NCERT Exemplar Solutions
Class 11 - Mathematics

Chapter 14: MATHEMATICAL REASONING

(i) to (v) and (viii) to (x) are statements.

(i) p: Number 7 is prime; q: Number 7 is odd

(ii) p: Chennai is in India; q: Chennai is capital of Tamil Nadu

(iii) p: 100 is divisible by 3; q: 100 is divisible by 11; r: 100 is divisible by 5

(iv) p: Chandigarh is capital of Haryana; q: Chandigarh is the capital of U.P

(v) p: \(\sqrt{7}\) is a rational number; q: \(\sqrt{7}\) is an irrational number

(vi) p: 0 is less than every positive integer; q: 0 is less than every negative integer

(vii) p: plants use sunlight for photosynthesis; q: plants use water for photosynthesis; r: plants use carbon dioxide for photosynthesis

(viii) p: two lines in a plane intersect at one point; q: two lines in a plane are parallel

(ix) p: a rectangle is a quadrilateral; q: a rectangle is a 5-sided polygon

(i) True: p: 57 is divisible by 2; q: 57 is divisible by 3

(ii) True: p: 24 is multiple of 4; q: 24 is multiple of 6

(iii) False: p: all living things have two eyes; q: all living things have two legs

(iv) True: p: 2 is an even number; q: 2 is a prime number

(i) The number 17 is not prime

(ii) 2 + 7 ≠ 6

(iii) Violets are not blue

(iv) \(\sqrt{5}\) is not a rational number

(v) 2 is a prime number

(vi) There exists a real number which is not an irrational number

(vii) Cow has not four legs

(viii) A leap year has not 366 days

(ix) There exist similar triangles which are not congruent

(x) Area of a circle is not same as the perimeter of the circle

(i) \(p \land q\) where p: Rahul passed in Hindi; q: Rahul passed in English

(ii) \(p \land q\) where p: x is an even integer; q: y is an even integer

(iii) \(p \land q \land r\) where p: 2 is factor of 12; q: 3 is factor of 12; r: 6 is factor of 12

(iv) \(p \lor q\) where p: x is an odd integer; q: x + 1 is an odd integer

(v) \(p \lor q\) where p: a number is divisible by 2; q: it is divisible by 3

(vi) \(p \lor q\) where p: x = 2 is a root of \(3x^2 - x - 10 = 0\); q: x = 3 is a root of \(3x^2 - x - 10 = 0\)

(i) It is false that all rational numbers are real and complex

(ii) It is false that all real numbers are rational or irrational

(iii) x = 2 is not a root of \(x^2 - 5x + 6 = 0\) or x = 3 is not a root of \(x^2 - 5x + 6 = 0\)

(iv) A triangle has neither 3-sides nor 4-sides

(v) 35 is not a prime number and it is not a complex number

(vi) It is false that all prime integers are either even or odd

(vii) \(|x|\) is not equal to x and not equal to −x

(viii) 6 is not divisible by 2 or it is not divisible by 3

(i) If the number is odd then its square is odd

(ii) If you take dinner then you will get sweet dish

(iii) If you will not study then you will fail

(iv) If an integer is divisible by 5 then its unit digits are 0 or 5

(v) If the number is prime then its square is not prime

(vi) If a, b, c are in A.P then \(2b = a + c\)

(i) The unit digit of an integer is zero if and only if it is divisible by 5

(ii) A natural number n is odd if and only if it is not divisible by 2

(iii) A triangle is an equilateral triangle if and only if all three sides of the triangle are equal

(i) If x ≠ y or y ≠ 3 then x ≠ 3

(ii) If n is not an integer then it is not a natural number

(iii) If the triangle is not equilateral then all three sides of the triangle are not equal

(iv) If xy is not positive integer then either x or y is not negative integer

(v) If natural number n is not divisible by 2 and 3 then n is not divisible by 6

(vi) The weather will not be cold if it does not snow

(i) If the rectangle R is rhombus then it is square

(ii) If tomorrow is Tuesday then today is Monday

(iii) If you must visit Taj Mahal then you go to Agra

(iv) If the triangle is right angle then sum of squares of two sides equals square of third side

(v) If the triangle is equilateral then all three angles of the triangle are equal

(vi) If 2x = 3y then x : y = 3 : 2

(vii) If opposite angles of a quadrilateral are supplementary then S is cyclic

(viii) If x is neither positive nor negative then x is 0

(ix) If ratio of corresponding sides of two triangles are equal then triangles are similar

(i) There exists

(ii) For all

(iii) There exists

(iv) For every

(v) For all

(vi) There exists

(vii) For all

(viii) There exists

(ix) There exists

(x) There exists

Two cases: if n is even, \(n = 2k\), then \(n^3 - n = 2(4k^3 - k)\) is even. If n is odd, \(n = 2k + 1\), then \(n^3 - n = 2(4k^3 + 6k^2 + 2k)\) is even.

(i) 125 is divisible by 5 and 7 → False

(ii) 131 is a multiple of 3 or 11 → False

Let r be rational and i be irrational. Assume r + i is rational. Then i = (r + i) − r, a difference of rationals, implying i is rational, contradiction. Thus r + i is irrational.

If x = y, then multiply both sides by x to get \(x^2 = xy\). But since x = y, xy = \(y^2\). Hence \(x^2 = y^2\).

Contrapositive: If n is odd, then \(n^2\) is odd. Let n = 2k + 1. Then \(n^2 = 4k^2 + 4k + 1\) which is odd. Thus if \(n^2\) is even, n must be even.

Statement. It is a declarative sentence that can be judged true (it is a true mathematical fact).

Statement. It is a declarative sentence that can be judged true.

Not a statement. It is an exclamation/wish and not a declarative sentence that can be assigned a truth value.

Statement. It is a declarative sentence that can be judged true.

Not a statement. It is a question and therefore not assigned a truth value.