Bank Math

PrepBank · পাতা ১৩৩ / ১৬১ · ১৩,২০১–১৩,৩০০ / ১৬,১২৪

১৩,২০১.

The base of a right-angled triangle is 7 m and hypotenuse is 25 m. What is its area?

ক
65 sq. meters
খ
72 sq. meters
গ
84 sq. meters
ঘ
108 sq. meters
ঙ
None

ব্যাখ্যা

Question: The base of a right-angled triangle is 7 m and hypotenuse is 25 m. What is its area?

Solution:
Given,
Base = 7 m, Hypotenuse = 25 m

By Pythagoras' Theorem
Height² = Hypotenuse² - Base²
⇒ Height²= 25² - 7²
⇒ Height²= 625 - 49
⇒ Height² = 576
∴ Height = 24

We know,
Area = (1/2) × base × height
= (1/2) × 7 × 24
= 84 sq. meters

১৩,২০২.

If the volume of a sphere is divided by its surface area, the result is 27 cm. the radius of the sphere is-

ক
81 cm
খ
64 cm
গ
54 cm
ঘ
49 cm

ব্যাখ্যা

Question: If the volume of a sphere is divided by its surface area, the result is 27 cm. the radius of the sphere is-

Solution:
Let,
the radius of the sphere is r cm
∴ the volume of a sphere = (4/3)πr³
∴ the surface area of a sphere = 4πr²

ATQ,
{(4/3)πr³}/(4πr²) = 27
⇒ r/3 = 27
∴ r = 81

∴ the radius of the sphere is 81 cm

১৩,২০৩.

What is the H.C.F. of 20/48, 28/72, and 36/90?

ক
2/75
খ
1/120
গ
3/75
ঘ
1/180

ব্যাখ্যা

Question: What is the H.C.F. of 20/48, 28/72, and 36/90?

Solution:
We know, H.C.F. of fractions = (H.C.F. of numerators)/(L.C.M of denominators)

H.C.F of numerators: H.C.F. of 20, 28 and 36 = 4
& L.C.M of denominators: L.C.M. of 48, 72 and 90 = 720

∴ Required H.C.F. = 4/720 = 1/180

১৩,২০৪.

If the difference between the circumference and diameter of a circle is 90 cm, then the diameter of the circle is-

ক
48 cm
খ
42 cm
গ
56 cm
ঘ
36 cm

ব্যাখ্যা

Question: If the difference between the circumference and diameter of a circle is 90 cm, then the diameter of the circle is-

Solution:
Let,
Radius of the circle = r
Diameter of the circle = 2r
Circumference of the circle = 2πr

According to the question,
2πr - 2r = 90
⇒ 2r(π - 1) = 90
⇒ r(π - 1) = 45
⇒ r = 45/(π - 1)
⇒ r = 45/(22/7 - 1) ; [π = 22/7]
⇒ r = 45/{(22 - 7)/7}
⇒ r = 45/(15/7)
⇒ r = 45 × (7/15)
∴ r = 21 cm

∴ Diameter of the circle = 2r = 2 × 21 = 42 cm

So the diameter of the circle is 42 cm.

১৩,২০৫.

A shopkeeper has sufficient money to buy 50 books. On reduction in the price of each book by Tk. 8, he could buy 10 books more. How much money does he has?

ক
Tk. 2200
খ
Tk. 2800
গ
Tk. 2400
ঘ
Tk. 3000

ব্যাখ্যা

Question: A shopkeeper has sufficient money to buy 50 books. On reduction in the price of each book by Tk. 8, he could buy 10 books more. How much money does he has?

Solution:
১টি বইয়ে দাম কমে ৮ টাকা
∴ ৫০টি বইয়ে দাম কমে (৫০ × ৮) টাকা
= ৪০০ টাকা

সে মোট বই কিনে (৫০ + ১০) টি
= ৬০টি

১০টি বইয়ের দাম ৪০০ টাকা
∴ ৬০টি বইয়ের দাম (৪০০ × ৬০)/১০ টাকা
= ২৪০০ টাকা

∴ তার কাছে ২৪০০ টাকা আছে।

১৩,২০৬.

There are 120 chickens and goats on a farm. There are 70 more chickens than goats. How many goats are there?

ক
25
খ
30
গ
45
ঘ
50

ব্যাখ্যা

Question: There are 120 chickens and goats on a farm. There are 70 more chickens than goats. How many goats are there?

Solution:
Let,
The number of goats be x
∴ The number of chickens x + 70

ATQ,
x + x + 70 = 120
⇒ 2x = 50
∴ x = 25

১৩,২০৭.

The operation @ is defined for all integers x and y as x @ y=xy-y. If x and y are positive integers, which of the following cannot be Zero?

ক
ক) x@y
খ
খ) y@x
গ
গ) (x-y)@y
ঘ
ঘ) (x+1)@y
ঙ
ঙ) None

ব্যাখ্যা

x@y = xy-y is zero, if x = y = 1
y@x = yx-x is zero if x = y = 1
(x-1)@y = xy-y-y = xy-2y is zero, if x = 2 and y = 1
(x+1)@y = xy+y-y = xy cannot be zero if both x and y are positive integers.

১৩,২০৮.

The ratio of copper to gold in an ornament weighing 16 grams is 1 : 3. How much more gold must be added to it to make the ratio 1 : 4?

ক
ক) 3 grams
খ
খ) 4 grams
গ
গ) 5 grams
ঘ
ঘ) 8 grams

ব্যাখ্যা

Question: The ratio of copper to gold in an ornament weighing 16 grams is 1 : 3. How much more gold must be added to it to make the ratio 1 : 4?

Solution:
The ratio of copper to gold in an ornament weighing 16 grams is 3 : 1
gold weight = (16 × 3)/4 = 12 grams
copper weight = 16 - 12 = 4 grams

let, x grams gold to be added

So,
4/(12 + x) = 1/4
⇒ 12 + x = 16
∴ x = 4

So, 4 grams gold to be added

১৩,২০৯.

A trader buys some goods for Tk. 150. If the overhead expenses be 12% of cost price, then at what price should it be sold to earn 10%?

ক
ক) 184.80
খ
খ) 185.80
গ
গ) 187.80
ঘ
ঘ) 188.80

ব্যাখ্যা

Total C.P. = Cost + Overhead Expenses
= Tk. (150 + 12% of 150)
= Tk. (150 + 18)
= Tk. 168
∴S.P.=Tk.(110/100×168)=Tk. 184.80

১৩,২১০.

A reduction of 21% in the price of pulse enables a person to buy 10.5 kg more for TK. 100. What is the reduced rate per kg of pulse?

ক
ΤΚ. 2.50
খ
ΤΚ. 2.25
গ
TK. 2
ঘ
Tk. 2.30
ঙ
None of these

ব্যাখ্যা

Question: A reduction of 21% in the price of pulse enables a person to buy 10.5 kg more for TK. 100. How much is reduced per kg of pulse?

Solution:
21% মূল্য হ্রাসে,
পূর্বের 100 টাকার দ্রব্য বর্তমানে পাওয়া যায় 79 টাকায়

100 টাকায় এখন পূর্বাপেক্ষা 10.5 কেজি দ্রব্য বেশি পাওয়া যায়।

10.5 কেজির বর্তমান মূল্য 21 টাকা
1 কেজির বর্তমান মূল্য 21/10.5 টাকা
= 2 টাকা

১৩,২১১.

∠P and ∠Q are complementary to each other. If ∠P = 3x + 9° and ∠Q = 2x - 4°, find the value of ∠Q.

ক
42°
খ
30°
গ
62°
ঘ
70°

ব্যাখ্যা

Question: ∠P and ∠Q are complementary to each other. If ∠P = 3x + 9° and ∠Q = 2x - 4°, find the value of ∠Q.

Solution:
Here,
∠P = 3x + 9° and ∠Q = 2x - 4°

For complementary angles,
∠P + ∠Q = 90°
⇒ (3x + 9°) + (2x - 4°) = 90°
⇒ 3x + 9° + 2x - 4° = 90°
⇒ 5x + 5° = 90°
⇒ 5x = 85°
∴ x = 17°

So, ∠Q = 2x - 4°
= (2 × 17) - 4°
= 30°

১৩,২১২.

At a club, 70% of the members are women and 60% of the members are married. If ⅔ of the men are single, what fraction of the women are married?

ক
ক) 5/7
খ
খ) 7/10
গ
গ) 1/3
ঘ
ঘ) 7/30

ব্যাখ্যা

ধরি,
মোট member = 100
∴ মহিলা member = 70 এবং পুরুষ member = 30 জন।
30 জন পুরুষের মধ্যে অবিবাহিত = 30 × 2/3 = 20 জন।
সুতরাং বিবাহিত = 10 জন।
মোট বিবাহিত = 60 জন।
∴ মহিলাদের মধ্যে বিবাহিত = 60 - 10 = 50 জন।
∴ নির্ণেয় fraction = 50/70
= 5/7

১৩,২১৩.

The cost price of 8 apples equals the selling price of 7 apples. The profit or loss percent in the transaction is?

ক
29.29%
খ
24.29%
গ
15.29%
ঘ
14.29%

ব্যাখ্যা

Question: The cost price of 8 apples equals the selling price of 7 apples. The profit or loss percent in the transaction is?

Solution:
দেওয়া আছে,
8টি আপেলের ক্রয়মূল্য = 7টি আপেলের বিক্রয়মূল্য

ধরি, 1টি আপেলের ক্রয়মূল্য = x টাকা
তাহলে,
8টি আপেলের ক্রয়মূল্য = 8x টাকা
7টি আপেলের বিক্রয়মূল্য = 8x টাকা

∴ 1টি আপেলের বিক্রয়মূল্য = 8x / 7 টাকা

∴ লাভ = বিক্রয়মূল্য − ক্রয়মূল্য
= (8x/7) − x
= (8x − 7x)/7
= x/7

∴ x টাকায় লাভ হয় x/7 টাকা
1 টাকায় লাভ হয় = (x/7)/x = 1/7
100 টাকায় লাভ হয় = (1/7) × 100 ≈ 14.29 টাকা

∴ লাভ হয় ≈ 14.29%

১৩,২১৪.

ক
1/2
খ
3/4
গ
7/3
ঘ
4/3

ব্যাখ্যা

Question:

Solution:

১৩,২১৫.

A sum of money at simple interest amounts to TK. 5600 in 2 years and Tk. 6500 in 5 years at the rate of -

ক
ক) 4%
খ
খ) 3%
গ
গ) 5%
ঘ
ঘ) 6%

ব্যাখ্যা

S.I. in 3 years = 6500 - 5600 = 900
Interest every year = 900/3 = 300
So, principal = 5600 - 2 × 300 = 5000

We know, I = pnr
Or, Interest rate, r = I/pn
= 600/(5000×2) × 100
= 6%

১৩,২১৬.

The difference between two positive numbers is 15 and the ratio between them is 7 : 4. Find the sum of the two numbers.

ক
ক) 40
খ
খ) 50
গ
গ) 55
ঘ
ঘ) 60

ব্যাখ্যা

Question: The difference between two positive numbers is 15 and the ratio between them is 7 : 4. Find the sum of the two numbers.

Solution:
Let the two positive numbers be 7x and 4x respectively

According to the question,
7x - 4x = 15
⇒ 3x = 15
⇒ x = 5

Then numbers are 7 × 5 = 35 and 4 × 5 = 20
So, sum of numbers = 35 + 20 = 55

১৩,২১৭.

A trader sells his goods at a discount 20%. He still makes a profit of 25%. If he sells the goods at the marked price only, What will be his profit?

ক
56.25%
খ
25.56%
গ
50.25%
ঘ
54.25%
ঙ
55.50%

ব্যাখ্যা

Let the cost price of goods be 100 taka.
At 25% profit,
if cost price of goods is 100 taka, selling price of goods is (100 + 25) taka or 125 taka.

At a discount 20%,
80% of goods' price = 125 taka.
100% of goods' price shall be (125×100/80) taka or 156.25 taka.
profit percentage would be (156.25 - 100) taka or 56.25 taka.

১৩,২১৮.

The expression x + 3 is______

ক
ক) one variable
খ
খ) two variables
গ
গ) no variable
ঘ
ঘ) none of these

ব্যাখ্যা

As ‘x’ from x + 3 is the lone variable here, x + 3 is one variable expression.

১৩,২১৯.

Present ages of Rakib and Tanvir are in the ratio of 7 : 5 respectively. Two years hence, the ratio of their ages will become 15 : 11 respectively. What is Rakib's present age in years?

ক
20 years
খ
27 years
গ
28 years
ঘ
32 years

ব্যাখ্যা

Question: Present ages of Rakib and Tanvir are in the ratio of 7 : 5 respectively. Two years hence, the ratio of their ages will become 15 : 11 respectively. What is Rakib's present age in years?

Solution:
Let the present ages of Rakib and Tanvir be 7x and 5x respectively.
ATQ,
(7x + 2)/(5x + 2) = 15/11
⇒ 77x + 22 = 75x + 30
⇒ 2x = 8
⇒ x = 4

So, Rakib's present age = 7 × 4 = 28 years

১৩,২২০.

Find the speed of the stream, if a boat takes 8 hours to travel a distance of 192 km upstream, and the speed of the boat in still water is 32 km/h.

ক
8 km/h
খ
6 km/h
গ
4 km/h
ঘ
10 km/h

ব্যাখ্যা

Question: Find the speed of the stream, if a boat takes 8 hours to travel a distance of 192 km upstream, and the speed of the boat in still water is 32 km/h.

Solution:
স্থির পানিতে নৌকার বেগ 32 km/h
ধরি,
স্রোতের বেগ x km/h

স্রোতের প্রতিকূলে কার্যকরি বেগ = 192/8 = 24 km/h

∴ 32 - x = 24
⇒ x = 32 - 24
∴ x = 8

∴ স্রোতের বেগ 8 km/h

১৩,২২১.

If a + b : b + c : c + a = 6 : 7 : 8 and a + b + c = 14, Then the value of c is-

ক
ক) 8
খ
খ) 6
গ
গ) 4
ঘ
ঘ) 2

ব্যাখ্যা

Question: If a + b : b + c : c + a = 6 : 7 : 8 and a + b + c = 14, Then the value of c is-

Solution:
a + b : b + c : c + a = 6 : 7 : 8

Let
a + b = 6k
b + c = 7k
c + a = 8k

Now,
a + b + b + c + c + a = 6k + 7k + 8k
⇒ 2a + 2b + 2c = 6k + 7k + 8k
⇒ 2(a + b + c) = 21k
⇒ 2 × 14 = 21k
⇒ 21k = 28
⇒ k = 28/21
⇒ k =4/3

Again,
a + b = 6k
= 6 × (4/3)
= 8

∴ c = ( a + b + c) - (a + b)
= 14 - 8
= 6

১৩,২২২.

If tanA = 3/4, then sec A =

ক
5/4
খ
4/5
গ
3/5
ঘ
5/3

ব্যাখ্যা

Question: If tanA = 3/4, then sec A =

Solution:
দেওয়া আছে,
tanA = 3/4
∴ অতিভুজ = √(3² + 4²) = √(9 + 16) = 5

আমরা জানি,
secA = অতিভুজ/ভূমি
= 5/4

১৩,২২৩.

If x³ < x² < x, then the value of x could be

ক
ক) 0
খ
খ) 1
গ
গ) 1/3
ঘ
ঘ) Square root of 3
ঙ
ঙ) 2

ব্যাখ্যা

Question: If x³< x²< x, then the value of x could be

Solution:
For, x = 0
x³= x² = x = 0

For, x = 1
x³ = x² = x = 1

For, x = 1/3
x³ = 1/27 and x² = 1/9
∴ x > x²> x³

For, x = √3
x³ = 3√3 and x² = 3
∴ x < x² < x³

For, x = 2
x³ = 8 and x² = 4
∴ x < x² < x³

১৩,২২৪.

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

ক
19
খ
15
গ
30
ঘ
38

ব্যাখ্যা

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

Solution:
There are 6 candidates and a voter has to vote for any two of them.
So, the required number of ways is = ⁶C₂
= 15

১৩,২২৫.

The price of an article was first increased by 10% and then again by 20%. If the last increased price was Tk. 33, the original price was:

ক
ক) Tk. 20
খ
খ) Tk. 25.50
গ
গ) Tk. 25
ঘ
ঘ) Tk. 22

ব্যাখ্যা

Question: The price of an article was first increased by 10% and then again by 20%. If the last increased price was Tk. 33, the original price was:

Solution:
Let the original price = 100x
Article price 1st increased by 10%.
Price of article become = 100x + 100x of 10%
= 100x + (100x × 10)/100
= 110x

And then the article price increased by 20%
Price of article become = 110x + 110x of 20%
= 110x + (110x × 20)/100
= 132x

ATQ,
132x = 33
⇒ x = 33/132
∴ x = 1/4

Therefore Original price of Article is
= 100 × 1/4
= Tk. 25

১৩,২২৬.

If x = 3 + 2√2 then, what is the value of x^-1?

ক
3 + 2√2
খ
√3 - √2
গ
3 - 2√2
ঘ
√3 - 2√2

ব্যাখ্যা

Question: If x = 3 + 2√2 then, what is the value of x^-1?

Solution:
Given,
x = 3 + 2√2
∴ 1/x = 1/(3 + 2√2)
⇒ x^-1 = (3 - 2√2)/(3 + 2√2)(3 - 2√2)
= (3 - 2√2)/(3)² - (2√2)²
= (3 - 2√2)/(9 - 8)
∴ x^-1 = (3 - 2√2)

১৩,২২৭.

The ratio between the perimeter and the breadth of a rectangle is 5 : 1. If the area of the rectangle is 216 sq. cm, what is the length of the rectangle?

ক
16 cm
খ
18 cm
গ
24 cm
ঘ
None of these

ব্যাখ্যা

Question: The ratio between the perimeter and the breadth of a rectangle is 5 : 1. If the area of the rectangle is 216 sq. cm, what is the length of the rectangle?

Solution:
et the breadth of the rectangle be B cm.
Perimeter of a rectangle = 2(Length + Breadth) = 2(L + B)
Given the Ratio of perimeter to breadth = 5 : 1
Now,
2(L + B)/B = 5/1
⇒ 2L + 2B = 5B
⇒ 3B = 2L
∴ B = (2/3)L

Area of the rectangle = L . B = 216 sq. cm.
⇒ L . (2/3)l = 216
⇒ L² = (216 × 3)/2
⇒ L² = 324 = 18²
∴ l = 18 cm

So the length of the rectangle is 18 cm.

১৩,২২৮.

The greatest number which when divides 989 and 1327 leaves the remainder 5 and 7 respectively =?

ক
16
খ
32
গ
8
ঘ
24

ব্যাখ্যা

Question: The greatest number which when divides 989 and 1327 leaves the remainder 5 and 7 respectively =?

Solution:
Let us subtract the remainder from the number.
989 − 5 = 984
1327 − 7 = 1320

For the greatest number, let us take the HCF of the numbers.
HCF = (984, 1320) = 24

∴ The greatest number is 24

১৩,২২৯.

In a family, the father took 1/4 of the cake and had 3 times as much as each of the other members had. The total number of family member is-

ক
9
খ
10
গ
11
ঘ
12

ব্যাখ্যা

Question: In a family, the father took 1/4 of the cake and had 3 times as much as each of the other members had. The total number of family member is-

Solution:
Let the total number of the family member excluding father be 'x' and the amount of cake be 100 units

Share of cake taken by father = 100 × 1/4 = 25 units
Remaining share of cake taken by family member = (100 - 25)/x
Now,
25 = 3 × 75/x
⇒ 25x = 3 × 75
⇒ x =(3 × 75)/25
⇒ x = 9

∴ Total number of the family members including father = 9 + 1 = 10

১৩,২৩০.

The H.C.F. of two numbers is 23 and the other two factors of their L.C.M. are 13 and 14. The larger of the two numbers is:

ক
ক) 276
খ
খ) 299
গ
গ) 322
ঘ
ঘ) 345

ব্যাখ্যা

Clearly, the numbers are (23 x 13) and (23 x 14).
Larger number = (23 x 14) = 322.

১৩,২৩১.

The simple interest on a certain sum at 10% per annum for 2 years is Tk. 2,000. Find the compound interest on the same sum at the same rate for 2 years.

ক
Tk. 2,050
খ
Tk. 2,100
গ
Tk. 2,150
ঘ
Tk. 2,200

ব্যাখ্যা

Question: The simple interest on a certain sum at 10% per annum for 2 years is Tk. 2,000. Find the compound interest on the same sum at the same rate for 2 years.

Solution:
We know,
I = Pnr

Given,
I = 2,000 Tk.
r = 10%
n = 2 years

∴ P = I/nr
= (2,000 × 100)/(2 × 10)
= 10000

Now,
Compound Principal = 10,000 × (1 + 10/100)²
= 10,000 × (110/100)²
= 10,000 × (11/10)²
= 10,000 × (121/100)
= 12,100

So, Compound interest = 12,100 - 10,000
= 2,100 Tk.

১৩,২৩২.

, then what is the value of m?

ক
1
খ
0
গ
- 1
ঘ
1/2
ঙ
1/4

ব্যাখ্যা

Question: , then what is the value of m?

Solution:

We have 4^m > 1
Now, if m = -1, then 4^m = 4^-1 = 1/4 = 0.25 < 1, so incorrect.
if m = 1, then 4^m = 4¹ = 4 > 1, correct.

∴ m = 1

১৩,২৩৩.

A square carpet with an area of 169 cm² must have 2 cm cut off one of its edges in order to be a perfect fit for a rectangular room. What is the area (in cm) of this rectangular room?

ক
ক) 117
খ
খ) 143
গ
গ) 145
ঘ
ঘ) 165

ব্যাখ্যা

Let, the carpet's length = x
Given, x² = 169 cm²
∴ carpet's length = √169 = 13 cm
Width of the room = (13 - 2) cm = 11 cm
So, area of the rectangular room = (13 × 11) = 143 cm²

১৩,২৩৪.

A tank is filled in 5 hours by three pipes A, B and C. The pipe C is twice as fast as B and B is twice as fast as A. How much time will pipe A alone take to fill the tank?

ক
30 hours
খ
32 hours
গ
35 hours
ঘ
40 hours

ব্যাখ্যা

প্রশ্ন: A tank is filled in 5 hours by three pipes A, B and C. The pipe C is twice as fast as B and B is twice as fast as A. How much time will pipe A alone take to fill the tank?

সমাধান:
Suppose,
pipe A alone takes x hours to fill the tank.
Then, pipes B and C will take x/2 and x/4 hours respectively to fill the tank.

Now,
1/x + 2/x + 4/x = 1/5
⇒ 7/x = 1/5
∴ x = 35

∴ pipe A alone takes 35 hours to fill the tank.

১৩,২৩৫.

1 বিলিয়ন = কত কোটি?

ক
10
খ
100
গ
1000
ঘ
10000

ব্যাখ্যা

প্রশ্ন: 1 বিলিয়ন = কত কোটি?

সমাধান:
• 1 বিলিয়ন = 100 কোটি

• 1 মিলিয়ন = 10 লক্ষ
• 10 মিলিয়ন = 1 কোটি
• 1000 বিলিয়ন = 1 ট্রিলিয়ন
• 100000 কোটি বা, 1 লক্ষ কোটি = 1 ট্রিলিয়ন

১৩,২৩৬.

In a mixture, the ratio of milk and water is 6 : 5. When 22 liter mixture is replaced by water, the ratio become 9 : 13. Find the quantity of water after replacement.

ক
ক) 40 liters
খ
খ) 48 liters
গ
গ) 57 liters
ঘ
ঘ) 52 liters

ব্যাখ্যা

Question: In a mixture, the ratio of milk and water is 6 : 5. When 22 liter mixture is replaced by water, the ratio become 9 : 13. Find the quantity of water after replacement.

Solution:
Let milk = 6X, water = 5X
so,
total mixture = 6X + 5X = 11X

When 22 liter of mixture is replaced with water, then remaining milk is = 6X - (22 × 6X/11X) = 6X - 12

remaining water = 5X - (22 × 5X/11X) + 22 = 5X + 12

ATQ,
6X - 12 : 5X + 12 = 9 : 13
13(6X - 12) = 9(5X + 12)
78X - 156 = 45X + 108
78X - 45X = 156 + 108
33X = 164
X = 8

∴ water after replacement = 5X + 12 = 40 + 12 = 52 liters

১৩,২৩৭.

2 men and 7 boys can do a piece of work in 14 days; 3 men and 8 boys can do the same in 11 days. Then, 8 men and 6 boys can do three times the amount of this work in-

ক
18 days
খ
21 days
গ
24 days
ঘ
25 days
ঙ
30 days

ব্যাখ্যা

Question: 2 men and 7 boys can do a piece of work in 14 days; 3 men and 8 boys can do the same in 11 days. Then, 8 men and 6 boys can do three times the amount of this work in-

Solution:
(2 × 14) men + (7 × 14) boys = (3 × 11) men + (8 × 11) boys
⇒ 28 men + 98 boys = 33 men + 88 boys
⇒ 5 men= 10 boys
∴ 1man= 2 boys

Therefore,
(2 men + 7 boys) = (2 × 2 +7) boys = 11 boys
(8 men + 6 boys) = (8 × 2 + 6) boys = 22 boys.

Let the required number of days be x.
More boys , Less days (Indirect proportion)
More work , More days (Direct proportion)

Therefore,
(22 × 1 × x) = (11 × 3 × 14)
⇒ x = 21

Hence, the required number of days = 21

১৩,২৩৮.

P, Q, R enter into partnership. P invests 3 times as much as Q invests and Q invests two third of what R invests. At the end of the year, the profit earned is TK. 6600. What is the share of R?

ক
Tk. 1200
খ
Tk. 1400
গ
Tk. 1800
ঘ
Tk. 2000

ব্যাখ্যা

Question: P, Q, R enter into partnership. P invests 3 times as much as Q invests and Q invests two third of what R invests. At the end of the year, the profit earned is TK. 6600. What is the share of R?

Solution:
P invests 3 times as much as Q invests and Q invests two third of what R invests.

Let,
R invests = x
Q invests = 2x/3
P invests = 3 × (2x/3) = 2x

The ratio of invests or profit 2x: 2x/3: x
= 6x : 2x : 3x
= 6 : 2 : 3

∴ The share of R = (3/11) × 6600
= 1800

১৩,২৩৯.

A box has coins worth 50, 25, and 10 paisa in a 1 : 2 : 5 ratio. If all the coins add up to 36 Taka, how much of that is from the 50 paisa coins alone?

ক
Tk. 11
খ
Tk. 12
গ
Tk. 33
ঘ
Tk. 24

ব্যাখ্যা

Question: A box has coins worth 50, 25, and 10 paisa in a 1 : 2 : 5 ratio. If all the coins add up to 36 Taka, how much of that is from the 50 paisa coins alone?

Solution:

১৩,২৪০.

If the ratios A : B = 1 : 2, B : C = 4 : 3, hold true, and their total sum is 630, determine C.

ক
270
খ
140
গ
210
ঘ
70

ব্যাখ্যা

Question: If the ratios A : B = 1 : 2, B : C = 4 : 3, hold true, and their total sum is 630, determine C.

Solution:
Given,
A : B = 1 : 2 = 2 : 4
B : C = 4 : 3

∴ A : B : C = 2 : 4 : 3

∴ Value of C = 630 × 3/9
= 210

১৩,২৪১.

Three lights blink at intervals of 18 sec, 24 sec, and 32 sec. If they blink together now, how many times will they blink together in 6 hours?

ক
70
খ
76
গ
86
ঘ
105

ব্যাখ্যা

Question: Three lights blink at intervals of 18 sec, 24 sec, and 32 sec. If they blink together now, how many times will they blink together in 6 hours?

Solution:
The lights will blink together at LCM of their intervals.
Intervals: 18, 24, 32 seconds

Factorize:
18 = 2 × 3²
24 = 2³ × 3
32 = 2⁵

LCM = product of highest powers of all primes:
LCM = 2⁵× 3²=3² × 9 = 288 seconds
Convert 6 hours to seconds:
6 hours = 6 × 60 × 60 = 21600 seconds

Number of times they blink together:
Times = 21600/288 + 1 (+1 because they blink together now, at time 0)
= 75 + 1
= 76

Note: If you only count after the first blink at time 0, it is 75 times; including the initial moment, it is 76 times.

১৩,২৪২.

A cyclist moving on a circular track of a radius of 100 meters completes one revolution in 2 minutes. What is the average speed of a cyclist (approx.)?

ক
310 meters/minute
খ
328 meters/minute
গ
336 meters/minute
ঘ
320 meters/minute
ঙ
None of the above

ব্যাখ্যা

Question: A cyclist moving on a circular track of a radius of 100 meters completes one revolution in 2 minutes. What is the average speed of a cyclist (approx.)?

Solution:
Distance covered in one complete revolution
= 2πr
= 2 × 100 × (22/7)
= 628.571 meters
≈ 628 meters

Hence,
Average speed
= 628/2
= 314 meters/minute

১৩,২৪৩.

When 4 is added to 1/2 of a number, the result is 14. What is the number?

ক
10
খ
16
গ
20
ঘ
24

ব্যাখ্যা

Question: When 4 is added to 1/2 of a number, the result is 14. What is the number?

Solution:
let the number be = x

ATQ,
(x/2) + 4 = 14
⇒ x/2 = 10
∴ x = 20

১৩,২৪৪.

A vessel is full of mixture of spirit and water in which there is 20% of spirit. 5 litres are drawn off and the vessel is filled up with water. If the spirit is now 12%, find the total quantity in the vessel (in ltrs).

ক
72
খ
12.5
গ
60
ঘ
7.2

ব্যাখ্যা

Question: A vessel is full of mixture of spirit and water in which there is 20% of spirit. 5 litres are drawn off and the vessel is filled up with water. If the spirit is now 12%, find the total quantity in the vessel (in ltrs).

Solution:
Let the total mixture be x liters.
It contains 20% spirit i.e. 0.2 of the total mixture.
When 5 liters are drawn off, quantity of spirit removed = 0.2(5) = 1 liters.
Remaining spirit = 0.2x - 1.
Also remaining spirit = 0.12x.

∴ 0.2x - 1 = 0.12x
⇒ 0.08x = 1
⇒ x = 1/0.08
⇒ x = 12.5

১৩,২৪৫.

A salesperson earns a 7.5% commission on sales. If he earned Tk. 1125 in commission, what was the total value of the goods he sold?

ক
Tk. 7500
খ
Tk. 18000
গ
Tk. 13500
ঘ
Tk. 15000

ব্যাখ্যা

Question: A salesperson earns a 7.5% commission on sales. If he earned Tk. 1125 in commission, what was the total value of the goods he sold?

Solution:
Given that,
Commission rate = 7.5%
Commission earned = Tk. 1125

Let the total sales amount be x.
Then,
(7.5/100) × x = 1125
⇒ x = (1125 × 100)/7.5
⇒ x = 15000

So the total value of the goods he sold was Tk. 15000

১৩,২৪৬.

A boat takes 12 hours to cover 64 km downstream and 8 hours to cover 32 km upstream. Then the speed of the boat in still water is

ক
ক) 14/3 km/hr
খ
খ) 3/2 km/hr
গ
গ) 6/5 km/hr
ঘ
ঘ) 8/7 km/hr

ব্যাখ্যা

Distance covered downstream = 64 km
Time is taken in downstream = 12 hours.
Rate of downstream = distance/time = a = 64 km/12 hours
= 16/3 km/hr.
Distance covered in upstream = 32km
Time is taken upstream = 8 hours.
Rate of upstream = distance/time = b = 32km/8 hours
= 4 km/hr.
Speed in still water = (a + b)/2 = (1/2){(16/3) + 4} km/hr
= (1/2)(28/3)
= 14/3 km/hr.

১৩,২৪৭.

A person lends Tk. 1540 for five year and Tk. 1800 for four year. If he gets Tk. 1788 as interest on both amounts, what is the rate of interest?

ক
ক) 14.5%
খ
খ) 11%
গ
গ) 12%
ঘ
ঘ) 10.5%

ব্যাখ্যা

Let the interest rate be r%
We know that,
S.I = PTR/100
=> (1540 x 5 x r)/100 + (1800 x 4 x r)/100 = 1788
=> r = 178800/14900 = 12%

১৩,২৪৮.

A motorcycle at a speed of 60km/hr, it covers a certain distance in 3 hours and the train can cover the same distance in 1 and half hours then the speed of the train is -

ক
80km/hr
খ
120km/hr
গ
160km/hr
ঘ
50km/hr

ব্যাখ্যা

The Distance covered by the motorcycle with speed 60km/hr in 3 hours = (60 x 3)
= 180 km
Now,
Speed = Distance/Time
Since the train covers the same 180 km in 3/2 hours (we can write 1 and half hours as 3/2hours)
Then the speed of the train = 180/ (3/2)
= 180 × (2/3)
= 120 km/hr.
Hence, the train travelled at the speed of 120km/hr to cross 180km in 1 and half hour.

১৩,২৪৯.

A square garden is surrounded by a path of uniform width 2 meters. If the area of the path is 48 square meters, find the side length of the garden.

ক
8 meters
খ
6 meters
গ
3 meters
ঘ
4 meters

ব্যাখ্যা

Question: A square garden is surrounded by a path of uniform width 2 meters. If the area of the path is 48 square meters, find the side length of the garden.

Solution:
Let the side of the garden = x meters.
Then, the side of the garden including the path = x + (2 × 2)
= x + 4 meters.

Area of path = Area of garden with path - Area of garden
⇒ 48 = (x + 4)² - x²
⇒ 48 = x² + 8x + 16 - x²
⇒ 8x + 16 = 48
⇒ 8x = 48 - 16
⇒ x = 32/8
∴ x = 4 meters

∴ Therefore, side length of the garden is 4 meters.

১৩,২৫০.

45% of a number is 1350. What is 75% of that number?

ক
2120
খ
2250
গ
2340
ঘ
2410

ব্যাখ্যা

Question: 45% of a number is 1350. What is 75% of that number?

Solution:
Let the number be x
Then,
45% of x = 1350
45x/100 = 1350
x = (1350 × 100)/45
x = 3000

75% of 3000 = 3000 × (75/100)
= 2250

১৩,২৫১.

If a + b + c = 0, (a + b)(b + c)(c + a) =?

ক
ক) -3/abc
খ
খ) 1/abc
গ
গ) abc
ঘ
ঘ) - abc

ব্যাখ্যা

Given that,
a + b + c = 0
a + b = - c
b + c = - a
c + a = - b

Now
(a + b)(b + c)(c + a) =(- c)(- a)(- b) = -abc

১৩,২৫২.

If A = 30°, then what is the value of (1 - tan²A)/(1 + tan²A)?

ক
1
খ
1/2
গ
1/√3
ঘ
√3/2

ব্যাখ্যা

Question: If A = 30°, then what is the value of (1 - tan²A)/(1 + tan²A)?

Solution:
Here, A = 30°

Now,
(1 - tan²A)/(1 + tan²A)
= {1 - (tan30°)²}/{1 + (tan30°)²}
= {1 - (1/√3)²}/{1 + (1/√3)²}
= (1 - 1/3)/(1 + 1/3)
= (2/3)/(4/3)
= 1/2

১৩,২৫৩.

A watch is sold for Tk. 840, and the seller makes a profit of 20% on the cost price. What will be the new selling price if he reduces the profit to 5%?

ক
795 Tk.
খ
772 Tk.
গ
750 Tk.
ঘ
735 Tk.

ব্যাখ্যা

Question: A watch is sold for Tk. 840, and the seller makes a profit of 20% on the cost price. What will be the new selling price if he reduces the profit to 5%?

Solution:
At 20% profit,
Selling Price = 120% of Cost Price

∴ Cost Price= (100/120) × Selling Price
= (100/120) × 840
= 700 Tk.

Now, if the seller wants 5% profit,
∴ New Selling Price = 700 + (5% of 700)
= 700 + {(5/100) × 700}
= 700 + 35
= 735 Tk.

১৩,২৫৪.

3.003/2.002 = ?

ক
ক) 1.05
খ
খ) 1.50015
গ
গ) 1, 501
ঘ
ঘ) 1.5015
ঙ
ঙ) 1.5

ব্যাখ্যা

Question: 3.003 / 2.002 =?

Solution:
3.003 / 2.002
= 3003/2002
= (2002 + 1001)/2002
= (2002/2002) + (1001/2002)
= 1 + (1/2)
= 1 + 0.5
= 1.5

১৩,২৫৫.

If A : B = 3 : 2, B : C = 5 : 4, C : D = 3 : 7 then A : B : C : D =?

ক
ক) 45 : 30 : 56 : 24
খ
খ) 30 : 45 : 56 : 24
গ
গ) 45 : 30 : 24 : 56
ঘ
ঘ) 30 : 56 : 45 : 24

ব্যাখ্যা

According to the question,
A : B = (3 : 2) × 5 = 15 : 10
B : C = (5 : 4) × 2 = 10 : 8
A : B : C = 15 : 10 : 8

Again,
A : B : C = (15 : 10 : 8) × 3 = 45 : 30 : 24
C : D = (3 : 7) × 8 = 24 : 56

A : B : C : D = 45 : 30 : 24 : 56

∴ The value of A : B : C : D is 45 : 30 : 24 : 56.

১৩,২৫৬.

A picture is copied onto a sheet of paper 8.5 inches by 10 inches. A 1.5 inch margin is left all around. What area in square inches does the picture cover?

ক
ক) 76
খ
খ) 65
গ
গ) 59.5
ঘ
ঘ) 38.5

ব্যাখ্যা

Question: A picture is copied onto a sheet of paper 8.5 inches by 10 inches. A 1.5 inch margin is left all around. What area in square inches does the picture cover?

Solution:
কপি করা কাগজের আকার = 8.5 × 10 বর্গ ইঞ্চি
চারপাশে মার্জিন আছে = 1.5 ইঞ্চি

প্রস্থ = 8.5 - (1.5 × 2)
= 8.5 - 3
= 5.5

দৈর্ঘ্য = 10 - (1.5 × 2)
= 10 - 3
= 7

দখলকৃত স্থানের কাগজের এলাকা = (5.5 × 7)বর্গ ইঞ্চি
= 38.5 বর্গ ইঞ্চি

১৩,২৫৭.

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

ক
35 years
খ
40 years
গ
45 years
ঘ
50 years
ঙ
None

ব্যাখ্যা

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

Solution:
Let the mother's present age be x years.
Then, the person's present age = (2/5)x years

ATQ,
(2x/5) + 8 = (1/2)(x + 8)
⇒ (2x + 40)/8 = (x + 8)/2
⇒ 2(2x + 40) = 5(x + 8)
⇒ 4x + 80 = 5x + 40
∴ x = 40

১৩,২৫৮.

Two men, A and B, run a 4 km race on a circular course of 1/4 km. If their speeds are in the ratio of 5:4. How often does the winner pass the other?

ক
ক) Thrice
খ
খ) Four times
গ
গ) Once
ঘ
ঘ) Twice

ব্যাখ্যা

Number of rounds to be completed = 4 / 1/4 = 16 times
Since the ratio of speeds of A and B is 5 : 4
⇒ When A covers 5 rounds, B will complete 4 rounds and they will meet.
⇒ A will pass B after 5^th, 10^th & 15^th round.
∴ The winner passes the other thrice.

১৩,২৫৯.

If the area and circumference of a circle are numerically equal, then the diameter of the circle is:

ক
ক) 2 units
খ
খ) 4 units
গ
গ) 6 units
ঘ
ঘ) 8 units

ব্যাখ্যা

Let r be the radius of the circle
Now
Area of circle = circumference of circle
πr²= 2πr
r²/r =2π/π
r = 2

Diameter of circle = 2r
= 2 × 2
= 4 units

১৩,২৬০.

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

ক
20080
খ
18220
গ
30240
ঘ
17190

ব্যাখ্যা

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

Solution:
Total no. of letters in the word 'INSTITUTE' = 9
Repeating letters:
I = 2 times
T = 3 times
∴ Required no. of ways = 9!/(2! × 3!)
= (9 × 8 × 7 × 6 × 5 × 4)/2
= 30240

১৩,২৬১.

Akbar and Vijay undertake to do a piece of work for Tk. 480. Akbar alone can do it in 75 days while Vijay alone can do it in 40 days. With the help of Polash, they finish the work in 25 days. How much should Polash get for his work?

ক
Tk. 40
খ
Tk. 20
গ
Tk. 360
ঘ
Tk. 100
ঙ
Tk. 60

ব্যাখ্যা

Question: Akbar and Vijay undertake to do a piece of work for Tk. 480. Akbar alone can do it in 75 days while Vijay alone can do it in 40 days. With the help of Polash, they finish the work in 25 days. How much should Polash get for his work?

Solution:
In 25 days, Akbar and Vijay would have done 1/3 and 5/8 of the work.
The remaining work is 1 - (1/3 + 5/8) = 1/24.
This means Pradeep has done 1/24th of the work, so he should be paid 1/24th of the amount i.e. 480 × (1/24) = Tk. 20

১৩,২৬২.

The area of a square is equal to the area of a parallelogram. If the base of the parallelogram is 49 meters and its height is 4 meters, what is the length of one side of the square?

ক
18 meters
খ
14 meters
গ
7 meters
ঘ
12 meters

ব্যাখ্যা

Question: The area of a square is equal to the area of a parallelogram. If the base of the parallelogram is 49 meters and its height is 4 meters, what is the length of one side of the square?

Solution:
Area of the parallelogram = Base × Height
= 49 × 4
= 196 square meters

Let,
Side of the square = k meters
∴ Area of the square = k² square meters

ATQ,
k² = 196
⇒ k = √196
∴ k = 14
Therefore, the length of one side of the square = 14 meters

১৩,২৬৩.

Find the side of the largest square slab which can be paved on the floor of a room 5 meters 44cm long and 3 meters 74 cm broad.

ক
42 cm
খ
38 cm
গ
34 cm
ঘ
48 cm

ব্যাখ্যা

Question: Find the side of the largest square slab which can be paved on the floor of a room 5 meters 44 cm long and 3 meters 74 cm broad.

Solution:
long = 5 meters 44 cm = 500 + 44 cm = 544 cm
broad = 3 meters 74 cm = 300 + 74 = 374 cm

∴ The side of the square slab is the H.C.F. of 544 and 374 cm i.e. 34.

১৩,২৬৪.

A can do a piece of work in certain number of days. While B takes three days more than A to complete the same work. Working together, A and B can complete the work in two days. How many days does B take to complete the work?

ক
ক) 4
খ
খ) 6
গ
গ) 7
ঘ
ঘ) 8

ব্যাখ্যা

Let, A alone can complete the whole work in x days, then
Hence, B alone can complete the whole work in = (x + 3) days

According to the question
⇒ 1/x + 1/(x + 3) = 1/2
⇒ (x + 3 + x)/x(x + 3) = 1/2
⇒ (2x + 3)/(x² + 3x) = 1/2
⇒ 4x + 6 = x² + 3x
⇒ x² - x - 6 = 0
⇒ x² - 3x - 2x - 6 = 0
⇒ x (x - 3) + 2 (x - 3) = 0
⇒ (x - 3) (x + 2) = 0

Taking,
x - 3 = 0 (x can’t be -2)
x = 3

Hence, x + 3 = 3 + 3 = 6
∴ B alone can complete the whole work in 6 days.

Short Trick:
Let, B alone can compete the whole work in 6 days
A alone can complete the whole work in = 6 - 3 = 3 days

Total work = 6
A and B together can complete the whole work in = 6/(2 + 1) = 2 days (satisfied)

So, we can say B alone can complete the whole work in 6 days.

১৩,২৬৫.

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

ক
24 days
খ
36 days
গ
42 days
ঘ
48 days

ব্যাখ্যা

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

Solution:
(Jahir + Kabir)'s one-day work = 1/16
As per the question, Jahir can finish twice as much work as finished by Kabir in a given duration of time.

Therefore, 2/3 of their one-day work will be completed by Jahir and 1/3 of their one day work will be completed by Kabir.

So, Kabir's one day work will be = (1/16) × (1/3) = 1/48
So, Kabir's will take 48 days to complete the task.

১৩,২৬৬.

Two students appeared at an examination. One of them secured 7 marks more than the other and his marks was 52% of the sum of their marks. The marks obtained by them are-

ক
54, 61
খ
61, 68
গ
71, 78
ঘ
84, 91

ব্যাখ্যা

Question: Two students appeared at an examination. One of them secured 7 marks more than the other and his marks was 52% of the sum of their marks. The marks obtained by them are-

Solution:

ধরি,
তাদের প্রাপ্ত নাম্বার যথাক্রমে x এবং x + 7

প্রশ্নমতে,
x + 7 = (52/100) × (x + x + 7)
⇒ x + 7 = (13/25) × (2x + 7)
⇒ 25x + 175 = 26x + 91
⇒ 26x - 25x = 175 - 91
⇒ x = 84

∴ অপর জনের প্রাপ্ত নাম্বার = 84 + 7
= 91

১৩,২৬৭.

{(2^{n + 3} - 2 × 2^{n + 1})/(2 × 2^{n + 2})} - 2^-1 is equal to -

ক
ক) 0
খ
খ) 1/2
গ
গ) 1
ঘ
ঘ) 3

ব্যাখ্যা

Question: {(2^{n + 3} - 2 × 2^{n + 1})/(2 × 2^{n + 2})} - 2^-1 is equal to -

Solution:
{(2ⁿ⁺³ - 2 × 2ⁿ⁺¹)/(2 × 2ⁿ⁺²)} - 2^-1
= {(2ⁿ2³ - 2ⁿ2²)/2ⁿ2³} - 2^-1
= {2ⁿ(2³ - 2²)}/2ⁿ2³} - 1/2
= {(2³ -2²)/2³} - 1/2
= {(8 - 4)/8} - 1/2
= 4/8 - 1/2
= 1/2 -1/2
= 0

১৩,২৬৮.

Pipes A and B can fill a tank in 8 hours and 12 hours respectively and pipe C can empty the full tank in 24 hours. If all the pipes are opened together, how much time will be needed to make the tank full?

ক
5 hours
খ
6 hours
গ
9 hours
ঘ
10 hours

ব্যাখ্যা

Question: Pipes A and B can fill a tank in 8 hours and 12 hours respectively and pipe C can empty the full tank in 24 hours. If all the pipes are opened together, how much time will be needed to make the tank full?

Solution:
Time taken by pipe A to fill the tank = 8 hours
Pipe A can fill in 1 hour = 1/8 part

Time taken by pipe B to fill the tank = 12 hours
Pipe B can fill in 1 hour = 1/12 part

Time taken by pipe C to empty the tank = 24 hours
Pipe C can empty in 1 hour = 1/24 part

Now, Portion of tank filled by all three pipes together in 1 hour =(1/8 + 1/12 - 1/24)
=(3 + 2 - 1)/24
= 4/24
= 1/6

Time taken to filled the tank when all three pipes are opened together = 6 hours

১৩,২৬৯.

An observer who is 1.6 meters tall is standing 25 meters away from a tower. If the angle of elevation from his eye to the top of the tower is 45°, what is the height of the tower?

ক
23.4 meters
খ
27.5 meters
গ
26.6 meters
ঘ
25 meters

ব্যাখ্যা

Question: An observer who is 1.6 meters tall is standing 25 meters away from a tower. If the angle of elevation from his eye to the top of the tower is 45°, what is the height of the tower?

Solution:

পর্যবেক্ষকের উচ্চতা, CD = 1.6 মিটার
এখানে, CD = EB
টাওয়ারের উচ্চতা = AB
এখন,
tan∠C = AE/CE
⇒ tan45° = AE/25
⇒ 1 = AE/25
∴ AE = 25

∴ AB = AE + BE
= 25 + 1.6
= 26.6
∴ টাওয়ারটির উচ্চতা 26.6 মিটার।

১৩,২৭০.

There is provision of food in fort for 1200 soldiers for 60 days. After 15 days, 200 soldiers leave the fort. Remaining food will last for how many days?

ক
ক) 54 days
খ
খ) 58 days
গ
গ) 64 days
ঘ
ঘ) 48 days

ব্যাখ্যা

1200 × 45 = 1000 × x
Hence, x = 54 days

১৩,২৭১.

The simple interest on a certain sum at the rate of 5% per annum for 3 years and 7 years differs by Tk. 400. What is the principal amount?

ক
Tk. 4000
খ
Tk. 3500
গ
Tk. 2500
ঘ
Tk. 2000

ব্যাখ্যা

Question: The simple interest on a certain sum at the rate of 5% per annum for 3 years and 7 years differs by Tk. 400. What is the principal amount?

Solution:
Let the principal amount be Tk. P
Then, ATQ
400 = {(P × 5 × 7)/100} - {(P × 5 × 3)/100}
⇒ 400 = (35P - 15P)/100
⇒ 40000 = 20P
⇒ P = 40000/20
∴ P = 2000

So the principal amount is Tk. 2000

১৩,২৭২.

Which of the following is a leap year?

ক
1900
খ
2000
গ
2022
ঘ
2010

ব্যাখ্যা

Question: Which of the following is a leap year?

Solution:
অধিবর্ষ বা লিপ ইয়ার নির্ণয়ের দুটি প্রধান নিয়ম রয়েছে:
১. সাধারণ বছর: বছরটি 4 দ্বারা নিঃশেষে বিভাজ্য হতে হবে।
২. শতাব্দী বছর (100 দ্বারা বিভাজ্য): বছরটি 400 দ্বারা নিঃশেষে বিভাজ্য হতে হবে।

এখন,
ক) 1900 সাল (শতাব্দী বছর): 1900 ÷ 400 দ্বারা বিভাজ্য নয় (ভাগশেষ 300)। ⇒ অধিবর্ষ নয়।

খ) 2000 সাল (শতাব্দী বছর): 2000 ÷ 400 = 5 (ভাগশেষ 0)। ⇒ অধিবর্ষ।

গ) 2022 সাল: এটি 4 দ্বারা বিভাজ্য নয়। (2022 ÷ 4 ⇒ ভাগশেষ 2)। ⇒ অধিবর্ষ নয়।

ঘ) 2010 সাল: এটি 4 দ্বারা বিভাজ্য নয়। (2010 ÷ 4 ⇒ ভাগশেষ 2)। ⇒ অধিবর্ষ নয়।

অতএব, 2000 সালটি অধিবর্ষ।

১৩,২৭৩.

A bicycle marked at Tk. 4,000, is sold with two successive discount of 20% and 10%. An additional discount of 5% is offered for cash payment. The selling price of the bicycle at cash payment is:

ক
ক) Tk. 3444
খ
খ) Tk. 2842
গ
গ) Tk. 2736
ঘ
ঘ) Tk. 2632

ব্যাখ্যা

Question: A bicycle marked at Tk. 4,000, is sold with two successive discount of 20% and 10%.An additional discount of 5% is offered for cash payment. The selling price of the bicycle at cash payment is:

Solution:
Marked Price = 4000
SP after first Discount of 20% = 4000 - 20% of 4000 = 3200
SP after second Discount of 10% = 3200 - 10% of 3200 = 2880
Now, the final selling price at cash = 2880 - 5% of 2880 = Tk. 2736

১৩,২৭৪.

What is the probability of getting a sum of 7 when two dice are thrown?

ক
0
খ
1
গ
1/2
ঘ
1/6
ঙ
5/36

ব্যাখ্যা

Question: What is the probability of getting a sum of 7 when two dice are thrown?

Solution:
Total number of ways = 6 × 6 = 36 ways.
Favorable cases = (1, 6) (6, 1) (2, 5) (5, 2) (3, 4) (4, 3) = 6 ways.

∴ Probability = 6/36 = 1/6

১৩,২৭৫.

Today is Friday. After 61 days, it will be:

ক
Saturday
খ
Sunday
গ
Tuesday
ঘ
Wednesday
ঙ
None

ব্যাখ্যা

Question: Today is Friday. After 61 days, it will be:

Solution:
Each day of the week is repeated after 7 days.
So, after 63 days, it will be Friday.
After 62 days, it will be Thursday.
∴ After 61 days, it will be Wednesday.

১৩,২৭৬.

There is a 29-meter ladder resting against a tall wall. The point where it touches the wall is 21 meters above the ground. What is the horizontal distance between the base of the ladder and the wall?

ক
25 m
খ
12 m
গ
15 m
ঘ
20 m

ব্যাখ্যা

Question: There is a 29-meter ladder resting against a tall wall. The point where it touches the wall is 21 meters above the ground. What is the horizontal distance between the base of the ladder and the wall?
(একটি 29 মিটার লম্বা মই একটি খাড়া দেয়ালের সাথে হেলান দেওয়া আছে। মইয়ের একপ্রান্ত মাটি হতে 21 মিটার উঁচুতে দেয়ালকে স্পর্শ করে। মই এর অপরপ্রান্ত হতে দেয়ালের দূরত্ব কত মিটার?)

Solution:
মনে করি,
দেয়ালের পাদদেশ হতে মইয়ের পাদদেশের দূরত্ব = x মিটার
দেয়াল মাটির সাথে সমকোণ উৎপন্ন করে।

∴ পিথাগোরাসের উপপাদ্য অনুসারে,
(মইয়ের উচ্চতা)² = (দেয়ালের উচ্চতা)² + (x)²
⇒ (29)² = (21)² + (x)²
⇒ 841 = 441 + (x)²
⇒ (x)² = 841 - 441
⇒ (x)² = 400
⇒ (x)² = (20)²
∴ x = 20

∴ দেয়ালের পাদদেশ হতে মইয়ের পাদদেশের দূরত্ব = 20 মিটার।

১৩,২৭৭.

The ratio of two numbers is 7:4. If 8 is added to both the numbers ratio becomes 13:8. What is the smaller number?

ক
ক) 52
খ
খ) 56
গ
গ) 38
ঘ
ঘ) 40

ব্যাখ্যা

Let the numbers be 7x and 4x
ATQ,
(7x + 8) : (4x + 8) = 13 : 8
Or, 56x + 64 = 52x + 104
Or, 4x = 40

The smaller number is 40

১৩,২৭৮.

log₁₀1000 =?

ক
ক) 3
খ
খ) 6
গ
গ) 8
ঘ
ঘ) 10

ব্যাখ্যা

log₁₀1000
= log₁₀10³= 3log₁₀10
= 3×1=3

১৩,২৭৯.

If x = y = 3z and xyz = 576 than x =?

ক
12
খ
8
গ
6
ঘ
14

ব্যাখ্যা

Question: If x = y = 3z and xyz = 576 than x =?

Solution:
Given that,
x = y = 3z
Now,
xyz = 576
⇒ x × x × (x/3) = 576
⇒ x³/3 = 576
⇒ x³ = 576 × 3
⇒ x³ = 1728
⇒ x³ = 12³
∴ x = 12

১৩,২৮০.

In a camp, food was sufficient for 2000 people for 54 days. After 15 days, more people came and the food lasted only for 20 more days. How many people came?

ক
ক) 1900
খ
খ) 1800
গ
গ) 2000
ঘ
ঘ) 1940

ব্যাখ্যা

Given that food was sufficient for 2000 people for 54 days
Hence, after 15 days, the remaining food was sufficient for 2000 people for 39 days (∵ 54 - 15 = 39)

Let x number of people come after 15 days.
Then, total number of people after 15 days = (2000 + x)
Then, the remaining food was sufficient for (2000 + x) people for 20 days

More men, Less days (Indirect Proportion)

(men) 2000 : (2000 + x) :: 20 : 39
⇒ 2000 × 39 = (2000 + x) 20
⇒ 100 × 39 = (2000 + x)
⇒ 3900 = 2000 + x
⇒ x = 3900 − 2000
= 1900

১৩,২৮১.

The total surface area of a hemisphere of radius 3r is

ক
4πr²
খ
2πr²
গ
3πr²
ঘ
None of these

ব্যাখ্যা

The total surface area of a hemisphere of radius 3r is 3π(3r)² = 3π(9r²) = 27πr²

১৩,২৮২.

An amount of Tk. 625 becomes Tk 1296 in 2 years if the interest is compounded half-yearly. What is the rate of compound interest?

ক
20%
খ
30%
গ
40%
ঘ
50%

ব্যাখ্যা

Question: An amount of Tk. 625 becomes Tk 1296 in 2 years if the interest is compounded half-yearly. What is the rate of compound interest?

solution:
let the rate be R%

Then,
625 {1 + R/(2 × 100)} ^{2 × 2} = 1296
⇒ (1 + R/200)⁴ = 1296/625
⇒ (1 + R/200)⁴ = (6/5)⁴
⇒ (1 + R/200) = 6/5
⇒ R/200 = 1/5
⇒ R = 40
∴ R = 40%

১৩,২৮৩.

A hostel had provisions for 250 men for 40 days. If 50 men left the hostel, how long will the food last at the same rate?

ক
ক) 45 days
খ
খ) 48 days
গ
গ) 50 days
ঘ
ঘ) 60 days

ব্যাখ্যা

A hostel had provisions for 250 men for 40 days

If 50 men leaves the hostel, remaining men = 250 - 50 = 200
We need to find out how long the food will last for these 200 men.

Let the required number of days = x days.

More men, Less days (Indirect Proportion)
(men) 250 : 200 :: x : 40
⇒ 250 × 40 = 200x
⇒ 5 × 40 = 4x
⇒ x = 5 × 10
= 50

১৩,২৮৪.

If ⁷P_r = 210, what is the value of r?

ক
6
খ
3
গ
4
ঘ
5

ব্যাখ্যা

Question: If ⁷P_r = 210, what is the value of r?

Solution:
We know,
ⁿP_r = n!/(n - r)!

Given that,
⁷P_r = 210
⇒ 7!/(7 - r)! = 210
⇒ 5040/(7 - r)! = 210 (7! = 5040)
⇒ (7 - r)! = 5040/210
⇒ (7 - r)! = 24
⇒ (7 - r)! = 4! ;(4! = 4 × 3 × 2 × 1 = 24)
⇒ 7 - r = 4
⇒ r = 7 - 4
∴ r = 3

১৩,২৮৫.

The graph of the equations 25x + 75y = 225 and x = 9 meet at the point:

ক
(0, 5)
খ
(3, 2)
গ
(9, 0)
ঘ
(5, 9)
ঙ
None

ব্যাখ্যা

Question: The graph of the equations 25x + 75y = 225 and x = 9 meet at the point:

Solution:
Given,
25x + 75y = 225 . . . . . . (i)
x = 9 . . . . . . (ii)

To find the point of intersection, we substitute x = 9 into the first equation:
⇒ 25 × 9 + 75y = 225
⇒ 75y = 225 - 225
∴ y = 0

∴ Meeting point = (9, 0)

১৩,২৮৬.

What is the length of AB?

ক
ক) 4/√3 m
খ
খ) 4√3 m
গ
গ) 8√3 m
ঘ
ঘ) 8/√3 m

ব্যাখ্যা

Question: What is the length of AB?

Solution:
Given, AB = 8 m and ∠ACB = 60°.

From ΔABC,
Sin60° = AB/AC
Or, √3/2 = AB/8
Or, 2AB = 8√3
∴ AB = 4√3

∴ The length of AB = 4√3 m.

১৩,২৮৭.

A student scored 70, 80, 75, and 85 in four subjects. What score must he get in the fifth subject to have an average of 80?

ক
90
খ
80
গ
85
ঘ
88

ব্যাখ্যা

Question: A student scored 70, 80, 75, and 85 in four subjects. What score must he get in the fifth subject to have an average of 80?

Solution:
- If the average is 80, then
- Total marks needed = 80 × 5 = 400
- Marks so far obtained in 4 subjects = 70 + 80 + 75 + 85 = 310
- Marks Required to obtain desired average = 400 - 310 = 90

১৩,২৮৮.

The average of 5 quantities is 6. The average of 3 of them is 8. What is the average of the remaining two numbers?

ক
ক) 6.5
খ
খ) 4
গ
গ) 3
ঘ
ঘ) 3.5
ঙ
ঙ) None of these

ব্যাখ্যা

Question: The average of 5 quantities is 6. The average of 3 of them is 8. What is the average of the remaining two numbers?

Solution:
The average of 5 quantities is 6
The sum of 5 quantities = 6 × 5
= 30

The average of 3 of them is 8
The sum of 3 = (3 × 8)
= 24

The sum of the remaining two numbers = 30 - 24
= 6
∴ the average of the remaining two numbers is = 6/2
= 3

১৩,২৮৯.

The length, breadth, and height of a brick are 10 cm, 4 cm, and 3 cm, respectively. Find the surface area of the brick?

ক
154 cm²
খ
156 cm²
গ
160 cm²
ঘ
164 cm²

ব্যাখ্যা

Question: The length, breadth, and height of a brick are 10 cm, 4 cm, and 3 cm, respectively. Find the surface area of the brick?

Solution:
Surface area of a Cuboid = 2(lb + bh + hl) cm²
So,
Surface area of a brick = 2(10 × 4 + 4 × 3 + 3 × 10) cm²
= 2(82) cm² = 164 cm²

১৩,২৯০.

Rafi went to the stationers and bought things worth Tk. 25 out of which 30 paisa went on sales tax on taxable purchases. if the tax rate was 6%, then what was the cost of the tax-free items?

ক
Tk. 20
খ
Tk. 19.7
গ
Tk. 15.5
ঘ
Tk. 15

ব্যাখ্যা

Question: Rafi went to the stationers and bought things worth Tk. 25 out of which 30 paisa went on sales tax on taxable purchases. if the tax rate was 6%, then what was the cost of the tax-free items?

Solution:
Let, cost of taxable purchases is x taka

ATQ,
0.06x = 30/100
⇒ x = 30/(100 × 0.06)
= 5 taka

∴ The cost of the tax-free items was = 25 - 5 - 0.3
= Tk. 19.7

১৩,২৯১.

Find the area of a triangle with side lengths of 5 meters, 6 meters, and 7 meters.

ক
18 square meters
খ
6√6 square meters
গ
12√2 square meters
ঘ
24 square meters

ব্যাখ্যা

Question: Find the area of a triangle with side lengths of 5 meters, 6 meters, and 7 meters.

Solution:
দেওয়া আছে,
ত্রিভুজের তিন বাহুর দৈর্ঘ্য, a = 5 মি., b = 6 মি. এবং c = 7 মি.

আমরা জানি,
ত্রিভুজের অর্ধপরিসীমা, s = (a + b + c)/2
= (5 + 6 + 7)/2
= 18/2
= 9 মি.

∴ ত্রিভুজের ক্ষেত্রফল = √{s(s - a)(s - b)(s - c)}
= √{9(9 - 5)(9 - 6)(9 - 7)}
= √(9 × 4 × 3 × 2)
= √(36 × 6)
= 6√6 বর্গ মি.

অতএব, ত্রিভুজটির ক্ষেত্রফল 6√6 বর্গ মি.

১৩,২৯২.

Three numbers A, B and C are in the ratio of 12 : 15 : 25. If sum of these numbers is 312, ratio between the difference of B and A and the difference of C and B is –

ক
ক) 3 : 7
খ
খ) 10 : 3
গ
গ) 3 : 10
ঘ
ঘ) 7 : 3

ব্যাখ্যা

Let the number A, B and C be 12a , 15a and 25a respectively
Then from the question 12a + 15a + 25a = 312
⇒ 52a = 312
⇒ a = 312/52
⇒ a = 6
Required ratio = {(15 × 6) - (12 × 6)}/{(25 × 6) - (15 × 6)}
= 3/10
= 3 : 10.

১৩,২৯৩.

A man takes 3 hours 45 minutes to row a boat 22.5 km downstream of a river and 2 hours 30 minutes to cover a distance of 10 km up stream. Find the speed of the river current in km/hr.

ক
ক) 1 km/hr
খ
খ) 2 km/hr
গ
গ) 3 km/hr
ঘ
ঘ) 4 km/hr

ব্যাখ্যা

এখানে
3 ঘণ্টা 45 মিনিট = 3(3/4) ঘণ্টা = 15/4 ঘণ্টা
2 ঘণ্টা 30 মিনিট = 2(1/2) ঘণ্টা = 5/2 ঘণ্টা
ধরি
নৌকার বেগ = x কি.মি./ঘণ্টা
স্রোতের বেগ= y কি.মি./ঘণ্টা

স্রোতের অনুকূলে নৌকার বেগ = 22.5/(15/4)
x + y = (22.5 × 4)/15
x + y = 6 কি.মি./ঘণ্টা ................(1)

স্রোতের প্রতিকূলে নৌকার বেগ = 10/(5/2) কি.মি./ঘণ্টা
x - y = (10 × 2)/5 কি.মি./ঘণ্টা
x - y = 4 কি.মি./ঘণ্টা ................(2)

(1) থেকে (2 বিয়োগ করে পাই,
x + y - x + y = (6 - 4)কি.মি./ঘণ্টা
2y = 2
y = 1 কি.মি./ঘণ্টা

১৩,২৯৪.

A person's present age is two-fifth 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
খ
35
গ
40
ঘ
50

ব্যাখ্যা

Question: A person's present age is two-fifth 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

ATQ,

(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

১৩,২৯৫.

A team of workers was employed by a contractor who undertook to finish 360 pieces of an article in a certain number of days. Making four more pieces per day than was planned, they could complete the job a day ahead of schedule. How many days did they take to complete the job ?

ক
10 days
খ
12 days
গ
8 days
ঘ
14 days

ব্যাখ্যা

Question: A team of workers was employed by a contractor who undertook to finish 360 pieces of an article in a certain number of days. Making four more pieces per day than was planned, they could complete the job a day ahead of schedule. How many days did they take to complete the job?

Solution:
Let the team take x days to finish 360 pieces
Then, number of pieces made each day = 360/x
More number of pieces per day, Less days (Indirect proportion)

১৩,২৯৬.

Two friends invested TK.1500 and TK. 3500 in a business. They earned a profit of TK. 1000. One-half of the profit was divided equally between them and the other half was divided in proportion to their capitals. How much did each of them receive ?

ক
TK. 350 and TK. 450
খ
TK. 450 and TK. 650
গ
TK. 400 and TK. 600
ঘ
TK. 600 and TK. 400

ব্যাখ্যা

Question: Two friends invested TK.1500 and TK. 3500 in a business. They earned a profit of TK. 1000. One-half of the profit was divided equally between them and the other half was divided in proportion to their capitals. How much did each of them receive ?

Solution:
Ratio of shares
= 1500 : 3500
= 3 : 7

Share of first friend
= TK. [(500/2) + {500 × (3/10)}]
= TK.[250 + 150]
= TK. 400

Share of second friend
= TK. [(500/2) + {500 × (7/10)}]
= TK.[250 + 350]
= TK. 600

১৩,২৯৭.

What is the average of natural numbers from 1 to 67?

ক
33.5
খ
33
গ
35
ঘ
34
ঙ
36.5

ব্যাখ্যা

Question: What is the average of natural numbers from 1 to 67?

Solution:
Given, natural numbers 1 to 67.

Average of n natural numbers = (n + 1)/2

Here, n = 67

Average = (67 + 1)/2 = 68/2 = 34

১৩,২৯৮.

Two containers contain milk and water in the ratios 5:2 and 9:5. What ratio should the mixtures be combined in to achieve a final ratio of 2:1 milk to water?

ক
1 : 3
খ
3 : 2
গ
3 : 7
ঘ
5 : 8
ঙ
None of the above

ব্যাখ্যা

Question: Two containers contain milk and water in the ratios 5:2 and 9:5. What ratio should the mixtures be combined in to achieve a final ratio of 2:1 milk to water?

Solution:
Let,
P unit of the first mixture is added to Q unit of the second mixture.

So, in the P unit of the first mixture,
Amount of milk present = (5/7) × P = 5P/7
Amount of water present = (2/7) × P = 2P/7

In the Q unit of the second mixture,
Amount of milk present = (9/14) × Q = 9Q/14
Amount of water present = (5/14) × Q = 5Q/14

ATQ,
{(5P/7) + (9Q/14)}/{(2P/7) + (5Q/14)} = 2/1
⇒ {(10P + 9Q)/14}/{(4P + 5Q)/14} = 2
⇒ 10P + 9Q = 8P + 10Q
⇒ 2P = Q

∴ P : Q = 1 : 2

১৩,২৯৯.

Which of the following is prime number?

ক
ক) 143
খ
খ) 289
গ
গ) 359
ঘ
ঘ) 117

ব্যাখ্যা

Question: Which of the following is prime number?

Solution:
১৪৩ সংখ্যাটি ১১ দ্বারা বিভাজ্য।
২৮৯ সংখ্যাটি ১৭ দ্বারা বিভাজ্য।
১১৭ সংখ্যাটি ৩ দ্বারা বিভাজ্য।
৩৫৯ সংখ্যাটি ১ ও ৩৫৯ ছাড়া অন্য কোনো সংখ্যা দ্বারা বিভাজ্য নয়। তাই ৩৫৯ একটি মৌলিক সংখ্যা।

১৩,৩০০.

A dishonest shopkeeper sells his grocery using weights 10% less than true weights and makes a profit of 30%. Find his total gain percentage.

ক
ক) 49.4%
খ
খ) 44.44%
গ
গ) 55.55%
ঘ
ঘ) 39.88%

ব্যাখ্যা

Let the weight of the grocery bag be 1000 gm.
Now, the shopkeeper sells his grocery using weights 10% less than true weights.
Hence, actual weight of bag = 90% of 1000 gm = 900 gm
If each gram = Tk.1, C.P. of each bag containing 900 gm = Tk. 900
The shopkeeper sells with a gain of 30% on true C.P.

Calculate the S.P.
Selling price = 100 + gain% × C.P.
Therefore,
Selling price = (130/100) × Tk. 1000
= Tk. 1300
Gain = S.P. - C.P. = 1300 - 900 = Tk. 400
Gain% = (400/900) × 100
= 44.44%