Bank Math Master

Question: If the difference between the circumference and diameter of a circle is 90 cm, then the diameter of the circle is -

Solution:
ধরি,
বৃত্তের ব্যাসার্ধ = r
বৃত্তের ব্যাস = 2r
বৃত্তের পরিধি = 2πr

প্রশ্নমতে,
2πr - 2r = 90
⇒ 2r(π - 1) = 90
⇒ r = (90/2){(22/7) - 1}
⇒ r = 45/(22 - 7)/7
⇒ r = (45 × 7)/15
∴ r = 21

∴ বৃত্তের ব্যাস = 2r = 2 × 21 = 42 সে.মি.

Question: If the one third of one fourth of a number is 15, then what is the 3/10 of the number?

Solution:
Let the number be x

ATQ,
(1/3) × (1/4) × x = 15
⇒ 1/12 × x = 15
x = 180

Now,
3x/10 = (3/10) × 180
∴ 3x/10 = 54

Question: A, B and C are partners of a company. During a particular year A received one-third of the profit, B received one-fourth of the profit and C received the remaining Tk. 5000. How much did B receive?

Solution:
ধরি,
মোট লাভ = x টাকা

প্রশ্নমতে,
x - (x/3 + x/4) = 5000
⇒ x - (4x + 3x)/12 = 5000
⇒ x - 7x/12 = 5000
⇒ (12x - 7x)/12 = 5000
⇒ 5x/12 = 5000
⇒ 5x = 5000 × 12
⇒ x = (5000 × 12)/5
∴ x = 12000

∴ B পায় = (12000 × 1/4) টাকা = 3000 টাকা

Question: What is the probability of getting a sum 9 from two throws of a dice?

Solution:
Total events = 6 × 6 = 36
Set of expected events = {(3, 6), (4, 5), (5, 4), (6, 3)}

∴ Probability = 4/36 = 1/9

Question: Having scored 98 runs in the 19th innings a cricketer increase his average score by 4.what will be his average score after the 19th innings?

Solution:
Let, his average score is x
His total score after 19th innings = 19x

ATQ,
(19x - 98)/18 = x - 4
⇒ 19x - 98 = 18x - 72
⇒ 19x - 18x = 98 - 72
∴ x = 26

His average score is 26

Question: A tank is 30% full with water. If 18 liters of water is added the tank become 3/4 full. What is the capacity of the tank?

Solution:
Let, Capacity of tank is x Liters.

ATQ,
30% of x + 18 = (3/4) × x
⇒ (30x/100) + 18 = 3x/4
⇒ (3x/10) + 18 = 3x/4
⇒ (3x/4) - (3x/10) = 18
⇒ (15x - 6x)/20 = 18
⇒ 9x = 18 × 20
⇒ 9x = 360
∴ x = 40

∴ Capacity of tank is 40 Liters.

Question: The difference of two numbers is 12. If 1 is added to the greater number, it becomes twice the smaller number. Calculate the two numbers?

Solution:
ধরি,
বড় সংখ্যাটি = x
ছোট সংখ্যাটি = y

১ম শর্তমতে,
x - y = 12 ............................. (1)

২য় শর্তমতে,
x + 1 = 2y ............................. (2)

(1) নং হতে (2) নং সমীকরণ বিয়োগ করে পাই,
x - y - x - 1 = 12 - 2y
বা, - y = - 13
∴ y = 13

(1) নং হতে পাই,
x - 13 = 12
বা, x = 12 + 13
∴ x = 25

∴ সংখ্যা দুটি 25 ও 13

Question: A man can row at the rate of 4 km/hr in still water. If the time taken to row a certain distance upstream is 3 times as much as to row the same distance downstream, find the speed of the current?

Solution:
Let,
speed of stream x
Time taken in downstream = t
Time taken in downstream = 3t

ATQ.
Distance in upstream = Distance in downstream
(4 - x)3t = (4 + x)t
⇒ (4 - x)3 = (4 + x)
⇒ 12 - 3x = 4 + x
⇒ 4x = 8
∴ x = 2

∴ Speed of stream = 2 km/hr

Question: 400 grams of sugar solution has 30% sugar in it. How much sugar should be added to make 50% in the solution?

Solution:
Sugar = 400 × 30% = 120 grams
Let,
We have to add x grams 

ATQ,
120 + x = 50% (400 + x)
⇒ 120 + x = (50/100) × (400 + x)
⇒ 120 + x = (1/2) × (400 + x)
⇒ 240 + 2x = 400 + x
∴ x = 160

Question: After paying 10 % tax on all income over 3000 Kawser had a net income of Tk 12000. Kawser's income before tax was.

Solution:
Let Kawser income was x.
Taxable income = x - 3000
∴ Tax paid = 10%(x - 3000)

ATQ,
x - 10%(x - 3000) = 12000
⇒ x - {10(x - 3000)/100} = 12000
⇒ (10x - x + 3000)10 = 12000
⇒ 9x + 3000 = 120000
⇒ 9x = 120000 - 3000
⇒ 9x = 117000
⇒ x = 117000/9
∴ x = 13000

Kawser income was Tk.13000

Question: ∠B is the right angle of a right angles triangle ABC. If tanA = 1, then 4sinACosA = ?

Solution:
 
দেওয়া আছে,
tanA = 1
ধরি,
বিপরীত বাহু = সন্নিহিত বাহু = a
অতিভুজ = √(a² + a²) = √2 a

∴ sinA = a/√2a = 1/√2
cosA =  a/√2 a = 1/√2

∴ 4sinACosA = 4 × (1/√2) × (1/√2)
= 4 × 1/2
= 2

Question: sin(A + 12°) = 1/2, find the value of A?

Solution:
sin(A + 12°) = 1/2
⇒ sin(A + 12°) = sin30°
⇒ A + 12° = 30°
⇒ A = 30° - 12°
∴ A = 18°

Question: If P = {x ∈ N : 2 < x ≤ 6 and x is natural number} and Q = {x ∈ N : x even number and x ≤ 8} then (P - Q) = ?

Solution:
দেওয়া আছে,
P = {x ∈ N : 2 < x ≤ 6 এবং x স্বাভাবিক সংখ্যা}
∴ P = {3, 4, 5, 6}

Q = {x ∈ N : x জোড় সংখ্যা এবং x ≤ 8}
∴ Q = {2, 4, 6, 8}

এখন,
(P - Q) = {3, 4, 5, 6} - {2, 4, 6, 8}
= {3, 5}

Question:  then x = ?

Solution:
x/√128 = √162/x
⇒ x² = √(128 × 162)
⇒ x² = √(2 × 64 × 2 × 81)
⇒ x² = 2 × 8 × 9
⇒ x² = 144
⇒ x = √144
∴ x = 12

Question: A tree leaned due to storm. The stick with height of 7 meter from its foot was leaned against the tree to make it straight. If the angle of depression at the point of contacting with the stick on the ground is 30°, find the length of the stick.

Solution:

মনে করি,
খুঁটিটির দৈর্ঘ্য BC = x মিটার,
গাছের গোড়া থেকে AB = 7 মিটার উচ্চতায় খুঁটিটি ঠেস দিয়ে আছে এবং অবনতি ∠DBC = 30°
∠ACB = ∠DBC = 30° [একান্তর কোণ বলে]

সমকোণী ΔABC থেকে পাই,
sin∠ACB = AB/BC
বা, sin30° = 7/x
বা, 1/2 = 7/x
∴ x = 14

∴ খুঁটিটির দৈর্ঘ্য 14 মিটার।

Question: A committee of 5 members is to be formed by selecting out of 8 men and 6 women. In how many different ways the committee can be formed if it should have at least 4 men? 

Solution:
      Men (8)       Women (6)
1)    4                     1
2)    5                     0

From (1) Number of ways = ⁸C₄ × ⁶C₁ = 70 × 6 = 420
From (2) Number of ways = ⁸C₅ × ⁶C₀ = 56 × 1 = 56

Total number of ways = 420 + 56 = 476

Question: If secθ + tanθ = x, then cotθ is - 

Solution:
দেওয়া আছে,
secθ + tanθ = x ................. (1)

আমরা জানি,
sec²θ - tan²θ = 1
বা, (secθ + tanθ)(secθ - tanθ) = 1
বা, x(secθ - tanθ) = 1
বা, secθ - tanθ = 1/x ................ (2)

(1) - (2) হতে পাই,
(secθ + tanθ) - (secθ - tanθ) = x - (1/x)
বা, 2tanθ = (x² - 1)/x
বা, tanθ = (x² - 1)/2x
∴ cotθ = 2x/(x² - 1)

Question: The volume of a cube is numerically equal to the sum of its edges. What is its total surface area in square units?

Solution: 
Let,
Length of one side = a
Volume = a³
Total sides = 12

ATQ
a³ = 12a
∴ a² = 12

∴ Total surface = 6a²
= 6 × 12
= 72

Question: If a pole 6 m high casts a shadow 2√3 m long on the ground, then the elevation of the sun is -

Solution:

ধরি,
AB = 6, BC = 2√3

ABC সমকোণী ত্রিভুজ হতে পাই,
tanθ = AB/BC
⇒ tanθ = 6/2√3
⇒ tanθ = 3/√3
⇒ tanθ = √3
⇒ tanθ = tan60°
∴ θ = 60°