Mathematical Reasoning Test

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Mathematical Reasoning Test - 16

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Question 1
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The contrapositive of "if in a triangle $$ABC, AB=AC$$, then $$\angle B=\angle C$$", is

A
If in a triangle $$ABC, \angle B=\angle C$$, then $$AB=AC$$.

B
If in a triangle $$ABC, AB\neq AC$$, then $$\angle B\neq\angle C$$.

C
If in a triangle $$ABC, \angle B\neq\angle C$$, then $$AB\neq AC$$.

D
If in a triangle $$ABC, \angle B\neq\angle C$$, then $$AB=AC$$.

Solution

Take $$p:AB=AC$$
and $$q:\angle B=\angle C$$
So the given statement is symbolically represented as $$p\rightarrow q$$
Now by definition, Contrapositive of a conditional statement $$p\rightarrow q$$ is $$\sim q\rightarrow \sim p$$
$$\sim q:\angle B\neq \angle C$$
and $$\sim p:AB\neq AC$$
Thus $$\sim q\rightarrow \sim p$$ is given by
If in a $$\triangle ABC, \angle B\neq \angle C$$ then $$AB\neq AC$$.
Question 2
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$$p$$: He is hard working.
$$q$$: He is intelligent.
Then $$ \sim q\Rightarrow\sim p$$, represents

A
If he is hard working, then he is not intelligent.

B
If he is not hard working, then he is intelligent.

C
If he is not intelligent, then he is not had working.

D
If he is not intelligent, then he is hard working.

Solution

p:she is hardworking
q:she is intelligent

~p:she is not hardworking
~q:she is not intelligent

~q=>~p means She is not intelligent implies she is not hardworking
Hence, Option C
Question 3
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$$p:$$ He is hard working.
$$q:$$ He will win.
The symbolic form of "If he will not win then he is not hard working", is

A
$$ p\Rightarrow q$$

B
$$ (\sim p)\Rightarrow (\sim q)$$

C
$$ (\sim q)\Rightarrow (\sim p)$$

D
$$ (\sim q)\Rightarrow p$$

Solution

Given $$p:$$ He is hard working
and $$q:$$ He will win
we get $$\sim p:$$ He is not hard working
and $$\sim q:$$ He will not win
Now the given statement in the question is "If he will not win then he is not hard working"
For this conditional statement, the symbolic form is $$\left( \sim q \right) \Rightarrow \left( \sim p \right) $$
Question 4
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The converse of: "If two triangles are congruent then they are similar" is

A
If two triangles are similar then they are congruent.

B
If two triangles are not congruent then they are not similar.

C
If two triangles are not similar then they are not congruent.

D
None

Solution

The converse of "If P then Q" is "If Q then P".
Question 5
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$$p\wedge (\sim p) \Rightarrow p$$ is

A
a tautology

B
a contradiction

C
neither tautology nor contradiction

D
none of these

Solution

A proposition is said to be a $$contradiction$$ if its truth value is $$F$$ for any assignment of truth values to its components.
From the above truth table, we can clearly see that the answer is $$B$$.
Question 6
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The contrapositive of $$ p\Rightarrow q$$, is

A
$$ p\Rightarrow q$$

B
$$ q\Rightarrow p$$

C
$$ \sim p\Rightarrow \sim q$$

D
$$ \sim q\Rightarrow \sim p$$

Solution

The contrapositive of "If $$p$$ then $$q$$" is "If not $$q$$ then not $$p$$" which is represented by $$\sim q\Rightarrow \sim p$$.
Question 7
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The inverse of "If $$x$$ has courage, then $$x$$ will win", is

A
If $$x$$ will win, then $$x$$ has courage.

B
If $$x$$ has no courage, then $$x$$ will not win.

C
If $$x$$ will not win, then $$x$$ has no courage.

D
If $$x$$ will not win, then $$x$$ has courage.

Solution

Take $$p:x$$ has courage
and $$q:x$$ will win
So the given conjugation is $$p\Rightarrow q$$
Now we need to find Inverse of this.
Be definition, Inverse of $$p\Rightarrow q$$ is $$\sim p\longrightarrow \sim q$$
$$\sim p:x$$ has no courage and
$$\sim q:x$$ will not win
Thus the inverse $$\sim p\longrightarrow \sim q$$ is symbolic form of "If $$x$$ has no courage, then $$x$$ will not win.
Question 8
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$$p:$$ He is hard working.
$$q:$$ He will win.
The symbolic form of "He is hard working then he will win", is

A
$$p\vee q$$

B
$$p\wedge q$$

C
$$p \Rightarrow q$$

D
$$q \Rightarrow p$$

Solution

Given $$p:$$ He is hard working
and $$q:$$ He will win
He is hard working then he will win is a conditional statement.
If $$p$$ happens then $$q$$ will happen.
The symbolic form of this condition is $$p\Rightarrow q$$
Question 9
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The converse of $$ p\Rightarrow q$$, is

A
$$ p\Rightarrow q$$

B
$$ q\Rightarrow p$$

C
$$ \sim p\Rightarrow \sim q$$

D
$$ \sim q\Rightarrow \sim p$$

Solution

Given conditional statement is "If p then q".
The converse is "If q then p."
Symbolically, the converse of $$p\Rightarrow q$$ is $$q\Rightarrow p$$.
A conditional statement is not logically equivalent to its converse.
Hence, option B.
Question 10
1 / -0

The converse of "If in a triangle $$ABC, AB=AC$$, then $$\angle B=\angle C$$", is

A
lf in a triangle $$ABC, \angle B=\angle C$$, then $$AB=AC$$.

B
lf in a triangle$$ABC, AB\neq AC$$, then $$\angle B\neq\angle C$$.

C
lf in a triangle $$ABC, \angle B\neq\angle C$$, then $$AB\neq AC$$.

D
lf in a triangle $$ABC, \angle B\neq\angle C$$, then $$AB=AC$$ .

Solution

Take $$p:AB=AC$$
and $$q:\angle B=\angle C$$
So the given statement is symbolically represented as $$p\rightarrow q$$
Now by definition, Converse of a conditional statement $$p\rightarrow q$$ is $$q\rightarrow p$$
So $$q\rightarrow p$$ is given by
"If in a triangle $$ABC, \angle B=\angle C,$$ then $$AB=AC.$$"

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Mathematical Reasoning Test - 16

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Mathematical Reasoning Test - 16

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Question 1

If in a triangle $$ABC, \angle B=\angle C$$, then $$AB=AC$$.

If in a triangle $$ABC, AB\neq AC$$, then $$\angle B\neq\angle C$$.

If in a triangle $$ABC, \angle B\neq\angle C$$, then $$AB\neq AC$$.

If in a triangle $$ABC, \angle B\neq\angle C$$, then $$AB=AC$$.

Solution

Question 2

If he is hard working, then he is not intelligent.

If he is not hard working, then he is intelligent.

If he is not intelligent, then he is not had working.

If he is not intelligent, then he is hard working.

Solution

Question 3

$$ p\Rightarrow q$$

$$ (\sim p)\Rightarrow (\sim q)$$

$$ (\sim q)\Rightarrow (\sim p)$$

$$ (\sim q)\Rightarrow p$$

Solution

Question 4

If two triangles are similar then they are congruent.

If two triangles are not congruent then they are not similar.

If two triangles are not similar then they are not congruent.

None

Solution

Question 5

a tautology

a contradiction

neither tautology nor contradiction

none of these

Solution

Question 6

$$ p\Rightarrow q$$

$$ q\Rightarrow p$$

$$ \sim p\Rightarrow \sim q$$

$$ \sim q\Rightarrow \sim p$$

Solution

Question 7

If $$x$$ will win, then $$x$$ has courage.

If $$x$$ has no courage, then $$x$$ will not win.

If $$x$$ will not win, then $$x$$ has no courage.

If $$x$$ will not win, then $$x$$ has courage.

Solution

Question 8

$$p\vee q$$

$$p\wedge q$$

$$p \Rightarrow q$$

$$q \Rightarrow p$$

Solution

Question 9

$$ p\Rightarrow q$$

$$ q\Rightarrow p$$

$$ \sim p\Rightarrow \sim q$$

$$ \sim q\Rightarrow \sim p$$

Solution

Question 10

lf in a triangle $$ABC, \angle B=\angle C$$, then $$AB=AC$$.

lf in a triangle$$ABC, AB\neq AC$$, then $$\angle B\neq\angle C$$.

lf in a triangle $$ABC, \angle B\neq\angle C$$, then $$AB\neq AC$$.

lf in a triangle $$ABC, \angle B\neq\angle C$$, then $$AB=AC$$ .

Solution

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