##### Section A contains 4 questions of 1 mark each. Scroll down for explanatory answer and video solution to all four 1-mark questions.

#### The probability of selecting a rotten apple randomly from a heap of 900 apples is 0.18. What is the number of rotten apples in the heap?

#### If a tower 30 m high, casts a shadow 10\\sqrt3) m long on the ground, then what is the angle of elevation of the sun?

#### If the angle between two tangents drawn from an external point P to a circle of radius 'a' and centre 'O', is 60

^{o}, then find the length of OP.#### What is the common difference of an A.P. in which a

_{21}– a_{7}= 84?

#### Question 1: The probability of selecting a rotten apple randomly from a heap of 900 apples is 0.18. What is the number of rotten apples in the heap?

#### Video Explanation

#### Explanatory Answer

The probability of selecting a rotten apple = 0.18

Total number of apples = 900

The probability of selecting a rotten apple = \\frac{\text{Number of rotten apples}}{\text{Total number of apples}})

Let the number of rotten apples be x.

Therefore, \\frac{x}{900}) = 0.18

x = 900 × 0.18 = 162

#### Question 2: If a tower 30 m high, casts a shadow 10\\sqrt3) m long on the ground, then what is the angle of elevation of the sun?

#### Video Explanation

#### Explanatory Answer

In the adjacent figure, AB is the tower and BC is the shadow.

θ is the angle of elevation.

tan θ = \\frac{\text{opposite side}}{\text{adjacent side}}) = \\frac{AB}{BC}) = \\frac{30}{10{\sqrt3}}) = \\frac{3}{\sqrt {3}} = {\sqrt {3}})

If tan θ = \\sqrt3), θ = 60°

The angle of elevation of the tower = 60°

#### Question 3: If the angle between two tangents drawn from an external point P to a circle of radius 'a' and centre 'O', is 60^{o}, then find the length of OP.

#### Video Explanation

#### Explanatory Answer

OAP and OBP are congruent triangles by RHS rule.

Explanation of the rule: OP is the common hypotenuse to both the right triangles OAP and OBP. One of the perpendicular sides of each of the triangles (OA and OB) are the radii to the circle. So, the third side of the two triangles, PA and PB will be equal. By SSS, the two triangles are congruent.

So, ∠APO = ∠BPO

Therefore, ∠APO = \\frac{∠APB}{2}) = \\frac{60}{2}) = 30^{o}

In right triangle OAP

sin 30^{o} = \\frac{\text{opposite side}}{\text{hypotenuse}}) = \\frac{OA}{OP})

OA is the radius = ‘a’

Therefore, sin 30^{o} = \\frac{1}{2}) = \\frac{a}{OP})

Or the length of OP = 2a

#### Question 4: What is the common difference of an A.P. in which a_{21} – a_{7} = 84?

#### Video Explanation

#### Explanatory Answer

Key Data: a_{21} – a_{7} = 84. The numbers are in an AP.

For an AP, the nth term a_{n} = a_{1} + (n – 1)d, where a_{1} is the first term of the AP and 'd' is its common difference.

So, a_{21} = a_{1} + (21 – 1)d = a_{1} + 20d and

a_{7} = a_{1} + (7 – 1)d = a_{1} + 6d

a_{21} – a_{7} = a_{1} + 20d – (a_{1} + 6d) = 84

Or 14d = 84

d = \\frac{84}{14}) = 6