Mathematical Reasoning Test

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Mathematical Reasoning Test - 23

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Weekly Quiz Competition

Question 1
1 / -0

Which of the following statements is the contrapositive of "If a polygon has four sides, then it is called a quadrilateral."?

A
If a polygon is called a quadrilateral, then it has four sides.

B
If a polygon is not called a quadrilateral, then it does not have four sides

C
If a polygon does not have four sides, then it is not called a quadrilateral.

D
None of these

Solution

Contrapositive will switch if and then and also add not to both parts.
Option B is correct.
Question 2
1 / -0

Which of the following statements is the contrapositive of "If you understand logic, you will be a good consumer."?

A
If you are a good consumer, then you do not understand logic.

B
If you are not a good consumer, then you do not understand logic.

C
If you do not understand logic, then you will not be a good consumer.

D
None of these

Solution

Contrapositive of $$p\rightarrow q$$ is given by $$\sim q\rightarrow \sim p$$
So contrapostive of given statement is given by,
"If you will not be a good consumer, then you don't understand logic"
Hence, option B is correct.
Question 3
1 / -0

I had Rs. 200 with me. I gave Rs. x to Anwar, Rs. $$\displaystyle \frac{x}{2}$$ to Vidhu and I am left with Rs $$\displaystyle \frac{x}{2}$$. The amount I gave to vidhu is ..........

A
Rs. 50

B
Rs. 40

C
Rs. 30

D
Rs. 15

Solution

Let us consider the problem:
$$\dfrac{{x + x}}{2} + \dfrac{x}{2} = 200$$
implies that,
$$\dfrac{{\left( {2x + x + x} \right)}}{2} = 200$$
implies that,
$$\dfrac{{4x}}{2} = 200$$
implies that,
$$2x = 200$$
implies that,
$$x = 100$$
Hence,the Vidhi got $$\dfrac{{100}}{2} = Rs50$$.
Question 4
1 / -0

"If Tom buys a red skateboard then Amanda buys green in-line skates". Which statement below is logically equivalent?

A
If Amanda does not buy green in-line skates then Tom does not buy a red skateboard

B
If Tom does not buy a red skateboard then Amanda does not buy green in-line skates

C
If Amanda buys green in-line skates then Tom buys a red skateboard

D
If Tom buys a red skateboard then Amanda does not buy green in-line skates

Solution

The original statement and its contrapositive are logically equivalent Remember that the contrapositive is the "converse of the inverse" --flip the "If....then" sections AND insert "NOTs"
Question 5
1 / -0

The statement "If x is divisible by 8 then it is divisible by 6" is false if x equals

A
6

B
14

C
32

D
48

Solution

A sentence in "If....then..." form is called an implication. The only time when an implication is false is when the "If" part of the sentence is true and the "then" part of the sentence is false. The number $$32$$ makes the first part of the statement true and the second part of the statement false.
Question 6
1 / -0

Which of the following is the negation of the statement, For all odd primes $$p < q$$ there exists positive non-primes $$r < s$$ such that $$p^2 + q^2 = r^2 + s^2$$.

A
For all odd primes $$p < q$$ there exists positive non-primes $$r < s$$ such that $$p^2 + q^2 = r^2 + s^2$$.

B
There exists odd primes $$p < q$$ such that for all positive non-primes $$r < s$$, $$p^2 + q^2 = r^2 + s^2$$.

C
There exists odd primes $$p < q$$ such that for all positive non-primes $$r < s$$, $$p^2 + q^2 \neq r^2 + s^2$$.

D
For all odd primes $$p < q$$ and for all positive non-primes $$r < s$$, $$p^2 + q^2 \neq r^2 + s^2$$.

Solution

For the inverse of a statement, "for all" and "there exists" are interchanged, and "$$=$$" changes to "$$\neq$$".

Therefore, the negative of the given statement will be:
There exists odd primes $$p<q$$ such that for all positive non-primes $$r<s$$, $$p^2+q^2 \neq r^2+s^2$$
Question 7
1 / -0

Negation of "Ram is in class $$X$$ or Rashmi is in class $$XII$$" is

A
Ram is not in class $$X$$ but Rashmi is in class $$XII$$

B
Ram is not in class $$X$$ and Rashmi is not in class $$XII$$

C
Either Ram is not in class $$X$$ or Rashmi is not in class $$XII$$

D
None of the above

Solution

Let $$p:$$ Ram is in class $$X$$

$$q:$$ Rashmi is in class $$XII$$

Given proposition is $$p\vee q$$

Its negation is $$\sim (p\vee q)=-p\wedge \sim q$$, Ram is not in class $$X$$ and Rashmi is not in class $$XII$$
Question 8
1 / -0

Consider the statement, if $$n$$ is divisible by $$30$$ then $$n$$ is divisible by $$2, 3$$ and by $$5$$. Which of the following statements is equivalent to this statement?

A
If $$n$$ is not divisible by $$30$$ then $$n$$ is divisible by $$2$$ or divisible by $$3$$ or divisible by $$5$$.

B
If $$n$$ is not divisible by $$30$$ then $$n$$ is not divisible by $$2$$ or not divisible by $$3$$ or not divisible by $$5$$.

C
If $$n$$ is divisible by $$2$$ and divisible by $$3$$ and divisible by $$5$$ then $$n$$ is divisible by $$30$$.

D
If $$n$$ is not divisible by $$2$$ or not divisible by $$3$$ or not divisible by $$5$$ then $$n$$ is not divisible by $$30$$.

Solution

If $$n$$ is not divisible by $$2$$ or divisible by $$3$$ or not divisible by $$5$$ then $$n$$ is divisible by $$30$$.
Question 9
1 / -0

Let p : A triangle is equilateral, q : A triangle is equiangular then inverse of $$\displaystyle q\rightarrow p$$ is

A
If a triangle is not equilateral then it is not equiangular

B
If a triangle is not equiangular then it is not equilateral

C
If a triangle is equiangular then it is not equilateral

D
If a triangle is equiangular then it is equilateral

Solution

Here, p is A triangle is equilateral and q is A triangle is equiangular.
The inverse of $$q \rightarrow p$$ is $$\sim q \rightarrow \sim p$$
On the same lines, the inverse statement becomes If a triangle is not equiangular then it is not equilateral.

Question 10

1 / -0

The statement form $$(p \Leftrightarrow r) \Rightarrow (q \Leftrightarrow r)$$ is equivalent to

$$[(\sim p \vee r)\wedge (p \vee \sim r)] \vee \sim [(\sim q \vee r)\wedge (q \vee \sim r)]$$

$$\sim [(\sim p\vee r)\wedge (p\vee \sim r)]\wedge [(\sim q\vee r) \vee(q\vee \sim r)]$$

$$[(\sim p\vee r)\wedge (\sim p \vee \sim r)] \wedge [(\sim q\vee r)\wedge (q\vee \sim r)]$$

$$\sim[(\sim p\vee r) \wedge (p\vee \sim r)]\wedge [(\sim q\vee r)\wedge (q \vee \sim r)]$$

Solution

$$p$$	$$q$$	$$r$$	$$p\leftrightarrow r$$	$$q\leftrightarrow r$$	$$\left( p\leftrightarrow r \right) \rightarrow \left( q\leftrightarrow r \right) $$
T	T	T	T	T	T
T	T	F	F	F	T
T	F	T	T	F	F
T	F	F	F	T	T
F	T	T	F	T	T
F	T	F	T	F	F
F	F	T	F	F	T
F	F	F	T	T	T

$$p$$

$$q$$

$$r$$

$$\sim p$$

$$\sim q$$

$$\sim r$$

$$\sim p\vee r$$

$$\left( p\vee \sim r \right) $$

$$\left[ \left( \sim p\vee r \right) \wedge \left( p\vee \sim r \right) \right]=a $$

$$\sim q\vee r$$

$$q\vee \sim r$$

$$[\left( \sim q\vee r \right) \wedge \left( q\vee \sim r \right)]=b$$

$$a\wedge b$$

$$\sim \left( a\wedge b \right) $$

The statement form,

$$\left( p\leftrightarrow r \right) \rightarrow \left( q\leftrightarrow r \right) $$

$$\therefore $$ The statement $$\left( p\leftrightarrow r \right) \rightarrow \left( q\leftrightarrow r \right) $$ is not equivalent to $$\sim \left[ \left( \sim p\vee r \right) \wedge \left( p\vee \sim r \right) \right] \wedge \left[ \left( \sim q\vee r \right) \wedge \left( q\vee \sim r \right) \right] $$

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Mathematical Reasoning Test - 23

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Mathematical Reasoning Test - 23

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Question 1

If a polygon is called a quadrilateral, then it has four sides.

If a polygon is not called a quadrilateral, then it does not have four sides

If a polygon does not have four sides, then it is not called a quadrilateral.

None of these

Solution

Question 2

If you are a good consumer, then you do not understand logic.

If you are not a good consumer, then you do not understand logic.

If you do not understand logic, then you will not be a good consumer.

None of these

Solution

Question 3

Rs. 50

Rs. 40

Rs. 30

Rs. 15

Solution

Question 4

If Amanda does not buy green in-line skates then Tom does not buy a red skateboard

If Tom does not buy a red skateboard then Amanda does not buy green in-line skates

If Amanda buys green in-line skates then Tom buys a red skateboard

If Tom buys a red skateboard then Amanda does not buy green in-line skates

Solution

Question 5

6

14

32

48

Solution

Question 6

For all odd primes $$p < q$$ there exists positive non-primes $$r < s$$ such that $$p^2 + q^2 = r^2 + s^2$$.

There exists odd primes $$p < q$$ such that for all positive non-primes $$r < s$$, $$p^2 + q^2 = r^2 + s^2$$.

There exists odd primes $$p < q$$ such that for all positive non-primes $$r < s$$, $$p^2 + q^2 \neq r^2 + s^2$$.

For all odd primes $$p < q$$ and for all positive non-primes $$r < s$$, $$p^2 + q^2 \neq r^2 + s^2$$.

Solution

Question 7

Ram is not in class $$X$$ but Rashmi is in class $$XII$$

Ram is not in class $$X$$ and Rashmi is not in class $$XII$$

Either Ram is not in class $$X$$ or Rashmi is not in class $$XII$$

None of the above

Solution

Question 8

If $$n$$ is not divisible by $$30$$ then $$n$$ is divisible by $$2$$ or divisible by $$3$$ or divisible by $$5$$.

If $$n$$ is not divisible by $$30$$ then $$n$$ is not divisible by $$2$$ or not divisible by $$3$$ or not divisible by $$5$$.

If $$n$$ is divisible by $$2$$ and divisible by $$3$$ and divisible by $$5$$ then $$n$$ is divisible by $$30$$.

If $$n$$ is not divisible by $$2$$ or not divisible by $$3$$ or not divisible by $$5$$ then $$n$$ is not divisible by $$30$$.

Solution

Question 9

If a triangle is not equilateral then it is not equiangular

If a triangle is not equiangular then it is not equilateral

If a triangle is equiangular then it is not equilateral

If a triangle is equiangular then it is equilateral

Solution

Question 10

$$[(\sim p \vee r)\wedge (p \vee \sim r)] \vee \sim [(\sim q \vee r)\wedge (q \vee \sim r)]$$

$$\sim [(\sim p\vee r)\wedge (p\vee \sim r)]\wedge [(\sim q\vee r) \vee(q\vee \sim r)]$$

$$[(\sim p\vee r)\wedge (\sim p \vee \sim r)] \wedge [(\sim q\vee r)\wedge (q\vee \sim r)]$$

$$\sim[(\sim p\vee r) \wedge (p\vee \sim r)]\wedge [(\sim q\vee r)\wedge (q \vee \sim r)]$$

Solution

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