Correct Answer: 6

Solution : The car runs in the same direction. So, the relative speed is (35 – 25) = 10 km/h.
Use the formula, $\text{Time}=\frac{\text{Distance}}{\text{speed}}$ to find the required time that two cars take to meet.
${\text{Time}}=\frac{60}{10}=6$ hours
Hence, the correct answer is 6 hours.

Correct Answer: inscriptions

Solution : The first option is correct.

• Let's have a look at the 7th sentence from the given paragraph:
  • "Documentary sources include mainly inscriptions and papyri."
• Upon perusal of the above statement, it can be concluded that inscriptions are included in the documentary sources of the history of the Roman Empire.

Hence, the correct answer is inscriptions.

Correct Answer: (3)

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

The preposition "to" is needed after "answers" to correctly convey the idea that there are many answers related to the topic. The preposition "about" doesn't convey the intended meaning of the sentence. In this context, the preposition "to" is used to show the relationship between the noun "answers" and the rest of the sentence.

Therefore, the correct sentence is: According to scientists, there are a lot of answers to it.

Correct Answer: (19, 13, 4)

Solution : Given:
(15, 9, 4); (7, 2, 3)

Here, (15, 9, 4)→(9 + 4) + 2 = 13 + 2 = 15
(7, 2, 3)→(2 + 3) + 2 = 5 + 2 = 7

Let's check the options –
First option: (21, 12, 6)→(12 + 6) + 2 = 18 + 2 = 20 ≠ 21
Second option: (19, 13, 4)→(13 + 4) + 2 = 17 + 2 = 19
Third option: (24, 9, 11)→(9 + 11) + 2 = 20 + 2 = 22 ≠ 24
Fourth option: (18, 7, 7)→(7 + 7) + 2 = 14 + 2 = 16 ≠ 18

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

Correct Answer: 2

Solution : Given:
$a+b+c=0$
$\therefore b+c=-a$
Now, $a+b+c=0$
By squaring both sides, we get,
$(a+b+c)^2=0$
⇒ $a^2+b^2+c^2+2(ab+bc+ca)=0$
⇒ $a^2+b^2+c^2+2[a(b+c)+bc]=0$
⇒ $a^2+b^2+c^2+2[a(-a)+bc]=0$
⇒ $a^2+b^2+c^2-2[a^2-bc]=0$
⇒ $a^2+b^2+c^2=2[a^2-bc]$
$\therefore\frac{a^2+b^2+c^2}{a^2-bc}=2$
Hence, the correct answer is $2$.

Correct Answer: 2

Solution : The correct improvement for the sentence is: has belonged.

Explanation: The original phrase "has been belonging" is not grammatically correct in this context. When expressing ownership or possession that has continued over a period of time, we typically use the present perfect tense. "Has belonged" is the correct form in this case.

has belonging

• It is not grammatically correct; it should be "has belonged."

belonged

• It is in the past tense and does not convey the ongoing nature of the family's ownership over generations as effectively as the present perfect tense.

Correct Answer: $\frac{19}{5}$

Solution : $\frac{\sec \theta-\tan \theta}{\sec \theta+\tan \theta}=\frac{1}{7}$
Using the identity $\sec \theta = \frac{1}{\cos \theta}$ and $\tan \theta = \frac{\sin \theta}{\cos \theta}$,
$\frac{\frac{1}{cos\theta}-\frac{\sin \theta}{\cos \theta}}{\frac{1}{cos\theta}+\frac{\sin\theta}{\cos\theta}}=\frac{1}{7}$
$⇒\frac{1-\sin \theta}{1+\sin \theta}=\frac{1}{7}$
On applying componendo and dividendo, 
$⇒\sin \theta = \frac{3}{4}$
Using the identity $\sin^2 \theta + \cos^2 \theta = 1$,
$⇒\cos \theta = \frac{\sqrt{7}}{4}$
Substituting these values into the expression $\frac{\operatorname{cosec} \theta+\cot ^2 \theta}{\operatorname{cosec} \theta-\cot ^2 \theta}$,
$=\frac{\frac{4}{3}+\left(\frac{\sqrt{7}}{3}\right)^2}{\frac{4}{3}-\left(\frac{\sqrt{7}}{3}\right)^2} =\frac{\frac{4}{3}+\frac{7}{9}}{\frac{4}{3}-\frac{7}{9}}
= \frac{19}{5}$
Hence, the correct answer is $\frac{19}{5}$.

Correct Answer: 24 metre east

Solution : Given: 

1. X and Y both start from the same point; X walks 17 m west. Then turns to his right and walks 13 meters.
2. At the same time. Y walks 9 m North, then turns East and walks 9 m north, then turns to his left and walks 4 m.

Where is Y now with respect to the position of X? 

So, the direction of Abhinav with respect to his house is North-West.

Hence, fourth option is correct.

Correct Answer: MKXU

Solution : Given:
OILS : UNLS :: PAIN : PKDT :: QUIK : ?

Add the consecutive natural numbers, in decreasing order (starting from 4) to the place value of the first three letters and 2 to the place value of the fourth letter of OILS and PAIN, and then reverse the order of the letters to get the required code.
OILS→O + 4 = S; I + 3 = L; L + 2 = N; S + 2 = U→SLNU
Now, reverse the order of the letters, SLNU→UNLS
So, OILS is related to UNLS.
PAIN→P + 4 = T; A + 3 = D; I + 2 = K; N + 2 = P→TDKP
Now, reverse the order of the letters, TDKP→PKDT
So, PAIN is related to PKDT.
Similarly, follow the same patter for QUIK→Q + 4 = U; U + 3 = X; I + 2 = K; K + 2 = M→UXKM
Now, reverse the order of the letters, UXKM→MKXU

So, QUIK is related to MKXU. Hence, the fourth option is correct.

Correct Answer:

Solution : Since the rotation of the figure is not allowed, we will check where the question figure can exactly fit itself in the given option figures. By comparison of all the option figures, the given question figure is embedded only in the fourth option figure.

Hence, the fourth option is correct.