Implication

Q. Negation of the statement $p\to (q\vee r)$ is

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

Q. Three schools A, B and C organized a mela for collecting funds for helping the rehabilitation of flood victims. They sold hand made fans, mats and plates from recycled material at a cost of Rs. 25, Rs.100, and Rs. 50 each. The number of articles sold are given below:


Find the funds collected by each school separetly by selling the above articles. Also find the total funds collected for the purpose.
Write one value generated by the above situation.

Q. Divide sum of $\frac{65}{12}$ and $\frac{8}{3}$ by their differences.

Q. Negation of the compound proposition : If I study hard, then I will pass.

1. I will study hard or I shall pass.
2. I will study hard or I shall not pass.
3. I will study hard and I will not pass.
4. I will study hard and I will pass.

Q. If truth value for $p\to (q\vee r)$ is false, then the truth values of $p,q,r$ are respectively :

1. $F,F,F$
2. $T,F,F$
3. $F,F,T$
4. $T,T,F$

Q.

Which of the following is the inverse of the proposition: “If a number is a prime then it is odd”?

1. If a number is not a prime then it is odd

2. If a number is not a prime then it is not odd

3. If a number is not odd then it is not a prime

4. If a number is not odd then it is a prime

Q. The negation of the statement "if $5>7$ then, $6<4$"

1. $(5>7)\wedge (6\ge 4)$
2. $(5>7)\vee (6\le 4)$
3. $(5\le 7)\vee (6\le 4)$
4. $(5\le 7)\wedge (6\le 4)$

Q. The irrational number among the following is

1. $3.5$
2. $\sqrt{3}$
3. $\frac{5}{9}$
4. $0.007$

Q.

Which of the following is a contradiction?

1. $\left(p\wedge q\right)\wedge ~\left(p\vee q\right)$

2. $p\vee \left(~p\wedge q\right)$

3. $\left(p\Rightarrow q\right)\Rightarrow p$

4. None of these

Q.

Negation of the compound proposition: If the examination is difficult, then I shall pass if I study hard

1. The examination is difficult and I study hard and I shall pass

2. The examination is difficult and I study hard but I shall not pass

3. The examination is not difficult and I study hard and I shall pass

4. None of these

Q.

The proposition $\left(p\Rightarrow ~p\right)\wedge \left(~p\Rightarrow p\right)$ is a

1. Tautology and contradiction

2. Neither tautology nor contradiction

3. Contradiction

4. Tautology

Q. $f\left(x\right)=\frac{1}{q}$ if $x=\frac{p}{q}$ where p and q are integer and $q\ne 0$, G.C.D of (p, q) = 1 and f(x) = 0 if x is irrational then set of continuous points of f(x) is

1. All integers
2. All real numbers
3. All rational numbers
4. All irrational number

Q. Let $p$ and $q$ be any two logical statements and $r$ be the statement $p\to (\sim p\vee q).$ If $r$ has the truth value $\text{F},$ then the truth values of $p$ and $q$ are, respectively

1. $\text{F, F}$
2. $\text{T, T}$
3. $\text{F, T}$
4. $\text{T, F}$

Q. If the equation ax+2x-2=0 has real and distinct roots then value of a is?

Q. If $p,q,r$ have truth values $T,F,T$ respectively, then which among the following will have truth value as TRUE.

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

Q. The negation of the proposition: "If a number is divisible by $15$, then it is divisible by $5$ or $3$" is

1. If a number is divisible by $15$ then it is not divisible by $5$ and $3$
2. A number is divisible by $15$ and it is not divisible by $5$ and $3$.
3. A number is not divisible by $15$ or it is divisible by $5$ or $3$.
4. A number is not divisible by $15$ or it is not divisible by $5$ and $3$

Q. Let A be the set of two positive integers. Let f : A --> set of positive integers be defined by
f(n) = p, where p is the highest prime factor of n
If range of f = {3}. Find set A. Is A uniquely determined?

Q. If truth value for $p\to (q\vee r)$ is false, then the truth values of $p,q,r$ are respectively :

1. $T,F,F$
2. $F,F,F$
3. $F,F,T$
4. $T,T,F$

Q.

Which of the following is not true for any two statements $p$ and $q$?

1. $~\left[p\vee \left(~q\right)\right]\equiv \left(~p\right)\wedge q$

2. $~(p\vee q)=(~p)\vee (~q)$

3. $q\wedge ~q$ is contradiction

4. $~[p\wedge (~p\left)\right]$ is a tautology

Q. Negation of the statement "If $△ABC$ is isosceles then the base angle's $A$ and $B$ are equal" , is

1. $△ABC$ is isosceles and the base angles $A$ and $B$ are not equal.
2. $△ABC$ is not isosceles and the base angles $A$ and $B$ are not equal.
3. $△ABC$ is not isosceles and the base angles $A$ and $B$ are equal.
4. $△ABC$ is not isosceles or the base angles $A$ and $B$ are not equal.

Q. The logical statement $(p\Rightarrow q)\wedge (q\Rightarrow \sim p)$ is equivalent to

1. $\sim p$
2. $p$
3. $q$
4. $\sim q$

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

1. a fallacy
2. a tautology
3. logically equivalent to $p\wedge q$
4. neither a fallacy nor a tautology

Q. If $p\text{and}q$ are substatements "A natural number is odd " and "A natural number is not divisible by $2$"respectively, then the biconditional statement $p\iff q$ is

1. A natural number is odd, if and only if it is divisible by $2$.
2. A natural number is odd, if and only if it is not divisible by $2$.
3. If a natural number is odd, then it is not divisible by $2$.
4. If a natural number is odd, then it is divisible by $2$.

Q.

What is the abstract noun of the word Need?

Q. Which of the following is not equivalent form of compound statement formed by the logical connective "if and only if" using the sub statements $p$ and $q$

1. if $p,$ then $q$
2. $q$ if and only if $p$
3. $p$ if and only if $q$
4. $p$ is necessary and sufficient condition for $q$ and vice - versa

Q. Considering truth table for $p\to \sim (p\wedge \sim q)$ as follows,
$\begin{array}{ccc}p& q& p\to \sim (p\wedge \sim q)\\ T& T& A\\ T& F& B\\ F& T& C\\ F& F& D\end{array}$
The truth value will be false for which among the following

1. $A$
2. $B$
3. $C$
4. $D$

Q. Let p : Kiran passed the examination,
q : Kiran is sad
The symbolic form of a statement "It is not true that Kiran passed therefore he is sad' is

1. $(\sim p\to q)$
2. $(p\to q)$
3. $\sim (p\to \sim q)$
4. $\sim (p\leftrightarrow q)$

Q.

Show that area of the prallelogram whose diagonals are given by $\overrightarrow{a}and\overrightarrow{b}is\frac{\overrightarrow{a}\times \overrightarrow{b}}{2}$. Also, find the area of the parallelogram, whose diagonals are $2\hat{i}-\hat{j}+kand\hat{i}+3\hat{j}-\hat{k}.$

Q.

Can l' hospital rule be applied to function giving (infinity/infinity) form?

Q. ntFind the Bi-quadratic equation with rational coefficients whose one of the root is 2+-3n