Direct Method of Validation

Q. Consider the statement : ''$P\left(n\right):n^2-n+41$ is prime.'' Then which of the following is true ?

1. Both $P\left(3\right)$ and $P\left(5\right)$ are false.
2. $P\left(3\right)$ is false but $P\left(5\right)$ is true.
3. $P\left(5\right)$ is false but $P\left(3\right)$ is true.
4. Both $P\left(3\right)$ and $P\left(5\right)$ are true.

Q. The Boolean expression $(p\wedge q)\Rightarrow \left(\right(r\wedge q)\wedge p)$ is equivalent to

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

Q.

Let f: W → W be defined as f(n) = n − 1, if is odd and f(n) = n + 1, if n is even. Show that f is invertible. Find the inverse of f. Here, W is the set of all whole numbers.

Q.

If $x$ varies directly as a square of $y$ for $y=6,x=72$. Find $x$ for $y=9$.

Q.

Using properties of determinants, prove that:

Q. If the difference between two prime numbers is $59,$ then sum of their squares is

1. $3253$
2. $3485$
3. $3725$
4. $3965$

Q. If $(p\wedge \sim q)\wedge (p\wedge r)\to \sim p\vee q$ is false, then the truth values of $p,q$ and $r$ are, repectively :

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

Q. There are two prime numbers $p$ so that $5p$ can be expressed in the form of $\left[\frac{n^2}{5}\right]$ for some positive integer $n,$ where $[.]$ denotes the greatest integer function. The sum of these two prime numbers is

1. $52$
2. $50$
3. $48$
4. $54$

Q. If $(p\wedge \sim q)\wedge (p\wedge r)\to \sim p\vee q$ is false, then the truth values of $p,q$ and $r$ are, repectively :

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

Q. Let g (x) be the inverse of an invertible function f (x) which is derivable at x = 3. If f (3) = 9 and f' (3) = 9, write the value of g' (9).

Q.

Show that the following statement is true by the method of contrapositive.

p: If x is an integer and x² is even, then x is also even.

Q.

Using standard identities, evaluate the following : $10.2\times 99.6$

Q.

Given AB = 0. Which of the following options is correct

1. A = 0

2. B = 0

3. None of these

4. Either A or B should be 0

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

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

Q.

If $f\left(x\right)=$$\{\begin{array}{cc}x,& \text{when x is rational}\\ 0,& \text{when x is irrational}\end{array}$; $g\left(x\right)=$$\{\begin{array}{cc}0,& \text{when x is rational}\\ x,& \text{when x is irrational}\end{array}$then $(f-g)$ is

1. One-one onto
2. One-one not onto
3. Not one-one but onto
4. Not one-one not onto

Q.

Check whether the following ststement are true or not :

(i) p : If x and y are odd integers, then x + y is an even integer.

(ii) q : If x, y are integers such that xy is even, then at least one of x and y is an even integer.

Q.

Show that the statement

p : " if x is a real number such that $x^3+x=0$ then x is 0 " is true by

(i) direct method

(ii) method of contrapositive

(iii) method of contradition.

Q.

Using properties of determinants, prove that:

Q. If $n$ is a prime number greater than $5$, show that $n^4-1$ is divisible by $240$.

Q. For any $2\times 2$ square matrix $A$, if $A\left(adjA\right)=\left[\begin{array}{cc}10& 0\\ 0& 10\end{array}\right]$, then |A| is

Q.

If $(p\wedge \sim r)\Rightarrow (q\wedge r)$ is false and q and r are both false, then p is ___.

1. True

2. False

3. May be true or false

4. Cannot conclude from given the information

Q. Let $x\in Q,y\in Q^c$, which of the following statement is always WRONG ?

1. $xy\in Q^c$
2. $\sqrt{2}x+y\in Q$
3. $y/x\in Q$, whenever defined
4. $x/y\in Q^c$, whenever defined

Q.

Let f: W → W be defined as f(n) = n − 1, if is odd and f(n) = n + 1, if n is even. Show that f is invertible. Find the inverse of f. Here, W is the set of all whole numbers.

Q.

Find the truth value of the following statement : "If x, y $\in Z$ are such that x and y are odd, then xy is odd."

1. False

2. True

Q. Prove:
$p⟷q\equiv \sim (p\wedge \sim q)\wedge (q\wedge \sim p)$

Q. Given "If | x + 1 | = 4 , then x=3, -5 "
$\left(x+1\right|=4x+1=4\mathrm{or}x+1=-4x=3\mathrm{or}x=-5$
Which method has been used for proving the above statement ?

1. Contrapositive
2. Direct method
3. Contradiction
4. Counter example

Q. Consider the statement : ''$P\left(n\right):n^2-n+41$ is prime.'' Then which of the following is true?

1. Both $P\left(3\right)$ and $P\left(5\right)$ are false.
2. $P\left(3\right)$ is false but $P\left(5\right)$ is true.
3. $P\left(5\right)$ is false but $P\left(3\right)$ is true.
4. Both $P\left(3\right)$ and $P\left(5\right)$ are true.

Q. The value of $\sqrt{2+\sqrt{2+\sqrt{\mathrm{2............}}}}$ is :

Q. The inverse of "If $x\in A\cup B$, then $x\in A$ or $x\in B$'', is

1. If $x\in A$ or $x\in B$, then $x\in A\cup B$.
2. If $x\notin A\cup B$, then $x\notin A$ and $x\notin B$.
3. If $x\notin A$ and $x\notin B$, then $x\notin A\cup B$.
4. If $x\notin A$ and $x\notin B$, then $x\in A\cup B$.

Q.

Let p be the statement “x is an irrational number”, q be the statement “y is a transcendental number”, and r be the statement “x is a rational number if and only if y is a transcendental number”.
Statement-1 : $risequivalenttoeitherqorp$
Statememt-2 : $risequivalentto(p\leftrightarrow \sim q)$

1. Statement – 1 is false, Statement – 2 is true

2. Statement – 1 is true, Statement – 2 is true; Statement – 2 is a correct explanation for Statement -1

3. Statement – 1 is true, Statement – 2 is true; Statement -2 is not a correct explanation for Statement -1

4. Statement – 1 is true. Statement – 2 is false