Identities MCQ Quiz - Objective Question with Answer for Identities - Download Free PDF

Concept used:

In question there are 4 variables (a, b, x and y) and only 2 equation. Hence we put any value to a and b to get value of x and y

Calculation:

Let a = 0 and b = 1

⇒ 

⇒ 

Again,

⇒ 

⇒ 

Now,

(x - a)2 - (y - b)2 = 

⇒ 

Also, b² = (1)² = 1

∴ The required value is b2.

Given:

a = 8√6 - 8√5

b = 8√6 + 8√5

c = 6√6 + 6√5

d = 4√6 + 4√5

e = √6 + √5

Formula used:

A number is rational if it does not contain any square root (√) term.

Calculations:

Check each pair:

1. a × b:

a × b = (8√6 - 8√5)(8√6 + 8√5)

⇒ a × b = (8√6)² - (8√5)²

⇒ a × b = 64 × 6 - 64 × 5

⇒ a × b = 384 - 320

⇒ a × b = 64 (Rational)

2. c × d:

c × d = (6√6 + 6√5)(4√6 + 4√5)

⇒ c × d = (6√6)(4√6) + (6√6)(4√5) + (6√5)(4√6) + (6√5)(4√5)

⇒ c × d = 24 × 6 + 24√30 + 24√30 + 24 × 5

⇒ c × d = 144 + 120 + 48√30

⇒ c × d contains √30 (Irrational)

3. d × e:

d × e = (4√6 + 4√5)(√6 + √5)

⇒ d × e = (4√6)(√6) + (4√6)(√5) + (4√5)(√6) + (4√5)(√5)

⇒ d × e = 4 × 6 + 4√30 + 4√30 + 4 × 5

⇒ d × e = 24 + 20 + 8√30

⇒ d × e contains √30 (Irrational)

Conclusion:

Among the options, only a × b is rational.

∴ The correct answer is option (3).

Given:

Formula used:

(a + b)(a - b) = a² - b²

Calculations:

⇒ 1375 - 1125

⇒ 250

∴ The result is a rational number. The correct answer is option (3).

Given:

a = √11 + √3

b = √12 + √2

c = √6 + √4

Formula used:

Compare the numerical values of a, b, and c.

Calculation:

Calculate a:

√11 ≈ 3.3166, √3 ≈ 1.732

⇒ a = 3.3166 + 1.732 = 5.0486

Calculate b:

√12 ≈ 3.4641, √2 ≈ 1.414

⇒ b = 3.4641 + 1.414 = 4.8781

Calculate c:

√6 ≈ 2.4495, √4 = 2

⇒ c = 2.4495 + 2 = 4.4495

Compare values:

a = 5.0486, b = 4.8781, c = 4.4495

⇒ a > b > c

∴ The correct answer is option (3).

Given:

If x is any natural number, then x³ - (1/x³) will always be greater than or equal to:

Option 1: 3 × (x - (1/x))

Option 2: 3 × (x + (1/x))

Option 3: x + (1/x)

Option 4: x³ + (1/x³)

Formula used:

For any natural number x, the inequality to verify is:

x³ - (1/x³) ≥ 3 × (x - (1/x))

Calculation:

Let x³ - (1/x³) = (x - (1/x)) × ((x² + (1/x²)) + 1)

⇒ x³ - (1/x³) = (x - (1/x)) × (x² + (1/x²) + 1)

Now, observe that x² + (1/x²) ≥ 2 for all x > 0 (AM-GM inequality).

⇒ x² + (1/x²) + 1 ≥ 3

⇒ (x - (1/x)) × (x² + (1/x²) + 1) ≥ 3 × (x - (1/x))

⇒ x³ - (1/x³) ≥ 3 × (x - (1/x))

∴ The correct answer is option 1.

Given:

x - 1/x = 3

Concept used:

a³ - b³ = (a - b)³ + 3ab(a - b)

Calculation:

x³ - 1/x³ = (x - 1/x)³ + 3 × x × 1/x × (x - 1/x)

⇒ (x - 1/x)3 + 3(x - 1/x)

⇒ (3)³ + 3 × (3)

⇒ 27 + 9 = 36

∴ The value of x³ - 1/x³ is 36.

Alternate Method If x - 1/x = a, then x³ - 1/x³ = a³ + 3a

Here a = 3

x - 1/x³ = 3³ + 3 × 3

= 27 + 9

= 36

Given:

x = √10 + 3

Formula used: 

a² - b² = (a + b)(a - b)

a³ - b³ = (a - b)(a² + ab + b²)

Calculation:

⇒ 1/x = √10 - 3

     ----(1)

Squaring both side of (1),

    -----(2)

∴ The required value is 234.

 Shortcut TrickGiven:

x = √10 + 3

Formula used: 

⇒ 

Calculation:

x = √10 + 3

⇒ 1/x = √10 - 3

⇒  

⇒ 

⇒ 

∴ The required value is 234.

Given:

x - (1/x) = (- 6)

Formula used:

If x - (1/x) = P, then

x + (1/x) = √(P² + 4) 

If x + (1/x) = P, then

x3 + (1/x3) = (P³ - 3P)

x⁵ - (1/x⁵) = {x³ + (1/x³)} × {x² - 1/x²} + {x - (1/x)}

Calculation:

x - (1/x) = (- 6)

x + (1/x) = √{(- 6)2 + 4} = √40 = 2√10

So, x² - 1/x² = (x + 1/x) (x - 1/x) = 2√10 × (-6) = -12√10

and x3 + (1/x3) =  (√40)³ - 3√40

⇒ 40√40 - 3√40 = 37 × 2√10 = 74√10

Now,

x5 - (1/x5) = {x3 + (1/x3)} × {x2 - 1/x2} + {x - (1/x)}

⇒ {74√10 × (-12√10)} + (- 6)

⇒ - 74 × 12 × (√10 × √10) - 6

⇒ (- 8880) - 6 = - 8886

∴ The correct answer is  - 8886.

Given:

p – 1/p = √7

Formula:

P³ – 1/p³ = (p – 1/p)³ + 3(p – 1/p)

Calculation:

P³ – 1/p³ = (p – 1/p)³ + 3 (p – 1/p)

⇒ p³ – 1/p³ = (√7)³ + 3√7

⇒ p³ – 1/p³ = 7√7 + 3√7

⇒ p³ – 1/p³ = 10√7

Shortcut Trick x - 1/x = a, then x³ - 1/x³ = a³ + 3a

Here, a = √7                                                          ( put the value in required eqⁿ )

⇒p³ – 1/p³ = (√7)³ + 3 × √7 = 7√7 + 3√7

 ⇒p3 – 1/p3  = 10√7.

Hence; option 4) is correct.

Given:

a + b + c = 14, ab + bc + ca = 47 and abc = 15

Concept used:

a³ + b³ + c³ - 3abc = (a + b + c) × [(a + b + c)² - 3(ab + bc + ca)]

Calculations:

a³ + b³ + c³ - 3abc = 14 × [(14)² - 3 × 47]

⇒ a³ + b³ + c³ – 3 × 15 = 14(196 – 141)

⇒ a³ + b³ + c³ = 14(55) + 45

⇒ 770 + 45

⇒ 815

∴ The correct choice is option 1.

Given:

Formula used:

(a + 1/a) = P ; then

(a² + 1/a²) = P² - 2

(a3 + 1/a3) = P³ - 3P

 = (a² + 1/a²) × (a³ + 1/a³) - (a + 1/a)

Calculation:

a + (1/a) = 7

⇒ (a2 + 1/a2) = (7)2 - 2 = 49 - 2 = 47

⇒ (a3 + 1/a3) = (7)³ - (3 × 7) = 343 - 21 = 322

a⁵ + (1/a⁵) = (a2 + 1/a2) × (a3 + 1/a3) - (a + 1/a)

⇒ 47 × 322 - 7

⇒ 15134 - 7 = 15127

 ∴ The correct answer is 15127.

Formula used:

(a + b)³ = a³ + b³ + 3ab(a + b)

Calculation:

⇒ x^2/3 + x^1/3 = 2

⇒ (x^2/3 + x^1/3)³ = 2³

⇒ x² + x + 3x(x^2/3 + x^1/3) = 8

⇒ x² + 7x - 8 = 0

⇒ x² + 8x - x - 8 = 0

⇒ x (x + 8) - 1 (x + 8) = 0

⇒ x = - 8 or x = 1

Formula used:

a3 + b3 + c3 - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca)

Calculation:

a + b + c = 0

a3 + b3 + c3 - 3abc = (a + b + c)(a2 + b2 + c2 - ab - bc - ca)

⇒ a3 + b3 + c3 - 3abc = 0 × (a2 + b2 + c2 - ab - bc - ca) = 0

⇒ a3 + b3 + c3 - 3abc = 0

⇒ a3 + b3 + c3 = 3abc 

Now, (a3 + b3 + c3)2 = (3abc)² = 9a2b2c2 

Given:

(a + b + c) = 19

(a2 + b2 + c2) = 155

Formula used:

a² + b² + c² - (ab + bc + ca) = (1/2) × [(a - b)2 + (b - c)2 + (c - a)2]

Calculation:

a + b + c = 19

Squaring both sides

⇒ (a + b + c)² = (19)²

⇒ a2 + b2 + c2 + 2 × (ab + bc + ca) = 361

⇒ 155 + 2 × (ab + bc + ca) = 361

⇒ 2 × (ab + bc + ca) = (361 - 155)

⇒ (ab + bc + ca) = 206/2 = 103

Now,

a2 + b2 + c2 - (ab + bc + ca) = (1/2) × [(a - b)2 + (b - c)2 + (c - a)2]

⇒ 2 × (155 - 103) = (a - b)2 + (b - c)2 + (c - a)2

⇒ (a - b)2 + (b - c)2 + (c - a)2 = 104

∴ The correct answer is 104.

Given:

x2 + (1/x2) = 7

Formula used:

x² + (1/x²) = P

then x + (1/x) = √(P + 2)

and x - (1/x) = √(P - 2)

⇒ x2 - (1/x2) = {x + (1/x)} × {x - (1/x)}

Calculation:

x2 + (1/x2) = 7

⇒ x + (1/x) = √(7 + 2) = √9

⇒ x + (1/x) = 3

⇒ x - (1/x) = -√(7 - 2)

⇒ x - (1/x) = - √5 {0

x2 - (1/x2) = {x + (1/x)} × {x - (1/x)}

⇒ 3 × (- √5)

∴ The correct answer is - 3√5.

Mistake Points

Please note that 

0

so

1/x > 1

so

x + 1/x > 1

and

x - 1/x 1 so x - 1/x

so

(x - 1/x)(x + 1/x)