Correct Answer: 8 cm

Solution :
Height $=$ AD $=\frac{\sqrt3}{2}\times \text{side} = \frac{\sqrt3}{2}\times 16\sqrt{3} = 24$ cm
Now, AP : PD = 2 : 1
Let AP = $2x$ and PD = $x$, Then, PD = $3x$
Here, $3x=24$
⇒ $x=8$ cm
Hence, the correct answer is 8 cm.

Correct Answer: Incidentally

Solution : The word with the spelling error is Incidently.

Explanation: The correct spelling is incidentally, which means happening by chance or unexpectedly. The incorrect options are misspellings and do not adhere to the standard spelling rules.

So, the correct sentence would be: "Incidentally

Correct Answer: naturally

Solution : The correct answer is the fourth option.

The adverb naturally is the most suitable choice, as it accurately conveys the idea that sugar occurs as a natural component in these food items.

Therefore, the correct answer is naturally.

Correct Answer: Both I and II

Solution : The correct option is Both I and II.

Papaya is indeed a good source of vitamin A. Vitamin A is a fat-soluble vitamin that plays a crucial role in maintaining healthy skin, vision, and immune function.

Spinach is a good source of

Correct Answer: Calcutta

Solution : The correct answer is Calcutta.

In 1756, Nawab Siraj-ud-Daulah was annoyed with the unfair trade practices of the British. He was also infuriated by the construction of Fort William without his permission. In retaliation to this, he seized Fort William in Calcutta and imprisoned

Correct Answer: September 16th

Solution : The correct answer is September 16th.

Annually, on September 16th, the world observes the International Day for the Preservation of the Ozone Layer, commemorating the signing of the Montreal Protocol in 1987. The ozone layer is crucial in shielding the Earth from harmful ultraviolet

Correct Answer: $x^{2}y^{2}\left ( x^{2}+y^{2}+3 \right )=1$

Solution : $x=\operatorname{cosec \theta}-\sin\theta$
⇒ $x=\frac{1}{\sin\theta}-\sin\theta=\frac{1-\sin^2\theta} {\sin\theta}=\frac{\cos^2\theta}{\sin\theta}$
Also, $x^2=(\frac{\cos^2\theta}{\sin\theta})^2$_____ (i)
$y=\sec\theta-\cos\theta$
⇒ $y=\frac{1}{\cos\theta}-\cos\theta=\frac{1-\cos^2\theta}{\cos\theta}=\frac{\sin^2\theta}{\cos\theta}$
⇒ $y^2=(\frac{\sin^2\theta}{\cos\theta})^2$_____ (ii)
So, $x^2+y^2=(\frac{\cos^2\theta}{\sin\theta})^2+(\frac{\sin^2\theta}{\cos\theta})^2$
⇒ $x^2+y^2=\frac{\cos^4\theta}{\sin^2\theta}+\frac{\sin^4\theta}{\cos^2\theta}$
⇒ $x^2+y^2=\frac{\cos^6\theta+\sin^6\theta}{\sin^2\theta\cos^2\theta}$
Add 3 to both sides, we get,
⇒ $x^2+y^2+3=\frac{\cos^6\theta+\sin^6\theta}{\sin^2\theta\cos^2\theta}+3$
⇒ $x^2+y^2+3=\frac{\cos^6\theta+\sin^6\theta+3\sin^2\theta\cos^2\theta(\sin^2\theta+\cos^2\theta)}{\sin^2\theta\cos^2\theta}$
⇒ $x^2+y^2+3=\frac{(\cos^2\theta+\sin^2\theta)^3}{\sin^2\theta\cos^2\theta}$
⇒ $x^2+y^2+3=\frac{1}{\sin^2\theta\cos^2\theta}$
⇒ $x^2+y^2+3=\frac{\sin^2\theta\cos^2\theta}{\sin^4\theta\cos^4\theta}$
⇒ $x^2+y^2+3=\frac{1}{(\frac{\cos^2\theta}{\sin\theta})^2×(\frac{\sin^2\theta}{\cos\theta})^2}$
From equation

Correct Answer: Two

Solution : Given:
WORKING

Arrange the letters of WORKING in the alphabetical order –
W O R K I N G→G I K N O R W

Now, according to the new letter cluster formed, the third letter from the left is K and the fourth letter

Correct Answer: Chetan Bhagat

Solution : The correct option is Chetan Bhagat.

The first novel that Indian author Chetan Bhagat released was titled Five Point Someone: What Not to Do at IIT. While he was still employed, Bhagat began writing his first book towards the beginning of the 2000s.

Correct Answer: 28

Solution : The given figure can be labelled as shown below –

Now, in the above figure, there are 28 triangles. They are ABD, DBC, ABC, BCH, BAI, EJI, EBF, FEV, FJW, FKM, FGL, GHK, GFB, EFG, TSY, TRa, TRU, RXS, RNQ, ROP, PQZ, PRa, TRP, abd,