Bank Math

ধরি, f(x) = 3x² + ax + a +3
যেহেতু, x + 2 দ্বারা বিভাজ্য সেহেতু,
f(-2) = 0 হবে
⇒ 3(-2)² + a(-2) + a + 3 = 0
⇒ 12 - 2a + a + 3 = 0
⇒ - a + 15 = 0
⇒ a = 15

The length of arc, s = πrθ/180 = π × 6 × 30/180 = π cm

Selling price = 50 - (50 × 20%)
= 50 - 10
= 40
Cost Price = (100/100+25) × 40
= (100/125) × 40
= 32

Curved surface এর ক্ষেত্রফল = 2Πrh
ব্যাসার্ধ হলে নতুন ব্যাসার্ধ = r/2
এবং দৈর্ঘ্য দ্বিগুণ হলে নতুন দৈর্ঘ্য = 2h
∴ নতুন ক্ষেত্রফল = 2Π(r/2)2h = 2Πrh.
সুতরাং ক্ষেত্রফল একই থাকবে।

Question: If p is a positive integer, what is the smallest possible value of p such that 2160 × p is a perfect square?

Solution:
আমরা জানি, একটি সংখ্যা পূর্ণবর্গ হতে হলে এর মৌলিক গুণনীয়কের ঘাতসমূহ জোড় সংখ্যা হতে হবে।

2160 = 2 × 2 × 2 × 2 × 3 × 3 × 3 × 5
= 2⁴ × 3³ × 5¹

2160p = 2⁴ × 3³ × 5¹ × p
এখানে, 2-এর ঘাত = 4 (জোড়), 3-এর ঘাত = 3 (বিজোড়), 5-এর ঘাত = 1 (বিজোড়)

পূর্ণবর্গ করতে হলে সব ঘাত জোড় হতে হবে।
তাই p = 3 × 5 = 15 হলে,

2160 × 15 = 2⁴ × 3³ × 5¹ × (3 × 5) = 2⁴ × 3⁴ × 5²
যেহেতু সব মৌলিক উৎপাদকের ঘাত জোড়, তাই এটি একটি পূর্ণবর্গ সংখ্যা।

সুতরাং, p = 15 হলে, 2160 × p পূর্ণবর্গ সংখ্যা হয়।

Question: A worker earns Tk. 250 on the first day and spends Tk. 200 on the second day, earns Tk. 250 on the third day and again spends Tk. 200 on the fourth day and so on. On which day would he have had Tk. 1000?

Solution:
১ম দিনে আয় করে ২৫০ টাকা।
২য় দিনে ব্যয় করে ২০০ টাকা।

∴ ২ দিনে তার জমা থাকে (২৫০ - ২০০) = ৫০ টাকা।

এখন, (১০০০ - ২৫০) = ৭৫০ টাকা।

৫০ টাকা জমা থাকে ২ দিনে
১ টাকা জমা থাকে ২/৫০ দিনে
৭৫০  টাকা জমা থাকে (২ × ৭৫০)/৫০ দিনে
= ৩০ দিন।

৩০ দিন পর তার হাতে থাকে ৭৫০ টাকা
এবং ৩১ তম দিনে সে আয় করে ২৫০ টাকা।
তাহলে মোট টাকা হয় (৭৫০ + ২৫০) = ১০০০ টাকা,

সুতরাং ৩১ দিনে তার কাছে ১০০০ টাকা ছিল।

Let the ten's digit be x.
Then, the unit's digit = x + 2.
Number = 10x + (x + 2) = 11x + 2
Sum of digits = x + (x + 2) = 2x + 2.
⇒ (11x + 2)(2x + 2) = 144
⇒ 22x² + 26x- 140 = 0
⇒ 11x² + 13x - 70 = 0
⇒ (x - 2)(11x + 35) = 0
⇒ x = 2
Hence, Required Number = 11x + 2 = 24

Question: If ⁿC₇ = ⁿC₅ , then find the value of n.

Solution:
We know, 
If ⁿC_r = ⁿC_s , then, n = r + s

Now, ⁿC₇ = ⁿC₅ 
∴ n = 7 + 5
= 12

Question: The present age of a son is one-fourth of the present age of his father. Eight years ago,  the father's age was seven times the age of his son. What is the present age of the son?

Solution: Let the father's age be 4x years
Then, Son's age = x years

ATQ,
4x - 8 = 7(x - 8)
or, 4x - 8 = 7x - 56
or, -8 + 56 = 7x - 4x
or, 48 = 3x
or, x = 16

∴ The present age of the son = 16 years.

Question: P is a girl and has the same number of brothers as sisters. Q is a boy and has thrice as many sisters as brothers. P and Q are the children of R. How many children does R have?

Solution: Let d and s represent the number of daughters and sons respectively.
Then, we have
d - 1 = s
⇒ d = s + 1
and
3(s - 1) = d
⇒ 3s - 3 = s + 1
⇒ 3s - s = 1 + 3
⇒ 2s = 4
⇒ s = 2

∴ d = 2 + 1 = 3 

∴ Total children = 2 + 3 = 5 

Here the points line are (1, 13) and (-3, 6)
∴ Slope, m = Y₁ - Y₂ / X₁ - X₂
= (13 - 6)/(1 + 3)
= 7/4
= 1.75

Let x be the number of boys and y be the number of girls.
Given total number of boys and girls = 100

x + y = 100 --------- (i)
A boy gets Tk. 3.60 and a girl gets Tk. 2.40
The amount given to 100 boys and girls = Tk. 312

3.60x + 2.40y = 312 ---------- (ii)

Solving (i) and (ii)
3.60x + 3.60y = 360 ---------- Multiply (i) by 3.60
3.60x + 2.40y = 312 ------------ (ii)

Equation (i) - (ii)
1.20y = 48
⇒ y = 48/1.20
⇒ y = 40

Number of girls = 40.

Question: What is the total interest on Tk. 1,200 at 10% per annum for 9 months?

Solution: 
Given,
Principal (P) = Tk. 1200
Rate (R) = 10%
Time (T) = 9 months
= 9/12 year
= 3/4 year

By Formula,
SI = PRT/100
= {1200 × 10 × (3/4)}/100
= (1200 × 10 × 3)/400
= 90 Tk

Question: Given that a square and a rectangle share the same area, and the square’s perimeter is 32 meters, while the rectangle’s length is 4 meters, calculate the rectangle’s width.

Solution:
Given that,
Perimeter of square = 32 m
Area of rectangle = Area of square
And length of rectangle = 4 m

Now, 
Perimeter of square,
4s = 32
⇒ s = 32/4 = 8
∴ s = 8 m
∴ Area of square = s² = 82 = 64 m²

 According to the Question,
Area of rectangle = Area of square
∴ Area of rectangle = 64 m²

∴ Area of rectangle = length × width
64 = 4 × w
⇒ w = 64/4
∴ w = 16 m

So the width of the rectangle = 16 meters

Let, the total capacity of the tank be 96 units (LCM of 24 & 32).

A fills = 96/24 = 4 units / minute.
B fills = 96/32 = 3 units / minute

Let the time for which B is opened is x minutes.
According to the question, 
4 × 9 + 3 × x = 96
3x = 96 - 36
3x = 60
x = 20 minutes.
The pipe B should be closed after 20 minutes

C.I. when interest compound yearly = Tk. [5000 × (1 + 4/100) (1 + 1/2 × 4/100)]
= Tk. 5304.
C.I. when interest is compounded half-yearly = Tk. 5000 (1 + 2/100)³
= Tk. 5306.04
Difference = Tk. (5306.04 - 5304)
= Tk. 2.04

Question: The ratio of total surface area to curved surface area of a cone whose radius is 5cm and height 12 cm is :

Solution:
দেওয়া আছে,
কোণকের ব্যাসার্ধ, r = 5 cm
কোণকের উচ্চতা, h = 12 cm

এখন,কোণকের হেলানো উচ্চতা (slant height) বের করতে হবে।
কোণকের হেলানো উচ্চতা ((slant height), l = √(r² + h²)
= √(5² + 12²)
= √(25 + 144)
= √(169)
= 13 cm

  এখন, কোণকের সমগ্র পৃষ্ঠতলের ক্ষেত্রফল ও বক্রতলের ক্ষেত্রফলের অনুপাত নির্ণয় করি।
  সমগ্র পৃষ্ঠতলের ক্ষেত্রফল (total surface area) = πrl + πr²
  বক্রতলের ক্ষেত্রফল (curved surface area) = πrl

  অনুপাত = ( πrl + πr² ) : ( πrl )
 =  πr(l + r) : πrl
= (l + r) : l   (উভয় পক্ষকে πr দ্বারা ভাগ করে)
= (13 + 5) : 13
= 18 : 13

∴ কোণকের সমগ্র পৃষ্ঠতলের ক্ষেত্রফল ও বক্রতলের ক্ষেত্রফলের অনুপাত = 18 : 13.

Total marks obtained by the student
= 55% of 800 = (55/100)×800 = 440

∴ Marks scored in English
= 15% of 440 = (15/100)×440 = 66

Question: A worker earns Tk. 300 on the first day and spends Tk. 150 on the second day, earns Tk. 300 on the third day and again spends Tk. 150 on the fourth day, and so on. On which day would he have had Tk. 1500?

Solution:
1ম দিনে আয় = 300 টাকা
2য় দিনে ব্যয় = 150 টাকা
∴ প্রতি 2 দিনে জমা হয় = 300 - 150 = 150 টাকা

শুধু 1ম দিনে আয় করায় হাতে থাকে = 300 টাকা
অতএব, 1500 - 300 = 1200 টাকা আরও জমা করতে হবে।

150 টাকা জমা হয় 2 দিনে,
∴ 1200 টাকা জমা হয় = (2 × 1200)/150 = 16 দিনে

অর্থাৎ, 16 দিনের শেষে জমা থাকবে = 1200 টাকা

17-তম দিনে আবার আয় হবে = 300 টাকা
∴ মোট সঞ্চয় হবে = 1200 + 300 = 1500 টাকা

∴ 17-তম দিনে তার কাছে মোট 1500 টাকা জমা থাকবে।

Question: Find the angle of elevation of the sun if the length of a tree's shadow is √3 times its actual height.

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

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

Question: It was Friday on January 1, 2016. What was the day of the week on January 1, 2017?

Solution: 
The year 2016 is a leap year. So, it has 2 odd days.
Given,
1st day of the year 2016 is Friday
So, 1st day of the year 2017 is 2 days beyond Friday.
Friday + 1 day = Saturday
Saturday + 1 day = Sunday
Hence, it will be Sunday.

Question: What is the sum of all two-digit numbers that gives a remainder of 3 when they are divided by 7?

Solution: 
general formula for that number = 7n + 3  

n = 1, then the number is = 7 + 3 = 10 
n = 2, then the number is =14 + 3 = 17
.
.
.
n= 13,  then the number is = 94

sum  = 10 + 17 + ... + 94 
= 13 (10 + 94)/2 
= 676 

Question: Equation of the straight line parallel to X-axis and also 4 units below X-axis is:

Solution: 
Equation of the straight line parallel to X-axis and also 4 units below X-axis is y = - 4

Question: The sum of three consecutive even integers is 30 more than the first of the numbers. What is the middle number?

Solution:
ধরি, তিনটি ক্রমিক জোড় সংখ্যা হলো x, x + 2, x + 4.

প্রশ্নমতে,
x + (x + 2) + (x + 4) = x + 30
⇒ 3x + 6 = x + 30
⇒ 3x − x = 30 − 6
⇒ 2x = 24
⇒ x = 24/2
⇒ x = 12

∴ প্রথম সংখ্যা = x = 12
দ্বিতীয় সংখ্যা (মাঝের সংখ্যা) = x + 2 = 14
তৃতীয় সংখ্যা = x + 4 = 16

∴ মাঝের সংখ্যাটি হলো = 14

Question: If f(y) = y³ + ky² - 4y - 8 , then for what value of k will f(- 2) = 0 ?

Solution: 
Given that, 
f(y) = y³ + ky² - 4y - 8 
f(- 2) = (- 2)³ + k(- 2)² - 4(- 2) - 8
= - 8 + 4k + 8 - 8
∴ f(- 2) = 4k - 8

Set f(- 2) = 0
⇒ 4k - 8 = 0
⇒ 4k = 8
⇒ k = 8/4
∴ k = 2

So the value of k is 2.

Question: The difference between two numbers is 960. When the larger number is divided by the smaller, the quotient is 5 and the remainder is 12. What is the smaller number?

Solution:
Given that,
The difference of two numbers = 960
Quotient when the larger number is divided by the smaller number = 5
Remainder when the larger number is divided by the smaller number = 12

Now,
Let the smaller number be x.
Larger number = 5x + 12

ATQ,
⇒ 5x + 12 - x = 960
⇒ 4x + 12 = 960
⇒ 4x = 960 - 12
⇒ 4x = 948
⇒ x = 948/4
∴ x = 237

So the smaller number is 237.

Let work is done by Rizvi in 1 day = 1/R & Shafi in 1/S day
Both complete work in 24 days. So in 1 day, together they complete = 1/24 = 1/R + 1/S
For 20 days both work together, so work done by them = 20(1/R + 1/S) = 5/6
Remaining work = {1 - (5/6)= 1/6} is done by Rizvi alone in 6 days
Work done by Rizvi in 6 days = 1/6 = 6(1/R)
∴ R = 36 = days needed by Rizvi to complete the work alone

According to the question,
∴ (1/36) + 1/S = 1/24
1/S = (1/24) - (1/36)
1/S = (3 - 2)/72
1/S = 1/72
S = 72 days needed by Shafi to complete the work alone.

Question: Which is the larger between two numbers if they are in the ratio of 6 : 13 and their least common multiple is 312?

Solution: 
Let, the numbers be 6x, 13x 
HCF is x and LCM is 312 

6x × 13x = 312 × x
⇒ x = 312/(6 × 13)
= 4 

Larger number = 13 × 4 = 52 

x + 2y = 2
y = (2 - x)/2

If x = 0, then,
y = (2 - 0)/2
= 2/2
= 1

Hence, x + 2y = 2 cuts the y-axis at (0, 1).

When A’s income is Tk 80 B’s Income is Tk 100
when A’s income is Tk 1 B’s Income is Tk 100/80
when A’s income is Tk 100 B’s Income is Tk (100/80) × 100 = 125%

So, A’s income is 125 - 100 = 25% more

Short-cut,
A’s income is 20% less than B’s
r = 20%
(r/100 - r) × 100
(20/100) × 100
= 25%

Let the new S.P. be Tk. x
Then,80:9=105:x
⇒x=(9×105/ 80)=11.81

Question: Abir is paid an hourly wage totalling Tk 500 for 50 hours of work in a week. If his hourly wage increases by 20% and he decides to work 20% fewer hours each week, how much will Abir be paid in a week ? 

Solution: 
Here, 
Hourly wage = 500/50 = 10

The hourly wage increases by 20%; 
Then, the present hourly wage = 10 + (10 × 20%)
= 10 + [10 × (20/100)] 
= (10 + 2) Tk
= 12 Tk/hour

The work hour decreases by 20%;
Then, the present work hour = 50 - (50 × 20%)
= 50 - [50 × (20/100)]
= (50 - 10) hour
= 40 hour

∴ Present weekly income = 12 × 40 = 480 Tk.

Let the number be P.
So,
P ÷ 44 = 432
⇒ P = 432 * 44
= 19008

P/31 = 19008 / 31
= 613, Remainder = 5

Question: What decimal of an hour is 45 seconds?

Solution:
1 hour = 60 × 60 = 3600 seconds

∴ Required decimal
= 45/3600
= 1/80
= 0.0125

Fathers age when his son was born= son's present age
Fathers age when his son was born + son's present age = father's present age
Therefore, 2 (Son's present age) = 38
=> Son's present age = 19
=> Son's age five years back = 19 − 5 = 14

Question: The least number, which when divided by 12, 15, 20 and 54 leaves in each case a remainder of 8 is-

Solution:
The number leaves a remainder 8 when divided by 12, 15, 20 and 54.
So the required number = LCM(12, 15, 20, 54) + 8

Now, 
12 = 2 × 2 × 3
15 = 3 × 5
20 = 2 × 2 × 5
54 = 2 × 3 × 3 × 3

∴ LCM(12, 15, 20, 54) = 540

∴ Required Number = 540 + 8 = 548 

Let S be the sample space
Then,
n(S) = ⁵²C₂
= 52 × 51/2 × 1
= 1326
Let E = event of getting 2 kings out of 4.
n(E) = 4C2
=> 4 × 3/2 × 1
P(E) = n(E)/n(S)
= 6/1326
= 1/221

Question: If x is doubled and y is tripled in the expression z = (4x/y), then the value of z is _____.

Solution: 
x এর দ্বিগুণ = 2x
y এর দিগুণ = 3y

এখন
z = (4 × 2x/3y)
 = 8x/3y
= (4x/y) × (2/3)

The value of z is multiplied by a factor 2/3.

Question: Find the value of 3(p + 5) - 2(2p - 3) + p

Solution: Given that,
3(p + 5) - 2(2p - 3) + p
= 3p + 15 - 4p + 6 + p
= (3p - 4p + p) + (15 + 6)
= 0 + 21
= 21

Question: Ashik buys a field of agricultural land for Tk. 3,60,000. He sells one-third at a loss of 20% and two-fifths at a gain of 25%. At what price must he sell the remaining field so as to make an overall profit of 10%?

Solution:
 Selling price of total agricultural field at a profit at 10% = 3,60,000 + 10% of 3,60,000
= 3,60,000 + 36000
= 396000

 Selling price of 1/3 agricultural field at a loss of 20% = (3,60,000/3) × (80/100)
= 120000 × (80/100)
= 96000

Selling price of 2/5 agricultural field at a profit of 25% = {(2 × 3,60,000)/5} × (125/100)
= 144000 × (125/100)
= 180000

Sell price of the remaining field = {396000 - (96000 + 180000)}
= 120000 tk 

Relative speed = (45 + 30) km/hr.
= 75 km/hr.
= 75 × (5/18) km/hr.
= (125/6) m/s.
We are calculating the time taken by the slower train to pass the driver of the faster one.
Hence, distance = length of the slower train = 500 meter.
Time = 500/(125/6)
= 500 × (6/125)
= 24 seconds.

Question: - 5x - [4y - {9x - (3y - 7x)}] simplifies to 

Solution:
- 5x - [4y - {9x - (3y - 7x)}]
= - 5x - [4y - {9x - 3y + 7x}]
= - 5x - [4y - 9x + 3y - 7x]
=  - 5x - [7y - 16x]
= - 5x - 7y + 16x
= 11x - 7y

Question: In a class of 92 students, 40 are taking English, 24 are taking Arabic and 10 are taking both courses. How many students are not enrolled in either course?

Solution:
Total students = 92
Students taking English n(E) = 40
Students taking Arabic n(A) = 24
Students taking both English and Arabic = 10

We know,
n(E ∪ A) = n(E) + n(A) - n(E ∩ A)
n(E ∪ A) = 40 + 24 - 10 = 54

∴ Not enrolled = Total students - n(E ∪ A) = 92 - 54 = 38