Bank Math

PrepBank · পাতা ১১৯ / ১৬১ · ১১,৮০১–১১,৯০০ / ১৬,১২৪

১১,৮০১.

If a and b are odd numbers. Which number is even?

ক
ab
খ
a + 2b + 2
গ
a + b + 1
ঘ
2a + 4b

ব্যাখ্যা

Question: If a and b are odd numbers. Which number is even?

Solution:
Let
a = 1
b = 3

ab = 1 × 3 = 3, which is odd.

a + 2b + 2 = 1 + 2 × 3 + 2 = 1 + 6 + 2 = 9, which is odd.

a + b + 1 = 1 + 3 + 1 = 5, which is odd.

2a + 4b = 2 × 1 + 4 × 3 = 2 + 12 = 14, which is even.

১১,৮০২.

The angle of elevation of the sun, when the length of the shadow of a tree √3 times the height of the tree, is:

ক
ক) 30°
খ
খ) 45°
গ
গ) 60°
ঘ
ঘ) 90°

ব্যাখ্যা

Question: The angle of elevation of the sun, when the length of the shadow of a tree √3 times the height of the tree is:

Solution:

Let,
AB = height of tree
BC= Shadow of tree
angle of elevation = C
∴ BC = √3 AB

We know,
tanC = AB/BC
⇒ tanC = AB/√3AB
⇒ tanC = 1/√3
⇒ tanC = tan30°
∴ C = 30°

১১,৮০৩.

If a + 1/a = √3, then what is the value of a³⁶ + a³⁰ + a⁶ + 2?

ক
3
খ
2
গ
0
ঘ
1

ব্যাখ্যা

Question: If a + 1/a = √3, then what is the value of a³⁶ + a³⁰ + a⁶ + 2?

Solution:
Given, a + 1/a = √3
Now,
a³ + 1/a³ = (a + 1/a)³ - 3a × (1/a)(a + 1/a)
⇒ a³ + 1/a³ = (√3)³ - 3(√3) [∵ a + 1/a = √3]
⇒ a³ + 1/a³ = 3(√3) - 3(√3)
⇒ a³ + 1/a³ = 0
⇒ a⁶ + 1 = 0 [Multiplying both sides by a³]

Then,
a³⁶ + a³⁰ + a⁶ + 2
= a³⁶ (a⁶ + 1) + (a⁶ + 1) + 1
= (a³⁶ × 0) + 0 + 1
= 0 + 1
= 1

১১,৮০৪.

If tanθ = 1, then the the value of

ক
3/2
খ
9/4
গ
1
ঘ
9/2
ঙ
2

ব্যাখ্যা

Question: If tanθ = 1, then the the value of

Solution:
Given,
tanθ = 1
⇒ tanθ = tan45°
∴ θ = 45°

Now,

১১,৮০৫.

Which one is the complementary angle of 90°?

ক
ক) 0°
খ
খ) 180°
গ
গ) 270°
ঘ
ঘ) 90°

ব্যাখ্যা

Question: Which one is the complementary angle of 90°?

Solution:
দুটি কোণের সমষ্টি এক সমকোণ হলে, একটিকে অপরটির পূরক কোণ বলে।

৯০° এর পূরক কোণ = ৯০° - ৯০°
= ০°

১১,৮০৬.

If ³√32 = 2^x then x is equal to -

ক
ক) 5/3
খ
খ) 3
গ
গ) 3/5
ঘ
ঘ) 5

ব্যাখ্যা

³√32 = 2^x
⇒ 2^5/3 = 2^x
⇒ x = 5/3

১১,৮০৭.

Jamal started a partnership by investing Tk. 18,000. After 4 months, Hasan joined by investing Tk. x. If at the end of the year the profits are divided equally, find Hasan’s investment.

ক
Tk. 25000
খ
Tk. 27000
গ
Tk. 29500
ঘ
Tk. 31500

ব্যাখ্যা

Question: Jamal started a partnership by investing Tk. 18,000. After 4 months, Hasan joined by investing Tk. x. If at the end of the year the profits are divided equally, find Hasan’s investment.

Solution:
Given,
Jamal invests Tk. 18,000 for 12 months
∴ Jamal’s investment = 18,000 × 12 = 2,16,000

Hasan invests Tk. x for 8 months
∴ Hasan’s investment = 8x

ATQ,
216,000 : 8x = 1 : 1
⇒ 216,000/8x = 1/1
⇒ 8x = 216000
⇒ x = 216000/8
∴ x = 27000

১১,৮০৮.

A car travels at 50 km/h. If it had traveled 10 km/h faster, it would have reached its destination 30 minutes earlier. What is the distance of the journey?

ক
150 km
খ
180 km
গ
200 km
ঘ
250 km

ব্যাখ্যা

Question: A car travels at 50 km/h. If it had traveled 10 km/h faster, it would have reached its destination 30 minutes earlier. What is the distance of the journey?

Solution:
Let the distance be x km.

Time at 50 km/h = x/50 hours
Time at 60 km/h = x/60 hours

According to the question, the difference in time = 0.5 hours
⇒ x/50 − x/60 = 0.5
⇒ (6x − 5x) / 300 = 0.5
⇒ x / 300 = 0.5
⇒ x = 0.5 × 300
⇒ x = 150 km

∴ Distance of the journey = 150 km

১১,৮০৯.

A father's age was 5 times his son's age 5 years ago and will be 3 times son's age after 2 years. The ratio of their present ages is -

ক
ক) 5 : 2
খ
খ) 5 : 3
গ
গ) 10 : 3
ঘ
ঘ) 11 : 5

ব্যাখ্যা

Question: A father's age was 5 times his son's age 5 years ago and will be 3 times son's age after 2 years. The ratio of their present ages is -

Solution:
ধরি,
৫ বছর আগে, পুত্রের বয়স x বছর
পিতার বয়স 5x বছর

বর্তমানে, পুত্রের বয়স x + 5 বছর
পিতার বয়স (5x + 5) বছর

২ বছর পর, পুত্রের বয়স x + 5 + 2 বছর
= x + 7 বছর

২ বছর পর পিতার বয়স = 5x + 5 + 2
= 5x + 7 বছর

প্রশ্নমতে,
5x + 7 = 3 (x + 7)
⇒ 5x + 7 = 3x + 21
⇒ 5x - 3x = 21 - 7
⇒ 2x = 14
∴ x = 7

পুত্রের বর্তমান বয়স = 7 + 5
= 12 বছর
পিতার বর্তমান বয়স = 5 × 7 + 5
= 35 + 5
= 40 বছর

∴ তাদের বর্তমান বয়সের অনুপাত = 40 : 12
= 10 : 3

১১,৮১০.

The area of a rectangle R with width 4 ft is equal to the area of a square S, which has a perimeter of 24 ft. the perimeter of the rectangle R, in feet, is

ক
ক) 9
খ
খ) 16
গ
গ) 24
ঘ
ঘ) 26

ব্যাখ্যা

Question: The area of a rectangle R with width 4 ft is equal to the area of a square S, which has a perimeter of 24 ft. the perimeter of the rectangle R, in feet, is -

Solution:
ধরি,
চতুর্ভুজ, R এর দৈর্ঘ্য এবং প্রস্থ যথাক্রমে l, b.
বর্গের এক বাহু = a

প্রশ্নমতে,
4a = 24
a = 6

∴ চতুর্ভুজের ক্ষেত্রফল = বর্গের ক্ষেত্রফল
l × b = a²
l = a²/b
= 36/4
= 9

∴ চতুর্ভুজের পরিসীমা = 2(l + b)
= 2(9 + 4)
= 26 feet

১১,৮১১.

Which of the following numbers is divisible by 3?

ক
177
খ
200
গ
331
ঘ
457
ঙ
None of these

ব্যাখ্যা

প্রশ্ন: Which of the following numbers is divisible by 3?

সমাধান:
ক. 177/3 = 59; যা 3 দ্ধারা বিভাজ্য।

খ. 200/3 = 66.67; যা 3 দ্ধারা বিভাজ্য নয়।

গ. 331/3 = 110.33; যা 3 দ্ধারা বিভাজ্য নয়।

ঘ. 457/3 = 152.33; যা 3 দ্ধারা বিভাজ্য নয়।

বিকল্প সমাধান:
কোনো সংখ্যার অঙ্কগুলোর যোগফল ৩ দ্বারা বিভাজ্য হলে, ঐ সংখ্যাটি ৩ দ্বারা বিভাজ্য হবে।
ক. 1 + 7 + 7 = 15; যা 3 দ্ধারা বিভাজ্য।
খ. 2 + 0 + 0 = 2; যা 3 দ্ধারা বিভাজ্য নয়।
গ. 3 + 3 + 1 = 7; যা 3 দ্ধারা বিভাজ্য নয়।
ঘ. 4 + 5 + 7 = 16; যা 3 দ্ধারা বিভাজ্য নয়।

১১,৮১২.

In the figure above, (a + b + c)/(s + t + u) = ?

ক
1/2
খ
1/3
গ
2/3
ঘ
1

ব্যাখ্যা

Question:

In the figure above, (a + b + c)/(s + t + u) = ?

Solution:
Here,
u = a + b
t = a + c
s = b + c

∴ s + t + u = b + c + a + c + a + b = 2(a + b + c)

Here,
(a + b + c)/(s + t + u)
= (a + b + c)/{2(a + b + c)}
= 1/2

১১,৮১৩.

A square field is surrounded by a path of uniform width 5 meters. If the area of the path is 220 square meters, find the side length of the field.

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

ব্যাখ্যা

Question: A square field is surrounded by a path of uniform width 5 meters. If the area of the path is 220 square meters, find the side length of the field.

Solution:
Let the side of the field = x meters.
Then, the side of the field including the path = x + (2 × 5)
= x + 10 meters.

Area of path = Area of field with path - Area of field
⇒ 220 = (x + 10)² - x²
⇒ 220 = x² + 20x + 100 - x²
⇒ 220 = 20x + 100
⇒ 20x = 220 - 100
⇒ 20x = 120
⇒ x = 120/20
⇒ x = 6 meters

∴ Therefore, the side length of the field is 6 meters.

১১,৮১৪.

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

ক
ক) 11 hours
খ
খ) 13 hours
গ
গ) 14 hours
ঘ
ঘ) 16 hours

ব্যাখ্যা

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

Solution:
Work done by the leak in 1 hour
= ( 1/2 − 3/7 ) = 1/14

∴ Leak will empty the tank in 14 hours

১১,৮১৫.

If (2000)¹⁰ = 1.024 × 10^k, then the value of k is

ক
27
খ
30
গ
33
ঘ
36

ব্যাখ্যা

Question: If (2000)¹⁰ = 1.024 × 10^k, then the value of k is

Solution:
(2000)¹⁰ = 1.024 × 10^k
⇒ (2 × 10³)¹⁰ = (1024/1000) × 10^k
⇒ 2¹⁰ × 10³⁰ = 1024 × 10^{k - 3}
⇒ 2¹⁰ × 10³⁰ = 2¹⁰ × 10^{k - 3}
⇒ 10³⁰ = 10^{k - 3}
⇒ k - 3 = 30
∴ k = 33

১১,৮১৬.

Karim completes one-fourth of a piece of work in 4 hours. Rahim works at three times Karim’s speed and finishes the remaining portion of the work. How many hours does Rahim take to complete it?

ক
3 hours
খ
4 hours
গ
2 hours
ঘ
5 hours

ব্যাখ্যা

Question: Karim completes one-fourth of a piece of work in 4 hours. Rahim works at three times Karim’s speed and finishes the remaining portion of the work. How many hours does Rahim take to complete it?

Solution:
Let, Whole work = 1
Karim completes one-fourth (1/4) of a piece of work in 4 hours
∴ Karim’s rate = (1/4)/4 = 1/16 per hour.

∴ Rahim works at three times Karim’s speed
∴ Rahim’s rate = 3 × 1/16 = 3/16 per hour.
∴ Remaining work = (1 - 1/4) = 3/4
∴ Rahim complets 3/4 part work in = (3/4) × (16/3)
= 4 hours

১১,৮১৭.

At what angle the hands of a clock are inclined at 15 minutes past 5?

ক
48°
খ
54.5°
গ
67.5°
ঘ
71.5°

ব্যাখ্যা

Question: At what angle the hands of a clock are inclined at 15 minutes past 5?

Solution:
Hours hand moves in 15 past.
5 from 12 p.m = (5 + 15/60) hours = 21/4 hours
Angle of hours hand = (360/12) × (21/4)
= 157.5°

Minutes hands makes angle of = (360/60) × 15
= 90°

Angle between hours and minutes hands = (157.5° - 90°)
= 67.5°

১১,৮১৮.

If a bus travels 160 km in 4 hours and a train travels 320 km in 5 hours at uniform speeds, then the ratio of the distances traveled by them in one hour is

ক
5 : 7
খ
5 : 8
গ
3 : 8
ঘ
None of these

ব্যাখ্যা

Question: If a bus travels 160 km in 4 hours and a train travels 320 km in 5 hours at uniform speeds, then the ratio of the distances traveled by them in one hour is

Solution:
Distance traveled by bus in one hour= 160 / 4
= 40

Distance traveled by bus in one hour= 320 / 5
= 64

Ratio = 40 : 64
= 5 : 8

১১,৮১৯.

A frog is sitting on vertex A of a square ABCD. It starts jumping to the immediately adjacent vertex on either side in random fashion and stops when it reaches point C. In how many ways can it reach point C if it makes exactly 7 jumps?

ক
ক) 1
খ
খ) 3
গ
গ) 5
ঘ
ঘ) 0

ব্যাখ্যা

In every odd jump (1st, 3rd, 5th, 7th etc) it will reach either to B or D. So, it is not possible for the frog to reach point C in 7th Jump.
The logic behind asking this question is to check if the student gets the point that if the frog takes odd number of jumps he can never reach vertex C.
Since the question asks the number of ways the frog can get to point C with 7 jumps which is an odd number, the answer to this question then becomes 0 as the frog can never reach point C with 7 jumps.

১১,৮২০.

The value of tan60° is:

ক
√3
খ
√3/2
গ
1
ঘ
1/2

ব্যাখ্যা

Question: The value of tan60° is:
(Officer General 2022 অনুযায়ী)

Solution:
cos60° = 1/2
sin60° = √3/2

Hence, tan60°
= sin60°/cos60°
= (√3/2)/(1/2)
= √3

১১,৮২১.

The sum of Moni and Mimi's ages is 50 years and the difference between their ages is 6 years. What is the product of their ages?

ক
ক) 524 years
খ
খ) 616 years
গ
গ) 480 years
ঘ
ঘ) 300 years

ব্যাখ্যা

Question: The sum of Moni and Mimi's ages is 50 years and the difference between their ages is 6 years. What is the product of their ages?

Solution:
Let, Moni's age be = x years
and Mimi's age be = y years

ATQ,
x + y = 50.........(i)
x - y = 6.............(ii)

Now,
(i) + (ii),
x + y + x - y = 50 + 6
⇒ 2x = 56
⇒ x = 28

So, y = 50 - 28 = 22

∴ product of their ages = 28 × 22 = 616 years

১১,৮২২.

5 men and 2 boys working together can do four times as much as work as a man and a boy. Working capacities of a man and a boy are in the ratio -

ক
ক) 1 : 2
খ
খ) 2 : 1
গ
গ) 1 : 3
ঘ
ঘ) 3 : 1

ব্যাখ্যা

Question: 5 men and 2 boys working together can do four times as much as work as a man and a boy. Working capacities of a man and a boy are in the ratio -

Solution:
Let, 1 man's 1 days's work = x
and 1 boy's 1 day's work = y

ATQ,
5x + 2y = 4(x + y)
⇒ 5x + 2y = 4x + 4y
⇒ x = 2y
⇒ x/y = 2/1
⇒ x : y = 2 : 1

১১,৮২৩.

A mixture contains two liquids 'A' and 'B' are in the ratio 4 : 1. If 10 litres of mixture is withdrawn and replaced with 10 litres of 'B', then the ratio becomes 2 : 3. What was the initial quantity of A?

ক
12 liters
খ
16 liters
গ
11 liters
ঘ
10 liters

ব্যাখ্যা

Question: A mixture contains two liquids 'A' and 'B' are in the ratio 4 : 1. If 10 litres of mixture is withdrawn and replaced with 10 litres of 'B', then the ratio becomes 2 : 3. What was the initial quantity of A?

Solution:
ধরি, মিশ্রণের প্রাথমিক পরিমাণ = 5x লিটার

A এর পরিমাণ = 4x লিটার
B এর পরিমাণ = x লিটার

∴ 10 লিটার মিশ্রণ তুলে নেওয়ার পর,
A এর পরিমাণ = 4x - (4/5) × 10 = 4x - 8 লিটার
B এর পরিমাণ = x - (1/5) × 10 = x - 2 লিটার

আবার,
B তে 10 লিটার যোগ করার পর,
B এর পরিমাণ = x - 2 + 10 = x + 8 লিটার

∴ প্রদত্ত অনুপাত,
⇒ (4x - 8)/(x + 8) = 2/3
⇒ 12x - 24 = 2x + 16
⇒ 10x = 16 + 24
⇒ x = 40/10
⇒ x = 4

∴ A এর পরিমাণ = 4 × 4 = 16 লিটার

১১,৮২৪.

40% of 650 + 80% 720 - 600 = ?

ক
236
খ
264
গ
196
ঘ
476

ব্যাখ্যা

Question: 40% of 650 + 80% 720 - 600 = ?

Solution:
40% of 650 + 80% 720 - 600
= (40/100) × 650 + (80/100) × 720 - 600
= (26000/100) + (57600/100) - 600
= 260 + 576 - 600
= 236

১১,৮২৫.

The ratio of red balls, to yellow balls, to green balls in a basket is 2 : 3 : 4. What is the probability that a ball chosen at random from the basket is a yellow ball?

ক
ক) 2/9
খ
খ) 1/9
গ
গ) 1/3
ঘ
ঘ) 2/3

ব্যাখ্যা

Question: The ratio of red balls, to yellow balls, to green balls in a basket is 2 : 3 : 4. What is the probability that a ball chosen at random from the basket is a yellow ball?

Solution:
The ratio of red balls, to yellow balls, to green balls in a basket is 2 : 3 : 4
let, there are 2x red balls, 3x yellow balls and 4x green balls
total balls = 2x + 3x + 4x
= 9x

∴ probability that a ball chosen at random from the basket is a yellow ball = 3x/9x
= 3/9
= 1/3

১১,৮২৬.

A vessel contains a mixture of P and Q in the ratio of 5 : 3. 16 liters of this mixture is taken out and 5 liters of P is poured in. The new mixture has a ratio of P to Q as 11 : 6. Find the total original quantity of mixture.

ক
80 liters
খ
98 liters
গ
92liters
ঘ
96 liters
ঙ
None of these

ব্যাখ্যা

Question: A vessel contains a mixture of P and Q in the ratio of 5 : 3. 16 liters of this mixture is taken out and 5 liters of P is poured in. The new mixture has a ratio of P to Q as 11 : 6. Find the total original quantity of mixture.

Solution:
Let,
Original Quantity of P = 5x
Original Quantity of Q = 3x

The quantity of P and Q in 16 liters of the mixture,
Quantity of P = (16 × 5x)/8x = 10 liters
Quantity of Q = (16 × 3x)/8x = 6 liters

Now,
5 liters of P poured in so the Quantity of P will be = 5x - 10 + 5 liters
= 5x - 5 liters

ATQ,
(5x - 5)/(3x - 6) = 11/6
⇒ 6(5x - 5) = 11(3x - 6)
⇒ 30x - 30 = 33x - 66
⇒ 3x = 36
∴ x = 12

So, total mixture originally = 8x
= (8 × 12) = 96 liters

১১,৮২৭.

Find the missing number
4, ?, 144, 400, 900, 1764

ক
100
খ
49
গ
36
ঘ
25

ব্যাখ্যা

Question: Find the missing number
4, ?, 144, 400, 900, 1764

Solution:
1764 = 42²
900 = (42 - 12)² = 30²
400 = (30 - 10)² = 20²
144 = (20 - 8)² = 12²
? = (12 - 6)² = 6² = 36
4 = (6 - 4)² = 22

১ম বার 12 বিয়োগ দেয়া হয়েছে, (42 - 12),
তারপর 2 কমিয়ে 10 বিয়োগ দেয়া হয়েছে (30 - 10),
অনুরূপ প্রক্রিয়ায় সমাধান করে উত্তর হবে 36।

১১,৮২৮.

If WORD = DROW, then PALE = ?

ক
LEAP
খ
PEAL
গ
APLE
ঘ
ELAP

ব্যাখ্যা

Question: If WORD = DROW, then PALE = ?

Solution:

এখানে,
WORD = DROW এর W আর D এবং O আর R নিজেদের মধ্যে স্থান পরিবর্তন করেছে।

একইভাবে,
PALE শব্দের P আর E এবং A আর L নিজেদের মধ্যে স্থান পরিবর্তন করবে।

∴ PALE = ELAP

১১,৮২৯.

What values of x satisfy the inequality 2 - 3x > 1?

ক
x < -1/3
খ
x < 1/3
গ
x > -1/3
ঘ
x > 1/3

ব্যাখ্যা

Question: What values of x satisfy the inequality 2 - 3x > 1?

Solution:
দেওয়া আছে,
2 - 3x > 1
⇒ 2 - 3x - 2 > 1 - 2 ; [অসমতার উভয় পক্ষ থেকে ২ বিয়োগ করে পাই]
⇒ - 3x > - 1
⇒ - 3x/- 3 < - 1/- 3 ; [একটি ঋণাত্মক সংখ্যা দ্বারা অসমতাকে ভাগ করলে, তখন অসমতার চিহ্নটি উল্টে যায়।]
∴ x < 1/3

১১,৮৩০.

The L.C.M and ratio of four numbers are 630 and 2:3 and their 2:3:5:7 respectively. The difference between the greatest and least number is:

ক
ক) 6
খ
খ) 14
গ
গ) 15
ঘ
ঘ) 21

ব্যাখ্যা

Let the numbers be 2x, 3x, 5x and 7x respectively.
Then, their L.C.M = (2 × 3 × 5 × 7)x = 210x
[∵ 2, 3, 5, 7 are prime numbers ]
So, 20x = 630
or x = 3
∵ The numbers are 6, 9, 15 and 21.
Required difference = 21 - 6 = 15.
Answer : 15

১১,৮৩১.

ক
10
খ
2
গ
5
ঘ
4

ব্যাখ্যা

Question:

Solution:

১১,৮৩২.

A man can do a work in 20 days and a woman in 15 days. If they work on it together for 6 days, then the fraction of the work that is left is-

ক
1/10
খ
1/12
গ
3/10
ঘ
7/10
ঙ
7/12

ব্যাখ্যা

Question: A man can do a work in 20 days and a woman in 15 days. If they work on it together for 6 days, then the fraction of the work that is left is-

Solution:
Man’s 1 day’s work = 1/20
Woman’s 1 day’s work = 1/15

(Man + woman)’s 1 day’s work = (1/20 + 1/15) = 7/60
(Man + woman)’s 6 day’s work = (7/60 × 6) = 7/10

Thus, Remaining work = 1 - 7/10 = 3/10

১১,৮৩৩.

A can do a work in 15 days, and B in 25 days. They work together for 5 days. How much of the work is left?

ক
1/3
খ
7/15
গ
3/10
ঘ
5/8

ব্যাখ্যা

Question: A can do a work in 15 days, and B in 25 days. They work together for 5 days. How much of the work is left?

Solution:
A, 15 দিনে করতে পারে কাজটির 1 অংশ
A, 1 দিনে করতে পারে কাজটির 1/15 অংশ

B, 25 দিনে করতে পারে কাজটির 1 অংশ
B, 1 দিনে করতে পারে কাজটির 1/25 অংশ

A ও B 1 দিনে করতে পারে কাজটির = (1/15) + (1/25) অংশ
= (5 + 3)/75 অংশ
= 8/75 অংশ

A ও B 5 দিনে করতে পারে কাজটির (5 × 8)/75 অংশ
= 8/15 অংশ

∴ কাজ বাকি থাকে = 1 - 8/15 অংশ
= (15 - 8)/15 অংশ
= 7/15 অংশ

১১,৮৩৪.

A man takes 8 hours in walking a distance and riding back to starting place. He could walk both ways in 10 hours. The time taken by him to ride back both ways is:

ক
6 hours
খ
5 hours
গ
4.5 hours
ঘ
5.5 hours

ব্যাখ্যা

Question: A man takes 8 hours in walking a distance and riding back to starting place. He could walk both ways in 10 hours. The time taken by him to ride back both ways is:

Solution:
Time taken in walking both the ways = 10 hours.................(i)
Time taken in walking one way and riding back = 8 hours...................(ii)

By the equation (ii) × 2 - (i), we have,
Time taken by the man in riding both ways,
= 16 hours - 10 hours
= 6 hours

১১,৮৩৫.

What is the sum of the reciprocals of the values of zeroes of the polynomial 6x² + 3x² - 5x + 1?

ক
ক) 3
খ
খ) 4
গ
গ) 5
ঘ
ঘ) 6

ব্যাখ্যা

Question: What is the sum of the reciprocals of the values of zeroes of the polynomial 6x² + 3x² - 5x + 1?

Solution:
6x² + 3x² - 5x + 1
⇒ 9x² - 5x + 1
Let α and β are two roots of the equations

As we know,
Sum of roots (α + β) = 5/9
Product of roots (αβ) = 1/9
According to the question :
⇒ 1/α + 1/β = (α + β)/αβ
= (5/9)/(1/9)
= (5/9) × (9/1)
= 5

১১,৮৩৬.

On a 12% discount sale, an article costs Tk. 704. What was the original price of the article?

ক
Tk. 800
খ
Tk. 850
গ
Tk. 700
ঘ
Tk. 840

ব্যাখ্যা

Question: : On a 12% discount sale, an article costs Tk. 704. What was the original price of the article?

Solution:
Let the original price be Tk. x.

According to the question,
x - 12% of x = 704
⇒ x - (12x/100) = 704
⇒ (100x - 12x)/100 = 704
⇒ 88x = 70400
∴ x = 800

So, the original price of the article was Tk. 800.

১১,৮৩৭.

A, B, and C share Tk. 2,400 in such a way that A has 3 times as much as B, and B has 5 times as much as C. How much money (in taka) does C receive?

ক
108.67 tk
খ
114.29 tk
গ
123.37 tk
ঘ
129.72 tk

ব্যাখ্যা

Question: A, B, and C share Tk. 2,400 in such a way that A has 3 times as much as B, and B has 5 times as much as C. How much money (in taka) does C receive?

Solution:
Given,
A = 3B
and, B = 5C

Now, substituting B = 5C into A = 3B
A = 3 × 5C = 15C

So, the ratio of A : B : C becomes A : B : C = 15C : 5 C : C = 15 : 5 : 1
The total parts of the ratio are: 15 + 5 + 1 = 21

C's share = (1/21)× 2400
= 114.29 (approximately)

∴ C receives approximately Tk. 114.29.

১১,৮৩৮.

A clock is set at 4 am. It loses 16 minutes in 24 hours. What will be the correct time when the clock indicates 9 pm on the 4th day?

ক
7 pm
খ
8 pm
গ
10 pm
ঘ
11 pm

ব্যাখ্যা

Question: A clock is set at 4 am. It loses 16 minutes in 24 hours. What will be the correct time when the clock indicates 9 pm on the 4th day?

Solution:
Time from 4 am on a day to 9 pm on the 4th day = 89 hours

As per the question, 23 hrs 44 minutes of this clock = 24 hours of the correct clock as this clock loses 16 minutes in 24 hours.

23 hrs 44 minutes = 356/15 Hours

Now, 356/15 Hrs of this clock = 24 hours of correct clock
89 hours of this clock = (15 × 24 × 89)/356 = 90 hours of the correct clock

∴ the correct clock gains one hour over the incorrect clock.

∴ The correct time on the fourth day would be 10 pm.

১১,৮৩৯.

A sum of money at a simple interest doubles in 5 years. It will become four times in:

ক
ক) 10 years
খ
খ) 12 years
গ
গ) 15 years
ঘ
ঘ) 20 years

ব্যাখ্যা

Case I:
Let principal be x
So, Amount after 5 years = 2x
Simple interest = Amount - Principal
= 2x - x
= x
Interest = Principal
So x = (x × R × 5)/100
⇒ R = 100/5

Case II:
Amount = 4x
Interest = 4x - x = 3x
Interest = 3 × Principal
Now
3x = [x × (100/5) × T₂]/100
T₂/5 = 3
⇒ T₂ = 15 years

১১,৮৪০.

A company blends two varieties of tea from two different tea gardens, one variety costing Tk. 30 per kg and the other Tk. 20 per kg in the ratio 4 : 3. He sells the blended tea at Tk. 27 per kg. Find the profit or loss percent.

ক
ক) 5%
খ
খ) 7%
গ
গ) 8%
ঘ
ঘ) 10%

ব্যাখ্যা

Question: A company blends two varieties of tea from two different tea gardens, one variety costing Tk. 30 per kg and the other Tk. 20 per kg in the ratio 4 : 3. He sells the blended tea at Tk. 27 per kg. Find the profit or loss percent.

Solution:
Let 4 kg of the first variety be mixed with 3 kg of second variety.
Then total cost price of 7 kg of tea = (30 × 4) + (20 × 3) = Tk. 180
Selling price of 7 kg of tea = (27 × 7) = Tk. 189

∴ Profit = Tk. (189 - 180) = Tk. 9

∴ Percentage of profit = (9 × 100)/180 = 5%

১১,৮৪১.

A number reduced by 25% becomes 225. What percent should it be increased so that it becomes 390?

ক
15%
খ
20%
গ
25%
ঘ
30%

ব্যাখ্যা

Question: A number reduced by 25% becomes 225. What percent should it be increased so that it becomes 390?

Solution:
Let
the number be x

Now
x - 25% of x = 225
⇒ x - 25x/100 = 225
⇒ (100x - 25x)/100 = 225
⇒ 75x/100 = 225
⇒ 75x = (225 × 100)
⇒ x = (225 × 100)/75
∴ x = 300

∴ Required increase
= (390 - 300)
= 90

∴ Increase percentage :
= (90/300) × 100%
= 30%

১১,৮৪২.

In how many ways 2 students can be chosen from the class of 18 students?

ক
ক) 153
খ
খ) 190
গ
গ) 162
ঘ
ঘ) 180

ব্যাখ্যা

Question: In how many ways 2 students can be chosen from the class of 18 students?

Solution:
Number of ways =¹⁸C₂ = 153

১১,৮৪৩.

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

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

ব্যাখ্যা

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

Solution:
Let the initial investments of A and B be 3x and 5x.
A : B : C = (3x × 12) : (5x × 12) : (5x × 6) = 36 : 60 : 30 = 6 : 10 : 5.

১১,৮৪৪.

If the length of a rectangle is increased by 30% and the width is decreased by 30%, then the area will be-

ক
Decreased by 9%
খ
Increased by 9%
গ
Decreased by 16%
ঘ
Increased by 16%

ব্যাখ্যা

Question: If the length of a rectangle is increased by 30% and the width is decreased by 30%, then the area will be-

Solution:
Let,
The length and breadth be 10 unit each.
∴ Area of rectangle = 10 × 10 = 100 unit²

New length = 10 + 30% of 10 = 10 + 3 unit
= 13 unit

New breadth = 10 - 30% of 10 = 10 - 3 unit
= 7 unit

∴ New area of rectangle = 13 × 7 = 91 unit²
Percentage decrease in area = 100 - 91 = 9 unit²

∴ The decrease percent is 9%.

১১,৮৪৫.

In a 100m race, Rakib defeats Tanvir by 5 seconds. If the speed of Rakib is 18 kmph, then the speed of Tanvir is:

ক
18.4 km/h
খ
14.4 km/h
গ
14.9 km/h
ঘ
15.6 km/h

ব্যাখ্যা

Question: In a 100m race, Rakib defeats Tanvir by 5 seconds. If the speed of Rakib is 18 kmph, then the speed of Tanvir is:
Solution:
Hints: (km/h)/3.6 = m/s
Time taken by Rakib = 100 / (18/3.6) seconds
= 20 seconds

Hence,
Time taken by Tanvir is = 20 + 5 = 25 seconds
So, Tanvir's speed=100/25 m/s
= 4 m/s
= 4 × 3.6 km/h
=14.4 km/h

১১,৮৪৬.

What percentage of numbers from 1 to 90 have 3 or 7 in the unit's digit?

ক
20%
খ
18%
গ
10%
ঘ
25%

ব্যাখ্যা

Question: What percentage of numbers from 1 to 90 have 3 or 7 in the unit's digit?

Solution:
Numbers from 1 to 90 with units digit 3 is-
3, 13, 23, 33, 43, 53, 63, 73, 83 = 9 numbers
And
Numbers from 1 to 90 with units digit 7is-
7, 17, 27, 37, 47, 57, 67, 77, 87 = 9 numbers

∴ Total numbers with units digit 3 or 7 = 9 + 9 = 18
And total numbers = 90

∴ Percentage = (18/90) × 100 = 20%

১১,৮৪৭.

The external and internal diameters of a hemispherical bowl are 10 cm and 8 cm respectively. What is the total surface area of the bowl?

ক
284 cm²
খ
286 cm²
গ
274 cm²
ঘ
296 cm²

ব্যাখ্যা

Question: The external and internal diameters of a hemispherical bowl are 10 cm and 8 cm respectively. What is the total surface area of the bowl?

Solution:

here,
R = 5cm
r = 4cm

total surface area:
S = 2πR² + 2πr² + π(R² - r²)
= 2πR² + 2πr² + πR² - πr²
= 3πR² + πr²
= π {3 × (5)² + (4)²}
= (22/7) (75 + 16)
= (22/7) × 91
= 286 cm²

১১,৮৪৮.

Which greatest number will divide 1304 and 1869 leaving remainders 8 and 9 respectively?

ক
ক) 12
খ
খ) 14
গ
গ) 16
ঘ
ঘ) 18

ব্যাখ্যা

Question: Which greatest number will divide 1304 and 1869 leaving remainders 8 and 9 respectively?

Solution:
The required number is = HCF of (1304 - 8) and (1869 - 9)
= HCF of 1296 and 1860
= 12

১১,৮৪৯.

asinθ = 1, acosθ = √3 then the value of √3tanθ - 1 = ?

ক
0
খ
1
গ
2
ঘ
√3 - 1

ব্যাখ্যা

Question: asinθ = 1, acosθ = √3 then the value of √3tanθ - 1 = ?

Solution:
asinθ = 1
acosθ = √3

Now,
asinθ/acosθ = 1/√3
⇒ tanθ = 1/√3
⇒ √3tanθ = 1
∴ √3tanθ - 1 = 0

১১,৮৫০.

If |x - 2| > 1, then what is the following should be correct?

ক
ক) x > 1 or x < 2
খ
খ) x > 3 or x < 1
গ
গ) x > 2 or x < 1
ঘ
ঘ) x > - 2 or x < - 1

ব্যাখ্যা

Question: If |x - 2| > 1, then what is the following should be correct?

Solution:
|x - 2| > 1

If (x - 2) is positive then,
x - 2 >1
⇒ x > 1 + 2
∴ x > 3

If (x - 2) is negative then,
- (x - 2) > 1
⇒ x - 2 < - 1
⇒ x < - 1 + 2
∴ x < 1

∴ x > 3 or x < 1

১১,৮৫১.

Given that an office opens at 9 a.m. and closes at 5:30 p.m., with a 17-minute lunch break, what is the proportion of the lunch break to the total workday?

ক
1 : 15
খ
1 : 30
গ
1 : 20
ঘ
1 : 25
ঙ
None of the above

ব্যাখ্যা

Question: Given that an office opens at 9 a.m. and closes at 5:30 p.m., with a 17-minute lunch break, what is the proportion of the lunch break to the total workday?

Solution:
The ratio of lunch breaks to the total period in the office
= 17/{(8 × 60) + 30}
= 17/510
= 1/30
= 1 : 30

১১,৮৫২.

Solve the quadratic equation: x² - 2x - 8 = 0

ক
x = 4 or x = - 2
খ
x = 5 or x = - 3
গ
x = 4 or x = - 4
ঘ
x = 2 or x = - 4

ব্যাখ্যা

Question: Solve the quadratic equation: x² - 2x - 8 = 0

Solution:
x² - 2x - 8 = 0
⇒ x² - 4x + 2x - 8 = 0
⇒ x(x - 4) + 2(x - 4) = 0
⇒ (x - 4)(x + 2) = 0
∴ x - 4 = 0
⇒ x = 4

or,
x + 2 = 0
⇒ x = - 2

১১,৮৫৩.

The difference between the two numbers is 20% of the larger number. If the smaller number is 12, find the larger number.

ক
25
খ
20
গ
15
ঘ
10

ব্যাখ্যা

Question: The difference between the two numbers is 20% of the larger number. If the smaller number is 12, find the larger number.

Solution:
Let the number be x
ATQ,
x - 12 = 20% of x
⇒ x - 12 = x/5
⇒ x - x/5 = 12
⇒ 4x/5 = 12
⇒ x = (12 × 5)/4
∴ x = 15

১১,৮৫৪.

The average of A and B is 45 and the sum of B & C is 78. What is the value of A - C?

ক
16
খ
12
গ
10
ঘ
22

ব্যাখ্যা

Question: The average of A and B is 45 and the sum of B & C is 78. What is the value of A - C?

Solution:
Given that,
Average of A and B = 45
Sum of B and C = 78

Now,
Average of A and B = 40, so-
⇒ (A + B)/2 = 45
∴ A + B = 90 ........(1)
And, B + C = 78 .........(2)

Subtract (2) from (1) than we get,
⇒ A + B - (B + C) = 90 - 78
⇒ A + B - B - C = 12
⇒ A - C = 12

So the value of A - C is 12.

১১,৮৫৫.

When an integer m is divided by 6, the remainder is 4. What is the remainder when 7m is divided by 3?

ক
3
খ
0
গ
2
ঘ
1

ব্যাখ্যা

Question: When an integer m is divided by 6, the remainder is 4. What is the remainder when 7m is divided by 3?

Solution:
m = 6 × quotient + remainder
or, m = 6n + 4 [Let, quotient = n]
or, 7m = 7 × 6n + 7 × 4
or, 7m = 7 × 6n + 28

since remainder can not be greater than or equal to divisor
= 6 ×(7n) + 27 + 1
= 3(14n + 9) + 1

if this number is divided by 3 the remainder will be 1

∴ Remainder is 1

১১,৮৫৬.

2 log 4 - 3 log 2 + (1/3) log 1000 =?

ক
log 4
খ
log 20
গ
log 25
ঘ
log 2

ব্যাখ্যা

Question:2 log 4 - 3 log 2 + (1/3) log 1000 =?

Solution:
Given that,
2 log 4 - 3 log 2 + (1/3) log 1000
= log 4² - log 2³ + log (10³)^1/3
= log 16 + log 10 - log 8
= log (16 × 10)/8
= log 20

১১,৮৫৭.

Two pipes A & B can fill the tank in 12 hours and 36 hours respectively. If both the pipes are opened simultaneously, how much time will be required to fill the tank?

ক
6 hours
খ
9 hours
গ
12 hours
ঘ
15 hours
ঙ
None of these

ব্যাখ্যা

Question: Two pipes A & B can fill the tank in 12 hours and 36 hours respectively. If both the pipes are opened simultaneously, how much time will be required to fill the tank?

Solution:
If pipe A requires 12 hrs to fill the tank, then part filled by pipe A in 1 hr = 1/12
If pipe B requires 36 hrs to fill the tank, then part filled by pipe B 1 hr = 1/36

Hence, part filled by (A + B) together in 1 hr = 1/12 + 1/36
= 4/36 = 1/9

In 1 hr both pipes together fill 1/9th part of the tank. This means, together they fill the tank in 9 hrs.

১১,৮৫৮.

Find the remainder when 7¹¹ + 7¹¹¹ + 7¹¹¹¹ is divided by 8.

ক
1
খ
2
গ
3
ঘ
5
ঙ
7

ব্যাখ্যা

Question: Find the remainder when 7¹¹ + 7¹¹¹ + 7¹¹¹¹ is divided by 8.

Solution:
When, 7 is divided by 8 then remainder = 7
When, 7² is divided by 8 then remainder = 1
When, 7³ is divided by 8 then remainder = 7
When, 7⁴ is divided by 8 then remainder = 1
Odd exponents give remainder = 7
Even exponents give remainder = 1

In 1st term, exponent is 11, which is odd so remainder = 7
In 2nd term, exponent is 111, which is also odd so remainder = 7
In 3rd term, exponent is 1111, which is odd so remainder = 7
So, (7¹¹ + 7¹¹¹ + 7¹¹¹¹) mod 8 = (7 + 7 + 7) mod 8 = 21 mod 8 = 5

১১,৮৫৯.

The average of runs of a cricket player in 10 innings was 42. How many runes must be made in his next innings to increase his average of runs by 3?

ক
85
খ
75
গ
69
ঘ
68

ব্যাখ্যা

Question: The average of runs of a cricket player in 10 innings was 42. How many runes must be made in his next innings to increase his average of runs by 3?

Solution:
after increasing 3 runs the average will be 45

so, run required
= (45 × 11) - (42 × 10)
= 495 - 420
= 75

১১,৮৬০.

Four pipes can fill a tank in 15, 20, 30 and 60 hours respectively. The first pipe was opened at 8 a.m, second at 9 a.m, third at 10 a.m. and fourth at 11 a.m. When will the tank be full-

ক
1 p.m.
খ
2 p.m.
গ
3 p.m.
ঘ
4 p.m.

ব্যাখ্যা

Question: Four pipes can fill a tank in 15, 20, 30 and 60 hours respectively. The first pipe was opened at 8 a.m, second at 9 a.m, third at 10 a.m. and fourth at 11 a.m. When will the tank be full-

Solution:
Let,
the time be x hours after 8 am.
Then, the first pipe worked for x hours
Second pipe for (x - 1) hours;
Third pipe for (x - 2) hours;
Fourth pipe for (x - 3) hours.

ATQ,
(x/15) + {(x - 1)/20} + {(x - 2)/30} + {(x - 3)/60} = 1
⇒ (4x + 3x - 3 + 2x - 4 + x - 3)/60 = 1
⇒ 10x - 10 = 60
⇒ 10x = 70
∴ x = 7

So, the tank will be full 7 hours after 8 am = 8 + 7 = 15 = 3 p.m.

১১,৮৬১.

The area of a parallelogram is 72 square centimetre and its altitude is twice the corresponding base. What is the length of the base?

ক
ক) 8 centimetre
খ
খ) 7 centimetre
গ
গ) 12 centimetre
ঘ
ঘ) 6 centimetre

ব্যাখ্যা

Let, base = x
Then, height = 2x
Area = base × height
= x × 2x
= 2x²
Area is given as 72 cm²2x² = 72 cm²
⇒ x² = 36 cm²
⇒ x = 6 cm
Hence, the length of the base is 6 cm.

১১,৮৬২.

Robin spends 70% of his salary and deposits 15% of his salary in the bank. If he is left with Tk. 1500, what is his monthly salary?

ক
Tk. 9000
খ
Tk. 9500
গ
Tk. 10000
ঘ
Tk. 10500

ব্যাখ্যা

Question: Robin spends 70% of his salary and deposits 15% of his salary in the bank. If he is left with Tk. 1500, what is his monthly salary?

Solution:
Let salary of Robin is Rs. x

As per question;
x - (70 % of x + 15 % of x) = 1500
⇒ x - 85% of x = 1500
⇒ 15% of x = 1500
⇒ (15x)/100= 1500
⇒ x = (1500 × 100)/15
∴ x = 10000

১১,৮৬৩.

A, B, C and D are four consecutive even numbers respectively and their average is 75. What is the product of A and C?

ক
5472
খ
5078
গ
4476
ঘ
3464

ব্যাখ্যা

Question: A, B, C and D are four consecutive even numbers respectively and their average is 75. What is the product of A and C?

Solution:
Let's define the four consecutive even numbers as A, B, C, and D. The average of these numbers is given as 75.
Than the sum is,
75 × 4 = 300
Now let the 1^stnumber is x. The next three will be x + 2, x + 4, x + 6

According to the question,
⇒ x + x + 2 + x + 4 + x + 6 = 300
⇒ 4x + 12 = 300
⇒ 4x = 300 - 12
⇒ 4x = 288
⇒ x = 288/4
∴ x = 72
So the 1^st number is A = 72
The next three numbers are B = 72 + 2 = 74, C = 72 + 4 = 76, D = 72 + 6 = 78

∴ Finally, the product of A and C is,
A × C = 72 × 76 = 5472
So, the product of A and C is 5472.

১১,৮৬৪.

Sue planted 4 times as many apple seeds as she planted orange seeds 15% of the apple seeds grow into trees and 10% of the orange seeds grew into trees. If a total of 420 apple trees and orange trees grew from the seeds, how many orange seeds did Sue plant?

ক
ক) 540
খ
খ) 600
গ
গ) 660
ঘ
ঘ) 720

ব্যাখ্যা

ধরি,
Sue কমলার চারা বুনেছিল x টি
Sue আপেলের চারা বুনেছিল 4x টি
প্রশ্নমতে,
4x এর 15% + x এর 10%= 420
4x এর 15/100 + x এর 10/100= 420
60x/100 + 10x/100 = 420
70x/100 = 420
70x = 42000
x = 42000/70
x = 600

১১,৮৬৫.

In triangle ABC, if AB = BC and ∠B = 90, ∠ A will be :

ক
ক) 40°
খ
খ) 45°
গ
গ) 110°
ঘ
ঘ) 130°

ব্যাখ্যা

Since, AB = BC
∠A = ∠C
let ∠A = x
so, ∠C is also equal to x
Now, ∠A + ∠B + ∠C = 180°
Or, 2x + 90 = 180
Or, x = (180-90)/2
Or, x = 45°

১১,৮৬৬.

What is the angle between the hour hand and minute hand at 1 : 20 pm?

ক
ক) 80°
খ
খ) 90°
গ
গ) 120°
ঘ
ঘ) 45°

ব্যাখ্যা

Question: What is the angle between the hour hand and minute hand at 1 : 20 pm?

Solution:
কোণ = |১১ × মিনিট - ৬০ × ঘণ্টা|°/২
= |১১ × ২০ - ৬০ × ১|°/২
= |২২০ - ৬০|°/২
= ১৬০°/২
= ৮০°

১১,৮৬৭.

If a ladder touches the roof of a wall and makes an angle of 30° with the 15 metre long wall, then the length of the ladder is-

ক
30 m
খ
7.5 m
গ
15 m
ঘ
17.3 m

ব্যাখ্যা

Question: If a ladder touches the roof of a wall and makes an angle of 30° with the 15 metre long wall, then the length of the ladder is-

Solution:

দেয়ালের উচ্চতা, h = 15 মিটার
দেয়ালের সাথে মইয়ের কোণ, θ = 30°
মইয়ের দৈর্ঘ্য, L = ?

আমরা জানি,
⇒ cos⁡θ = দেয়ালের উচ্চতা/মইয়ের দৈর্ঘ্য
⇒ cos30° = 15/L
⇒ √3/2 = 15/L
⇒ L = 30/√3
∴ L = 17.3 মিটার

১১,৮৬৮.

Which of the following is not a quadratic equation?

ক
x² + 3x - 5 = 0
খ
x² + x³ + 2 = 0
গ
3 + x + x² = 0
ঘ
x² - 9 = 0

ব্যাখ্যা

Question: Which of the following is not a quadratic equation?

Solution:
Option B is not a quadratic equation Since it has degree 3.

১১,৮৬৯.

The total ages of father and son is 42 years and the difference of thier ages is 22 years. What is son's age now?

ক
ক) 10 years
খ
খ) 11 years
গ
গ) 12 years
ঘ
ঘ) 13 years
ঙ
ঙ) 14 years

ব্যাখ্যা

Let's father's age x and son's age is y
ATQ, x + y = 42 ....(i)
x - y = 22 .... (ii)
(i) - (ii) = 2y = 20
∴ y = 10
∴ son's age is 10 years

১১,৮৭০.

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

ক
Sunday
খ
Saturday
গ
Wednesday
ঘ
Tuesday

ব্যাখ্যা

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

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

১১,৮৭১.

A metallic cone of radius 18 cm and height 25 cm is melted and made into spheres of radius 3 cm each. How many spheres are there?

ক
ক) 36
খ
খ) 50
গ
গ) 75
ঘ
ঘ) 125

ব্যাখ্যা

Question: A metallic cone of radius 18 cm and height 25 cm is melted and made into spheres of radius 3 cm each. How many spheres are there?

Solution:
We know that,
Volume of the cone = (1/3)π × r² × h
and volume of the sphere = (4/3)π × r³

Here,
Radius of cone, r₁ = 18 cm
Height of cone, h = 25 cm
Radius of sphere, r₂ = 3 cm

Number of spheres = Volume of the cone/Volume of the sphere
= {(1/3)π × (r₁)² × h}/{(4/3)π × (r₂)³}
= {(1/3)π × 18 × 18 × 25}/{(4/3)π × 3 × 3 × 3}
= 75

১১,৮৭২.

Find the remainder when 4⁹⁶ is divided by 8.

ক
ক) 0
খ
খ) 1
গ
গ) 2
ঘ
ঘ) 3

ব্যাখ্যা

Question: Find the remainder when 4⁹⁶ is divided by 8.

Solution:
4⁹⁶/8
= (4² × 4⁹⁴)/8
= (16 × 4⁹⁴)/8
= 2 × 4⁹⁴

যেহেতু 4⁹⁶, 8 দ্বারা নিঃশেষে বিভাজ্য। অতএব, ভাগশেষ শূন্য হবে।

১১,৮৭৩.

A motor boat takes 12 hours to go downstream and it takes 24 hours to return the same distance. What is the time taken by boat in still water?

ক
ক) 16 h
খ
খ) 18 h
গ
গ) 21 h
ঘ
ঘ) 24 h

ব্যাখ্যা

If t₁ and t₂ are the upstream and down stream times.
Then time taken in still water is given by
(2 × t₁ × t₂) / (t₁ + t₂)
= (2 × 12 × 24) / 36
= 16h

১১,৮৭৪.

What is the value of log_√3 81?

ক
1/√3
খ
4
গ
√3/2
ঘ
8

ব্যাখ্যা

Question: What is the value of log_√3 81?

Solution:
log_√3 81
= log_√3 3⁴
= 4 log_√3 3
= 4 log_√3 (√3)²
= 2 ⋅ 4 log_√3 √3
= 2 ⋅ 4 ⋅ 1
= 8

১১,৮৭৫.

∠X and ∠Y are complementary to each other. If ∠X = 25° + 2y and ∠Y = 3y, find the value of ∠Y.

ক
25°
খ
35°
গ
39°
ঘ
45°

ব্যাখ্যা

Question: ∠X and ∠Y are complementary to each other. If ∠X = 25° + 2y and ∠Y = 3y, find the value of ∠Y.

Solution:
Here,
∠X = 25° + 2y and ∠Y = 3y
For complementary angles,
∠X + ∠Y = 90°
⇒ (25° + 2y) + 3y = 90°
⇒ 25° + 5y = 90°
⇒ 5y = 65°
∴ y = 13°

So, ∠Y = 3 × 13° = 39°

১১,৮৭৬.

A question paper has two parts, A and B, each containing 10 questions. If a student has to choose 7 from part A and 6 from part B, in how many ways can he choose the questions?

ক
ক) 21,200
খ
খ) 22,200
গ
গ) 25,200
ঘ
ঘ) 24,200

ব্যাখ্যা

Question: A question paper has two parts, A and B, each containing 10 questions. If a student has to choose 7 from part A and 6 from part B, in how many ways can he choose the questions?

Solution:
No. of questions in part A = 10
No. of questions in part B = 10

A student has to choose 8 from part A and 6 from part B

Required number of ways = (¹⁰C₇ × ¹⁰C₆) =120 × 210 = 25,200

১১,৮৭৭.

⁶P_m = 360, ⁶C_m = 15, what is the value of m?

ক
ক) 4
খ
খ) 5
গ
গ) 6
ঘ
ঘ) 7

ব্যাখ্যা

Question: ⁶P_m = 360, ⁶C_m = 15, what is the value of m?

Solution:
⁶P_m = 360
⇒ 6!/(6 - m)! = 360........(1)

⁶C_m = 15
⇒ 6!/m! (6 - m)! = 15..........(2)

(1) ÷ (2) ,
{6!/(6 - m)!} / {6!/m! (6 - m)! } = 360/15
⇒ m! = 24
= 4 × 3 × 2 × 1

∴ m = 4

১১,৮৭৮.

A man rows downstream 54 km and upstream 42 km, taking 6 hours each time. The speed of the man is?

ক
8 km/hr
খ
9 km/hr
গ
12 km/hr
ঘ
6 km/hr

ব্যাখ্যা

Question: A man rows downstream 54 km and upstream 42 km, taking 6 hours each time. The speed of the man is?

Solution:
Speed of upstream = 42/6 = 7 km/hr
Speed of downstream = 54/6 = 9 km/hr

Speed of man in still water = (7 + 9)/2 = 16/2 = 8 km/hr

১১,৮৭৯.

If θ is a positive angle and 9sin²θ - 9 = 0, then the value of tan(θ - 30°) is equal to?

ক
1
খ
√3
গ
1/√3
ঘ
0

ব্যাখ্যা

Question: If θ is a positive angle and 9sin²θ - 9 = 0, then the value of tan(θ - 30°) is equal to?

Solution:
Given,
9sin²θ - 9 = 0
⇒ 9sin²θ = 9
⇒ sin²θ = 1
⇒ sinθ = 1
⇒ sinθ = sin90°
∴ θ = 90°

Now,
tan(θ - 30°) = tan(90° - 30°)
= tan60°
= √3

১১,৮৮০.

Mr. Tamim travelled from A to B at a speed of 3 km/hr and returned from B to A at a speed of 5 km/hr. If the entire trip required 8 hours what is the distance between A and B in km?

ক
15km
খ
10km
গ
14km
ঘ
18km
ঙ
40km

ব্যাখ্যা

Question: Mr. Tamim travelled from A to B at a speed of 3 km/hr and returned from B to A at a speed of 5 km/hr. If the entire trip required 8 hours what is the distance between A and B in km?

Solution:
Let, the distance between A to B be x km.

According to the question,
⇒ (x/3) + (x/5) = 8
⇒ (5x + 3x)/15 = 8
⇒ (8x)/15 = 8
⇒ x = (15 × 8)/8
∴ x = 15

∴ Distance between A to B is = 15km

১১,৮৮১.

Tom, Dick and Harry went for lunch to a restaurant. Tom had $100 with him, Dick had $60 and Harry had $40. They got a bill for $104 and decided to give a tip of $16. They further decided to share the total expenses in the ratio of the amounts of money each carried. The amount of money which Tom paid more than what Harry paid is-

ক
ক) 120
খ
খ) 200
গ
গ) 60
ঘ
ঘ) 24
ঙ
ঙ) 36

ব্যাখ্যা

Question: Tom, Dick and Harry went for lunch to a restaurant. Tom had $100 with him, Dick had $60 and Harry had $40. They got a bill for $104 and decided to give a tip of $16. They further decided to share the total expenses in the ratio of the amounts of money each carried. The amount of money which Tom paid more than what Harry paid is-

Solution:
Tom : Dick : Harry = 100 : 60 : 40
= 10 : 6 : 4
= 5 : 3 : 2

Total bill = 104 + 16
= 120

Tom paid = 120 × (5/10) = 60
Harry paid = 120 × (2/10) = 24

Tom paid ( 60 - 24) = 36 more than Harry paid.

১১,৮৮২.

Siddik has a new set of golf clubs. Using a club 8, 7, and 6 the ball travels an average distance of 100m, 108m, 114m respectively. How far will the ball go if he uses a club 5?

ক
ক) 122m
খ
খ) 120m
গ
গ) 118m
ঘ
ঘ) 116m

ব্যাখ্যা

এখানে,
দূরত্বকে একটি ধারা মনে করে পাই,
ধারাঃ 100m 108m 114m 118m
পার্থক্যঃ 8m 6m 4m
সুতরাং সঠিক উত্তর 118m

১১,৮৮৩.

A train passes a stationary pole in 8 seconds. The train also passes a 200 m long bridge in 28 seconds. What is the length of the train?

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

ব্যাখ্যা

Question: A train passes a stationary pole in 8 seconds. The train also passes a 200 m long bridge in 28 seconds. What is the length of the train?

Solution:
Given that,
Time to pass a pole = 8 s
Time to pass a 200 m bridge = 28 s

Let the length of the train = L meters
When passing a pole, the train covers distance = L in 8 s
And when passing a bridge, distance = (L + 200) in 28 s

Now,
From pole,
Speed = Distance/Time = L/8 m/s

And,
From bridge,
Speed = Distance/Time = (L + 200)/28 m/s

ATQ,
L/8 = (L + 200)/28
⇒ 28L = 8L + 1600
⇒ 28L - 8L = 1600
⇒ 20L = 1600
⇒ L = 1600/20
∴ L = 80 m

So the length of the train is 80 meters.

১১,৮৮৪.

The difference between the length and the perimeter of a rectangle is 100 cm. What is the breadth of the rectangle?

ক
80 cm
খ
60 cm
গ
100 cm
ঘ
Data Inadequate

ব্যাখ্যা

Question: The difference between the length and the perimeter of a rectangle is 100 cm. What is the breadth of the rectangle?

Solution:
Let the length of the rectangle be 'x' and breadth of the rectangle be 'y'

According to the question:
2(x + y) - x = 100
⇒ 2x + 2y - x = 100
⇒ x + 2y = 100

From this we cannot find 'y' (breadth), so the given data is inadequate.

১১,৮৮৫.

At the rate of 8.5% p. a. simple interest, sum of Tk. 4800 will earn how much interest in 2 years 6 months?

ক
ক) Tk.918
খ
খ) Tk.1020
গ
গ) Tk.980
ঘ
ঘ) Tk.1080

ব্যাখ্যা

Principal = Tk. 4800
Rate = 8.5%
Time = 2 years + 6 months
= 2 + (6/12)
= 2 + (1/2)
= 5/2
= 2.5 years

Simple Interest = (4800 × 8.5 × 2.5)/100
= Tk.1020

১১,৮৮৬.

A shopkeeper expects a gain of 22.5% on his cost price. If in a week, his sale was of Tk 392, what was his profit?

ক
ক) 62
খ
খ) 70
গ
গ) 72
ঘ
ঘ) 76

ব্যাখ্যা

C.P. = (100/122.5)x 392
= (1000/1225)x 392
= 320
Profit = (392 - 320) = 72.

১১,৮৮৭.

Two trains A and B start running together from the same point in the same direction, at the speed of 60 kmph and 72 kmph respectively. If the length of each of the trains is 210 meters, how long will it take for B to cross train A?

ক
ক) 124 sec
খ
খ) 128 sec
গ
গ) 126 sec
ঘ
ঘ) 144 sec

ব্যাখ্যা

Relative speed = (72 - 60) km/hr
= 12 km/hr
= 12 × (5/18) m/sec
= 10/3 m/sec

Total distance covered = Sum of lengths of trains = (210 + 210) m = 420 m
Time taken = (420 × 3)/10sec=126sec

১১,৮৮৮.

It is between 3 pm and 4 pm and the distance between the hour hand and the minute hand of clock is 18 minutes spaces. What time does the clock show ?

ক
3 : 36 pm
খ
3 : 21 pm
গ
3 : 12 pm
ঘ
3 : 24 pm

ব্যাখ্যা

Question: It is between 3 pm and 4 pm and the distance between the hour hand and the minute hand of clock is 18 minutes spaces. What time does the clock show?

Solution:
At 3 o'clock, the minute hand is 15 minute spaces behind the hour hand.
Thus, the minute hand has to gain (15 + 18) = 33 minute spaces
55 minutes are gained in 60 minutes.
33 minutes are gained in = (60/55) × 33 minutes
= 36 minutes

∴ The hands will be 18 minutes spaces apart at 3 : 36 pm.

১১,৮৮৯.

If I would have been twice as efficient as today, I would have finished work in 12 days. If my efficiency is reduced to one-third of what it is at present, in how many days, I would be able to finish the work?

ক
18
খ
8
গ
52
ঘ
72

ব্যাখ্যা

Question: If I would have been twice as efficient as today, I would have finished work in 12 days. If my efficiency is reduced to one-third of what it is at present, in how many days, I would be able to finish the work?

Solution:
Let I finish work in x days.
With double efficiency, the time taken = x/2 days
That means when the efficiency was double (2x),
then the time taken to finish the work is 12 days
Now,
with the present efficiency time taken to complete the work = 12 × 2 =24 days
With one-third efficiency, the days required to finish the work = 3 × 24 = 72 days,
as efficiency is inversely proportional to days.
Hence, when the efficiency gets one-third, then the work will be finished in 72 days.

১১,৮৯০.

If a/b + b/a = 2, then the value of (a - b) is -

ক
ক) 1
খ
খ) 2
গ
গ) -1
ঘ
ঘ) 0

ব্যাখ্যা

Given,
a/b + b/a = 2
⇒ (a² + b²)/ab = 2
⇒ (a² + b²) = 2ab
⇒ a² + b² - 2ab = 0
⇒ (a - b)² = 0
⇒ a - b = 0.

১১,৮৯১.

Eight years ago, Karim was five times as old as his son. Now, Karim is 28 years older than his son. What is the present age of the son?

ক
16 years
খ
17 years
গ
18 years
ঘ
15 years
ঙ
14 years

ব্যাখ্যা

Question: Eight years ago, Karim was five times as old as his son. Now, Karim is 28 years older than his son. What is the present age of the son?

Solution:
ধরি,
বর্তমানে Karim-এর বয়স = x বছর এবং তার ছেলের বয়স = y বছর

৮ বছর আগে, Karim তার ছেলের বয়সের পাঁচ গুণ ছিল। অর্থাৎ
⇒ x - 8 = 5(y - 8).......... (১)

আবার,
বর্তমানে Karim তার ছেলের চেয়ে ২৮ বছর বড়।
x = y + 28

x এর মান (১) এ বসিয়ে পাই,
⇒ (y + 28) - 8 = 5(y - 8)
⇒ y + 20 = 5y - 40
⇒ 20 + 40 = 5y - y
⇒ 60 = 4y
∴ y = 15

∴ ছেলের বর্তমান বয়স = 15 বছর।

১১,৮৯২.

A student erroneously multiplied a number by 2/5 instead of 5/2.What is the percentage error in the calculation?

ক
24%
খ
54%
গ
74%
ঘ
84%

ব্যাখ্যা

Question: A student erroneously multiplied a number by 2/5 instead of 5/2.What is the percentage error in the calculation?

Solution:
Let the number be 100.
2/5 of 100 is 40 while 5/2 of 100 is 250.
Now the difference is 210 on a base of 250.
Therefore, percentage difference is (210/250) × 100 = 84%.

১১,৮৯৩.

0.004 × ? = 0.000016

ক
ক) 0.04
খ
খ) 0.004
গ
গ) 0.4
ঘ
ঘ) 0.0004

ব্যাখ্যা

0.004 × a = 0.000016
a = 0.000016/0.004
a = 0.004

১১,৮৯৪.

50 people consume 350 kg of rice in 30 days. In how many days will 35 people consume 50 kg of rice?

ক
ক) 6
খ
খ) 8
গ
গ) 12
ঘ
ঘ) None of these

ব্যাখ্যা

Let the required number of days be x
Less people, More days (Indirect proportion)
Less quantity, Less days ( Direct proportion)
People 35 : 50
⟩ :: 30 : x
Quantity 350 : 50
⇒ 35 × 350 × x = 50 × 50 × 30
⇒ x = 300/49

১১,৮৯৫.

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
10
খ
12
গ
15
ঘ
18

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:
ways can one cast his vote =⁶C₂
= 15

১১,৮৯৬.

The average of 15 numbers is zero. Of them, at the most, how many may be greater than zero?

ক
ক) 0
খ
খ) 1
গ
গ) 10
ঘ
ঘ) 14

ব্যাখ্যা

প্রশ্ন: The average of 15 numbers is zero. Of them, at the most, how many may be greater than zero?

সমাধান: ধরি, সংখ্যাগুলি হল a₁, a₂, ..., a₁₅
দেয়া আছে, সংখ্যাগুলোর গড় শূন্য

তাহলে, (a₁ + a₂ +...+ a₁₅)/15 = 0
⇒ a₁ + a₂ +...+ a₁₅ = 0
⇒ a₁ + a₂ +...+ a₁₄= −a₁₅
∴ সর্বোচ্চ ১৪ টি সংখ্যার মান শুন্য থেকে বড় হতে পারে।

১১,৮৯৭.

In a class, the number of girls is 5/6 of the number of boys. The total students of the class is 121. If 5 more girls join the class, what will be the new ratio of girls to boys?

ক
11 : 10
খ
5 : 6
গ
7 : 5
ঘ
10 : 11

ব্যাখ্যা

Question: In a class, the number of girls is 5/6 of the number of boys. The total students of the class is 121. If 5 more girls join the class, what will be the new ratio of girls to boys?

Solution:
Let the number of boys be x
Then, number of girls = 5/6 of x = 5x/6

ATC,
x + (5x/6) = 121
⇒ (6x + 5x)/6 = 121
⇒ 6x + 5x = 6 × 121
⇒ 11x = 6 × 121
⇒ x = (6 × 121)/11
∴ x = 66

So, number of boys = x = 66
number of girls = 5/6 of x = 5/6 of 66 = 55

Now, 5 more girls join,
∴ New number of girls = 55 + 5 = 60

∴ ratio (girls : boys) = 60/66
= 10/11
= 10 : 11

১১,৮৯৮.

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
280 ways
ঘ
260 ways

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

১১,৮৯৯.

The contributions made by Rasel and Sejan are in the ratio of 3 : 2. If 5% of total profit is donated and Rasel gets 8550 as his share of profit, what is the total profit?

ক
14250
খ
14500
গ
15000
ঘ
15500

ব্যাখ্যা

Question: The contributions made by Rasel and Sejan are in the ratio of 3 : 2. If 5% of total profit is donated and Rasel gets 8550 as his share of profit, what is the total profit?

Solution:
Let the total profit after doneted = x
The ratio of contribution by Rasel and Sejan = 3 : 2
Sum of ratios = 5

Rasel's share = (3/5) × x = 8550
⇒ x = (8550 × 5)/3
∴ Total profit after doneted x = 14250

ATQ,
Rasel gets 8550 after 5 % donated
Now,
95% = 14250
1% = 14250/ 95
And, 100% = (14250/95) × 100 = 15000

১১,৯০০.

A 180 meter long train running at the speed of 72 km/h crosses another train running in the opposite direction at the speed of 108 km/h in 10 seconds. What is the length of the other train?

ক
120 meters
খ
320 meters
গ
220 meters
ঘ
420 meters

ব্যাখ্যা

Question: A 180 meter long train running at the speed of 72 km/h crosses another train running in the opposite direction at the speed of 108 km/h in 10 seconds. What is the length of the other train?

Solution:
Relative speed = (72 + 108) km/h
= 180 × (5/18) m/sec
= 50 m/sec

Let,
Length of the other train = x metres

Then,
(x + 180)/10 = 50
⇒ x + 180 = 500
⇒ x = 500 - 180
∴ x = 320

∴ The length of the other train is 320 meters.