# NCERT Class 11-Math's: Chapter –14 Mathematical Reasoning Part 10

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**Question 9:**

Write down the contrapositive of the following statements:

(i) If and , then

(ii) If is a natural number, then is an integer.

(iii) If all three sides of a triangle are equal, then the triangle is equilateral.

(iv) If and *y* are negative integers, then is positive.

(v) If natural number is divisible by , then *n* is divisible by and .

(vi) If it snows, then the weather will be cold.

(vii) If *x* is a real number such that , then .

**Answer:**

(i) If then or

(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 is not positive integer then either or is not negative integer

(v) If natural number is not divisible by and then *n* is not divisible by

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

**Question 10:**

Write down the converse of following statements :

(i) If a rectangle is a square, then is a rhombus.

(ii) If today is Monday, then tomorrow is Tuesday.

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

(iv) If the sum of squares of two sides of a triangle is equal to the square of third side of a triangle, then the triangle is right angled.

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

(vi) If , then

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

(viii) If *x* is zero, then *x* is neither positive nor negative.

(ix) If two triangles are similar, then the ratio of their corresponding sides are equal.

**Answer:**

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

(ii) If tomorrow is Tuesday then today is Monday.

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

(iv) If the triangle is right angle then sum of squares of two sides of a triangle is equal to the square of third side.

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

(vi) If then

(vii) If the opposite angles of a quadrilaterals are supplementary then S is cyclic.

(viii) If *x* is neither positive nor negative than *x* is .

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

**Question 11:**

Identify the Quantifiers in the following statements.

(i) There exists a triangle which is not equilateral.

(ii) For all real numbers and ,

(iii) There exists a real number which is not a rational number

(iv) For every natural number is also a natural number

(v) For all real numbers *x* with is greater than

(vi) There exists a triangle which is not an isosceles triangle

(vii) For all negative integers *x*, *x*3 is also a negative integers

(viii) There exists a statement in above statements which is not true.

(ix) There exists a even prime number other than

(x) There exists a real number *x* such that

**Answer:**

(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