Correct Answer: Only (i) and (iii)

Solution :
Given: In $\triangle PQR, PQ=QR$ and $O$ is an interior point of $\triangle PQR$ such that $\angle OPR=\angle ORP$.
Since $PQ = QR$, so $\triangle PQR$ is an isosceles triangle.
From the given data we cannot say that $O$ is the centroid of the $\triangle PQR$.
Also, $PQ=QR, OP=OR$ and $OQ=OQ$
So, $\triangle PQO$ and $\triangle RQO$ are congruent.
Hence, the correct answer is Only (i) and (iii).

Correct Answer: 100%

Solution : The foreign exchange reserves in 1997–1998 were $5040$ million, as seen in the bar graph.
2520 million was the amount of foreign exchange reserves in 1993–94.
Change in the number of reserves = $5040 - 2520= 2520$
Percentage increase = $\frac{\text{Difference}}{\text{The foreign exchange reserves in 1993–1994}} \times 100$.
= $\frac{2520}{2520}\times 100=100$%
Hence, the correct answer is 100%.

Correct Answer: didn’t they?

Solution : The third option is the correct answer.

The sentence is a statement followed by a question tag. In English, when the main clause is in the past tense, the tag question should also be in the past tense. So, "didn't they?" is the correct form.

A question tag is a very short clause at the end of a statement that changes the statement into a question. A question tag is formed by inverting the subject and verb of a statement.

Therefore, the correct sentence should be: "They met yesterday, didn't they?"

Correct Answer: Not a member of the Parliament

Solution : The correct option is Not a member of the Parliament.

The Vice-President of India is not a member of either house of Parliament. The Vice-President is a separate constitutional office and does not sit in the Rajya Sabha (Council of States) or the Lok Sabha (House of People). Instead, the Vice-President's primary responsibility is to preside over the Rajya Sabha and undertake other constitutionally mandated duties. The Vice-President is chosen by an electoral college comprised of members of both houses of Parliament, as well as representatives of state and union territory legislatures. Once elected, the Vice-President's major role is to serve as the Rajya Sabha's ex-officio Chairman.
 

Correct Answer: CADB

Solution : The correct choice is the third option.

The logical flow of sentences are as follows:

• In sentence C, the idea is introduced that God constructed the human body with forethought and solicitude.
• Sentence A then brings attention to a potential vulnerability as it mentions that the person seems to have become weary, particularly when it comes to the ears.
• Following this, sentence D explains that he left the ears as the most vulnerable part of a human being.
• Finally, sentence B provides the result or consequence of this vulnerability, stating that individuals spend their time hankering after something unattainable, namely silence.

The flow of information follows a logical sequence, starting with the construction of the human body, addressing a vulnerability, and concluding with the consequence of that vulnerability.

Therefore, the correct order is CADB.

Correct Answer: (16, 26, 31)

Solution : Given:
(3,13,18), (29, 39, 44)
Add 10 to the first number to get the second number and add 5 to the second number to get the third number –
⇒ (3,13,18)→3 + 10 = 13, 13 + 5 = 18
⇒ (29, 39, 44)→29 + 10 = 39, 39 + 5 = 44

Let's check each option –
First option: (17, 27, 42)→17 + 10 = 27, 27 + 5 = 32 ≠ 42
Second option: (25, 30, 55)→25 + 10 = 35 ≠ 30, 30 + 5 = 35 ≠ 55
Third option: (26, 30, 47)→26 + 10 = 36 ≠ 30, 30 + 5 = 35 ≠ 47
Fourth option: (16, 26, 31)→16 + 10 = 26, 26 + 5 = 31

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

Correct Answer: (1)

Solution : The correct answer is the first option.

Farther should be replaced with further to convey the intended meaning that if he requires additional help, he should be directed to the speaker.

Farther is used to signify physical distance, whereas further is used to refer to something additional to what already exists. Hence, further would be a more appropriate choice for this sentence. The other options, much and best do not go contextually with the sentence.

Therefore, the correct sentence is, "If he wants further help, send him to me".

Correct Answer: 3

Solution : $(a + b + c)^{2} = a^{2}  + b^{2} + c^{2} + 2(ab + bc + ca)$
$⇒(a + b + c)^{2} -3(ab + bc + ca) = a^{2} + b^{2} + c^{2} + 2(ab + bc + ca) –3(ab + bc + ca)$
Putting the values of $a, b,$ and $c$, we get,
$⇒(101 + 102 + 103)^{2}-3(101 \times 102 + 102 \times 103 + 103\times 101) = a^{2} + b^{2} + c^{2} - (ab + bc + ca)$
$⇒a^{2} + b^{2} + c^{2} -ab - bc - ca = 306^2 -3(10302 + 10506 + 10403)$
$⇒a^{2} + b^{2} + c^{2} -ab - bc - ca = 93636 -3(31211)$
$⇒a^{2} + b^{2} + c^{2} -ab - bc - ca = 93636 -93633$
$\therefore a^{2} + b^{2} + c^{2} -ab - bc - ca= 3$
Hence, the correct answer is 3.

Correct Answer:

Haliotis

Solution : The correct option is Haliotis.

"Haliotis" refers to the phylum Mollusca and the genus of sea mollusks usually known as abalones. Abalones are gastropod mollusks prized for their iridescent shells, which are frequently utilised in food and ornamental arts. The term "hyla" typically refers to a genus of tree frogs, and these animals belong to the animal kingdom (Kingdom: Animalia). "Hydra" belongs to the phylum Cnidaria. Hares are mammals, specifically from the family Leporidae, and they are closely related to rabbits.

Correct Answer: 11 cm

Solution :
TP = 15 cm
PB = 2$x$ + 3 cm
PA = 9 cm
If a secant AB and tangent at the point T of a circle intersect at a point T of a circle intersect at a point P outside the circle, then PA × PB = PT²
9 × (2$x$ + 3) = 15²
⇒ 2$x$ + 3 = $\frac{225}{9}$ = 25
⇒ $x$ = $\frac{25-3}{2}$ = 11
Hence, the correct answer is 11 cm.