Correct Answer: 16 cm

Solution : If two triangles are similar, then their perimeters are directly proportional to the length of their corresponding sides.
So, $\frac{\text{Perimeter of $\triangle$ABC}}{\text{Perimeter of $\triangle$ABC}}= \frac{AB}{DE} = \frac{32}{12}$
⇒ $AB = \frac{32}{12}\times 6$
⇒ $AB = 16\ \mathrm{cm}$
Hence, the correct answer is 16 cm.

Correct Answer: 2

Solution : Given: $a+\frac{1}{a-2}=4$
Subtracting 2 from both sides,
⇒ $a-2+\frac{1}{a-2}=2$
Squaring both sides,
⇒ $( a-2)^2+\frac{1}{( a-2)^2} + 2=2^2$
⇒ $( a-2)^2+\frac{1}{( a-2)^2}=2^2-2$
⇒ $( a-2)^2+\frac{1}{( a-2)^2}=2$
Hence, the correct answer is 2.

Correct Answer: Odisha

Solution : The correct option is Odisha.

One of India's biggest steel production sites is the Rourkela Steel Plant. It is located in the city of Rourkela in the Sundargarh district of Odisha. One of India's earliest integrated steel factories, the Rourkela Steel Plant was founded in 1959. It was a significant step toward India's industrialisation and self-sufficiency in steel production. Steel Authority of India Limited (SAIL), is the owner and operator of the plant.

Correct Answer: 217.50 m

Solution : In 1500 m race, Anil beats Bakul by 150 m.
When Anil completes the race, Bakul covers 1500 – 150 = 1350 m
Bakul beats Charles by 75 m in 1500 m race.
So in a 1350 m race, Bakul is ahead of Charles by $\frac{75}{1500}\times 1350=67.50$ m
So, Anil beats Charles by (150 + 67.5) = 217.50 m
Hence, the correct answer is 217.50 m.

Correct Answer: May success be achieved by you in all your endeavours.

Solution : The correct choice is the third option.

Explanation:
In a passive voice construction, the subject of the sentence receives the action, and the focus is on the action rather than the doer.
The verb "achieve" becomes "be achieved" (to be + past participle).
The subject "you" becomes the agent in the passive voice construction and is placed at the end of the sentence with "by you."
The passive construction shifts the focus from the doer (you) to the action (success being achieved).

Therefore, the passive voice would be: May success be achieved by you in all your endeavours.

Correct Answer: 9

Solution : Given: $4x^2-12x+k$ is a perfect square.
$4x^2-12x+k=(2x)^2-2×2x×3+k$
Since this a perfect,
So, $(2x)^2-2×2x×3+k=(2x)^2-2×2x×3+3^2=(2x–3)^2$
Value of $k$ is 3² = 9
Hence, the correct answer is 9.

Correct Answer: Sedimentary rocks

Solution : The correct option is Sedimentary rocks.

The type of rock that may contain fossils of plants, animals, and other microorganisms that once lived on or in them is called sedimentary rock. Sedimentary rocks are formed from the accumulation and cementation of sediments, which include the remains of organisms such as shells, bones, and plant material.

Correct Answer: at the station, it take

Solution : The segment that contains a grammatical error is the second option.

Explanation: The error lies in the tense consistency as per zero conditional. The correct form should be "it took" instead of "it take". In this context, took is the correct verb form to indicate that the action took place in the past as the subordinate clause is also in the past tense to maintain tense consistency.

So, the correct sentence would be: "When the train pulled in at the station, it takes Joseph some time to get off."

Correct Answer: $4\sqrt{3}\ \text{cm}$

Solution : Given: Side of the equilateral triangle = 12 cm
We know that,
The radius of a circle circumscribing in an equilateral triangle
= $\frac{\text{side}}{\sqrt3}$
= $\frac{12}{\sqrt3}$
= $4\sqrt3\ \text{cm}$
Hence, the correct answer is $4\sqrt3\ \text{cm}$.

Correct Answer:

Solution : In a 3D view of dice,
1. Each face out of the three faces in the 3D view should be from all the pairs.
2. Two opposite faces cannot appear adjacently in the 3D view of a folded cube figure. So, the opposite pairs of faces are as follows –

Based on the above diagram and rules, the following image can be formed –

Hence, the first option is correct.