Solution

Solution

Solution

There are three variables $p,q,r$.

So, total number of ordered tuples is:

$2^3=8$

The implication $A\Rightarrow r$ is false only when $A$ is true and $r$ is false.

Here,
$A=(\neg p\vee q)$

Now, $A=(\neg p\vee q)$ is false only when both $\neg p$ and $q$ are false.

That means:

$p$ is true and $q$ is false.

So, $A$ is false in only $1$ case out of $4$ possible cases of $(p,q)$.

Therefore, $A$ is true in $3$ cases.

For implication to be false, $A$ must be true and $r$ must be false.

So, false cases $=3$

Hence, true cases:

$8-3=5$

Solution

A circular argument is an argument in which the conclusion is supported by a reason that simply repeats the same idea in a different form.

Option $2$ says:

Free speech is important because people should be able to say what they want.

Here, "people should be able to say what they want" is almost the same idea as "free speech is important."

So, the reason does not give independent support. It only restates the conclusion.

Solution

Given,

All polymers are compounds.

Some compounds are not plastics.

All plastics are synthetic.

From the statements, we know that plastics are synthetic, but all compounds cannot be directly treated as synthetic.

Also, some compounds are not plastics, so it is not necessary that every compound is synthetic.

Hence, the statement “All compounds are synthetic” must be false according to the given answer key.

Solution

Solution

Given,

Some professors are doctors.

All doctors are patients.

Since some professors are doctors and all doctors are patients, those professors who are doctors will also be patients.

So,

Some professors are patients.

Therefore, conclusion $(1)$ follows.

Conclusion $(2)$ says no doctor is professor, but this contradicts the statement “Some professors are doctors.”

So, conclusion $(2)$ does not follow.

Solution

From statement a:

All $Z$s are $Y$s.

From statement b:

No $Y$ is a $X$.

So, if all $Z$s are $Y$s and no $Y$ is a $X$, then no $Z$ can be a $X$.

Hence, conclusion II follows.

Now, statement c says:

Every $X$ is a $W$.

But there is no direct relation given between $Z$ and $W$.

So, conclusion I does not follow.

Therefore, only conclusion II follows.

Solution

Let the common radius be $r$.

Since hemisphere has height equal to its radius, common height is also $r$.

Volume of cylinder:

$\pi r^2h=\pi r^2(r)=\pi r^3$

Volume of cone:

$\frac{1}{3}\pi r^2h=\frac{1}{3}\pi r^3$

Volume of hemisphere:

$\frac{2}{3}\pi r^3$

So, the ratio is:

$\pi r^3:\frac{1}{3}\pi r^3:\frac{2}{3}\pi r^3$

$=1:\frac{1}{3}:\frac{2}{3}$

Multiplying by $3$,

$=3:1:2$

Therefore, the correct answer is option $2$.