And

Q. The compound statement $(P\vee Q)\wedge (\sim P)\Rightarrow Q$ is equivalent to :

1. $P\vee Q$
2. $\sim (P\Rightarrow Q)\iff P\wedge \sim Q$
3. $P\wedge \sim Q$
4. $\sim (P\Rightarrow Q)$

Q.

How many types of logical operators are there?

Q. Consider the following three statements:
$p:4$ and $7$ are coprimes.
$q:$ Every function on a set $A$ is a relation on $A$.
$r:6$ divides $3$.
Then the truth value of which of the following Boolean expressions is false?

1. $p\wedge (q\vee r)$
2. $\sim p\vee (\sim q\vee \sim r)$
3. $\sim p\wedge (q\vee \sim r)$
4. $(p\vee \sim q)\vee r$

Q. Negation of the Boolean expression $p\iff (q\Rightarrow p)$ is

Q.

Let $p$ be the proposition: Mathematics is interesting and let $q$ be the proposition that Mathematics is difficult, then the symbol $p\wedge q$ means

1. Mathematics is interesting implies that Mathematics is difficult

2. Mathematics is interesting implies and is implied by Mathematics is difficult

3. Mathematics is interesting and Mathematics is difficult

4. Mathematics is interesting or Mathematics is difficult

Q. Truth value for the statement "$5+7=10$ and $4+3=7$" is

1. Depends upon the truth values of sub statement.
2. $5+7=10⟺4+3=7$ can not be determined.
3. $T$
4. $F$

Q. The statement $(p\vee q)\wedge (\sim p\wedge \sim q)$ is

1. a contradiction
2. a contingency
3. logically equivalent to $x\vee (\sim x)$
4. a tautology

Q. Which among the following options represent correct usage of logical connectives in statements $p,q$
$p$ : A physical quantity is scalar or vector
$q$ : A natural number is even or odd.

1. ($p$) Inclusive OR, ($q$) Exclusive OR
2. ($p$) Exclusive OR, ($q$) Inclusive OR
3. ($p$) Inclusive OR , ($q$) Inclusive OR
4. ($p$) Exclusive OR, ($q$)Exclusive OR

Q.

The truth values of $p,q$ and$r$ for which $(p\wedge q)\vee (~r)$ has truth value $F$ are respectively

1. $F,T,F$

2. $F,F,F$

3. $T,T,T$

4. $T,F,F$

5. $F,F,T$

Q. Which among the following options correctly represnts truth table for $(\sim p)\wedge q$

1. $\begin{array}{ccc}p& q& (\sim p)\wedge q\\ T& T& F\\ T& F& F\\ F& T& T\\ F& F& F\end{array}$
2. $\begin{array}{ccc}p& q& (\sim p)\wedge q\\ T& T& F\\ T& F& T\\ F& T& F\\ F& F& F\end{array}$
3. $\begin{array}{ccc}p& q& (\sim p)\wedge q\\ T& T& T\\ T& F& F\\ F& T& F\\ F& F& F\end{array}$
4. $\begin{array}{ccc}p& q& (\sim p)\wedge q\\ T& T& F\\ T& F& F\\ F& T& F\\ F& F& F\end{array}$

Q. In the question below, a sentence has been given in Direct/Indirect speech. From the given alternatives, choose the one which best expresses the given sentence in Indirect/Direct speech.
''I will be twenty five tomorrow'', said Vidushi.

1. Vidushi is saying that she will be twenty five by tomorrow.
2. Vidushi said that she would be twenty five the next day.
3. Vidushi will say that she would be twenty five tomorrow.
4. Vidushi had told that she would be twenty five the next day.

Q. Let
$p:2$ is a prime number
$q:\cos {30}^{\circ }=\frac{1}{2}$
$r:{\sec }^{2}x+{\tan }^{2}x=1$
$s:\sqrt{7}$ is an irrational number
$u:{\pi }^2$ is greater than $10$

Then which among the following statements are valid

1. $p\vee q,q\vee s,r\vee s,s\vee u$
2. $p\wedge q,q\wedge s,r\wedge s,s\wedge u$
3. $p\vee q,q\vee r,r\vee s,s\vee u$
4. $p\vee q,q\wedge r,r\vee s,s\wedge u$

Q. The truth value of "$20+10=2$" and
"$20\times 10=200$" is

1. $F$
2. $T$
3. $T$ or $F$ based on sub statements truth value
4. cannot be determined

Q. Let $p$ be the proposition ''Mathematics is intresting'' and $q$ be the proposition that ''Mathematics is difficult'', then which among the following represents $p\wedge q$

1. Mathematics is interesting or mathematics is difficult.
2. Mathematics is interesting and mathematics is difficult.
3. Mathematics is interesting and mathematics is not difficult.
4. Mathematics is not interesting and mathematics is not difficult.

Q. Consider the following statements:
$p:2$ is a prime number
$q:\cos {30}^{\circ }=\frac{1}{2}$
$r:{\sec }^{2}x+{\tan }^{2}x=1$
$s:\sqrt{7}$ is an irrational number
$u:{\pi }^2$ is greater than $10$

Then which among the following contain only valid statements?

1. $p\vee q,q\vee r,r\vee s,s\vee u$
2. $p\vee q,q\wedge r,r\vee s,s\wedge u$
3. $p\vee q,q\vee s,r\vee s,s\vee u$
4. $p\wedge q,q\wedge s,r\wedge s,s\wedge u$

Q. The disjunction statement that can be formed from statements, " It is raining"; "The sun is shining", is

1. It is raining and the sun is shining
2. It is raining or the sun is shining
3. It is raining and the sun is not shining
4. It is not raining or the sun is not shining

Q. The statement among the following that is a tautology is:

1. $A\wedge (A\vee B)$
2. $B\to [A\wedge (A\to B\left)\right]$
3. $A\vee (A\wedge B)$
4. $[A\vee (A\to B\left)\right]\to B$

Q. The value of the determinant $\left|\begin{array}{ccc}bc& ca& ab\\ p& q& r\\ 1& 1& 1\end{array}\right|,$ where a, b, c are the $p$th, $q$th and $r$th terms of a $HP$, is

1. $ap+bq+cr$
2. $(a+b+c)(p+q+r)$
3. none of these
4. $0$

Q. If $B=A$, $|P|=|Q|=1$, then $adj(Q^{-1}{BP}^{-1)}$ is

1. $P.Q$
2. $Q.adj\left(A\right).P$
3. $P.adj\left(A\right).Q$
4. $P.A^{-1}.Q$

Q. Let $\overrightarrow{p}$ and $\overrightarrow{q}$ be the position vectors of $P$ and $Q$ respectively with respect to $O$ and $|\overrightarrow{p}|=p,|\overrightarrow{q}|=q$. $R,S$ divide $P,Q$ Internally and externally in the ratio $2:3$ respectively. If $OR$ and $OS$ are perpendicular, then

1. $9p^2=4q^2$
2. $4p^2=9q^2$
3. $9p=4q$
4. $4p=9q$

Q. The truth value of "$20+10=2$" and
"$20\times 10=200$" is

1. $T$
2. $F$
3. $T$ or $F$ based on sub statements truth value
4. cannot be determined

Q. The contrapositive of $p\to (\sim q\to \sim r)$ is

1. $(\sim q\wedge r)\to \sim p$
2. $(q\to r)\to \sim p$
3. none of these.
4. $(q\vee \sim r)\to \sim p$

Q. The statement $(p\vee q)\wedge (\sim p\wedge \sim q)$ is

1. a contradiction
2. logically equivalent to $x\vee (\sim x)$
3. a tautology
4. a contingency

Q. Let $p:2$ is a prime number
$q:\cos {30}^{\circ }=\frac{1}{2}$
$r:{\sec }^{2}x+{\tan }^{2}x=1$
$s:\sqrt{7}$ is an irrational number
$u:{\pi }^2$ is greater than $10$

The statement which are all false is

1. $p\vee q,r\vee s,r\vee u$
2. $p\wedge q,q\vee r,p\vee u$
3. $p\wedge q,q\wedge s,s\wedge u$
4. $p\wedge q,q\vee r,r\vee s$

Q. If points $\hat{i}+\hat{j},\hat{i}-\hat{j}$ and $p\hat{i}+q\hat{j}+r\hat{k}$ are collinear, then

1. $r=0$
2. $p=1$
3. $q\in R$
4. $q\ne 1$

Q.

$(p\wedge \sim q)\wedge (\sim p\wedge q)$ is ___.

1. A tautology

2. A fallacy

3. Both a tautology and a fallacy

4. Neither a tautology nor a fallacy

Q. I: The total number of ways in which a selection (one or more) can be made of $p+q+r$ things of which $p$ are alike, $q$ are alike, $r$ are alike is $(p+1)(q+1)(r+1)-1$
II: The number of permutations of n things taken together when $p$ of the things are alike of one kind, $q$ of them alike of a second kind, $r$ of them alike of a third kind and the rest all different is $\frac{n!}{p!q!r!3!}$
Which of the above statements is true?

1. only I
2. only II
3. Both I and II
4. Neither I nor II

Q. If A is a $2\times 3$ matrix and B is $3\times 2$ matrix then the order of ${\left(AB\right)}^T$ is equal to the order of _______

1. $A^T{B}^T$
2. $AB$
3. BA
4. All of these

Q. $\left|\begin{array}{ccc}0& p-q& p-r\\ q-p& 0& q-r\\ r-p& r-q& 0\end{array}\right|$ is equal to

1. $0$
2. $p+q+r$
3. $p-q-r$
4. $-p+q+r$

Q. If $p:x$ is odd; $q:x^2$ is odd. Then "$x$ is odd and $x^2$ is not odd", is represented as

1. $p\wedge q$
2. $(\sim p)\vee q$
3. $p\wedge (\sim q)$
4. $p\vee q$