Explanation

When two circles touch each other externally, they have a single point of contact.

The distance between their centers (C1C2) is equal to the sum of their radii (r1 + r2).

Explanation

When an object is placed at the principal focus (F) of a convex lens, the rays emerge parallel after refraction.

Parallel rays appear to meet at infinity, so the image is formed at infinity (very far away).

Explanation

Arrive : Depart is a pair of opposites.

Similarly, Use and Misuse are opposites — one is proper use, the other is wrong use.

Explanation

Given y = 2/x

Let's find the first and second derivatives:

dy/dx = -2/x²

d²y/dx² = 4/x³

Now, let's evaluate y + 1/2 * d²y/dx²:

y + 1/2 * d²y/dx² = 2/x + 1/2 * 4/x³

= 2/x + 2/x³

= (2x² + 2)/x³

= 2(x² + 1)/x³

= 2(1 + x²)/x³

Explanation

Networks are typically classified into four main types based on distance (geographical coverage):

1. PAN (Personal Area Network) – Very short distance (e.g., Bluetooth, USB connections).
2. LAN (Local Area Network) – Limited area (e.g., home, office, or school network).
3. MAN (Metropolitan Area Network) – City-wide coverage (e.g., ISP networks in a city).
4. WAN (Wide Area Network) – Large-scale, global reach (e.g., the Internet).

Explanation

In indirect speech, "Will you" becomes "if you would" after a reporting verb like "asked."

"Tomorrow" changes to "next day" in indirect speech.

Explanation

The phrase "might be" shows uncertainty or a chance that something could happen — a key marker of possibility.

The sentence implies that while it looks nice, there’s a possibility it is expensive.

Explanation

Begum Jahan Ara Shah Nawaz focused on the rights and upliftment of women in her writings.

She addressed educational and social reforms to promote gender equality and empowerment.

Born: 1896 | Died: 1979

Explanation

"In small cages" describes the noun "beast" by providing additional information about where the beast is.

This makes it an adjective phrase that modifies the noun "beast."

Explanation

At the peak of the Pakistan Movement, Gilgit-Baltistan was under the control of the Maharaja of Kashmir, as it was a part of the princely state of Jammu and Kashmir.

In 1947, following the partition, the region was later administered by Pakistan after the Kashmir conflict.