Q12 of 69 Page 261

Prove by direct method that for any integer ‘n’, n3 – n is always even.

We have given, n3-n

Let us Assume, n is even


Let n=2k, where k is natural number


n3-n=(2k)3-(2k)


n3-n=2k(4k2-1)


let k(4k2-1)=m


n3-n=2m


Therefore, (n3-n) is even.


Now, Let us Assume n is odd


Let n=(2k+1), where k is natural number


n3-n=(2k+1)3-(2k+1)


n3-n= (2k+1)[(2k+1)2-1]


n3-n= (2k+1)[(4k2+4k+1-1)]


n3-n= (2k+1)[(4k2+4k)]


n3-n= 4k(2k+1)(k+1)


n2-n= 2.2k(2k+1)(k+1)


let λ=2k(2k+1)(k+1)


n3-n=2λ


therefore, n3-n is even.


Hence, n3-n is always even


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10

Write down the converse of following statements :

(i) If a rectangle ‘R’ is a square, then R 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 x : y = 3 : 2, then 2x = 3y.


(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.

 11

Identify the Quantifiers in the following statements.

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


(ii) For all real numbers x and y, xy = yx.


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


(iv) For every natural number x, x + 1 is also a natural number.


(v) For all real numbers x with x > 3, x2 is greater than 9.


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


(vii) For all negative integers x, x3 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 2.


(x) There exists a real number x such that x2 + 1 = 0.

 13

Check the validity of the following statement.

p : 125 is divisible by 5 and 7.

 13

Check the validity of the following statement.

q : 131 is a multiple of 3 or 11.