Correct Answer: Formation of lather with soap

Solution : The correct option is the Formation of lather with soap.

To determine the hardness of water, the lather or soap test is widely employed. Hard water combines with soap to generate soap scum or curd, an intractable material. When you use soap in hard water, it produces less lather and instead forms a scum. This can make cleaning harder, as more soap is needed to produce the desired cleansing effect. Soft water, on the other hand, readily forms a lather with soap.

Correct Answer: 207

Solution : Given:
12, 25, 51, 103, ?, 415, 831

Multiply the previous term by 2 and add 1 to the resultant term to get the next term.
(12 × 2) + 1 = 25; (25 × 2) + 1 = 51; (51 × 2) + 1 = 103; (103 × 2) + 1 = 207; (207 × 2) + 1 = 415; (415 × 2) + 1 = 831

So, 207 is the missing term in the series. Hence, the second option is correct.

Correct Answer: Pronounciation

Solution : The incorrectly spelt word is the fourth option.

Explanation: The correct spelling is pronunciation. It refers to the way in which a word is spoken with emphasis on the correct sounds and articulation. The use of ou instead of u constitutes a spelling error.

The meanings of the other options are as follows:

• Believe: It means to accept something as true or to have faith or confidence in something or someone.

• Audible: It means capable of being heard or loud enough to be heard.

• Throat: It refers to the passage in the neck that leads to the stomach and the lungs, used for swallowing and breathing.

Correct Answer: Alliteration

Solution : The fourth option is the correct answer.

Alliteration is the appropriate term. It specifically refers to the repetition of initial consonant sounds, adding a rhythmic and melodic quality to language.

The meanings of the other options are as follows:

• A metaphor is a figure of speech in which a word or phrase is applied to an object or action to which it is not literally applicable.
• Pun is a play on words that have similar sounds but different meanings.
• Hyperbole means exaggerated statements or claims not meant to be taken literally.

Correct Answer: Only conclusions I and II follow

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

Let's analyse the conclusions –
Conclusion I: Some stones being bats is a possibility – As some stones are cubes and all bats are cubes. So some stones being bats is possible. Therefore, this conclusion follows.
Conclusion II: No bat is a glass – Since all the bats are cubes and no cube is a glass. So, no bat is a glass. Therefore, this conclusion follows.
Conclusion III: All cubes are bats – From the Venn diagram, it is evident that some parts of the circles representing cubes lie outside the circle representing bats while some lie inside. So, from this, it can be concluded that some cubes are bats. Therefore, this conclusion does not follow.
Conclusion IV: No stone is a glass – From the Venn diagram, it is evident that there is no direct relation between stone and glass. So, no definite relation can be drawn. Therefore, this conclusion does not follow.

So, only conclusions I and II follow. Hence, the third option is correct.

Correct Answer: 16

Solution : Given:

'+' means '×'

'–' means '÷'

'×' means '–'

'÷' means '+'

16 ÷ 64 – 8 × 4 + 2 = ?

Now, replacing the symbols as per the code language given in the question, we get –

16 + 64 ÷ 8 – 4 × 2

Solving this equation using the BODMAS rule, we get –

= 16 + 8 – 4 × 2

= 16 + 8 – 8

= 16

Hence, option third is correct.

Correct Answer: $3$

Solution :
$\tan A=\frac{1}{2}$
$⇒\frac{BC}{AB}=\frac{1}{2}$
Let $BC=x$ units, then $AB=2x$ units
Using Pythagoras theorem,
$⇒AC^2=AB^2+BC^2$
$⇒AC^2=x^2+(2x)^2$
$⇒AC^2=5x^2$
$⇒AC =x\sqrt5 $ units
Now, we have $\triangle ABC$ right-angled at B, with $\tan A=\frac{1}{2}$, and we've found that $BC=x$, $AB=2x$, and $AC=x\sqrt{5}$.
$⇒\sin A = \frac{BC}{AC} = \frac{x}{x\sqrt{5}} = \frac{1}{\sqrt{5}}$
$⇒\cos A = \frac{AB}{AC} = \frac{2x}{x\sqrt{5}} = \frac{2}{\sqrt{5}}$
$⇒\sin C = \frac{AB}{AC} = \frac{2x}{x\sqrt{5}} = \frac{2}{\sqrt{5}}$
$⇒\cos C = \frac{BC}{AC} = \frac{x}{x\sqrt{5}} = \frac{1}{\sqrt{5}}$
Substituting these values into the given expression, we get:
$⇒\frac{\sin A(\cos C+\cos A)}{\cos C(\sin C-\sin A)} = \frac{\frac{1}{\sqrt{5}}(\frac{1}{\sqrt{5}}+\frac{2}{\sqrt{5}})}{\frac{1}{\sqrt{5}}(\frac{2}{\sqrt{5}}-\frac{1}{\sqrt{5}})} = \frac{\frac{3}{5}}{\frac{1}{5}} = 3$
Hence, the correct answer is 3.

Correct Answer: 70,000

Solution : As per the given graph:
The number of students enrolled at State and Central Universities in 1980 = 8000 + 10000 = 18000
The number of students enrolled at State and Central Universities in 1990 = 14000 + 12000 = 26000
The number of students enrolled at State and Central Universities in 2000 = 12000 + 14000 = 26000
Required total enrolment = 18000 + 26000 + 26000 = 70000
Hence, the correct answer is 70,000.

Correct Answer: $4\sqrt{6}\;\mathrm{cm}$ 

Solution :

We have, $\mathrm{AX:BX = 3:2}$ and the radius of the circle is $5\;\mathrm{cm}$.
$\mathrm{AX=\frac{3}{5} \times 10 = 6}\;\mathrm{cm}$ 
$\mathrm{BX=\frac{2}{5} \times 10 = 4}\;\mathrm{cm}$ 
We know the theorem states that the two chords of a circle intersect within the circle, the product of the segments of one chord is equal to the product of the segments of the other chord.
$\mathrm{PX\times QX = AX \times XB}$
$\mathrm{PX^2=6 \times 4}$ [Since $\mathrm{PX =QX}$]
$\mathrm{PX=2\sqrt{6}}$
$\mathrm{PQ=2PX=4\sqrt{6}}$
Hence, the correct answer is $4\sqrt{6}\;\mathrm{cm}$.

Correct Answer: Porcupine

Solution : The correct option is Porcupine.

Among the options given, the "porcupine" is protected in India under the Wildlife (Protection) Act of 1972. The porcupine is a Schedule II species under the Act, which means it is legally protected and killing or catching them without valid authorization is illegal. The purpose of the Act is to conserve and safeguard diverse wildlife species and their habitats in India.