Correct Answer: Four

Solution : The correct answer is Four.

India had four sites selected for the 2019 UNESCO Asia-Pacific Awards for Cultural Heritage Conservation. Among them, three are located in Mumbai—Flora Mountain, Keneseth Synagogue, and Gloria Church. The fourth site is the Vikram Sarabhai Library in Ahmedabad. The announcement of these awards took place in Malaysia in 2019.

Correct Answer: 64°

Solution :

$\angle APB =128^{\circ}$
Let $\angle OAB = x$
$\angle PAB = 90^{\circ} - x$
Since PA and PB are tangents from the same external point P,
⇒ PA = PB
$\therefore \triangle APB$ is an isosceles triangle.
⇒ $\angle PAB = \angle PBA = 90^{\circ} - x$
In $ \triangle APB$,
$\angle APB + \angle PAB + \angle PBA = 180^{\circ}$
⇒ $128^{\circ} + (90^{\circ} - x) + (90^{\circ} - x) = 180^{\circ}$
⇒ $128^{\circ}  = 2x$
⇒ $x = 64^{\circ}$
Hence, the correct answer is $64^{\circ}$.

Correct Answer: 2005

Solution : The percentage increase in the export of apples in 2003 = $\frac{6.5-5.2}{5.2}$ × 100 = 25%
The percentage increase in the export of apples in 2005 = $\frac{9.9-7.8}{7.8}$ × 100 = 26.92%
The percentage increase in the export of apples in 2006 = $\frac{10.8-9.9}{9.9}$ × 100 = 9.09%
The percentage increase in the export of apples in 2008 = $\frac{11.4-9.5}{9.5}$ × 100 = 20%
The maximum percentage increase in the export of apples to that of the previous years is in 2005.
Hence, the correct answer is 2005.

Correct Answer: Only conclusion I follows

Solution : The possible Venn diagram according to the given statement is as follows –

Let's analyse the conclusions –
Conclusion I: Some kings are queens – From the Venn diagram, it is evident that some parts of the circle representing kings lie inside the circle representing queens while some lie outside. So, from this, it can be concluded that some kings are queens. Thus, this conclusion does not follow.
Conclusion II: All queens are aces – From the Venn diagram, it is evident that there is no direct relation between queens and aces. So, no definite relation can be drawn. Thus, this conclusion does not follow.

So, the only conclusion I follows. Hence, the third option is correct.

Correct Answer: All of the conclusions follow

Solution : The possible Venn diagram according to the given statements is as follows –

Let's analyse the conclusions –
Conclusion (I): No pomegranate is a plum – From the given Venn diagram, the circle of the pomegranate is completely inside the circle of a tree, implying that all pomegranates are trees and no tree is a plum. Therefore, no pomegranate is a plum is a true conclusion.
Conclusion (II): No plum is a tree – From the given Venn diagram, no tree is a plum. Therefore, no plum is a tree is also a true conclusion.
Conclusion (III): Some trees are pomegranates – From the given Venn diagram, all pomegranates are trees. Therefore, this conclusion is true.

So, all of the conclusions follow. Hence, the first option is correct.

Correct Answer: Smiling broadly

Solution : The correct choice is the second option.

The idiom beaming from ear to ear is used to describe someone who is pleased, and the context indicates a positive emotion. The word "beaming" itself conveys a sense of radiant happiness. In this case, the person is not playing, working, or criticising bitterly; instead, the idiom suggests a wide, happy smile.

Therefore, the right answer is, smiling broadly.

Correct Answer:

Solution : As per the mirror image properties, closer things appear close to the mirror in the reflection.
Here, according to the information provided, the mirror is placed on the right side of the figure. So, the left side of the reflected image will appear as the right side, and the right side will appear as the left side. However, the top and bottom of the reflected image will remain the same.
Thus, the correct mirror image of the given figure will be –

Hence, the third option is correct.

Correct Answer: 3

Solution : The error is in part (R) marked by the third option.

Explanation:

In this sentence, "such as" is used instead of "like" to convey the idea that luck is an example or type of thing, rather than a similarity. Such as is used for giving examples whereas like is used for exhibiting similarity.

So, the correct sentence would be: "The old man told his sons that there was no such thing as luck."

Correct Answer: $1$

Solution : Given: $x+y = 2z$
Now, $\frac{x}{x-z}+\frac{z}{y-z}$
$= \frac{x(y-z)+z(x-z)}{(x-z)(y-z)}$
$= \frac{xy−xz+zx−z^2}{xy−xz−zy+z^2}$
$= \frac{xy−z^2}{xy−z(x+y)+z^2}$
$= \frac{xy−z^2}{xy−z(2z)+z^2}$
$= \frac{xy−z^2}{xy−z^2}$
$= 1$
Hence, the correct answer is $1$.

Correct Answer: 77088

Solution : The prime factorization of the given numbers is:
$96  = 2^5 \times 3 $
$132  = 2^2 \times 3^1 \times 11^1 \\$
$438  = 2^1 \times 3^1 \times 73^1$
Now, identify the highest powers of each prime factor:
It is $2^5$,$3^1$,$11^1$, and $73^1$
Multiply these highest powers to find the LCM:
$\text{LCM of}\;(96, 132, 438) = 2^5 \times 3^1 \times 11^1 \times 73^1 = 77088$
Hence, the correct answer is 77088.