Correct Answer: To express an idea with great precision

Solution : The correct choice is the fourth option.

Explanation: The idiom "to hit the nail on the head" means expressing an idea with great precision or accurately addressing the main issue. It is used to describe someone who gets to the core of a problem or situation without unnecessary elaboration.

Therefore, the correct answer is to express an idea with great precision.

Correct Answer: $\sqrt{2} \cos \theta$

Solution : According to the question,
⇒  $\cos \theta-\sin \theta=\sqrt{2} \sin \theta$
⇒ $(\cos\theta + \sin\theta)^{2} = 2 \sin^{2}\theta$
⇒ $\cos^{2}\theta + \sin^{2}\theta + 2\sin\theta\cos~\theta = 2\sin^{2}\theta$
⇒ $\sin^{2}\theta - \cos^{2}\theta - 2\sin~\theta~cos\theta = 0$
⇒ $\sin^{2}\theta + \cos^{2}\theta - 2\sin\theta\cos\theta = 2 \cos^{2}\theta$
⇒ $(\sin~\theta - \cos\theta)^{2}=2\cos^2\theta$
⇒ $\sin\theta - \cos \theta$ = $\sqrt{2}\cos\theta$
Hence, the correct answer is $\sqrt{2}\cos\theta$.

Correct Answer: 4

Solution :

LCM of 2, 3, 4, 5, 6 and 7 is 420.
The number should be between 2000 and 2500.
So, the number is = 420 × 5 = 2100
The remainder will be 1 in each case so the number = 2100 + 1 = 2101
Sum of digit of the number 2101 is = 2 + 1 + 0 + 1 = 4
Hence, the correct answer is 4.

Correct Answer: Puri

Solution : The correct option is Puri.

Puri is not traditionally associated with the Kumbh Mela. The Kumbh Mela is a major Hindu festival and pilgrimage that rotates among four locations: Prayagraj (formerly Allahabad), Haridwar, Nashik and Ujjain. Puri,  in the eastern Indian state of Odisha, is not one of the designated locations for the Kumbh Mela.

Correct Answer: trusteeship

Solution : The correct answer is trusteeship.

Mahatma Gandhi introduced the socio-economic philosophy of trusteeship. It is founded on the idea that everyone who owns money or property holds it in trust for society. It offers a way for the wealthy to serve as trustees of trusts that look out for the welfare of the entire population. His religious conviction that everything belonged to god, and came from god served as its foundation.

Correct Answer: Ammonia

Solution : The correct option is Ammonia.

Ammonia (NH₃) does not belong to the family of organic compounds. Organic compounds are generally defined as compounds containing carbon-hydrogen (C-H) bonds, and ammonia does not contain carbon. Instead, it is an inorganic compound composed of nitrogen and hydrogen atoms.

Correct Answer: 1, 3, 5, 4, 2

Solution : Given: 
1. Measurement 2. Methodology 3. Melanin 4. Metaphor 5. Meritorious

Step 1: The first two letters are common in all so we will start with the third letter. The third letter of each word is a, t, r, l. Based on the alphabetical order we can arrange them as – Measurement, Melanin, Meritorious, Methodology, Metaphor
Step 2: Compare the fourth letter of (Methodology, Metaphor). Metaphor will come before Methodology in the sequence as a comes before h.

So, the sequence will be Measurement, Melanin, Meritorious, Metaphor, Methodology, or 1, 3, 5, 4, 2. Hence, the second option is correct.

Correct Answer: Vinoba Bhave

Solution : The correct answer is Vinoba Bhave.

Vinoba Bhave, a follower of Mahatma Gandhi, founded the Bhoodan Movement, also known as the Land Donation Movement, in April 1951. The campaign began impromptu during an evening gathering in a village in Andhra Pradesh, where landowners donated land to the landless. Over the following 20 years, the campaign gave rise to a law known as the Bhoodan Act, which gave the government authority over the distribution of land from the land bank.

Correct Answer: Samudragupta

Solution : The correct answer is Samudragupta.

Samudragupta is known as "India's Napoleon". He was the son of Chandragupta I. He is believed to be the real founder of the Gupta dynasty. Historian A. V. Smith called him an "Indian Napoleon" because of his great military spirit.

Correct Answer: 2

Solution : Given that the average age of 5 members = 17
$\therefore$ Total age of 5 members = 17 × 5 = 85
Now, let the present age of the baby be $x$ years
Present total age of the family $= 85 + 5\times3 + x = 100 + x$
According to the question,
$\frac{100+x}{6}=17$
$⇒ 100 + x = 102$
$\therefore x = 2$ years
Hence, the correct answer is 2.