Bank Math

PrepBank · পাতা ৫৩ / ১৬১ · ৫,২০১–৫,৩০০ / ১৬,১২৪

৫,২০১.

Four unbiased coins are tossed. What is the probability of getting at most two heads?

ক
ক) 1/2
খ
খ) 5/8
গ
গ) 11/16
ঘ
ঘ) 5/16

সঠিক উত্তর: গ

গ) 11/16

উত্তর

সঠিক উত্তর: গ

গ) 11/16

গ

ব্যাখ্যা

Question: Four unbiased coins are tossed. What is the probability of getting at most two heads?

Solution:

The total number of events = 16
The numbers of event with at most two heads = 11
∴ The probability of getting at most two heads = 11/16

৫,২০২.

A man invested Tk 1552 in a stock at 97 to obtain an income of Tk. 128 The divided from the stock is -

ক
ক) 7.5%
খ
খ) 8%
গ
গ) 8.5%
ঘ
ঘ) 9.7%

সঠিক উত্তর: খ

খ) 8%

উত্তর

সঠিক উত্তর: খ

খ) 8%

খ

ব্যাখ্যা

By investing Tk. 1552, income = Tk. 128
By investing Tk. 97, income
= (128/1552) × 97
= Tk 8
∴ Dividend 8%.

৫,২০৩.

If a quarter kg of carrots costs 60 poysa, how many poysa will 200 gms cost?

ক
ক) 78 poysa
খ
খ) 48 poysa
গ
গ) 54 poysa
ঘ
ঘ) 62 poysa
ঙ
ঙ) 65 poysa

সঠিক উত্তর: খ

খ) 48 poysa

উত্তর

সঠিক উত্তর: খ

খ) 48 poysa

খ

ব্যাখ্যা

Quarter of Kg means 250 gm
Less weight, less price (Direct Proportion)
So, 250 : 200 :: 60 : x
=> x = 48
So 200 gm will cost 48 poysa.

৫,২০৪.

ক
a + b = 1
খ
a - b = 1
গ
a = b
ঘ
a.a = b.b

সঠিক উত্তর: ক

a + b = 1

উত্তর

সঠিক উত্তর: ক

a + b = 1

ক

ব্যাখ্যা

log(a/b) + log(b/a) = log(a + b)
or, log(a + b) = log{(ab)/(ba)}
or, log(a + b) = log1
or, a + b = 1

৫,২০৫.

If x⁶ + x⁵ + x⁴ + x³ + x² + x + 1 = 0, then, find the value of x³⁵ + x⁶³.

ক
ক) 0
খ
খ) 1
গ
গ) 2
ঘ
ঘ) 4

সঠিক উত্তর: গ

গ) 2

উত্তর

সঠিক উত্তর: গ

গ) 2

গ

ব্যাখ্যা

Question: If x⁶ + x⁵ + x⁴ + x³ + x² + x + 1 = 0, then, find the value of x³⁵+ x⁶³.

Solution:
Given that
x⁶ + x⁵ + x⁴ + x³ + x² + x + 1 = 0
x⁶ + x⁵ + x⁴ + x³ + x² + x = - 1..........(1)

Multiplying by x,
we have
x⁷+ x⁶ + x⁵ + x⁴ + x³ + x² + x = 0 ........(2)
⇒ x⁷ - 1 = 0
⇒ x⁷ = 1

According to question
x³⁵+ x⁶³=(x⁷)⁵ + (x⁷)⁹= 1⁵ + 1⁹ = 1 + 1 = 2

৫,২০৬.

A, B and C started a business with initial investments of Tk. 1250, Tk. 1750 and Tk. 2000 respectively. After one year A, B and C made additional investments of Tk. x + 650, Tk. x +500, Tk. x + 850 respectively. Find the profit share of B out of the total profit of Tk. 4350, after two years.

ক
Tk. 1350
খ
Tk. 1450
গ
Tk. 1550
ঘ
Can’t be determined

সঠিক উত্তর: ঘ

Can’t be determined

উত্তর

সঠিক উত্তর: ঘ

Can’t be determined

ঘ

ব্যাখ্যা

Question: A, B and C started a business with initial investments of Tk. 1250, Tk. 1750 and Tk. 2000 respectively. After one year A, B and C made additional investments of Tk. x + 650, Tk. x +500, Tk. x + 850 respectively. Find the profit share of B out of the total profit of Tk. 4350, after two years.

Solution:
A : B : C = {(1250 × 24) + (x + 650) × 12} : {(1750 × 24) + (x +500) × 12} : {(2000 × 24) + (x + 850) × 12}
= (30000 + 7800 + 12x) : (42000 + 6000 + 12x) : (48000 + 10200 + 12x)
= (37800 + 12x) : (48000 + 12x) : (58200 + 12x)
= {12(3150 + x)} : {12(4000 + x)} : {12(4850 + x)}
= (3150 + x) : (4000 + x) : (4850 + x)

Here x is not defined so the share of B can’t be determined.

৫,২০৭.

In a class of 60 students, 20 students like Math, 25 students like English, and 30 students like Science. If 5 students like both Math and English, 7 students like both Math and Science, 8 students like both English and Science, and 3 students like neither of these subjects, how many students like all three subjects?

ক
2
খ
4
গ
6
ঘ
5
ঙ
1

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: In a class of 60 students, 20 students like Math, 25 students like English, and 30 students like Science. If 5 students like both Math and English, 7 students like both Math and Science, 8 students like both English and Science, and 3 students like neither of these subjects, how many students like all three subjects?

Solution:
Total students, n(U) = 60
Number who like Math, n(M) = 20
Number who like English, n(E) = 25
Number who like Science, n(S) = 30
Number who like both Math and English, n(M ∩ E) = 5
Number who like both Math and Science, n(M ∩ S) = 7
Number who like both English and Science, n(E ∩ S) = 8
Number who like neither subject = 3

n(M ∪ E ∪ S) = n(U) - neither
= 60 - 3 = 57

∴ n(M ∪ E ∪ S) = n(M) + n(E) + n(S) - n(M ∩ E) - n(M ∩ S) - n(E ∩ S) + n(M ∩ E ∩ S)
⇒ 57 = 20 + 25 + 30 - 5 - 7 - 8 + n(M ∩ E ∩ S)
⇒ 57 = 75 - 20 + n(M ∩ E ∩ S)
⇒ 57 = 55 + n(M ∩ E ∩ S)
⇒ n(M ∩ E ∩ S) = 57 - 55
⇒ n(M ∩ E ∩ S) = 2

∴ 2 Students like all three subjects.

৫,২০৮.

If x² + 2xy + y² = 25 and xy = 6, then x + y is:

ক
± 5
খ
± 10
গ
± 13
ঘ
± 4

সঠিক উত্তর: ক

± 5

উত্তর

সঠিক উত্তর: ক

± 5

ক

ব্যাখ্যা

Question: If x² + 2xy + y² = 25 and xy = 6, then x + y is:

Solution:
Given that,
x²+ 2xy + y² = 25 and xy = 6
⇒ (x + y)² = 25
⇒ x + y = ± 5

৫,২০৯.

A train travels 10 miles at a speed of 50 miles per hour. How fast must the train travel on the return trip if the round-trip travel time is to be 20 minutes?

ক
ক) 55 miles/hours
খ
খ) 60 miles/hours
গ
গ) 65 miles/hours
ঘ
ঘ) 75 miles/hours
ঙ
ঙ) None

সঠিক উত্তর: ঘ

ঘ) 75 miles/hours

উত্তর

সঠিক উত্তর: ঘ

ঘ) 75 miles/hours

ঘ

ব্যাখ্যা

To travel 10 miles at a speed of 50 mph, the train needs = 10/50 × 60 = 12 minutes
As the given time is 20 minutes
Then for the round trip the train has to travel 10 miles within 8 minutes
∴ Required speed = 10 × 8/60 = 75 mph

৫,২১০.

How many degrees are between the hands of a clock at 4:30?

ক
75°
খ
55°
গ
45°
ঘ
35°

সঠিক উত্তর: গ

45°

উত্তর

সঠিক উত্তর: গ

45°

গ

ব্যাখ্যা

Question: How many degrees are between the hands of a clock at 4:30?

Solution:
Value of angle = {(11×30) - (60 × 4)}/2
= (330 - 240)/2
= 90/2
= 45°

৫,২১১.

A sum of 1750 taka is divided into two parts such that the interests on the first part at 8% simple interest per annum and that on the other part at 6% simple interest per annum are equal. The interest on each part ( in taka) is ?

ক
55
খ
60
গ
65
ঘ
70
ঙ
75

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Let the 1st part be p
Therefore, the 2nd part = 1750 - p

∴ p × (8/100) × 1 = (1750−p)×(6/100)×1
or, p = 750

The interest on each part ( in taka) is 750 × 8/100 = 60 taka

৫,২১২.

Rectangular Floors X and Y have equal area. If Floor X is 12 feet by 18 feet and Floor Y is 9 feet wide, what is the length of Floor Y, in feet?

ক
ক) 13.5
খ
খ) 18
গ
গ) 21
ঘ
ঘ) 24

সঠিক উত্তর: ঘ

ঘ) 24

উত্তর

সঠিক উত্তর: ঘ

ঘ) 24

ঘ

ব্যাখ্যা

Given: area = 12×18 = 9×length [As, x floor area = y floor area]
∴ length = 12×18 / 9 = 24 feet

৫,২১৩.

Two trains start at the same time from Dhaka and Cumilla and proceed towards each other at 70 kmph and 90 kmph respectively. When they meet, it is found that one train has travelled 60 km more than the other. Find the distance between Dhaka and Cumilla.

ক
390 km
খ
480 km
গ
350 km
ঘ
450 km

সঠিক উত্তর: খ

480 km

উত্তর

সঠিক উত্তর: খ

480 km

খ

ব্যাখ্যা

Question: Two trains start at the same time from Dhaka and Cumilla and proceed towards each other at 70 kmph and 90 kmph respectively. When they meet, it is found that one train has travelled 60 km more than the other. Find the distance between Dhaka and Cumilla.

Solution:
Let,
They meet after t time.

ATQ,
(90t) = (70t) + 60
⇒ (90t) - (70t) = 60
⇒ 20t = 60
∴ t = 3 hours

So the distance between Dhaka and Cumilla = (90 × 3) + (70 × 3)
= 270 + 210
= 480 km

৫,২১৪.

How many integers from 1 to 150 are divisible by 5 but not by 6?

ক
20
খ
25
গ
28
ঘ
30

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: How many integers from 1 to 150 are divisible by 5 but not by 6?

Solution:
150 পর্যন্ত সংখ্যাগুলোর মধ্যে-
5 দ্বারা বিভাজ্য সংখ্যা = ( 150 ÷ 5) = 30 টি
5এবং 6 এর লসাগু = 30
এখন, 150 ÷ 30 = 5
∴ 5 এবং 6 উভয় দ্বারা বিভাজ্য সংখ্যা = 5টি
সুতরাং, 5 দ্বারা বিভাজ্য কিন্তু 6 দ্বারা বিভাজ্য নয় এমন সংখ্যা = (30 - 5) = 25 টি সংখ্যা

৫,২১৫.

What is the next number in the following sequence?
12, 41, 169, 850, ?

ক
5100
খ
5105
গ
5130
ঘ
5205

সঠিক উত্তর: খ

5105

উত্তর

সঠিক উত্তর: খ

5105

খ

ব্যাখ্যা

Question: What is the next number in the following sequence?
12, 41, 169, 850, ?

Solution:
(12 × 3) + 5 = 41
(41 × 4) + 5 =169
(169 × 5) + 5 = 850
(850 × 6) + 5 = 5105

৫,২১৬.

Rakib correctly remembers that his father's birthday is before 20th January but after 16th January, whereas his sister correctly remembers that their father’s birthday is after 18th January but before 23rd January. On which date in January is definitely their father's birthday?

ক
ক) 18
খ
খ) 19
গ
গ) 20
ঘ
ঘ) Missing data

সঠিক উত্তর: খ

খ) 19

উত্তর

সঠিক উত্তর: খ

খ) 19

খ

ব্যাখ্যা

According to Rakib 17th, 18th or 19th ...... (i)
According to her sister 19th, 20th, 21st or 22nd ......(ii)
From (i) and (ii) ⇒ 19th
Answer: 19

৫,২১৭.

In a certain language PENSION is coded as NEISNOP, how is FOLLAGE coded in that code?

ক
ক) EOALLGF
খ
খ) OFILGAE
গ
গ) FGLIAOE
ঘ
ঘ) None of these

সঠিক উত্তর: ক

ক) EOALLGF

উত্তর

সঠিক উত্তর: ক

ক) EOALLGF

ক

ব্যাখ্যা

Question: In a certain language PENSION is coded as NEISNOP, how is FOLLAGE coded in that code?

Solution:
PENSION ⇔ NEISNOP
P ⇒ E ⇒ N ⇒ S ⇒ I ⇒ O ⇒ N
1 ⇒ 2 ⇒ 3 ⇒ 4 ⇒ 5 ⇒ 6 ⇒ 7
N ⇒ E ⇒ I ⇒ S ⇒ N⇒ O ⇒ P
Here,
Letters of 1st and 7th position inter-change their position.
Letters of 3rd and 5th position also inter-change their position.
And the letters of even number position remain same.

F ⇒ O ⇒ L ⇒ L ⇒ A ⇒ G ⇒ E
1 ⇒ 2 ⇒ 3 ⇒ 4 ⇒ 5 ⇒ 6 ⇒ 7
E ⇒ O ⇒ A ⇒ L ⇒ L ⇒ G ⇒ F

∴ FOLLAGE ⇔ EOALLGF

৫,২১৮.

If p and q are even numbers, which of the following is always even?

ক
p + q + 3
খ
pq + 5
গ
3p + q
ঘ
p² + q + 3

সঠিক উত্তর: গ

3p + q

উত্তর

সঠিক উত্তর: গ

3p + q

গ

ব্যাখ্যা

Question: If p and q are even numbers, which of the following is always even?

Solution:
Take p = 2 and q = 6 (both even)
a) p + q + 3 = 2 + 6 + 3 = 11 → Odd
b) pq + 5 = (2 × 6) + 5 = 12 + 5 = 17 → Odd
c) 3p + q = (3 × 2) + 6 = 6 + 6 = 12 → Even
d) p² + q + 3 = (2)² + 6 + 3 = 4 + 6 + 3 = 13 → Odd

Answer: c) 3p + q is always even.

৫,২১৯.

In a shower, 5 cm of rain falls. The volume of water that falls on 1.5 hectares of ground is-

ক
75 m³
খ
750 m³
গ
7500 m³
ঘ
75000 m³

সঠিক উত্তর: খ

750 m³

উত্তর

সঠিক উত্তর: খ

750 m³

খ

ব্যাখ্যা

Question: In a shower, 5 cm of rain falls. The volume of water that falls on 1.5 hectares of ground is-

Solution:

1 hectare = 10,000 m²

So, Area = (1.5 × 10000) m² = 15000 m².

Depth = 5 cm = 5/100 m = 1/20 m.

Volume = (Area × Depth) = 15000 × 1/20 m³= 750 m³.

৫,২২০.

Amir and Hannan working together can finish a work in 3 hours. If they work alone, Hannan takes 3 times as long as Amir. How long does Hannan take to finish the job alone?

ক
6
খ
12
গ
14
ঘ
16
ঙ
None

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: Amir and Hannan working together can finish a work in 3 hours. If they work alone, Hannan takes 3 times as long as Amir. How long does Hannan take to finish the job alone?

Solution:
ধরি,
আমির একা কাজটি শেষ করতে সময় নেয় = x ঘণ্টা
হান্নান একা কাজটি শেষ করতে সময় নেয় = 3x ঘণ্টা

এখন
আমিরের 1 ঘণ্টায় কাজের পরিমাণ = 1/x
হান্নানের 1 ঘণ্টায় কাজের পরিমাণ = 1/3x

তারা একসাথে 1 ঘণ্টায় যতটুকু কাজ করতে পারে = (1/x) + (1/3x) = 4/3x

একসাথে কাজটি শেষ করতে সময় লাগে = 3 ঘণ্টা

⇒ 3 ঘণ্টায় তারা পুরো ১টা কাজ শেষ করে

তাহলে,
3 × (4/3x) = 1
⇒ 4/x = 1
⇒ x = 4

সুতরাং,
আমির একা কাজটি শেষ করতে সময় নেয় = ৪ ঘণ্টা
হান্নান একা কাজটি শেষ করতে সময় নেয় = 3 × 4 = 12 ঘণ্টা

৫,২২১.

If a is 10% less than b and c is 40% less than d, then ac is what percent less than bd?

ক
37%
খ
40%
গ
46%
ঘ
54%

সঠিক উত্তর: গ

46%

উত্তর

সঠিক উত্তর: গ

46%

গ

ব্যাখ্যা

Suppose, b = 100
a = 100 - 10 = 90

Suppose, d = 100
c = 100 - 40 = 60

ac = 90 × 60
= 5400;

bd
= 100 × 100
= 10,000

ac is less than bd in percentage
= (10,000 - 5400)/10,000
= 4600/10,000 × 100%
= 46%

৫,২২২.

What is the LCM of 12, 15, and 20?

ক
60
খ
120
গ
180
ঘ
240

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: What is the LCM of 12, 15, and 20?

Solution:
12 = 2 × 2 × 3
15 = 3 × 5
20 = 2 × 2 × 5

∴ LCM of 12, 15, and 20 is = 2 × 2 × 3 × 5 = 60

৫,২২৩.

Mr. Rahman deposited Tk. 40,000 in a scheme at 8% annual simple interest. He closed the account after 6 months and incurred an early withdrawal fee of Tk. 320. Calculate his net earnings.

ক
Tk. 960
খ
Tk. 1280
গ
Tk. 1600
ঘ
Tk. 1120

সঠিক উত্তর: খ

Tk. 1280

উত্তর

সঠিক উত্তর: খ

Tk. 1280

খ

ব্যাখ্যা

Question: Mr. Rahman deposited Tk. 40,000 in a scheme at 8% annual simple interest. He closed the account after 6 months and incurred an early withdrawal fee of Tk. 320. Calculate his net earnings.

Solution:
Given that,
Principal, P = Tk. 40000
Rate of interest, r = 8% per annum
Time, n = 6 months = 6/12 = 1/2 year
Early withdrawal fee = Tk. 320

We know,
Simple interest = Prn/100
= (40000 × 8 × 0.5)/100
= Tk. 1600

∴ Net earnings = Interest earned - Early withdrawal fee
= 1600 - 320
= Tk. 1280

Therefore, his net earnings from the investment are Tk. 1280.

৫,২২৪.

A certain machine produces 1,000 units of product P per hour. Working continuously at this constant rate, this machine will produce how many units of product P in 7 days?

ক
24,000
খ
40,000
গ
7,000
ঘ
168,000

সঠিক উত্তর: ঘ

168,000

উত্তর

সঠিক উত্তর: ঘ

168,000

ঘ

ব্যাখ্যা

Question: A certain machine produces 1,000 units of product P per hour. Working continuously at this constant rate, this machine will produce how many units of product P in 7 days?

Solution:
7 days = 7 × 24 hours = 168 hours
In 1 hour, the machine produces 1,000 units
In 168 hours, the machine produces 1,000 × 168 units = 168000 units

৫,২২৫.

ক
8
খ
32
গ
64
ঘ
256

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question:

Solution:

৫,২২৬.

If the sum of the 3 consecutive integers is 240, then the sum of the two larger integers is:

ক
ক) 79
খ
খ) 159
গ
গ) 169
ঘ
ঘ) 161
ঙ
ঙ) None of these

সঠিক উত্তর: ঘ

ঘ) 161

উত্তর

সঠিক উত্তর: ঘ

ঘ) 161

ঘ

ব্যাখ্যা

Question: If the sum of the 3 consecutive integers is 240, then the sum of the two larger integers is:

Solution:
Let,
Three numbers are , x - 1, x , x + 1

ATQ,
x - 1 + x + x + 1 = 240
⇒ 3x = 240
⇒ x = 240/3
∴ x = 80

∴ the sum of the two larger integers is = x + x + 1
= 80 + 80 + 1
= 161

৫,২২৭.

Which of the following fractions is greater than 3/4 and less than 5/6?

ক
1/2
খ
2/3
গ
4/5
ঘ
9/10

সঠিক উত্তর: গ

4/5

উত্তর

সঠিক উত্তর: গ

4/5

গ

ব্যাখ্যা

Question: Which of the following fractions is greater than 3/4 and less than 5/6?

Solution:
3/4 = 0.75,
5/6 = 0.833,

1/2 = 0.5,
2/3 = 0.66,
4/5 = 0.8,
9/10 = 0.9

Clearly, 0.8 lies between 0.75 and 0.833.

৫,২২৮.

If n = 2.5m + 4 and 15m - 2 = 40 then n = ?

ক
22
খ
11
গ
28
ঘ
36
ঙ
None

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: If n = 2.5m + 4 and 15m - 2 = 40 then n = ?

Solution:
Given, 15m - 2 = 40
15m = 42
m = 42/15
m = 2.8

Now, n = 2.5m + 4
= (2.5 × 2.8) + 4
∴ n = 11

৫,২২৯.

What is the sum of the first 110 natural numbers?

ক
ক) 5010
খ
খ) 5515
গ
গ) 6050
ঘ
ঘ) 6105

সঠিক উত্তর: ঘ

ঘ) 6105

উত্তর

সঠিক উত্তর: ঘ

ঘ) 6105

ঘ

ব্যাখ্যা

আমরা জানি,
n সংখ্যক স্বাভাবিক সংখ্যার যোগফল = {n(n + 1)/2}
∴ 110 টি স্বাভাবিক সংখ্যার যোগফল = {110(110 + 1)/2} = 6105

৫,২৩০.

The average of 7 consecutive numbers is 20. The largest of these number is -

ক
ক) 23
খ
খ) 22
গ
গ) 20
ঘ
ঘ) 24

সঠিক উত্তর: ক

ক) 23

উত্তর

সঠিক উত্তর: ক

ক) 23

ক

ব্যাখ্যা

ধরি, সংখ্যাগুলো, x - 3, x - 2, x - 1, x + 1, x + 2, x + 3
(x - 3 + x - 2 + x - 1 + x + x + 1 + x + 2 + x + 3)/7 = 20
Or, 7x/7 = 20
Or, x = 20
∴ বৃহত্তম সংখ্যাটি = x + 3 = 20 + 3 = 23

৫,২৩১.

Seven men can complete a work in 12 days. They started the work and after 5 days, two men left. In how many days will the work be completed by the remaining men?

ক
ক) 5(1/4)
খ
খ) 6(1/2)
গ
গ) 7(2/3)
ঘ
ঘ) 9(4/5)

সঠিক উত্তর: ঘ

ঘ) 9(4/5)

উত্তর

সঠিক উত্তর: ঘ

ঘ) 9(4/5)

ঘ

ব্যাখ্যা

(7 × 12) men can complete the work in 1 day.
∴ 1 man's 1 day's work = 1/84.
7 men's 5 day's work = (1/12) × 5
= 5/12.
Remaining work = 1 - (5/12)
= 7/12
5 men's 1 day's work = (1/84) × 5
= 5/84.
5/84 work is done by them in 1 days.
7/12 work is done by them in (84/5) × (7/12)
= 49/5 days.
= 9(4/5) days.

৫,২৩২.

(1/2) × (3/4) × (8/4) ÷ (8/2) × (1/2) =?

ক
ক) 3/32
খ
খ) 3/8
গ
গ) 2/3
ঘ
ঘ) 1
ঙ
ঙ) 3/2

সঠিক উত্তর: ক

ক) 3/32

উত্তর

সঠিক উত্তর: ক

ক) 3/32

ক

ব্যাখ্যা

Question: (1/2) × (3/4) × (8/4) ÷ (8/2) × (1/2) =?

Solution:
(1/2) × (3/4) × (8/4) ÷ (8/2) × (1/2)
= (1/2) × (3/4) × (8/4) × (2/8) × (1/2)
= 3/32

৫,২৩৩.

A 107 digit number is formed by writing first 58 natural numbers next to each other. Find the remainder when number is divided by 8.

ক
4
খ
6
গ
7
ঘ
9
ঙ
10

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: A 107 digit number is formed by writing first 58 natural numbers next to each other. Find the remainder when number is divided by 8.

Solution:
Given that the 107-digit number is formed by writing the first 58 natural numbers.
The last few natural numbers are 56, 57, 58.
So, the last three digits of the number are: 56,57,58 which form the number 5758
Thus, the last three digits of the number are 758.
Now, 758 is divided by 8: 758 ÷ 8 = 94 remainder 6

৫,২৩৪.

If the average of 'm' numbers is n² and that of 'n' numbers is m², then the average of (m + n) numbers is -

ক
m + n
খ
(m + n)/mn
গ
mn
ঘ
mn(n + m)

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: If the average of 'm' numbers is n² and that of 'n' numbers is m², then the average of (m + n) numbers is -

Solution:
Sum of m numbers = mn²
Sum of n numbers = nm²

∴ Sum of m and n numbers = mn² + nm²
= mn(n + m)

∴ Average of (m + n) numbers = mn(n + m)/(m + n)
= mn

৫,২৩৫.

The average of ten numbers is 10. If each number is multiplied by 8, what is the new average?

ক
100
খ
72
গ
64
ঘ
80

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: The average of ten numbers is 10. If each number is multiplied by 8, what is the new average?

Solution:
The sum of ten numbers = 10 × 10 = 100

Now,
If each number is multiplied by 8, the new average is
= (The sum of ten numbers × 8) / 10
= (100 × 8) / 10
= 80

৫,২৩৬.

A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle 30° with it. The distance between the foot of the tree to the point where to top touches the ground is 12 m then the height of the tree is-

ক
14√2 m
খ
12√3 m
গ
24√2 m
ঘ
18√3 m

সঠিক উত্তর: খ

12√3 m

উত্তর

সঠিক উত্তর: খ

12√3 m

খ

ব্যাখ্যা

Question: A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle 30° with it. The distance between the foot of the tree to the point where to top touches the ground is 12 m then the height of the tree is-

Solution:

Let the height of the tree be h. Let the part that is still standing on the ground be x, and that part which has fallen be y.
Hence, we have
sin θ = x/y
⇒ sin θ = 1/2 ;[ θ = 30°]
⇒ x/y = 1/2
⇒ y = 2x ...............(1)

Also, it is given that the distance of the tip of the fallen tree to that of the base of the tree is 12 m
⇒ cos θ = 12/y
⇒ cos 30° = 12/y
⇒ √3/2 = 12/y
⇒ y = 24/√3 = (8 × √3 × √3)/√3
⇒ y = 8√3

From (1), x = 8√3/2 = 4√3

∴ Height of tree, h = x + y = 4√3 + 8√3 = 12√3 m

৫,২৩৭.

In a mixture of 30 liters, the ratio of acid and water is 7 : 3. How much more water is to be added to get a new mixture containing acid and water in the ratio of 3 : 7?

ক
30 liters
খ
35 liters
গ
40 liters
ঘ
50 liters

সঠিক উত্তর: গ

40 liters

উত্তর

সঠিক উত্তর: গ

40 liters

গ

ব্যাখ্যা

Question: In a mixture of 30 liters, the ratio of acid and water is 7 : 3. How much more water is to be added to get a new mixture containing acid and water in the ratio of 3 : 7?

Solution:
৩০ লিটার দ্রবণে এসিড ও পানির অনুপাত ৭ : ৩
এসিডের পরিমাণ = (৭/১০) × ৩০
= ২১ লিটার

পানির পরিমাণ = ৩০ - ২১ লিটার
= ৯ লিটার

ধরি, ক লিটার পানি মেশাতে হবে।

প্রশ্নমতে,
২১/(ক + ৯) = ৩ / ৭
⇒ ১৪৭ = ৩ক + ২৭
⇒ ৩ক = ১৪৭ - ২৭
⇒ ৩ক = ১২০
∴ ক = ৪০

অর্থাৎ, ৪০ লিটার পানি মেশাতে হবে।

৫,২৩৮.

What is the solution of the inequality: - 6 ≤ 3x + 3 < 27?

ক
(- 3, 8)
খ
[- 3, 8]
গ
[- 3, 8)
ঘ
(- 3, 8]
ঙ
None of these

সঠিক উত্তর: গ

[- 3, 8)

উত্তর

সঠিক উত্তর: গ

[- 3, 8)

গ

ব্যাখ্যা

Question: What is the solution of the inequality: - 6 ≤ 3x + 3 < 27?

Solution:
- 6 ≤ 3x + 3 < 27
⇒ - 6 - 3 ≤ 3x + 3 - 3 < 27 - 3
⇒ - 9 ≤ 3x < 24
⇒ - 9/3 ≤ 3x/3 < 24/3
⇒ - 3 ≤ x < 8

∴ solution of the inequality: [-3, 8)

৫,২৩৯.

If 5 years ago, the ratio of age of Hasan and Rina was 1 : 2 and after 15 years from present their ratio would be 5 : 6. Find the age of Rina after 20 years.

ক
30 years
খ
45 years
গ
35 years
ঘ
38 years

সঠিক উত্তর: গ

35 years

উত্তর

সঠিক উত্তর: গ

35 years

গ

ব্যাখ্যা

Question: If 5 years ago, the ratio of age of Hasan and Rina was 1 : 2 and after 15 years from present their ratio would be 5 : 6. Find the age of Rina after 20 years.

Solution:
Let, present age of Hasan be x
and present age of Rina be y.

Then, according to question
(x - 5)/(y - 5) = 1/2
⇒ 2x - 10 = y - 5
⇒ x = (y + 5)/2 .............(1)

Also,
(x + 15)/(y + 15) = 5/6
⇒ 6x + 90 = 5y + 75
⇒ 6x + 15 = 5y

Putting value of x from equation 1, we get
3y + 15 + 15 = 5y
⇒ 2y = 30
⇒ y = 15

∴ Age of Rina after 20 years = 15 + 20 = 35 years.

৫,২৪০.

Today is Monday. After 100 days, what day of the week will it be?

ক
Tuesday
খ
Wednesday
গ
Thursday
ঘ
Friday

সঠিক উত্তর: খ

Wednesday

উত্তর

সঠিক উত্তর: খ

Wednesday

খ

ব্যাখ্যা

Question: Today is Monday. After 100 days, what day of the week will it be?

Solution:
Each day of the week is repeated after 7 days.
So, after (7 × 14) = 98 days, it will be Monday.
After 99 days, it will be Tuesday.
∴ After 100 days, it will be Wednesday.

৫,২৪১.

The area of a rectangle is 40 cm² and one of its sides is 5 cm long. What will be its perimeter?

ক
13 cm
খ
20 cm
গ
26 cm
ঘ
33 cm

সঠিক উত্তর: গ

26 cm

উত্তর

সঠিক উত্তর: গ

26 cm

গ

ব্যাখ্যা

Question: The area of a rectangle is 40 cm² and one of its sides is 5 cm long. What will be its perimeter?

Solution:
another side = 40/5
= 8 cm

perimeter = 2 (8 + 5)
= 2 × 13
= 26 cm

৫,২৪২.

If the equation 4x² + x(p + 1) + 1 = 0 has exactly two equal roots, one of the value of p is-

ক
5
খ
- 3
গ
3
ঘ
0

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: If the equation 4x² + x(p + 1) + 1 = 0 has exactly two equal roots, one of the value of p is-

Solution:
Roots are equal if discriminant
b² - 4ac = 0

In given equation:
(p + 1)² - 4.4.1 = 0
⇒ (p + 1)² = 16 = (4)²
⇒ p +1 = ± 4

If take positive value
p = 3

If take negative value
p = - 5

৫,২৪৩.

A collection of books went on sale and 2/3 of them was sold for Taka 2.30 each. If none of the 36 remaining books were sold, what was the total amount received for the books that were sold?

ক
ক) 165.6
খ
খ) 180
গ
গ) 135.6
ঘ
ঘ) 90

সঠিক উত্তর: ক

ক) 165.6

উত্তর

সঠিক উত্তর: ক

ক) 165.6

ক

ব্যাখ্যা

Question: A collection of books went on sale and 2/3 of them was sold for Taka 2.30 each. If none of the 36 remaining books were sold, what was the total amount received for the books that were sold?

Solution:
মনেকরি
মোট বই = x টি
বিক্রিত বই = 2x/3 অংশ
বই বিক্রয় হয়নি = x - (2x/3) অংশ
= (3x - 2x)/3
= x/3

এখন
x/3 অংশ বই = 36
2x/3 অংশ বই = 36 × 2
= 72 টি

বিক্রয়কৃত বইয়ের মূল্য= (72 × 2.30) টাকা
= 165.6 টাকা

৫,২৪৪.

Today is Friday. After 60 days, what day of the week will it be?

ক
Sunday
খ
Tuesday
গ
Monday
ঘ
Thursday

সঠিক উত্তর: খ

Tuesday

উত্তর

সঠিক উত্তর: খ

Tuesday

খ

ব্যাখ্যা

Question: Today is Friday. After 60 days, what day of the week will it be?

Solution:
Each day of the week is repeated after 7 days.
So, after (7 × 8) = 56 days, it will be Friday.
After 57 days, it will be Saturday.
After 58 days, it will be Sunday.
After 59 days, it will be Monday.

∴ After 60 days, it will be Tuesday.

৫,২৪৫.

A person swimming in a stream that flows 3 km/hr finds that in a given time, he can swim four times as far with the stream as he can against it. At what rate does he swim?

ক
3 km/hr
খ
5 km/hr
গ
4.5 km/hr
ঘ
6 km/hr

সঠিক উত্তর: খ

5 km/hr

উত্তর

সঠিক উত্তর: খ

5 km/hr

খ

ব্যাখ্যা

Question: A person swimming in a stream that flows 3 km/hr finds that in a given time, he can swim four times as far with the stream as he can against it. At what rate does he swim?

Solution:
ধরি,
স্রোতের প্রতিকূলে গতিবেগ = x কিমি/ঘন্টা
এবং স্রোতের অনুকূলে গতিবেগ = 4x কিমি/ঘন্টা।

স্রোতের গতিবেগ = (স্রোতের অনুকূলে গতিবেগ - স্রোতের প্রতিকূলে গতিবেগ)/2
= (4x - x)/2
= 3x/2 কিমি/ঘন্টা

প্রশ্নমতে, স্রোতের গতিবেগ 3 কিমি/ঘন্টা।
⇒ 3x/2 = 3
⇒ 3x = 6
⇒ x = 2

স্থির পানিতে সাঁতার কাটার গতিবেগ = (স্রোতের অনুকূলে গতিবেগ + স্রোতের প্রতিকূলে গতিবেগ)/2
= (4x + x)/2
= 5x/2
= (5 × 2)/2 [x এর মান বসিয়ে]
= 10/2
= 5 কিমি/ঘন্টা

সুতরাং, লোকটি 5 কিমি/ঘন্টা গতিতে সাঁতার কাটে।

৫,২৪৬.

What is the mean of the range, mode and median of the data given below?
5, 10, 3, 6, 4, 8, 9, 3, 15, 2, 9, 4, 19, 11, 4

ক
8.33
খ
9.33
গ
10.63
ঘ
11.63

সঠিক উত্তর: খ

9.33

উত্তর

সঠিক উত্তর: খ

9.33

খ

ব্যাখ্যা

Question: What is the mean of the range, mode and median of the data given below?
5, 10, 3, 6, 4, 8, 9, 3, 15, 2, 9, 4, 19, 11, 4

Solution:
Given data = 5, 10, 3, 6, 4, 8, 9, 3, 15, 2, 9, 4, 19, 11, 4
Arranging in ascending order = 2, 3, 3, 4, 4, 4, 5, 6, 8, 9, 9, 10, 11, 15, 19
Here,
Most frequent data is 4
So, Mode = 4

Total terms in the given data, (n) = 15 (It is odd)
∴ Median = {(n + 1)/2}^th term
= {(15 + 1)/2}^th term
= (8)^th term
= 6

Now,
Range = (Maximum value - Minimum value) + 1
= (19 - 2) + 1
= 18

∴ Mean of Range, Mode and Median = (Range + Mode + Median)/3
= (18 + 4 + 6)/3 = 28/3 = 9.33

৫,২৪৭.

is equal to?

ক
ক) 2cosecθ
খ
খ) secθ
গ
গ) 1/2secθ
ঘ
ঘ) 2secθ

সঠিক উত্তর: ঘ

ঘ) 2secθ

উত্তর

সঠিক উত্তর: ঘ

ঘ) 2secθ

ঘ

ব্যাখ্যা

Question: is equal to?

Solution:
2secθ

৫,২৪৮.

ক
ক) 5
খ
খ) 3
গ
গ) 2
ঘ
ঘ) 1
ঙ
ঙ) 7

সঠিক উত্তর: খ

খ) 3

উত্তর

সঠিক উত্তর: খ

খ) 3

খ

ব্যাখ্যা

Hence,
we can write, √(6 + x) = x
⇒ 6 + x = x²
⇒ x² - x - 6 = 0
⇒ x2 - 3x + 2x - 6 = 0
⇒ x(x - 3) + 2(x - 3) = 0
⇒ (x + 2) (x - 3) = 0
⇒ x = -2, 3

Since, we cannot be negative, therefore, x = 3.

৫,২৪৯.

A train 240 m long is running at a speed of 90 km/hr. In what time will it pass a bridge 260 m long?

ক
20 seconds
খ
30 seconds
গ
40 seconds
ঘ
50 seconds
ঙ
55 seconds

সঠিক উত্তর: ক

20 seconds

উত্তর

সঠিক উত্তর: ক

20 seconds

ক

ব্যাখ্যা

Speed = 90 km/hr
= 90 × 5/18 m/s
= 25 m/s
Total distance = (240 + 260) m
= 500 m
Required time = (500m) / (25m/s)
= 20 seconds
[ Time = Distance / Speed ]

৫,২৫০.

Sami took a loan of Tk. 2400 with simple interest for as many years as the rate of interest. If he paid Tk. 864 as interest at the end of the loan period, what was the rate of interest?

ক
3.6
খ
6
গ
18
ঘ
4

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: Sami took a loan of Tk. 2400 with simple interest for as many years as the rate of interest. If he paid Tk. 864 as interest at the end of the loan period, what was the rate of interest?

Solution:
আসল, P = 2400
সুদের হার = r%
সময়, n = r
সুদ, I = 864

∴ r = (I × 100)/(P × n)
⇒ r(2400 × r)= (864 × 100)
⇒ 24r² = 864
⇒ r²= 864/24
⇒ r² = 36
∴ r = 6

৫,২৫১.

If 8 men or 12 women can build a 240-meter road in 10 days, how many days will 6 men and 6 women take to build a 360-meter road?

ক
12 days
খ
8 days
গ
15 days
ঘ
18 days

সঠিক উত্তর: ক

12 days

উত্তর

সঠিক উত্তর: ক

12 days

ক

ব্যাখ্যা

Question: If 8 men or 12 women can build a 240-meter road in 10 days, how many days will 6 men and 6 women take to build a 360-meter road?

Solution:
দেওয়া আছে, ৮ জন পুরুষের কাজের ক্ষমতা = ১২ জন মহিলার কাজের ক্ষমতা
অর্থাৎ, ২ জন পুরুষ = ৩ জন মহিলা

এখন,
৩ জন মহিলা = ২ জন পুরুষ
∴ ১ জন মহিলা = (২/৩) জন পুরুষ
∴ ৬ জন মহিলা = ৬ × (২/৩) = ৪ জন পুরুষ
সুতরাং, মোট শ্রমিক সংখ্যা = ৬ জন পুরুষ + ৪ জন পুরুষ = ১০ জন পুরুষ।

৮ জন পুরুষ ২৪০ মিটার রাস্তা তৈরি করে ১০ দিনে।
∴ ১ জন পুরুষ ২৪০ মিটার রাস্তা তৈরি করে = (১০ × ৮) = ৮০ দিনে
∴ ১ জন পুরুষ ১ মিটার রাস্তা তৈরি করে = (৮০ ÷ ২৪০) = ১/৩ দিনে
∴ ১০ জন পুরুষ ১ মিটার রাস্তা তৈরি করে = (১/৩ ÷ ১০) = ১/৩০ দিনে
∴ ১০ জন পুরুষ ৩৬০ মিটার রাস্তা তৈরি করে = (১/৩০) × ৩৬০ = ১২ দিনে।

∴ ৬ জন পুরুষ এবং ৬ জন মহিলা ৩৬০ মিটার রাস্তা তৈরি করতে ১২ দিন সময় নেবে।

৫,২৫২.

In a quiz competition, Ms. Fatima is the 20th highest and the 15th lowest in the rankings. How many contestants were in the competition?

ক
36
খ
35
গ
32
ঘ
34

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: In a quiz competition, Ms. Fatima is the 20th highest and the 15th lowest in the rankings. How many contestants were in the competition?

Solution:
Given that,
Ms. Fatima is the 20th highest,
and the 15th lowest ranked contestant.

∴ Total = people above + Ms. Fatima + people below = 19 + 1 + 14 = 34

৫,২৫৩.

Out of three numbers, the first is twice the second and is half of the third. If the average of the three numbers is 56, then the average of the first and the third number is -

ক
52
খ
66
গ
72
ঘ
94

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: Out of three numbers, the first is twice the second and is half of the third. If the average of the three numbers is 56, then the average of the first and the third number is -

Solution:
Let,
the second number be = x
So, the first number is = 2x
and the third number is = 2x × 2 = 4x (since the first is half of the third)

ATQ,
(2x + x + 4x)/3 = 56
⇒ 2x + x + 4x = 56 × 3
⇒ 7x = 168
∴ x = 24

∴ the second number = 24
So, the first number is = 2 × 24 = 48
and the third number is = 4 × 24 = 96

Then, the average of the first and the third number is = (48 + 96)/2 = 72

৫,২৫৪.

The LCM of two numbers is 1920 and their HCF is 16. If one of the number is 128, find the other number.

ক
216
খ
224
গ
230
ঘ
240
ঙ
260

সঠিক উত্তর: ঘ

240

উত্তর

সঠিক উত্তর: ঘ

240

ঘ

ব্যাখ্যা

Question: The LCM of two numbers is 1920 and their HCF is 16. If one of the number is 128, find the other number.

Solution:
1st number × 2nd number = L.C. M. × H.C.F

We have,
First number × second number = LCM × HCF
∴ Second number = (1920 × 16)/128
= 240

৫,২৫৫.

The perimeter of an isosceles triangle is 100 cm. If the base is 36 cm, find the length of the equal sides.

ক
64 cm
খ
24 cm
গ
36 cm
ঘ
32 cm

সঠিক উত্তর: ঘ

32 cm

উত্তর

সঠিক উত্তর: ঘ

32 cm

ঘ

ব্যাখ্যা

Question: The perimeter of an isosceles triangle is 100 cm. If the base is 36 cm, find the length of the equal sides.

Solution:
Let the length of equal side = x.
∴ x + x + 36 = 100
⇒ 2x = 64
∴ x = 32cm.

৫,২৫৬.

A town had 10,000 residents in 2000. Its population declines at a rate of 10% per annum. What will be its total population in 2005?

ক
ক) 5400.50
খ
খ) 5604.71
গ
গ) 5804.81
ঘ
ঘ) 5904.90

সঠিক উত্তর: ঘ

ঘ) 5904.90

উত্তর

সঠিক উত্তর: ঘ

ঘ) 5904.90

ঘ

ব্যাখ্যা

The population of the town decreases by 10% every year.
Thus, it has a new population every year. So the population for the next year is calculated on the current year population.
For the decrease, we have the formula A = P(1 - R/100)ⁿ

Therefore,
the population at the end of 5 years
= 10000(1 - 10/100)⁵= 10000(1 - 0.1)⁵
= 10000 x 0.9⁵
= 5904.9

৫,২৫৭.

Rifat, Joy, Banik subscribe to Tk. 50,000 for a business. Rifat subscribes Tk. 4000 more than Joy and Joy subscribes Tk. 5000 more than Banik. Out of a total profit of Tk. 35000, Rifat receives-

ক
ক) Tk. 8400
খ
খ) Tk. 11900
গ
গ) Tk. 15200
ঘ
ঘ) Tk. 14700

সঠিক উত্তর: ঘ

ঘ) Tk. 14700

উত্তর

সঠিক উত্তর: ঘ

ঘ) Tk. 14700

ঘ

ব্যাখ্যা

প্রশ্ন: Rifat, Joy, Banik subscribe to Tk. 50000 for a business. Rifat subscribes Tk. 4000 more than Joy and Joy subscribes Tk. 5000 more than Banik. Out of a total profit of Tk. 35000, Rifat receives-

সমাধান:
Let,
Banik subscribes = x
Then,
Joy subscribe = x + 5000
Rifat subscribe = x + 5000 + 4000 = x + 9000
∴ x + x + 5000 + x + 9000 = 50000
⇒ 3x = 36000
⇒ x = 12000

Rifat : Joy : Banik = 21000 : 17000 : 12000
= 21 : 17 : 12

∴ Rifat’s share of profit = 35000 × (21/50) = 14700

৫,২৫৮.

In a camp of soldiers there was a stock of food for 90 days for 6000 soldiers. After 30 days, 2400 soldiers left the barracks. For how many days shall the leftover food last for the remaining soldiers?

ক
120 days
খ
100 days
গ
150 days
ঘ
160 days

সঠিক উত্তর: খ

100 days

উত্তর

সঠিক উত্তর: খ

100 days

খ

ব্যাখ্যা

Question: In a camp of soldiers there was a stock of food for 90 days for 6000 soldiers. After 30 days, 2400 soldiers left the barracks. For how many days shall the leftover food last for the remaining soldiers?
Let,
the remaining food last for P days
6000 soldiers had provision for 90 days
3600 soldiers had provision for P days

ATQ,
3600/6000 = 60/P
⇒ 3/5 = 60/P
⇒ 3P = 300
⇒ P = 300/3
∴ P = 100

The remaining food last for 100 days for the remaining soldiers.

৫,২৫৯.

The equation of the given curve is:

ক
y = 7x
খ
x²= 9y
গ
x² + y² = 25
ঘ
f(t) = (A + Bt)e^{- Ct}

সঠিক উত্তর: ঘ

f(t) = (A + Bt)e^{- Ct}

উত্তর

সঠিক উত্তর: ঘ

f(t) = (A + Bt)e^{- Ct}

ঘ

ব্যাখ্যা

The equation of the given curve is:
f(t) = (A + Bt)e^{- Ct}
y = 7x is the equation of the straight line.
x² = 9y is the equation of conic
x² + y² = 25 is the equation of circle.

৫,২৬০.

In how many different ways can the letters of the word 'BINARY' be arranged so that the vowels always come together?

ক
120 ways
খ
240 ways
গ
660 ways
ঘ
720 ways

সঠিক উত্তর: খ

240 ways

উত্তর

সঠিক উত্তর: খ

240 ways

খ

ব্যাখ্যা

Question: In how many different ways can the letters of the word "BINARY" be arranged so that the vowels always come together?

Solution:
the given words contain 6 different letters.
When the vowels "ia" are taken together, we may treat them as 1 letter.

5 numbers can be arranged in = 5! ways
= 120 ways

two vowels can be arranged = 2! ways
= 2 ways

∴ Total number of arrangement = (120 × 2) ways
= 240 ways

৫,২৬১.

A watch is 1 minute slow at 1 pm on Tuesday and 2 minutes fast at 1 pm on Thursday. When did it show the correct time = ?

ক
5 : 00 pm on Wednesday
খ
5 : 00 am on Wednesday
গ
1 : 00 pm on Wednesday
ঘ
1 : 00 am on Wednesday

সঠিক উত্তর: খ

5 : 00 am on Wednesday

উত্তর

সঠিক উত্তর: খ

5 : 00 am on Wednesday

খ

ব্যাখ্যা

Question: A watch is 1 minute slow at 1 pm on Tuesday and 2 minutes fast at 1 pm on Thursday. When did it show the correct time = ?

Solution:
৩ মিনিট পরিবর্তন হয় ২ × ২৪ ×৬০ মিনিটে
১ মিনিট পরিবর্তন হয় (২ × ২৪ ×৬০)/৩ মিনিটে = (২ × ২৪ ×৬০)/৩ × ৬০ ঘণ্টায়
= ১৬ ঘণ্টায়

অতএব, ঘড়িটি সঠিক সময় দিবে বুধবার ভোর ৫ টায়।

৫,২৬২.

In how many different ways can the letters of the word BANANA be arranged where all A will be together?

ক
12
খ
16
গ
20
ঘ
24

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: In how many different ways can the letters of the word BANANA be arranged where all A will be together?

Solution:
If all the A remain together than we count them as 1 letter.
Three A can be arranged in 3!/3! = 1 way
∴ Number of letter in the word where three A's are counted as 1,
So, total letters will be = 4
Repeated letters N = 2

∴ The number of arrangement = 4!/2! = 12

৫,২৬৩.

What is the difference between the compound interest on Tk. 10000 for 1 year at 8% per annum compounded yearly and half yearly?

ক
ক) 13
খ
খ) 16
গ
গ) 18
ঘ
ঘ) 19

সঠিক উত্তর: খ

খ) 16

উত্তর

সঠিক উত্তর: খ

খ) 16

খ

ব্যাখ্যা

Question: What is the difference between the compound interest on Tk. 10000 for 1 year at 8% per annum compounded yearly and half yearly?

Solution:
Interest compounded half yearly = [10000 × (1 + (4/100))²] - 10000
= (10000 × (26/25) × (26/25)) - 10000
= 816
Interest compunded yearly = [10000 × (1 + (8/100))] - 10000
= (10000 × (27/25)) - 10000
= 800
So, the difference of interest = 816 - 800 = 16

৫,২৬৪.

The sum of first 40 natural numbers is -

ক
800
খ
810
গ
820
ঘ
840

সঠিক উত্তর: গ

820

উত্তর

সঠিক উত্তর: গ

820

গ

ব্যাখ্যা

Question: The sum of first 40 natural numbers is -

Solution:
We know that,
The sum of the first n natural numbers is n(n+1)/2

Sum of first 40 natural numbers is = 40(40 + 1)/2
= 820

৫,২৬৫.

Maximum value of (2sinθ + 3cosθ) is?

ক
√13
খ
√5
গ
√13/2
ঘ
√7

সঠিক উত্তর: ক

√13

উত্তর

সঠিক উত্তর: ক

√13

ক

ব্যাখ্যা

Maximum value of (2sinθ+3cosθ) = √(2² + 3²) = √13

৫,২৬৬.

A book vendor sold a book at loss of 10%. Had he sold it for Tk. 108 more, he would have earned a profit of 10%. Find the cost of the book = ?

ক
ক) Tk. 442
খ
খ) Tk. 540
গ
গ) Tk. 648
ঘ
ঘ) Tk. 740

সঠিক উত্তর: খ

খ) Tk. 540

উত্তর

সঠিক উত্তর: খ

খ) Tk. 540

খ

ব্যাখ্যা

Given loss = 10% profit = 10%
Difference of overall profit and loss = 10 - (-10) = 20%
20% of cp = sp
20/ 100×cp = 108
20 × cp = 108 × 100
cp = 10800/20
∴ cp = 540

৫,২৬৭.

In what proportion water must be added to spirit to gain 20% by selling it at the cost price?

ক
3 : 8
খ
2 : 7
গ
1 : 5
ঘ
none of these

সঠিক উত্তর: গ

1 : 5

উত্তর

সঠিক উত্তর: গ

1 : 5

গ

ব্যাখ্যা

Question: In what proportion water must be added to spirit to gain 20% by selling it at the cost price?

Solution:
The percentage gain is essentially the ratio of pure spirit to water in the diluted solution. This gain is made because the addition of water increases the volume without increasing the cost, allowing the adulterated spirit to be sold at the original price (the cost price for the pure spirit).
Since the selling price equals the cost price (CP) in this case, the 20% gain represents the proportion of water in the solution.

Therefore, the proportion of spirit to water is 100% : 20% or 5 : 1.

Hence, water should be added to spirit in a 1 ∶ 5 proportion to gain 20% by selling it at the cost price.

৫,২৬৮.

In a business venture, Humayun and Kabir contribute in the ratio of 5:4. With 10% of the total profit allocated to charity and Humayun’s share being Tk 945, what is the overall profit?

ক
1720 tk
খ
1800 tk
গ
1890 tk
ঘ
1960 tk

সঠিক উত্তর: গ

1890 tk

উত্তর

সঠিক উত্তর: গ

1890 tk

গ

ব্যাখ্যা

Question: In a business venture, Humayun and Kabir contribute in the ratio of 5:4. With 10% of the total profit allocated to charity and Humayun’s share being Tk 945, what is the overall profit?

Solution:
Let,
The total profit = 100 tk

After paying 10% to charity,
Humayun’s share = 90 × (5/9)
= 50 tk

Now,
If Humayun’s share is tk 50, total profit = 100 tk
If Humayun’s share is tk 1, total profit = 100/50 tk
If Humayun’s share is tk 945, total profit = (100/50) × 945
= 1890 tk

So, the total profit is 1890 tk.

৫,২৬৯.

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

ক
Tk. 20000
খ
Tk. 22000
গ
Tk. 18000
ঘ
Tk. 23500

সঠিক উত্তর: ক

Tk. 20000

উত্তর

সঠিক উত্তর: ক

Tk. 20000

ক

ব্যাখ্যা

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

Solution:
Let the Principal = Tk. 100
When compounded annually,
A = 100 [1 + 20/100]²
= 100 × 1.2 × 1.2
= 144

When compounded half yearly,
A = 100[1 + 10/100]⁴
= 100 × 1.1 × 1.1 × 1.1 × 1.1
= 146.41

Difference, 146.41 - 144 = 2.41

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

৫,২৭০.

Sima started a software business by investing Tk. 50,000. After six months, Rajon joined her with a capital of Tk. 80000. After 3 years, they earned a profit of Tk. Tk. 24500. What was Sima’s share in the profit?

ক
ক) Tk. 9423
খ
খ) Tk. 10250
গ
গ) Tk. 10500
ঘ
ঘ) Tk. 14000

সঠিক উত্তর: গ

গ) Tk. 10500

উত্তর

সঠিক উত্তর: গ

গ) Tk. 10500

গ

ব্যাখ্যা

Sima : Rajon = (50000 × 36) : (80000 × 30)
1800000 : 2400000
= 3 : 4
∴ Sima's share = Tk. (24500 × 3/7)
= Tk. 10500.

৫,২৭১.

৫০ কি.মি./ঘণ্টা গতিতে চলে, একটি ট্রেন সঠিক সময়ে তার গন্তব্যে পৌঁছে। ট্রেনটি ৪০কি.মি. /ঘণ্টা গতিবেগে চললে তাহলে ২৪ মিনিট দেরি হয়। মোট দূরত্ব কত কিলোমিটার?

ক
ক) ৬০ কিলোমিটার
খ
খ) ৮০ কিলোমিটার
গ
গ) ৮৫ কিলোমিটার
ঘ
ঘ) ৯০ কিলোমিটার

সঠিক উত্তর: খ

খ) ৮০ কিলোমিটার

উত্তর

সঠিক উত্তর: খ

খ) ৮০ কিলোমিটার

খ

ব্যাখ্যা

মনেকরি
মোট দূরত্ব = ক কিলোমিটার

প্রশ্নমতে,
ক /৪০ - ক/৫০ = ২৪/৬০
(৫ক - ৪ক)/২০০ = ২৪/৬০
ক/২০০ = ২৪/৬০
ক = (২৪/৬০) × ২০০
ক = ৮০ কিলোমিটার

৫,২৭২.

We reverse a number and form a new one. The old number is 45 less than the new number. The sum of the digits of the old number is 9. What is the new number?

ক
a. 36
খ
b. 54
গ
c. 72
ঘ
d. 81

সঠিক উত্তর: গ

c. 72

উত্তর

সঠিক উত্তর: গ

c. 72

গ

ব্যাখ্যা

Let the two digits be X and Y.
Let the older number be A and the newer one be B.
A = 10X + Y
∴ B = 10Y + X

From given, B = 45 + A = 45 + 10X + Y
10Y + X = 45 + 10X + Y
⇒ 9Y - 9X = 45
⇒ 9(Y - X) = 45
⇒ Y - X = 45/9
⇒ Y - X = 5

Y - X = 5 ----------------- (1)
X + Y = 9 ---------------- (2)

Solving (1) and (2),
Y - X + X + Y = 5 + 9
⇒ 2Y = 14
⇒ Y = 7
∴ X = 9 - Y
= 9 - 7 = 2

So, A = 27; B = 72

৫,২৭৩.

666 ÷ 6 ÷ 3 = ?

ক
37
খ
333
গ
111
ঘ
84

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: 666 ÷ 6 ÷ 3 = ?

Solution:
666 ÷ 6 ÷ 3
=(666/6) ÷ 3
= 111 ÷ 3
= 37

৫,২৭৪.

A can write 100 pages in 25 hours. A and B together can write 210 pages in 30 hours. In what time can B write 42 pages?

ক
10 hours
খ
12 hours
গ
14 hours
ঘ
18 hours
ঙ
None

সঠিক উত্তর: গ

14 hours

উত্তর

সঠিক উত্তর: গ

14 hours

গ

ব্যাখ্যা

Question: A can write 100 pages in 25 hours. A and B together can write 210 pages in 30 hours. In what time can B write 42 pages?

Solution:
Given,
In 25 hours A can write 100 pages
∴ In 1 hour A can write 100/25 pages
= 4 pages

Here,
∴ In 1 hour A and B together can write 210/30 pages
= 7 pages
and,
B's 1 hour work = (A + B)'s 1 hour work - A's 1 hour work
= 7 - 4
= 3 pages/hour

B's time
3 pages in 1 hour
∴ 1 page in 1/3 hour
∴ 42 pages in = (1 × 42)/3 hour
= 14 hours

৫,২৭৫.

45% of 750 - 25% of 480 = ?

ক
217.50
খ
376.21
গ
120
ঘ
337.50

সঠিক উত্তর: ক

217.50

উত্তর

সঠিক উত্তর: ক

217.50

ক

ব্যাখ্যা

Question: 45% of 750 - 25% of 480 = ?

Solution:
Given that,
= 45 × (750/100) - 25 × (480/100)
= (45 × 7.5) - (25 × 4.8)
= 337.5 - 120
= 217.50

৫,২৭৬.

If log₃(x² - 4x) - log₃(x - 4) = 4, Than what is the value of x.

ক
36
খ
405
গ
64
ঘ
81

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: If log₃(x² - 4x) - log₃(x - 4) = 4, Than what is the value of x.

Solution:
Given that,
log₃(x² - 4x) - log₃(x - 4) = 4
⇒ log₃[x(x - 4)/(x - 4)] = 4 ; [log_aM - log_aN = log_a(M/N)]
⇒ log₃x = 4
⇒ x = 3⁴
∴ x = 81

৫,২৭৭.

A square has an area of 25 m². If a circle has a radius equal to the length of the square’s diagonal, what is the area of the circle?

ক
75π sq. m.
খ
50π sq. m.
গ
100π sq. m.
ঘ
65π sq. m.

সঠিক উত্তর: খ

50π sq. m.

উত্তর

সঠিক উত্তর: খ

50π sq. m.

খ

ব্যাখ্যা

Question: A square has an area of 25 m². If a circle has a radius equal to the length of the square’s diagonal, what is the area of the circle?

Solution:
Area of square = 25
Side of square = √25 = 5

Diagonal of square = 5√2
So, the radius of the circle is 5√2 m

Area of circle = πr²
= π(5√2)²
= 50π m²

∴ The area of the circle is 50π sq. m.

৫,২৭৮.

If Adnan’s salary is 60% higher than Bina’s salary, by what percentage is Bina’s salary less than Adnan’s?

ক
27.5% lower
খ
35.5% lower
গ
30.5% lower
ঘ
37.5% lower

সঠিক উত্তর: ঘ

37.5% lower

উত্তর

সঠিক উত্তর: ঘ

37.5% lower

ঘ

ব্যাখ্যা

Question: If Adnan’s salary is 60% higher than Bina’s salary, by what percentage is Bina’s salary less than Adnan’s?

Solution:
Let,
Bina’s Salary is Tk. 100.
Then,
∴ Adnan’s Salary = (100 + 60% of 100)
= 100 + (60× 100)/100
= 100 + 60
= 160

∴ Difference between Adnan’s Salary and Bina’s Salary = 160 - 100 = 60

∴ lower = (60/160) × 100 = 37.5%

∴ Bina’s salary is 37.5% lower than Adnan’s salary.

৫,২৭৯.

Two vessels P and Q contain 62.5% and 87.5% of alcohol respectively. If 2 litres from vessel P is mixed with 4 litres from vessel Q, the ratio of alcohol and water in the resulting mixture is?

ক
ক) 16 : 5
খ
খ) 14 : 5
গ
গ) 16 : 7
ঘ
ঘ) 19 : 5

সঠিক উত্তর: ঘ

ঘ) 19 : 5

উত্তর

সঠিক উত্তর: ঘ

ঘ) 19 : 5

ঘ

ব্যাখ্যা

Quantity of alcohol in vessel P = 62.5/100 × 2 = 5/4 litres
Quantity of alcohol in vessel Q = 87.5/100 × 4 = 7/2 litres
Quantity of alcohol in the mixture formed = 5/4 + 7/2 = 19/4 = 4.75 litres
As 6 litres of mixture is formed, ratio of alcohol and water in the mixture formed
= 4.75 : 1.25 = 19 : 5.

৫,২৮০.

A bicycle marked at Tk. 2000 is sold for Tk. 1200 after two successive discounts. The first discount was 20%. Find the second discount percentage.

ক
35.75%
খ
25%
গ
30%
ঘ
20.25%

সঠিক উত্তর: খ

25%

উত্তর

সঠিক উত্তর: খ

25%

খ

ব্যাখ্যা

Question: A bicycle marked at Tk. 2000 is sold for Tk. 1200 after two successive discounts. The first discount was 20%. Find the second discount percentage.

Solution:
Given that,
Original marked price = Tk. 2000
Final selling price after two successive discounts = Tk. 1200
First discount = 20%

Now,
Price after first discount = 2000 - 20% of 2000
= 2000 - (20/100) × 2000
= 2000 - 400
= Tk. 1600

Let the second discount be x%.
This discount is applied on Tk. 1600, and after the second discount, the selling price becomes Tk. 1200.
Discount amount in second discount = 1600 - 1200 = Tk. 400

So the second discount percentage ia,
⇒ x% of 1600 = 400
⇒ x = (400/1600) × 100
⇒ x = (1/4) × 100
∴ x = 25%

So the second discount is 25%.

৫,২৮১.

A boat travel 240 km downstream in 3 hours and the time is taken by the boat to travel the same distance in upstream in 6 hours. Find the speed of stream (in kmph).

ক
28 km/h.
খ
24 km/h.
গ
10 km/h.
ঘ
12 km/h.
ঙ
20 km/h.

সঠিক উত্তর: ঙ

20 km/h.

উত্তর

সঠিক উত্তর: ঙ

20 km/h.

ঙ

ব্যাখ্যা

Question: A boat travel 240 km downstream in 3 hours and the time is taken by the boat to travel the same distance in upstream in 6 hours. Find the speed of stream (in kmph).

Solution:
Given that,
Distance (both ways) = 240 km
Time downstream = 3 hours
Time upstream = 6 hours

Now,
Downstream speed = 240/3 = 80 km/h
Upstream speed = 240/6 = 40 km/h

Let,
Speed of boat in still water = B
Speed of stream = S

Then,
B + S = 80 .........(1)
B - S = 40 ...........(2)
Now, (1) - (2) then we get,
⇒ B + S - B + S = 80 - 40
⇒ 2S = 40
∴ S = 40/2 = 20 km/h

Therefore, the speed of the stream is 20 km/h.

৫,২৮২.

A mother is twice as old as her son. If 20 years ago, the age of the mother was 10 times the age of the son, what is the present age of the mother?

ক
38 years
খ
40 years
গ
43 years
ঘ
45 years

সঠিক উত্তর: ঘ

45 years

উত্তর

সঠিক উত্তর: ঘ

45 years

ঘ

ব্যাখ্যা

Question: A mother is twice as old as her son. If 20 years ago, the age of the mother was 10 times the age of the son, what is the present age of the mother?

Solution:
Let the age of son = x years
∴ Age of mother would be = 2x years

As per question 20 years ago;
10 (x - 20) = 2x - 20
⇒ 10x - 200 = 2x - 20
⇒ 10x - 2x = - 20 + 200
⇒ 8x = 180
⇒ x = 180/8
∴ x = 22.5

∴ Mother's age is = 2 × 22.5 = 45 years

৫,২৮৩.

If x : y = y : z = 1.5 and z = 2 , What is the value of x?

ক
ক) 3
খ
খ) 4
গ
গ) 4.5
ঘ
ঘ) 3.5
ঙ
ঙ) 5

সঠিক উত্তর: গ

গ) 4.5

উত্তর

সঠিক উত্তর: গ

গ) 4.5

গ

ব্যাখ্যা

Given, y : z = 1.5
∴ y = 1.5 × 2 = 3
And, x : y = 1.5
∴ x = 1.5 × 3 = 4.5

৫,২৮৪.

A man walk a certain distance and rides back in 4 hours 30 minutes. He could ride both ways in 3 hours. The time required by the man to walk both ways is:

ক
ক) 5 hours
খ
খ) 6 hours 30 minutes.
গ
গ) 5 hours 45 minutes.
ঘ
ঘ) 6 hours

সঠিক উত্তর: ঘ

ঘ) 6 hours

উত্তর

সঠিক উত্তর: ঘ

ঘ) 6 hours

ঘ

ব্যাখ্যা

Question: A man walk a certain distance and rides back in 4 hours 30 minutes. He could ride both ways in 3 hours. The time required by the man to walk both ways is:

Solution:
Time taken to ride one way = 3/2
= 1.5 hrs
Time taken to walk one way = 4.5 - 1.5
= 3 hrs
Time taken to walk both way = 3 × 2
= 6 hours

৫,২৮৫.

The diameter of the base of a cone is 10.5 cm and its slant height is 10cm. The curved surface area is:

ক
ক) 134 cm²
খ
খ) 165 cm²
গ
গ) 154 cm²
ঘ
ঘ) 195 cm²

সঠিক উত্তর: খ

খ) 165 cm²

উত্তর

সঠিক উত্তর: খ

খ) 165 cm²

খ

ব্যাখ্যা

Question: The diameter of the base of a cone is 10.5 cm and its slant height is 10cm. The curved surface area is:

Solution:
কোণকের ভূমির ব্যাস = 10.5 cm
কোণকের ভূমির ব্যাসার্ধ r = 10.5/2 = 5.25 cm
কোণকের তীর্যক উচ্চতা l = 10 cm

কোণকের বক্রতলের ক্ষেত্রফল = πrl = (22/7) × 5.25 × 10 = 165 cm²

৫,২৮৬.

What is the area of the largest sphere that can be placed inside a cube of volume 216 cm³?

ক
ক) 49π/3
খ
খ) 36π
গ
গ) 15√2π
ঘ
ঘ) 3√2π
ঙ
ঙ) None

সঠিক উত্তর: খ

খ) 36π

উত্তর

সঠিক উত্তর: খ

খ) 36π

খ

ব্যাখ্যা

Let the length of each side of the cube = a
So, the volume of the cube is a³
∴ a = 6

We know, area of a sphere is 4/3πr³
= 4/3π3³
= 4/3π.27
= 36π

৫,২৮৭.

tanθ.√(1 - cos²θ) = ?

ক
sinθ
খ
cosθ
গ
sin²θ / cosθ
ঘ
tanθ.cosθ

সঠিক উত্তর: গ

sin²θ / cosθ

উত্তর

সঠিক উত্তর: গ

sin²θ / cosθ

গ

ব্যাখ্যা

Question: tanθ.√(1 - cos²θ) = ?

Solution:
tanθ.√(1 - cos²θ)
= (sinθ/cosθ)√(sin²θ)
= (sinθ/cosθ)(sinθ)
= sin²θ/cosθ

৫,২৮৮.

A bag contains 4 white, 5 red and 6 blue balls. Three balls are drawn at random from the bag. The probability that all of them are red, is

ক
ক) 1/22
খ
খ) 2/91
গ
গ) 3/22
ঘ
ঘ) 2/77

সঠিক উত্তর: খ

খ) 2/91

উত্তর

সঠিক উত্তর: খ

খ) 2/91

খ

ব্যাখ্যা

Let S be the sample space
⇒n(S) = number of ways of drawing 3 balls out of 15
= ¹⁵C₃
= 455

Let E be the event of getting all of the 3 red balls.
∴n(E) = ⁵C₃
=10

∴P(E) = n(E)/n(S)
=10/455
=2/91

৫,২৮৯.

The simple interest on a sum of money at 8% per annum for 6 years is half the sum. the sum is-

ক
Tk. 4800
খ
Tk. 6000
গ
Tk. 8000
ঘ
Data is inadequate

সঠিক উত্তর: ঘ

Data is inadequate

উত্তর

সঠিক উত্তর: ঘ

Data is inadequate

ঘ

ব্যাখ্যা

Question: The simple interest on a sum of money at 8% per annum for 6 years is half the sum. the sum is-

Solution:
ধরি, আসল = P
মুনাফার হার, r = 8%
সময়, n = 6 বছর
মুনাফা, I = P/2

আমরা জানি,
I = Pnr
বা, P/2 = P × 6 × 8/100
বা, P/2 = 12P/25
বা, (P/2) - (12P/25) = 0
বা, (25P - 24P)/50 = 0
বা, P = 0

∴ ডেটা অপর্যাপ্ত।

৫,২৯০.

Anik alone can do a piece of work in 6 days and Bishal alone in 8 days. Anik and Bishal undertook to do it for Tk. 5400. With the help of Dinesh, they completed the work in 3 days. How much is to be paid to Dinesh?

ক
Tk. 675
খ
Tk. 900
গ
Tk. 870
ঘ
Tk. 780

সঠিক উত্তর: ক

Tk. 675

উত্তর

সঠিক উত্তর: ক

Tk. 675

ক

ব্যাখ্যা

Question: Anik alone can do a piece of work in 6 days and Bishal alone in 8 days. Anik and Bishal undertook to do it for Tk. 5400. With the help of Dinesh, they completed the work in 3 days. How much is to be paid to Dinesh?

Solution:
Anik's 1day work = 1/6
Bishal's 1 day work = 1/8

∴ (Anik + Bishal + Dinesh)'s 1 day work = 1/3

∴ Dinesh's 1 day work = (1/3) - (1/6) - (1/8) = (8 - 4 - 3)/24 = 1/24

So, Dinesh's 3 day work = 3 × (1/24) = 1/8
If Dinesh contributed 8th part of work then he will receive 8th part of total payment

∴ Dinesh should be paid = 5400 × (1/8) = Tk. 675

৫,২৯১.

If ab + bc + ca = 0, then what is the value of {1/(a² - bc)} + {1/(b² - ca)} + {1/(c² - ab)}?

ক
ক) 0
খ
খ) 1
গ
গ) abc
ঘ
ঘ) 1/abc

সঠিক উত্তর: ক

ক) 0

উত্তর

সঠিক উত্তর: ক

ক) 0

ক

ব্যাখ্যা

Question: If ab + bc + ca = 0, then what is the value of {1/(a² - bc)} + {1/(b² - ca)} + {1/(c² - ab)}?

Solution:
Given that
ab + bc + ca = 0

ab = - bc - ca
bc = - ab - ca
ca = - ab - bc

{1/(a² - bc)} + {1/(b² - ca)} + {1/(c² - ab)}
= {1/(a² + ab + ca)} + {1/(b² + ab + bc)} + {1/(c² + bc + ca)}
= [1/{a(a + b + c)}] + [1/{b(a + b + c)}] + [1/{c(a + b + c)}]
= (bc + ac + ab)/{abc(a + b + c)}
= 0//{abc(a + b + c)}
= 0

৫,২৯২.

Two horses start trotting towards each other, one from A to B and another from B to A. They cross each other after one hour and the first horse reaches B, 5/6 hour before the second horse reaches A. If the distance between A and B is 50 km. what is the speed of the slower horse?

ক
30 km/h
খ
20 km/h
গ
25 km/h
ঘ
15 km/h

সঠিক উত্তর: খ

20 km/h

উত্তর

সঠিক উত্তর: খ

20 km/h

খ

ব্যাখ্যা

Question: Two horses start trotting towards each other, one from A to B and another from B to A. They cross each other after one hour and the first horse reaches B, 5/6 hour before the second horse reaches A. If the distance between A and B is 50 km. what is the speed of the slower horse?

Solution:
If the speed of the faster horse be f and that of slower horse be s
Then,
f + s = 50/1 = 50 km/h
∴ f = 50 - s

ATQ,
50/s - 50/f = 5/6
⇒ (50f - 50s)/(sf) = 5/6
⇒ 50(f - s) = (5/6)(sf)
⇒ 50(50 - s - s) = (5/6)(sf)
⇒ 6(2500 - 100s) = 5 × s × (50 - s)
⇒ 15000 - 600s = 250s - 5s²
⇒ 5s² - 600s - 250s + 15000 = 0
⇒ 5s² - 850s + 15000 = 0
⇒ s² - 170s + 3000 = 0
⇒ s² - 150s - 20s + 3000 = 0
⇒ s(s - 150) - 20(s - 150) = 0
⇒ (s - 150)(s - 20) = 0
∴ s = 150, 20 [ 150 not be acceptablr]

∴ The speed of the slower horse is 20 km/h

৫,২৯৩.

Rafiq took a loan of Tk. 1600 with simple interest for as many years as the rate of interest. If he paid Tk. 256 as interest at the end of the loan period, what was the rate of interest?

ক
ক) 9%
খ
খ) 5%
গ
গ) 4%
ঘ
ঘ) 6%

সঠিক উত্তর: গ

গ) 4%

উত্তর

সঠিক উত্তর: গ

গ) 4%

গ

ব্যাখ্যা

Simple interest is the same as the rate of interest.
Hence,
Rate of interest = R% and Time = R years

S.I. = (P × R × R)/100
⇒ 256 = (1600 × R²)/100
⇒ 16R²= 256
⇒ R²= 16
⇒ R=4 %

Rate of Interest = 4%.

৫,২৯৪.

Six years ago Anika was P times as old as Benu was. If Anika is now 17 years old, how old is Benu now in terms of P?

ক
ক) (11/P) + 6
খ
খ) (P/11) + 6
গ
গ) 17 - (P/6)
ঘ
ঘ) 17/P
ঙ
ঙ) 17p - 6

সঠিক উত্তর: ক

ক) (11/P) + 6

উত্তর

সঠিক উত্তর: ক

ক) (11/P) + 6

ক

ব্যাখ্যা

Let Benu’s age now be B.
Anika’s age now is A.
(A - 6) = P(B - 6)
But A is 17 and therefore 11 = P(B - 6)
11/P = B - 6
(11/P) + 6 = B

৫,২৯৫.

If x : y = 5 : 2 then the value of (xy + y²)/(x² - y²) ?

ক
2/3
খ
3/7
গ
5/9
ঘ
7/11

সঠিক উত্তর: ক

2/3

উত্তর

সঠিক উত্তর: ক

2/3

ক

ব্যাখ্যা

Question: If x : y = 5 : 2 then the value of (xy + y²)/(x² - y²) ?

Solution: Let x = 5k and y = 2k

Now,
(xy + y²)/(x² - y²)
= {(5k)(2k)+(2k)²}/{(5k)² - (2k)²}
= (10k² + 4k²)/(25k² - 4k²)
= 14k²/21k²
= 14/21
= 2/3

৫,২৯৬.

If √3ⁿ= 729, then the value of n is ?

ক
15
খ
6
গ
9
ঘ
12

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: If √3ⁿ= 729, then the value of n is ?

Solution:
Given that,
√3ⁿ = 729
⇒ √3ⁿ = 3⁶
⇒ (√3ⁿ)² = (3⁶)²
⇒ 3ⁿ = 3¹²
∴ n = 12

৫,২৯৭.

The compound interest on Tk. 2000 for 2 years at 10% is-

ক
ক) 400 Tk.
খ
খ) 240 Tk.
গ
গ) 420 Tk.
ঘ
ঘ) 480 Tk.

সঠিক উত্তর: গ

গ) 420 Tk.

উত্তর

সঠিক উত্তর: গ

গ) 420 Tk.

গ

ব্যাখ্যা

Question: The compound interest on Tk. 2000 for 2 years at 10% is-

Solution:
Givent that, P = Tk. 2000
n = 2 years
r = 10%
Required amount = P(1 + r) n
= [2000 × {1 + (10/100)²}]
= [2000 × (11/10) × (11/10)]
= 2420

Compound Interest = 2420 - 2000 = 420 Tk.

৫,২৯৮.

A glass when full of milk, weighs 0.75 kg. It weighs 0.5 kg when the glass is half full. What is weight of the empty glass?

ক
0.25 kg
খ
0.35 kg
গ
0.45 kg
ঘ
0.5 kg

সঠিক উত্তর: ক

0.25 kg

উত্তর

সঠিক উত্তর: ক

0.25 kg

ক

ব্যাখ্যা

Question: A glass when full of milk, weighs 0.75 kg. It weighs 0.5 kg when the glass is half full. What is weight of the empty glass?

Solution:
Let,
Weight of Glass = x kg
Weight of Milk = y kg

Now,
x + y = 0.75..................(1)

And
x + y/2 = 0.5
⇒ (2x + y)/2 = 0.5
⇒ 2x +y = 1.00..................(2)

(2) - (1) ⇒
2x + y - (x + y) = 1.00 - 0.75
⇒ 2x + y - x - y = 0.25
∴ x = 0.25

∴ Weight of Glass is 0.25 kg

৫,২৯৯.

The least multiple of 7, which leaves a remainder of 4, when divided by 6, 9, 15 and 18 is:

ক
ক) 74
খ
খ) 94
গ
গ) 184
ঘ
ঘ) 364

সঠিক উত্তর: ঘ

ঘ) 364

উত্তর

সঠিক উত্তর: ঘ

ঘ) 364

ঘ

ব্যাখ্যা

L.C.M. of 6, 9, 15 and 18 is 90.
Let required number be 90k + 4, which is multiple of 7.
Least value of k for which (90k + 4) is divisible by 7 is k = 4.
Required number = (90 x 4) + 4 = 364.

৫,৩০০.

In a class, there are 12 boys and 10 girls. Three students are selected at random. The probability that 1 girl and 2 boys are selected is-

ক
1/7
খ
2/5
গ
3/7
ঘ
1/5
ঙ
None of these

সঠিক উত্তর: গ

3/7

উত্তর

সঠিক উত্তর: গ

3/7

গ

ব্যাখ্যা

Question: In a class, there are 12 boys and 10 girls. Three students are selected at random. The probability that 1 girl and 2 boys are selected is-

Solution:

Let S be the sample space and E be the event of selecting 1 girl and 2 boys.

Then, n(S) = Number ways of selecting 3 students out of 22
= ²²C₃
= (22 x 21 x 20)/(3 x 2 x 1)
= 1540

n(E) = ¹⁰C₁ x ¹²C₂
= (10 x 12 x 11)/(2 x 1)
= 660

∴ P(E) = n(E)/n(S)
= 660/1540
= 3/7