Question

QUE 1 :- State whether the following statements are true or false:

(i) The sum of the interior angles of a triangle is 180°.

(ii) Every odd number greater than 1 is prime.

(iii) For any real number x, 4x + x = 5x.

(iv) For every real number x, 2x > x.

(v) For every real number x, x 2 ≥ x.

(vi) If a quadrilateral has all its sides equal, then it is a square.

(vii) All prime numbers are odd.

(vii) Two times a real number is always even.

(xi) For any x, 3x +1 > 4.

(x) For any x, x 3 ≥ 0.​

Answer 1

a) True , according to angle sum property of a triangle

b) False , for example : 9 is an odd composite number

c) True , For example , x=-1

4(-1) +(-1)=5(-1)

-4-1=-5

-5 = -5

d) False , for example : If the real number is negative , taken -1 , so 2x = 2(-1) = -2 and x =-1 . Here 2x is smaller than x .

e) True , If the square is done , the number becomes positive , so it would be eventually greater than negative integer.

x : negative integer , x²>0

x : positive integer. , x²>0

x : 0 , x² = 0

f) False , a rhombus too has all four sides equal .

g) False , 2 is the only even prime number

h) True , because it then becomes a multiple of 2 .

i) False , if x = any negative integer , 0 or 1 .

j) False , x³ will be smaller than zero if x = negative integer

Answer 2

$\huge\tt\underline{Answer:}$

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i) True

Reason : Sum of angles of triangle property states that sum of all the angles inside the triangle is equal to 180°.

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ii) False

Reason : 9 is an odd number, greater than one, but it is not a prime number as it is a multiple of 3.

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iii) True

Reason : If x is considered to be a real number, let it be equal to any real number say 2, thus x = 2

Substituting the value of x in the equation x + 4x = 5x, we get,

2 + (4 × 2) = (5 × 2)

2 + 8 = 10

10 = 10

•°• L.H.S. = R.H.S.

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iv) False

Reason : Since, x is said to be a real number, if it turns out to be a negative integer, then, 2x < x

E.g., let x = -2

Substituting it in 2x, we get,

(2 × -2) = -4 and x = -2

-4 < -2

•°• 2x < x

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v) True

Reason : Square of any real number, whether it is positive or negative, is greater than that real number.

E.g. If x = -2, then,

(-2)² = 4

•°• 4 > -2

And, if x = 2, then,

(2)² = 4

•°• 4 > 2

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vi) False

Reason : A quadrilateral whose all the four sides are equal is said to be rhombus, and if all the angles of rhombus is 90°, then only it is said to be a square.

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vii) False

Reason : Two is an even prime number.

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viii) True

Reason : Twice any real number is divided by two, and hence, is said to be the multiple of two.

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ix) False

Reason : If x is a negative integer, or a whole number like zero or one, then 3x + 1 is not equal to 4.

E.g., Let x = 0

Substituting it in the equation 3x + 1 = 4, we get,

(3 × 0) + 1 = 4

0 + 1 = 4

1 < 4

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x) False

Reason : If x is a negative integer, cube of x will be less than zero.

E.g., Let x = -2

•°• (-2)³ = -8, which is less than zero

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