Bank Math

PrepBank · পাতা ২৮ / ১৬১ · ২,৭০১–২,৮০০ / ১৬,১২৪

২,৭০১.

A school has 10 classes with the same number of students in each class. One day, the weather was bad and many students were absent. 3 classes were half full. 5 classes were 3/4 full and 2 classes were 1/6 empty. A total of 74 students were absent. What is the total number of students in this school?

ক
ক) 200
খ
খ) 220
গ
গ) 240
ঘ
ঘ) 400
ঙ
ঙ) None of these

ব্যাখ্যা

ATQ, 3x/2 + 5x/4 + 2x/6 = 74
⇒ (18x + 15x + 4x)/12 = 74
⇒ 37x = 888
⇒x = 24
∴ স্কুলের ছাত্রছাত্রী সংখ্যা = 24×10 = 240

২,৭০২.

A trapezium ABCD has side AD parallel to BC, ∠BAD = 90°, BC = 3 cm and AD = 8 cm. If the perimeter of the trapezium is 36 cm, then its area is

ক
44 sq. cm.
খ
50 sq. cm.
গ
66 sq. cm.
ঘ
72 sq. cm.

ব্যাখ্যা

Question: A trapezium ABCD has side AD parallel to BC, ∠BAD = 90°, BC = 3 cm and AD = 8 cm. If the perimeter of the trapezium is 36 cm, then its area is

Solution:

AB + CD = 36 - 8 - 3 = 25
CD = 25 - AB = 25 - x

CD² = x² + 5²
⇒ (25 - x)² = x² + 25
⇒ 625 - 50x + x² = x² + 25
⇒ 50x = 625 - 25 - 600
⇒ x = 12

Area = (1/2) (8 + 3) 12
= 66 sq. cm.

২,৭০৩.

The difference between a number and its square is 72. What is the number?

ক
ক) 19
খ
খ) 18
গ
গ) 30
ঘ
ঘ) 9

ব্যাখ্যা

ধরি,
সংখ্যাটি x
প্রশ্নমতে,
x² - x = 72
x²- x - 72 = 0
x² - 9x + 8x - 72 = 0
x(x - 9) + 8(x - 9) = 0
(x - 9)(x + 8) = 0

হয়
x - 9 = 0
x = 9

অথবা
x + 8 = 0
x = - 8 [গ্রহণযোগ্য নয় ]

২,৭০৪.

A salesman usually makes 45% profit on every TV he sells. During a sale, he reduces his margin of profit to 40%, while his sales increase by 10%, What is the ratio of his new profit to his usual profit?

ক
9 : 8
খ
9 : 10
গ
44 : 45
ঘ
10 : 11
ঙ
.

ব্যাখ্যা

Question: A salesman usually makes 45% profit on every TV he sells. During a sale, he reduces his margin of profit to 40%, while his sales increase by 10%, What is the ratio of his new profit to his usual profit?

Solution:
প্রথমে টিভি বিক্রয় করেন = x টি

45% লাভে মোট লাভ = 45x/100
10% বিক্রি বৃদ্ধিতে নতুন বিক্রির পরিমাণ = x + x এর 10%
= x + 10x/100
= 1.1x

নতুন লাভ = 1.1x × 40/100
= 44x/100

নতুন লাভ : পুরাতন লাভ = 44x/100 : 45x/100
= 44 : 45

২,৭০৫.

Mannan and Hannan rent a pasture for 12 months. Mannan puts in 200 cows for 8 months. How many cows can Hannan put in the pasture for the remaining 4 months if he pays 1.5 as much as Mannan?

ক
450 cows
খ
600 cows
গ
300 cows
ঘ
520 cows

ব্যাখ্যা

Question: Mannan and Hannan rent a pasture for 12 months. Mannan puts in 200 cows for 8 months. How many cows can Hannan put in the pasture for the remaining 4 months if he pays 1.5 as much as Mannan?

Solution:
Let Mannan's cow (C₁) = 200 cows, and time spend T₁ = 8 months
Hannan's cow (C₂) = y cows, and time spend T₂ = 4 months
Let the profit of Mannan = x
Then the profit of Hannan = 1.5x = (3/2)x = (3x/2)

(C₁ × T₁)/ (C₂ × T₂) = p₁/p₂
⇒ (200 × 8)/ (y × 4) = x/ (3x/2)
⇒ 400/y = 2x/3x
⇒ 400/y = 2/3
⇒ y = (400 × 3)/2
∴ y = 600

Hence, Hannan can put 600 cows for the remaining 4 months.

২,৭০৬.

If log_x(125/8) = - 3, what is the value of x?

ক
3/5
খ
2/5
গ
5/3
ঘ
3/8

ব্যাখ্যা

Question: If log_x(125/8) = - 3, what is the value of x?

Solution:
log_x(125/8) = - 3
⇒ x^{- 3} = 125/8 [log_b(a) = c ⇒ b^c = a]
⇒ x^{- 3} = 5³/2³
⇒ x^{- 3} = (5/2)³
⇒ x^{- 3} = (2/5)^{- 3}
∴ x = 2/5

২,৭০৭.

A and B can do a piece of work in 45 and 40 days respectively. They began the work together but A leaves after some days and B finished the remaning work in 23 days. After how many days did A leave?

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

ব্যাখ্যা

Question: A and B can do a piece of work in 45 and 40 days respectively. They began the work together but A leaves after some days and B finished the remaning work in 23 days. After how many days did A leave?

Solution:
Work done by B in 23 days {(1/40) × 23} = 23/40 part

Remaining work = (1 - 23/40) part
= 17/40 part

(A + B)'s 1 day's work = 1/45 + 1/40 = 17/360 part

Now,
17/360 part has done by (A + B) in 1 day
∴ 1 part has done by (A + B) in 360/17 day
∴ 17/40 part has done by (A + B) in {(360/17)(17/40)} days
= 9 days

২,৭০৮.

The ratio between the length and breadth of a rectangular park is 3 : 2. If a man cycling along the boundary of the park at the speed of 12 km/hr completes one round in 8 minutes, then the area of the park.

ক
ক) 153600 m²
খ
খ) 150035 m²
গ
গ) 133636 m²
ঘ
ঘ) 148600 m²

ব্যাখ্যা

Perimeter = Distance covered in 8 min.
Perimeter = (12000/60)×8 m
Perimeter = 1600 m

Let length = 3x metres and
breadth = 2x metres.
Then,
2(3x + 2x) = 1600
5x = 800
x = 160
Therefore Length = 480 m and Breadth = 320 m

Therefore Area = (480 x 320) m²
= 153600 m²

২,৭০৯.

Difference of numerator and denominator of a proper fraction is 1; if 2 is subtracted from numerator and 2 is added to denominator of the fraction, it will be equal to 1/6. Find the fraction.

ক
1/4
খ
3/4
গ
3/5
ঘ
3/7

ব্যাখ্যা

Solution: Difference of numerator and denominator of a proper fraction is 1; if 2 is subtracted from numerator and 2 is added to denominator of the fraction, it will be equal to 1/6. Find the fraction.

Solution:
ধরি,
ভগ্নাংশটির লব = x
ভগ্নাংশটির হর = x + 1
∴ ভগ্নাংশটি = x/x + 1

প্রশ্নমতে,
(x - 2)/(x + 1 + 2) = 1/6
(x - 2)/(x + 3) = 1/6
6x - 12 = x + 3
6x - x = 12 +3
5x = 15
x = 3

∴ ভগ্নাংশটি = 3/4

২,৭১০.

a, b, c, d and e are five consecutive integers in increasing order of size. Which of the following is always even?

ক
ac + e
খ
ac + d
গ
a + b + c
ঘ
ab + c
ঙ
None of the above

ব্যাখ্যা

Question: a, b, c, d and e are five consecutive integers in increasing order of size. Which of the following is always even?

Solution:
ধরি
a = 1, b = 2, c = 3, d = 4, and e = 5,

অপশন (ক) ac + e = 1 × 3 + 5 = 3 + 5 = 8

অপশন (খ) ac + d = 1 × 3 + 4 = 3 + 4 = 7

অপশন (গ) a + b + c = 1 + 2 + 3 = 6

অপশন (ঘ) ab + c = 1 × 2 + 3 = 2 + 3 = 5
............................
...........................................
আবার
ধরি
a = 2, b = 3, c = 4, d = 5, and e = 6,

অপশন (ক) ac + e = 2 × 4 + 6 = 8 + 6 = 14

অপশন (খ) ac + d = 2 × 4 + 5 = 8 + 5 = 13

অপশন (গ) a + b + c = 2 + 3 + 4 = 9

অপশন (ঘ) ab + c = 2 × 3 + 4 = 6 + 4 = 10

উভয় ক্ষেত্রে অপশন (ক) জোড় সংখ্যা। তাই সঠিক উত্তর হিসেবে অপশন (ক) নেওয়া হয়েছে।

২,৭১১.

A right triangle with sides 3 cm, 4 cm and 5 cm is rotated the side of 3 cm to form a cone. The volume of the cone so formed is-

ক
12π cm³
খ
15π cm³
গ
16π cm³
ঘ
20π cm³

ব্যাখ্যা

Question: A right triangle with sides 3 cm, 4 cm and 5 cm is rotated the side of 3 cm to form a cone. The volume of the cone so formed is-

Solution:

Clearly, we have r = 3 cm and h = 4 cm.

∴ Volume = (1/3)πr²h = (1/3) × π × 3² × 4 cm³= 12 cm³.

২,৭১২.

Find the ratio of the areas of the incircle and the circumcircle of a square.

ক
3 : 2
খ
1 : 2
গ
5 : 2
ঘ
1 : 1

ব্যাখ্যা

Question: Find the ratio of the areas of the incircle and circumcircle of a square.

Solution:
ধরি, বর্গক্ষেত্রের বাহুর দৈর্ঘ্য r

বর্গের অন্তর্বৃত্তের ব্যাস বর্গের বাহুর সমান।
অন্তর্বৃত্তের ক্ষেত্রফল = π(r/2)²
= πr²/4

বর্গের বহিঃবৃত্তের ব্যাস বর্গের কর্ণের সমান।
বহিঃবৃত্তের ক্ষেত্রফল = π(√2 r/2)²
= πr²/2

অনুপাত = πr²/4 : πr²/2
= (1/4) : (1/2)
= 1 : 2

২,৭১৩.

A solution containing 10% sugar by weight is made sweeter by doubling the amount of sugar. The percent of sugar, by weight, in the new solution is-

ক
13.12%
খ
18.18%
গ
19.19%
ঘ
28.13%

ব্যাখ্যা

Question: A solution containing 10% sugar by weight is made sweeter by doubling the amount of sugar. The percent of sugar, by weight, in the new solution is-

Solution:
let, solution is 100 unit
amount of sugar = 100 × 10%
= 100 × 10/100
= 10 unit

by doubling, amount of sweet = 20 unit
solutin = 100 + 10 = 110 unit

percent of sugar = 20 × 100%/110
= 18.18%

২,৭১৪.

If set A = {5, 15, 20, 30} and B = {5, 15, 20} which of the following represents (A ∩ B) ?

ক
ক) {5, 15, 20, 30}
খ
খ) {20}
গ
গ) {5, 15, 20}
ঘ
ঘ) {5, 20, 30}

ব্যাখ্যা

Question: If set A = {5, 15, 20, 30} and B = {5, 15, 20} which of the following respresents (A ∩ B) ?

Solution:
দেওয়া আছে,
A = {5, 15, 20, 30}
B = {5, 15, 20}
এখন,
A ∩ B = {5, 15, 20, 30} ∩ {5, 15, 20}
= {5, 15, 20}

২,৭১৫.

Which of the following is a prime number?

ক
ক) 187
খ
খ) 149
গ
গ) 171
ঘ
ঘ) 153

ব্যাখ্যা

প্রশ্ন : Which of the following is a prime number?

সমাধান :

১০০ থেকে ২০০ পর্যন্ত মৌলিক সংখ্যা :

101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199

২,৭১৬.

The price of commodity X increases by 40 paisa every year, while the price of commodity Y increases by 15 paisa every year. If in 2001, the price of commodity X was Tk 4.20 and that of Y was Tk. 6.30, in which year commodity X will cost 40 paisa more than the commodity Y?

ক
2010
খ
2013
গ
2012
ঘ
2011
ঙ
.

ব্যাখ্যা

Question: The price of commodity X increases by 40 paisa every year, while the price of commodity Y increases by 15 paisa every year. If in 2001, the price of commodity X was Tk 4.20 and that of Y was Tk. 6.30, in which year commodity X will cost 40 paisa more than the commodity Y?

Solution:
Suppose commodity X will cost 40 paise more than Y after P years.

Then,
(4.20 + 0.40P) - (6.30 + 0.15P) = 0.40
⇒ 4.20 + 0.40P - 6.30 - 0.15P = 0.40
⇒ 0.25P - 2.10 = .40
⇒ 0.25P = 0.40 + 2.10
⇒ 0.25P = 2.5
⇒ P = 2.5/0.25
∴ P = 10

X will cost 40 paise more than Y 10 years after 2001 which is 2011.

২,৭১৭.

Which of the following is the least number which will leave the remainder 5, when divided by 8, 12, 16, and 20?

ক
235
খ
245
গ
255
ঘ
265
ঙ
None of the above

ব্যাখ্যা

Question: Which of the following is the least number which will leave the remainder 5, when divided by 8, 12, 16, and 20?

Solution:
First we need to find the least number, so we have to find out the LCM of 8, 12, 16, and 20.
8 = 2 x 2 x 2
12 = 2 x 2 x 3
16 = 2 x 2 x 2 x 2
20 = 2 x 2 x 5

LCM = 2 x 2 x 2 x 2 x 3 x 5 = 240

240 is the least number that is exactly divisible by 8, 12, 16, and 20.

So, the required number that will leave remainder 5 is -
240 + 5 = 245

২,৭১৮.

The average of 30 numbers is 45. When 5 more numbers are included, the average of 35 numbers becomes 50. Find the average of the 5 new numbers.

ক
105
খ
80
গ
70
ঘ
65.5
ঙ
95

ব্যাখ্যা

Question: The average of 30 numbers is 45. When 5 more numbers are included, the average of 35 numbers becomes 50. Find the average of the 5 new numbers.

Solution:
Total of 50 numbers = 30 × 45 = 1350
Now,
total of 35 numbers = 35 × 50 = 1750
Hence, sum of 5 numbers = 1750 - 1350 = 400

∴ Average of five numbers = 400/5
= 80

২,৭১৯.

Two trains running in opposite directions cross a man standing on the platform in 27 seconds and 12 seconds respectively and they cross each other in 24 seconds. The ratio of their speeds is:

ক
3 : 1
খ
4 : 1
গ
3 : 2
ঘ
4 : 3

ব্যাখ্যা

Question: Two trains running in opposite directions cross a man standing on the platform in 27 seconds and 12 seconds respectively and they cross each other in 24 seconds. The ratio of their speeds is:

Solution:
Let,
the speeds of the two trains be x m/sec and y m/sec respectively.

Then,
length of the first train = 27x metres,
and length of the second train = 12y metres.

ATQ,
(27x + 12y)/(x + y) = 24
⇒ 27x + 12y = 24x + 24y
⇒ 27x - 24x = 24y - 12y
⇒ 3x = 12y
⇒ x/y = 12/3
∴ x : y = 4 : 1

২,৭২০.

Ali, Liton, and Karim pooled their funds to buy a gift for a friend. Ali contributed Tk. 200 less than 1/3 of the cost of the gift and Liton contributed Tk. 200 more than 1/4 of the cost. If Karim contributed the remaining Tk. 1500, what was the cost of the gift?

ক
Tk. 4300
খ
Tk. 3600
গ
Tk. 3300
ঘ
Tk. 2400

ব্যাখ্যা

Question: Ali, Liton, and Karim pooled their funds to buy a gift for a friend. Ali contributed Tk. 200 less than 1/3 of the cost of the gift and Liton contributed Tk. 200 more than 1/4 of the cost. If Karim contributed the remaining Tk. 1500, what was the cost of the gift?

Solution:
Let,
The cost of the gift be Tk. x
ALi contributed = x/3 - 200
Liton contributed = x/4 + 200
Karim contributed = 1500
all together equal to x.
x/3 + x/4 + 1500 = x
⇒ 7x/12 +1500 = x
⇒ 5x/12 = 1500
⇒ x = (12 × 1500)/5
⇒ x = 12 × 300
∴ x = 3600

২,৭২১.

The compound interest on 300 tk. at 7% per annum is 21 taka. The period is:

ক
1 year
খ
2 years
গ
3 years
ঘ
4 years

ব্যাখ্যা

Question: The compound interest on 300 tk. at 7% per annum is 21 taka. The period is:

Solution:
চক্রবৃদ্ধি সুদাসল = P (1 + r)ⁿ
= 300 + 21
= 321 tk.

প্রশ্নমতে,
321 = 300 × (1 + 7/100)ⁿ
⇒ (1 + 7/100)ⁿ = 321/300
⇒ (107/100)ⁿ = 107/100
⇒ (107/100)ⁿ = (107/100)¹
∴ n = 1 year

২,৭২২.

If each child is given 7 pencils, there are 2 pencils left over. But if each is given 8 pencils, then 5 more pencils are needed. How many pencils are there in total?

ক
44
খ
51
গ
58
ঘ
65

ব্যাখ্যা

Question: If each child is given 7 pencils, there are 2 pencils left over. But if each is given 8 pencils, then 5 more pencils are needed. How many pencils are there in total?

Solution:
Let,
the number of children be x.

ATQ,
7x + 2 = 8x - 5
⇒ 8x - 7x = 5 + 2
∴ x = 7

So, number of pencils = (7 × 7) + 2 = 51

২,৭২৩.

The ratio 5 : 4 expressed as a percent equals-

ক
12.5%
খ
100%
গ
80%
ঘ
125%

ব্যাখ্যা

Question: The ratio 5 : 4 expressed as a percent equals-

Solution:
Given,
The ratio = 5 : 4
= (5/4) × 100%
= 125%

২,৭২৪.

The total surface area of a cube is 96 cm². The volume of the cube is:

ক
ক) 8 cm³
খ
খ) 512 cm³
গ
গ) 64 cm³
ঘ
ঘ) 27 cm³
ঙ
ঙ) 32 cm³

ব্যাখ্যা

We know that the Total Surface Area of the cone = 6a².
6a² = 96 cm²
a² = 96/6 = 16
a = 4 cm

The volume of cone = a³ cubic units
V = 4³ = 64cm³.

২,৭২৫.

If log_a2 = a and log_a5 = b, then log_a50 =

ক
ক) a + b
খ
খ) a + b²
গ
গ) ab²
ঘ
ঘ) a + 2b

ব্যাখ্যা

Question: If log_a2 = a and log_a5 = b, then log_a50 =

Solution:
log_a2 = a
log_a5 = b

log_a50 = log_a(2 × 5²)
= log_a2 + log_a5²
= log_a2 + 2log_a5
= a + 2b

২,৭২৬.

Sarah finishes 60% of a painting in 12 days. She then asks David to help and together they complete the remaining work in 4 days. How many days would David take to complete the whole painting alone?

ক
20 days
খ
18 days
গ
28 days
ঘ
25 days

ব্যাখ্যা

Question: Sarah finishes 60% of a painting in 12 days. She then asks David to help and together they complete the remaining work in 4 days. How many days would David take to complete the whole painting alone?

Solution:
Sarah does 60% or 3/5 work in 12 days
in one day, she does 1/20 work
in 4 days, she does 1/5 work
work remaining = (2/5) - (1/5)
= 1/5 work
David does 1/5 parts in 4 days
he will complete the work in 20 days

২,৭২৭.

If the mean of a, b, c is P and ab + bc + ca = 0, then the mean of a², b², c² is-

ক
p
খ
9P²
গ
P²
ঘ
3P²

ব্যাখ্যা

Question: If the mean of a, b, c is P and ab + bc + ca = 0, then the mean of a², b², c² is-

Solution :
Given,
ab + bc + ca = 0
and, the mean of a, b, c is P
⇒ (a + b + c)/3 = P
⇒ (a + b + c) = 3P ..............(1)

Now,
(a + b + c)² = a²+ b² + c² + 2(ab + bc + ca)
⇒ (3P)² = a² + b² + c²+ 2 × 0
⇒ 9P² = a² + b² + c²...

∴ Mean = (a² + b² + c²)/3
⇒ (9P²)/3
⇒ 3P²

২,৭২৮.

An employee pays 3 workers X, Y and Z a total of Tk. 610 in a week. X is paid 125% of the amount Y is paid and 80% of the amount Z is paid. Hou much does X make a week?

ক
ক) Tk. 150
খ
খ) Tk. 175
গ
গ) Tk. 180
ঘ
ঘ) Tk. 200

ব্যাখ্যা

Question: An employee pays 3 workers X, Y and Z a total of Tk. 610 in a week. X is paid 125% of the amount Y is paid and 80% of the amount Z is paid. Hou much does X make a week?

Solution:
X = 125% Y
⇒ X = 125Y/100 = 5Y/4
∴ Y = 4X/5

X = 80% of Z
⇒ X = 80Z/100
⇒ X = 4Z/5
∴ Z = 5X/4

X + Y + Z = 610
⇒ X + 4X/5 + 5X/4 = 610
⇒ (20X + 16X + 25X)/20 = 610
⇒ 61X = 610 × 20
⇒ X = (610 × 20)/61
= 200 taka

২,৭২৯.

The HCF of two numbers is 13 and their LCM is 546. What is the sum of the numbers?

ক
ক) 225
খ
খ) 256
গ
গ) 169
ঘ
ঘ) 338

ব্যাখ্যা

Question: The HCF of two numbers is 13 and their LCM is 546. What is the sum of the numbers?

Solution:
Let,
The numbers be 13x and 13y where x and y are co-prime.
∴ LCM = 13xy

Now, 13xy = 546
⇒ xy = 546/13
⇒ xy = 42
⇒ xy = 6 × 7
x = 6 and y = 7
Or, x = 7 and y = 6

1st number = 13 × 6 = 78
2nd number = 13 × 7 = 91
Sum = 78 + 91 = 169

২,৭৩০.

Jisan cycled halfway at 3 km/h and the rest at 6 km/h. What was his average speed for the whole trip?

ক
2 km/h
খ
3 km/h
গ
4 km/h
ঘ
4.5 km/h

ব্যাখ্যা

Question: Jisan cycled halfway at 3 km/h and the rest at 6 km/h. What was his average speed for the whole trip?

Solution:
ধরি,
3 কি.মি./ঘণ্টা বেগে অতিক্রম করে = x কি.মি.
এবং 6 km/h বেগে অতিক্রম করে = x কি.মি.
∴ পথের মোট দূরত্ব = x + x = 2x কি.মি.

যাত্রাপথের অর্ধেক দূরত্বে প্রয়োজনীয় সময় = x/3 ঘণ্টা
বাকি অর্ধেক দূরত্বে প্রয়োজনীয় সময় = x/6 ঘণ্টা

∴ সম্পূর্ণ যাত্রায় গড় গতিবেগ = মোট দূরত্ব/মোট সময়
= 2x/{(x/3) + (x/6)}
= 2x/{(2x + x)/6}
= 2x/(3x/6)
= 2x/(x/2)
= 2x × (2/x)
= 4 কি.মি./ঘণ্টা

২,৭৩১.

Fifty percent of a number is 30 less than three-fourth of the same number. Find the number.

ক
210
খ
180
গ
150
ঘ
120

ব্যাখ্যা

Question: Fifty percent of a number is 30 less than three-fourth of the same number. Find the number.

Solution:
Let,
the number = x.

ATQ,
50% of x = (3/4) of x - 30
⇒ 50x/100 = (3x/4) - 30
⇒ x/2 = (3x/4) - 30
⇒ (3x/4) - (x/2) = 30
⇒ (3x - 2x)/4 = 30
⇒ x/4 = 30
∴ x = 120

So, the number is 120.

২,৭৩২.

Suppose that a person rows a boat in still water at the speed of 10 km/hr and the water runs at the speed of 4 km/hr. This person travels a certain distance & then returns. If it takes 4 hrs more for him to travel upstream than that of downstream then what will be the distance?

ক
16 km
খ
30 km
গ
42 km
ঘ
70 km

ব্যাখ্যা

If a boat takes time 't' hours more in upstream than to move downstream for the same distance, then the distance is given by,
Distance = [(x²– y²) (t)]/(2y) km

Given parameters are:
Speed of a boat in still water = 10 km/hr
Speed of running water = 4 km/hr
Required time = 4 hrs to travel upstream more than downstream

Therefore, we obtain,
D = 4 x (10²– 4²)/(2 x 4)
= 42 km.

২,৭৩৩.

√(0.0288 × 0.08) = ?

ক
ক) 0.024
খ
খ) 0.48
গ
গ) 0.048
ঘ
ঘ) 0.0048

ব্যাখ্যা

√(0.0288 × 0.08)
= √0.002304
= √(0.048)²
= 0.048

২,৭৩৪.

The difference of two numbers is 2 and the difference of their squares is 28, then sum of the numbers -

ক
ক) 18
খ
খ) 14
গ
গ) 16
ঘ
ঘ) 12

ব্যাখ্যা

Question: The difference of two numbers is 2 and the difference of their squares is 28, then sum of the numbers -
Solution:
ধরি সংখ্যা দুটি যথাক্রমে x and y
⇒ x - y = 2 - - (i)

সংখ্যাদ্বয়ের বর্গের পার্থক্য
⇒ x²- y² = 28 - - (ii)

সূত্র ব্যবহার করে পাই,
x² - y² = (x + y)(x - y)
⇒ 28 = 2 (x + y)
⇒ x + y = 28/2
⇒ x + y = 14

∴ The sum of the number is 14.

২,৭৩৫.

What number should replace the question mark?

ক
18
খ
44
গ
64
ঘ
68

ব্যাখ্যা

Question: What number should replace the question mark?

Solution:
In first figure:
4 × 8 + 7 = 32 + 7 = 39

In second figure:
6 × 3 + 9 = 18 + 9 = 27

So, in third figure:
9 × 7 + 5 = 63 + 5 = 68

∴ The number is 68

২,৭৩৬.

If sin x + cos x = 1, then x = ?

ক
30°
খ
45°
গ
60°
ঘ
90°

ব্যাখ্যা

Question: If sin x + cos x = 1, then x = ?

Solution:
অপশন টেস্ট করে পাই,
x = 30° হলে,
sin 30° + cos 30°
= (1/2) + (√3/2)
= (1 + √3)/2 ;[যা সত্য নয়]

x = 45° হলে,
sin 45° + cos 45°
= (1/√2) + (1/√2)
= 2/√2
= √2 ;[যা সত্য নয়]

x = 60° হলে,
sin 60° + cos 60°
= (√3/2) + (1/2)
= (1 + √3)/2 ;[যা সত্য নয়]

x = 90° হলে,
sin 90° + cos 90°
= 1 + 0
= 1 ;[যা সত্য]

∴ x = 90° হলে, sin x + cos x = 1 হবে।

২,৭৩৭.

A and B together can dive a trench in 12 days, which A alone can dive in 30 days. In how long B alone can burrow it?

ক
18 days
খ
19 days
গ
20 days
ঘ
21 days

ব্যাখ্যা

Question: A and B together can dive a trench in 12 days, which A alone can dive in 30 days. In how long B alone can burrow it?

Solution:
(A+B)'s 1 day work = 1/12,
A's 1 day work =1/30
∴ B's 1 day work = (1/12 - 1/30) = 3/60 = 1/20

Henceforth, B alone can dive the trench in 20 days.

২,৭৩৮.

Asim, Raju and Rokon agree to pay their total electricity bill in the proportion 3 : 4 : 5. Asim pays the first day's bill of Tk. 50, Raju pays the second day's bill of Tk. 55 and Rokon pays the third day's bill of Tk. 75. How much should Asim pay/get to settle the accounts?

ক
15
খ
17
গ
12
ঘ
5

ব্যাখ্যা

Total bill paid by Asim, Raju and Rokon = ( 50 + 55 +75 ) = Tk. 180

Let the amount paid by Asim, Raju, and Rokon be Tk. 3x, 4x and 5x respectively.
Therefore, (3x + 4x + 5x ) = 180
12x = 180
x = 15

Therefore, the amount to be paid by,
Asim = Tk. 45
Raju = Tk. 60
Rokon = Tk. 75

But actually as given in the question, Asim pays Tk. 50, Raju pays Tk. 55 and Rokon pays Tk. 80.
Hence, Asim pays Tk. 5 more and Raju 5 Tk less than the actual amount to be paid.
Hence Raju needs to pay Tk. 5 to Asim to settle the amount.

২,৭৩৯.

The present ages of A and B are in the ratio 3 : 5. After 10 years, the ratio of their ages will be 4 : 6. What is the difference in their present ages?

ক
13 years
খ
15 years
গ
16 years
ঘ
20 years

ব্যাখ্যা

Question: The present ages of A and B are in the ratio 3 : 5. After 10 years, the ratio of their ages will be 4 : 6. What is the difference in their present ages?

Solution:
Let the present ages be,
A = 3x and B = 5x

Ages after 10 years,
A = 3x + 10
B = 5x + 10

According to the problem, the ratio becomes 4 : 6
(3x + 10) : (5x + 10) = 4 : 6
⇒ (3x + 10)/(5x + 10) = 4/6
⇒ 3(3x + 10) =2(5x + 10)
⇒ 9x + 30 = 10x + 20
⇒ 10x - 9x = 30 - 20
∴ x = 10

A = 3 × 10 = 30 years
B = 5 × 10 = 50 years

∴ Difference = 50 - 30 = 20 years

২,৭৪০.

A man on tour travels first 160 km at 64 km/hr and the next 160 km at 80 km/hr. The average speed for the first 320 km of the tour is-

ক
71 km/hr
খ
35.55 km/hr
গ
36 km/hr
ঘ
71.11 km/hr
ঙ
None of these

ব্যাখ্যা

Question: A man on tour travels first 160 km at 64 km/hr and the next 160 km at 80 km/hr. The average speed for the first 320 km of the tour is-

Solution:
Total time taken = (160/64 + 160/80)hrs
= (5/2 + 2)
= 9/2 hrs.

∴ Average speed = (320 × (2/9) km.hr
= 71.11 km/hr.

২,৭৪১.

For 9 innings, Tanveer has an average of 75 runs. In the tenth inning, he scores 100 runs, thus increasing his average. His new average is-

ক
75
খ
100
গ
72
ঘ
77.5
ঙ
79.5

ব্যাখ্যা

Question: For 9 innings, Tanveer has an average of 75 runs. In the tenth inning, he scores 100 runs, thus increasing his average. His new average is-

Solution:

Total score for 9 innings is 75 × 9 = 675

Total score after 10th innings = 675 + 100 = 775

So, average = 775/10 = 77.5

২,৭৪২.

P and Q started a business in the ratio of 2 : 3. After, 1 year P left the business but Q continued. After 2 years they had the profit of Tk. 26,000. What is the profit of Q?

ক
Tk. 5000.
খ
Tk. 20500.
গ
Tk. 19500.
ঘ
Tk. 10500.

ব্যাখ্যা

Question: P and Q started a business in the ratio of 2 : 3. After, 1 year P left the business but Q continued. After 2 years they had the profit of Tk. 26,000. What is the profit of Q?

Solution:
Let the initial capital of P and Q be 2x and 3x, respectively.

Then, ratio of profits
= 2x × 12 : 3x × 24
= 24x : 72x
= 1 : 3

∴ Q's share of profit
= 26000 × (3/4)
= Tk. 19500.

২,৭৪৩.

The King and Queen of black color are taken out from a deck of 52 playing cards. A card is drawn from the remaining well-shuffled cards. The probability of getting a spade card is-

ক
11/48
খ
13/48
গ
11/52
ঘ
13/52

ব্যাখ্যা

Question: The King and Queen of black color are taken out from a deck of 52 playing cards. A card is drawn from the remaining well-shuffled cards. The probability of getting a spade card is-

Solution:
Number of Kings and Queens of black color = 4
∴ Remaining cards = 52 - 4
= 48

Spade card is black, hence 1 queen and 1 king of spade is removed, remaining spade card = 13 - (1 + 1)
= 11

∴ Probability = 11/48

২,৭৪৪.

What is the solution set of the inequality, - 2x + 11 ≥ 5?

ক
(- ∞, 3]
খ
(3, - ∞)
গ
(- ∞, 4]
ঘ
[3, ∞)

ব্যাখ্যা

Question: What is the solution set of the inequality, - 2x + 11 ≥ 5?

Solution:
Given that,
- 2x + 11 ≥ 5
⇒ - 2x + 11 - 11 ≥ 5 - 11 ; [Subtract 11 from both sides]
⇒ - 2x ≥ - 6
∴ x ≤ 3 ; [Divide both sides by - 2]

Solution set: x ≤ 3
or in interval notation: (- ∞, 3]

২,৭৪৫.

Two numbers are in the raton 3 : 4. If the difference of their squares is 63, then what is the smaller number?

ক
ক) 6
খ
খ) 9
গ
গ) 12
ঘ
ঘ) 15

ব্যাখ্যা

Question: Two numbers are in the raton 3 : 4. If the difference of their squares is 63, then what is the smaller number?

Solution:
Let the numbers be 3x and 4x.
Then, (4x)² - (3x)² = 63
⇒ 16x² - 9x² = 63
⇒ 7x² = 63
⇒ x² = 9
∴ x = 3

∴ The smaller number (3 × 3) = 9

২,৭৪৬.

A, B, and C can complete a piece of work in 8, 12, and 24 days respectively. Working together, they will complete the same work in -

ক
6 days
খ
3 days
গ
8 days
ঘ
4 days

ব্যাখ্যা

Question: A, B, and C can complete a piece of work in 8, 12, and 24 days respectively. Working together, they will complete the same work in -

Solution:
(A + B + C)'s 1 days work = 1/8 + 1/12 + 1/24
= (3 + 2 + 1)/24
= 6/24
= 1/4

∴ A, B, and C together can complete the work in = 4 days

২,৭৪৭.

Write in terms of indices: log₂₇81= 4/3

ক
ক) 3⁴
খ
খ) 3³
গ
গ) 27^1/3
ঘ
ঘ) 27^2/3

ব্যাখ্যা

log₂₇81 = 4/3
81 = (27)^4/3 = (3³)^4/3 = 3⁴

২,৭৪৮.

A certain distance is covered at a certain speed. If half the distance is covered in double the time, the ratio of the two speeds is-

ক
ক) 2 : 1
খ
খ) 3 : 1
গ
গ) 2 : 4
ঘ
ঘ) 4 : 1

ব্যাখ্যা

Question: A certain distance is covered at a certain speed. If half the distance is covered in double the time, the ratio of the two speeds is-

Solution:
Let x km be covered in y hours.
Then, speed = x/y km/hr

In the second case, x/2 km is covered in 2y hours.
∴ New speed = (x/2)/2y km/hr
= x/4y

∴ Ratio of speeds = x/y : x/4y
= 1 : 1/4
= 4 : 1

২,৭৪৯.

A, B and C can complete a piece of work in 6, 8 and 12 days respectively. Working together, they will complete the work in ____ .

ক
ক) 5/3 days
খ
খ) 7/3 days
গ
গ) 10/3 days
ঘ
ঘ) 8/3 days

ব্যাখ্যা

Question: A, B and C can complete a piece of work in 6, 8 and 12 days respectively. Working together, they will complete the work in ____ .

Solution:
A 1 দিনে করে কাজটির =1/6 অংশ
B 1 দিনে করে কাজটির =1/8 অংশ
C 1 দিনে করে কাজটির =1/12 অংশ

A + B + C 1 দিনে করে কাজটির =(1/6) + (1/8) + (1/12) অংশ
= (4 + 3 + 2 )/24 অংশ
= 9/24 অংশ
= 3/8 অংশ
A + B + C 3/8 অংশ কাজ করে 1 দিনে
A + B + C 1 (সম্পূর্ণ)অংশ কাজ করে (1 × 8)/3 দিনে
= 8/3 দিনে

২,৭৫০.

A tank can be filled in 9 minutes by two pipes together. After keeping both pipes open for 6 minutes, the first pipe is closed. If it then takes another 7 minutes to completely fill the tank, how many minutes will the second pipe alone take to fill the tank?

ক
14 min
খ
18 min
গ
21 min
ঘ
28 min

ব্যাখ্যা

Question: A tank can be filled in 9 minutes by two pipes together. After keeping both pipes open for 6 minutes, the first pipe is closed. If it then takes another 7 minutes to completely fill the tank, how many minutes will the second pipe alone take to fill the tank?

Solution:
দুইটি পাইপ একত্রে 9 মিনিটে পূর্ণ করে 1 অংশ
∴ দুইটি পাইপ একত্রে 1 মিনিটে পূর্ণ করে (1/9) অংশ
∴ দুইটি পাইপ একত্রে 6 মিনিটে পূর্ণ করে (6/9) অংশ
= 2/3 অংশ

∴ অবশিষ্ট অংশ = {1 - (2/3)} অংশ
= (3 - 2)/3 অংশ
= 1/3 অংশ

২য় পাইপ দ্বারা 1/3 অংশ পূর্ণ হয় 7 মিনিটে
∴ ২য় পাইপ দ্বারা 1 অংশ পূর্ণ হয় (3 × 7) মিনিটে
= 21 মিনিটে

২,৭৫১.

If the average of the four numbers M, 2M +3, 3M - 5 and 5M + 1 is 63, what is the value of M?

ক
ক) 23
খ
খ) 22
গ
গ) 11
ঘ
ঘ) 32

ব্যাখ্যা

প্রশ্নমতে,
(M + 2M + 3 + 3M - 5 + 5M + 1)/4 = 63
⇒ (11M - 1) = 63 × 4
⇒ 11M = 252 + 1
⇒ 11M = 253
⇒ M = 253 / 11
⇒ M = 23

২,৭৫২.

Find the equation of the line with x-intercept = -3 and y-intercept = 2.

ক
2x - 5y + 10 = 0
খ
3x - 2y - 6 = 0
গ
4x - 3y + 12 = 0
ঘ
2x - 3y + 6 = 0

ব্যাখ্যা

Question: Find the equation of the line with x-intercept = -3 and y-intercept = 2.

Solution:
Given, x-intercept = - 3,
So, the line passes through (-3, 0).
y-intercept = 2,
So, the line passes through (0, 2).

We know, the intercept form of a line is:
(x/a) + (y/b) = 1, where a = x-intercept, b = y-intercept.
⇒ x/(- 3) + (y/2) = 1
⇒ (- 2x + 3y)/6 = 1
⇒ - 2x + 3y = 6
⇒ 2x - 3y + 6 = 0

∴ The equation of the line is 2x - 3y + 6 = 0

২,৭৫৩.

The present ratio of students to teachers at a certain school is 30 to 1. If the student enrollment were to increase by 50 students and the numbers of teachers were to increase by 5, the ratio of students would then be 25 to 1. What is the present number of teachers?

ক
ক) 8
খ
খ) 10
গ
গ) 12
ঘ
ঘ) 15

ব্যাখ্যা

Question: The present ratio of students to teachers at a certain school is 30 to 1. If the student enrollment were to increase by 50 students and the numbers of teachers were to increase by 5, the ratio of students would then be 25 to 1. What is the present number of teachers?

Solution:
Let the number of student and teacher be represented by S and T respectively.
S : T = 30 : 1
S/T = 30/1
S = 30T.............(1)

If number of students increases by 50 and number of teachers increases by 5
50 + S : T + 5 = 25 : 1
⇒ (50 + S)/(T + 5) = 25/1
⇒ S + 50 = 25(T + 5)
⇒ S + 50 = 25T + 125
⇒ 30T + 50 = 25T + 125 [From (1)]
⇒ 30T - 25T = 125 - 50
⇒ 5T = 75
⇒ T= 75/5 =15

∴ No of teachers=T=15

২,৭৫৪.

A rectangular field is 3.2 yards long. A fence marking the boundary is 11.2 yards in length. What is the area of the field in square yards?

ক
4.68
খ
7.68
গ
9.28
ঘ
11.28

ব্যাখ্যা

Question: A rectangular field is 3.2 yards long. A fence marking the boundary Is 11.2 yards in length. What is the area of the field in square yards?

Solution:
দেওয়া আছে,
আয়তাকার মাঠের দৈর্ঘ্য = 3.2 গজ
মাঠের চতুর্দিকে বেড়ার দৈর্ঘ্য = মাঠের পরিসীমা = 2(দৈর্ঘ্য + প্রস্থ) = 11.2 গজ

ধরি,
মাঠের প্রস্থ = x গজ

প্রশ্নমতে,
2(3.2 + x) = 11.2
⇒ 6.4 + 2x = 11.2
⇒ 2x = 11.2 - 6.4
⇒ 2x = 4.8
⇒ x = 4.8/2
⇒ x = 2.4

∴ আয়তাকার মাঠের ক্ষেত্রফল = দৈর্ঘ্য × প্রস্থ = (3.2 × 2.4) বর্গগজ = 7.68 বর্গগজ

২,৭৫৫.

A 9% stock yields 8%. The market value of the stock is-

ক
Tk. 72
খ
Tk. 116.50
গ
Tk. 90
ঘ
Tk. 112.50

ব্যাখ্যা

Question: A 9% stock yields 8%. The market value of the stock is-

Solution:
Earn Tk. 8 when market value Tk. 100
Earn Tk. 1 when market value Tk. 100/8
Earn Tk. 9 when market value Tk. (100 × 9)/8
= Tk. 112.5

২,৭৫৬.

The 3rd term of a geometric sequence is 48, and the 6th term is 384. What is the common ratio is?

ক
2
খ
6
গ
3
ঘ
4
ঙ
5

ব্যাখ্যা

Question: The 3rd term of a geometric sequence is 48, and the 6th term is 384. What is the common ratio is?

Solution:
Given,
The 3rd term of a geometric sequence, a₃ = 48
The 6th term of the same sequence, a₆ = 384

In a geometric sequence we know,
a_n = ar^{n - 1}

So,
a₃ = ar^{3 - 1}= ar² .......(1)
And
a₆ = ar^{6 - 1}= ar⁵........ (2)

Now (2) ÷ (1),
ar⁵/ar² = 384/48
⇒ r³ = 8 = 2³
∴ r = 2

So the common ratio is 2.

২,৭৫৭.

An individual is cycling at a speed of 37.5 km per hour. He catches his predecessor who had started earlier in two hours. What is the speed of his predecessor who had started 3 hours earlier?

ক
ক) 8 kmph.
খ
খ) 10 kmph.
গ
গ) 15 kmph.
ঘ
ঘ) 12 kmph.

ব্যাখ্যা

Question: An individual is cycling at a speed of 37.5 km per hour. He catches his predecessor who had started earlier in two hours. What is the speed of his predecessor who had started 3 hours earlier?

Question:
The distance covered in two hour,
= 2 × 37.5= 75 km
Time taken by first individual = (3h + 2h) = 5h

Then, the speed of predecessor,
= 75/5
= 15 kmph.

২,৭৫৮.

January 1, 2008 is Tuesday. What day of the week lies on Jan 1, 2009?

ক
ক) Monday
খ
খ) Wednesday
গ
গ) Thursday
ঘ
ঘ) Sunday

ব্যাখ্যা

The year 2008 is a leap year. So, it has 2 odd days.
1st day of the year 2008 is Tuesday (Given)
So, 1st day of the year 2009 is 2 days beyond Tuesday.
Hence, it will be Thursday.

২,৭৫৯.

The number 89712∗ is divisible by 4. The unknown non zero digit marked as ∗ will be -

ক
ক) 6
খ
খ) 2
গ
গ) 3
ঘ
ঘ) 4

ব্যাখ্যা

Question: The number 89712∗ is divisible by 4. The unknown non zero digit marked as ∗ will be -
Solution:
কোনো সংখ্যা ৪ দ্বারা বিভাজ্য কিনা সেটা বুঝা যায় যদি সংখ্যার শেষ দুটি অংক ৪ দ্বারা বিভাজ্য হয়।
∗ এর স্থানে 4 হলে দশক ও একক স্থানীয় অঙ্ক দ্বারা গঠিত সংখ্যা 24 যা 4 দ্বারা বিভাজ্য।

২,৭৬০.

A pupil's marks were wrongly entered as 63 instead of 43. Due to that the average marks for the class got increased by half. The number of pupils in the class is

ক
ক) 30
খ
খ) 40
গ
গ) 50
ঘ
ঘ) 60

ব্যাখ্যা

প্রশ্ন: A pupil's marks were wrongly entered as 63 instead of 43. Due to that the average marks for the class got increased by half. The number of pupils in the class is

সমাধান:
ধরি, মোট ছাত্র সংখ্যা x
একজন বাদে বাকি ছাত্রদের নম্বরের সমষ্টি y

(y + 63)/ x = a + 0.5 ....(Ⅰ)
(y + 43)/ x = a.........( Ⅱ)

(Ⅰ) - ( Ⅱ) থেকে পাই,
{(y + 63)/ x} - {(y + 43)/ x } = a + 0.5 - a
⇒ (y + 63 - y - 43)/x = 0.5
⇒ 20/x = 0.5
⇒ x = 40

∴ ক্লাসে ছাত্র সংখ্যা ৪০ জন।

২,৭৬১.

A man bought cookies at 3 for Tk. 1. How many for Tk. 1 should he sell to make a profit a 50%.

ক
1
খ
2
গ
1.5
ঘ
None of these

ব্যাখ্যা

Question: A man bought cookies at 3 for Tk. 1. How many for Tk. 1 should he sell to make a profit a 50%.

Solution:
CP of 3 cookies = Tk. 1
SP of 3 cookies = 150% of Tk. 1 = 3/2

For Tk. 3/2, the man sells 3 cookies.
Hence for Tk. 1, number of cookies sold = (3 × 2)/3 = 2

২,৭৬২.

Bus fares were recently increased from Taka 12 to Taka 16. What was the approximate percentage of increase?

ক
4%
খ
15%
গ
21%
ঘ
33%

ব্যাখ্যা

Question: Bus fares were recently increased from Taka 12 to Taka 16. What was the approximate percentage of increase?

Solution:
বাস ভাড়া বাড়ে = (16 - 12) টাকা
= 4 টাকা

বাস ভাড়া শতকরা বাড়ে = (4/12) × 100%
= 33.33%
≈ 33%

২,৭৬৩.

The driver of an ambulance sees a school bus 40 m ahead of him after 20 seconds, the school bus is 60 meter behind. If the speed of the ambulance is 30 km/h, what is the speed of the school bus?

ক
ক) 10 kmph
খ
খ) 12 kmph
গ
গ) 15 kmph
ঘ
ঘ) 22 kmph
ঙ
ঙ) 21 kmph

২,৭৬৪.

Tofail, Hasib, Hemal enter into a partnership. Tofail initially invests 25 lakh & adds another 10 lakhs after one year. Hasib initially invests 35 lakh & withdrawal 10 lakh after 2 years and Hemal invests 30 Lakh. In what ratio should the profit be divided at the end of 3 years?

ক
ক) 19 : 19 : 18
খ
খ) 17 : 18 : 19
গ
গ) 18 : 18 : 19
ঘ
ঘ) 17 : 17: 19

ব্যাখ্যা

প্রশ্ন: Tofail, Hasib, Hemal enter into a partnership. Tofail initially invests 25 lakh & adds another 10 lakhs after one year. Hasib initially invests 35 lakh & withdrawal 10 lakh after 2 years and Hemal invests 30 Lakh. In what ratio should the profit be divided at the end of 3 years?

সমাধান:
Tofail : Hasib : Hemal = (25 × 1 + 35 × 2) : (35 × 2 + 25 × 1) : (30 × 3)
= (25 + 70) : (70 + 25) : 90
= 95 : 95 : 90
= 19 : 19: 18

২,৭৬৫.

The total cost of flooring a room at Tk. 6.50 per square meter is Tk. 390. If the length of the room is 8m, what is its breadth?

ক
7.5 m
খ
8.5 m
গ
10.5 m
ঘ
12.5 m

ব্যাখ্যা

Question: The total cost of flooring a room at Tk. 6.50 per square meter is Tk. 390. If the length of the room is 8m, what is its breadth?

Solution:
ঘরের ক্ষেত্রফল = 390/6.50
= 60 বর্গমিটার

প্রশ্নমতে
8 × প্রস্থ = 60
বা, প্রস্থ = 60/8
∴ প্রস্থ =7.5

আয়তাকার ক্ষেত্রের প্রস্থ =7.5 মিটার

২,৭৬৬.

P, Q, R have the total money of Tk. 2800. Q have half of P and R have half of Q. The amount of R is

ক
ক) 400 Tk.
খ
খ) 600 Tk.
গ
গ) 800 Tk.
ঘ
ঘ) 1600 Tk.

ব্যাখ্যা

question: P, Q, R have the total money of Tk. 2800. Q have half of P and R have half of Q. The amount of R is

Solution:
Let P have x

then,
Q have x/2 and
R have x/4

Now
x + (x/2) + (x/4) = 2800
7x/4 = 2800
x = 1600

Hence,
The amount of R is (1600/4) = 400

২,৭৬৭.

The value of log₃2 ⋅ log₄3 ⋅ log₅4 ⋅ log₆5 ⋅ log₇6 ⋅ log₈7 is-

ক
1/2
খ
2
গ
1/3
ঘ
1
ঙ
1/4

ব্যাখ্যা

Question: The value of log₃2 ⋅ log₄3 ⋅ log₅4 ⋅ log₆5 ⋅ log₇6 ⋅ log₈7 is-

Solution:
log₃2 ⋅ log₄3 ⋅ log₅4 ⋅ log₆5 ⋅ log₇6 ⋅ log₈7
= (log₃2 ⋅ log₄3) (log₅4 ⋅ log₆5) (log₇6 ⋅ log₈7)
= log₄2 · log₆4 · log₈6 [log_bM × log_ab = log_aM]
= (log₄2 · log₆4) log₈6
= log₆2 ⋅ log₈6
= log₈2
= 1/log₂8
= 1/log₂2³
= 1/(3log²2)
= 1/(3 × 1) [∵ log₂2 = 1]
= 1/3

২,৭৬৮.

What is the distance covered by a car traveling at a speed of 40 kmph in 15 minutes?

ক
15 km
খ
30 km
গ
20 km
ঘ
10 km

ব্যাখ্যা

Question: What is the distance covered by a car traveling at a speed of 40 kmph in 15 minutes?

Solution:
converting speed into km/min, we get
40 kmph = 40/60 km/min = 2/3 km/min.

Therefore, distance traveled = 15 × (2/3) = 10 km.

২,৭৬৯.

If cosA sinA = 1,then (cosA + sinA)² =?

ক
2
খ
3
গ
4
ঘ
6

ব্যাখ্যা

প্রশ্ন: If cosA sinA = 1,then (cosA + sinA)² =?

সমাধান:
(cosA + sinA)²
= cos²A + 2 cosA sinA + sin²A
= 1 + 2.1 [sin²A + cos²A = 1]
= 1 + 2
= 3

২,৭৭০.

Three numbers are in the ratio 2 : 3 : 5 and their H.C.F is 6. The numbers are-

ক
ক) 12, 18, 30
খ
খ) 10, 12, 20
গ
গ) 12, 30, 32
ঘ
ঘ) 42, 30, 36

ব্যাখ্যা

Let
the numbers be2x, 3x and 5x
Then, Their H.C.F = x
x = 6

The numbers are (2 × 6), (3 × 6) and (5 × 6) = 12, 18 and 30

২,৭৭১.

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

ক
5
খ
5/9
গ
9
ঘ
8

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

ATQ,
1/α + 1/β = (α + β)/αβ
= (5/9)/(1/9)
= (5/9) × (9/1)
= 5

২,৭৭২.

In how many ways 5 students can be chosen from the class of 10 students?

ক
ক) 152
খ
খ) 252
গ
গ) 352
ঘ
ঘ) 452

ব্যাখ্যা

Question: In how many ways 5 students can be chosen from the class of 10 students?

Solution:
ways 5 students can be chosen from the class of 10 students is = ¹⁰C₅
= 10!/(5! 5!)
= 252

২,৭৭৩.

Mr. Pollob can do a job in 120 minutes, while his colleague can do the same job in 3 hours. How long will it take them to work together to finish the same job?

ক
70 minutes
খ
72 minutes
গ
75 minutes
ঘ
78 minutes

ব্যাখ্যা

Question: Mr. Pollob can do a job in 120 minutes, while his colleague can do the same job in 3 hours. How long will it take them to work together to finish the same job?

Solution:
Mr. Pollob can do a job in 120 minutes = 2 hours
∴ Mr. Pollob can do in 1 hour = 1/2 portion of the job

His colleague can do in 1 hour = 1/3 portion of the job

Together they can do in 1 hour = 1/2 + 1/3 = 5/6 portion of the job

∴ Together they can do full job in (6/5) hour = (6 × 60)/5 minutes
= 72 minutes

২,৭৭৪.

Read the following questions carefully and choose the right answer. (33-56):
৩৩) The length of a rectangle is 20% more than its breadth. Find the ratio of the area of the rectangle to that of the square whose side is equal to the breadth of the rectangle.

ক
ক) 8:9
খ
খ) 3:2
গ
গ) 6:5
ঘ
ঘ) 2:1

ব্যাখ্যা

Let the breadth of the rectangle be x
∴ According to question,
Length = 1.20x
∴ Area of rectangle = 1.20x × x = 1.20x²
Area of square = x × x = x²
∴ Required ratio = 1.20x²/x²
= 12/10
= 6/5

২,৭৭৫.

An 80L solution of alcohol and water has 45% alcohol in it. If you want the mixture to be 75% alcohol, how much alcohol would you add to it?

ক
30 litres
খ
75 litres
গ
96 litres
ঘ
110 litres

ব্যাখ্যা

Currently alcohol quantity = (45/100) × 80 = 36 litres.
Let A be alcohol added.

So,
36 + A = (75/100) × (80 + A)
⇒ 36 + A = (3/4) × (80 + A)
⇒ 144 + 4A = 240 + 3A
⇒ A = 240 - 144 = 96

∴ A = 96 Litres = This is the additional quantity of alcohol to be added.

২,৭৭৬.

30 men are engaged to a work by a contractor for 6 days. They work for 4 days but due to some reasons, only 50% of the work will be completed. How many more men should be engaged to work to complete the work in a given period of time?

ক
ক) 30 men
খ
খ) 35 men
গ
গ) 40 men
ঘ
ঘ) 60 men

ব্যাখ্যা

Question: 30 men are engaged to a work by a contractor for 6 days. They work for 4 days but due to some reasons, only 50% of the work will be completed. How many more men should be engaged to work to complete the work in a given period of time?

Solution:
১/২ অংশ কাজ ৪ দিনে করে ৩০ জন
∴ ১/২ অংশ কাজ ১ দিনে করে ৩০ × ৪ জন
∴ ১/২ অংশ কাজ ২ দিনে করে (৩০ × ৪)/২ জন
= ৬০ জন

অতিরিক্ত লোক লাগবে ৬০ - ৩০ জন
= ৩০ জন

২,৭৭৭.

What is the difference between the biggest and the smallest fraction among 2/3, 3/4, 4/5 and 5/6 ?

ক
1/6
খ
1/20
গ
1/12
ঘ
1/30

ব্যাখ্যা

Question: What is the difference between the biggest and the smallest fraction among 2/3, 3/4, 4/5 and 5/6 ?

Solution:
Converting each of the given fractions into decimal form, we get,
2/3 = 0.66
3/4 = 0.75
4/5 = 0.8
5/6 = 0.833

Since 0.833 > 0.8 > 0.75 > 0.66
So, 5/6 > 4/5 > 3/4 > 2/3

∴ Required difference = (5/6) - (2/3) = 1/6

২,৭৭৮.

A thief steals a car at 1.30 pm and drive it off 40 km/hr. The theft is discovered at 2 pm and the owner sets off in another car at 50 km/hr he will catch the thief at-

ক
2 : 30 pm
খ
3 pm
গ
3 : 30 pm
ঘ
4 pm

ব্যাখ্যা

Question: A thief steals a car at 1.30 pm and drive it off 40 km/hr. The theft is discovered at 2 pm and the owner sets off in another car at 50 km/hr he will catch the thief at-

Solution:
Distance covered by thief in (2 pm - 1.30 pm) = 1/2 hours

1/2 hours at speed of 40 km/h = 40 × (1/2) = 20 km

Their relative speed in same direction = (50 - 40) km/h
= 10 km/h

ATQ,
20 km, is the distance that has to be covered by owner to catch the thief.

Required time = (20/10) hours
=2 hours

Therefore, he will over take the thief at :
= 2 pm + 2 hours
= 4 pm

২,৭৭৯.

When Rakib, Piyash and Niloy complete a task, Rakib and Piyash together do 70% of the work and Piyash and Niloy together do 50% of the work. who is the most efficient?

ক
ক) Rakib
খ
খ) Piyash
গ
গ) Niloy
ঘ
ঘ) Can't be determined

ব্যাখ্যা

Question: When Rakib, Piyash and Niloy complete a task, Rakib and Piyash together do 70% of the work and Piyash and Niloy together do 50% of the work. who is the most efficient?

Solution:
রাকিব, পিয়াস একসাথে ৭০% কাজ করে।
পিয়াস, নিলয় একসাথে করে ৫০%
রাকিব, পিয়াস ও নিলয় মিলে করে ১০০% কাজ

পিয়াস করে = ৭০% + ৫০% - ১০০% কাজ
= ২০% লাজ

নিলয় করে = ৫০% - ২০%
= ৩০%

রাকিব করে = ৭০% - ২০%
= ৫০%

অর্থাৎ, রাকিব সবচেয়ে দক্ষ।

২,৭৮০.

A positive number, when decreased by 4, is equal to 21 times the reciprocal of the number. The number is:

ক
3
খ
5
গ
7
ঘ
9

ব্যাখ্যা

Question: A positive number, when decreased by 4, is equal to 21 times the reciprocal of the number. The number is:

Solution:
মনেকরি
সংখ্যাটি = x

প্রশ্নমতে
x - 4 = 21/x
বা, x(x - 4) = 21
বা, x² - 4x - 21 = 0
বা, x² - 7x + 3x - 21 = 0
বা, x(x - 7) + 3(x - 7) = 0
(x - 7)(x + 3) = 0

হয়
x - 7 = 0
x = 7

অথবা
x + 3 = 0
x = - 3

সংখ্যাটি = 7

২,৭৮১.

ক
14
খ
6
গ
12
ঘ
49
ঙ
7

ব্যাখ্যা

Question:

Solution:

২,৭৮২.

A question paper has two parts, A and B, each containing 10 questions. If a student has to choose 5 from part A and 8 from part B, in how many ways can he choose the questions?

ক
10340
খ
11240
গ
11340
ঘ
11360

ব্যাখ্যা

Question: A question paper has two parts, A and B, each containing 10 questions. If a student has to choose 5 from part A and 8 from part B, in how many ways can he choose the questions?

Solution:
ways to choose 5 from part A = ¹⁰C₅
ways to choose 8 from part B = ¹⁰C₈

choose 5 from part A and 8 from part B = ¹⁰C₅ × ¹⁰C₈
= {10!/(5! 5!)} × {10!/(2! 8!)}
= 11340

২,৭৮৩.

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

ক
Tk. 3500
খ
Tk. 3812
গ
Tk. 3972
ঘ
Tk. 4100

ব্যাখ্যা

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

Solution: let simple interest r%

I = pnr/100
⇒ I/p = nr/100
⇒ 0.6 = 6r/100
⇒ r = 10%

The compound interest = 12000 (1.1)³ - 12000
= 12 × 1331 - 12000
= 15972 - 12000
= Tk. 3972

২,৭৮৪.

A sum of Tk. 20,000 yields a compound interest of Tk. 4200 when invested at 10% per annum. What is the investment period in years?

ক
2 years
খ
3 years
গ
3.5 years
ঘ
4 years

ব্যাখ্যা

Question: A sum of Tk. 20,000 yields a compound interest of Tk. 4200 when invested at 10% per annum. What is the investment period in years?

Solution:
দেওয়া আছে,
আসল, P = 20000 টাকা
সুদের হার, r = 10% বার্ষিক
চক্রবৃদ্ধি সুদ, CI = 4200 টাকা

আমরা জানি,
চক্রবৃদ্ধি মূল, A = P + CI
= 20000 + 4200 = 24200 টাকা

চক্রবৃদ্ধি মূলের সূত্র ব্যবহার করে পাই,
A = P × (1 + r/100)ⁿ
⇒ 24200 = 20000 × (1 + 10/100)ⁿ
⇒ 24200 = 20000 × (110/100)ⁿ
⇒ (110/100)ⁿ = 24200/20000
⇒ (1.10)ⁿ = 1.21
⇒ (1.10)ⁿ = (1.10)²
∴ n = 2

∴ বিনিয়োগের সময়কাল 2 বছর।

২,৭৮৫.

The marked price of a Footwear is Tk. 200, and it is sold after applying two successive 20% discounts. What is the final price at which it is sold?

ক
Tk. 500
খ
Tk. 600
গ
Tk. 900
ঘ
Tk. 128
ঙ
None

ব্যাখ্যা

Question: The marked price of a Footwear is Tk. 200, and it is sold after applying two successive 20% discounts. What is the final price at which it is sold?

Solution:
Discount 1 = 200 × (20/100) = Tk. 40

Selling price after 1st discount = 200 - 40 = Tk. 160

Discount 2 = 160 × (20/100) = Tk. 32

∴ Selling price after 2nd discount = 160 - 32 = Tk. 128

২,৭৮৬.

The present age of Amir and his mother are in the ratio 2 : 5 respectively. Four years hence, the ratio of their ages become 5 : 11 respectively. What was Amir's present age in years?

ক
18 years
খ
16 years
গ
15 years
ঘ
12 years

ব্যাখ্যা

Question: The present age of Amir and his mother are in the ratio 2 : 5 respectively. Four years hence, the ratio of their ages become 5 : 11 respectively. What was Amir's present age in years?

Solution:
Let
the present ages of Amir and his mother be 2x and 5x respectively.

ATQ,
(2x + 4) : (5x + 4) = 5 : 11
(2x + 4)/(5x + 4) = 5/11
⇒ 22x + 44 = 25x + 20
⇒ 25x - 22x = 44 - 20
⇒ 3x = 24
∴ x = 8

Amir's present age = (2 × 8) = 16 years

২,৭৮৭.

A train crossing a station in 30 seconds with a speed of 72kmph. If the length of the station is 300m, what is the size of the train?

ক
200m
খ
250m
গ
350m
ঘ
400m
ঙ
None of the above

ব্যাখ্যা

Question: A train crossing a station in 30 seconds with a speed of 72kmph. If the length of the station is 300m, what is the size of the train?

Solution:
Speed = 72kmph
= (72 × 1000)/3600 mps
= 20 mps

total distance covered by the train is = 20 × 30 = 600m

∴ size of the train is = (600 - 300) = 300m

২,৭৮৮.

If y = sinx then for which value of x, y has the highest value?

ক
45°
খ
90°
গ
60°
ঘ
0°

ব্যাখ্যা

Question: If y = sinx then for which value of x, y has the highest value?

Solution:
y = sinx
y এর মান সর্বোচ্চ হবে যদি sinx এর মান সর্বোচ্চ হয়।
sinx এর সর্বোচ্চ মান 1
তাহলে,
x = 90°

২,৭৮৯.

In how many ways can 5 people from a group of 8 people be seated around a circular table?

ক
1200
খ
560
গ
2520
ঘ
1344

ব্যাখ্যা

Question: In how many ways can 5 people from a group of 8 people be seated around a circular table?

Solution:
5 people out of 8 = ⁸C₅
= 8!/5!(8 - 5)!
= 8!/(3! × 5!)
= (8 × 7 × 6 × 5!)/(6 × 5!)
= 56

And 5 people around a circular table = (5 - 1)! = 4! = 24

∴ Total ways = 24 × 56 = 1344

২,৭৯০.

Compute the surface area of a cuboid with length 10 cm, width 6 cm, and height 4 cm.

ক
332 square cm
খ
240 square cm
গ
166 square cm
ঘ
248 square cm

ব্যাখ্যা

Question: Compute the surface area of a cuboid with length 10 cm, width 6 cm, and height 4 cm.

Solution:
the surface area of a cuboid = 2(ab + bc + ca)

Surface Area= 2 × (10 × 6 + 10 × 4 + 6 × 4) = 248 square cm

২,৭৯১.

Sayed started a software business by investing Tk. 20000. After six months, Rafi joined him with a capital of Tk. 30000. After 3 years, they earned a profit of Tk. 13950. What was Sayed’s share in the profit?

ক
Tk. 6200
খ
Tk. 6400
গ
Tk. 4200
ঘ
Tk. 7750

ব্যাখ্যা

Question: Sayed started a software business by investing Tk. 20000. After six months, Rafi joined him with a capital of Tk. 30000. After 3 years, they earned a profit of Tk. 13950. What was Sayed’s share in the profit?

Solution:
Ratio of capitals of Sayed and Rafi
= (20000 × 36) : (30000 × 30)
= 720000 : 900000
= 4 : 5.

Sayed’s share is = Tk. (13950 × 4)/9 = Tk. 6200.

২,৭৯২.

Two numbers are respectively 20% and 25% lower than a third number. By how much percentage is the second number lower than the first?

ক
ক) (25/2)%
খ
খ) (25/4)%
গ
গ) 25%
ঘ
ঘ) (25/6)%

ব্যাখ্যা

Let
The third number be x

First number = 80% of x
= 80x/100
= 4x/5

Second number = 75% of x
= 75x/100
= 3x/4

Difference = (4x/5) - (3x/4)
= (16x - 15x)/20
= x/20

Required Percentage (x/20) × (5/4x) × 100% = (25/4)%

২,৭৯৩.

cot³30° - 2sin60° = ?

ক
√3
খ
3√3
গ
2√3
ঘ
1/√3

ব্যাখ্যা

Question: cot³30° - 2sin60° = ?

Solution:
Given that,
cot³30° - 2sin60°
= (√3)³ - 2(√3/2)
= 3√3 - √3
= 2√3

২,৭৯৪.

What is the distance of the origin from the line 12x - 5y + 26 = 0?

ক
ক) 2
খ
খ) 3
গ
গ) 4
ঘ
ঘ) 5

ব্যাখ্যা

প্রদত্ত রেখার সমীকরণ 12x - 5y + 26 = 0
মূলবিন্দুর স্থানাংক (0,0)
এখানে,
A = 12 ,B = - 5 C = 26

নির্ণেয় দূরুত্ব = । Ax + By + C /√(A² + B²)।
= ।(12 × 0) - (5 × 0) + 26 / √{12²+ (- 5)²}।
= । 26/√169।
=। 26/13।
= 2

২,৭৯৫.

Which of the following is not equivalent to 25x² = y² - 4?

ক
x² = (y² - 4)/25
খ
25x² = (y + 2)(y - 2)
গ
75x² = 3y² - 12
ঘ
5x = y - 2

ব্যাখ্যা

Question: Which of the following is not equivalent to 25x² = y² - 4?

Solution:

25x² = y² - 4

= y² - 2²

= (y + 2) (y - 2)

25x² = y² - 4

⇒ 3 × 25x² = 3(y² - 4)

⇒ 75x² = 3y² - 12

25x² = y² - 4

⇒ x² = (y² - 4)/25

So, last option is not equivalent to 25x² = y² - 4

২,৭৯৬.

The area of the right angle triangle is 184 square cm one of its leg is 16 cm long. Find the length of the other leg.

ক
23 cm
খ
22 cm
গ
24 cm
ঘ
20 cm
ঙ
18 cm

ব্যাখ্যা

Area of the triangle = 1/2 × base × height
⇒ 184 = 1/2 × 16 × other leg
So,
other leg = (184 × 2)/16
= 23 cm

২,৭৯৭.

Two trains start at the same time from Chittagong and Sylhet and proceed towards each other at 80 km/h and 100 km/h, respectively. When they meet, it is found that one train has travelled 80 km more than the other. Find the distance between Chittagong and Sylhet.

ক
720 km
খ
520 km
গ
620 km
ঘ
600 km

ব্যাখ্যা

Question: Two trains start at the same time from Chittagong and Sylhet and proceed towards each other at 80 km/h and 100 km/h, respectively. When they meet, it is found that one train has travelled 80 km more than the other. Find the distance between Chittagong and Sylhet.

Solution:
Let the trains meet after t hours.

ATQ,
(100 × t) = (80 × t) + 80
⇒ 100t - 80t = 80
⇒ 20t = 80
∴ t = 4 hours

∴ Distance between Chittagong and Sylhet = (100 × 4) + (80 × 4)
= 400 + 320
= 720 km

∴ The distance between Chittagong and Sylhet is 720 km.

২,৭৯৮.

4.036 divided by 0.04 gives :

ক
ক) 1.009
খ
খ) 10.09
গ
গ) 100.9
ঘ
ঘ) 10.90

ব্যাখ্যা

4.036/ 0.04 =403.6/4= 100.9

২,৭৯৯.

A man purchased a shirt at Tk. 450 after availing a discount of 25%. What is the catalog price of the shirt?

ক
Tk. 490
খ
Tk. 530
গ
Tk. 560
ঘ
Tk. 590
ঙ
Tk. 600

ব্যাখ্যা

Question: A man purchased a shirt at Tk. 450 after availing a discount of 25%. What is the catalog price of the shirt?

Solution:
25% discount,
If the catalog price is Tk. 100 then the purchased = (100 - 25)
= Tk. 75

If the purchase price is Tk. 75, the catalog price = 100 taka.
If the purchase price is Tk. 1, the catalog price = 100/75 taka
If the purchase price is Tk. 450, the catalog price = (100 × 450)/75 taka
= Tk. 600

২,৮০০.

A rectangular block 6 cm by 12 cm by 15 cm is cut up into an exact number of equal cubes. Find the least possible number of cubes.

ক
ক) 30
খ
খ) 35
গ
গ) 40
ঘ
ঘ) 45

ব্যাখ্যা

Volume of the block = (6 × 12 × 15) cm³
= 1080 cm³

Side of the largest cube = H.C.F of 6 cm, 12 cm, 15 cm
= 3 cm.

Volume of this cube = (3 × 3 × 3) cm³
= 27 cm³

Number of cubes = 1080/27
= 40.