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.’