ব্যাংক নিয়োগ প্রস্তুতি ⎯ লং কোর্স

Question: Find the maximum distance between two points on the perimeter of a rectangular garden whose length and breadth are 12 m and 5 m. 

Solution: 
 maximum distance = diagonal length 
= √(12² + 5² )
= √(144 + 25) 
= √169 
= 13 m 

Question: For which value of the angle of elevation, length of a stick and length of its' shadow is equal?

Solution: 

ধরি, খুঁটিটির দৈর্ঘ্য AB, ছায়ার দৈর্ঘ্য BC, উন্নতি কোণ θ
AB = BC

চিত্র হতে,
tanθ = AB/BC = 1
⇒ tanθ = tan45°
⇒ θ = 45°

অর্থাৎ, উন্নতি কোণ  45° হলে, খুঁটির দৈর্ঘ্য ও ছায়ার দৈর্ঘ্য সমান হবে।

Question: The angle of elevation of a tower becomes 60° from 45° by moving 60 metres towards a Minar. Find the height of the Minar.

Solution: 

let, height AB 
In triangle ABD, 
tan60 = AB/BD
⇒ √3 = AB/BD
⇒ BD = AB/√3

In triangle ABC, 
tan45 = AB/BC 
⇒ 1 = AB/BC
⇒ AB = BC 
⇒ AB = BD + DC
⇒  AB = BD + 60 
⇒ AB = (AB/√3) + 60 
⇒ AB - (AB/√3)  = 60 
⇒ AB . (√3 - 1)/√3 = 60
⇒ AB = 60√3/(√3 - 1)
= 60√3 (√3 + 1)/(√3 - 1)(√3 + 1)
= 30 (3 + √3) m

Question: In the following figure, ∠CAB = 90°, then ∠CDB =? 


Solution: 
বৃত্তস্থ চতুর্ভুজের বিপরীত কোণদ্বয়ের সমষ্টি ১৮০
⇒ ∠CAB + ∠CDB = 180 
⇒ 90 + ∠CDB  = 180 
⇒ ∠CDB = 180 - 90 
 ⇒ ∠CDB = 90°

Question: The area of a circle is increased by 22 square cm if its radius is increased by 1 cm. The original radius of the circle is -

Solution:
Let the original radius of the circle be r cm.

ATQ,
π(r + 1)² - πr² = 22
⇒ π{(r + 1)² - r²} = 22
⇒ π(r² + 2r + 1 -r²) = 22
⇒ 2r + 1 = 22/π
⇒ 2r + 1 = (22 × 7)/22
⇒ 2r + 1 = 7
⇒ 2r = 6
⇒ r = 3 cm

Question: What is the length of the diagonal of a square whose area is 4 times of another square with diagonal as 5√2 cm?

Solution:
Area of square = (1/2) × (length of diagonal)²

Area of square₂ =(1/2) × (5√2)²  
= 25 cm²

Area of square₁ = 4 × 25 = 100 cm²

Length of diagonal of square₁ = √(2 × area)
= √(2 × 100)
= 10√2 cm

Question: From a circular sheet of paper with a radius of 20 cm, four circles of radius 5 cm each are cut out. What is the ratio of the uncut to the cut portion?

Solution:
Area of the sheet of paper with a radius of 20 cm. = π(20)² = 400π cm²
Area of 4 circles of radius 5 cm. = 4 × π(5)²=100π cm²
Area of remaining portion = 400π - 100π = 300π cm²
Therefore, the required ratio = 300π : 100π = 3 : 1

Question: A pole of 66 metre long breaks into two parts without complete separation and makes an angle 30° with the ground. Find the length of the broken part of the pole.

Solution: 


sin30 = x/(66 - x)
⇒ 1/2 = x/(66 - x) 
⇒ 66 - x = 2x 
⇒ 3x = 66
⇒ x = 66/3 = 22

the length of the broken part of the pole = 66 - 22 = 44 m

Question: A circle touches all four sides of a quadrilateral PQRS. If PQ = 11 cm. QR = 12 cm and PS = 8 cm. Then what is the length of RS?

Solution: 

If a circle touches all four sides of quadrilateral PQRS then,
PQ+ RS = SP+ RQ

So,
11 + RS = 8+ 12
⇒ RS = 20 - 11
⇒ RS = 9

Question: A parallelogram has a base of 30m and  height is 10m. Then its area is-

Solution: 
area = base × height
= 30 × 10
= 300 m²

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: The length of a rectangle is twice its breadth. If its length is decreased by 5 cm and breadth is increased by 5 cm, the area of the rectangle is increased by 75 sq. cm. Find the length of the rectangle. 

Solution: 
let, breadth x, length is 2x 
Area = 2x × x 
= 2x² 

New area = (2x - 5) × (x + 5)
= 2x² - 5x + 10x - 25 
= 2x² + 5x - 25 

2x² + 5x - 25 - 2x² = 75 
⇒ 5x = 75 + 25 = 100 
⇒ x = 100/5 = 20 

length = 2 × 20 
= 40 m

Question: The distance between two parallel tangents of a circle is 20 cm, then the radius of the circle is-

Solution: 
Distance between two parallel tangents = 20 cm
That means, diameter = 20 cm
Therefore, the radius of the circle = 20/2
= 10 cm

Question: The length of two parallel sides of a trapezium are 30 cm and 60 cm respectively, and the distance between the parallel sides is 8 cm. Find the area of the trapezium.

Solution: 
Area of the Trapezium = (1/2) × (Sum of the parallel sides) × (Distance between parallel sides)
= (1/2) × (30 + 60) × 8
= (1/2) × 90 × 8
= 360 cm²

∴ Area of the Trapezium = 360 cm²

Question: The area of a rectangle is 252 cm² and its length and breadth are in the ratio of 9 : 7 respectively. What is its perimeter?

Solution: 
length, breadth 9x, 7x 

9x × 7x = 252
⇒ 63x² = 252
⇒ x² = 252/63 = 4
⇒ x = 2

perimeter = 2 (9x + 7x)
= 2 × 16x
= 32x
= 32 × 2
= 64 cm

Question: The area of a square inscribed in a circle is 140 cm². What is the area of the semi-circle?

Solution:
The area of a square inscribed in a circle is 140 cm²
side of square = √140 cm
= 2√35 cm

diagonal of the square = √2 × 2√35
= 2√70 cm

diameter of circle = 2√70 cm
radius of the circle = √70 cm
∴ area of the circle = π (√70)2 cm²
= (22/7) × 70 cm²
= 220 cm²

area of semi-circle = 220/2 
= 110 cm²

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

Solution: 

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

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

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 

Question: Angles of a quadrilateral are in the ratio 3 : 4 : 5 : 8. The largest angle is -

Solution:
Let First angle = 3x
Second angle = 4x
Third angle = 5x
and fourth angle = 8x
We know 3x + 4x + 5x + 8x = 360°
⇒ 20x = 360°
⇒ x = 18°

∴ Measure of largest angle = 8x
= (8 × 18°)
= 144°

Question: The angle of elevation of the top of a building from a point on the ground, which is 30 m away from the foot of the building, is 30°. The height of the building is:

Solution: 

let, x is the height of the building.

Hence,
tan 30° = perpendicular/base = x/30
⇒ 1/√3 = x/30
⇒ x = 30/√3 m
⇒ x = (30√3)/(√3 × √3)
⇒ x = (30√3)/3
⇒ x = 10√3 m

Question: The difference between the circumference and the radius of a circle is 185 cm. Find the diameter of the circle.

Solution:
Let r be the radius of circle

Given that,
2πr - r = 185
⇒ r(2π - 1) = 185
⇒ r{(44/7) - 1} = 185
⇒ r (44 - 7)/7 }= 185
⇒ r(37/7) = 185
⇒ r = 185 (7/37)
∴ r = 35

The radius of the circle is 35 cm.
∴ Diameter = 2 × 35 
= 70 cm

প্রশ্ন: A wire can be bent in the form of a circle of radius 7cm. If it is bent in the form of a square, then what will be its area?

সমাধান: 
দেওয়া আছে,
বৃত্তের ব্যাসার্ধ r = 7 cm 
বৃত্তের পরিধি = 2πr 
= 2 × (22/7) × 7 
= 2 × 22 × 1
= 44 cm 

বর্গের এক বাহুর দৈর্ঘ্য = 44/4 cm 
= 11 cm 

∴ বর্গের ক্ষেত্রফল = (11)² cm² 
= 121 cm²