Correct Answer: $21-14\sqrt{2}$

Solution :

We have $PQRS$ as a rectangle and a semicircle with $SR$ as diameter and $QR = 7\;\mathrm{cm}$.
Construction: Draw a line from $Q$ on line $ST$ (where $T$ is the centre of the semicircle) and pass through point $U$ (where $U$ is the point of contact and $O$ is the centre of the circle).
Now, $RT$ and $QR$ are the radius of the bigger circle and equal to $7\;\mathrm{cm}$.
In $\triangle QRT$,
$QT^2=QR^2+TR^2$
$QT^2 = 49 + 49$
$ QT = 7\sqrt{2}$ ____(i)
Let the radius of the smaller circle $=r\;\mathrm{cm}$
$QO = r\sqrt{2}$
From the figure,
$QT = TU + UO + OQ$
$QT = 7 + r + r\sqrt{2}$ ____(ii)
From equation (i) and (ii)
$7\sqrt{2} = 7 + r (\sqrt{2} + 1)$
⇒ $r = \frac{7 (\sqrt{2} - 1)}{(\sqrt{2} + 1)}$
⇒ $r = 7 (3 - 2\sqrt{2})$
Hence, the correct answer is $21 - 14\sqrt{2}$.

Correct Answer: 67, 42, 66, 00

Solution : Given:
BARK

Number representations of each letter:
B → 32, 67, 78
A → 42, 03, 55
R → 13, 23, 66
K → 00, 24

First option: 32, 55, 69, 24
R cannot be represented by 69. Hence, it's incorrect.

Second option: 32, 03, 66, 57
K cannot be represented by 57. Hence, it's incorrect.

Third option: 67, 42, 66, 00
All the letters of the word BARK can be represented through this option. Hence, it's correct.

Fourth option: 23, 30, 66, 97
B cannot be represented as 23; A cannot be represented as 30; K cannot be represented as 97. Hence, it's incorrect.

Hence, the third option is correct.

Correct Answer: Kerala

Solution : The correct option is Kerala.

Kerala was the first state in India to introduce a fat tax on junk food. This tax was proposed in 2016 as a part of the Kerala State Budget. The tax was levied on foods with high fat and sugar content, including fast food items like burgers, pizzas and sugary beverages. The 14.5% fat tax was enacted to curb the rising addiction to processed foods.

Correct Answer: contests

Solution : The fourth option is correct.

We can say that the implied meaning of the sentence is that the Twins Days Festival is a three-day festival with talent shows and look-alike competitions. According to the above-mentioned points, the most appropriate option to fill in the blank is "contests" to make the sentence meaningfully correct.

Correct Answer: 1960s

Solution : The correct option is the 1960s.

The Green Revolution in India started in the mid-1960s with the introduction of high-yielding varieties of wheat and rice. It is often associated with the work of agricultural scientists such as Norman Borlaug and the use of modern practices and technology to increase food production in the country.

Correct Answer: Halite

Solution : The correct answer is Halite.

The sedimentary mineral commonly known as "salt" that usually forms in arid climates where seawater evaporates is halite. Halite is the mineral form of sodium chloride (NaCl), and it is a significant component of salt deposits found in such environments. These deposits can be found in regions like deserts or arid coastal areas where there is a history of the evaporation of seawater over time, leaving behind salt deposits.

Correct Answer: Coelenterates

Solution : The correct option is Coelenterates.

Coelenterates, also known as Cnidarians, are not triploblastic animals. Triploblastic animals have three germ layers during embryonic development: the endoderm, mesoderm, and ectoderm. Cnidarians are diploblastic, which means they have only two germ layers, the endoderm and ectoderm.

Correct Answer: An lion

Solution : The correct choice is the third option.

The article an is used before a word starting with a vowel. In this sentence, an should be replaced with the, as the lion refers to the class of lions.

The correct sentence is: "The lion is the most ferocious of all animals."

Correct Answer: 214

Solution : Given: 
< means –, ! means ×, > means +, and @ means ÷
205 > 210 @ 15 ! 2 < 19

After replacing the symbols, the equation is as follows –
= 205 + 210 ÷ 15 × 2 – 19
= 205 + 14 × 2 – 19
= 205 + 28 – 19
= 233 – 19
= 214

So, 214 is the required answer to the given equation. Hence, the third option is correct. 

Correct Answer: before

Solution : The correct choice is the third option.

In the context of the sentence, it discusses the order in which students have listened to different teachers. The sentence suggests that the students have sometimes listened to three or four teachers before the teacher in question. Before indicates the order of events or actions, which is the most suitable option here.