Bank Math

PrepBank · পাতা ১২৪ / ১৬১ · ১২,৩০১–১২,৪০০ / ১৬,১২৪

১২,৩০১.

In how many ways can 5 balls can be chosen from 9 different balls?

ক
144
খ
186
গ
126
ঘ
220

সঠিক উত্তর: গ

126

উত্তর

সঠিক উত্তর: গ

126

গ

ব্যাখ্যা

Question: In how many ways can 5 balls can be chosen from 9 different balls?

Solution:

Here,
Total number of different balls, n = 9
Chosen balls from different balls, r = 5

The number of ways 5 balls can be chosen is
= ⁿC_r= n!/r!(n - r)!
= 9!/5!(9 - 5)!
= 9!/(5! × 4!)
= (9 × 8 × 7 × 6 × 5!)/(5! × 4!)
= (9 × 8 × 7 × 6)/4!
= 3024/(4 × 3 × 2 × 1)
= 3024/24
= 126

∴ 5 balls can be chosen from 9 different balls in 126 ways.

১২,৩০২.

The supplement of an angle exceeds twice the angle by 30°. Then the angle is equal to-

ক
30°
খ
45°
গ
50°
ঘ
60°

সঠিক উত্তর: গ

50°

উত্তর

সঠিক উত্তর: গ

50°

গ

ব্যাখ্যা

Question: The supplement of an angle exceeds twice the angle by 30°. Then the angle is equal to-

Solution:
Let the angle be x
Then, its supplement = 180 - x

According to the question,
180 - x = 2x + 30
⇒ 180 - 30 = 3x
⇒ 150 = 3x
⇒ x = 50°

১২,৩০৩.

If ⁶P_r = 360 and If ⁶C_r = 15, then r =?

ক
3
খ
4
গ
5
ঘ
6

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: If ⁶P_r = 360 and If ⁶C_r = 15, then r =?

Solution:
ⁿP_r = ⁿC_r × r!
⇒ ⁶P_r = 15 × r!
⇒ 360 = 15 × r!
⇒ r! = 360/15
⇒ r! = 24
⇒ r! = 4 × 3 × 2 × 1
⇒ r! = 4!
∴ r = 4

১২,৩০৪.

In how many ways can six different rings be worn on four fingers of one hand?

ক
ক) 10
খ
খ) 12
গ
গ) 15
ঘ
ঘ) 16

সঠিক উত্তর: গ

গ) 15

উত্তর

সঠিক উত্তর: গ

গ) 15

গ

ব্যাখ্যা

Required number of ways,
= ⁶C₄= 6×5×4! /2!4!
= 15 ways

১২,৩০৫.

By working 3 hours a day, P can complete a work in 12 days, and working 4 hours a day, Q can complete the same work in 9 days. Working 6 hours a day, they can jointly complete the work in -

ক
2 days
খ
3 days
গ
4 days
ঘ
5 days
ঙ
None

সঠিক উত্তর: খ

3 days

উত্তর

সঠিক উত্তর: খ

3 days

খ

ব্যাখ্যা

Question: By working 3 hours a day, P can complete a work in 12 days, and working 4 hours a day, Q can complete the same work in 9 days. Working 6 hours a day, they can jointly complete the work in -

Solution:
P can complete the work in = 3 × 12 = 36 hours
Q can complete the work in = 4 × 9 = 36 hours

(P + Q)'s 1 hour's work = (1/36) + (1/36)
= 2/36
= 1/18

Hence, P and Q can complete the work in 18 hours
So, working 6 hours a day they require = 18/6 = 3 days to complete the work

১২,৩০৬.

If B = 45° , then what is the value of (1 - cot²B)/(1 + cot²B)?

ক
1/2
খ
1
গ
2
ঘ
0

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: If B = 45° , then what is the value of (1 - cot²B)/(1 + cot²B)?

Solution:
Here, B = 45°

Now,
(1 - cot²B)/(1 + cot²B)
= {1 - (cot45°)²}/{1 + (cot45°)²}
= (1 - 1²)/(1 + 1²)
= 0/2
= 0

১২,৩০৭.

ক
1/1000
খ
1/506
গ
253/500
ঘ
None of these

সঠিক উত্তর: ক

1/1000

উত্তর

সঠিক উত্তর: ক

1/1000

ক

ব্যাখ্যা

Question:

Solution:

১২,৩০৮.

If you divided 50 by half and add 5 with the resulting figure, then what is the final result?

ক
ক) 55
খ
খ) 105
গ
গ) 95
ঘ
ঘ) 125

সঠিক উত্তর: খ

খ) 105

উত্তর

সঠিক উত্তর: খ

খ) 105

খ

ব্যাখ্যা

Quetion: If you divided 50 by half and add 5 with the resulting figure, then what is the final result?

Solution:
50/0.5+5
=(50 ×10)/5 + 5
= 100 + 5
=105

১২,৩০৯.

In 60 litres of mixture, milk and water are in 3 : 2. How much water to add to make it 1 : 1?

ক
6 litres
খ
9 litres
গ
12 litres
ঘ
15 litres

সঠিক উত্তর: গ

12 litres

উত্তর

সঠিক উত্তর: গ

12 litres

গ

ব্যাখ্যা

Question: In 60 litres of mixture, milk and water are in 3 : 2. How much water to add to make it 1 : 1?

Solution:
Find the quantity of milk and water in the initial mixture:
→ Total parts = 3 + 2 = 5 parts
→ Milk = (3/5) × 60 = 36 L
→ Water = (2/5) × 60 = 24 L

Let x liters of water be added.
New amount of water = 24 + x
Milk remains the same = 36 L

According to the question:
We want the final ratio to be 1:1, i.e.
→ Milk = Water
⇒ 36 = 24 + x
⇒ x = 36 − 24 = 12 L

১২,৩১০.

The sides of a triangle are in the ratio 10 : 24 : 26 and its perimeter is 300 m. What is its area?

ক
2500 m²
খ
3000 m²
গ
3500 m²
ঘ
4000 m²
ঙ
None of these

সঠিক উত্তর: খ

3000 m²

উত্তর

সঠিক উত্তর: খ

3000 m²

খ

ব্যাখ্যা

Question: The sides of a triangle are in the ratio 10 : 24 : 26 and its perimeter is 300 m. What is its area?

Solution:
Let the sides are 10x, 24x, and 26x.
The perimeter is 300 m.
So, 10x + 24x + 26x = 300
⇒ 60x = 300
∴ x = 5

So, the sides are 10 × 5 = 50 meters
24 × 5 = 120 meters
26 × 5 = 130 meters

10² + 24² = 26²
⇒ 100 + 576 = 676
⇒ 676 = 676
so, it is a right triangle.

The area of a right triangle is = (1/2) × base × height
= (1/2) × 50 × 120
= 3000 m²

১২,৩১১.

The average of 20 numbers is zero. Of them, at the most, how many may be greater than zero?

ক
0
খ
1
গ
10
ঘ
19

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: The average of 20 numbers is zero. Of them, at the most, how many may be greater than zero?

Solution:
ধরি,
সংখ্যাগুলি হল a₁, a₂, ..., a₂₀

দেয়া আছে,
সংখ্যাগুলোর গড় শূন্য

তাহলে,
(a₁ + a₂ +...+ a₂₀)/20 = 0
⇒ a₁ + a₂ +...+ a₂₀ = 0
⇒ a₁ + a₂ +...+ a₁₉ = - a₂₀

∴ সর্বোচ্চ 19 টি সংখ্যার মান শুন্য থেকে বড় হতে পারে।

১২,৩১২.

The investment ratio of two partners, A and B, is 11 : 12, while their profit ratio is 2 : 3. Given that A invested his funds for 8 months, determine the duration of B's investment.

ক
12 months
খ
9 months
গ
10 months
ঘ
11 months

সঠিক উত্তর: ঘ

11 months

উত্তর

সঠিক উত্তর: ঘ

11 months

ঘ

ব্যাখ্যা

Question: The investment ratio of two partners, A and B, is 11 : 12, while their profit ratio is 2:3. Given that A invested his funds for 8 months, determine the duration of B's investment.

Solution:
Let, A invested Tk 11x for 8 months
and B invested Tk 12x for y months

Now,
(11x × 8) : (12x × y) = 2 : 3
⇒ (11x × 8) / (12x × y) = 2/3
⇒ 88x/12xy = 2/3
⇒ 24y = 264
⇒ y = 11

So, B invested for 11 months.

১২,৩১৩.

If 4xy = 4, what is the value of log_yx -

ক
ক) -1
খ
খ) 2
গ
গ) 0
ঘ
ঘ) 1/2

সঠিক উত্তর: ক

ক) -1

উত্তর

সঠিক উত্তর: ক

ক) -1

ক

ব্যাখ্যা

Question: If 4xy = 4, what is the value of log_yx -

Solution:
4xy = 4
⇒ xy = 1
∴ x = 1/y

log_yx
=log_y(1/y)
= log_yy^-1
= -1 log_yy
= -1

১২,৩১৪.

If 3^{(a + 2)} = 9^{(3a - 4)} then the value of a is = ?

ক
8
খ
1
গ
2
ঘ
6

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: If 3^{(a + 2)} = 9^{(3a - 4)} then the value of a is = ?

Solution:
3^{(a + 2)} = 9^{(3a - 4)}
⇒ 3^{(a + 2)} = (3²)^{(3a - 4)}
⇒ 3^{(a + 2)} = 3^{(6a - 8)}
⇒ a + 2 = 6a - 8
⇒ 5a = 10
∴ a = 2

১২,৩১৫.

Which of the following is equivalent to the pair of inequalities x + 7 > 11 and x - 2 ≤ 5?

ক
ক) 3 < x ≤ 8
খ
খ) 4 < x ≤ 7
গ
গ) 2 < x ≤ 6
ঘ
ঘ) 4 < x ≤ 8

সঠিক উত্তর: খ

খ) 4 < x ≤ 7

উত্তর

সঠিক উত্তর: খ

খ) 4 < x ≤ 7

খ

ব্যাখ্যা

প্রশ্ন : Which of the following is equivalent to the pair of inequalities x + 7 > 11 and x - 2 ≤ 5?

সমাধান:

x + 7 > 11

⇒ x > 11 - 7

⇒ x > 4

x - 2 ≤ 5

⇒ x ≤ 5 + 2

⇒ x ≤ 7

x > 4 and x ≤ 7

⇒ 4 < x ≤ 7

১২,৩১৬.

In a word jumble, there are 8 consonants and 5 vowels given. Find out in how many ways can we form a 5-letter word having three consonants and 2 vowels?

ক
720
খ
8540
গ
67200
ঘ
None of these

সঠিক উত্তর: গ

67200

উত্তর

সঠিক উত্তর: গ

67200

গ

ব্যাখ্যা

Question: In a word jumble, there are 8 consonants and 5 vowels given. Find out in how many ways can we form a 5-letter word having three consonants and 2 vowels?

Solution:
Number of ways of selecting 3 consonants from 8 is ⁸C₃
Number of ways of selecting 2 vowels from 5 is ⁵C₂

Number of ways of selecting 3 consonants from 8 and 2 vowels from 5 is ⁸C₃ × ⁵C₂
= {(8 × 7 × 6)/(3 × 2 × 1)} × {(5 × 4)/(2 × 1)}
= 56 × 10
= 560
It means we can have 560 groups where each group contains total 5 letters (3 consonants and 2 vowels).
Number of ways of arranging 5 letters among themselves 5! = 5 × 4 × 3 × 2 × 1 = 120

∴ Total number of words will be formed = 560 × 120
= 67200

∴ Required number of ways is 67200

১২,৩১৭.

4^{x + 2} = 2^{2x + 1} + 14 , Find the value of x.

ক
ক) 0
খ
খ) 1
গ
গ) 2
ঘ
ঘ) 4

সঠিক উত্তর: ক

ক) 0

উত্তর

সঠিক উত্তর: ক

ক) 0

ক

ব্যাখ্যা

Question: 4^{x + 2} = 2^{2x + 1} + 14, Find the value of x.

Solution:
4^{x + 2} = 2^{2x + 1} + 14
⇒ 4².4^x - 2^2x.2¹ = 14
⇒ 16.4^x - 4^x.2 = 14
⇒ 4^x(16 - 2) = 14
⇒ 4^x.14 = 14
⇒ 4^x= 1
⇒ 4^x = 4⁰
⇒ x = 0

১২,৩১৮.

What is Rasel's present age, if after 20 years his age will be 10 times his age 10 years back?

ক
6.2 years
খ
7.7 years
গ
10 years
ঘ
13.3 years
ঙ
None of these

সঠিক উত্তর: ঘ

13.3 years

উত্তর

সঠিক উত্তর: ঘ

13.3 years

ঘ

ব্যাখ্যা

Question: What is Rasel's present age, if after 20 years his age will be 10 times his age 10 years back?

Solution:

Let Rasel's present age be x

Rasel's age before 10 years = x - 10)

Rasel's age after 20 years = (x + 20)

We are given that,
Rasel's age after 20 years (x + 20) is 10 times his age 10 years back (x - 10)

Therefore,
(x + 20) = 10(x - 10)

⇒ x + 20 = 10x - 100

⇒ 9x = 120
∴ x = 13.3 years

১২,৩১৯.

What is the greatest number of three digits, which, when divided by 6, 9, and 12, leaves a remainder of 3 in each case?

ক
991
খ
981
গ
975
ঘ
973

সঠিক উত্তর: গ

975

উত্তর

সঠিক উত্তর: গ

975

গ

ব্যাখ্যা

Question: What is the greatest number of three digits, which, when divided by 6, 9, and 12, leaves a remainder of 3 in each case?

Solution:
the largest three-digit number is = 999
the L.C.M of 6, 9, 12 is = 36
dividing 999 by 36 we get the remainder of 27

so, the number is = (999 -27) + 3 = 975

১২,৩২০.

The HCF and LCM of two numbers are 8 and 48 respectively. If one of the number is 24, then the other number is = ?

ক
ক) 16
খ
খ) 32
গ
গ) 48
ঘ
ঘ) 64

সঠিক উত্তর: ক

ক) 16

উত্তর

সঠিক উত্তর: ক

ক) 16

ক

ব্যাখ্যা

HCF = 8
LCM = 48
One number = 24
Let other number be = y
∴ 24y = 48 × 8
⇔ y = 16
---------------------------------
প্রশ্নে বলা হয়েছে যে, দুইটি সংখ্যার গসাগু ও লসাগু যথাক্রমে ৮ ও ৪৮। একটি সংখ্যা ২৪ হলে, অপর সংখ্যা কত?
আমরা জানি, দুইটি সংখ্যার গসাগু × লসাগু = একটি সংখ্যা × ২৪
একটি সংখ্যা = ৮ × ৪৮/২৪ = ১৬

১২,৩২১.

Two fifth of one fourth of three-seventh of a number is 15. What is the half of the number?

ক
ক) 57
খ
খ) 175
গ
গ) 157
ঘ
ঘ) 350

সঠিক উত্তর: খ

খ) 175

উত্তর

সঠিক উত্তর: খ

খ) 175

খ

ব্যাখ্যা

ধরি,
সংখ্যাটি x
প্রশ্নমতে,
(2x/5) এর (1/4) এর (3/7) = 15
6x/140 = 15
3x/70 = 15
x = 15(70/3)
x = 350

সংখ্যাটির অর্ধেক = 350/2 = 175

১২,৩২২.

If 1 - 4x ≤ 5, then-

ক
ক) x ≥ - 2
খ
খ) x ≥ - 3
গ
গ) x ≥ - 4
ঘ
ঘ) x ≥ - 1

সঠিক উত্তর: ঘ

ঘ) x ≥ - 1

উত্তর

সঠিক উত্তর: ঘ

ঘ) x ≥ - 1

ঘ

ব্যাখ্যা

Question: If 1 - 4x ≤ 5, then-

Solution:
দেয়া আছে,
1 - 4x ≤ 5
বা,1 - 4x - 1 ≤ 5 - 1
বা,- 4x ≤ 4
বা,- 4x/4 ≤ 4/4
বা,- x ≤ 1
বা,(- x ) ( - 1) ≥ 1(- 1)
x ≥ - 1

১২,৩২৩.

The product of two consecutive positive integers is 20. To find the integers, this can be represented in the form of quadratic equation as-

ক
ক) 2x² - x + 20= 0
খ
খ) x² + x – 20= 0
গ
গ) x² - x + 20= 0
ঘ
ঘ) x² - 2x + 20= 0

সঠিক উত্তর: খ

খ) x² + x – 20= 0

উত্তর

সঠিক উত্তর: খ

খ) x² + x – 20= 0

খ

ব্যাখ্যা

Let x and (x + 1) be the two consecutive integers.

According to the given,
x(x + 1) = 20
x²+ x = 20
x²+ x – 20= 0

১২,৩২৪.

A trader mixes 26 kg of fertilizer at Tk. 20 per kg with 30 kg of fertilizer of other variety at Tk. 36 per kg and sells the mixture at Tk. 30 per kg. His profit percent is:

ক
4%
খ
5%
গ
6%
ঘ
8%

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: A trader mixes 26 kg of fertilizer at Tk. 20 per kg with 30 kg of fertilizer of other variety at Tk. 36 per kg and sells the mixture at Tk. 30 per kg. His profit percent is:

Solution:
Cost price of 56 kg fertilizer = Tk. (26 × 20) + (30 × 36)
= Tk. (520 + 1080)
= Tk. 1600

∴ Selling price of 56 kg rice = Tk. (56 × 30)
= Tk.1680

Profit = Tk. (1680 - 1600) = Tk. 80
∴ Gain = {(80/1600) × 100}
= 5%

১২,৩২৫.

Two pipes A and B can fill a tank separately in 12 and 16 hours respectively. If both of them are opened together when the tank is initially empty, how much time will it take to completely fill the tank?

ক
7/48 hours
খ
48/7 hours
গ
13/2 hours
ঘ
6 hours

সঠিক উত্তর: খ

48/7 hours

উত্তর

সঠিক উত্তর: খ

48/7 hours

খ

ব্যাখ্যা

Question: Two pipes A and B can fill a tank separately in 12 and 16 hours respectively. If both of them are opened together when the tank is initially empty, how much time will it take to completely fill the tank?

Solution:
Part of tank filled by pipe A in one hour working alone = 1/12
Part of tank filled by pipe B in one hour working alone = 1/16

∴ Part of tank filled by pipe A and pipe B in one hour working together = (1/12) + (1/16) = 7/48
Therefore, time taken to completely fill the tank if both A and B work together = 48/7 hours

১২,৩২৬.

8 is 5% of what number?

ক
180
খ
140
গ
160
ঘ
148

সঠিক উত্তর: গ

160

উত্তর

সঠিক উত্তর: গ

160

গ

ব্যাখ্যা

Question: 8 is 5% of what number?

Solution:
ধরি, ৮, x এর ৫%

x এর ৫% = ৮
⇒ x × ৫/১০০ = ৮
⇒ x/২০ = ৮
⇒ x = ২০ × ৮
= ১৬০

১২,৩২৭.

Find the compound interest on Tk 8000 at 15% interest per annum for 2 years, compounded annually-

ক
2850 tk
খ
10580 tk
গ
2580 tk
ঘ
8580 tk

সঠিক উত্তর: গ

2580 tk

উত্তর

সঠিক উত্তর: গ

2580 tk

গ

ব্যাখ্যা

Question: Find the compound interest on Tk 8000 at 15% interest per annum for 2 years, compounded annually-

Solution:
Compound Principal for 2 years = 8000{1 + (15/100)²
= 8000(115/100)²
= (8000 × 115 × 115)/(100 × 100)
= 10580 taka

Compound interest = 10580 - 8000 tk
= 2580 tk.

১২,৩২৮.

A sum of Tk. 3300 is divided among A, B and C such that A gets 2/5 of what B gets and B gets 1/3 of what C gets. B’s share is:

ক
700 Tk.
খ
750 Tk.
গ
800 Tk.
ঘ
900 Tk.

সঠিক উত্তর: খ

750 Tk.

উত্তর

সঠিক উত্তর: খ

750 Tk.

খ

ব্যাখ্যা

Question: A sum of Tk. 3300 is divided among A, B and C such that A gets 2/5 of what B gets and B gets 1/3 of what C gets. B’s share is:

Solution:
Let,
C’s share = Tk. x
Then,
B’s share = Tk. x/3
A’s share = Tk. (2/5) × (x/3) = Tk. 2x/15

∴ 2x/15 + x/3 + x = 3300
⇒ (2x + 5x + 15x)/15 = 3300
⇒ 22x/15 = 3300
⇒ 22x = 3300 × 15
⇒ 22x = 49500
⇒ x = 49500/22
⇒ x = 2250

∴ B’s share = 2250/3 = 750 Tk.

১২,৩২৯.

The ratio of milk to water in a mixture is 5:3. When adding 4 liters of water, the ratio becomes 5:5.
What was the quantity of milk in the original mixture?

ক
10 liters
খ
15 liters
গ
20 liters
ঘ
25 liters

সঠিক উত্তর: ক

10 liters

উত্তর

সঠিক উত্তর: ক

10 liters

ক

ব্যাখ্যা

Question: The ratio of milk to water in a mixture is 5:3. When adding 4 liters of water, the ratio becomes 5:5. What was the quantity of milk in the original mixture?

Solution:
Let the initial quantity of
Milk = 5x liters
Water = 3x liters

When 4 liters of water are added, the new quantity of water becomes = (3x+4) liters
The new ratio becomes 5:5, which simplifies to 1:1. This means the amount of milk and water are now equal.
5x=3x+4
2x = 4
∴ x = 2

So, the initial quantity of Milk = 5 × 2 = 10 liters

১২,৩৩০.

A tradesman sold an article at a loss of 20%. If the selling price had been increased by 100 taka, there would have been a gain of 5%. What was the cost price of the article

ক
ক) 375 taka
খ
খ) 400 taka
গ
গ) 425 taka
ঘ
ঘ) 450 taka

সঠিক উত্তর: খ

খ) 400 taka

উত্তর

সঠিক উত্তর: খ

খ) 400 taka

খ

ব্যাখ্যা

Question: A tradesman sold an article at a loss of 20%. If the selling price had been increased by 100 taka, there would have been a gain of 5%. What was the cost price of the article.

সমাধান:
ধরি,
দ্রব্যটির ক্রয়মূল্য ক টাকা

২০% ক্ষতিতে,
ক্রয়মূল্য ১০০ টাকা হলে বিক্রয়মূল্য ৮০ টাকা
∴ ক্রয়মূল্য ক টাকা হলে বিক্রয়মূল্য (৮ক)/১০ টাকা

বিক্রয়মূল্য ১০০ টাকা বেশি হলে মোট বিক্রয়মূল্য = (৮ক)/১০ + ১০০ টাকা
= (৮ক + ১০০০)/১০ টাকা

৫% লাভে,
বিক্রয়মূল্য ১০৫ টাকা হলে ক্রয়মূল্য ১০০ টাকা
∴ বিক্রয়মূল্য (৮ক + ১০০০)/১০ টাকা হলে ক্রয়মূল্য {১০০(৮ক + ১০০০)}/(১০৫ × ১০) টাকা
= (৮০ক + ১০০০০)/১০৫ টাকা

শর্তমতে,
(৮০ক + ১০০০০)/১০৫ = ক
বা, ৮০ক + ১০০০০ = ১০৫ক
বা, ১০০০০ = ২৫ক
বা, ক = ১০০০০/২৫
∴ ক = ৪০০

∴ দ্রব্যটির ক্রয়মূল্য ৪০০ টাকা

১২,৩৩১.

If a + b + c = 6 and a² + b²+ c² = 40 then, a³ + b³ + c³ - 3abc = ?

ক
412
খ
232
গ
180
ঘ
252

সঠিক উত্তর: ঘ

252

উত্তর

সঠিক উত্তর: ঘ

252

ঘ

ব্যাখ্যা

Question: If a + b + c = 6 and a² + b² + c² = 40 then, a³ + b³ + c³ - 3abc = ?

Solution:
Given that,
a + b + c = 6
a² + b² + c² = 40

Now,
a + b + c = 6
⇒ (a + b + c)² = 6²
⇒ a² + b² + c² + 2ab + 2bc + 2ac = 36
⇒ 40 + 2(ab + bc + ca) = 36
⇒ 2(ab + bc + ca) = - 4
⇒ ab + bc + ca = - 2

Then,
a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ac)
= 6[40 - (-2)]
= 6[40 + 2]
= 6 × 42
= 252

∴ a³ + b³ + c³ - 3abc = 252

১২,৩৩২.

A ferry can carry 24 buses or 36 cars at a time. If there are 18 buses on the ferry, how many cars can be loaded onto it?

ক
12
খ
8
গ
6
ঘ
9

সঠিক উত্তর: ঘ

উত্তর

সঠিক উত্তর: ঘ

ঘ

ব্যাখ্যা

Question: A ferry can carry 24 buses or 36 cars at a time. If there are 18 buses on the ferry, how many cars can be loaded onto it?

Solution:
Here,
24 buses = 36 cars
∴ 1 bus = 36/24 cars
∴ 18 buses = (36 × 18)/24 cars
= 27 cars

∴ Required number of cars = 36 - 27 = 9 cars

১২,৩৩৩.

Which one is incorrect?

ক
ক) cosec²A = 1 + cot²A
খ
খ) sec²A = 1 + tan²A
গ
গ) tan²A = 1 - sec²A
ঘ
ঘ) cos²A = 1 - sin²A

সঠিক উত্তর: গ

গ) tan²A = 1 - sec²A

উত্তর

সঠিক উত্তর: গ

গ) tan²A = 1 - sec²A

গ

ব্যাখ্যা

Question: Which one is incorrect?

Solution:
আমরা জানি,
sin²A + cos²A = 1
⇒ sin²A = 1 - cos²A
⇒ cos²A = 1 - sin²A

আবার,
sec²A - tan²A = 1
⇒ sec²A = 1 + tan²A
⇒ tan²A = sec²A - 1

এবং
cosec²A - cot²A = 1
⇒ cosec²A = 1 + cot²A
⇒ cosec²A - 1 = cot²A

১২,৩৩৪.

Working 4 hours a day, A can Complete a work in 10 days and working 5 hours a day, B can complete the same work in 12 days. Working 8 hours a day, they can jointly complete the work in:

ক
ক) 2 days
খ
খ) 3 days
গ
গ) 4 days
ঘ
ঘ) 5 days

সঠিক উত্তর: খ

খ) 3 days

উত্তর

সঠিক উত্তর: খ

খ) 3 days

খ

ব্যাখ্যা

Question : Working 4 hours a day, A can Complete a work in 10 days and working 5 hours a day, B can complete the same work in 12 days. Working 8 hours a day, they can jointly complete the work in:

Solution:
Working 4 hours a day, A can complete the work in 10 days i.e.
= 4 × 10 = 40 hours
Working 5 hours a day, B can complete the work in 12 days i.e.
= 5 × 12 = 60 hours

(A + B)'s 1 hour's work,
(1/40) + (1/60)
= (6 + 4)/240
= 10/240
= 1/24

Hence, A and B can complete the work in 24 hours
they require 3 days to complete the work.

১২,৩৩৫.

A tree of height 4 meter casts a show of length 6.5 meter. What is the height of a house casting a shadow 26 meter long?

ক
ক) 14 meter
খ
খ) 17 meter
গ
গ) 15 meter
ঘ
ঘ) 16 meter

সঠিক উত্তর: ঘ

ঘ) 16 meter

উত্তর

সঠিক উত্তর: ঘ

ঘ) 16 meter

ঘ

ব্যাখ্যা

If shadows length is 6.5 meters then tree's height is 4 meters
So, if shadows length is 26 meters, then house's height = 4×26 / 6.5 = 16 meters

১২,৩৩৬.

In a survey, 30% of the people surveyed owned a personal computer and 75% owned a cellular telephone. If 25% owned both a cellular telephone and a personal computer, then the percentage of the people who does not have either of the instrument?

ক
ক) 40%
খ
খ) 30%
গ
গ) 25%
ঘ
ঘ) 20%

সঠিক উত্তর: ঘ

ঘ) 20%

উত্তর

সঠিক উত্তর: ঘ

ঘ) 20%

ঘ

ব্যাখ্যা

Question: In a survey, 30% of the people surveyed owned a personal computer and 75% owned a cellular telephone. If 25% owned both a cellular telephone and a personal computer, then the percentage of the people who does not have either of the instrument?

Solution:
people owned a personal computer n(p) = 30%
people owned a cellular computer n(c) = 75%
people owned both personal and cellular computer, n(p ∩ c) = 25%
people owned any one or both of them = n (p ∪ c)

n (p ∪ c) = n(p) + n(c) - n(p ∩ c)
= 30% + 75% - 25%
= 80%

∴ the percentage of the people who does not have either of the instrument = 100% - 80%
= 20%

১২,৩৩৭.

What is the banker’s discount, if the true discount on a bill of Tk. 840 is Tk.105?

ক
ক) 100
খ
খ) 110
গ
গ) 120
ঘ
ঘ) 130

সঠিক উত্তর: গ

গ) 120

উত্তর

সঠিক উত্তর: গ

গ) 120

গ

ব্যাখ্যা

Present Worth (P.W.) = 840 - 105
= Tk. 735
Therefore, S.I. on Tk. 735 = Tk. 105
S.I. on Tk. 8400
= 105 × 840/735
= Tk. 120

১২,৩৩৮.

If sinθ + cosθ = √3, then what is tanθ + cotθ equal to?

ক
1
খ
0
গ
2/√3
ঘ
1/√3

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: If sinθ + cosθ = √3, then what is tanθ + cotθ equal to?

Solution:
sinθ + cosθ = √3
⇒ (sinθ + cosθ)² = (√3)²
⇒ sin²θ+ 2sinθcosθ + cos²θ = 3

We know,
sin²θ + cos²θ = 1

∴ 1 + 2sinθcosθ = 3
⇒ 2sinθcosθ = 2
∴ sinθcosθ = 1

Now,
tanθ + cotθ
= sinθ/cosθ + cosθ/sinθ
= (sin²θ + cos²θ)/(sinθcosθ)
= 1/1
= 1

১২,৩৩৯.

A student scored 30% marks and failed by 15 marks. Another student scored 50% marks and secured 25 marks more than the pass marks. What is the pass percentage?

ক
33%
খ
37.5%
গ
40%
ঘ
45%

সঠিক উত্তর: খ

37.5%

উত্তর

সঠিক উত্তর: খ

37.5%

খ

ব্যাখ্যা

Question: A student scored 30% marks and failed by 15 marks. Another student scored 50% marks and secured 25 marks more than the pass marks. What is the pass percentage?

Solution:
Let total marks = x

According to the question,
30% of x +15 = 50% of x - 25
⇒ 0.30x + 15 = 0.50x - 25
⇒ 0.50x - 0.30x = 25 + 15
⇒ 0.20x = 40
⇒ x = 40/0.20
∴ x = 200

∴ Pass marks = 30% of x + 15
= 0.30 × 200 + 15
= 60 + 15
=75

∴ Pass percentage = (75/200) × 100
= 37.5%

১২,৩৪০.

Find the greatest number that will divide 43, 91, and 183 and leave the same remainder. What is the square root of this number?

ক
2
খ
3
গ
6
ঘ
7
ঙ
None of the above

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: Find the greatest number that will divide 43, 91, and 183 and leave the same remainder. What is the square root of this number?

Solution:
the number is the H.C.F of (91 - 43), (183 - 91) and (183 - 43)
= H.C.F of 48, 92 and 140
= 4

The square root of 4 is √4 = 2

১২,৩৪১.

If Mario was 32 years old 8 years ago, how old was he x years ago?

ক
x - 40
খ
x - 24
গ
40 - x
ঘ
24 - x
ঙ
24 + x

সঠিক উত্তর: গ

40 - x

উত্তর

সঠিক উত্তর: গ

40 - x

গ

ব্যাখ্যা

Question: If Mario was 32 years old 8 years ago, how old was he x years ago?

Solution:
If Mario's current age is M,
then Mario's age 8 years ago = M - 8;
Here,
M - 8 = 32;
therefore, M = 40 (Mario's current age)

∴ His age x years ago means M - x or 40 - x;

১২,৩৪২.

A room 6.2m × 8m is to be carpeted leaving a margin of 10 cm from each wall. If the cost of the carpet is Tk. 15 per sq. meter, the cost of carpeting the room will be:

ক
Tk. 650
খ
Tk. 702
গ
Tk. 799
ঘ
Tk. 852

সঠিক উত্তর: খ

Tk. 702

উত্তর

সঠিক উত্তর: খ

Tk. 702

খ

ব্যাখ্যা

Question: A room 6.2m × 8m is to be carpeted leaving a margin of 10 cm from each wall. If the cost of the carpet is Tk. 15 per sq. meter, the cost of carpeting the room will be:

Solution:
Area of the carpet :
= [(6.20 - 0.20) × (8 - 0.20)] m²
= (6 × 7.8) m²
= 46.8 m²

∴ Cost of carpeting :
= Tk. (46.8 × 15)
= Tk. 702

১২,৩৪৩.

The average age of a group of persons going for tour is 16 years. Twenty new persons with an average age of 15 years join the group on the spot due to which their average age become 15.5 years. The number of persons initially going for tour is -

ক
40
খ
20
গ
10
ঘ
5

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: The average age of a group of persons going for tour is 16 years. Twenty new persons with an average age of 15 years join the group on the spot due to which their average age become 15.5 years. The number of persons initially going for tour is -

Solution:
The number of persons initially going for tour is x
initial total age = 16x

new age added = 20 × 15 = 300 years

(16x + 300)/(x + 20) = 15.5
⇒ 16x + 300 = 15.5 (x + 20)
⇒ 16x + 300 = 15.5x + 310
⇒ 0.5x = 10
∴ x = 10/0.5 = 20

১২,৩৪৪.

(1331)^{- (2/3)} = ?

ক
ক) - 1/11
খ
খ) - 11/121
গ
গ) 1/121
ঘ
ঘ) 121/11

সঠিক উত্তর: গ

গ) 1/121

উত্তর

সঠিক উত্তর: গ

গ) 1/121

গ

ব্যাখ্যা

Question: (1331)^{- (2/3)} = ?

Solution:
(1331)^{- (2/3)}
= (11³)^-(2/3)
= 11^{- 2}= 1/11²
= 1/121

১২,৩৪৫.

If a + 1/a = 2, Then a³ + 1/a³ =?

ক
ক) 1/2
খ
খ) 2
গ
গ) 3/2
ঘ
ঘ) 7

সঠিক উত্তর: খ

খ) 2

উত্তর

সঠিক উত্তর: খ

খ) 2

খ

ব্যাখ্যা

Question: If a + 1/a = 2, Then a³ + 1/a³ =?

Solution:
Given that,
a + 1/a = 2

∴ a³ + 1/a³ = (a + 1/a)³ - 3. a. (1/a)(a + 1/a)
= 2³ - 3 × 2
= 8 - 6
= 2

১২,৩৪৬.

A right triangle has sides in the ratio of 5:12:13. What is the measure of the smallest angle in the triangle, in degrees?

ক
ক) 13.34
খ
খ) 22.62
গ
গ) 34.14
ঘ
ঘ) 42.71

সঠিক উত্তর: খ

খ) 22.62

উত্তর

সঠিক উত্তর: খ

খ) 22.62

খ

ব্যাখ্যা

We know that, sinθ = AB/AC
⇒ sinθ = 5/13
⇒ θ = sin^-1(5/13)
∴ θ = 22.62°

১২,৩৪৭.

How many positive integers less than 100 are multiples of both 2 and 3?

ক
12
খ
14
গ
16
ঘ
18

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: How many positive integers less than 100 are multiples of both 2 and 3?

Solution:
A number that is a multiple of both 2 and 3 is a multiple of their LCM:
LCM(2, 3) = 6

Find how many multiples of 6 are less than 100:
Multiples of 6: 6, 12, 18, …

Largest multiple less than 100: 96
Number of terms in this sequence (arithmetic progression with first term a₁=6, common difference d = 6, last term a_n= 96):
n = (a_n- a₁)/d + 1
= (96 - 6)/6+1
= 90/6 + 1
= 15 + 1
= 16

১২,৩৪৮.

ক
- 0.10
খ
- 0.12
গ
- 0.18
ঘ
- 0.15
ঙ
- 0.8

সঠিক উত্তর: খ

- 0.12

উত্তর

সঠিক উত্তর: খ

- 0.12

খ

ব্যাখ্যা

Question:

Solution:

১২,৩৪৯.

(√25 + √25)² = ?

ক
ক) 50
খ
খ) 20
গ
গ) 100
ঘ
ঘ) 125
ঙ
ঙ) 130

সঠিক উত্তর: গ

গ) 100

উত্তর

সঠিক উত্তর: গ

গ) 100

গ

ব্যাখ্যা

Question: (√25 + √25)² = ?

Solution:
(√25 + √25)²
=(2√25)²
= 4 × 25
= 100

১২,৩৫০.

Two cards are drawn together from a pack of 52 cards. The probability that one is a diamond and one is a heart, is?

ক
7/102
খ
9/52
গ
13/102
ঘ
15/52

সঠিক উত্তর: গ

13/102

উত্তর

সঠিক উত্তর: গ

13/102

গ

ব্যাখ্যা

Question: Two cards are drawn together from a pack of 52 cards. The probability that one is a diamond and one is a heart, is?

Solution:
Two cards can be drawn together from a standard deck in = ⁵²C₂ = 1326 ways.

One diamond can be drawn in = ¹³C₁ ways
and a heart too can be drawn in = ¹³C₁ ways.

Therefore, the number of ways that a diamond and a heart can be drawn = ¹³C₁ × ¹³C₁
= 13 × 13
= 169

Therefore, the required probability = 169/1326
= 13/102

১২,৩৫১.

A jar is filled with liquid, 2 parts of which are water and 4 parts milk. How much of the mixture must be drawn off and replaced with water so that the mixture may be half water and half milk?

ক
1/2
খ
1/4
গ
1/8
ঘ
None of the above

সঠিক উত্তর: খ

1/4

উত্তর

সঠিক উত্তর: খ

1/4

খ

ব্যাখ্যা

Question: A jar is filled with liquid, 2 parts of which are water and 4 parts milk. How much of the mixture must be drawn off and replaced with water so that the mixture may be half water and half milk?

Solution:
মনে করি,
পাত্রটিতে তরল পদার্থ আছে = 6 ইউনিট
এবং এই তরলের x ইউনিট পানি দ্বারা প্রতিস্থাপন করা হল।

∴ নতুন মিশ্রণে পানির পরিমাণ = (2 - (2x/6) + x)
= 2 - (x/3) + x
= (6 - x + 3x)/3
= (2x + 6)/3 ইউনিট

এবং, নতুন মিশ্রণে দুধের পরিমাণ = 4 - (4x/6)
= 4 - (2x/3)
= (12 - 2x)/3 ইউনিট

প্রশ্নমতে,
(2x + 6)/3 = (12 - 2x)/3
⇒ 2x + 6 = 12 - 2x
⇒ 2x + 2x = 12 - 6
⇒ 4x = 6
⇒ x = 3/2

∴ মিশ্রণের প্রতিস্থাপিত অংশের পরিমাণ = (3/2) × (1/6)
= 1/4

১২,৩৫২.

A man walking at the speed of 6 kmph crosses a square field diagonally in 4 minutes. What is the area of the field?

ক
ক) 40000 Square Meters
খ
খ) 50000 Square Meters
গ
গ) 60000 Square Meters
ঘ
ঘ) 80000 Square Meters

সঠিক উত্তর: ঘ

ঘ) 80000 Square Meters

উত্তর

সঠিক উত্তর: ঘ

ঘ) 80000 Square Meters

ঘ

ব্যাখ্যা

Question: A man walking at the speed of 6 kmph crosses a square field diagonally in 4 minutes. What is the area of the field?

Solution:
In 60 minutes he goes 6000 m
In 4 minutes he goes (6000 × 4)/60 m
= 400 m

Now,
√2 × length = 400
⇒ length = 400/√2
⇒ (length)² = (400/√2)²
⇒ Area = 160000/2
⇒ Area = 80000 Square Meters

১২,৩৫৩.

The difference between a two-digit number and the number obtained by interchanging the positions of its digits is 36. What is the difference between the two digits of that number?

ক
2
খ
3
গ
4
ঘ
5

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: The difference between a two-digit number and the number obtained by interchanging the positions of its digits is 36. What is the difference between the two digits of that number?

Solution:
ধরি, একক স্থানীয় অঙ্ক x, দশক স্থানীয় অঙ্ক y

প্রশ্নমতে,
10y + x - (10x + y) = 36
⇒ 10y + x - 10x - y = 36
⇒ 9y - 9x = 36
⇒ 9(y - x) = 36
⇒ y - x = 4

১২,৩৫৪.

If Titu walks at 14 km/hr instead of 10 km/hr, he can cover 20 km more. The actual distance travelled by him is -

ক
80 km
খ
70 km
গ
56 km
ঘ
50 km

সঠিক উত্তর: ঘ

50 km

উত্তর

সঠিক উত্তর: ঘ

50 km

ঘ

ব্যাখ্যা

Question: If Titu walks at 14 km/hr instead of 10 km/hr, he can cover 20 km more. The actual distance travelled by him is -

Solution:
টিটুর বেগ বৃদ্ধি পায় = ১৪ - ১০ কি.মি./ঘণ্টা
= ৪ কি.মি./ঘণ্টা

সে মোট হাটে = ২০/৪ ঘণ্টা = ৫ ঘণ্টা

প্রকৃত বেগে ১ ঘণ্টায় যায় ১০ কি.মি.
∴ প্রকৃত বেগে ৫ ঘণ্টায় যায় (১০ × ৫) কি.মি.
= ৫০ কি.মি.

প্রকৃতপক্ষে দূরত্ব হলো ৫০ কি.মি.

১২,৩৫৫.

A computer takes 50 nanoseconds to do an addition. How many additions can it do in 1 second?

ক
30 million
খ
25 million
গ
40 million
ঘ
5 million
ঙ
20 million

সঠিক উত্তর: ঙ

20 million

উত্তর

সঠিক উত্তর: ঙ

20 million

ঙ

ব্যাখ্যা

In 50 nano seconds, it can do an addition.

We know, 1 nanosecond = 10^-9 seconds

In 1 second, it can do = 1/ (50 * 10^-9) = 20000000 = 20 million

১২,৩৫৬.

If a, b and c are the lengths of the three sides of a triangle, then which of the following is true?

ক
ক) a + b < c
খ
খ) a - b < c
গ
গ) a + b = ৫
ঘ
ঘ) a + b ≥ c

সঠিক উত্তর: খ

খ) a - b < c

উত্তর

সঠিক উত্তর: খ

খ) a - b < c

খ

ব্যাখ্যা

আমরা জানি,
ত্রিভুজের যে কোন দুই বাহুর সমষ্টি তার তার তৃতীয় বাহু অপেক্ষা বৃহত্তর।
আবার, ত্রিভুজের যে কোন দুই বাহুর অন্তর বা ব্যবধান তৃতীয় বাহু অপেক্ষা ক্ষুদ্রতর।
২য় অনুসিদ্ধান্ত অনুসারে,
অপশন b তে প্রদত্ত a - b < c সঠিক।
অর্থাৎ দুটি বাহুর অন্তর তৃতীয় বাহু থেকে ছােট।

১২,৩৫৭.

A, B, and C enter into a partnership. A contributes one-third of the capital while C contributes as much as A and B together contribute. If the profit at the end of the year amounts to Tk.19800, what would B receive?

ক
9400 Tk.
খ
3300 Tk.
গ
5800 Tk.
ঘ
6100 Tk.

সঠিক উত্তর: খ

3300 Tk.

উত্তর

সঠিক উত্তর: খ

3300 Tk.

খ

ব্যাখ্যা

Question: A, B, and C enter into a partnership. A contributes one-third of the capital while C contributes as much as A and B together contribute. If the profit at the end of the year amounts to Tk.19800, what would B receive?

Solution:
As C contributes as much as A and B,
∴ C's share is = 1/2
A's share = 1/3
B's share = ( 1 - 1/3 - 1/2) = 1/6

∴ B will receive = 1/6(19800) = 3300 Tk.

১২,৩৫৮.

A single discount equivalent to discount series 20%, 20% and 10% is = ?

ক
ক) 50%
খ
খ) 48.4%
গ
গ) 42.4%
ঘ
ঘ) 40.4%

সঠিক উত্তর: গ

গ) 42.4%

উত্তর

সঠিক উত্তর: গ

গ) 42.4%

গ

ব্যাখ্যা

Single equivalent discount for 20% and 20%
= 20+20−20 × 20/100
= 40−4 = 36%
Single equivalent discount for 36% and 10%
= 36+10−36 × 10/100
= 46−3.6 =42.4%

১২,৩৫৯.

The ratio of orange and water in a juice mixture is 2 : 3, then the percentage of water in mixture is-

ক
35%
খ
20%
গ
30%
ঘ
60%

সঠিক উত্তর: ঘ

60%

উত্তর

সঠিক উত্তর: ঘ

60%

ঘ

ব্যাখ্যা

Question: The ratio of orange and water in a mixture is 2 : 3, then the percentage of water in mixture is-

Solution:
The ratio of orange and water in a mixture is 2 : 3
Total = 5

∴ percentage of water = (3/5) × 100%
= 60%

১২,৩৬০.

If cosB + cos²B = 1, then sin²B + sin⁴B =?

ক
0
খ
1
গ
2
ঘ
3

সঠিক উত্তর: খ

উত্তর

সঠিক উত্তর: খ

খ

ব্যাখ্যা

Question: If cosB + cos²B = 1, then sin²B + sin⁴B =?

Solution:
cosB + cos²B = 1
⇒ cosB = 1 - cos²B
⇒ cosB = sin²B

sin²B + sin⁴B
= sin²B + (sin²B)²
= sin²B + cos²B
= 1

১২,৩৬১.

A company pays rent of Tk. 1500 per month for office space to its owner. But if the company pays the annual rate at the beginning of the year the owner gives a discount of 10% on the total annual rent. What is the annual amount the company pays to the owner after the discount?

ক
ক) Tk. 15500
খ
খ) Tk. 16200
গ
গ) Tk. 17300
ঘ
ঘ) Tk. 18000

সঠিক উত্তর: খ

খ) Tk. 16200

উত্তর

সঠিক উত্তর: খ

খ) Tk. 16200

খ

ব্যাখ্যা

Question: A company pays rent of Tk. 1500 per month for office space to its owner. But if the company pays the annual rate at the beginning of the year the owner gives a discount of 10% on the total annual rent. What is the annual amount the company pays to the owner after the discount?

Solution:
Total annual rent = Tk. (1500 × 12) = Tk. 18000
Discount = 10% of Tk. 18000
= Tk. (10 × 18000)/100 = Tk. 1800
Annual rent paid after discount = Tk. (18000 - 1800) = Tk. 16200

১২,৩৬২.

Solve |5x + 5| - 8 ≤ 17

ক
- 5 ≤ x ≤ 5
খ
- 5 ≤ x ≤ 4
গ
0 ≤ x ≤ 4
ঘ
- 6 ≤ x ≤ 4
ঙ
None of these

সঠিক উত্তর: ঘ

- 6 ≤ x ≤ 4

উত্তর

সঠিক উত্তর: ঘ

- 6 ≤ x ≤ 4

ঘ

ব্যাখ্যা

Question: Solve |5x + 5| - 8 ≤ 17

Solution:
We have
|5x + 5| - 8 ≤ 17
⇒ |5x + 5| ≤ 25
⇒ - 25 < 5x + 5 ≤ 25
⇒ - 30 < 5x ≤ 20
⇒ - 6 ≤ x ≤ 4

১২,৩৬৩.

A dishonest shopkeeper, at the time of selling and purchasing, weighs 10% less and 20% more per kilogram respectively. Find the percentage profit earned by treachery. (Assuming he sells at Cost Price)

ক
30%
খ
31%
গ
32%
ঘ
33%
ঙ
33.33%

সঠিক উত্তর: ঙ

33.33%

উত্তর

সঠিক উত্তর: ঙ

33.33%

ঙ

ব্যাখ্যা

During purchase he got = 100 + 20 = 120
During sell he gave = 100 - 10 = 90
Profit % = (30/90) × 100
= 33.33%

১২,৩৬৪.

In a race, the speeds of A and B are in the ratio of 5 : 6. A takes 20 minutes more than B takes to reach the destination. What is the time taken by A to reach the destination in hours?

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

সঠিক উত্তর: খ

2 hours

উত্তর

সঠিক উত্তর: খ

2 hours

খ

ব্যাখ্যা

Question: In a race, the speeds of A and B are in the ratio of 5 : 6. A takes 20 minutes more than B takes to reach the destination. What is the time taken by A to reach the destination in hours?

Solution:
Ratio of speeds = 5/6
⇒ Ratio of times taken = 6/5

Let,
A and B take 6x and 5x hours to reach the destination.

ATQ,
6x - 5x = 20/60
∴ x = 1/3

∴ Time taken by A = 6x = (6 × 1/3) hours
= 2 hours.

১২,৩৬৫.

The total time taken by a motorboat to go 40 km downstream and return upstream is 4.5 hours. If the boat's speed in still water is 20 km/h, find the speed of the stream.

ক
6.67 km/hours
খ
5.67 km/hours
গ
4.99 km/hours
ঘ
8.22 km/hours
ঙ
None of the above

সঠিক উত্তর: ক

6.67 km/hours

উত্তর

সঠিক উত্তর: ক

6.67 km/hours

ক

ব্যাখ্যা

Question: The total time taken by a motorboat to go 40 km downstream and return upstream is 4.5 hours. If the boat's speed in still water is 20 km/h, find the speed of the stream.
(একটি মোটরবোটকে ৪০ কিলোমিটার প্রবাহের দিকে যেতে এবং ফেরত আসতে ৪.৫ ঘণ্টা সময় লাগে। যদি নৌকার গতির পরিমাণ স্থির পানিতে ২০ কিলোমিটার প্রতি ঘণ্টা হয়, তাহলে স্রোতের বেগ কত হবে?)

Solution:
ধরা যাক, স্রোতের বেগ = x km/h
তাহলে,
স্রোতের অনুকূলে বেগ = (20 + x) km/h
স্রোতের প্রতিকূলে বেগ = (20 - x) km/h

প্রশ্নমতে
40/(20 + x) + 40/(20 - x) = 4.5
⇒ 1600/(400 - x²) = 9/2
⇒ 9(400 - x²) = 2 × 1600
⇒ 3600 - 3200 = 9x²
⇒ (3x)² = 20²
⇒ 3x = 20
⇒ x = 20/3
∴ x = 6.67

∴ স্রোতের বেগ হল 6.67 km/h

১২,৩৬৬.

The average age of a group of 20 employees is 30 years. If 10 more employees join the group, the average age increases by 3 years. Find the average age of the new employees?

ক
35 years
খ
36 years
গ
39 years
ঘ
42 years

সঠিক উত্তর: গ

39 years

উত্তর

সঠিক উত্তর: গ

39 years

গ

ব্যাখ্যা

Question: The average age of a group of 20 employees is 30 years. If 10 more employees join the group, the average age increases by 3 years. Find the average age of the new employees?

Solution:
Here,
Total age of the 20 employees,
= 20 × 30=600 years

After joining 10 new employees, average age increases by 3 years.
So, New average age = 30 + 3 = 33 years

Total age of the 30 employees,
= 30 × 33 = 990 years

∴ Total age of the 10 new employees = (990 - 600) years
= 390 years

∴ Average age of the 10 new employees = 390/10 = 39 years

১২,৩৬৭.

A clock shows 12 : 00. Through how many degrees has the hour hand moved by 25 minutes past 5?

ক
150°
খ
155°
গ
160.5°
ঘ
165°
ঙ
162.5°

সঠিক উত্তর: ঙ

162.5°

উত্তর

সঠিক উত্তর: ঙ

162.5°

ঙ

ব্যাখ্যা

Question: A clock shows 12 : 00. Through how many degrees has the hour hand moved by 25 minutes past 5?

Solution:
Full circle = 360°

Clock has 12 hours; Hour hand moves per hour, 360°/12 = 30° per hour
And, 1 hour = 60 minutes
Hour hand moves per minute = 30°/60 = 0.5°

Now,
From 12 : 00 to 5 : 25 = 5 hours 25 minutes = (5 × 60) + 25 = 325 minutes

∴ Degrees moved by hour hand = 325 × 0.5° = 162.5°

By 25 minutes past 5, the hour hand has moved through 162.5° from the 12 : 00 position.

১২,৩৬৮.

A jar contains black and white marbles. If there are ten marbles in the jar, then which of the following could not be the ratio of black to white marbles?

ক
1 : 4
খ
1 : 10
গ
7 : 3
ঘ
9 : 1

সঠিক উত্তর: খ

1 : 10

উত্তর

সঠিক উত্তর: খ

1 : 10

খ

ব্যাখ্যা

Question: A jar contains black and white marbles. If there are ten marbles in the jar, then which of the following could not be the ratio of black to white marbles?

Solution:
Since the number of black and white marbles are whole numbers,
So the sum of the terms of the ratio must be a factor of 10.
ক) 1:4 = 1+4 = 5 a factor of 10.
খ) 1:10 = 1+10 = 11 not a factor of 10.
গ) 7:3 = 7+3 =10 a factor of 10.
ঘ) 9:1 = 9+1 = 10 a fac tor of 10.

Here we can see that the sum of only the second option is not a factor of 10

১২,৩৬৯.

The least number of complete years in which a sum of money put out at 21% compound interest will be more than doubled is:

ক
3 years
খ
4 years
গ
5 years
ঘ
6 years

সঠিক উত্তর: খ

4 years

উত্তর

সঠিক উত্তর: খ

4 years

খ

ব্যাখ্যা

Question: The least number of complete years in which a sum of money put out at 21% compound interest will be more than doubled is:

Solution:
ধরি, আসল (Principal) = P
বার্ষিক সুদের হার (Rate of Interest), r = 21%
সময় (Time in years) = t
শর্ত অনুযায়ী, চক্রবৃদ্ধি মূলধন আসলের দ্বিগুণের বেশি হবে।
অর্থাৎ, A > 2P

আমরা জানি, চক্রবৃদ্ধি সুদের সূত্র হলো,
A = P(1 + r/100)ⁿ
প্রশ্নমতে,
P(1 + 21/100)ⁿ > 2P
⇒ (1 + 0.21)ⁿ > 2
⇒ (1.21)ⁿ > 2
এখন, আমরা n-এর বিভিন্ন পূর্ণসংখ্যা মান বসিয়ে পরীক্ষা করে দেখি:
যদি n = 1 হয়, (1.21)¹ = 1.21 (যা 2-এর থেকে ছোট)
যদি n = 2 হয়, (1.21)² = 1.4641 (যা 2-এর থেকে ছোট)
যদি n = 3 হয়, (1.21)³ = 1.772 (যা 2-এর থেকে ছোট)
যদি n = 4 হয়, (1.21)⁴ = 2.144 (যা 2-এর থেকে বড়)

∴ সর্বনিম্ন 4 বছরে মূলধন দ্বিগুণের চেয়ে বেশি হবে।

১২,৩৭০.

A shopkeeper suffers a loss of 20% upon selling a shirt for Tk. 4000. If he wants to make 15% profit after giving an 8% discount on the marked price, what is the marked of the shirt in Tk.?

ক
6000
খ
6250
গ
5000
ঘ
5750
ঙ
None

সঠিক উত্তর: খ

6250

উত্তর

সঠিক উত্তর: খ

6250

খ

ব্যাখ্যা

Question: A shopkeeper suffers a loss of 20% upon selling a shirt for Tk. 4000. If he wants to make 15% profit after giving an 8% discount on the marked price, what is the marked of the shirt in Tk.?

Solution:
At 20% loss,
Selling price Tk. 80 when cost price Tk. 100
Selling price Tk. 1 when cost price Tk. 100/80
Selling price Tk. 4000 when cost price Tk. (100 × 4000)/80
= Tk. 5000

At 15% profit,
cost price Tk. 100 when selling price Tk. 115
cost price Tk. 1 when selling price Tk. 115/100
cost price Tk. 5000 when selling price Tk. (115 × 5000)/100
= Tk. 5750

At 8% discount,
Selling price Tk. 92 when marked price Tk. 100
Selling price Tk. 1 when marked price Tk. 100/92
Selling price Tk. 5750 when marked price Tk. (100 × 5750)/92
= Tk. 6250

১২,৩৭১.

On February 5, 1998, it was Thursday. The day of the week on February 5, 1997, was-

ক
Wednesday
খ
Monday
গ
Friday
ঘ
Sunday

সঠিক উত্তর: ক

Wednesday

উত্তর

সঠিক উত্তর: ক

Wednesday

ক

ব্যাখ্যা

Question: On February 5, 1998, it was Thursday. The day of the week on February 5, 1997, was-

Solution:
1997 was an ordinary year, it had 1 odd day. So, the day on February 5, 1998, would be one day beyond the day on February 5, 1997.

∴ Thursday on February 5, 1998, would be one day beyond the day on February 5, 1997, so the day on February 5, 1997, was Wednesday.

১২,৩৭২.

In a race of 200 m, A can beat B by 31 m and C by 18 m. In a race of 350 m, C will beat B by -

ক
15m
খ
17m
গ
22m
ঘ
25m

সঠিক উত্তর: ঘ

25m

উত্তর

সঠিক উত্তর: ঘ

25m

ঘ

ব্যাখ্যা

Question: In a race of 200 m, A can beat B by 31 m and C by 18 m. In a race of 350 m, C will beat B by -

Solution:
A : B = 200 : 169
A : C = 200 : 182

B/C = (B/A) × (A/C)
= 169/182

So, in a 350 race B will pass = (169/182) × 350
= 325m

Hence, C will beat B by (350 - 325) or, 25 metres

১২,৩৭৩.

In how many ways can a committee of 4 men and 3 women be formed from 6 men and 5 women?

ক
110
খ
90
গ
150
ঘ
124

সঠিক উত্তর: গ

150

উত্তর

সঠিক উত্তর: গ

150

গ

ব্যাখ্যা

Question: In how many ways can a committee of 4 men and 3 women be formed from 6 men and 5 women?

Solution:
We have 6 men and 5 women.
We need to choose 4 men from 6 and 3 women from 5.

∴ Number of ways = ⁶C₄ × ⁵C₃
= {6!/4!(6 - 4)!} × {5!/3!(5 - 3)!}
= {(6 × 5)/2} × {(5 × 4)/2}
= 15 × 10
= 150 ways

১২,৩৭৪.

The product of two numbers is 192 and the sum of these two numbers is 28. What is the smaller these two numbers ?

ক
18
খ
14
গ
12
ঘ
21

সঠিক উত্তর: গ

উত্তর

সঠিক উত্তর: গ

গ

ব্যাখ্যা

Question: The product of two numbers is 192 and the sum of these two numbers is 28. What is the smaller these two numbers ?

Solution:
Let the numbers be x and (28 - x)
Then,
x(28 - x) = 192
⇒ 28x - x2 = 192
⇒ x² - 28x + 192 = 0
⇒ x² - 16x - 12x + 192 = 0
⇒ x(x - 16) - 12(x - 16) = 0
⇒ (x - 16)(x - 12) = 0
Now,
x - 16 = 0
∴ x = 16
or
x - 12 = 0
∴ x = 12

So the numbers are 16 and 12. The smaller is 12.

১২,৩৭৫.

The average speed of a car is one-third the speed of an express train. The train covers 1800 km in 15 hours. How much distance will the car cover in 36 minutes?

ক
14 km
খ
18 km
গ
20 km
ঘ
24 km

সঠিক উত্তর: ঘ

24 km

উত্তর

সঠিক উত্তর: ঘ

24 km

ঘ

ব্যাখ্যা

Question: The average speed of a car is one-third the speed of an express train. The train covers 1800 km in 15 hours. How much distance will the car cover in 36 minutes?

Solution:
ট্রেনের গতিবেগ = অতিক্রান্ত দূরত্ব/সময়
= 1800 কিমি/15 ঘন্টা
= 120 কিমি/ঘন্টা

এখন, গাড়ির গতিবেগ ট্রেনের গতিবেগের এক-তৃতীয়াংশ।

∴ গাড়ির গতিবেগ = ট্রেনের গতিবেগ × 1/3
= 120 কিমি/ঘন্টা × 1/3
= 40 কিমি/ঘন্টা

∴ গাড়ির অতিক্রান্ত দূরত্ব = গাড়ির গতিবেগ × সময়
= 40 কিমি/ঘন্টা × 36 মিনিট
= 40 কিমি/ঘন্টা × (36/60) ঘন্টা
= 40 কিমি/ঘন্টা × 0.6 ঘন্টা
= 24 কিমি

সুতরাং, গাড়িটি 36 মিনিটে 24 কিমি দূরত্ব অতিক্রম করবে।

১২,৩৭৬.

By investing Tk. 3450 in a 4.5% stock , a man obtains an income of Tk. 150. Find the market price of the stock.

ক
ক) Tk. 98.5
খ
খ) Tk. 100.5
গ
গ) Tk. 101.5
ঘ
ঘ) Tk. 103.5

সঠিক উত্তর: ঘ

ঘ) Tk. 103.5

উত্তর

সঠিক উত্তর: ঘ

ঘ) Tk. 103.5

ঘ

ব্যাখ্যা

To earn Tk. 150 investment = Tk. 3450
To earn Tk. 4.5 investment = Tk. (3450 × 4.5)/150
= Tk. 103.5

১২,৩৭৭.

For three years, Tk 40,000 is invested at a 10% yearly interest rate.
What is the difference between the Compound Interest (CI) and the Simple Interest (SI)?

ক
Tk. 1,000
খ
Tk. 1,200
গ
Tk. 1,240
ঘ
Tk. 1,380

সঠিক উত্তর: গ

Tk. 1,240

উত্তর

সঠিক উত্তর: গ

Tk. 1,240

গ

ব্যাখ্যা

Question: For three years, Tk 40,000 is invested at a 10% yearly interest rate. What is the difference between the Compound Interest (CI) and the Simple Interest (SI)?

Solution:
Given:
Principal (P) = Tk. 40,000
Rate of Interest (r) = 10% per annum
Time (n) = 3 years

Simple Interest, I = Pnr
= 40000 × 3 × (10/100)
= 12000

Compound Principal = P × (1 + r)ⁿ
= 40000 × (1 + (10/100))³
= 40,000×(1.1)³
= 53240

Compound Interest, C = 53240−40000 = 13240

So, Difference = 13240 - 12000 = 1240 Taka

১২,৩৭৮.

Which of the following numbers cannot be the last digit of a squared number?

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

বর্গ সংখ্যার শেষের অংক 0, 1, 4, 5, 6 এবং 9 হতে পারে।
2 বর্গ সংখ্যার শেষের অংক হতে পারে না।

1 এর বর্গ = 1² = 1
2 এর বর্গ = 2² = 4
3 এর বর্গ = 3² = 9
4 এর বর্গ = 4² = 16
5 এর বর্গ = 5² = 25
6 এর বর্গ = 6² = 36
10 এর বর্গ= 10² = 100

১২,৩৭৯.

Sumi can type 75 pages in 25 minutes. Maria can type 5 pages in 15 minutes. Working together, how many pages can they type in 30 minutes?

ক
75 pages
খ
80 pages
গ
90 pages
ঘ
100 pages
ঙ
105 pages

সঠিক উত্তর: ঘ

100 pages

উত্তর

সঠিক উত্তর: ঘ

100 pages

ঘ

ব্যাখ্যা

Question: Sumi can type 75 pages in 25 minutes. Maria can type 5 pages in 15 minutes. Working together, how many pages can they type in 30 minutes?

Solution:
Sumi can type in 1 min = 75/25 = 3 pages
Maria can type in 1 min = 5/15 = 1/3 page

∴ Working together they can type in 1 min = (3 + 1/3) pages
= 10/3 pages

∴ They can type in 30 min = (10 × 30)/3 pages
= 100 pages

১২,৩৮০.

If the rate of interest is 10% per annum and is compounded half yearly, the principal of Tk. 5000 in 3/2 years will amount to -

ক
ক) Tk. 5630.025
খ
খ) Tk. 5750.50
গ
গ) Tk. 5788.125
ঘ
ঘ) Tk. 5835.05

সঠিক উত্তর: গ

গ) Tk. 5788.125

উত্তর

সঠিক উত্তর: গ

গ) Tk. 5788.125

গ

ব্যাখ্যা

Given, P = 5000
Interest rate per year = 10%, so per half year = 5%
Time = 3/2 years or 3 half year

C = 5000 × 105/100 × 105/100 × 105/100
= 5788.125

১২,৩৮১.

3x + (1/2x) = 5, then the value of 8x³ + (1/27x³) is?

ক
820/27
খ
620
গ
750/20
ঘ
None of these

সঠিক উত্তর: ক

820/27

উত্তর

সঠিক উত্তর: ক

820/27

ক

ব্যাখ্যা

Question: 3x + (1/2x) = 5, then the value of 8x³ + (1/27x³) is?

Solution:

১২,৩৮২.

If 0 ≤ x ≤ 5. What is the maximum value of cos(x/3)?

ক
ক) 1
খ
খ) 0
গ
গ) 1/2
ঘ
ঘ) √3/2

সঠিক উত্তর: ক

ক) 1

উত্তর

সঠিক উত্তর: ক

ক) 1

ক

ব্যাখ্যা

x= 0 হলে
cos(x/3) = cos(0/3)
= cos0
= 1
অন্য যেকোনো মানের জন্য cos এর মান 1 অপেক্ষা ছোট হবে।

১২,৩৮৩.

The largest 4 digit number exactly divisible by 88 is-

ক
9944
খ
9768
গ
9988
ঘ
8888

সঠিক উত্তর: ক

9944

উত্তর

সঠিক উত্তর: ক

9944

ক

ব্যাখ্যা

Question: The largest 4 digit number exactly divisible by 88 is-

Solution:
88 ) 9999 ( 113
88
_______
119
88
_______
319
264
_______
55

∴ Required number = (9999 - 55) = 9944

১২,৩৮৪.

An empty swimming pool with a capacity of 5,760 gallons is filled at the rate of 12 gallons per minute. How many hours does it take to fill the pool to capacity?

ক
8
খ
20
গ
96
ঘ
480
ঙ
720

সঠিক উত্তর: ক

উত্তর

সঠিক উত্তর: ক

ক

ব্যাখ্যা

Question: An empty swimming pool with a capacity of 5,760 gallons is filled at the rate of 12 gallons per minute. How many hours does it take to fill the pool to capacity?

Solution:
The pool to its capacity of 5,760 gallons.
The "rate of work" is 12 gallons per minute.

Fill in 1 min 12 gallon
Fill in 60 min (12 × 60) = 720 gallon

720 gallon is filled in 1 hour
5,760 gallons filled in (1 × 5760)/720 hours
= 8 hours

১২,৩৮৫.

The single discount which is equivalent to discount of 10%, 20% and 25% is -

ক
34%
খ
46%
গ
28%
ঘ
32%

সঠিক উত্তর: খ

46%

উত্তর

সঠিক উত্তর: খ

46%

খ

ব্যাখ্যা

Question: The single discount which is equivalent to discount of 10%, 20% and 25% is -

Solution:
Let marked price be 100%.
⇒ S.P = (90/100) × (80/100) × (75/100) × 100
⇒ S.P = 54%

Discount will be 100 – 54 = 46%

∴ The single discount will be 46%

১২,৩৮৬.

The speed of a boat in still water in 17 km/hr and the rate of current is 3 km/hr. The distance travelled downstream in 12 minutes is:

ক
ক) 3 km
খ
খ) 3.6 km
গ
গ) 4 km
ঘ
ঘ) 4.2 km

সঠিক উত্তর: গ

গ) 4 km

উত্তর

সঠিক উত্তর: গ

গ) 4 km

গ

ব্যাখ্যা

Question: The speed of a boat in still water in 17 km/hr and the rate of current is 3 km/hr. The distance travelled downstream in 12 minutes is:

Solution:
Speed downstream = (17 + 3) kmph = 20 kmph.
Distance travelled = 20 × (12/60) km = 4 km

১২,৩৮৭.

If 10 men or 15 boys can make 260 mats in 20 days, then how many mats will be made by 8 men and 12 boys in 20 days?

ক
562 mats
খ
416 mats
গ
386 mats
ঘ
360 mats

সঠিক উত্তর: খ

416 mats

উত্তর

সঠিক উত্তর: খ

416 mats

খ

ব্যাখ্যা

Question: If 10 men or 15 boys can make 260 mats in 20 days, then how many mats will be made by 8 men and 12 boys in 20 days?

Solution:
Here
10 men = 15 boys
∴ 1 men = 15/10 boys
∴ 8 men = (15 × 8)/10 boys
= 12 boys

∴ 8 men and 12 boys = (12 + 12) = 24 boys

15 boys can make in 20 days 260 mats
∴ 1 boys can make in 20 days 260/15 mats
∴ 24 boys can make 260 mats in (260 × 24)/15 mats
= 416 mats

১২,৩৮৮.

Idrak buys 80 kg sugar at taka 40 per kg. and 40 kg rice at taka 50 per kg. At what price approximately in taka per kg should he sell to make profit 10% of cost?

ক
ক) 42.3
খ
খ) 44.3
গ
গ) 47.7
ঘ
ঘ) 49.2

সঠিক উত্তর: গ

গ) 47.7

উত্তর

সঠিক উত্তর: গ

গ) 47.7

গ

ব্যাখ্যা

80 কেজি চিনির ক্রয়মূল্য = 80 × 40 = 3200 টাকা
40 কেজি চালের ক্রয়মূল্য = 40 ×50 = 2000 টাকা

চিনি ও চালের মোট পরিমাণ = 80 + 40 = 120 কেজি

১০% লাভে
মোট বিক্রয়মূল্য = 5200 + 5200 এর 10%
= (5200 + 520)
= 5720 টাকা

প্রতিকেজির বিক্রয়মূল্য = 5720/120 = 47.7 টাকা

১২,৩৮৯.

In a camp, there is a meal for 240 men or 400 children. If 300 children have taken the meal, how many men will be catered to with remaining meal?

ক
40 men
খ
30 men
গ
80 men
ঘ
60 men

সঠিক উত্তর: ঘ

60 men

উত্তর

সঠিক উত্তর: ঘ

60 men

ঘ

ব্যাখ্যা

Question: In a camp, there is a meal for 240 men or 400 children. If 300 children have taken the meal, how many men will be catered to with remaining meal?

Solution:
Camp has = 400 children
Already have taken meal = 300 children

Remaining children to take meal = 400 - 300 = 100 children

the camp has meal for 400 children = 240 men
the camp has meal for 1 children = 240/400 men
the camp has meal for 100 children = (240 × 100)/400 men
= 60 men

১২,৩৯০.

A can do a piece of work in 24 days, B in 32 days and C in 64 days. All begin to do it together, but A leaves after 6 days and B leaves 6 days before the completion of the work. How many days did the work last?

ক
ক) 15
খ
খ) 12
গ
গ) 17
ঘ
ঘ) 20

সঠিক উত্তর: ঘ

ঘ) 20

উত্তর

সঠিক উত্তর: ঘ

ঘ) 20

ঘ

ব্যাখ্যা

L.C.M. of 24, 32 and 64 is 192. Hence:
A’ s efficiency = 8
B’s efficiency = 6
C’s efficiency = 3.
In 6 days they produce 17 × 6 = 102 units. (1)
Also given that C works alone for 6 days. Number of units produced by C in those 6 days = 6 × 3 = 18 units (2)
Total (1) + (2) = 120.
Balance units 72 done by Band C :Their 1 days production: 9 units.
Hence they worked for 8 days for producing the 72 units.(3)
Total days of work: 6 + 8 + 6= 20 days.

১২,৩৯১.

If p and n are integers such that p > n >0 and p² - n² = 12, which of the following values of (p - n)?

ক
ক) -1
খ
খ) 2
গ
গ) 8
ঘ
ঘ) 18
ঙ
ঙ) None of the above

সঠিক উত্তর: খ

খ) 2

উত্তর

সঠিক উত্তর: খ

খ) 2

খ

ব্যাখ্যা

Since, p² - n² = 12 and p > n > 0
Best way to solve this question is putting assuming value,
let p = 4 and n = 2
So, 4² - 2² = 12, and the assumption values are correct. So, 4-2 is equal to 2

১২,৩৯২.

If two times A is equal to three times of B and also equal to four times of C, then A : B : C is -

ক
2 : 3 : 4
খ
4 : 3 : 2
গ
3 : 4 : 6
ঘ
6 : 4 : 3

সঠিক উত্তর: ঘ

6 : 4 : 3

উত্তর

সঠিক উত্তর: ঘ

6 : 4 : 3

ঘ

ব্যাখ্যা

Question: If two times A is equal to three times of B and also equal to four times of C, then A : B : C is -

Solution:
2A = 3B
Or, B = 2A/3
and 2A = 4C
Or, C = A/2

Hence, A : B : C = A : 2A/3 : A/2
= 1 : 2/3 : 1/2
= 6 : 4 : 3

১২,৩৯৩.

Each of the following questions has a group. Find out which one of the given alternatives will be another member of the group or of that class.
Apple, Grape, Orange

ক
ক) Vegetable
খ
খ) Fruits
গ
গ) Stems
ঘ
ঘ) Oats

সঠিক উত্তর: খ

খ) Fruits

উত্তর

সঠিক উত্তর: খ

খ) Fruits

খ

ব্যাখ্যা

Apple, Grape and Orange all these are fruits.

১২,৩৯৪.

The simple interest on a certain sum of money at 4% per annum for 3 years is 1,200 Taka. The compound interest on the same sum for the same period and at the same rate is -

ক
1,200 Taka
খ
1,225 Taka
গ
1,231 Taka
ঘ
1,249 Taka

সঠিক উত্তর: ঘ

1,249 Taka

উত্তর

সঠিক উত্তর: ঘ

1,249 Taka

ঘ

ব্যাখ্যা

Question: The simple interest on a certain sum of money at 4% per annum for 3 years is 1,200 Taka. The compound interest on the same sum for the same period and at the same rate is -

Solution:
Given, Simple Interest, I = 1,200 Taka
r = 4% = 0.04
n = 3 years

We know,
I = Pnr
P = I/nr
P = 1200/(3×0.04)
P = 10,000 Taka

For Compound Interest,

So, Compound Interest = 11248.64 - 10000
= 1248.64 Taka
= 1249 Taka

১২,৩৯৫.

If the sum of two numbers is 22 and their difference is 8. Find the product of these two numbers?

ক
110
খ
105
গ
115
ঘ
137

সঠিক উত্তর: খ

105

উত্তর

সঠিক উত্তর: খ

105

খ

ব্যাখ্যা

Question: If the sum of two numbers is 22 and their difference is 8. Find the product of these two numbers?

Solution:
Let, two numbers are, a and b
a + b = 22
a - b = 8

We know,
ab = {(a + b)/2}² - {(a - b)/2}²
= {22/2}² - {8/2}²
= (11)² - (4)²
= 121 - 16
= 105
∴ ab = 105

So the product of these two numbers is 105

১২,৩৯৬.

The average monthly income of P and Q is Tk. 5000. The average monthly income of Q and R is Tk. 6050 and the average monthly income of P and R is Tk. 5400. Calculate the monthly income of Q.

ক
TK. 6450
খ
TK. 5650
গ
TK. 4350
ঘ
TK. 3550

সঠিক উত্তর: খ

TK. 5650

উত্তর

সঠিক উত্তর: খ

TK. 5650

খ

ব্যাখ্যা

Question: The average monthly income of P and Q is Tk. 5000. The average monthly income of Q and R is Tk. 6050 and the average monthly income of P and R is Tk. 5400. Calculate the monthly income of Q.

Solution:
Let P, Q and R represent their respective monthly incomes. Then, we have:

P + Q = (5000 x 2) = 10000 .... (i)
Q + R = (6050 x 2) = 12100 .... (ii)
P + R = (5400 x 2) = 10800 .... (iii)

Adding (i), (ii) and (iii), we get:
2(P + Q + R) = 32900
∴ P + Q + R = 16450 .... (iv)

Subtracting (iii) from (iv),
We get Q = 5650

∴ Q's monthly income = TK. 5650

১২,৩৯৭.

P is 6 times greater than Q then by what percent is Q smaller than P?

ক
87%
খ
79.67%
গ
83.33%
ঘ
71%

সঠিক উত্তর: গ

83.33%

উত্তর

সঠিক উত্তর: গ

83.33%

গ

ব্যাখ্যা

Question: P is 6 times greater than Q then by what percent is Q smaller than P?

Solution:
Let Q = 10.
Then, P = 60.
Q is 50 less than P.
Q, % less than P = (50/60) × 100
= 83.33%

১২,৩৯৮.

If ⁿC₁₄ = ⁿC₇, what is the value of ⁿC₂?

ক
135
খ
180
গ
210
ঘ
360

সঠিক উত্তর: গ

210

উত্তর

সঠিক উত্তর: গ

210

গ

ব্যাখ্যা

Question: If ⁿC₁₄ = ⁿC₇, what is the value of ⁿC₂?

Solution:
আমরা জানি,
যদি ⁿC_a= ⁿC_b হয়, তবে হয় a = b অথবা a + b = n হবে।

এখানে,
ⁿC₁₄ = ⁿC₇
⇒ 14 + 7 = n
⇒ n = 21

∴ ⁿC₂ = ²¹C₂= 21!/(2! × (21 - 2)!)
= 21!/(2! × 19!)
= (21 × 20 × 19!)/(2 × 1 × 19!)
= (21 × 20)/2
= 21 × 10
= 210

১২,৩৯৯.

A train of length 200 meters crosses a man running at 10 km/hr in the same direction in 10 seconds. What is the speed of the train?

ক
72 km/hr
খ
62 km/hr
গ
80 km/hr
ঘ
85 km/hr
ঙ
82 km/hr

সঠিক উত্তর: ঙ

82 km/hr

উত্তর

সঠিক উত্তর: ঙ

82 km/hr

ঙ

ব্যাখ্যা

Question: A train of length 200 meters crosses a man running at 10 km/hr in the same direction in 10 seconds. What is the speed of the train?

Solution:
When the train and man are moving in same direction then relative speed will be the difference between their individual speeds. In this problem the other way to find the relative speed is to divide the distance covered (length of train) by the time taken by the train to cross the man.

Relative Speed = 200/10

We will convert it into Km/hr
(200/10) × (18/5) = 72 km/hr

Now, let the speed of the train is X km/hr.
So, the relative speed, 72 km/hr = X km/hr - 10 km/hr
⇒ X - 10 = 72
⇒ X = 72 + 10
∴ X = 82 km/hr

১২,৪০০.

After getting two successive discounts, a shirt with a list price of 150 taka available at 105 taka. If the second discount is 12.5%, find the first discount.

ক
ক) 30%
খ
খ) 25%
গ
গ) 20%
ঘ
ঘ) 35%

সঠিক উত্তর: গ

গ) 20%

উত্তর

সঠিক উত্তর: গ

গ) 20%

গ

ব্যাখ্যা

Question: After getting two successive discounts, a shirt with a list price of 150 taka available at 105 taka. If the second discount is 12.5%, find the first discount.

সমাধান:
Let,
The first discount be x%

Then, 87.5% of (100 - x)% of 150 = 105
⇒ 87.5/100 × (100 - x)/100 × 150 =105
⇒ 100 - x = (105 × 100 × 100)/(87.5 × 150)
⇒ 100 - x = 80
⇒ x = 20

∴ The first discount be 20%