Important Questions Class 11 Maths Chapter 14
Important Questions Class 11 Mathematics Chapter 14
Important Questions for CBSE Class 11 Mathematics Chapter 14 – Mathematical Reasoning
Class 11 Chapter 14 Mathematics is concerned with mathematical reasoning. The chapter teaches students how to analyse given statements or hypotheses and draw conclusions about whether they are true or false. Mathematicians are typically involved in two types of reasoning.
- Inductive Reasoning
- Deductive Reasoning.
This chapter goes through deductive reasoning. Deductive reasoning is a type of reasoning that starts with a hypothesis or a broad statement. Through logical reasoning, a valid conclusion is then deduced.
Extramarks Important Questions for Class 11 Mathematics Chapter 14 are useful for students who want to study in a question-answer format. These questions are compiled by subject matter experts and written in an easy-to-understand language. These questions will help students effectively prepare for the Class 11 Mathematics exam.
CBSE Class 11 Mathematics Chapter-14 Important Questions
Study Important Questions for Class 11 Mathematics Chapter 14 – Mathematical Reasoning
Given below are Extramarks’ Important Questions for Class 11 Mathematics Chapter 14 for 4 marks each. The complete set of questions can be accessed by clicking the link provided.
4 Marks Questions
Q1. Give three examples of non-statement sentences. Give reasons for your assertions.
A1. The three examples of non-statement sentences are discussed below:-
(i) “Rani is a lovely young lady” is not a statement. This sentence is dependent on one’s point of view. Rani may appear beautiful to some people while appearing unappealing to others. As a result, we cannot say that the sentence is true logically.
(ii) “Shut the door” is not a declarative statement. It is simply a sentence that directs someone. There is no doubt about whether it is true or false.
(iii) The sentence “Yesterday was Friday” is not a statement. This sentence is true only on Saturdays, and it is false on other days of the week. In this case, whether something is true or false is determined by the time it is said rather than by mathematical reasoning.
Q2. Write the following statement five times to convey the same meaning.
A2. An obtuse-angled triangle is a triangle that is equiangular.
“If a triangle is equiangular, it is an obtuse-angled triangle,” is given.
The above statement can be written as follows:
- “A triangle is only equiangular if it has an obtuse-angled triangle.”
- “A triangle is not an equiangular triangle if it is not an obtuse-angled triangle.”
- “Equi AngularityEquiangularity is a sufficient condition for an obtuse-angled triangle.”
- “Obtuse angles in a triangle are required for it to be equiangular.”
- “If a triangle is equiangular, it is obtuse-angled.”
Q3. Check to see if the following two statements are negative to each other. Give reasons for your response.
(i) x + y = y + x holds for all real numbers x and y.
(ii) There exists a real number x and a real number y for which x + y = y + x.
A3. The statements are as follows:
p: “x + y = y + x is true for all real numbers x and y”
q: “There exists a real number x and a real number y for which x + y = y + x.”
The negation of p is
p: “There are real numbers x and y for which x + yy + x”
As a result, the given statements are not mutually exclusive and can both be true.
It should be noted that p is always true regardless of the x and y values.
Q4. Demonstrate that the statement is correct.
“If x is a real number such that x3 + 4x = 0, then x is 0,” according to p, is true.
A4. “If x is a real number such that x3 + 4x = 0, then x is 0,” says p.
(i) The Direct Method-
Consider,
x² + 4x + = 0, x 𝜖 R
x (x2 + 4x) = 0, x R x = 0.
( If x R, then x2 + 4x 4 )
If the product of two numbers is zero, at least one of them must be zero.
As a result, p is a true statement.
(ii) Contradiction Method
Consider that x is a nonzero real number.
(The square of a non-zero real number is always positive)
⇒ x² > 0 (∴ square of non-zero real number is always positive)
⇒ x² + 4x > 4 ⇒ x² + 4x ≠ 0
⇒ x ( x² + 4x ) ≠ 0 (∴ x ≠ 0 and x² + 4x ≠ 0 )
⇒ x³ + 4x ≠ 0, which is a contradiction
Hence, x=0
(iii) Contra-Positive Method:
Consider the question: “x∈ R and x3 + 4x = 0.”
And r: “x = 0”
As a result, the given statement p is q ⇒ r
Its polar opposite is ∼r ⇒ ∼q
“If x is a non-zero real number, then x3 + 4x = 0 is also non-zero,” in other words.
Now,
x ≠ 0 , x 𝜖 R ⇒ x² > 0 ⇒ x² + 4 > 4 ⇒ x² + 4 ≠ 0
⇒ x( x² + 4 ) ≠ 0 ⇒ x³ + 4x ≠ 0 i,e ∼ r ⇒ ∼ q
As a result, the statement ∼r ⇒ ∼q is always true.
As a result, q ⇒ r is also true.
While proving an implication, the ‘method of contradiction’ is another form of the ‘contra-positive method.’
Q.1 Which of the following are mathematically acceptable statements?
(i) All irrational numbers are real numbers.
(ii) How wonderful
(iii) In 2004 the president of USA was a woman.
(iv) What are you doing?
Marks:4
Ans
(i) All irrational numbers are real numbers is a statement because it is true.
(ii) How wonderful! It is not a statement because it is an exclamation.
(iii) In 2004, the president of USA was a woman is a statement because it is false.
(iv) What are you doing? It is not a statement because it is a question.
Q.2 Verify by the method of contradiction:p: ?3 is irrational.
Marks:4
Ans
In this method, we assume that the given statement is false, i.e., we assume that 3 is a rational number. This means that there exists positive integers a and b such that 3 = a/b, where a and b have no common factors.
Squaring the equation, we get
3 = a2/b23b2 = a2 3 divides a.
Therefore, there exists an integer c such that a = 3c.
Then, a2 = 9c2 and 3 b2 = a2
Hence, 3b2 = 9c2 ⇒ b2 = 3c2 3 divides b. But, we have already shown that 3 divides a. It means 3 is a common factor of a and b. This shows that our assumption is wrong. Thus, the statement 3 is irrational is true.
Q.3 Write the contrapositive of the following statement:
(i) If a number is divisible by 4 then it is divisible by 2.
(ii) If you are born in Nepal then you are a citizen of Nepal.
(iii) If a number n is odd then n2 is odd.
(iv) Something is hot implies that it has a high temperature.
Marks:4
Ans
(i) If a number is not divisible by 2 then it is not divisible by 4.
(ii) If you are not a citizen of Nepal then you were not born in Nepal.
(iii) If n2 is not even, then n is not even.
(iv) If something has not high temperature, then it is not hot.
Q.4 Find the component statements of the following compound statements:
(i) The roof is white and the wall is red.
(ii) 0 is a positive number or a negative number.
Marks:2
Ans
(i) The component statements are:
p: The roof is white.
q: The wall is red.
(ii) The component statements are:
p: 0 is a positive number.
q: 0 is a negative number.
Q.5 Show by the method of contradiction that the following statement is true:p: If x is a real number such that x5 + 4x = 0, then x is 0.
Marks:4
Ans
Let q and r be the statements given by
q : x is a real number such that x5 + 4x = 0
r: x is 0
Then, p: if q, then r.
If possible, let p be not true.
Then p is true
(q r) is true.
q and r is true.
x is a real number such that x5 + 4x = 0 and x 0
x = 0 and x 0
This is a contradiction.
Hence, p is true.
FAQs (Frequently Asked Questions)
1. What are some real-time applications of mathematical reasoning?
Mathematical reasoning is an intriguing chapter that teaches students various concepts and skills required to make deductive inferences and conclusions. Such abilities enable students to hone their logical abilities. This enables them to conduct the mathematical investigations required to solve problems in Mathematics and other scientific fields, such as Physics. This type of reasoning is commonly employed by researchers and scientists in their respective fields.
2. What is a mathematically accepted statement? Give examples
A statement is a fundamental unit of mathematical reasoning. A mathematically accepted statement is one that is either true or false and cannot be interpreted. Given below are some examples to demonstrate this:
- Rain is beautiful: this is not a mathematically valid statement because it is subjective. Some people find rain to be beautiful, while others do not.
- Barack Obama was America’s first black president: this is a mathematically correct statement. It is true.
3. What are the types of reasoning?
There are two main types of reasoning, namely – inductive and deductive. Inductive reasoning occurs when a person uses past experiences to reach a conclusion. Deductive reasoning occurs when one or more statements are provided to reach a logical conclusion.
4. What are the types of quantifiers?
here are two kinds of quantifiers, namely – existential and universal. An existential quantifier is a formula that holds some or all of the values of the variable or domain. A Universal quantifier is an assertion of all the values of a given domain or variable.
