List of practice Questions

Consider the following logical statements:
R: If \(p \rightarrow q\) is false, then \(p \vee q\) is false.
S: If \(p \leftrightarrow q\) is false, then \(p \vee q\) is false.
Evaluate the truth values of R and S.

If the statement $(p \wedge q) \rightarrow (r \vee \neg s)$ is False (F), then the truth values of $p, q, r$ and $s$ are respectively

Identify the truth value of the statement pattern \( (p \land \sim q) \rightarrow r \).

Write the converse, inverse, and contrapositive of: "If voltage increases, then current decreases".

Write the dual of the statement \( p \land [\neg q \lor (p \land q) \lor \neg r] \).

For the propositions p and q, consider the following statements:
(I) \((p \vee q) \wedge (\neg p \vee \neg q)\) is a tautology.
(II) \((p \vee q) \wedge (\neg p \wedge \neg q)\) is a contradiction.
Choose the correct answer:

The correct simplified circuit diagram for the logical statement \[ \left[ (q \land ( \sim q \lor r)) \land (\sim p \lor (p \land r)) \right] \] is

If \( \{ (p \land \sim q) \land (p \land r) \} \rightarrow (p \lor q) \) has truth value false, then truth values of the statements \( p, q, r \) are respectively

Which of the following statement is a tautology?

\( \int_{0}^{1} \log (\frac{1}{x} - 1) dx = \)}

The differential equation whose solution is \( Ax^2 + By^2 = 1 \), where A and B are arbitrary constants is of}

$\int \frac{\sin 2x}{(a+b\cos x)^2} dx =$

If $3 \sin \alpha = 5 \sin \beta$, then $\tan \left( \frac{\alpha+\beta}{2} \right) + \tan \left( \frac{\alpha-\beta}{2} \right) =$

The circumradius of the triangle formed by the lines $xy + 2x + 2y + 4 = 0$ and $x + y + 2 = 0$ is

If $\bar{p} = 2\hat{\text{i}} + \hat{\text{k}}, \bar{q} = \hat{\text{i}} + \hat{\text{j}} + \hat{\text{k}}, \bar{r} = 4\hat{\text{i}} - 3\hat{\text{j}} + 7\hat{\text{k}}$ and a vector $\bar{\text{m}}$ is such that $\bar{\text{m}} \times \bar{q} = \bar{r} \times \bar{q}, \bar{\text{m}} \cdot \bar{p} = 0$, then $\bar{\text{m}} = .......$

The sum of the following infinite series is: \[ \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \frac{1}{5!} + \dots \]

Using the truth table, verify \( p \lor (q \land r) \equiv (p \lor q) \land (p \lor r) \).

Write the converse, inverse and contrapositive of the statement "If a triangle is equilateral then it is equiangular".

In the following figure, CD = 5 cm, BE = 10 cm, AE = 12 cm, \(\angle DAB = \angle DCB\), and \(\angle DAE = \angle DBC = 90^\circ\). Points AFCD create a rhombus. The length of BF (in cm) is:
\includegraphics[width=0.3\linewidth]{7.png}

The chart below shows the data of the number of cars bought by Millennials and Gen X people in a country from the year 2010 to 2020 as well as the yearly fuel consumption of the country (in Million liters). Considering the data presented in the chart, which one of the following options is true?
\includegraphics[width=0.5\linewidth]{8.png}

The assembly shown below has three teethed circular objects (Pinions) and two teethed flat objects (Racks), which are perfectly mating with each other. Pinions can only rotate clockwise or anti-clockwise staying at its own center. Racks can translate towards the left (\(\leftarrow\)) or the right (\(\rightarrow\)) direction. If the object \(A\) (Rack) is translating towards the right (\(\rightarrow\)) direction, the correct statement among the following is:
\includegraphics[width=0.5\linewidth]{9.png}