Plus One Maths Chapter Wise Questions and Answers Chapter 14 Mathematical Reasoning

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Kerala Plus One Maths Chapter Wise Questions and Answers Chapter 14 Mathematical Reasoning

Plus One Maths Mathematical Reasoning Three Mark Questions and Answers

Question 1.
Write the contrapositive and converse of the following statements: (3 score each)

1. If x is a prime number, then x is odd.
2. If the two lines are parallel, then they do not intersect in the same plane.
3. Something is cold implies that it has low temperature.
4. You cannot comprehend geometry if you do not know how to reason deductively.
5. x is an even number implies that x is divisible by 4.

Answer:
1. Here p: x is a prime number.
q: x is odd.
Now, ~ p: x is not a prime number.
~ q: x is not odd.
The contrapositive statement is:
If x is not odd x is not a prime number.
The converse statement is:
If x is an odd, then x is a prime number.

2. Here p: Two lines are parallel.
q: They do not intersect in the same plane.
Now, ~ p: Two lines are not parallel.
~ q: They intersect in the same plane.
The contrapositive statement is:
If two intersect in the same plane then they are not parallel.
The converse statement is:
If the two lines do not intersect in the same plane, then they are parallel.

3. Here p: Something is cold.
q: It has low temperature.
Now, ~ p: Something is not cold.
~ q: It does not have low temperature.
The contrapositive statement is:
If something does not have low temperature then it is not cold.
The converse statement is:
If something has low temperature then it is cold.

4. Here p: You cannot comprehend geometry.
q: You do not know how to reason deductively.
Now, ~ p: You can comprehend geometry.
~ q: You know how to reason deductively.
The contrapositive statement is:
If you know how to reason deductively, then you can comprehend geometry.
The converse statement is:
If you do not know how to reason deductively then you cannot comprehend geometry.

5. Here p: x is an even number.
q: x is divisible by 4.
Now, ~ p: x is not an even number.
~ q: x is not divisible by 4.
The contrapositive statement is:
If x is not divisible by 4 then x is not an even number.
The converse statement is:
If x is divisible by 4 then x is an even number.

Question 2.
Show that the statement: “For any real numbers a and b, a² = b² implies that a = b” is not true by giving a counter-example.
Answer:
The given compound statement is of the form “if p then q”.
We assume that p is true then
a, b ∈ R such that a² = b²
Let us take a = -3 and b = 3
Now a² = b² but a ≠ b
So when p is true, q is false.
Thus the given compound statement is not true.

Plus One Maths Mathematical Reasoning Four Mark Questions and Answers

Question 1.
State whether the or used in the following statements is “exclusive” or “inclusive”. Give reasons for your answer.

1. A rectangle is a quadrilateral or a pentagon. (2)
2. A square is a polygon or a parallelogram. (2)

Answer:
1. This statement make use of exclusive ‘or’. Since a geometrical figure cannot be a quadrilateral and also a pentagon.

2. This statement make use of inclusive ‘or’. Since a geometrical figure can be a polygon as well as a parallelogram.

Plus One Maths Mathematical Reasoning Six Mark Questions and Answers

Question 1.
Show that the statement p: “If x is a real number such that x³ + 4x = 0 then x is 0” is true by

1. Direct method. (2)
2. Method of contradiction. (2)
3. Method of contrapositive. (2)

Answer:
The given compound statement is of the from “if p then q”.
P: x ∈ R such that x³ + 4x = 0
q: x = 0

1. Direct method:
We assume that p is true then
x ∈ R such that x³ + 4x = 0
x ∈ R such that x(x² + 4) = 0
x ∈ R such that x = 0 or x² + 4 = 0
⇒ x = 0
⇒ q is true.
So when p is true, q is true.
Thus the given compound statement is true.

2. Method of contradiction:
We assume that p is true and q is false, then x ≠ 0
x ∈ R such that x³ + 4x = 0
x ∈ R such that x(x² + 4) = 0
x ∈ R such that x = 0 or x² + 4 = 0
⇒ x = 0
Which is a contradiction. So our assumption that x ≠ 0 is false. Thus the given compound statement is true.

3. Method of contrapositive.
We assume that q is false, then x ≠ 0
⇒ x ∈ R such that x³ + 4x ≠ 0
⇒ q is false
So when q is false, p is false.
Thus the given compound statement is true.

Plus One Maths Mathematical Reasoning Practice Problems Questions and Answers

Question 1.
Which of the following sentences are statements: (1 score each)

1. There are 35 days in a month.
2. Mathematics is difficult.
3. The sum of 5 and 7 is greater than 10.
4. The square of a number is an even number.
5. The sides of a quadrilateral have equal length.
6. Answer this question
7. The product of -1 and 8 is -8.
8. The sum interior angles of a triangle is 180.
9. Today is a windy day.
10. All real numbers are complex numbers.

Answer:

1. No month have 35 days. Hence it is a statement.
2. Here the correctness of the sentence depends upon the observer. So the sentence is not a statement.
3. The sentence is true so it is a statement.
4. Here the correctness of the sentence depends upon the observer. So the sentence is not a statement.
5. This sentence is sometimes true and some time false. So the sentence is not a statement.
6. This sentence is an order. So the sentence is not a statement.
7. The sentence is true so it is a statement.
8. The sentence is true so it is a statement.
9. It is not clear from the context which day is referred. Hence not a statement.
10. The sentence is true, so it is a statement.

Question 2.
Write the negation of the following statements: (1 score each)

1. Chennai is the capital of Tamil Nadu.
2. \(\sqrt{2}\) is not a complex number.
3. All triangles are not equilateral triangle.
4. The number 2 is greater than 7.
5. Every natural number is an integer.

Answer:

1. Negation: Chennai is not the capital of Tamil Nadu.
2. Negation: \(\sqrt{2}\) is a complex number.
3. Negation: All triangles are equilateral triangle.
4. Negation: The number 2 is not greater than 7.
5. Negation: Every natural number is not an integer.

Question 3.
Find the component statements of the following compound statements and check they are true or false. (1 score each)

1. Number 3 is prime or it is odd.
2. All integers are positive or negative.
3. 100 is divisible by 3,11 and 5.
4. The sun shines or it rains.
5. India is a democracy and a monarchy.

Answer:
1. The component statements are
p: Number 3 is prime
q: Number 3 is odd.
Both the component statements p and q are true.

2. The component statements are
p: All integers are positive
q: All integers are negative.
Both the component statements p and q are true.

3. The component statements are
p: 100 is divisible by 3.
q: 100 is divisible by 11.
r: 100 is divisible by 5.
The component statements p and q are false, whereas r is true.

4. The component statements are
p: The sun shines
q: It rains.
Both the component statements p and q are true.

5. The component statements are
p: India is a democracy.
q: India is a monarchy.
Both the component statements p is true, whereas the component statement q is false.

Question 4.
For each of the following compound statement first identify the connecting words and then break it into component statement: (2 score each)

1. All rational numbers are real and all real numbers are not complex.
2. Square of an integer is positive or negative.
3. The sand heats up quickly in the sun and does not cool down fast at night.
4. x = 2 and x = 3 are the roots of the equation 3x² – x – 10 = 0

Answer:
1. The component statement has the connecting word ‘and’ component statements are
p: All rational numbers are real
q: All real numbers are not complex.

2. The component statement has the connecting word ‘or’ component statements are
p: Square of an integer is positive.
q: Square of an integer is negative.

3. The component statement has the connecting word ‘and’ component statements are
p: The sand heats up quickly in the sun.
q: The sand does not cool down fast at night.

4. The component statement has the connecting word ‘and’ component statements are
p: x = 2 are the roots of the
equation 3x² – x – 10 = 0.
q: x = 3 are the roots of the equation 3x² – x – 10 = 0.