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

#### PQ is a tangent at a point C to a circle with centre O. If AB is a diameter and ∠CAB = 30°, find ∠PCA.

#### For what value of k will k + 9, 2k – 1 and 2k + 7 are the consecutive terms of an A.P. ?

#### A ladder, leaning against a wall, makes an angle of 60° with the horizontal. If the foot of the ladder is 2.5 m away from the wall, find the length of the ladder.

#### A card is drawn at random from a well shuffled pack of 52 playing cards. Find the probability of getting neither a red card nor a queen.

#### Question 1:PQ is a tangent at a point C to a circle with centre O. If AB is a diameter and ∠CAB = 30°, find ∠PCA.

#### Video Explanation

#### Explanatory Answer

AB is a diameter to the circle and O is the centre of the circle.

Join OC.

OA and OC are radii to the circle. So, OA = OC

∴ ∆ AOC is an isosceles triangle.

In an isosceles triangle, two sides and correspondingly two opposite angles are equal.

So, ∠OAC = ∠OCA = 30°

∠OCP is the angle between the tangent and the radius.

∴ ∠OCP = 90°

∠ACP = ∠OCP - ∠OCA = 90° – 30° = 60°

#### Question 2: For what value of k will k + 9, 2k – 1 and 2k + 7 be consecutive terms of an A.P?

#### Video Explanation

#### Explanatory Answer

k + 9, 2k - 1 and 2k + 7 are consecutive terms of an Arithmetic Progression.

Key Property: The difference between any two consecutive terms of an AP is the same.

So, (2k – 1) – (k + 9) = (2k + 7) – (2k -1)

k – 10 = 8 or k = 18

#### Question 3: A ladder, leaning against a wall, makes an angle of 60° with the horizontal. If the foot of the ladder is 2.5 m away from the wall, find the length of the ladder.

#### Video Explanation

#### Explanatory Answer

AC is the ladder and AB is the wall. C is the foot of the ladder which is 2.5 m away from the wall.

In right ∆ABC, AC is the hypotenuse

The trigonometric ratio of the adjacent side (BC) and hypotenuse (AC) is Cos θ

cos 60° = \\frac{BC}{AC})

cos 60° = \\frac{1}{2}) = \\frac{2.5}{AC})

or AC = 5 m

Length of the ladder = 5m

#### Question 4: A card is drawn at random from a well shuffled pack of 52 playing cards. Find the probability of getting neither a red card nor a queen.

#### Video Explanation

#### Explanatory Answer

Number of red cards in a pack = 26

Number of black queens = 2 (**Note**: We only have to remove black queens as red queens would have been removed along with the 26 red cards).

The remaining 52 – (26 + 2) = 24 cards are neither a red card nor a queen.

Probability = \\frac{\text{Number of ways of selecting neither a red nor a queen}}{\text{Number of ways of selecting 1 card from the pack}})

= \\frac{24}{52}) = \\frac{6}{13})