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Is |x - 1| less than 1 ?

1). (x - 1)^2 less than and equal to 1

2). x^2 - 1 greater than 0

Answer: E

|x-1| less than 1 is only true when 0 less than x less than 1

From statement (1): (x-1)^2<=1

True when 0<=x<=2

If x=0.5, then |x-1| less than 1 is true

If x=2, then |x-1| less than 1 is not true

Hence insufficient

From statement (2): x^2>1 means x>1 and x<-1

True when x=1.5, but not when x=3

Hence insufficient

Statement (1) and (2) together: 1 is less than x is less than and equal to 2

Taking x=1.5 and x=2

Hence insufficient

Which of the following is always equal to sqrt (9 + x^2 - 6x)?

a) x - 3

b) 3 + x

c)|3 - x|

d)|3 + x|

e) 3 - x

Answer: C

sqrt (9+x^2-6x)

= sqrt( (3-x)^2 )

= |3-x|
What is the value of x?

1) (-x)^3 = -x^3

2) (-x)^2 = -x^2

Answer: B

From statement (1):

if x = 0 both sides are equal

if x = 1 both sides are again equal {(-1)^3 = -1 & -1^3 = 1}

=> x = 0 or x = 1

Hence insufficient

From statement (2):

x can only be zero because the square of a number other than zero cannot be negative

{(-1)^2 = 1 which is not equal to -(1)^2)}

=> from above it is sufficient to say that x = 0

Hence sufficient

Is |x - 1| less than 1 ?

(1). (x - 1) ^2 less than and equal to 1

(2). (x^2) - 1 greater than 0

Answer: E

From statement (1): (x - 1) ^2 less than and equal to 1

Now this is true only if 0 is less than and equal to x and x is less than and equal to 2. (We know that (x - 1) ^2 less than 1 is only true when 0 is less than x and x is less than 2)When we take x = 0.5, then |x-1| less than 1 is true

When we take x = 2, then |x-1| less than1 does not holds true

Hence insufficient

From statement (2): x^2 greater than 1 => x is less than -1 and x is greater than 1

When we take x = 1.5, then x^2 greater than 1 is true

When we take x = 3, then x^2 greater than 1 does not holds true

Hence insufficient

*Taking both statements (1) and (2) together -- *We have 1 is less than x and x is less than and equal to 2

When we take x = 1.5, then 1 is less than x and x is less than and equal to 2 is true

When we take x = 2, then 1 is less than x and x is less than and equal to 2 does not holds true Hence insufficient