Explanation

When a pair of dice are tossed, there are 36 possible outcomes.

The combinations that result in a total of 7 or 11 are:

Total 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6 outcomes

Total 11: (5,6), (6,5) = 2 outcomes

Total favorable outcomes = 6 + 2 = 8

Probability = (Number of favorable outcomes) / (Total possible outcomes)

Probability = 8/36 = 2/9

Explanation

1. Arrange Data in Ascending Order:

   $$\mathrm{<b><font\; color="\#0000ff">2.5</font></b>}<b><font\; color="\#0000ff">,</font></b>\mathrm{<b><font\; color="\#0000ff">2.8</font></b>}<b><font\; color="\#0000ff">,</font></b>\mathrm{<b><font\; color="\#0000ff">2.8</font></b>}<b><font\; color="\#0000ff">,</font></b>\mathrm{<b><font\; color="\#0000ff">2.9</font></b>}<b><font\; color="\#0000ff">,</font></b>\mathrm{<b><font\; color="\#0000ff">3.0</font></b>}<b><font\; color="\#0000ff">,</font></b>\mathrm{<b><font\; color="\#0000ff">3.3</font></b>}<b><font\; color="\#0000ff">,</font></b>\mathrm{<b><font\; color="\#0000ff">3.4</font></b>}<b><font\; color="\#0000ff">,</font></b>\mathrm{<b><font\; color="\#0000ff">3.6</font></b>}<b><font\; color="\#0000ff">,</font></b>\mathrm{<b><font\; color="\#0000ff">3.7</font></b>}<b><font\; color="\#0000ff">,</font></b>\mathrm{<b><font\; color="\#0000ff">4.0</font></b>}<b><font\; color="\#0000ff">,</font></b>\mathrm{<b><font\; color="\#0000ff">4.4</font></b>}<b><font\; color="\#0000ff">,</font></b>\mathrm{<b><font\; color="\#0000ff">4.8</font></b>}<b><font\; color="\#0000ff">,</font></b>\mathrm{<b><font\; color="\#0000ff">4.8</font></b>}<b><font\; color="\#0000ff">,</font></b>\mathrm{<b><font\; color="\#0000ff">5.2</font></b>}<b><font\; color="\#0000ff">,</font></b>\mathrm{<b><font\; color="\#0000ff">5.6</font></b>}$$

2. Find Median:

   • There are 15 data points (odd number), so the median is the 8th value in the ordered list.

   • The 8th value is 3.6.

Explanation

A consistent system of linear equations has at least one solution.

If there is exactly one solution, the system is independent.

If there are infinitely many solutions, the system is dependent.

Explanation

1. Calculation: First, compute -2u + 2v:

   • $-2u=(-4,4,-6)$

   • $2v=(2,-6,8)$

   • $-2u+2v=(-4+2,4-6,-6+8)=(-2,-2,2)$

2. Norm Calculation: The norm (magnitude) of $-2u+2v$ is:

   $$\mathrm{<b><font\; color="\#0000ff">\mid </font></b>}\mathrm{<b><font\; color="\#0000ff">\mid </font></b>}<b><font\; color="\#0000ff">-</font></b>\mathrm{<b><font\; color="\#0000ff">2</font></b>}\mathrm{<b><font\; color="\#0000ff">u</font></b>}<b><font\; color="\#0000ff">+</font></b>\mathrm{<b><font\; color="\#0000ff">2</font></b>}\mathrm{<b><font\; color="\#0000ff">v</font></b>}\mathrm{<b><font\; color="\#0000ff">\mid </font></b>}\mathrm{<b><font\; color="\#0000ff">\mid </font></b>}<b><font\; color="\#0000ff">=</font></b>\sqrt{<b><font\; color="\#0000ff">(</font></b><b><font\; color="\#0000ff">-</font></b>\mathrm{<b><font\; color="\#0000ff">2</font></b>}{<b><font\; color="\#0000ff">)</font></b>}^{\mathrm{<b><font\; color="\#0000ff">2</font></b>}}<b><font\; color="\#0000ff">+</font></b><b><font\; color="\#0000ff">(</font></b><b><font\; color="\#0000ff">-</font></b>\mathrm{<b><font\; color="\#0000ff">2</font></b>}{<b><font\; color="\#0000ff">)</font></b>}^{\mathrm{<b><font\; color="\#0000ff">2</font></b>}}<b><font\; color="\#0000ff">+</font></b>{\mathrm{<b><font\; color="\#0000ff">2</font></b>}}^{\mathrm{<b><font\; color="\#0000ff">2</font></b>}}}<b><font\; color="\#0000ff">=</font></b>\sqrt{\mathrm{<b><font\; color="\#0000ff">4</font></b>}<b><font\; color="\#0000ff">+</font></b>\mathrm{<b><font\; color="\#0000ff">4</font></b>}<b><font\; color="\#0000ff">+</font></b>\mathrm{<b><font\; color="\#0000ff">4</font></b>}}<b><font\; color="\#0000ff">=</font></b>\sqrt{\mathrm{<b><font\; color="\#0000ff">12</font></b>}}<b><font\; color="\#0000ff">=</font></b>\mathrm{<b><font\; color="\#0000ff">2</font></b>}\sqrt{\mathrm{<b><font\; color="\#0000ff">3</font></b>}}$$

Answer: $$23

Explanation

An order on a set "S" is typically denoted by the symbol "≤" (less than or equal to).

It representing a relation that establishes a sense of ordering between elements within the set. 

Explanation

A compound proposition that is always true, regardless of the truth values of its components, is called a tautology.

The statement "P ∨ ¬P" (P or not P) is a tautology because it is always true, whether P is true or false.

Explanation

The phrase "Castles in the air" refers to fanciful schemes or unrealistic plans that are unlikely to be achieved.

It is often used to describe dreams or ideas that are imaginative but not grounded in reality.

Explanation

Eliminate means ختم کریں

Explanation

The correct verb tenses to use are "was stolen" and "was offered".

This indicates that the bike was stolen (past event) and the reward was offered (past event) to recover it.

"My bike which was stolen was brought back only when a 20-pound-reward was offered."

میری موٹرسائیکل جو چوری کی گئی تھی اسے صرف اس وقت واپس لایا گیا جب 20 پاؤنڈ کا بدلہ پیش کیا گیا تھا۔

Explanation