QUESTIONS ON SIMPLIFICATION (PART-VII)

QUERY 61

SHIV KISHOR
Let 1.331 = a
Then the given expression
= [(1/a + 1/a² + ….1/a⁷)/(1/a² +1/a³ + …..1/a⁸)]^2/3

= [(1/a +1/a² + ….1/a⁷)/{(1/a)(1/a + 1/a² + …..1/a⁷)}]^2/3

= [1/(1/a)]^2/3

= a^2/3

= (1.331)^2/3

= 1.21 (option ‘4’)

QUERY 62

If √x = √3 – √5, then the value of x² – 16x + 6

A) 0
B) -2
C) 2
D) 4

MAHA GUPTA
√x = √3 – √5

=> x = 8 – 2√15 —- squaring both sides

=> x – 8 = -2√15

=> x² + 64 – 16x = 60 —-again squaring both sides

=> x² – 16x = -4

Now the find expression
x² – 16x + 6
= -4 + 6               —-putting x² – 16x = -4
= 2 (option ‘C’)

QUERY 63

SHIV KISHOR

In such a question convert the expression on the very right under the root sign in whole square of an expression, and so on

(Option ‘b’)

QUERY 64

If x = b + c – 2a, y = c + a – 2b, z = a + b – 2c, then value of x2 + y2 – z2 + 2xy

A) 1
B) 3
C) 0
D) 7

MAHA GUPTA
Adding all the equations
x + y + z = b + c – 2a + c + a – 2b + a + b – 2c
=> x + y + z = 0

Now the given expression can be re-written as
(x + y)² – z²
= (x + y + z)(x + y – z)
= 0 (option ‘C’)               —-putting x + y + z = 0

QUERY 65

x + y + z = 0; then find x²/yz + y²/zx + z²/xy = ?

A) 3
B) 0
C) 5
D) 9

MAHA GUPTA
As x + y + z = 0, let x = 2, y = -1 and z = -1

Putting these values in the find expression
(2²)/(-1)(-1) + (-1)²/-1*2 + (-1)²/2(-1)
= 4 -1/2 – 1/2 = 3 (option ‘A’)

QUERY 66

x + y = 9; if x and y are real numbers find the maximum value of xy

A) 20
B) 81/4
C) 81
D) 21

MAHA GUPTA
Though this question can be solved by AM, GM method also, here we do need not that. If sum of two numbers is there, the maximum value of their multiplication can only be when if both of them are equal. Here the sum is 9; so both of them should be 9/2.

Therefore maximum value of xy = (9/2)*(9/2) = 81/4 (option ‘B’)

ADDITIONAL
If the numbers are integers, they must be 4 and 5 for xy to be the maximum as the numbers needs to have as near as possible. So xy = 4*5 = 20

QUERY 67

If 9√x = √12 + √147, find the value of x.

A) 2
B) 3
C) 4
D) 5

RAHUL VERMA
√12 + √147 = 2√3 + 7√3 = 9√3

Comparing it with LHS of the equation
x = 3 (option ‘B’)

QUERY 68

If √x = √3 – √5, then the value of x² – 16x + 6 = ?

A) 2
B) 10
C) 12
D) -2

AJAY SHEORAN

(Option ‘A’)

QUERY 69

A) 1
B) 0
C) 3
D) 2

SUBHASH CHANDRA

(option ‘A’)

QUERY 70

If 9x⁴ – 12x³ + 10x² + px + 1 is a perfect square, find the value of p

A) -2
B) 4
C) -4
D) 3

MAHA GUPTA
By looks we can see that the given expression can be converted into the form: [3x² – 2x + 1]²

See how
[3x² – 2x +1]²
= (3x²)² + 4x² + 1² + 2(3x²)(-2x) + 2(-2x)1 + 2*1*(3x²)             —- Applying (a+b+c)² = a² + b² + c² + 2ab + 2bc + 2ca
= 9x² -12x² + 10x² + (- 4x) + 1

Now comparing this with the given expression
p = -4 (option ‘C’)