Correct Answer: 1

Solution : Given:
$4x=\sqrt5+2$
⇒ $\frac{1}{4x}=\sqrt5-2$ (by rationalising)
⇒ $4x - \frac{1}{4x}=(\sqrt5+2)-(\sqrt5-2)$
⇒ $4x - \frac{1}{4x}=\sqrt5+2-\sqrt5+2$
⇒ $4x - \frac{1}{4x}=4$
Dividing both sides by 4, we get,
⇒ $\frac{4x}{4}-\frac{1}{4x×4}=\frac{4}{4}$
$\therefore x - \frac{1}{16x}=1$
Hence, the correct answer is 1.

Correct Answer: Father

Solution : As per the given information, the family tree will be as follows –

Here, the quadrilateral represents the male, and the circular figure represents the female in the figure.

So, from the above family tree, the man is the father of that woman. Hence, the fourth option is correct.

Correct Answer: (125, 25, 100)

Solution : Given:
(23, 14, 9); (37, 19, 18)

Here, (23, 14, 9)→14 + 9 = 23
(37, 19, 18)→19 + 18 = 37

Let's check the options –
First option: (125, 25, 5)→25 + 5 = 30 ≠ 125
Second option: (125, 100, 5)→100 + 5 = 105 ≠ 125
Third option: (125, 25, 100)→25 + 100 = 125
Fourth option: (125, 5, 25)→5 + 25 = 30 ≠ 125

So, only the third option follows the same pattern as followed by the given set of numbers. Hence, the third option is correct.

Correct Answer: $2 \sqrt{3}$ cm

Solution : Given: ABC is an equilateral triangle with a side of 12 cm.
The length of the inradius of the equilateral triangle = $\frac{\text{side}}{2\sqrt3}=\frac{12}{2\sqrt3}=2\sqrt3$ cm
Hence, the correct answer is $2\sqrt3$ cm.

Correct Answer: 15 : 9

Solution : Let's check the options –
First option: 20 : 11; (11 × 2) – 2 = 22 – 2 = 20
Second option: 6 : 4; (4 × 2) – 2 = 8 – 2 = 6
Third option: 15 : 9; (9 × 2) – 2 = 18 – 2 = 16 ≠ 15
Fourth option: 12 : 7; (7 × 2) – 2 = 14 – 2 = 12

So, only the third option is different from the other three. Hence, the third option is correct.

Correct Answer: Henry David Thoreau 

Solution : Correct Answer is Henry David Thoreau 

Civil disobedience began after the British ignored Gandhi’s 11 demands. In response, Gandhi allied himself with young activists to lead a movement known as the ‘Civil Disobedience’ Movement. During Gandhi’s anti-British protests, he called on all Indians to follow his lead. Gandhi’s inspiration for leading the civil disobedience movement came from the writings of the American writer Henry David Thoreau.

Correct Answer: Legacy

Solution : The second option is the correct answer.

A legacy is the property or money that is left to someone in a will. It is a bequest or inheritance that is passed down from one generation to another.

The meanings of the other options are as follows:

• A ledger is a book or a system for keeping financial records.
• A legend is a traditional story or myth with historical significance.
• Lexical means pertaining to words or vocabulary.

Therefore, the correct answer is legacy.

Correct Answer: Toes

Solution : Given:
Fingers : Hand :: ? : Feet

Like, fingers are the extensions of our hands.
Similarly, toes are the extensions of our feet.

Hence, the second option is correct.

Correct Answer: $\frac{\sqrt{b^2-a^2}}{a}$

Solution : Given,
$\operatorname{cosec} \theta=\frac{b}{a}$
⇒ $\sin\theta=\frac ab$
We have to find the value of $\frac{\sqrt{3} \cot \theta+1}{\tan \theta+\sqrt{3}}$
We know, $\cot\theta=\frac{\cos\theta}{\sin\theta}$ and $\tan\theta=\frac{\sin\theta}{\cos\theta}$
$\frac{\sqrt{3} \cot \theta+1}{\tan \theta+\sqrt{3}}=\frac{\sqrt3\times\frac{\cos\theta}{\sin\theta}+1}{\frac{\sin\theta}{\cos\theta}+\sqrt3}$
= $\frac{\cos\theta(\sqrt3\cos\theta+\sin\theta)}{\sin\theta(\sin\theta+\sqrt3\cos\theta)}$
= $\frac{\cos\theta}{\sin\theta}$
= $\frac{\sqrt{1-\sin^2\theta}}{\sin\theta}$ [As $\sin^2\theta+\cos^2\theta=1$]
= $\frac{\sqrt{1-(\frac ab)^2}}{\frac ab}$
= $\frac{\sqrt{b^2-a^2}}{a}$
Hence, the correct answer is $\frac{\sqrt{b^2-a^2}}{a}$.

Correct Answer: Two

Solution : Given:
Each vowel in the word CAPSULE is changed to the previous letter in the English alphabetical series and each consonant is changed to the following letter in the English alphabetical series.

In the newly formed word, Q is 3^rd from left and T is 3^rd from right.

There are two letters between Q (which is 3^rd from the left) and T (which is 3^rd from the right) in the English alphabetical series. Hence, the fourth option is correct.