Bank Math

PrepBank · পাতা ৭ / ১৬১ · ৬০১–৭০০ / ১৬,১২৪

৬০১.

A prize of 2000 tk is given to anyone who solves a hacker puzzle independently. The probability that Rafi will win the prize is 0.6, and the probability that kamol will win the prize is 0.7. What is the probability that both will win the prize?

ক
ক) 0.39
খ
খ) 0.48
গ
গ) 0.42
ঘ
ঘ) 0.52

সঠিক উত্তর: গ

গ) 0.42

উত্তর

সঠিক উত্তর: গ

গ) 0.42

গ

ব্যাখ্যা

Question: A prize of 2000 tk is given to anyone who solves a hacker puzzle independently. The probability that Rafi will win the prize is 0.6, and the probability that kamol will win the prize is 0.7. What is the probability that both will win the prize?

Solution:
probability that Rafi will win the prize is, P(A) = 0.6
probability that kamol will win the prize is, P(B) = 0.7
As events are independent,
P (A ∩ B) = P(A) × P (B)
= 0.6 × 0.7
= 0.42

৬০২.

A TV set is marked at Tk. 18,000. During a sale, a customer buys it at Tk. 16,200. what is the discount percentage.

ক
5%
খ
8%
গ
10%
ঘ
18%

সঠিক উত্তর: গ

10%

উত্তর

সঠিক উত্তর: গ

10%

গ

ব্যাখ্যা

Question: A TV set is marked at Tk. 18,000. During a sale, a customer buys it at Tk. 16,200. what is the discount percentage.

Solution:
Given,
Marked Price = Tk. 18000
Selling price = Tk. 16200

∴ Discount = (18,000 - 16,200) = Tk. 1,800

∴ Discount % = (1800/18000) × 100 = 10%

৬০৩.

The capacity of two pots is 12 liters and 48 liters respectively. Find the capacity of a which can exactly measure the content of the two pots.

ক
ক) 10000 cc
খ
খ) 12000 cc
গ
গ) 8000 cc
ঘ
ঘ) 16000 cc

সঠিক উত্তর: খ

খ) 12000 cc

উত্তর

সঠিক উত্তর: খ

খ) 12000 cc

খ

ব্যাখ্যা

Question: The capacity of two pots is 12 liters and 48 liters respectively. Find the capacity of a which can exactly measure the content of the two pots.

Solution:
Required capacity = H.C.F. of 12 liters and 48 liters = 12 liters

1 liter = 1000 cc
12 liters = 12000 cc

৬০৪.

The sum of the ages of 5 siblings born at the intervals of 3 years each is 100 years. What is the age of the youngest sibling?

ক
16 years
খ
14 years
গ
10 years
ঘ
4 years
ঙ
None of the above

সঠিক উত্তর: খ

14 years

উত্তর

সঠিক উত্তর: খ

14 years

খ

ব্যাখ্যা

Question: The sum of the ages of 5 siblings born at the intervals of 3 years each is 100 years. What is the age of the youngest sibling?

Solution:
Let the ages of siblings be x, (x + 3), (x + 6), (x + 9) and (x + 12) years.

Then,
x + (x + 3) + (x + 6) + (x + 9) + (x + 12) = 100
⇒ 5x = 100 - 30
⇒ 5x = 70
⇒ x = 14

∴ Age of the youngest sibling = x = 14 years.

৬০৫.

Calculate the compound interest at the rate of 6% per annum for 2 years on the principal, which in 2 years at the rate of 2% per annum gives Tk. 8000 as simple interest.

ক
27420
খ
24720
গ
42770
ঘ
47270

সঠিক উত্তর: খ

24720

উত্তর

সঠিক উত্তর: খ

24720

খ

ব্যাখ্যা

Question: Calculate the compound interest at the rate of 6% per annum for 2 years on the principal, which in 2 years at the rate of 2% per annum gives Tk. 8000 as simple interest.

Solution:
S.I. = PRT/100
⇒ P = (S.I. × 100) / (R × T)
∴ P = (8000 × 100)/(2 × 2) = 200000

A = 200000(1 + 6/100)² = 200000(53/50)² = 224720

C.I. = 224720 - 200000 = 24720.

৬০৬.

The average of the first five multiples of 9 is:

ক
20
খ
27
গ
28
ঘ
30
ঙ
35

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Required average = total sum of multiple of 9/5
= (9 + 18 + 27 + 36 + 45)/5
= 27

Note that, the average of 9 and 45 is also 27.
An average of 18 and 36 is also 27.

৬০৭.

If you roll a six-sided die, what is the probability of getting a 3 or a 4?

ক
1/2
খ
1/6
গ
1/3
ঘ
1/4

সঠিক উত্তর: গ

1/3

উত্তর

সঠিক উত্তর: গ

1/3

গ

ব্যাখ্যা

Question: If you roll a six-sided die, what is the probability of getting a 3 or a 4?

Solution:
On a six-sided die, the probability of throwing any number is 1 in 6.
The probability of throwing a 3 = 1/6
The probability of throwing a 4 = 1/6

Therefore, the probability of throwing either a 3 or 4 is 1/6 + 1/6 = (1+1)/6 = 2/6 = 1/3

৬০৮.

Find the value of the polynomial 6x³ - 2x² - x + 3 when x = - 1

ক
- 8
খ
2
গ
- 4
ঘ
0

সঠিক উত্তর: গ

- 4

উত্তর

সঠিক উত্তর: গ

- 4

গ

ব্যাখ্যা

Question: Find the value of the polynomial 6x³ - 2x² - x + 3 when x = - 1

Solution:
Given,
x = - 1

We get,
6x³ - 2x² - x + 3 = 6(- 1)³ - 2(- 1)² - (- 1) + 3
= - 6 - 2 + 1 + 3
= - 4

৬০৯.

A company's profits have doubled for each of the last 6 years. If the total profits for the last 6 years were Tk. 126 lacs, what were the profits in the first year?

ক
Tk. 1,00,000
খ
Tk. 2,00,000
গ
Tk. 3,00,000
ঘ
Tk. 4,00,000
ঙ
Tk. 6,00,000

সঠিক উত্তর: খ

Tk. 2,00,000

উত্তর

সঠিক উত্তর: খ

Tk. 2,00,000

খ

ব্যাখ্যা

Question: A company's profits have doubled for each of the last 6 years. If the total profits for the last 6 years were Tk. 126 lacs, what were the profits in the first year?

Solution:
মনে করি, প্রথম বছরে কোম্পানির লাভ ছিল x টাকা।
যেহেতু প্রতি বছর লাভ দ্বিগুণ হচ্ছে, তাই 6 বছরের লাভগুলো হবে যথাক্রমে: x, 2x, 4x, 8x, 16x এবং 32x

প্রশ্নমতে,
x + 2x + 4x + 8x + 16x + 32x = 1,26,00,000
⇒ 63x = 1,26,00,000
⇒ x = 1,26,00,000/63
⇒ x = 2,00,000

∴ প্রথম বছরে কোম্পানির লাভ ছিল 2,00,000 টাকা।

৬১০.

The retail price of a pen is 40 taka. Asif got a discount of 20% over the retail price and he eventually saved taka 240 on his total purchase of the pens. How many pens did he buy?

ক
ক) 20
খ
খ) 24
গ
গ) 12
ঘ
ঘ) 30

সঠিক উত্তর: ঘ

ঘ) 30

উত্তর

সঠিক উত্তর: ঘ

ঘ) 30

ঘ

ব্যাখ্যা

According to the question,
Retail price 40 Tk. (per pen).
20% discount = 40 × (20/100)
= 8 Tk.
অর্থ্যাৎ,
প্রতি pen এ save হয় = 8 Tk.
Total savings = 240 Tk.
∴ Number of pen = 240/8
= 30 pen.

৬১১.

What is the area of an isosceles triangle if two of its sides measure 6 and 12?

ক
8√5
খ
15√5
গ
9√5
ঘ
9√15

সঠিক উত্তর: ঘ

9√15

উত্তর

সঠিক উত্তর: ঘ

9√15

ঘ

ব্যাখ্যা

Question: What is the area of an isosceles triangle if two of its sides measure 6 and 12?

Solution:
The given triangle is an Isosceles triangle and hence, two of the three sides of the triangle are equal.
Hence, the third side of the triangle can either be 6 or be 12.

If the two equal sides of the triangle measure 6, the sides of the triangle become 6, 6, and 12.
However, the sum of the two smaller sides is not greater than the third side.
∴ 6 is not a possible value of the third side.

If the two equal sides of the triangle measure 12, the sides of the triangle become 6, 12, and 12.
The sum of the smaller two sides is greater than the third side, and hence, the value of the third side is 12.

Let, a = 12, b = 6
∴ Area = (b/4) × √(4a² - b²)
= (6/4) × √{4 × (12)² - (6)²}
= (3/2) × √(576 - 36)
= (3/2) ×√540
= (3/2) × √(36 × 15)
= (3/2) × 6 × √15
= 9√15

৬১২.

To make 5686 exactly divisible by 7 what number should be subtracted from it??

ক
3
খ
2
গ
1
ঘ
0

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: To make 5686 exactly divisible by 7 what number should be subtracted from it??

Solution:
if we divide 5686 by 7 the remainder is 2
to make it divisible by 7 we need to subtract 2 from it.

so the number is = 5686 - 2 = 5684
dividing 5684 by 7 we get the result = 812

৬১৩.

The present ages of Rajib and his elder brother are in the ratio 5 : 7. After five year the ratio of their ages will be 3 : 4. Find Rajib's present age.

ক
35 Years
খ
30 years
গ
25 years
ঘ
20 years

সঠিক উত্তর: গ

25 years

উত্তর

সঠিক উত্তর: গ

25 years

গ

ব্যাখ্যা

Question: The present ages of Rajib and his elder brother are in the ratio 5 : 7. After five year the ratio of their ages will be 3 : 4. Find Rajib's present age.

Solution:
Ratio of present ages of Rajib and his elder brother = 5 : 7
Let the present ages of Rajib and his elder brother be 5x and 7x.

After five years,
Age of Rajib = 5x + 5
and Age of elder Brother = 7x + 5
According to question, then the ratio of their ages = 3 : 4
⇒ (5x + 5) : (7x + 5) = 3 : 4
⇒ (5x + 5)/(7x + 5) = 3/4
⇒ 20x + 20 = 21x + 15
∴ x = 5

Thus, Present age of Rajib = 5x = 5 × 5 = 25 years

৬১৪.

If a + b = √11 and a - b = √5, what is the value of 8ab(a² + b²)?

ক
96
খ
100
গ
56
ঘ
84

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: If a + b = √11 and a - b = √5, what is the value of 8ab(a² + b²)?

Solution:
দেওয়া আছে, a + b = √11 এবং a - b = √5

আমরা জানি,
4ab = (a + b)² - (a - b)²2(a² + b²) = (a + b)² + (a - b)²

এখন,
8ab(a² + b²) = (4ab) × 2(a² + b²)
= [(a + b)² - (a - b)²] × [(a + b)² + (a - b)²]
= [(√11)² - (√5)²] × [(√11)² + (√5)²]
= (11 - 5) × (11 + 5)
= 6 × 16
= 96

৬১৫.

What is the length of the diagonal of a square whose area is 4 times of another square with diagonal as 5√2 cm?

ক
10 cm
খ
5√2 cm
গ
10√5 cm
ঘ
10√2 cm

সঠিক উত্তর: ঘ

10√2 cm

উত্তর

সঠিক উত্তর: ঘ

10√2 cm

ঘ

ব্যাখ্যা

Question: What is the length of the diagonal of a square whose area is 4 times of another square with diagonal as 5√2 cm?

Solution:
Area of square = (1/2) × (length of diagonal)²

Area of square₂ =(1/2) × (5√2)²
= 25 cm²

Area of square₁ = 4 × 25 = 100 cm²

Length of diagonal of square₁ = √(2 × area)
= √(2 × 100)
= 10√2 cm

৬১৬.

A and B together have taka 1,210. If 4/15 of A's amount is equal to 2/5 of B's amount. What is amount B has?

ক
Tk. 484
খ
Tk. 480
গ
Tk. 478
ঘ
Tk. 470

সঠিক উত্তর: ক

Tk. 484

উত্তর

সঠিক উত্তর: ক

Tk. 484

ক

ব্যাখ্যা

Question: A and B together have taka 1,210. If 4/15 of A's amount is equal to 2/5 of B's amount. What is amount B has?

Solution:
4A/15 = 2B/5
⇒ A = (2B × 15)/(5 × 4)
⇒ A = 3B/2
⇒ A/B = 3/2
⇒ A : B = 3 : 2

B's share = 1210 × (2/5) = 484 Tk

৬১৭.

A sum of Tk. 800 amounts to Tk. 920 in 3 years at a certain simple interest rate. If the rate of interest is increased by 3% per annum, what will be the total amount after 3 years?

ক
Tk 668
খ
Tk 844
গ
Tk 992
ঘ
Tk 998
ঙ
None

সঠিক উত্তর: গ

Tk 992

উত্তর

সঠিক উত্তর: গ

Tk 992

গ

ব্যাখ্যা

Question: A sum of Tk. 800 amounts to Tk. 920 in 3 years at a certain simple interest rate. If the rate of interest is increased by 3% per annum, what will be the total amount after 3 years?

Solution:
Here,
Sum = Tk 800,
Amount = Tk 920,

∴ Simple interest = Amount - Principal
= 920 - 800
= Tk 120

∴ Rate = (Interest × 100)/(Principal × Time)
= (120 × 100)/(800 × 3)
= 5% per annum

If the rate is increased by 3%,
∴ Rate = 8% per annum

∴ simple interest = (800 × 8 × 3)/(800 × 3)
= Tk 192

∴ Amount = Principal + S.I.
= (800 + 192)
= Tk 992

৬১৮.

A man can row 34/3 kmph in still water and finds that it takes him thrice as much time to row up than as to row down the same distance in the river. The speed of the current is:

ক
ক) 17/3 km/hr.
খ
খ) 34/3 km/hr.
গ
গ) 28/3 km/hr.
ঘ
ঘ) 11/3 km/hr.

সঠিক উত্তর: ক

ক) 17/3 km/hr.

উত্তর

সঠিক উত্তর: ক

ক) 17/3 km/hr.

ক

ব্যাখ্যা

Let speed upstream be x kmph
Then speed downstream = 3x kmph
Speed in still water = (1/2)(3x + x)kmph
=2x kmph

Now
2x = 34/3
x = (34/3) × 2
x = 17/3

So Speed upstream = 17/3 km/hr
Speed downstream = 17 km/hr

Hence speed of the current =(1/2){17 - (17/3)} km/hr.
= (1/2)(51 - 17)/3 km/hr.
= (1/2)(34/3) km/hr.
= 17/3 km/hr.

৬১৯.

There is 60% increase in an amount in 6 years at simple interest. What will be the compound interest of Tk. 12,000 after 3 years at the same rate?

ক
3970
খ
3742
গ
3982
ঘ
3972

সঠিক উত্তর: ঘ

3972

উত্তর

সঠিক উত্তর: ঘ

3972

ঘ

ব্যাখ্যা

Question: There is 60% increase in an amount in 6 years at simple interest. What will be the compound interest of Tk. 12,000 after 3 years at the same rate?

Solution:
Let,P = Tk.100
Then, I = Tk. 60
and, T = 6years
we know,
I = Pnr/100
r = (I × 100)/Pn
= (60 × 100)/ (100 × 6)
= 10%

the compound principle of Tk. 12,000 after 3 years at the same rate wii be
C = P(1 + r)n
= 12000 ( 1 + 10/100)3
= 12000 ( 11/10)3
= 12000 (1331/1000)
= 12 × 1331
= 15972
∴ compound interest will be = (15972 - 12000)
= 3972 Tk.

৬২০.

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

ক
ক) 1/2
খ
খ) 3/5
গ
গ) 1/9
ঘ
ঘ) 1/3

সঠিক উত্তর: গ

গ) 1/9

উত্তর

সঠিক উত্তর: গ

গ) 1/9

গ

ব্যাখ্যা

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

৬২১.

The present ages of Johny and Mukul are in the ratio of 6 : 4 , Five years ago their ages were in the ratio of 5 : 3, How old is Johny now?

ক
42 years
খ
36 years
গ
30 years
ঘ
24 years

সঠিক উত্তর: গ

30 years

উত্তর

সঠিক উত্তর: গ

30 years

গ

ব্যাখ্যা

বর্তমান বয়স , Johny = 6x এবং mukul = 4x
প্রশ্নমতে,
(6x-5)/(4x-5) = 5/3
20x - 25 = 18x - 15
20x - 18x = 25 -15
2x = 10
x = 5

∴ বর্তমানে Jonny এর বয়স = 6x = 6×5 = 30
উত্তরঃ 30 years

৬২২.

The sum of four consecutive multiples of 4 is 200. What is the largest number?

ক
48
খ
52
গ
56
ঘ
60

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: The sum of four consecutive multiples of 4 is 200. What is the largest number?

Solution:
ধরি,
চারটি ক্রমিক 4 এর গুণিতক যথাক্রমে x, (x + 4), (x + 8) এবং (x + 12)

প্রশ্নমতে,
x + (x + 4) + (x + 8) + (x + 12) = 200
⇒ 4x + 24 = 200
⇒ 4x = 200 - 24
⇒ 4x = 176
⇒ x = 176/4
⇒ x = 44

∴ বৃহত্তম সংখ্যা = x + 12
= 44 + 12
= 56

৬২৩.

The value of 1 + {(tan 15° - tan 60°)/(cot 30° - cot 75°)} is -

ক
2
খ
1
গ
- 1
ঘ
0

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: The value of 1 + {(tan 15° - tan 60°)/(cot 30° - cot 75°)} is -

Solution:
1 + (tan 15° - tan 60°)/(cot 30° - cot 75°)
= 1 + (tan 15° - tan 60°)/cot (90° - 60°) - cot (90° - 15°)
= 1 + (tan 15°- tan 60°)/(tan 60° - tan 15°)
= 1 + (tan 15°- tan 60°)/(- 1)(tan 15° - tan 60°)
= 1 - 1
= 0

৬২৪.

The equation of the given curve is :

ক
ক) y = -x
খ
খ) y = 1/x
গ
গ) y = x
ঘ
ঘ) y = -1

সঠিক উত্তর: খ

খ) y = 1/x

উত্তর

সঠিক উত্তর: খ

খ) y = 1/x

খ

ব্যাখ্যা

According to the figure x and y are in negative relation but as they are in (+ +) coordination, so they have built a positive correlation. Which is satisfied only by y = 1/x equation. And equation a, b and d are equation of straightline.

৬২৫.

From a point P on level ground, the angle of elevation of the top of a tower is 30°. If the tower is 80√3 m high, how far is p from the foot of the tower ?

ক
ক) 220 m
খ
খ) 80 m
গ
গ) 160 m
ঘ
ঘ) 240 m

সঠিক উত্তর: ঘ

ঘ) 240 m

উত্তর

সঠিক উত্তর: ঘ

ঘ) 240 m

ঘ

ব্যাখ্যা

The angle of elevation of the top of a tower = 30°
The height of the tower = AB = 80√3m
We have
tan 30° = AB/PB
⇒ 1/√3 = 80√3/PB
⇒ PB = 80√3 × √3
⇒ PB = 80 × 3 = 240 m

∴ The distance (in m) of point P from the foot of the tower is 240 m

৬২৬.

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
Tk. 689
খ
Tk. 869
গ
Tk. 968
ঘ
Tk. 698

সঠিক উত্তর: ঘ

Tk. 698

উত্তর

সঠিক উত্তর: ঘ

Tk. 698

ঘ

ব্যাখ্যা

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

Solution:
Simple interest for 1 year = Tk. (854 - 815)
= Tk. 39

∴ Simple interest for 3 years = Tk.(39 × 3)
= Tk. 117

∴ Sum = (815 - 117)
= Tk. 698

৬২৭.

How long will it take to row 20 km upstream if one can row 10 km in 10 minutes in still water and the same distance in 8 minutes with the stream?

ক
12 min
খ
13.33 min
গ
24 min
ঘ
26.67 min
ঙ
27 min

সঠিক উত্তর: ঘ

26.67 min

উত্তর

সঠিক উত্তর: ঘ

26.67 min

ঘ

ব্যাখ্যা

Question: How long will it take to row 20 km upstream if one can row 10 km in 10 minutes in still water and the same distance in 8 minutes with the stream?

Solution:
Let x be the speed of man in still water
and y be the speed of stream.
∴ Speed of man (x) = 60 km/hr
and speed of downstream = 75 km/hr.
∴ Speed of stream = 15 km/hr.

Hence upstream speed = 60 - 15 = 45 km/hr.

So time taken to cover 20 km = (20/45) × 60 min = 26.67min.

৬২৮.

The captain of a football team of 11 members is 26 years old and the goalkeeper is 3 years older than him. If the ages of these two are excluded, the average age of the remaining players is one year less than the average age of the whole team. What is the average age of the team?

ক
20 years
খ
21 years
গ
22 years
ঘ
23 years

সঠিক উত্তর: ঘ

23 years

উত্তর

সঠিক উত্তর: ঘ

23 years

ঘ

ব্যাখ্যা

Question: The captain of a football team of 11 members is 26 years old and the goalkeeper is 3 years older than him. If the ages of these two are excluded, the average age of the remaining players is one year less than the average age of the whole team. What is the average age of the team?

Solution:
ধরি, সম্পূর্ণ টিমের বয়সের গড় x বছর
মোট বয়স = 11x বছর

২ জনকে বাদ দিলে অবশিষ্ট থাকে ৯ জন
৯ জনের গড় বয়স = x - 1 বছর
৯ জনের মোট বয়স = 9 (x - 1) বছর

এখন,
∴ 11x - (26 + 29) = 9(x - 1)
⇒ 11x - 55 = 9x - 9
⇒ 11x - 9x = 55 - 9
⇒ 2x = 46
⇒ x = 23 years

∴ The average age of the team is 23 years.

৬২৯.

The difference between 4/5 of a number and 45% of the number is 56. What is 55% of the number?

ক
82
খ
88
গ
98
ঘ
96

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: The difference between 4/5 of a number and 45% of the number is 56. What is 55% of the number?

Solution:
Let the number be x.

According to the question,
4/5 of x - 45% of x = 56
⇒ (4x/5) − (45x/100) = 56
⇒ (4x/5) − (9x/20) = 56
⇒ (16x - 9x)/20 = 56
⇒ 7x/20 = 56
⇒ x = (56 × 20)/7
⇒ x = 8 × 20
∴ x = 160

∴ 55% of x = (55/100) × 160 = 88

৬৩০.

Two friends, Rina and Mina, start walking from the same point on a circular track in opposite directions at 7:00 AM. Rina walks at 1.5 rounds per hour, Mina walks at 2.5 rounds per hour. By what time will they have crossed each other 6 times?

ক
7 : 50 AM
খ
8 : 10 AM
গ
8 : 20 AM
ঘ
8 : 30 AM
ঙ
9 : 00 AM

সঠিক উত্তর: ঘ

8 : 30 AM

উত্তর

সঠিক উত্তর: ঘ

8 : 30 AM

ঘ

ব্যাখ্যা

Question: Two friends, Rina and Mina, start walking from the same point on a circular track in opposite directions at 7:00 AM. Rina walks at 1.5 rounds per hour, Mina walks at 2.5 rounds per hour. By what time will they have crossed each other 6 times?

Solution:
Given,
Rina's speed = 1.5 rounds/hour
Mina's speed = 2.5 rounds/hour
Starting time = 7:00 AM

Since they walk in opposite directions,
Relative speed = (1.5 + 2.5) = 4 rounds/hour

∴ Time to meet 6 times = 6/4 = 1.5 hours = 1 hour 30 minutes

∴ They have crossed each other 6 times = (7 : 00 AM + 1 hour 30 minutes)
= 8 : 30 AM

৬৩১.

Let N be the smallest positive integer that is divisible by both 6 and 9. How many distinct prime factors does N have?

ক
6
খ
5
গ
2
ঘ
3
ঙ
1

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: Let N be the smallest positive integer that is divisible by both 6 and 9. How many distinct prime factors does N have?

Solution:
We are told that N is divisible by both 6 and 9. So, the smallest such number is their LCM.

Now, LCM(6, 9) = 2 × 3 × 3 = 18
Therefore, N = 18
The prime factorization of 18 is 2 × 3². The distinct prime factors are 2 and 3.

∴ There are exactly 2 distinct prime factors.
∴ N has 2 distinct prime factors.

৬৩২.

A’s current age is four times B’s current age. 6 years ago, A’s age was seven times B’s age at that time. What is the difference between their present ages?

ক
32 years
খ
36 years
গ
39 years
ঘ
40 years
ঙ
None

সঠিক উত্তর: খ

36 years

উত্তর

সঠিক উত্তর: খ

36 years

খ

ব্যাখ্যা

Question: A’s current age is four times B’s current age. 6 years ago, A’s age was seven times B’s age at that time. What is the difference between their present ages?

Solution:
Let,
B's current age = x years
A's current age = 4x years (since A is 4 times B)

ATQ,
4x - 6 = 7(x - 6)
⇒ 4x - 6= 7x - 42
⇒ 7x - 4x= 42 - 6
⇒ 3x = 36
∴ x = 12

∴ B's current age = 12 years
∴ A's current age = (4 × 12) = 48 years

So the difference between their present ages = (48 - 12) = 36 years

৬৩৩.

At 8 am two trains started traveling towards each other from stations 300 Km apart. They passed each other at 12 noon. the same day. If the average speed of the faster train was 9 km more than that of the slower train, then what is the speed of the faster train in km/hr?

ক
ক) 40
খ
খ) 42
গ
গ) 45
ঘ
ঘ) None

সঠিক উত্তর: খ

খ) 42

উত্তর

সঠিক উত্তর: খ

খ) 42

খ

ব্যাখ্যা

Question: At 8 am two trains started traveling towards each other from stations 300 Km apart. They passed each other at 12 noon. the same day. If the average speed of the faster train was 9 km more than that of the slower train, then what is the speed of the faster train in km/hr?

Solution:
ধরি
দ্রুতগতির ট্রেন এর গতিবেগ = x কি.মি./ঘণ্টা
ধীরগতির ট্রেন এর গতিবেগ =y কি.মি./ঘণ্টা

যাত্রার মোট সময় = 12 - 8 = 4 ঘণ্টা
এখানে
x + y = 300/4 = 75...............(1)
আবার
x - y = 9 .................(2)

(1) + (2) ⇒
x + y + x - y = 75 + 9
2x = 84
x = 42 কি.মি./ঘণ্টা

৬৩৪.

A bike rider increases his speed by 1 km after every 30 minutes. If the distance travelled in the 1st 30 minutes was 30 km then what was the total distance travelled in 6 hours?

ক
426 km
খ
213 km
গ
639 km
ঘ
none of these

সঠিক উত্তর: ক

426 km

উত্তর

সঠিক উত্তর: ক

426 km

ক

ব্যাখ্যা

Given that,
Distance travelled in 1st 30 minutes = 30 km
Speed of the bike increases by 1 km after every 30 minutes
Distance travelled in 2^nd 30 minutes = 31 km
Distance travelled in 3^rd 30 minutes = 32km
6 hours contains 12 thirty minutes.
Total Distance Travelled = [30 + 31 + 32 + ... (12 terms)]
This is an Arithmetic Progression(AP) with
first term a = 35, number of terms n = 12 and common difference d = 1
Sum of the first n terms of an Arithmetic Progression(AP),
Sn = (n/2)[2a + (n-1)d]
where n =number of terms
Here, [30 + 31 + 32 +... (12 terms)]
S₁₂ = (12/2)[2 × 30 + (12-1)1]
= 6[60 + 11]
= 6 × 71
= 426
Hence, the total distance travelled = 426 km.

৬৩৫.

The area of a rectangle is 252 cm² and its length and breadth are in the ratio of 9 : 7 respectively. What is its perimeter?

ক
24 cm
খ
35 cm
গ
48 cm
ঘ
64 cm

সঠিক উত্তর: ঘ

64 cm

উত্তর

সঠিক উত্তর: ঘ

64 cm

ঘ

ব্যাখ্যা

Question: The area of a rectangle is 252 cm² and its length and breadth are in the ratio of 9 : 7 respectively. What is its perimeter?

Solution:
length, breadth 9x, 7x

9x × 7x = 252
⇒ 63x² = 252
⇒ x² = 252/63 = 4
⇒ x = 2

perimeter = 2 (9x + 7x)
= 2 × 16x
= 32x
= 32 × 2
= 64 cm

৬৩৬.

A man takes 6 hours 15 minutes walking a distance and riding back to starting place. He could walk both ways in 7 hours 45 minutes. The time taken by him to ride back both ways is -

ক
ক) 4 hours
খ
খ) 4 hours 30 minutes
গ
গ) 4 hours 45 minutes
ঘ
ঘ) 5 hours

সঠিক উত্তর: গ

গ) 4 hours 45 minutes

উত্তর

সঠিক উত্তর: গ

গ) 4 hours 45 minutes

গ

ব্যাখ্যা

Time is taken in walking both the ways = 7 hours 45 minutes -------- (i)

Time is taken in walking one way and riding back = 6 hours 15 minutes ----------- (ii)
By the equation (ii) × 2 - (i), we have,

Time is taken by the man in riding both ways,
= 12 hours 30 minutes - 7 hours 45 minutes
= 4 hours 45 minutes.

৬৩৭.

The total age of A and B is 14 years more than the total age of B and C. C is how many years younger than A ?

ক
ক) 16 years
খ
খ) 14 years
গ
গ) 12 years
ঘ
ঘ) 11 years

সঠিক উত্তর: খ

খ) 14 years

উত্তর

সঠিক উত্তর: খ

খ) 14 years

খ

ব্যাখ্যা

Question: The total age of A and B is 14 years more than the total age of B and C. C is how many years younger than A ?

Solution:
According to the question
Age of (A + B) = 14 + Age of (B + C)
⇒ A + B = 14 + B + C
⇒ A = 14 + C
∴ C is 14 years younger than A.

৬৩৮.

To fill a tank, 30 buckets of water are required. How many buckets of water will be required to fill the same tank if the capacity of the bucket is reduced to two-third of its present capacity?

ক
42.5
খ
38
গ
37.5
ঘ
45

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: To fill a tank, 30 buckets of water are required. How many buckets of water will be required to fill the same tank if the capacity of the bucket is reduced to two-third of its present capacity?

Solution:
Let,
The capacity of 1 bucket be n
∴ The capacity of tank be 30n
Capacity of the new bucket = 2n/3

∴ Required number of buckets = (30n)/(2n/3)
= (30n × 3)/2n
= 90/2
= 45

৬৩৯.

The difference between two integers is 6. If their product is 352, then the sum of two numbers is?

ক
48
খ
38
গ
32
ঘ
42

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: The difference between two integers is 6. If their product is 352, then the sum of two numbers is?

Solution:
Let the two integers be x and y
The difference between the two integers is 6,
∴ x - y = 6 and their product is, xy = 352

Now, We know
⇒ (x + y)² = (x - y)² + 4xy
⇒ (x + y)² = 6² + 4 × 352
⇒ (x + y)² = 36 + 1408
⇒ (x + y)² = 1444
⇒ x + y = √1444
∴ x + y = 38

∴ The sum of the two numbers is 38.

৬৪০.

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

ক
ক) 4
খ
খ) 5
গ
গ) 7
ঘ
ঘ) 9

সঠিক উত্তর: ঘ

ঘ) 9

উত্তর

সঠিক উত্তর: ঘ

ঘ) 9

ঘ

ব্যাখ্যা

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

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

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

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

৬৪১.

If 2cos²θ + 5sinθ = 4 where 0° < θ < 90°, what is the value of cotθ ?

ক
1
খ
1/√3
গ
√3
ঘ
2

সঠিক উত্তর: গ

√3

উত্তর

সঠিক উত্তর: গ

√3

গ

ব্যাখ্যা

Question: If 2cos²θ + 5sinθ = 4 where 0° < θ < 90°, what is the value of cotθ ?

Solution:
দেওয়া আছে,
2cos²θ + 5sinθ = 4
⇒ 2(1 - sin²θ) + 5sinθ = 4 [cos²θ = 1 - sin²θ]
⇒ 2 - 2sin²θ + 5sinθ - 4 = 0
⇒ - 2sin²θ + 5sinθ - 2 = 0
⇒ 2sin²θ - 5sinθ + 2 = 0
⇒ 2sin²θ - 4sinθ - sinθ + 2 = 0
⇒ 2sinθ(sinθ - 2) -1(sinθ - 2) = 0
⇒ (sinθ - 2)(2sinθ - 1) = 0

যেহেতু sinθ এর সর্বোচ্চ মান 1, তাই sinθ = 2 অসম্ভব।
∴ 2sinθ - 1=0
⇒ sinθ = 1/2
⇒ sinθ = sin30°
∴ θ = 30°

অতএব, cotθ = cot30° = √3

৬৪২.

How many prime numbers are there between 50 and 60?

ক
1
খ
2
গ
3
ঘ
4

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: How many prime numbers are there between 50 and 60?

Solution:
A prime number is a number that is divisible only by 1 and itself.

Numbers between 50 and 60 are:
51, 52, 53, 54, 55, 56, 57, 58, 59

Among these, the prime numbers are:
53, 59.

∴ Total number of prime numbers = 2.

৬৪৩.

In a class of 72 children are seated in rows and columns in such a way that the number of children in each row is 12.5% more than the number of children in each column. How many children are there in each row?

ক
10
খ
9
গ
7
ঘ
8

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Let the number of children in each column be x.
Thus, the number of children in each row is 1.125x

ATQ,
72 = x × (1.125x)
=> x²= 72/1.125
=> x²= 64
∴x=8

Thus, number of children in a row = 1.125x = 1.125 × 8 = 9

৬৪৪.

A sum of money amounts to Tk. 28000 in 2 years at 20% simple interest per annum. Find the sum.

ক
Tk. 20000
খ
Tk. 25000
গ
Tk. 22000
ঘ
Tk. 18000
ঙ
None of these

সঠিক উত্তর: ক

Tk. 20000

উত্তর

সঠিক উত্তর: ক

Tk. 20000

ক

ব্যাখ্যা

Question: A sum of money amounts to Tk. 28000 in 2 years at 20% simple interest per annum. Find the sum.

Solution:
Here,
A = 28000, T = 2, R = 20
⇒ A = P + SI
⇒ A = P + (P × R × T/100)
⇒ A = P [1 + (R × T/100)]
⇒ 28000 = P [1 + 0.4]
⇒ P = 28000/1.4
⇒ P = 20000
Thus, the required sum is Tk. 20000

৬৪৫.

A sum of money lent out at simple interest amounts Tk. 720 after 2 years and to Tk.1020 after a further period of 5 years. What is the sum ?

ক
ক) Tk. 450
খ
খ) Tk. 400
গ
গ) Tk. 480
ঘ
ঘ) Tk. 600

সঠিক উত্তর: ঘ

ঘ) Tk. 600

উত্তর

সঠিক উত্তর: ঘ

ঘ) Tk. 600

ঘ

ব্যাখ্যা

⇒ S.I. for 2 years = Tk. 720.
⇒ S.I. for 7 years = Tk. 1020
∴ S.I. for 5 years = Tk. 1020 - Tk. 720= Tk. 300.

S.I. for 2 years =(300/5) × 2= Tk..120
Principal=720 - 120 = Tk. 600
∴ The sum is Tk. 600.

৬৪৬.

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

ক
3 : 1
খ
2 : 1
গ
1 : 2
ঘ
3 : 2

সঠিক উত্তর: ঘ

3 : 2

উত্তর

সঠিক উত্তর: ঘ

3 : 2

ঘ

ব্যাখ্যা

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

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

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

৬৪৭.

A 4-digit number is formed using the digits 1 to 7 without repetition. What is the probability that the number is even?

ক
3/7
খ
3/8
গ
4/9
ঘ
5/8

সঠিক উত্তর: ক

3/7

উত্তর

সঠিক উত্তর: ক

3/7

ক

ব্যাখ্যা

Question: A 4-digit number is formed using the digits 1 to 7 without repetition. What is the probability that the number is even?

Solution:
Total available digits = 7 (1, 2, 3, 4, 5, 6, 7)
So, total 4-digit numbers = ⁷P₄ = 840

The last digit must be even (2, 4, 6) = 3 choices.
Choosing 3 digits from 6 digits = ⁶P₃ = 120

So, Number of even 4-digit numbers = 3 × 120 = 360

∴ Probability = 360/840 = 3/7

৬৪৮.

If a * b = 2a - 3b + ab, then 3 * 5 + 5 * 3 is equal to?

ক
18
খ
20
গ
22
ঘ
24

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

3 ∗ 5 + 5 ∗ 3
⇒ 3 ∗ 5 = 2 × 3 − 3 × 5 + 3 × 5 = 6 − 15 + 15 = 6
⇒ 5 ∗ 3 = 2 × 5 − 3 × 3 + 3 × 5 = 10 − 9 + 15 = 16
∴ 3 ∗ 5 + 5 ∗ 3 ⇒ 6 + 16 = 22

৬৪৯.

The base of a right-angled triangle is 6 m and the hypotenuse is 10 m. What is its area?

ক
54 sq. meters
খ
34 sq. meters
গ
24 sq. meters
ঘ
40 sq. meters

সঠিক উত্তর: গ

24 sq. meters

উত্তর

সঠিক উত্তর: গ

24 sq. meters

গ

ব্যাখ্যা

Question: The base of a right-angled triangle is 6 m and the hypotenuse is 10 m. What is its area?

Solution:
Given,
Base = 6 m, Hypotenuse = 10 m

By Pythagoras' Theorem
Height² = Hypotenuse² - Base²
⇒ Height²= 10² - 6²
⇒ Height²= 100 - 36
⇒ Height² = 64
∴ Height = 8

We know,
Area = (1/2) × base × height
= (1/2) × 6 × 8
= 24 sq. meters

৬৫০.

What is the distance between the points (- 1, - 4) and (4, 8)?

ক
10
খ
13
গ
17
ঘ
5

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: What is the distance between the points (- 1, - 4) and (4, 8)?

Solution:
আমরা জানি, দুটি বিন্দুর মধ্যবর্তী দূরত্ব নির্ণয়ের সূত্র হলো:
d = √{(x₂ - x₁)² + (y₂ - y₁)²}

দেওয়া আছে,
প্রথম বিন্দু A = (x₁, y₁) = (- 1, - 4)
দ্বিতীয় বিন্দু B = (x₂, y₂) = (4, 8)

∴ d = √[{4 - (-1)}² + {8 - (- 4)}²]
⇒ d = √[(4 + 1)² + (8 + 4)²]
⇒ d = √[5² + 12²]
⇒ d = √[25 + 144]
⇒ d = √169
⇒ d = 13

∴ দূরত্ব হলো 13 একক।

৬৫১.

One number is 3 more than 2 times of another number. Their sum is 21. Find the numbers.

ক
ক) 7, 14
খ
খ) 2, 19
গ
গ) 6, 15
ঘ
ঘ) 10, 11

সঠিক উত্তর: গ

গ) 6, 15

উত্তর

সঠিক উত্তর: গ

গ) 6, 15

গ

ব্যাখ্যা

প্রশ্ন: একটি সংখ্যা অপর একটি সংখ্যার দ্বিগুণের থেকে 3 বেশি। সংখ্যাদ্বয়ের সমষ্টি 21 হলে, সংখ্যাদ্বয়-

সমাধান:
ধরি,
একটি সংখ্যা = Y
∴ অপর সংখ্যাটি = 2Y + 3

প্রশ্নমতে,
Y + 2Y + 3 = 21
3Y = 18
Y = 6

অন্য সংখ্যাটি = 15

৬৫২.

The value of sin30° + cos60° = ?

ক
0
খ
1
গ
1/2
ঘ
2

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: The value of sin30° + cos60° = ?

Solution:
We know,
sin30° = 1/2
cos60° = 1/2

So,
sin30° + cos60° = 1/2 + 1/2
= 1

∴ The value of sin30° + cos60° is 1.

৬৫৩.

Three numbers are in the ratio 2 : 5 : 8 and their average is 75. What is the difference between larger and smaller numbers?

ক
90
খ
80
গ
95
ঘ
55

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: Three numbers are in the ratio 2 : 5 : 8 and their average is 75. What is the difference between larger and smaller numbers?

Solution:
Let the three numbers are 2x, 5x, and 8x, where x is a constant multiplier.
The average of the three numbers is given as 75.
The average of the three numbers is,
⇒ (2x + 5x + 8x)/3 = 75
⇒ 15x/3 = 75
⇒ 5x = 75
⇒ x = 75/5
∴ x = 15
The first number is = 2x = 2 × 15 =30
The third number is = 8x = 8 × 15 = 120

∴ The difference between the larger and smaller numbers is, 120 − 30 = 90

৬৫৪.

A rowing team rows a boat and covers a distance of 18 km in 6 hours while going upstream. When they go downstream, they cover a distance of 30 km in 5 hours. Find the speed of the boat in still water?

ক
ক) 4.5 km/h
খ
খ) 3.5 km/h
গ
গ) 2.5 km/h
ঘ
ঘ) 5.5 km/h

সঠিক উত্তর: ক

ক) 4.5 km/h

উত্তর

সঠিক উত্তর: ক

ক) 4.5 km/h

ক

ব্যাখ্যা

The boat covers 15 km upstream in 5 hours and 24 km downstream in 4 hours
According to the question,
Downstream speed = 30/5 = 6 km/h
Upstream speed = 18/6 = 3 km/h

∴ Speed of the Boat in still water = 1/2 × (6 + 3) = 4.5 km/h

৬৫৫.

A container contain 80 litres of mixture of Honey and water in the ratio 5:3. If 16 liters of this mixture is replaced by Honey, then the ratio of Honey to water in the resulting mixture becomes:

ক
ক) 5 : 2
খ
খ) 4 : 3
গ
গ) 7 : 3
ঘ
ঘ) 8 : 1

সঠিক উত্তর: গ

গ) 7 : 3

উত্তর

সঠিক উত্তর: গ

গ) 7 : 3

গ

ব্যাখ্যা

৮০ লিটার মিশ্রণে মধু ও পানির অনুপাত = ৫ঃ৩
মিশ্রণে,
মধুর পরিমাণ = (৫ × ৮০)/৮ = ৫০ লিটার।
পানির পরিমাণ = (৩ × ৮০)/৮ = ৩০ লিটার।

প্রশ্নে বলা হয়েছে, মধু দ্বারা ১৬ লিটার মিশ্রণ প্রতিস্থাপিত হয়েছে।

১৬ লিটার মিশ্রণেও মধু ও পানির অনুপাত একই থাকবে।
অতএব ১৬ লিটার মিশ্রণে,
মধুর পরিমাণ = (৫ × ১৬)/৮ = ১০ লিটার।
পানির পরিমাণ = ১৬ - ১০ = ৬ লিটার।

সুতরাং, মোট মধুর পরিমাণ = ৫০ - ১০ + ১৬ = ৫৬ লিটার।
মোট পানির পরিমাণ = ৩০ - ৬ = ২৪ লিটার

মধু ও পানির অনুপাত = ৫৬ঃ২৪ = ৭ঃ৩

৬৫৬.

If P = {1, 4, 9, 16, 25, 36, 49, 64}, the number of proper subsets of P is

ক
215
খ
272
গ
265
ঘ
255

সঠিক উত্তর: ঘ

255

উত্তর

সঠিক উত্তর: ঘ

255

ঘ

ব্যাখ্যা

Question: If P = {1, 4, 9, 16, 25, 36, 49, 64}, the number of proper subsets of P is
(Janata RC 2022 অনুযায়ী)

Solution:
দেওয়া আছে,
P = {1, 4, 9, 16, 25, 36, 49, 64}

সেটের উপাদান সংখ্যা = 8

∴ প্রকৃত উপসেট সংখ্যা = 2ⁿ - 1
= 2⁸ - 1
= 256 - 1
= 255

৬৫৭.

What is the average (arithmetic mean) of all the multiple of tens from 10 to 150 inclusive?

ক
ক) 100
খ
খ) 90
গ
গ) 80
ঘ
ঘ) 70

সঠিক উত্তর: গ

গ) 80

উত্তর

সঠিক উত্তর: গ

গ) 80

গ

ব্যাখ্যা

Question: What is the average (arithmetic mean) of all the multiple of tens from 10 to 150 inclusive?

Solution:
10 থেকে 150 এর মধ্যে 10 এর গুণিতক আছে 15টি
এগুলো হলো 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150

নির্ণেয় গড় = 1200/15 = 80

৬৫৮.

The ratio between the speeds of two trains is 7 : 8. If the second train runs 400 km in 4 hours, then the speed of the first train is-

ক
65.5 km/hr
খ
70 km/hr
গ
75 km/hr
ঘ
84 km/hr
ঙ
87.5 km/hr

সঠিক উত্তর: ঙ

87.5 km/hr

উত্তর

সঠিক উত্তর: ঙ

87.5 km/hr

ঙ

ব্যাখ্যা

Question: The ratio between the speeds of two trains is 7 : 8. If the second train runs 400 km in 4 hours, then the speed of the first train is-

Solution:
Let the speed of two trains be 7x and 8x km/hr.
Then,
8x = (400/4) = 100
∴ x = 100/8 = 12.5

Speed of first train = (7 × 12.5) km/hr = 87.5 km/hr.

৬৫৯.

Find an equation of the horizontal line containing the point (3, 2).

ক
y = 2
খ
y = 3
গ
x = 2
ঘ
x = 3

সঠিক উত্তর: ক

y = 2

উত্তর

সঠিক উত্তর: ক

y = 2

ক

ব্যাখ্যা

Question: Find an equation of the horizontal line containing the point (3, 2).

[২০২২ সাল ভিত্তিক সমন্বিত ৮ ব্যাংক ও ১ আর্থিক প্রতিষ্ঠান পদের নাম: অফিসার (জেনারেল)]
Solution:
প্রদত্ত বিন্দু (3, 2)
এখানে
x = 3, y = 2
অনুভূমিক রেখা সবসময় y-এর মান একই
বিন্দু (3, 2)-এর মধ্যে দিয়ে যে অনুভূমিক রেখাটি যাবে, সেটির প্রতিটি বিন্দুর y-এর মান হবে 2

নির্ণেয় সমীকরণ হবে: y = 2

৬৬০.

If , what is the value of x?

ক
1/4
খ
1/2
গ
3/4
ঘ
5/3

সঠিক উত্তর: গ

3/4

উত্তর

সঠিক উত্তর: গ

3/4

গ

ব্যাখ্যা

Question:
If , what is the value of x?

Solution:

৬৬১.

If (x/y) + (y/x) = √7, what is the value of (x⁴/y⁴) + (y⁴/x⁴)?

ক
22
খ
23
গ
25
ঘ
27

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: If x/y + y/x = √7, what is the value of (x⁴/y⁴) + (y⁴/x⁴)?

Solution:
Given that,
x/y + y/x = √7

Now,
(x⁴/y⁴) + (y⁴/x⁴)
= (x²/y²)² + (y²/x²)²
= (x²/y² + y²/x²)² - 2 ; [a²+ b² = (a + b)² - 2ab]
= {(x/y)² + (y/x)²}² - 2
= {(x/y + y/x)² - 2}² - 2 ; [a² + b² = (a + b)² - 2ab]
= {(√7)² - 2}² - 2
= (7 - 2)² - 2
= 25 - 2
= 23

৬৬২.

The ratio of copper and zinc in a 63 kg alloy is 4 : 3. Some amount of copper is extracted from the alloy, and the ratio becomes 10 : 9. How much copper is extracted?

ক
6 kg
খ
8 kg
গ
10 kg
ঘ
12 kg

সঠিক উত্তর: ক

6 kg

উত্তর

সঠিক উত্তর: ক

6 kg

ক

ব্যাখ্যা

Question: The ratio of copper and zinc in a 63 kg alloy is 4 : 3. Some amount of copper is extracted from the alloy, and the ratio becomes 10 : 9. How much copper is extracted?

Solution:
Amount of copper in alloy = [copper ratio/ sum of ratios] × total quantity of alloy
Copper = (4/7) × 63 = 36 kg
Similarly, the amount of zinc in alloy = (3/7) × 63 = 27 kg
Let the extracted copper from alloy = x kg
Remaining copper in alloy = 36 - x
The new ratio = 10 : 9
i.e., 36 - x : 27 = 10 : 9
⇒ 36 - x : 3 = 10 : 1
⇒ 36 - x = 10 × 3
⇒ 36 - x = 30
∴ x = 36 - 30 = 6 kg.
Hence, the extracted amount of copper is 6 kg.

৬৬৩.

A container contains 40 liters of milk. From this container 4 liters of milk was taken out and replaced by water. This process was repeated further two times. How much milk is now contained by the container?

ক
26 liters
খ
29.16 liters
গ
28 liters
ঘ
28.2 liters
ঙ
None of these

সঠিক উত্তর: খ

29.16 liters

উত্তর

সঠিক উত্তর: খ

29.16 liters

খ

ব্যাখ্যা

Question: A container contains 40 liters of milk. From this container 4 liters of milk was taken out and replaced by water. This process was repeated further two times. How much milk is now contained by the container?

Solution:
Milk contained by the container now
= 40(1 - 4/40)³
= 40(1 - 1/10)³
= 40 × 0.9 × 0.9 × 0.9
= 29.16

৬৬৪.

Pavel is twice as good as workman as Tanveer. When they work together they can finish a task in 16 days. If Tanveer works alone, in how many days he will complete the task?

ক
46 days
খ
48 days
গ
50 days
ঘ
52 days

সঠিক উত্তর: খ

48 days

উত্তর

সঠিক উত্তর: খ

48 days

খ

ব্যাখ্যা

Question: Pavel is twice as good as workman as Tanveer. When they work together they can finish a task in 16 days. If Tanveer works alone, in how many days he will complete the task?

Solution:
ধরি,
তানভীর কাজটি করতে পারে ২ক দিনে
পাভেল কাজটি করতে পারে ক দিনে

তাহলে,
তানভীর ১ দিনে করে ১/২ক অংশ
∴ তানভীর ১৬ দিনে করে ১৬/২ক অংশ = ৮/ক অংশ

পাভেল ১ দিনে করে ১/ক অংশ
∴ পাভেল ১ দিনে করে ১৬/ক অংশ

৮/ক + ১৬/ক = ১
বা, (৮ + ১৬)/ক = ১
বা, ২৪/ক = ১
∴ ক = ২৪

তানভীর ১ দিনে করে = ১/(২ × ২৪) = ১/৪৮ অংশ
∴ তানভীর একা ৪৮ দিনে কাজটি সম্পন্ন করবে।

৬৬৫.

If 7 workers can assemble a car in 8 hours, how long would it take 12 workers to assemble the same car?

ক
280 minutes
খ
290 minutes
গ
310 minutes
ঘ
330 minutes
ঙ
None

সঠিক উত্তর: ক

280 minutes

উত্তর

সঠিক উত্তর: ক

280 minutes

ক

ব্যাখ্যা

Question: If 7 workers can assemble a car in 8 hours, how long would it take 12 workers to assemble the same car?

Solution:
Given,
7 workers can assemble a car in 8 hours
∴ 1 workers can assemble a car in (8 × 7) hours
∴ 12 workers can assemble a car in (56/12) hours
= (56 × 60)/12 minutes
= 280 minutes

৬৬৬.

A and B can do a work in 12 days. B and C can do it in 15 days. A and C can do it in 20 days. If all of them work together, in how many days can they finish the work?

ক
4 days
খ
6 days
গ
8 days
ঘ
10 days

সঠিক উত্তর: ঘ

10 days

উত্তর

সঠিক উত্তর: ঘ

10 days

ঘ

ব্যাখ্যা

Question: A and B can do a work in 12 days. B and C can do it in 15 days. A and C can do it in 20 days. If all of them work together, in how many days can they finish the work?

Solution:
(A + B), 1days works = 1/12,
(B + C), 1days work = 1/15,
(C + A), 1 days work = 1/20

Now,
2(A + B + C) 1days work = (1/12) + (1/20) + (1/15)
= (5 + 3 + 4)/60
=12/60
=1/5

(A + B + C) 1 days works = 1/10
∴ A, B and C together can do it in = 10 days

৬৬৭.

A person's present age is two-fifths of the age of his mother. After 8 years, he will be one half of the age of his mother. What is the present age of the mother?

ক
30 years
খ
36 years
গ
40 years
ঘ
42 years

সঠিক উত্তর: গ

40 years

উত্তর

সঠিক উত্তর: গ

40 years

গ

ব্যাখ্যা

Question: A person's present age is two-fifths of the age of his mother. After 8 years, he will be one half of the age of his mother. What is the present age of the mother?

Solution:
Let the age of the person’s mother be ‘x years
Age of the person = (2x/5)

After 8 years, age of the person’s mother = ‘x + 8’ years
Age of the person after 8 years = (2x/5) + 8 years

According to question:
(2x/5) + 8 = (1/2)(x + 8)
⇒ (2x/5) + 8 = (x/2) + 4
⇒ (x/2) - (2x/5) = 8 - 4
⇒ (5x - 4x)/10 = 4
⇒ x/10 = 4
∴ x = 40

So, the present age of his mother is 40 years.

৬৬৮.

A truck covers a distance of 550 meters in 1 minute whereas a bus covers a distance of 44 kms in 60 minutes. The ratio of their speeds is-

ক
ক) 3 ∶ 7
খ
খ) 4 ∶ 5
গ
গ) 3 ∶ 5
ঘ
ঘ) 3 ∶ 4

সঠিক উত্তর: ঘ

ঘ) 3 ∶ 4

উত্তর

সঠিক উত্তর: ঘ

ঘ) 3 ∶ 4

ঘ

ব্যাখ্যা

Given that
Distance covered by truck = 550 m
Time taken by truck to cover the distance = 1 min
Distance covered by bus = 44 km = 44,000 m
Time taken by bus to cover the distance = 60 min

Speed of truck = 550/1 = 550 m/min
Speed of bus = 44,000/60 = 2,200/3 m/min
Speed of truck ∶ Speed of bus = 550 ∶ (2,200/3)
= 1 : 4/3
= 3 ∶ 4

∴ The ratio of the speed of the truck to the speed of the bus is 3 ∶ 4.

৬৬৯.

Rafi alone can complete a work in 10 days and Tareq alone can complete it in 15 days. Rafi and Tareq undertook to complete the work for Tk. 7500. With the help of Salman, they finished the work in 5 days. How much should Salman be paid?

ক
Tk. 1200
খ
Tk. 1250
গ
Tk. 1300
ঘ
Tk. 1380

সঠিক উত্তর: খ

Tk. 1250

উত্তর

সঠিক উত্তর: খ

Tk. 1250

খ

ব্যাখ্যা

Question: Rafi alone can complete a work in 10 days and Tareq alone can complete it in 15 days. Rafi and Tareq undertook to complete the work for Tk. 7500. With the help of Salman, they finished the work in 5 days. How much should Salman be paid?

Solution:
Rafi's 1 day work = 1/10
Tareq's 1 day work = 1/15
Rafi + Tareq + Salman's 1 day work = 1/5

∴ Salman's 1 day work = 1/5 - (1/10 + 1/15)
= (6 - 3 - 2)/30
= 1/30
Salman's 5 days work = 5 × 1/30 = 1/6

Salman completed 1/6 of the total work.
∴ He should be paid 1/6 of Tk. 7500
∴ Salman’s payment = 7500 × 1/6 = Tk. 1250

৬৭০.

A train 150 m long, takes 30 seconds to cross a bridge 500 m long. How much time will the train take to cross a platform 370 m long?

ক
ক) 20 sec
খ
খ) 22 sec
গ
গ) 23 sec
ঘ
ঘ) 24 sec

সঠিক উত্তর: ঘ

ঘ) 24 sec

উত্তর

সঠিক উত্তর: ঘ

ঘ) 24 sec

ঘ

ব্যাখ্যা

Question: A train 150 m long, takes 30 seconds to cross a bridge 500 m long. How much time will the train take to cross a platform 370 m long?

Solution:
ট্রেন + ব্রিজের মোট দূরত্ব = ১৫০ + ৫০০ = ৬৫০ মি
ট্রেন + প্লাটফর্ম = ১৫০ + ৩৭০ = ৫২০ মি

৬৫০ মি অতিক্রম করে ৩০ সেকেন্ডে
১ মি অতিক্রম করে ৩০/৬৫০ সেকেন্ডে
৫২০ মি অতিক্রম করে (৩০ × ৫২০)/৬৫০ সেকেন্ডে
= ২৪ সেকেন্ডে

৬৭১.

In the given figure ABCD is a Parallelogram then the find out of the value of x is?

ক
75°
খ
45°
গ
30°
ঘ
60°

সঠিক উত্তর: খ

45°

উত্তর

সঠিক উত্তর: খ

45°

খ

ব্যাখ্যা

Question: In the given figure ABCD is a Parallelogram then the find out of the value of x is?

Solution:
Given that,
ABCD is a parallelogram.
than,
∠C = (2x + 30)°
∠A = (5x - 105)°
Angle A and Angle C, Opposite angles in a parallelogram are equal.
∠A = ∠C
⇒ 5x - 105° = 2x + 30°
⇒ 5x - 2x = 30° + 105°
⇒ 3x = 135°
∴ x = 45°

৬৭২.

Find the maximum distance between two points on the perimeter of a rectangular garden whose length and breadth are 15 m and 8 m.

ক
35 m
খ
23 m
গ
21 m
ঘ
17 m

সঠিক উত্তর: ঘ

17 m

উত্তর

সঠিক উত্তর: ঘ

17 m

ঘ

ব্যাখ্যা

Question: Find the maximum distance between two points on the perimeter of a rectangular garden whose length and breadth are 15 m and 8 m.

Solution:
Given that,
Length of rectangle, L = 15 m
Breadth of rectangle, B = 8 m

Now, the maximum distance between two points on a rectangle is the diagonal of the rectangle.

∴ Diagonal = √(L² + B²)
= √(15² + 8²)
= √(225 + 64)
= √289
= 17 m

So the maximum distance between two points on the perimeter is 17 m.

৬৭৩.

A rectangular block 4 cm by 12 cm by 8 cm is cut up into an exact number of equal cubes. Find the least possible number of cubes.

ক
ক) 6
খ
খ) 12
গ
গ) 16
ঘ
ঘ) 40

সঠিক উত্তর: ক

ক) 6

উত্তর

সঠিক উত্তর: ক

ক) 6

ক

ব্যাখ্যা

Volume of the block
= (4 × 12 × 8) cm³
= 384 cm³

Side of the largest cube
= H.C.F of 4 cm, 12 cm, 8 cm
= 4 cm.

Volume of this cube = (4 × 4 × 4) cm³
= 64 cm³

Number of cubes
= 384/64
= 6

৬৭৪.

How long will a boy take to run round a square field of side 35 meters, If he runs at the rate of 9 km/hr?

ক
56 sec
খ
54 sec
গ
52 sec
ঘ
50 sec
ঙ
55 sec

সঠিক উত্তর: ক

56 sec

উত্তর

সঠিক উত্তর: ক

56 sec

ক

ব্যাখ্যা

Question: How long will a boy take to run round a square field of side 35 meters, If he runs at the rate of 9 km/hr?

Solution:
Speed = 9 km/hr = 9 × (5/18) m/sec = 5/2 m/sec

Distance = (35 × 4) m = 140 m.

Time taken = 140 × (2/5) sec= 56 sec

৬৭৫.

Rubel can do a job alone in 12 days. He works for 8 days and then leaves, and Rajib finishes the remaining work in 7 days. How long would it take Rajib to do 3/7 of the work alone?

ক
3 days
খ
6 days
গ
9 days
ঘ
12 days

সঠিক উত্তর: গ

9 days

উত্তর

সঠিক উত্তর: গ

9 days

গ

ব্যাখ্যা

Question: Rubel can do a job alone in 12 days. He works for 8 days and then leaves, and Rajib finishes the remaining work in 7 days. How long would it take Rajib to do 3/7 of the work alone?

Solution:
রুবেল একা,
12 দিনে করতে পারে = 1 অংশ
∴ 8 দিনে করতে পারে = 8/12 = 2/3 অংশ

∴ অবশিষ্ট কাজ = 1 - (2/3) = 1/3 অংশ

হাবিব একা,
1/3 অংশ কাজ করে = 7 দিনে
∴ হাবিব 1 অংশ কাজ করে = 3 × 7 দিনে
∴ হাবিব 3/7 অংশ কাজ করে = (3 × 7 × 3)/7 দিনে
= 9 দিনে

৬৭৬.

A person has Tk. 950 in 29 notes consisting of Tk. 500, 100, 50 and 10. How many notes should at least be in Tk. 50?

ক
ক) 1
খ
খ) 2
গ
গ) 5
ঘ
ঘ) None

সঠিক উত্তর: খ

খ) 2

উত্তর

সঠিক উত্তর: খ

খ) 2

খ

ব্যাখ্যা

Question: A person has Tk. 950 in 29 notes consisting of Tk. 500, 100, 50 and 10. How many notes should at least be in Tk. 50?

Solution:
১ টি ৫০০ টাকার নোট, ১ টি ১০০ টাকার নোট, ১ টি ৫০ টাকার নোট = ৫০০ + ১০০ + ৫০ টাকা
= ৬৫০ টাকা

১০ টাকার নোট লাগবে = ৯৫০ - ৬৫০/১০
= ৩০০/১০
= ৩০ টি ; যা সম্ভব নয় যেহেতু মোট নোট আছে ২৯ টি

আবার,
১ টি ৫০০ টাকার নোট, ১ টি ১০০ টাকার নোট, ২ টি ৫০ টাকার নোট = ৫০০ + ১০০ + ১০০ টাকা
= ৭০০ টাকা

১০ টাকার নোট লাগবে = ৯৫০ - ৭০০/১০
= ২৫০/১০
= ২৫ টি ;

মোট নোট = ২৫ + ১ + ১ + ২ টি
= ২৯ টি

অতএব, ৫০ টাকার কমপক্ষে ২ টি নোট লাগবে।

৬৭৭.

Choose the one that will be best fit in the blank position?
ELFA, GLHA, ILJA,______, MLNA

ক
ক) OLPA
খ
খ) LLMA
গ
গ) KLMA
ঘ
ঘ) KLLA

সঠিক উত্তর: ঘ

ঘ) KLLA

উত্তর

সঠিক উত্তর: ঘ

ঘ) KLLA

ঘ

ব্যাখ্যা

Question: Choose the one that will be best fit in the blank position? ELFA, GLHA, ILJA,______, MLNA

Solution:
প্রথম বর্ণে একটি বর্ণ বাদ দিয়ে পরের টি বসানো হয়েছে।
E (F) G (H) I (J) K (L) M

প্রতিটি অংশে ২য় ও ৪র্থ বর্ণ L ও A।

৩য় অংশে একটি বর্ণ বাদ দিয়ে পরের টি বসানো হয়েছে।
F (G) H (I) J (k) L (M) N

অতএব, শূন্যস্থানে বসবে KLLA

৬৭৮.

Sohan bought some oranges of which 10% are rotten. He gives 80% of the remaining to his neighbor. Now he is left out with 36 oranges. How many oranges he bought?

ক
188 oranges
খ
200 oranges
গ
176 oranges
ঘ
220 oranges
ঙ
None of these

সঠিক উত্তর: খ

200 oranges

উত্তর

সঠিক উত্তর: খ

200 oranges

খ

ব্যাখ্যা

Question: Sohan bought some oranges of which 10% are rotten. He gives 80% of the remaining to his neighbor. Now he is left out with 36 oranges. How many oranges he bought?

Solution:
Let,
Sohanr bought = 100 oranges.
Oranges after removing rotten ones = 90.
Oranges given to neighbor = 80% of 90 = 72 Oranges.

Now,
He left with eggs = (90 - 72) Oranges
= 18 Oranges.

Now,
Comparing,
Remaining 18 when total oranges = 100
Remaining 18 when total oranges = 100/18
Remaining 36 when total oranges = (100 × 36)/18
= 200

৬৭৯.

There are 2 shirts, 3 jeans, 3 socks and 2 skirts. In how many ways shopkeeper can arrange these things so that all the socks come together and all the skirts come together?

ক
ক) 4690
খ
খ) 3260
গ
গ) 2520
ঘ
ঘ) 5040

সঠিক উত্তর: ঘ

ঘ) 5040

উত্তর

সঠিক উত্তর: ঘ

ঘ) 5040

ঘ

ব্যাখ্যা

৩টি মোজাকে ১টি এবং ২টি স্কার্টকে ১টি ধরে মোট = 1 + 1 + 2 + 3 = 7টি পণ্য পাওয়া যাবে।
দোকানদার সকল মোজা এবং সকল স্কার্টগুলো একত্রে রেখে 7টি পণ্যকে সাজাতে পারেন = 7! ভাবে ।
= 5040

৬৮০.

Chose the pair of numbers which comes next: 75, 65, 85, 55, 45, 85, 35, _______

ক
ক) 25, 15
খ
খ) 35, 25
গ
গ) 35, 85
ঘ
ঘ) 25, 85

সঠিক উত্তর: ঘ

ঘ) 25, 85

উত্তর

সঠিক উত্তর: ঘ

ঘ) 25, 85

ঘ

ব্যাখ্যা

প্রশ্ন: Chose the pair of numbers which comes next: 75, 65, 85, 55, 45, 85, 35, _______

সমাধান:
এই সিরিজে ১০ বিয়োগ করে পরবর্তী সংখ্যা নেওয়া হয়েছে।
প্রতি দুই ঘর পরপর ৮৫ সংখ্যাটি পুনরাবৃত্তি হয়েছে।

পরবর্তী সংখ্যাটি 35 - 10 = 25
যেহেতু, দুই সংখ্যা পরপর 85 সংখ্যাটি পুনরাবৃত্তি হচ্ছে।
অতএব, 25 এর পরবর্তী সংখ্যাটি হবে 85

∴পরবর্তী জোড়াটি হবে 25, 85

৬৮১.

The ratio of ducks and frogs in a pond is 37 : 39 respectively. The average number of ducks and frogs in the pond is 152. What is the number of frogs in the pond?

ক
ক) 144
খ
খ) 148
গ
গ) 152
ঘ
ঘ) 156

সঠিক উত্তর: ঘ

ঘ) 156

উত্তর

সঠিক উত্তর: ঘ

ঘ) 156

ঘ

ব্যাখ্যা

Question: The ratio of ducks and frogs in a pond is 37 : 39 respectively. The average number of ducks and frogs in the pond is 152. What is the number of frogs in the pond?

Solution:
The ratio of Ducks and Frogs in the Pond = 37 : 39
Average of Ducks and Frogs in the Pond = 152
So, the total number of Ducks and Frogs in the Pond = 2 × 152 = 304
∴ Number of Frogs = (304 × 39)/76 = 156

৬৮২.

13, 17, 19, 23, 29, ?
Which number replace the Question mark?<

ক
30
খ
31
গ
33
ঘ
34

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: 13, 17, 19, 23, 29, ?
Which number replace the Question mark?

Solution:
13, 17, 19, 23, 29
All the numbers are prime number
∴ Next number will be 31

৬৮৩.

ক
ক) 16
খ
খ) 12
গ
গ) 8
ঘ
ঘ) 4

সঠিক উত্তর: ক

ক) 16

উত্তর

সঠিক উত্তর: ক

ক) 16

ক

ব্যাখ্যা

(4/5) × (√x/5) × (16/25) = 256/625
√x (64/625) = 256/625
√x × 64 = 256
√x = 256/64
√x = 4
(√x )²= 4²
x = 16

৬৮৪.

If cosθ + sinθ = 1, then cosθsinθ = ?

ক
ক) 0
খ
খ) 1/2
গ
গ) 1
ঘ
ঘ) 1/4

সঠিক উত্তর: ক

ক) 0

উত্তর

সঠিক উত্তর: ক

ক) 0

ক

ব্যাখ্যা

Given, cosθ + sinθ = 1
Or, (cosθ + sin)² = 1²
Or, cos²θ + sin²θ + 2cosθsinθ = 1
Or, 2cosθsinθ = 0 [As, cos²θ + sin²θ = 1]
Or, cosθsinθ = 0

৬৮৫.

The average salary of all the workers in a workshop is Tk. 8000. The average salary of 7 technicians is Tk. 12000 and the average salary of the rest is Tk. 6000. What is the total number of workers in the workshop?

ক
ক) 18
খ
খ) 21
গ
গ) 23
ঘ
ঘ) 25

সঠিক উত্তর: খ

খ) 21

উত্তর

সঠিক উত্তর: খ

খ) 21

খ

ব্যাখ্যা

Question: The average salary of all the workers in a workshop is Tk. 8000. The average salary of 7 technicians is Tk. 12000 and the average salary of the rest is Tk. 6000. What is the total number of workers in the workshop?

Solution:
Let the total number of workers be x
Then,
8000x = (12000 × 7) + 6000(x - 7)
⇒ 8000x - 6000x = 84000 - 42000
⇒ 2000x = 42000
⇒ x = 42000/2000
⇒ x = 42/2
∴ x = 21

∴ The total number of workers in the workshop is 21

৬৮৬.

A and B are in the ratio of 6 : 5 and B and C are in the ratio of 4 : 3. What is the ratio of A : C?

ক
12 : 15
খ
7 : 15
গ
8 : 5
ঘ
13 : 15

সঠিক উত্তর: গ

8 : 5

উত্তর

সঠিক উত্তর: গ

8 : 5

গ

ব্যাখ্যা

Question: A and B are in the ratio of 6 : 5 and B and C are in the ratio of 4 : 3. What is the ratio of A : C?

Solution:
Given the ratio of,
A : B = 6 : 5 = (6 × 4) : (5 × 4) = 24 : 20

And,
B : C = 4 : 3 = (4 × 5) : (3 × 5) = 20 : 15

∴ A : C = 24 : 15 = 8 : 5

৬৮৭.

The least number by which 180 must be multiplied to make it a perfect square is:

ক
3
খ
4
গ
2
ঘ
5

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: The least number by which 180 must be multiplied to make it a perfect square is:

Solution:
Prime factorization of 180 = 2 × 2 × 3 × 3 × 5
= 2² × 3² × 5¹
For a number to be a perfect square, every exponent in the prime factorization must be even.
2² ; even power
3² ; even power
5¹ ; odd power
The only odd exponent is 5. To make it even, multiply by 5.
= 180 × 5
= 900
= 30²

So the least number by which 180 must be multiplied to make it a perfect square is 5.

৬৮৮.

A truck travels the first 120 km at an average speed of 60 km/h and the next 120 km at an average speed of 40 km/h. What is its average speed for the entire journey in km per hour?

ক
48 km/hr
খ
54 km/hr
গ
60 km/hr
ঘ
72 km/hr

সঠিক উত্তর: ক

48 km/hr

উত্তর

সঠিক উত্তর: ক

48 km/hr

ক

ব্যাখ্যা

Question: A truck travels the first 120 km at an average speed of 60 km/h and the next 120 km at an average speed of 40 km/h. What is its average speed for the entire journey in km per hour?

Solution:
প্রথম অংশের জন্য সময়:
সময় = দূরত্ব/গতিবেগ
= 120 কিমি / 60 কিমি/ঘন্টা
= 2 ঘন্টা

দ্বিতীয় অংশের জন্য সময়:
সময় = দূরত্ব/গতিবেগ
= 120 কিমি / 40 কিমি/ঘন্টা
= 3 ঘন্টা

মোট অতিক্রান্ত দূরত্ব = (120 + 120) কিমি = 240 কিমি
মোট সময় = (2 + 3) ঘন্টা = 5 ঘন্টা

গড় গতিবেগ = মোট অতিক্রান্ত দূরত্ব/মোট সময়
= 240 কিমি/5 ঘন্টা
= 48 কিমি/ঘন্টা

∴ পুরো যাত্রায় ট্রাকটির গড় গতিবেগ ছিল 48 কিমি/ঘন্টা।

৬৮৯.

A company employs 12 persons working 55 hours a week to complete a project. If the working hours turned into 44 hours a week. How many additional persons will be required to complete the project in time?

ক
ক) 3
খ
খ) 4
গ
গ) 5
ঘ
ঘ) 6

সঠিক উত্তর: ক

ক) 3

উত্তর

সঠিক উত্তর: ক

ক) 3

ক

ব্যাখ্যা

Question: A company employs 12 persons working 55 hours a week to complete a project. If the working hours turned into 44 hours a week. How many additional persons will be required to complete the project in time?

Solution:
Total working hours is (55 × 12) = 660 hours

Total persons needed for 44 hours a week = 660/44 = 15

∴ Additional persons will be required = 15 - 12 = 3

৬৯০.

A discount of 10% on one article is equal in amount to a discount of 25% on a second article. What can be the ratio of their marked prices?

ক
5 : 2
খ
2 : 5
গ
1 : 5
ঘ
2 : 1

সঠিক উত্তর: ক

5 : 2

উত্তর

সঠিক উত্তর: ক

5 : 2

ক

ব্যাখ্যা

Question: A discount of 10% on one article is equal in amount to a discount of 25% on a second article. What can be the ratio of their marked prices?

Solution:
Let
the marked prices of the two articles be X and Y

ATQ,
10% of X = 25% of Y
⇒ 10X/100 = 25Y/100
⇒ 10X = 25Y
⇒ X/Y = 25/10
∴ X : Y = 5 : 2

The ratio of their marked prices is = 5 : 2

৬৯১.

A team started a work where the work ratio of a day is twice the previous day. If the work is fully finished in 10 days, how many days it took to finish 1/8th of the work?

ক
ক) 2 days
খ
খ) 3 days
গ
গ) 5 days
ঘ
ঘ) 7 days

সঠিক উত্তর: ঘ

ঘ) 7 days

উত্তর

সঠিক উত্তর: ঘ

ঘ) 7 days

ঘ

ব্যাখ্যা

Question: A team started a work where the work ratio of a day is twice the previous day. If the work is fully finished in 10 days, how many days it took to finish 1/8th of the work?

Solution:
প্রতিদিনের কাজ পূর্বের দিনের দ্বিগুণ হয়।

১০ দিনে যদি সম্পূর্ণ শেষ হয় তাহলে,
৯ দিনে শেষ হবে ১/২ কাজ
৮ দিনে শেষ হবে (১/২)/২ বা, ১/৪ কাজ
৭ দিনে শেষ হবে (১/৪)/২ বা, ১/৮ কাজ

shortcut:
১ অংশ কাজ হয় ১০ দিনে,
১/৮ অংশ বা ১/২^৩অংশ কাজ হয় = (১০ - ৩) দিনে = ৭ দিনে।
যত অংশ দেয়া থাকবে তার হরকে ২ এর ভিত্তি আকারে প্রকাশ করে ২ এর সূচককে মোট দিন থেকে বাদ দিয়ে দিন বের করতে হবে।

একই ভাবে যদি ১/১৬ অংশ বা ১/২^৪ অংশ কাজ হবে = (১০ -৪) = ৬ দিনে।

৬৯২.

P and Q started a business investing Tk. 57000 and Tk. 33000 respectively. In what ratio the profit earned after 2 years be divided between P and Q respectively.

ক
ক) 19 : 33
খ
খ) 17 : 31
গ
গ) 19 : 11
ঘ
ঘ) 12 : 31

সঠিক উত্তর: গ

গ) 19 : 11

উত্তর

সঠিক উত্তর: গ

গ) 19 : 11

গ

ব্যাখ্যা

In this question as time frame for both investors is same as then just get the ratio of their investments.

P : Q
= 57000 : 33000
= 57 : 33
= 19 : 11

৬৯৩.

একটি পণ্যের বিক্রয়মূল্য দ্বিগুণ হলে বিক্রেতার মুনাফা বেড়ে তিনগুণ হয়। মূল্য বৃদ্ধি না করে পণ্যটি বিক্রয় করলে শতকরা কত লাভ হবে?

ক
৫০%
খ
১০০%
গ
১৫০%
ঘ
১২৫%
ঙ
কোনোটিই নয়

সঠিক উত্তর: খ

১০০%

উত্তর

সঠিক উত্তর: খ

১০০%

খ

ব্যাখ্যা

প্রশ্ন: একটি পণ্যের বিক্রয়মূল্য দ্বিগুণ হলে বিক্রেতার মুনাফা বেড়ে তিনগুণ হয়। মূল্য বৃদ্ধি না করে পণ্যটি বিক্রয় করলে শতকরা কত লাভ হবে?

সমাধান:
ধরি,
ক্রয়মূল্য ১০০ টাকা
বিক্রয়মূল্য = ১০০ + ক টাকা

লাভ হয় = (১০০ + ক - ১০০) = ক টাকা
বিক্রয়মূল্য দ্বিগুণ= ২(১০০ + ক)

প্রশ্নমতে
৩ক = ২(১০০ + ক) - ১০০
বা, ৩ক = ২০০ + ২ক - ১০০
বা, ৩ক - ২ক = ১০০
ক = ১০০

বিক্রয়মূল্য = ১০০ + ১০০ = ২০০ টাকা
লাভ = ১০০ টাকা
শতকরা লাভ = (১০০/১০০) × ১০০ = ১০০%

৬৯৪.

The three sides of a triangle are x + 1, 2x - 1 and 3x + 1 respectively and the perimeter is 25cm. The length of the smallest side is-

ক
5cm
খ
3cm
গ
4cm
ঘ
7cm

সঠিক উত্তর: ক

5cm

উত্তর

সঠিক উত্তর: ক

5cm

ক

ব্যাখ্যা

Question: The three sides of a triangle are x + 1, 2x - 1 and 3x + 1 respectively and the perimeter is 25cm. The length of the smallest side is-

Solution:
এখানে
x + 1 + 2x - 1 + 3x + 1 = 25
6x + 1 = 25
6x = 25 - 1
6x = 24
x = 4

১ম বাহু = x + 1 = 4 + 1 = 5
২য় বাহু = 2x - 1 = 2 × 4 - 1 = 7
৩য় বাহু = 3x + 1 = 3 × 4 + 1 = 13

৬৯৫.

A company makes a profit of 5% on its first TK. 1000 of sales each day, and 4% on all sales in excess of 1000 for that day. What would be the profit of the company in a day when sales are TK. 6000?

ক
Tk. 300
খ
Tk. 250
গ
Tk. 275
ঘ
Tk. 325

সঠিক উত্তর: খ

Tk. 250

উত্তর

সঠিক উত্তর: খ

Tk. 250

খ

ব্যাখ্যা

Question: A company makes a profit of 5% on its first TK. 1000 of sales each day, and 4% on all sales in excess of 1000 for that day. What would be the profit of the company in a day when sales are TK. 6000?

Solution:
Profit for first TK. 1000 = 1000 × (5/100) = TK. 50

Total sales excess of Tk. 1000 = (6000 - 1000) = Tk. 5000

∴ Profit for excess of Tk. 1000 = 5000 × (4/100) = Tk. 200

∴ Total Profit (50+ 200) = Tk. 250

৬৯৬.

A is three times as old as B. C was twice-as old as A four years ago. In four years' time. A will be 31. What are the present ages of B and C ?

ক
ক) 45,10
খ
খ) 9,50
গ
গ) 10,50
ঘ
ঘ) 50,15

সঠিক উত্তর: খ

খ) 9,50

উত্তর

সঠিক উত্তর: খ

খ) 9,50

খ

ব্যাখ্যা

ধরি
B এর বর্তমান বয়স = x বছর
A এর বর্তমান বয়স 3x বছর
A এর বর্তমান বয়স হলো 31 - 4 = 27 বছর

প্রশ্নমতে,
3x = 27
x = 9 বছর
B এর বর্তমান বয়স = 9 বছর
C এর বর্তমান বয়স হলো = 27 × 2 - 4 = 54 - 4 = 50 বছর

৬৯৭.

Payel bought an item at 20% discount on its original price. he sold it with 40% increase on the price he bought it. The new sale price is by what percent more than the original price?

ক
10%
খ
12%
গ
14%
ঘ
18%

সঠিক উত্তর: খ

12%

উত্তর

সঠিক উত্তর: খ

12%

খ

ব্যাখ্যা

Question: Payel bought an item at 20% discount on its original price. he sold it with 40% increase on the price he bought it. The new sale price is by what percent more than the original price?

Solution:
Let, the original price be 100
After 20% discount, the price is = 100 - 20
= 80 taka

Selling price = 80 + 80 × 0.4
= 80 + 32
= 112 taka

%increase = (112 - 100)%
= 12%

৬৯৮.

By selling an article at 1/2 of the marked price, there is a 25% loss. Find the profit percent if the article is sold at the marked price.

ক
58%
খ
70%
গ
80%
ঘ
50%
ঙ
40%

সঠিক উত্তর: ঘ

50%

উত্তর

সঠিক উত্তর: ঘ

50%

ঘ

ব্যাখ্যা

Question: By selling an article at 1/2 of the marked price, there is a 25% loss. Find the profit percent if the article is sold at the marked price.

Solution:
Let the marked price be x TK

Then,
Selling price = (1/2)x , loss = 25%

∴ Cost price = TK. (100/75) × (x/2)
= TK. (2x/3)

If an article is sold at marked price.
Then profit = TK. {x - (2x/3)}
= TK. (x/3)

∴ profit%
= {(x/3) × (3/2x) × 100}%
= 50%

৬৯৯.

What is the angle between the hour and minute hand of a clock when it is 8 : 20 am?

ক
105°
খ
110°
গ
120°
ঘ
130°

সঠিক উত্তর: ঘ

130°

উত্তর

সঠিক উত্তর: ঘ

130°

ঘ

ব্যাখ্যা

Question: What is the angle between the hour and minute hand of a clock when it is 8 : 20 am?

Solution:
8টা 20 মিনিট = 8 + (20/60) ঘণ্টা
= 8 + (1/3) = 25/3 ঘণ্টা

আমরা জানি,
ঘণ্টার কাঁটা 12 ঘণ্টায় 360° ঘোরে।
∴ 1 ঘণ্টায় ঘোরে = 360°/12 = 30°
∴ 25/3 ঘণ্টায় ঘোরে = (30° × 25)/3 = 250°

আবার,
মিনিটের কাঁটা 60 মিনিটে 360° ঘোরে।
∴ 1 মিনিটে ঘোরে = 360°/60 = 6°
∴ 20 মিনিটে ঘোরে = 20 × 6° = 120°

∴ ঘড়ির কাঁটা দুটির মধ্যবর্তী কোণ = |250° - 120°| = 130°

৭০০.

A sum of money becomes 7/6 of itself in 3 years at a certain rate of simple interest. What is the rate of interest per annum?

ক
(50/9)%
খ
(60/9)%
গ
(70/9)%
ঘ
(55/9)%
ঙ
(51/9)%

সঠিক উত্তর: ক

(50/9)%

উত্তর

সঠিক উত্তর: ক

(50/9)%

ক

ব্যাখ্যা

Let principal = 6y
Hence, Amount = 6y × 7/6 = 7y
simple interest = 7y - 6y = y
If time 3 years, then simple interest,
y = 6yr × 3/100 [r = rate of simple interest]
or, r = 50/9