Correct Answer: $1$

Solution : The given equation is $k\left(\tan 45^{\circ} \sin 60^{\circ}\right)=\cos 60^{\circ} \cot 30^{\circ}$.
$⇒k\left(1 \times \frac{\sqrt{3}}{2}\right) = \frac{1}{2} \times \sqrt{3}$
$⇒k = \frac{\frac{1}{2} \times \sqrt{3}}{\frac{\sqrt{3}}{2}} = \frac{1}{2} \times \frac{2}{\sqrt{3}} \times \sqrt{3} = 1$
Hence, the correct answer is $1$.

Correct Answer: The sun is shining while it rains

Solution : The third option is the correct answer.

This idiom is a regional expression used to describe a weather phenomenon where it is raining while the sun is shining. The phrase, the devil is beating his wife is a colloquial way of referring to this specific weather condition. It's a folk saying that doesn't have a literal meaning but is used to describe the simultaneous occurrence of rain and sunshine.

Therefore, the correct answer is: The sun is shining while it rains.

Correct Answer: 6 km

Solution : Let $x$ be the required distance.
Time taken in the first case = $\frac{x}{3}$ hours
Time taken in the second case = $\frac{x}{4}$ hours
Time difference = $\frac{x}{3} - \frac{x}{4}$ hours
So, $\frac{x}{3} - \frac{x}{4}$ = $\frac{(20+10)}{60}$
⇒ $\frac{x}{12}$ = $\frac{1}{2}$
⇒ $x$ = 6 km
Hence, the correct answer is 6 km.

Correct Answer: very intimate terms

Solution : The correct choice will be the first option.

The idiom hand in glove means to be in a very close or friendly association or partnership with someone. It suggests a relationship where individuals work closely together, cooperate seamlessly or share a strong connection.

For example, if two colleagues are said to be hand in glove, it implies that they work closely together, coordinating their efforts seamlessly.

Therefore, the answer will be very intimate terms.

Correct Answer: Sachin Tendulkar

Solution : The correct option is Sachin Tendulkar

"Playing It My Way" is the autobiography of the former Indian cricketer Sachin Tendulkar. The book, co-authored with Boria Majumdar, was published in 2014 and provides insights into Tendulkar's life, cricket career, and personal experiences. Tendulkar shares anecdotes and behind-the-scenes stories from his cricketing career in the autobiography.

Correct Answer: (3)

Solution : The error lies in the third part of the sentence.

When one is used in a sentence, it is always followed by the indefinite pronoun one to make the sentence grammatically accurate. Thus, he should be replaced with one.

Therefore, the correct sentence is: "When one hears of the incident about the plane crash, one feels very sorry."

Correct Answer: bats

Solution : The correct option is bats.

The scientific name for the spotted bat is "Euderma maculatum". The spotted bat is a species found in North America that is distinguished by the big white dots on its black fur. Because of its distinctive colouring, it is among the most recognisable bat species. The western regions of North America, extending from southern Canada to Mexico, are home to the species.

Correct Answer: 6

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

There are 6 quadrilaterals in the above figure. They are ADFE, FEBC, ADCE, BCFA, AFCE, ABCD.

Hence, the second option is correct.

Correct Answer: N16R

Solution : Given:
J2Z, K4X, L7V, M11T, ?

Add 1 to the place value of the first letter and add consecutive natural numbers starting from 2 to the number of the previous term respectively, and subtract 2 from the third letter to obtain the next term in the series –
J2Z→J + 1 = K; 2 + 2 = 4; Z – 2 = X→K4X
K4X→K + 1 = L; 4 + 3 = 7; X – 2 = V→L7V
L7V→L + 1 = M; 7 + 4 = 11; V – 2 = T→M11T
M11T→M + 1 = N; 11 + 5 = 16; T – 2 = R→N16R

So, N16R is the missing term of the series. Hence, the fourth option is correct.

Correct Answer: 1

Solution : $\cot \theta=\frac{1}{\sqrt{3}}$
⇒ $\theta=60^{\circ}$
So, $\frac{2-\sin ^2 \theta}{1-\cos ^2 \theta}+\left(\operatorname{cosec}^2 \theta-\sec \theta\right)$
= $\frac{2-\sin ^2 60^{\circ}}{1-\cos ^2 60^{\circ}}+\left(\operatorname{cosec}^2 60^{\circ} -\sec 60^{\circ}\right)$
= $\frac{2- \frac{3}{4}}{1-\frac{1}{4}}+\frac{4}{3} -2$
= $\frac{\frac{5}{4}}{\frac{3}{4}}+\frac{4}{3} -2$
= $\frac{5}{3}+\frac{4}{3} -2$
= $\frac{9}{3} -2$
= 3 – 2
= 1
Hence, the correct answer is 1.
 

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