Correct Answer: 90°

Solution :
In $\triangle$PQR, PQ = PR
⇒ $\angle$Q = $\angle$R = b
PM is the angle bisector of $\angle$P.
$\angle$QPM = $\angle$RPM = a
Apply angle sum property in $\triangle$PQR,
⇒ $\angle$P + $\angle$Q + $\angle$R = 180°
⇒ $\angle$P + b + b = 180°
⇒ $\angle$P = 180° – 2b
$\angle$QPM = $\frac{180° – 2b}{2}$ = 90° – b
In $\triangle$PQM,
Let $\angle$PMQ = $\theta$
⇒ $\angle$QPM + $\angle$Q + $\angle$M = 180°
⇒ 90° – b + b + $\theta$ = 180°
⇒ $\theta$ = 90°
Hence, the correct answer is 90°.

Correct Answer: 68%

Solution : The number of students who passed
= 80% of 20 + 60% of 30
= $\frac{80 × 20}{100}$ + $\frac{60 × 30}{100}$
= 16 + 18
= 34
Now, the pass percentage of the entire class
= $\frac{\text{Number of students passed}}{\text{Total students}}$ × 100
= $\frac{34}{(20 + 30)}$ × 100
= 68%
Hence, the correct answer is 68%.

Correct Answer: GF

Solution : Let's check the options –
First option: AG; A→1, G→7, A + G = 1 + 7 = 8
Second option: CE; C→3, E→5, C + E = 3 + 5 = 8
Third option: BF; B→2, F→6, B + F = 2 + 6 = 8
Fourth option: GF; G→7, F→6, G + F = 7 + 6 = 13

The fourth option does not follow the pattern of the other options. Hence, the fourth option is correct.

Correct Answer: It contains the maximum content of water vapour.

Solution : The correct option is It contains the maximum content of water vapour.

Nitrogen makes up 78% of the gas in the air, oxygen 21%, followed by argon and carbon dioxide in trace amounts. Water vapour is present in the air in varying amounts, along with dust particles. At a specific pressure and temperature, the air is considered to be saturated when it contains the maximum amount of water vapour. It is found that the relative humidity is 100% in such a condition.

Correct Answer:

Solution : Since the rotation is restricted. So, the embedded figure will have the same orientation as the main figure. By comparison of all the option figures, the given question figure is embedded only in the third option figure.

Hence, the third option is correct.

Correct Answer: Friday

Solution : Given:
The 20th of March 2001 is Tuesday.

Number of days from 21st of March 2001 to 20th of March 2021 –
365 × 20 = 7300; add 5 days for leap years (2004, 2008, 2012, 2016, and 2020) = 7300 + 5 = 7305
Number of days from 21st of March 2021 to 23rd of April 2021 –
March = 11; April = 23
Total number of days: 7305 + 11 + 23 = 7339
The remainder after dividing 7339 by 7 is 3. 
Tuesday + 3 days = Friday

So, the 23rd of April 2021 will be Friday. Hence, the first option is correct.

Correct Answer: 9 : 13

Solution : Total number of toys sold by shops A, B, and C in the first week = 20 + 40 + 30 = 90
Total number of toys sold by shops C, D, and E in the second week = 20 + 60 + 50 = 130
Ratio = 90 : 130 = 9 : 13
Hence, the correct answer is 9 : 13.

Correct Answer: No Improvement

Solution : The correct choice is the first option.

The given sentence is in the past tense. It doesn't need any improvement, as it uses the correct verb form to maintain grammatical accuracy and report the event correctly that has happened in the past.

Therefore, the given sentence is correct and requires no improvement.

Correct Answer: 3

Solution : $9\sqrt{x} = \sqrt{12} + \sqrt{147}$
⇒ $9\sqrt{x} = 2\sqrt{3} + 7\sqrt{3}$
⇒ $9\sqrt{x} = 9\sqrt{3}$
$\therefore x = 3$
Hence, the correct answer is 3.

Correct Answer: Consider

Solution : The correct choice is the fourth option.

Consider means to think about something, typically before making a decision or forming an opinion. It involves examining or reflecting on various aspects of a situation.

The meanings of the other options are as follows:

• Supplicate means to earnestly and humbly ask for something, typically in the form of a request.
• Dramatise refers to the act of turning something, such as a story, event, or situation, into a dramatic form.
• Formerly display seems like an incomplete or unclear phrase. "Formerly" generally means in the past or previously, while "display" refers to exhibiting or showing something. Together, they do not form a typical phrase with a specific meaning.

Therefore, the correct sentence is: Consider what happens when you walk into a fast-food restaurant.