Maharashtra Board Class 12 Maths Part 1 Chapter 1 Mathematical Logic 1.1 Solutions

State Which Of The Following Sentences Are Statements. Justify Your Answer. In Case Of Statements, Write Down The Truth Value:

Question (i). A triangle has ‘\( n \)’ sides.
Answer: It is a statement that is false, hence its truth value is 'F'. A triangle always has exactly three sides, which is a constant number rather than a variable.
In simple words: A triangle always has 3 sides, not an unknown number 'n' of sides, so this statement is false.

🎯 Exam Tip: Always state whether the sentence is a statement first, and then clearly write its truth value as 'T' or 'F'.

Question (ii). The sum of interior angles of a triangle is \( 180^\circ \).
Answer: It is a statement which is true, hence its truth value is 'T'. This is a fundamental geometric property of all Euclidean triangles.
In simple words: If you add up all three inside angles of any triangle, they always equal 180 degrees, making this a true statement.

🎯 Exam Tip: Mathematical facts and established geometric theorems are always statements, and their truth value is 'T'.

Question (iii). You are amazing!
Answer: It is an exclamatory sentence, hence it is not a statement.
In simple words: An exclamatory sentence expresses strong emotion and cannot be labeled as true or false, so it is not a mathematical statement.

🎯 Exam Tip: Sentences ending with an exclamation mark (!) are exclamatory and can never be classified as mathematical statements.

Question (iv). Please grant me a loan.
Answer: It is an imperative sentence, hence it is not a statement.
In simple words: A sentence that makes a request or gives a command is imperative and cannot be true or false, so it is not a mathematical statement.

🎯 Exam Tip: Look for polite request words like "Please" or direct commands; these indicate imperative sentences which are not statements.

Question (v). \( \sqrt{-4} \) is an irrational number.
Answer: It is a statement that is false, hence its truth value is 'F'.
In simple words: This is a statement because we can clearly say it is false, as the square root of a negative number is an imaginary number, not an irrational one.

🎯 Exam Tip: Even if a mathematical assertion is incorrect, it is still considered a statement as long as we can definitely label it as false.

Question (vi). \( x^2 - 6x + 8 = 0 \)
\( \implies x = -4 \) or \( x = -2 \).
Answer: It is a statement that is false, hence its truth value is 'F'.
In simple words: Solving the quadratic equation gives positive values 4 and 2, not negative ones, making this statement false.

🎯 Exam Tip: Always solve the given equation to verify if the implied values are correct before determining the truth value.

Question (vii). He is an actor.
Answer: It is an open sentence, hence it is not a statement.
In simple words: Since we do not know who "He" refers to, we cannot decide if this is true or false, making it an open sentence.

🎯 Exam Tip: Sentences containing pronouns like "he", "she", "it", or variables without specified values are open sentences and not statements.

Question (viii). Did you eat lunch yet?
Answer: It is an interrogative sentence, hence it is not a statement.
In simple words: A question cannot be answered with a simple true or false, so it is not considered a mathematical statement.

🎯 Exam Tip: Any sentence ending with a question mark is interrogative and can never be a mathematical statement.

Question (ix). Have a cup of cappuccino.
Answer: It is an imperative sentence, hence it is not a statement.
In simple words: This sentence is an invitation or request, which cannot be true or false, so it is not a statement.

🎯 Exam Tip: Sentences that offer, suggest, or command are imperative and do not have a truth value.

Question (x). \( (x + y)^2 = x^2 + 2xy + y^2 \) for all \( x, y \in \mathbb{R} \).
Answer: It is a mathematical identity that is true for all real numbers, hence its truth value is ‘T’.
In simple words: This algebraic formula is always true no matter what real numbers you choose for x and y. Since it is always correct, its truth value is True.

🎯 Exam Tip: Remember that standard algebraic identities are always true statements with a truth value of T.

Question (xi). Every real number is a complex number.
Answer: It is a statement that is true, hence its truth value is ‘T’. This is because any real number 'a' can be written in the complex form a + 0i.
In simple words: Every real number can be written as a complex number with a zero imaginary part. Therefore, the statement is completely true.

🎯 Exam Tip: Keep in mind that the set of real numbers is a subset of complex numbers, making this statement always true.

Question (xii). 1 is a prime number.
Answer: It is a statement that is false, hence its truth value is ‘F’. By definition, a prime number must be greater than 1 and have exactly two distinct factors.
In simple words: A prime number must have exactly two factors: 1 and itself. Since 1 only has one factor, it is not considered a prime number.

🎯 Exam Tip: Never mistake 1 for a prime number; it is neither prime nor composite, which is a very common trap in exams.

Question (xiii). With the sunset, the day ends.
Answer: It is a statement that is true, hence its truth value is ‘T’. This represents a universally accepted natural phenomenon that occurs daily.
In simple words: This is a simple fact of nature that everyone agrees on. Since it is a true fact, its truth value is T.

🎯 Exam Tip: Universal truths and scientific facts are always classified as statements with a truth value of T.

Question (xiv). \( 1! = 0 \).
Answer: It is a statement that is false, hence its truth value is 'F'. In mathematics, the factorial of 1 is defined as 1, not 0.
In simple words: The factorial of 1 is equal to 1. Since the equation says it equals 0, the statement is false.

🎯 Exam Tip: Be careful with factorials; remember that \( 0! = 1 \) and \( 1! = 1 \), which are common points of confusion.

Question (xv). \( 3 + 5 > 11 \).
Answer: It is a statement that is false, hence its truth value is ‘F’. Adding 3 and 5 gives 8, which is clearly less than 11.
In simple words: When you add 3 and 5, you get 8. Since 8 is not greater than 11, this mathematical statement is false.

🎯 Exam Tip: Always simplify the mathematical expression on both sides before determining the truth value.

Question (xvi). The number \( \pi \) is an irrational number.
Answer: It is a statement that is true, hence its truth value is ‘T’. The value of pi is a non-terminating and non-recurring decimal.
In simple words: The number pi cannot be written as a simple fraction, and its decimal goes on forever without repeating. This makes it an irrational number, so the statement is true.

🎯 Exam Tip: Remember that \( \pi \) and \( e \) are classic examples of irrational numbers, so any statement asserting this is always true.

Question (xvii). \( x^2 - y^2 = (x + y)(x - y) \) for all \( x, y \in R \).
Answer: It is a mathematical identity that is true, hence its truth value is ‘T’. This algebraic identity holds true for any real numbers substituted into it.
In simple words: This is a standard math formula that is always correct for any real numbers. Since it is a true fact, its truth value is T.

🎯 Exam Tip: Remember that algebraic identities are always true statements for all values in their domain, so their truth value is always T.

Question (xviii). The number 2 is only even a prime number.
Answer: It is a statement that is true, hence its truth value is ‘T’. This makes 2 the unique even prime number in the entire set of integers.
In simple words: The number 2 is the only number that is both even and prime. Since this is a true fact, its truth value is T.

🎯 Exam Tip: Pay close attention to unique mathematical properties like 2 being the only even prime, as these are common true statements in logic exams.

Question (xix). Two coplanar lines are either parallel or intersecting.
Answer: It is a statement that is true, hence its truth value is ‘T’. In a single flat plane, two lines must either cross each other or run parallel forever.
In simple words: If two lines are on the same flat surface, they will either cross each other at some point or never meet at all. Since this is always true, its truth value is T.

🎯 Exam Tip: Make sure to note the word 'coplanar' because lines in different planes (skew lines) can be neither parallel nor intersecting.

Question (xx). The number of arrangements of 7 girls in a row for a photograph is 7!
Answer: It is a statement that is true, hence its truth value is ‘T’. The factorial notation correctly represents the total number of permutations for arranging distinct individuals.
In simple words: When arranging 7 different people in a line, the number of ways to do it is indeed 7 factorial (7!). Since this mathematical rule is correct, the statement is true.

🎯 Exam Tip: Standard permutation formulas for arranging 'n' distinct objects in a row always yield n! arrangements, which is a mathematically true statement.

Question (xxi). Give me a compass box.
Answer: It is an imperative sentence, hence it is not a statement. Sentences that express requests or commands do not have a truth value of true or false.
In simple words: This sentence is a request or command, not a statement of fact. Since we cannot say it is "true" or "false", it is not a mathematical statement.

🎯 Exam Tip: Commands, requests, questions, and exclamations are never considered mathematical statements because they cannot be assigned a truth value.

Question (xxii). Bring the motor car here.
Answer: It is an imperative sentence, hence it is not a statement. This sentence functions as an order rather than asserting a declarative fact.
In simple words: This is an order telling someone to do something. Because it is a command and cannot be true or false, it is not a statement.

🎯 Exam Tip: Always identify the type of sentence first; if it is imperative (giving an order), it is immediately classified as not a statement.

Question (xxiii). It may rain today.
Answer: It is an open sentence, hence it is not a statement. This is because the truth of the sentence depends on the specific day and weather conditions, which are not specified.
In simple words: A sentence is not a mathematical statement if its truth changes depending on the situation. Since "today" is not specific, we cannot say for sure if it is true or false.

🎯 Exam Tip: Remember that sentences involving subjective terms, future predictions, or unspecified times are classified as open sentences and are not mathematical statements.

Question (xxiv). If \( a + b < 7 \), where \( a \ge 0 \) and \( b \ge 0 \), then \( a < 7 \) and \( b < 7 \).
Answer: It is a statement that is true, hence its truth value is ‘T’. Since both numbers are non-negative, neither individual number can be equal to or greater than 7 if their sum is strictly less than 7.
In simple words: If you add two positive numbers together and get a sum less than 7, then each of those numbers must individually be less than 7. Since this is always true, its truth value is T.

🎯 Exam Tip: When dealing with inequalities, test with boundary values like 0 to quickly verify if the statement holds true under all given conditions.

Question (xxv). Can you speak English?
Answer: It is an interrogative sentence, hence it is not a statement. Questions do not assert a fact that can be declared true or false.
In simple words: A question cannot be answered with a simple "true" or "false" as a statement of fact. Therefore, any sentence ending with a question mark is not a mathematical statement.

🎯 Exam Tip: Identify interrogative sentences immediately by looking for a question mark; these can never be mathematical statements.