > a > a² > a³
ঘ
a² > a > a⁰ > a³
ঙ
None of these

সঠিক উত্তর: ক

a⁰ > a² > a³ > a

উত্তর

সঠিক উত্তর: ক

a⁰ > a² > a³ > a

ক

ব্যাখ্যা

Question: If a = m - 1, then which of the following is true when m = 1/2

Solution:
a = m - 1
⇒ a = 1/2 - 1
∴ a = - (1/2)

a⁰ = 1
a = - (1/2)
a² = 1/4
a³ = - (1/8)

∴ a⁰ > a²> a³ > a

৩০৮.

If a man can swim at 8 km/hr in still water and the river flows at 2 km/hr, taking 80 minutes for a round trip, what is the distance to the location?

ক
3 km
খ
4 km
গ
5 km
ঘ
7 km

সঠিক উত্তর: গ

5 km

উত্তর

সঠিক উত্তর: গ

5 km

গ

ব্যাখ্যা

Question: If a man can swim at 8 km/hr in still water and the river flows at 2 km/hr, taking 80 minutes for a round trip, what is the distance to the location?

Solution:
Let the place be p km far.

Speed downstream = 8 + 2 = 10 km/hr

Speed upstream = 8 - 2 = 6 km/hr

As per the question;
p/6 + p/10 = 80/60
⇒ (5p + 3p)/30 = 4/3
⇒ 24p = 120
∴ p = 120/24 = 5 km

৩০৯.

There are 200 questions on a 3-hour examination. Among these questions 50 are mathematics problems. It is suggested that twice as much time be spent on each math problem as for each other question. How many minutes should be spent on mathematics problems?

ক
32 minutes
খ
60 minutes
গ
72 minutes
ঘ
100 minutes

সঠিক উত্তর: গ

72 minutes

উত্তর

সঠিক উত্তর: গ

72 minutes

গ

ব্যাখ্যা

Question: There are 200 questions on a 3-hour examination. Among these questions 50 are mathematics problems. It is suggested that twice as much time be spent on each math problem as for each other question. How many minutes should be spent on mathematics problems?

Solution:
Total exam time 3 hours = 3 × 60 minutes = 180 minutes
Total number of questions 200

Let
x minutes be the time spent on each non-math question.
∴ the time spent on each math problem would be 2x minutes.

∴ the total time spent on non-math questions (200 - 50) × x minutes
= 150x minutes

The total time spent on math problems 50 × 2x minutes
= 100x minutes

ATQ,
150x + 100x = 180
⇒ 250 x = 180
⇒ x = 180 / 250
∴ x = 0.72 minutes

∴ Time for each math problem = 2 × x = 2 × 0.72 = 1.44 minutes
∴ Time for 50 math problems = 1.44 × 50 = 72 minutes.

৩১০.

A book was sold at a loss of 15%. If the selling price had been increased by 75 Taka, there would have been a profit of 10%. What is the cost price of the book?

ক
Tk. 350
খ
Tk. 300
গ
Tk. 280
ঘ
Tk. 400
ঙ
None of these

সঠিক উত্তর: খ

Tk. 300

উত্তর

সঠিক উত্তর: খ

Tk. 300

খ

ব্যাখ্যা

Question: A book was sold at a loss of 15%. If the selling price had been increased by 75 Taka, there would have been a profit of 10%. What is the cost price of the book?

Solution:
ধরি,
বইটির ক্রয়মূল্য ১০০ টাকা হলে,

১৫% ক্ষতিতে বিক্রয়মূল্য = (১০০ - ১৫) টাকা = ৮৫ টাকা
১০% লাভে বিক্রয়মূল্য = (১০০ + ১০) টাকা = ১১০ টাকা

∴ বিক্রয়মূল্যের পার্থক্য = (১১০ - ৮৫) = ২৫ টাকা

এখন,
বিক্রয়মূল্যের পার্থক্য ২৫ টাকা হলে ক্রয়মূল্য = ১০০ টাকা
∴ বিক্রয়মূল্যের পার্থক্য ১ টাকা হলে ক্রয়মূল্য = ১০০/২৫ টাকা
∴ বিক্রয়মূল্যের পার্থক্য ৭৫ টাকা হলে ক্রয়মূল্য = (১০০ × ৭৫)/২৫ টাকা
= ৩০০ টাকা

∴ বইটির ক্রয়মূল্য = ৩০০ টাকা।

৩১১.

The price of a loaf of bread was increased by 20%. How many that used to buy 300 loaves?

ক
ক) 240
খ
খ) 250
গ
গ) 280
ঘ
ঘ) 320

সঠিক উত্তর: খ

খ) 250

উত্তর

সঠিক উত্তর: খ

খ) 250

খ

ব্যাখ্যা

Let the price of one loaf of bread is 100x
So, price of 300 loaves of bread is 30000x

At 20% increase,
Price of one loaf of bread is 120x
So, with 30000x taka = 30000x/120x = 250 loaves of bread can be purchased

৩১২.

The ratio of the cost price and selling price is 5 : 4, the loss percent is =?

ক
20%
খ
22%
গ
25%
ঘ
30%

সঠিক উত্তর: ক

20%

উত্তর

সঠিক উত্তর: ক

20%

ক

ব্যাখ্যা

Question: The ratio of the cost price and selling price is 5 : 4, the loss percent is =?

Solution:
let, cost price 5x
selling price 4x taka

loss = {(5x - 4x)/5x} × 100%
= (1/5) × 100%
= 20%

৩১৩.

A sum of money is put on C.I. for 2 years at 20%. It would fetch Tk. 482 more if the interest is payable half yearly than if it were payable yearly. Find the sum.

ক
Tk. 20000
খ
Tk. 15000
গ
Tk. 22000
ঘ
Tk. 28000

সঠিক উত্তর: ক

Tk. 20000

উত্তর

সঠিক উত্তর: ক

Tk. 20000

ক

ব্যাখ্যা

Question: A sum of money is put on C.I. for 2 years at 20%. It would fetch Tk. 482 more if the interest is payable half yearly than if it were payable yearly. Find the sum.

Solution:
Let the Principal = Tk. 100
When compounded annually,
A = 100 [1+20/100]² = 144

When compounded half yearly,
A = 100[1+10/100]⁴ = 146.41

Difference, 146.41 - 144 = 2.41
If difference is 2.41, then Principal = Tk. 100
If difference is 482, then Principal = (100 × 482)/2.41 = 20000

৩১৪.

The mean of 16 items was found to be 30. On rechecking, it was found that two items were wrongly taken as 22 and 18 instead of 32 and 28 respectively. Find the correct mean.

ক
31.25
খ
40
গ
33.5
ঘ
35

সঠিক উত্তর: ক

31.25

উত্তর

সঠিক উত্তর: ক

31.25

ক

ব্যাখ্যা

Question: The mean of 16 items was found to be 30. On rechecking, it was found that two items were wrongly taken as 22 and 18 instead of 32 and 28 respectively. Find the correct mean.

Solution:
Calculated mean of 16 items = 30.
Incorrect sum of these 16 items = (30 × 16) = 480.

Correct sum of these 16 items
= (incorrect sum) - (sum of incorrect items) + (sum of actual items)
= [480 - (22 + 18) + (32 + 28)]
= 500.

Therefore, correct mean = 500/16 = 31.25.
Hence, the correct mean is 31.25.

৩১৫.

If α, β are the roots of the equation 2x² + 10x - 12 = 0, then α + β equals to:

ক
24
খ
- 8
গ
6
ঘ
- 5

সঠিক উত্তর: ঘ

- 5

উত্তর

সঠিক উত্তর: ঘ

- 5

ঘ

ব্যাখ্যা

Question: If α, β are the roots of the equation 2x² + 10x - 12 = 0, then α + β equals to:

Solution:
Given that,
2x² + 10x - 12 = 0
Where, a = 2, b = 10, c = - 12

Now, For a quadratic equation ax2 + bx + c = 0, the sum of roots α + β = - b/a
sum of roots α + β = - b/a = - 10/2 = - 5

৩১৬.

If sec(x - 30°) = 2, then sin x = ?

ক
1/2
খ
0
গ
1
ঘ
None of the above

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: If sec(x − 30°) = 2, then sin x = ?

Solution:
sec (x - 30°) = 2
⇒ sec (x - 30°) = sec 60°
⇒ x - 30° = 60°
⇒ x = 90°
∴ sin 90° = 1

৩১৭.

The area of a square park is 1600 sq. meters. If fencing costs Tk. 5 per meter, what is the total cost to fence the park?

ক
Tk. 800
খ
Tk. 1200
গ
Tk. 1600
ঘ
Tk. 1050

সঠিক উত্তর: ক

Tk. 800

উত্তর

সঠিক উত্তর: ক

Tk. 800

ক

ব্যাখ্যা

Question: The area of a square park is 1600 sq. meters. If fencing costs Tk. 5 per meter, what is the total cost to fence the park?

Solution:
Given that,
Area of square park = 1600 m²
∴ Side length of the square = √1600 = 40 meters

Perimeter of the square = 4 × side = 4 × 40 = 160 meters
And cost of fencing = Tk. 5 per meter

∴ Total cost = perimeter × cost per meter
= 160 × 5
= Tk. 800

So the total cost to fence the park is Tk. 800.

৩১৮.

A merchant sells one table for Tk. 1200, making a profit of 25%, and another table for Tk. 1500, incurring a loss of 10%. What is his overall gain or loss percentage?

ক
3.14% gain
খ
2.78% gain
গ
4.82% loss
ঘ
1.73% loss
ঙ
None

সঠিক উত্তর: খ

2.78% gain

উত্তর

সঠিক উত্তর: খ

2.78% gain

খ

ব্যাখ্যা

Question: A merchant sells one table for Tk. 1200, making a profit of 25%, and another table for Tk. 1500, incurring a loss of 10%. What is his overall gain or loss percentage?

Solution:
C.P. of 1st table = Tk.(100 × 1200)/125 = Tk. 960
C.P. of 2nd table = Tk.(100 × 1500)/90 = Tk. 1667
So, total C.P. = Tk.(960 + 1667) = Tk. 2627
Total S.P. = Tk.(1200 + 1500) = Tk. 2700

∴ Gain% = {(2700 - 2627) × 100}/2627%
= (73 × 100)/2627%
= 2.78%

৩১৯.

Mamun bought 50 shares at Tk. 60, and 2 months later, he purchased 25 shares at Tk 56, at what price should he purchase 25 additional shares in order to have an average price of Tk 58 per share?

ক
ক) 53
খ
খ) 54
গ
গ) 55
ঘ
ঘ) 56

সঠিক উত্তর: ঘ

ঘ) 56

উত্তর

সঠিক উত্তর: ঘ

ঘ) 56

ঘ

ব্যাখ্যা

Question: Mamun bought 50 shares at Tk. 60, and 2 months later, he purchased 25 shares at Tk 56, at what price should he purchase 25 additional shares in order to have an average price of Tk 58 per share?

Solution:
ধরি
x টাকা দরে আরো ২৫ টি শেয়ার কিনেছিল

প্রশ্নমতে
(50 × 60) + (25 × 56) + (25 × x)/(50 + 25 + 25) = 58
(3000 + 1400 + 25x)/100 = 58
25x + 4400 = 5800
25x = 5800 - 4400
25x = 1400
x = 1400/25
x = 56

৩২০.

A dishonest trader mixes 2 kg of vegetable ghee costing Tk. 45 a kg with 3 kg of standard ghee costing Tk. 70 per kg. He sells the mixed ghee at Tk. 65 per kg. What is his percentage of profit?

ক
ক) 16.67%
খ
খ) 8.33%
গ
গ) 6%
ঘ
ঘ) 21%

সঠিক উত্তর: খ

খ) 8.33%

উত্তর

সঠিক উত্তর: খ

খ) 8.33%

খ

ব্যাখ্যা

Question: A dishonest trader mixes 2 kg of vegetable ghee costing Tk 45 a kg with 3 kg of standard ghee costing Tk. 70 per kg. He sells the mixed ghee at Tk. 65 per kg. What is his percentage of profit?

Solution:
2 কেজি Vegetable ঘি'র ক্রয়মূল্য = 2 × 45 = 90 টাকা
আবার, 3 কেজি Standard ঘি'র ক্রয়মূল্য
= (3 × 70) = 210 টাকা

∴ (2 + 3) = 5 কেজি ঘি'র মোট ক্রয়মূল্য = 90 + 210 = 300 টাকা।

আবার, 5 কেজি ঘি'র বিক্রয়মূল্য = 5 × 65 = 325 টাকা

∴ লাভ = (325 - 300) টাকা = 25 টাকা
300 টাকায় লাভ হয় 25 টাকা
100 টাকায় লাভ হয় (25 × 100)/300 টাকা
= 8.33%

৩২১.

The curved surface area and the diameter of a right circular cylinder are 660 sq.cm and 21 cm respectively. Find its height (in cm).

ক
9
খ
10
গ
12
ঘ
8

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: The curved surface area and the diameter of a right circular cylinder are 660 sq.cm and 21 cm respectively. Find its height (in cm).

Solution:
Diameter of cylinder = 21 cm
Radius of cylinder = 21/2 cm

The curved Surface area of cylinder = 2πrh,
Where,
r = radius,
h = height

According to the question
660 = 2 × (22/7) × (21/2) × h
⇒ 660 = 66 × h
∴ h = 10 cm

৩২২.

If n(U) = 100, n(A) = 40, n(B) = 35, and n(A ∩ B) = 15, then what is n(A ∪ B)′?

ক
56
খ
40
গ
45
ঘ
60
ঙ
30

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: If n(U) = 100, n(A) = 40, n(B) = 35, and n(A ∩ B) = 15, then what is n(A ∪ B)′?

Solution:
Given that,
n(U) = 100
n(A) = 40
n(B) = 35
and n(A ∩ B) = 15

We know,
n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
= 40 + 35 - 15
= 60
∴ n(A ∪ B) = 60

Now,
n(A ∪ B)′ = n(U) - n(A ∪ B)
= 100 - 60 = 40

∴ n(A ∪ B)′ = 40

৩২৩.

How many liters of water should be added to a 30 liters mixture of milk and water containing milk and water in the ratio of 7 : 3 such that the resultant mixture has 40% water in it?

ক
ক) 3
খ
খ) 5
গ
গ) 9
ঘ
ঘ) 12

সঠিক উত্তর: খ

খ) 5

উত্তর

সঠিক উত্তর: খ

খ) 5

খ

ব্যাখ্যা

Question: How many liters of water should be added to a 30 liters mixture of milk and water containing milk and water in the ratio of 7 : 3 such that the resultant mixture has 40% water in it?

Solution:
Water in the mixture = 30 × (3/10) = 9 liters

ATQ,
9 + x = 40% of (30 + x)
⇒ 9 + x = (2/5)(30 +x)
⇒ 45 + 5x = 60 + 2x
⇒ 3x = 15
⇒ x = 5

৩২৪.

Dipok drives from City A to B at 40 km per hour and returns over the same road at 30 km per hour and spends 8 hours away from home including a one hour stop for lunch. What is the distance (in km) between City A and City B?

ক
ক) 100 km
খ
খ) 120 km
গ
গ) 60 km
ঘ
ঘ) 80 km

সঠিক উত্তর: খ

খ) 120 km

উত্তর

সঠিক উত্তর: খ

খ) 120 km

খ

ব্যাখ্যা

Question: Dipok drives from City A to B at 40 km per hour and returns over the same road at 30 km per hour and spends 8 hours away from home including a one hour stop for lunch. What is the distance (in km) between City A and City B?

Solution:
ধরি,
A থেকে B এর দূরত্ব = x কি.মি.
প্রশ্নমতে,
(x/40) + (x/30) + 1 = 8
⇒ (x/40) + (x/30) = 8 - 1
⇒ (4x + 3x)/120 = 7
⇒ 7x/120 = 7
⇒ 7x = 7 × 120
⇒ x = (7 × 120)/7
x = 120

৩২৫.

x^1/2/27 = 12/x^3/2, What is the value of x is-

ক
15
খ
4
গ
25
ঘ
18

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: x^1/2/27 = 12/x^3/2, What is the value of x is-

Solution:
Given that,
⇒ x^1/2/27 = 12/x^3/2
⇒ x^1/2 × x^3/2 = 27 × 12
⇒ x^{{(1/2) + (3/2)}} = 324
⇒ x^4/2 = 324
⇒ x² = 324
⇒ x = √324
∴ x = 18

৩২৬.

Two fair dice are thrown together. What is the probability that the product of the two numbers that appear is 20?

ক
3/4
খ
1/18
গ
2/9
ঘ
1/2

সঠিক উত্তর: খ

1/18

উত্তর

সঠিক উত্তর: খ

1/18

খ

ব্যাখ্যা

Question: Two fair dice are thrown together. What is the probability that the product of the two numbers that appear is 20?

Solution:
Total number of possible outcomes when throwing two dice = 6 × 6 = 36
Favorable outcomes where the product is 20.
The possible pairs (first die, second die) are, (4, 5) and (5, 4)
∴ favorable outcomes = 2
∴ Probability = Number of favorable outcomes/Total outcomes
= 2/36
= 1/18

∴ The probability is 1/18.

৩২৭.

A man's regular pay is Taka 30 per hour up to 40 hours. Overtime is twice the payment for regular time. If he was paid Taka 1680, how many hours of overtime did he work?

ক
8
খ
12
গ
9
ঘ
15
ঙ
14

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

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

৩২৮.

What is the volume of a cone if its radius is 3 cm and its height is 14 cm?

ক
132 cm³
খ
244 cm³
গ
164 cm³
ঘ
42 cm³

সঠিক উত্তর: ক

132 cm³

উত্তর

সঠিক উত্তর: ক

132 cm³

ক

ব্যাখ্যা

Question: What is the volume of a cone if its radius is 3 cm and its height is 14 cm?

Solution:
Given that,
radius, r = 3 cm
height, h = 14 cm

Now,
Volume = (1/3) × π × r² × h
= (1/3) × (22/7) × 3² × 14
= 132 cm³

৩২৯.

The sum of the ages of mother and her daughter is 84 years. The mother’s age is 5 times the daughter’s age. What is the age of mother?

ক
70 years
খ
55 years
গ
60 years
ঘ
68 years
ঙ
75 years

সঠিক উত্তর: ক

70 years

উত্তর

সঠিক উত্তর: ক

70 years

ক

ব্যাখ্যা

Question: The sum of the ages of mother and her daughter is 84 years. The mother’s age is 5 times the daughter’s age. What is the age of mother?

Solution:
Let the daughter’s age = x
Then mother’s age = 5x

ATQ,
⇒ x + 5x = 84
⇒ 6x = 84
⇒ x = 84/6
∴ x = 14

∴ Mother’s age = 5 × 14 = 70 years

৩৩০.

If a + b + c = 13 and a² + b² + c² = 69, then what is the value of ab + bc + ca?

ক
- 81
খ
50
গ
100
ঘ
- 80

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: If a + b + c = 13 and a² + b² + c² = 69, then what is the value of ab + bc + ca?

Solution:
We know,
(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
⇒ 2(ab + bc + ca) = (a + b + c)² - ( a² + b² + c²)
⇒ 2(ab + bc + ca) = 13² - 69 [given, a + b + c = 13 and a² + b² + c² = 69]
⇒ 2(ab + bc + ca) = 169 - 69 = 100
⇒ (ab + bc + ca) = 100/2
∴ (ab + bc + ca) = 50

৩৩১.

If 738A6A is divisible by 11, then the value of A is:

ক
ক) 7
খ
খ) 12
গ
গ) 9
ঘ
ঘ) 10

সঠিক উত্তর: গ

গ) 9

উত্তর

সঠিক উত্তর: গ

গ) 9

গ

ব্যাখ্যা

A number is divisible by 11 when the difference between the total number of odd places and the total number of even places is equal to zero or multiple of 11

Therefore, A + A + 3 = 7 + 8 + 6
⇒ 2A = 18
⇒ A = 9

৩৩২.

If selling price is doubled, the profit triples. Find the profit percent?

ক
400%
খ
300%
গ
200%
ঘ
100%

সঠিক উত্তর: ঘ

100%

উত্তর

সঠিক উত্তর: ঘ

100%

ঘ

ব্যাখ্যা

Question: If selling price is doubled, the profit triples. Find the profit percent?

Solution:
Let,
The cost price be Tk.100 and sell price be Tk .x,
Then
The profit is (x - 100)

Now The sell price is doubled, then the new sell price is 2x
New profit is (2x - 100)

ATQ,
3(x - 100) = 2x - 100
⇒ 3x - 300 = 2x - 100
⇒ 3x - 2x = 300 - 100
∴ x = 200

Then the Profit = (200 - 100) = 100
Hence the profit percentage is = (100 × 100)/100 %
= 100%

৩৩৩.

Two trains passing towards same direction with the speed of 72km/h and 108km/h. The length of one train is 200m and the overtaking takes 1 minute. How long is the another train?

ক
300m
খ
400m
গ
500m
ঘ
600m

সঠিক উত্তর: খ

400m

উত্তর

সঠিক উত্তর: খ

400m

খ

ব্যাখ্যা

Question: Two trains passing towards same direction with the speed of 72km/h and 108km/h. The length of one train is 200m and the overtaking takes 1 minute. How long is the another train?

Solution:
hints: kmph/3.6 = mps
1st train speed = 72km/h
= 20 m/s
second train speed = 108 km/h
= 30m/s

as moving towards same direction,
resultant speed = 30 - 20 = 10m/s
total distance = the length of two trains.
total time = 1 minute = 60s

∴ total distance covered is = speed × time
= 10 × 60 = 600m

length of the second train is = 600 - 200
= 400m

৩৩৪.

If tan(3x - 15°) = cot(3y + 15°), then (x + y) is:

ক
10°
খ
30°
গ
20°
ঘ
50°

সঠিক উত্তর: খ

30°

উত্তর

সঠিক উত্তর: খ

30°

খ

ব্যাখ্যা

Question: If tan(3x - 15°) = cot(3y + 15°), then (x + y) is:

Solution:
tan(3x - 15°) = cot(3y + 15°)
⇒ tan (3x - 15°) = tan {90° - (3y + 15°)}
⇒ 3x - 15° = 75° - 3y
⇒ 3x + 3y = 75° + 15°
⇒ 3(x + y) = 90°
⇒ x + y = 30°

∴ x + y = 30°

৩৩৫.

A's savings and expenditures are in the ratio of 1 : 4. His income increases by 10 % and expenditure also increases by 15%. How much percent does his savings increase or decrease?

ক
ক) increased by 10%
খ
খ) increased by 12%
গ
গ) decreased by 10%
ঘ
ঘ) decreased by 12%

সঠিক উত্তর: গ

গ) decreased by 10%

উত্তর

সঠিক উত্তর: গ

গ) decreased by 10%

গ

ব্যাখ্যা

Question: A's savings and expenditures are in the ratio of 1 : 4. His income increases by 10 % and expenditure also increases by 15%. How much percent does his savings increase or decrease?

Solution:
let A's savings = x
and expenditures = 4x

his total income = x + 4x = 5x

his increased income = 110% of 5x = 5.5x
his Increased expenditure = 115% of 4x = 4.6x
so, savings = 5.5x - 4.6x = 0.9x

now, decreased savings = x - 0.9x = 0.1x

∴ decreased % = (0.1 × 100)/1 = 10%

৩৩৬.

If 7 spiders make 7 webs in 7 days, then 1 spider will make 1 web in how many days?

ক
1
খ
3
গ
7
ঘ
14
ঙ
None of these

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: If 7 spiders make 7 webs in 7 days, then 1 spider will make 1 web in how many days?

Solution:
7 spiders make 7 webs in 7 days
1 spiders make 1 webs in (7 ×7)/7 days
= 7 days

৩৩৭.

Bananas are bought at Tk. 36 per dozen and sold at a profit of 20%. How much is the selling price of twenty bananas?

ক
ক) Tk. 75
খ
খ) Tk. 72
গ
গ) Tk. 68
ঘ
ঘ) None of the Above

সঠিক উত্তর: খ

খ) Tk. 72

উত্তর

সঠিক উত্তর: খ

খ) Tk. 72

খ

ব্যাখ্যা

Question: Bananas are bought at Tk. 36 per dozen and sold at a profit of 20%. How much is the selling price of twenty bananas?

Solution:
৩৬ টাকায় ক্রয় করে ১২ টি কলা
∴ ১ টাকায় ক্রয় করে ১২/৩৬ টি কলা
∴ ১০০ টাকায় ক্রয় করে (১২ × ১০০)/৩৬ টি কলা
= ১০০/৩ টি কলা

২০% লাভে বিক্রয়মূল্য = (১০০ + ২০) টাকা = ১২০ টাকা

২০% লাভ করতে হলে,
১০০/৩ টি কলার বিক্রয়মূল্য ১২০ টাকা
∴ ১ টি কলার বিক্রয়মূল্য (১২০ × ৩)/১০০ টাকা
∴ ২০ টি কলার বিক্রয়মূল্য (১২০ × ৩ × ২০)/১০০ টাকা
= ৭২ টাকা

৩৩৮.

In an election, the winning candidate secured 70% of the total votes and won by a majority of 60,000 votes. What is the total number of votes polled?

ক
145,000
খ
150,000
গ
155,000
ঘ
160,000

সঠিক উত্তর: খ

150,000

উত্তর

সঠিক উত্তর: খ

150,000

খ

ব্যাখ্যা

Question: In an election, the winning candidate secured 70% of the total votes and won by a majority of 60,000 votes. What is the total number of votes polled?

Solution:
Let, The winning candidate secured 0.70x votes,
and the losing candidate secured 0.30x votes.

ATQ,
0.70x - 0.30x = 60,000
⇒ 0.40x = 60,000
⇒ x = 60,000 / 0.40
= 150,000

Thus, the total number of votes polled is 150,000.

৩৩৯.

The speed of a boat in still water in 15 km/hr and the rate of current is 3 km/hr. The distance travelled downstream in 12 minutes is:

ক
ক) 1.2 km
খ
খ) 3.6 km
গ
গ) 1.8 km
ঘ
ঘ) 2.4 km

সঠিক উত্তর: খ

খ) 3.6 km

উত্তর

সঠিক উত্তর: খ

খ) 3.6 km

খ

ব্যাখ্যা

The speed of a boat in still water=15km/hr
Rate of current =3km/hr
Speed in downstream=15+3=18km/hr

Time given=12min= 12/60 hr= 1/5 hr
Distance travelled= Speed×Time=18× 1/5 =3.6km

৩৪০.

Determine the value of the 4^th term of the sequence: sin⁡(nπ/6)

ক
√3/2
খ
- 1/2
গ
1/√3
ঘ
1/2

সঠিক উত্তর: ক

√3/2

উত্তর

সঠিক উত্তর: ক

√3/2

ক

ব্যাখ্যা

Question: Determine the value of the 4^th term of the sequence: sin⁡(nπ/6)

Solution:
Here,
The 4^th term of sin(nπ/6) = {sin(4 × π)/6}
= {sin(4 × 180°)/6}
= sin 120°
= sin(90° + 30°)
= cos 30°
= √3/2

৩৪১.

The average age of the children in a tour group is 8 years, and that of the adults is 30 years. If the average age of the entire tour group is 15 years, find the ratio of children to adults in the group.

ক
3 : 7
খ
5 : 7
গ
9 : 7
ঘ
15 : 7

সঠিক উত্তর: ঘ

15 : 7

উত্তর

সঠিক উত্তর: ঘ

15 : 7

ঘ

ব্যাখ্যা

Question: The average age of the children in a tour group is 8 years, and that of the adults is 30 years. If the average age of the entire tour group is 15 years, find the ratio of children to adults in the group.

Solution:
Here,
Average age of children = 8 years
Average age of adults = 30 years
Average age of the entire group = 15 years

Let the number of children = m
and the number of adults = n

Then, the total number of people in the group is (m + n)

ATQ,
8m + 30n = 15(m + n)
⇒ 8m + 30n = 15m + 15n
⇒ 15m - 8m = 30n - 15n
⇒ 7m = 15n
⇒ m/n = 15/7
⇒ m : n = 15 : 7

∴ The ratio of children to adults in the group is 15 : 7

৩৪২.

একটি সভাতে মহিলা ও পুরুষের অনুপাত ৫ : ৪। যদি আরও ৩৬ জন পুরুষ সভাতে যোগ দেয় তাহলে অনুপাতটি উল্টে যাবে। সভাতে কতজন মহিলা ছিল?

ক
৬০ জন
খ
৬৪ জন
গ
৭৮ জন
ঘ
৮০ জন
ঙ
কোনোটিই নয়

সঠিক উত্তর: ঘ

৮০ জন

উত্তর

সঠিক উত্তর: ঘ

৮০ জন

ঘ

ব্যাখ্যা

প্রশ্ন: একটি সভাতে মহিলা ও পুরুষের অনুপাত ৫ : ৪। যদি আরও ৩৬ জন পুরুষ সভাতে যোগ দেয় তাহলে অনুপাতটি উল্টে যাবে। সভাতে কতজন মহিলা ছিল?

সমাধান:
ধরি,
সভাতে মহিলা ও পুরুষের সংখ্যা যথাক্রমে ৫ক এবং ৪ক।

প্রশ্নমতে,
৫ক/(৪ক + ৩৬) = ৪/৫
⇒ ২৫ক = ১৬ক + ১৪৪
⇒ ২৫ক - ১৬ক = ১৪৪
⇒ ৯ক = ১৪৪
⇒ ক = ১৪৪/৯
∴ ক = ১৬

∴ মহিলার সংখ্যা ছিল = (৫ × ১৬) = ৮০ জন

৩৪৩.

An aluminum sheet 27 cm long, 8 cm broad, and 1 cm thick is melted into a cube. The difference in the surface areas of the two solids would be

ক
276 cm²
খ
296 cm²
গ
282 cm²
ঘ
286 cm²

সঠিক উত্তর: ঘ

286 cm²

উত্তর

সঠিক উত্তর: ঘ

286 cm²

ঘ

ব্যাখ্যা

Question: An aluminum sheet 27 cm long, 8 cm broad, and 1 cm thick is melted into a cube. The difference in the surface areas of the two solids would be

Solution:
Volume of cube = Volume of sheet = (27 × 8 × 1) cm³
= (27 × 8) cm³
Length of the cube = a
∴ a³ = (27 × 8)
⇒ a³ = 3³ × 2³
⇒ a = (3 × 2)
∴ a = 6

Surface area of sheet =2(lb + bh + lh)
= 2(27 × 8 + 8 × 1 + 27 × 1) cm²
= 2(216 + 8 + 27) cm²
= 502 cm²
Surface area of cube = 6a² =(6 × 6²) cm²
= 216 cm²

∴ Required difference = (502 - 216) cm²
= 286 cm²

৩৪৪.

A ferry can travel twice as fast when empty as when it is full. It travels 20 miles with full load, spends 1 hour for unloading and returns to its original post empty. It took 11 hours to complete the journey. What is the speed of the ferry when it is empty?

ক
ক) 5
খ
খ) 6
গ
গ) 6.5
ঘ
ঘ) 8

সঠিক উত্তর: খ

খ) 6

উত্তর

সঠিক উত্তর: খ

খ) 6

খ

ব্যাখ্যা

ধরি,
ফেরিটির মাল বোঝাই অবস্থায় গতি বেগ x মাইল /ঘণ্টা
ফেরিটির খালি অবস্থায় গতি বেগ 2x মাইল/ঘণ্টা

মাল বোঝাই অবস্থায়
ফেরিটির 20 মাইল যেতে সময় লাগে = 20/x ঘণ্টা

খালি অবস্থায়
ফেরিটির 20 মাইল যেতে সময় লাগে =20/2x = 10/x ঘণ্টা

প্রশ্নমতে,
20/x + 10/x = 11 - 1
(20 + 10)/x = 10
30/x = 10
10x = 30
x = 3

ফেরিটির খালি অবস্থায় গতি বেগ 2 × 3 = 6 মাইল/ঘণ্টা

৩৪৫.

4, -8, 16, -32, 64, (....)

ক
ক) 128
খ
খ) -128
গ
গ) 192
ঘ
ঘ) -192

সঠিক উত্তর: খ

খ) -128

উত্তর

সঠিক উত্তর: খ

খ) -128

খ

ব্যাখ্যা

Each number is the proceeding number multiplied by -2. So, the required number is -128.

৩৪৬.

The price of 357 mangoes is Tk. 1517.25. Find the approximate price of 49 dozens of such mangoes?

ক
Tk. 3099
খ
Tk. 3599
গ
Tk. 4099
ঘ
Tk. 2499

সঠিক উত্তর: ঘ

Tk. 2499

উত্তর

সঠিক উত্তর: ঘ

Tk. 2499

ঘ

ব্যাখ্যা

Question: The price of 357 mangoes is Tk. 1517.25. Find the approximate price of 49 dozens of such mangoes?

Solution:
We know that 1 dozen = 12 piece
49 dozens = 49 × 12 = 588 mangoes

357 mangoes = Tk. 1517.25
∴ 1 mango = Tk. 1517.25/357
∴ 588 mangoes = Tk. (1517.25 × 588)/357
= Tk. 2499

৩৪৭.

4600 Taka is lent at compound interest of 10% per annum for 2 years. Find the amount after two years.

ক
Tk. 5656
খ
Tk. 5566
গ
Tk. 5556
ঘ
Tk. 6655

সঠিক উত্তর: খ

Tk. 5566

উত্তর

সঠিক উত্তর: খ

Tk. 5566

খ

ব্যাখ্যা

Question: 4600 Taka is lent at compound interest of 10% per annum for 2 years. Find the amount after two years.

Solution:
Compound amount = P (1 + r)ⁿ
= 4600 {1 + (10/100)}²
= 4600 × (110/100) × (110/100)
= 46 × 11 × 11
= 5566

৩৪৮.

Galib buys 5 dolls for his 5 nieces. The gifts include two identical Sun-and-Fun beach dolls, one Elegant Eddie dress-up doll, one G.I. Josie army doll, and one Tulip Troll doll. If the youngest niece doesn't want the G.I. Josie doll, in how many different ways can he give the gifts?

ক
96
খ
60
গ
48
ঘ
12

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: Galib buys 5 dolls for his 5 nieces. The gifts include two identical Sun-and-Fun beach dolls, one Elegant Eddie dress-up doll, one G.I. Josie army doll, and one Tulip Troll doll. If the youngest niece doesn't want the G.I. Josie doll, in how many different ways can he give the gifts?

Solution:
total ways = 5!/2! = 60

if the the youngest niece wants the G.I. Josie doll = 4!/2! = 12

the youngest niece doesn't want the G.I. Josie doll, different ways = 60 - 12
= 48

৩৪৯.

A manufacturer sells a pair of shoes to a wholesale dealer at a profit of 20 %. Wholesalers sell them to retailers at a profit of 25 %. The shoes are again sold to the customer for Tk. 50.50, thereby earning a profit of 30 %. Find the cost price of the manufacturer.

ক
Tk. 20.36
খ
Tk. 22.90
গ
Tk. 25.89
ঘ
Tk. 30.50

সঠিক উত্তর: গ

Tk. 25.89

উত্তর

সঠিক উত্তর: গ

Tk. 25.89

গ

ব্যাখ্যা

Profit earned by manufacturer = 20 %
Profit earned by wholesaler = 25 %
Profit earned by retailer = 30%
S.P. of shoes = Tk. 50

Therefore,
130 % of 125 % of 120 % of x = 50.50
⇒ (120/100) × (125/100) × (130/100) × x = 5050/100
⇒ (195/100) x = 5050/100
⇒ x = (5050 × 100)/(195 × 100)
⇒ x = 25.89

Cost price of shoes = Tk. 25.89.

৩৫০.

The ratio of income and expenditure of a person is 5 : 2, If he saves Tk. 3000 per month, what is his monthly income?

ক
Tk. 4000
খ
Tk. 5000
গ
Tk. 1000
ঘ
Tk. 12000

সঠিক উত্তর: খ

Tk. 5000

উত্তর

সঠিক উত্তর: খ

Tk. 5000

খ

ব্যাখ্যা

Question: The ratio of income and expenditure of a person is 5 : 2, If he saves Tk. 3000 per month, what is his monthly income?

Solution:
Let, income = 5x
and expenditure = 2x
Then,
Savings = 5x - 2x = 3000
⇒ 3x = 3000
∴ x = 1000

∴ monthly income = 5x = 5 × 1000 = Tk. 5000

৩৫১.

Find the degree of the polynomial 3x⁵ + 5x²y⁴ + 7y³ + 2.

ক
6
খ
4
গ
5
ঘ
14

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: Find the degree of the polynomial 3x⁵ + 5x²y⁴ + 7y³ + 2.

Solution:
একটি বহুপদীর (polynomial), ঘাত (degree) হলো সেই বহুপদীর পদগুলির মধ্যে সর্বোচ্চ ঘাত। প্রতিটি পদের ঘাত হলো সেই পদের চলকগুলির (variables) ঘাতগুলির যোগফল।

এখন, প্রতিটি পদের ঘাত:
3x⁵ এর ঘাত = 5
5x²y⁴ এর ঘাত = 2 + 4 = 6
7y³ এর ঘাত = 3
2 এর ঘাত = 0

সর্বোচ্চ ঘাত = 6
সুতরাং, বহুপদীটির ঘাত (degree) = 6

৩৫২.

A pump can fill a tank with water in 2 hours. Because of a leak, it took (7/3) hours to fill the tank. The leak can drain all the water of the tank in:

ক
(13/3)hrs.
খ
7 hrs.
গ
8 hrs.
ঘ
14 hrs.

সঠিক উত্তর: ঘ

14 hrs.

উত্তর

সঠিক উত্তর: ঘ

14 hrs.

ঘ

ব্যাখ্যা

Question: A pump can fill a tank with water in 2 hours. Because of a leak, it took (7/3) hours to fill the tank. The leak can drain all the water of the tank in:

Solution:
pump fills 1/2 tank in 1 hour

because of leak, pump fills 3/7 tank in 1 hour

leak drains (1/2) - (3/7) = 1/14 tank in 1 hour
leak drains full tank in 14 hours

৩৫৩.

How many different six digit numbers can be formed using all of the following digits 3, 3, 4, 4, 4, 5?

ক
ক) 50
খ
খ) 40
গ
গ) 60
ঘ
ঘ) 30

সঠিক উত্তর: গ

গ) 60

উত্তর

সঠিক উত্তর: গ

গ) 60

গ

ব্যাখ্যা

Question: How many different six digit numbers can be formed using all of the following digits 3, 3, 4, 4, 4, 5?

Solution:
প্রদত্ত অঙ্ক মোট 6টি যার মধ্যে 2টি 3 এবং 3টি 4 আছে।

∴ নির্ণেয় ছয় অঙ্কবিশিষ্ট মোট গঠিত সংখ্যা = 6!/(2! × 3!) টি
= 60 টি

৩৫৪.

Find the midpoint of the line segment joining the points P₁ = (- 2, 5) and P₂ = (8, - 1).

ক
(3, 2)
খ
(1, 4)
গ
(3, 8)
ঘ
(4, 2)

সঠিক উত্তর: ক

(3, 2)

উত্তর

সঠিক উত্তর: ক

(3, 2)

ক

ব্যাখ্যা

Question: Find the midpoint of the line segment joining the points P₁ = (- 2, 5) and P₂ = (8, - 1).

Solution:

৩৫৫.

A sum of Tk. 1800 has been divided among A, B and C such that A gets 1/2 of what B gets and B gets 1/3 of what C gets. C's share is-

ক
Tk. 600
খ
Tk. 900
গ
Tk. 1200
ঘ
Tk. 1400

সঠিক উত্তর: গ

Tk. 1200

উত্তর

সঠিক উত্তর: গ

Tk. 1200

গ

ব্যাখ্যা

Question: A sum of Tk. 1800 has been divided among A, B and C such that A gets 1/2 of what B gets and B gets 1/3 of what C gets. C's share is-

Solution:
Let, C's share = Tk. x
Then, B's share = Tk. x/3,
A's share = Tk. (1/2)(x/3) = Tk. x/6

∴ x/6 + x/3 + x = 1800
⇒ (x + 2x + 6x)/6 = 1800
⇒ 9x/6 = 1800
⇒ x = (1800 × 6)/9
⇒ x = Tk. 1200

∴ C's share = Tk. 1200

৩৫৬.

Rafsan rows to a place 48 km distant and comes back in 14 hours. He finds that he can row 4 km with the stream in the same time as 3 km against the stream. The rate of the stream is:

ক
0.5 km/hr
খ
1 km/hr
গ
1.8 km/hr
ঘ
2 km/hr

সঠিক উত্তর: খ

1 km/hr

উত্তর

সঠিক উত্তর: খ

1 km/hr

খ

ব্যাখ্যা

Question: Rafsan rows to a place 48 km distant and comes back in 14 hours. He finds that he can row 4 km with the stream in the same time as 3 km against the stream. The rate of the stream is:

Solution:
let, rate of stream y and rate of rafsan is x km/hr

ATQ,
4/(x + y) = 3/(x - y)
⇒ 4x - 4y = 3x + 3y
⇒ x = 7y

48/(x + y) + 48/(x - y) = 14
⇒ 1/(7y + y) + 1/(7y - y) = 14/48
⇒ (1/8y) + (1/6y) = 14/48
⇒ 7/24y = 14/48
⇒ y = 1 km/hr

৩৫৭.

A 220 meters long train running at the speed of 108 kmph crosses another train running in opposite direction at the speed of 90 kmph in 9 seconds. What is the length of the other train?

ক
275 meters
খ
250 meters
গ
225 meters
ঘ
210 meters

সঠিক উত্তর: ক

275 meters

উত্তর

সঠিক উত্তর: ক

275 meters

ক

ব্যাখ্যা

Question: A 220 meters long train running at the speed of 108 kmph crosses another train running in opposite direction at the speed of 90 kmph in 9 seconds. What is the length of the other train?

Solution:
Relative speed =(108 + 90) km/h
= (198 × 5/18) m/sec
= 55 m/sec

Let,
the length of the other train be x metres.
Then,
(x + 220)/9 = 55
⇒ x + 220 = 55 × 9
⇒ x = 495 - 220
∴ x = 275

৩৫৮.

Kavita bought land for Tk. 50000. She sold the land by offering a 30% discount on the land. Find how much did she lose.

ক
Tk. 15000
খ
Tk. 12000
গ
Tk. 18000
ঘ
Tk. 20000

সঠিক উত্তর: ক

Tk. 15000

উত্তর

সঠিক উত্তর: ক

Tk. 15000

ক

ব্যাখ্যা

Question: Kavita bought land for Tk. 50000. She sold the land by offering a 30% discount on the land. Find how much did she lose.

Solution:
Principal amount p = 50000
Discount rate, r = 30%

Total Discount = 50000 × (30/100)
= 15000

Thus, Selling price = 50000 - 15000 = 35000
Hence, Kavitha lost Tk. 15000

৩৫৯.

The product of 0.09 and 0.007 is :

ক
ক) 0.00630
খ
খ) 0.000663
গ
গ) 0.06300
ঘ
ঘ) 0.00063

সঠিক উত্তর: ঘ

ঘ) 0.00063

উত্তর

সঠিক উত্তর: ঘ

ঘ) 0.00063

ঘ

ব্যাখ্যা

9 × 7 = 63
Sum of decimal places = 5
∴ 0.09 × 0.007 = 0.00063

৩৬০.

A license plate begins with three letters. If the possible letters are P, Q, R, S, how many different permutations of these letters can be made if no letter is used more than once?

ক
ক) 20
খ
খ) 22
গ
গ) 24
ঘ
ঘ) 26

সঠিক উত্তর: গ

গ) 24

উত্তর

সঠিক উত্তর: গ

গ) 24

গ

ব্যাখ্যা

Question: A license plate begins with three letters. If the possible letters are P, Q, R, S, how many different permutations of these letters can be made if no letter is used more than once?

Solution:
For the first letter, there are 4 possible choices. After that letter is chosen, there are 3 possible choices. Finally, there are 2 possible choices.
∴ Total permutations = 4 × 3 × 2
= 24

৩৬১.

An aeroplane covers a certain distance at a speed of 250 kmph in 6 hours. To cover the same distance in 100 minutes, it must travel at a speed of-

ক
900 km/hr
খ
700 km/hr
গ
800 km/hr
ঘ
950 km/hr

সঠিক উত্তর: ক

900 km/hr

উত্তর

সঠিক উত্তর: ক

900 km/hr

ক

ব্যাখ্যা

Question: An aeroplane covers a certain distance at a speed of 250 kmph in 6 hours. To cover the same distance in 100 minutes, it must travel at a speed of-

Solution:
Total Distance = (250 × 6) = 1500 km.
Time 100 minutes = 100/60 hr
= 5/3 hr

We know that,
Speed = Distance/Time
∴ Required speed = 1500/(5/3) km/hr
= 900 km/hr.

৩৬২.

If 9 engines consume 24 metric tons of coal, when each is working 8 hours per day, how much coal should be available for 8 engines, each running 13 hours per day, it is given that 3 engines of the former type consume as much as 4 engines of later type.

ক
20 metric tons
খ
15 metric tons
গ
26 metric tons
ঘ
17 metric tons

সঠিক উত্তর: গ

26 metric tons

উত্তর

সঠিক উত্তর: গ

26 metric tons

গ

ব্যাখ্যা

Question: If 9 engines consume 24 metric tons of coal, when each is working 8 hours per day, how much coal should be available for 8 engines, each running 13 hours per day, it is given that 3 engines of the former type consume as much as 4 engines of later type.

Solution:
We have:
The lesser engines, less coal consumed
More working hours, more coal consumed
Both the cases are directly proportional.
If three engines of former type consume 1 unit, 1 engine will consume 1/3 unit.
If four engines of latter type consume 1 unit, 1 engine will consume ¼ units.
And, less rate of consumption, less coal consumed.
Now, Number of engines = 9: 8
Working hours = 8:13
Therefore, rate of consumption = 1/3 : 1/4

Let the coal consumed by 8 engines is x metric tones
9 × 8 × 1/3 : 8 × 13 × 1/4 = 24 : x
⇒ 9 × 8 × (1/3) × x = 8 × 13 × (1/4) × 24
⇒ 24x = 624
∴ x = 26

৩৬৩.

It takes eight hours for a 600 km journey, if 120 km is done by train and the rest by car. It takes 20 minutes more, if 200 km is done by train and the rest by car. The ratio of the speed of the train to that of the cars is-

ক
2 : 3
খ
3 : 4
গ
4 : 5
ঘ
3 : 5

সঠিক উত্তর: খ

3 : 4

উত্তর

সঠিক উত্তর: খ

3 : 4

খ

ব্যাখ্যা

Question: It takes eight hours for a 600 km journey, if 120 km is done by train and the rest by car. It takes 20 minutes more, if 200 km is done by train and the rest by car. The ratio of the speed of the train to that of the cars is-

Solution:
Let, Speed of train and car is T and C respectively

ATQ,
(120/T) + (600 - 120)/C = 8 hours or 480 minutes
⇒ (120/T) + (600 - 120)/C = 480
⇒ (120/T) + (480/C) = 480
⇒ (1/T) + (4/C) = 4 ........... (1)

Again, (200/T) + (600 - 200)/C = 480 + 20
⇒ (2/T) + (4/C) = 500
⇒ (2/T) + (4/C) = 5 ............. (2)

Now, (2) - (1) ⇒
(2/T) - (1/T) = 1
⇒ 1/T = 1
∴ T = 1

from, (1) ⇒ (1/T) + (4/C) = 4
⇒ 1 + (4/C) = 4
⇒ 4/C = 3
∴ C = 4/3

So, the ratio of the speed of the train to that of the cars is = 1 : (4/3)
= 3 : 4

৩৬৪.

The angles of a triangle are (x + 6)°, (2x - 4)° and (3x + 4)°. Then the value of x is?

ক
ক) 29°
খ
খ) 30°
গ
গ) 45°
ঘ
ঘ) 36°

সঠিক উত্তর: ক

ক) 29°

উত্তর

সঠিক উত্তর: ক

ক) 29°

ক

ব্যাখ্যা

Question: The angles of a triangle are (x + 6)°, (2x - 4)° and (3x + 4)°. Then the value of x is?

Solution:
We know
Sum of all angles in triangle is 180°
Now
(x + 6)° + (2x - 4)° + (3x + 4)° = 180°
⇒ 6x + 6° = 180°
⇒ (x + 1) = 30°
⇒ x = 29°

৩৬৫.

Reza sells 500 shares in a company via a stock broker who charges a flat Tk. 20 commission rate on all transactions under Tk. 1000. His bank account is credited with Tk. 692 from the sale of the shares. What price were his shares sold at?

ক
ক) 109
খ
খ) 131
গ
গ) 142.4
ঘ
ঘ) 168.9

সঠিক উত্তর: গ

গ) 142.4

উত্তর

সঠিক উত্তর: গ

গ) 142.4

গ

ব্যাখ্যা

কমিশনসহ রেজার শেয়ারের বিক্রয়মূল্য ৬৯২+২০ = ৭১২ টাকা।
তাহলে ৫০০ শেয়ারের বিক্রয়মূল্য ৭১২/৫০০ = ১.৪২৪ বা ১৪২.৪ সেন্টস।

৩৬৬.

0.57 expressed as a percent of 3.8 is-

ক
15%
খ
30%
গ
40%
ঘ
None of the above

সঠিক উত্তর: ক

15%

উত্তর

সঠিক উত্তর: ক

15%

ক

ব্যাখ্যা

Question: 0.57 expressed as a percent of 3.8 is-

Solution:
Required Percentage = (0.57 ×100)/3.8
= (57 × 100 × 10)/(38 × 100)
= 15%

৩৬৭.

If secθ + tanθ = x, then tanθ is

ক
x² + 1/x
খ
x² - 1/x
গ
x² + 1/2x
ঘ
x² - 1/2x

সঠিক উত্তর: ঘ

x² - 1/2x

উত্তর

সঠিক উত্তর: ঘ

x² - 1/2x

ঘ

ব্যাখ্যা

Question: If secθ + tanθ = x, then tanθ is

Solution:
দেওয়া আছে,
secθ + tanθ = x ................. (1)

আমরা জানি,
sec2θ - tan2θ = 1
বা, (secθ + tanθ)(secθ - tanθ) = 1
বা, x(secθ - tanθ) = 1
বা, secθ - tanθ = 1/x ................ (2)

(1) - (2) হতে পাই,
(secθ + tanθ) - (secθ - tanθ) = x - (1/x)
বা, 2tanθ = (x² - 1)/x
বা, tanθ = (x² - 1)/2x
∴

৩৬৮.

What is the ratio of the surface area of a cube to the surface area of a rectangular solid identical to the cube in all ways except that its length has been doubled?

ক
1/2
খ
3/5
গ
2
ঘ
None of these

সঠিক উত্তর: খ

3/5

উত্তর

সঠিক উত্তর: খ

3/5

খ

ব্যাখ্যা

Question: What is the ratio of the surface area of a cube to the surface area of a rectangular solid identical to the cube in all ways except that its length has been doubled?

Solution: let, each side of cube is x
surface area of cube = 6x²

length of a rectangular solid is 2x
surface area = (2x² + 4x² + 4x²)
= 10x²

ratio = 6x²/10x²
= 3/5

৩৬৯.

A retailer, while selling a watch, was asking for such a price that would allow him to offer a 10% discount and still make a profit of 20% on cost. If the cost of the watch was Tk. 1,200, what was his asking price?

ক
Tk. 1,400
খ
Tk. 1,600
গ
Tk. 1,800
ঘ
Tk. 1,700
ঙ
None of these

সঠিক উত্তর: খ

Tk. 1,600

উত্তর

সঠিক উত্তর: খ

Tk. 1,600

খ

ব্যাখ্যা

Question: A retailer, while selling a watch, was asking for such a price that would allow him to offer a 10% discount and still make a profit of 20% on cost. If the cost of the watch was Tk. 1,200, what was his asking price?

Solution:
প্রথম ধাপে, 20% লাভে ঘড়িটির বিক্রয়মূল্য (Selling Price) নির্ণয় করতে হবে।
ক্রয়মূল্য = 1,200 টাকা
বিক্রয়মূল্য = 1,200 + (1,200 এর 20%)
= 1,200 + 240 = 1,440 টাকা

এখন, দ্বিতীয় ধাপে, 10% ছাড়ের ওপর ভিত্তি করে ধার্যমূল্য (Asking Price) নির্ণয় করতে হবে।
যদি ধার্যমূল্য 100 টাকা হয়, তবে 10% ছাড়ে বিক্রয়মূল্য হবে (100 - 10) = 90 টাকা।

বিক্রয়মূল্য 90 টাকা হলে, ধার্যমূল্য = 100 টাকা
বিক্রয়মূল্য 1 টাকা হলে, ধার্যমূল্য = 100/90 টাকা
বিক্রয়মূল্য 1,440 টাকা হলে, ধার্যমূল্য = (100/90) × 1,440 = 1,600 টাকা

∴ ঘড়িটির ধার্যমূল্য বা Asking Price ছিল 1,600 টাকা।

৩৭০.

(0.1 × 0.01 × 0.001 × 10⁷) is equal to:

ক
1/10
খ
1/100
গ
10
ঘ
100

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: (0.1 × 0.01 × 0.001 × 10⁷) is equal to:

Solution:
Given expression,
(0.1 × 0.01 × 0.001 × 10⁷)
= (1/10) × (1/100) × (1/1000) × 10⁷
= 10⁷/10⁶
= 10 ^{(7 - 6)}
= 10

৩৭১.

In the figure given above, LM is parallel to QR. If LM divides the ΔPQR such that area of trapezium LMRQ is two times the area of ΔPLM, then what is PL/PO equal to?

ক
1/√2
খ
1/3
গ
1/√3
ঘ
1

সঠিক উত্তর: গ

1/√3

উত্তর

সঠিক উত্তর: গ

1/√3

গ

ব্যাখ্যা

Question:
In the figure given above, LM is parallel to QR. If LM divides the ΔPQR such that area of trapezium LMRQ is two times the area of ΔPLM, then what is PL/PO equal to?

Solution:
In the given figure.
MRQL = 2ΔPLM

Let, area of ΔPLM be x,

Then,
the area of trapezium = 2x
∴ ΔPQR = 2x + x = 3x
Here, it is clear from the given figure that ΔPQR ∼ ΔPLM

∴ ΔPQR/ΔPLM = 3x/x
⇒ PL²/PQ² = 1/3
∴ PL/PQ = 1/√3

৩৭২.

Two boats, travelling at 5 km/h and 10 km/h, head directly towards each other. They begin at a distance of 20 km from each other. How far apart are they (in km) one minute before they collide?

ক
1/4 km
খ
1/6 km
গ
1/3 km
ঘ
4/3 km
ঙ
None of the above

সঠিক উত্তর: ক

1/4 km

উত্তর

সঠিক উত্তর: ক

1/4 km

ক

ব্যাখ্যা

Question: Two boats, travelling at 5 km/h and 10 km/h, head directly towards each other. They begin at a distance of 20 km from each other. How far apart are they (in km) one minute before they collide?
(৫ কিমি/ঘণ্টা এবং ১০ কিমি/ঘণ্টা বেগে চলমান দুটি নৌকা পরস্পরের দিকে মুখোমুখি যাত্রা শুরু করে। তারা ২০ কিমি দূরত্ব থেকে এগোতে শুরু করে। সংঘর্ষের এক মিনিট পূর্বে তাদের মধ্যে কত কিলোমিটার দূরত্ব ছিল?)

Solution:
শেষ মুহূর্তে সংঘর্ষের আগে, দুইটি নৌকা যথাক্রমে দূরত্ব অতিক্রম করে 5/60 km and 10/60 km.

যেহেতু তারা একে অপরের দিকে অগ্রসর হচ্ছে, তাই 1 মিনিটে আপেক্ষিকভাবে অতিক্রান্ত হচ্ছে
1/12 + 1/6
= 1/4 km

৩৭৩.

The numbers of prime factors in the expression 6⁸× 7¹⁷ ×11²⁷ is equal to-

ক
ক) 60
খ
খ) 64
গ
গ) 66
ঘ
ঘ) 68

সঠিক উত্তর: ক

ক) 60

উত্তর

সঠিক উত্তর: ক

ক) 60

ক

ব্যাখ্যা

Question: The numbers of prime factors in the expression 6⁸× 7¹⁷ ×11²⁷ is equal to-

Solution:
6⁸ × 7¹⁷× 11²⁷
= (2 × 3)⁸ × 7¹⁷× 11²⁷
= 2⁸ × 3⁸× 7¹⁷× 11²⁷
Number of prime factors in the given expression
= (8 + 8 + 17 + 27)
= 60

৩৭৪.

If x/2 years ago Topu was 12, and x/2 years from now he will be 2x years old, how old will he be 3x years from now?

ক
54 years
খ
36 years
গ
48 years
ঘ
18 years

সঠিক উত্তর: ক

54 years

উত্তর

সঠিক উত্তর: ক

54 years

ক

ব্যাখ্যা

Question: If x/2 years ago Topu was 12 years old, and x/2 years from now he will be 2x years old, how old will he be 3x years from now?

Solution:
Given that,
x/2 years ago Topu was = 12 years old
∴ Now Topu is = 12+ (x/2) years

Again,
After x/2 years Topu will be = (12+ x/2 + x/2) = (12 + x) years

ATQ,
12 + x = 2x
⇒ x = 12

∴ Topu's present age = 12 + (12/2) = 12 + 6 years
= 18 years.

∴ After 3x = 3 × 12 = 36 years Topu will be 18 + 36 = 54 years.

৩৭৫.

85% of a number is added to 24, the result is the same number. Find the number?

ক
ক) 150
খ
খ) 140
গ
গ) 130
ঘ
ঘ) 160

সঠিক উত্তর: ঘ

ঘ) 160

উত্তর

সঠিক উত্তর: ঘ

ঘ) 160

ঘ

ব্যাখ্যা

Question: 85% of a number is added to 24, the result is the same number. Find the number?

Solution:
ধরি,
সংখ্যাটি x

প্রশ্নমতে,
x এর 85% + 24 = x
(85x/100) + 24 = x
(17x /20) + 24 = x
x - (17x /20) = 24
(20x - 17x)/20 = 24
3x/20 = 24
x/20 = 8
x = 160

৩৭৬.

A tap can fill a tank in 6 hours. After half the tank is filled three more similar taps are opened. What is the total time taken to fill the tank completely?

ক
ক) 4 hrs 15 min
খ
খ) 3 hrs 45 min
গ
গ) 3 hrs 24 min
ঘ
ঘ) 4 hrs 51 min

সঠিক উত্তর: খ

খ) 3 hrs 45 min

উত্তর

সঠিক উত্তর: খ

খ) 3 hrs 45 min

খ

ব্যাখ্যা

Time taken by one tap to fill half the tank = 3 hrs.
Part filled by one tap in 1 hour = 1/6
Part filled by four taps in 1 hour = (4×1/6) = 2/3
Remaining part = (1−1/2) = 1/2
2/3 of the tank is filled by four taps in 1 hour.
So, 1/2 of the tank is filled in = 3/2×1/2=3/4 hours
3/4 hours = 3/4 × 60 = 45 min
So, the total time taken = 3 hrs + 45 min = 3 hrs 45 min or 225 min
-------------------------------------------------
Alternative way:
A tap can fill a tank in 6 hours.
A tap can fill half of a tank in 6/2 or 3 hours.
4 tap can fill half of a tank in 3/4 hours = 45 min
Total time taken = 3 hours 45 min

৩৭৭.

A man on tour travels first 160 km at 64 km/hr and the next 160 km at 80 km/hr. The average speed for the first 320 km of the tour is-

ক
35.55 km/hr
খ
36 km/hr
গ
71.11 km/hr
ঘ
71 km/hr

সঠিক উত্তর: গ

71.11 km/hr

উত্তর

সঠিক উত্তর: গ

71.11 km/hr

গ

ব্যাখ্যা

Question: A man on tour travels first 160 km at 64 km/hr and the next 160 km at 80 km/hr. The average speed for the first 320 km of the tour is-

Solution:
Total time taken = (160/64 + 160/80) hrs
= (5/2 + 2) hrs
= (5 + 4)/2 hrs
= 9/2 hrs.

Average speed = 320 × (2/9) km.hr
= 71.11 km/hr.

৩৭৮.

If sinA = 5/13 then, cosA = ?

ক
13/5
খ
12/13
গ
7/13
ঘ
8/13
ঙ
11/13

সঠিক উত্তর: খ

12/13

উত্তর

সঠিক উত্তর: খ

12/13

খ

ব্যাখ্যা

Question: If sinA = 5/13 then, cosA = ?

Solution:
আমরা জানি,
sin²A + cos²A = 1
⇒ (5/13)² + cos²A = 1 [এখানে, sinA = 5/13]
⇒ 25/169 + cos²A = 1
⇒ cos²A = 1 - 25/169
⇒ cos²A = (169 - 25)/169
⇒ cos²A = 144/169
⇒ cosA = √(144/169)
∴ cosA = 12/13

৩৭৯.

A sum of money at simple interest amounts to Tk. 815 in 3 years and to Tk. 854 in 4 years. The sum is -

ক
ক) Tk.638
খ
খ) Tk.469
গ
গ) Tk.718
ঘ
ঘ) Tk. 698

সঠিক উত্তর: ঘ

ঘ) Tk. 698

উত্তর

সঠিক উত্তর: ঘ

ঘ) Tk. 698

ঘ

ব্যাখ্যা

S. I for 1 year = Tk. (854 - 815)
= Tk. 39
S. I for 3 years = Tk. (39 × 3)
= Tk. 117

Principal = Tk. (815 - 117) = Tk. 698

৩৮০.

Find the smallest number by which 5808 should be multiplied so that the product becomes a perfect square.

ক
ক) 3
খ
খ) 7
গ
গ) 11
ঘ
ঘ) 2

সঠিক উত্তর: ক

ক) 3

উত্তর

সঠিক উত্তর: ক

ক) 3

ক

ব্যাখ্যা

5808 = (4 × 4) × (11 × 11) × 3 = 4² × 11² × 3
∴ 5808 should be multiplied by another 3 to make it a full square.

৩৮১.

A man rows downstream at 32 km/h and rows upstream at 22 km/h. At what speed can he row in still water?

ক
27 km/h
খ
5 km/h
গ
54 km/h
ঘ
15 km/h

সঠিক উত্তর: ক

27 km/h

উত্তর

সঠিক উত্তর: ক

27 km/h

ক

ব্যাখ্যা

Question: A man rows downstream at 32 km/h and rows upstream at 22 km/h. At what speed can he row in still water?

Solution:
Given,
Man rows downstream = 32 km/h
Man rows upstream = 22 km/h
We know that,
Speed in still water = (Downstream speed + Upstream speed) ÷ 2
= (32 + 22) ÷ 2
= 54 ÷ 2
= 27 km/h

৩৮২.

For a motorboat that covers a certain distance downstream in 2 hours & returns in 3 hours, what would be its speed in still water if the speed of stream is 6 km/hr?

ক
ক) 9 km/hr
খ
খ) 15 km/hr
গ
গ) 30 km/hr
ঘ
ঘ) 36 km/hr

সঠিক উত্তর: গ

গ) 30 km/hr

উত্তর

সঠিক উত্তর: গ

গ) 30 km/hr

গ

ব্যাখ্যা

Hint: If a boat moves to a certain distance downstream in 't1 ' hours & returns the same distance upstream in time 't2' hours, then
Speed of boat in still water = y{(t2+t1)/(t2–t1)} km/hr.
With the given parameters,
y = 6 km/hr, t1 = 3 hrs, t2 = 2 hrs
We can find, Speed of boat in still water
(x) = 6{(3+2)/(3–2)}= 30 km/hr

৩৮৩.

An old man is walking on a foggy road at a speed of x km/h. Due to low visibility, the old man see only up to 600 meters. If a car overtakes the man from behind with the speed of 15 km/hr then the man can see the car for 216 seconds. Find the speed of the man?

ক
2 km/h
খ
3 km/h
গ
5 km/h
ঘ
6 km/h

সঠিক উত্তর: গ

5 km/h

উত্তর

সঠিক উত্তর: গ

5 km/h

গ

ব্যাখ্যা

Question: An old man is walking on a foggy road at a speed of x km/h. Due to low visibility, the old man see only up to 600 meters. If a car overtakes the man from behind with the speed of 15 km/hr then the man can see the car for 216 seconds. Find the speed of the man?

Solution:
Distance up to old man see = 600/1000 = 0.6km
Time for which the man can see the car = 216/(60 × 60) = 0.06 hour

ATQ,
0.6/(15 - x) = 0.06
⇒ 15 - x = 10
∴ x = 5

So the old man is walking at a speed of 5 km/h

৩৮৪.

It was Saturday on January 1, 2015. What was the day of the week on Jan 1, 2016?

ক
Sunday
খ
Monday
গ
Friday
ঘ
Saturday

সঠিক উত্তর: ক

Sunday

উত্তর

সঠিক উত্তর: ক

Sunday

ক

ব্যাখ্যা

Question: It was Saturday on January 1, 2015. What was the day of the week on Jan 1, 2016?

Solution:
2015 সালটি 4 দ্বারা বিভাজ্য নয়। (2016 ÷ 4 = 503.75), তাই এটি একটি অধিবর্ষ নয়।

• অতিরিক্ত দিন নির্ণয়:
365 ÷ 7 = 52 সপ্তাহ এবং 1 দিন অতিরিক্ত হয়।

এখন,
1 লা জানুয়ারি 2015 দিনটি ছিল Saturday।
যেহেতু 2015 সালটি অধিবর্ষ নয়, তাই 1লা জানুয়ারি 2016 দিনটি হবে Saturday এর থেকে 1 দিন বেশি।
⇒ Saturday + 1 দিন = Sunday

∴ 1লা জানুয়ারি 2016 দিনটি হবে Sunday।

৩৮৫.

Three numbers are in the ratio 2 : 3 : 4, If their LCM is 240 the smaller of the three numbers is = ?

ক
ক) 40
খ
খ) 60
গ
গ) 30
ঘ
ঘ) 70
ঙ
ঙ) 80

সঠিক উত্তর: ক

ক) 40

উত্তর

সঠিক উত্তর: ক

ক) 40

ক

ব্যাখ্যা

Let number are = 2x, 3x, 4x
given,
LCM of (2×3×2)x = 12x
12x = 240
x = 20
∴ numbers are 2×20 = 40
3×20 = 60
4×20 = 80
∴ Smaller is 40

৩৮৬.

What is the radius of a circle if the length of its largest chord is 30 cm?

ক
15 cm
খ
60 cm
গ
15√2 cm
ঘ
30√2 cm

সঠিক উত্তর: ক

15 cm

উত্তর

সঠিক উত্তর: ক

15 cm

ক

ব্যাখ্যা

Question: What is the radius of a circle if the length of its largest chord is 30 cm?

Solution:
দেওয়া আছে,
বৃত্তের বৃহত্তম জ্যা-এর দৈর্ঘ্য = 30 সে.মি.

আমরা জানি,
বৃত্তের বৃহত্তম জ্যা হলো বৃত্তের ব্যাস।
এবং, বৃত্তের ব্যাস = 2 × ব্যাসার্ধ

∴ ব্যাসার্ধ = ব্যাস/2
= 30/2 সে.মি.
= 15 সে.মি.

অতএব, বৃত্তটির ব্যাসার্ধ 15 সে.মি.

৩৮৭.

If one number is 20% smaller than another number, and that second number is 10% greater than 150, what is the value of the first number?

ক
133
খ
132
গ
152
ঘ
140.55
ঙ
None

সঠিক উত্তর: খ

132

উত্তর

সঠিক উত্তর: খ

132

খ

ব্যাখ্যা

Question: If one number is 20% smaller than another number, and that second number is 10% greater than 150, what is the value of the first number?

Solution:
Let,
First number x
Second number y

ATQ,
y is 10% more than 150
⇒ y = 150 + [(150 × 10)/100]
= 165

and x is 20% less than y
⇒ x = 165 - [(165 × 20)/100]
= 165 - 33
= 132

∴ First number 132

৩৮৮.

Solve for x, log₃(2x + 1) = 4

ক
45
খ
37
গ
81
ঘ
40

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: Solve for x, log₃(2x + 1) = 4

Solution:
Given that,
⇒ log₃(2x + 1) = 4

⇒ 2x + 1 = 3⁴
⇒ 2x + 1 = 81
⇒ 2x = 81 - 1
⇒ 2x = 80

৩৮৯.

Rajib can do a work in 5 days while Farez can do the same work in 3 days. Both of them finish the work together and get Tk. 240. What is Rajib's share?

ক
ক) Tk. 75
খ
খ) Tk. 80
গ
গ) Tk. 85
ঘ
ঘ) Tk. 90

সঠিক উত্তর: ঘ

ঘ) Tk. 90

উত্তর

সঠিক উত্তর: ঘ

ঘ) Tk. 90

ঘ

ব্যাখ্যা

Question: Rajib can do a work in 5 days while Farez can do the same work in 3 days. Both of them finish the work together and get Tk. 240. What is Rajib's share?

Solution:
Rajib's wages : Farez's wages = Rajib's 1 day's work : Farez's 1 day's work
= 1/5 : 1/3
= 3 : 5

Sum of the ratio = 3 + 5 = 8

So, Rajib's share = (3/8) × 240 = Tk. 90

৩৯০.

According to meteorological records, it rained on 21 days in the month of June last year. What is the probability that it will rain on fourth of June this year?

ক
ক) 1/21
খ
খ) 21/31
গ
গ) 7/10
ঘ
ঘ) 1/2

সঠিক উত্তর: গ

গ) 7/10

উত্তর

সঠিক উত্তর: গ

গ) 7/10

গ

ব্যাখ্যা

Question: According to meteorological records, it rained on 21 days in the month of June last year. What is the probability that it will rain on fourth of June this year?

Solution:
June month has 30 days
favorable events = 21 days

∴ the probability that it will rain on fourth of June this year = 21/30
= 7/10

৩৯১.

In 1 minute 3/7 of a bucket is filled. The rest of the bucket can be filled in -

ক
4/7 minutes
খ
4/3 minutes
গ
7/4 minutes
ঘ
None

সঠিক উত্তর: খ

4/3 minutes

উত্তর

সঠিক উত্তর: খ

4/3 minutes

খ

ব্যাখ্যা

Question: In 1 minute 3/7 of a bucket is filled. The rest of the bucket can be filled in -

Solution:
Filled of the bucket 3/7 part.
Remaining = 1 - (3/7) = 4/7 part.

3/7 part is filled in = 1 min
∴ 1 part is filled in = 7/3 min
So, 4/7 part is filled in = (7/3) × (4/7) min
= 4/3 minutes

৩৯২.

The average (arithmetic mean) of x and y is 18. If z = 12, what is the average of x, y, and z.

ক
ক) 14
খ
খ) 15
গ
গ) 16
ঘ
ঘ) 18

সঠিক উত্তর: গ

গ) 16

উত্তর

সঠিক উত্তর: গ

গ) 16

গ

ব্যাখ্যা

x + y = 2 × 18 = 36
x + y + z = 36 + 12 = 48
Average: 48/3 = 16

৩৯৩.

The area of the four walls of a room is 120 square meters and the length is twice the breadth. If the height of the room is 4 m, then the area of the floor is -

ক
ক) 20 square meters
খ
খ) 30 square meters
গ
গ) 50 square meters
ঘ
ঘ) 60 square meters

সঠিক উত্তর: গ

গ) 50 square meters

উত্তর

সঠিক উত্তর: গ

গ) 50 square meters

গ

ব্যাখ্যা

Question: The area of the four walls of a room is 120 square meters and the length is twice the breadth. If the height of the room is 4 m, then the area of the floor is -

Solution:
Let the breadth = x meters and
length = (2x) metres

Area of 4 walls = 2(2x + x) × 4
= 24x

ATQ,
∴ 24x = 120
⇒ x = 5

So, length = 10 m, and breadth = 5 m

Area of the floor = 10 × 5 = 50 square meters

৩৯৪.

A tap can fill a tank in 30 minutes and another tap can fill the same tank in 60 minutes. lf both the taps are opened simultaneously when the tank is half empty, in how many minutes will the tank be full?

ক
ক) 7 minutes
খ
খ) 10 minutes
গ
গ) 18 minutes
ঘ
ঘ) 20 minutes
ঙ
ঙ) 15 minutes

সঠিক উত্তর: খ

খ) 10 minutes

উত্তর

সঠিক উত্তর: খ

খ) 10 minutes

খ

ব্যাখ্যা

Tap A can fill the tank in 30 minutes. Therefore, it fills 1/30th of the tank every minute.
Tap B can fill the tank in 60 minutes. Therefore, it fills 1/60th of the tank every minute.
Together, the two taps will fill
1/30 + 1/60
= (2 + 1)/60
= 3/60
= 1/20th of the tank every minute.

Therefore, when both the taps are opened simultaneously, they will fill the tank in 20 minutes. As the tank is already half full, they need to fill only half the tank.
Therefore, the tank will overflow 10 minutes after both the taps are opened.

৩৯৫.

Two workers A and B are engaged to do a work. A working alone takes 8 hours more to complete the job than if both worked together. If B worked alone, he would need 9/2 hours more to complete the job than they both working together. What time would they take to do the work together?

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

সঠিক উত্তর: খ

6 hours

উত্তর

সঠিক উত্তর: খ

6 hours

খ

ব্যাখ্যা

Question: Two workers A and B are engaged to do a work. A working alone takes 8 hours more to complete the job than if both worked together. If B worked alone, he would need 9/2 hours more to complete the job than they both working together. What time would they take to do the work together?

Solution:
Let,
both together can do the work in x hours.
A can do it in = (x + 8) hour
B can do it in = {x + (9/2)} hour
= (2x + 9)/2 hour

so,
{1/(x + 8)} + {2/(2x + 9)} = 1/x
or, (2x + 9 + 2x + 16)/{(x + 8)(2x + 9)} = 1/x
or, x(4x + 25) = (x + 8)(2x + 9)
or, 4x² + 25x = 2x² + 25x + 72
or, 2x² = 72
or, x² = 36
∴ x = 6

৩৯৬.

One-third of Ratan's investment in National Savings Certificate is equal to one-half of his investment in FDR. If he has Tk. 1,50,000 as total investment. how much he invested in savings certificate?

ক
ক) Tk. 30,000
খ
খ) Tk. 50,000
গ
গ) Tk. 60,000
ঘ
ঘ) Tk. 90,000

সঠিক উত্তর: ঘ

ঘ) Tk. 90,000

উত্তর

সঠিক উত্তর: ঘ

ঘ) Tk. 90,000

ঘ

ব্যাখ্যা

Question: One-third of Ratan's investment in National Savings Certificate is equal to one-half of his investment in FDR. If he has Tk. 1,50,000 as total investment. how much he invested in savings certificate?

Solution:
Let saving in N.S.C and P.P.F. be Tk. x and Tk.(150000 - x) respectively

Now
x/3 = (1/2)(150000 - x)
x/3 = 75000 - x/2
(x/3) + (x/2) = 75000
(2x + 3x)/6= 75000
5x/6 = 75000
x = (75000 × 6)/5
x = 90000

৩৯৭.

A and B invest in a business in the ratio 3 : 2. If 5% of the total profit goes to charity and A's share is Tk 1425, the total profit is-

ক
ক) 1500 Tk
খ
খ) 2000 Tk
গ
গ) 2500 Tk
ঘ
ঘ) 3000 Tk

সঠিক উত্তর: গ

গ) 2500 Tk

উত্তর

সঠিক উত্তর: গ

গ) 2500 Tk

গ

ব্যাখ্যা

Question: A and B invest in a business in the ratio 3 : 2. If 5% of the total profit goes to charity and A's share is Tk 1425, the total profit is-

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

If A's share is Tk 57, Total profit = Tk 100
If A's share Tk 1425 ,Total profit = (100/57) × 1425 = 2500

৩৯৮.

If 4^{x + y} = 1 and 4^{x - y} = 4, then the values of x and y respectively are-

ক
- 1/2 and 1/2
খ
- 1/2 and - 1/2
গ
1/2 and - 1/2
ঘ
1/2 and 1/2

সঠিক উত্তর: গ

1/2 and - 1/2

উত্তর

সঠিক উত্তর: গ

1/2 and - 1/2

গ

ব্যাখ্যা

Question: If 4^{x + y} = 1 and 4^{x - y} = 4, then the values of x and y respectively are-

Solution:
4^{x + y} = 1
⇒ 4^{x + y} = 4⁰
⇒ x + y = 0 ..............(1)

4^{x - y} = 4
⇒ 4^{x - y} = 4¹
⇒ x - y = 1 ..................(2)

x + y + x - y = 1
⇒ 2x = 1
∴ x = 1/2

1/2 - y = 1
⇒ y = (1/2) - 1
y = (1 - 2)/2
= - 1/2

৩৯৯.

A car running at a speed of 140 km/hr reached its destination in 2 hours. If the car wants to reach at its destination in 1 hour, at what speed it needs to travel?

ক
300 km/hr
খ
280 km/hr
গ
250 km/hr
ঘ
240 km/hr

সঠিক উত্তর: খ

280 km/hr

উত্তর

সঠিক উত্তর: খ

280 km/hr

খ

ব্যাখ্যা

Question: A car running at a speed of 140 km/hr reached its destination in 2 hours. If the car wants to reach at its destination in 1 hour, at what speed it needs to travel?

Solution:
Distance to be covered = Speed × Time = 140 × 2 = 280 km

Time = 1 hour

Required Speed = 280/1 = 280 km/hr

৪০০.

A is 3 years older than B and 3 years younger than C. B and D are twins. How older is C than D?

ক
ক) 12 years
খ
খ) 6 years
গ
গ) 3 years
ঘ
ঘ) Cannot be Determined

সঠিক উত্তর: খ

খ) 6 years

উত্তর

সঠিক উত্তর: খ

খ) 6 years

খ

ব্যাখ্যা

Question: A is 3 years older than B and 3 years younger than C. B and D are twins. How older is C than D?

Solution:
A এর বয়স = B + 3 বছর
A এর বয়স = C - 3 বছর

যেহেতু
B এবং D যমজ
B = D

B + 3 = C - 3
D + 3 = C - 3
D = C - 6
C = D + 6

C, D এর চেয়ে 6 বছরের বড়।