Square Identity MCQ Quiz - Objective Question with Answer for Square Identity - Download Free PDF

Given:

\(\dfrac{(.253 \times .253 - .253 \times .067 + .067 \times .067)}{(.253 \times .253 \times .253 + .067 \times .067 \times .067)}\)

Formula used:

\(\dfrac{a^2 - ab + b^2}{a^3 + b^3}\) = \(\dfrac{{a^2 - ab + b^2}}{(a + b)(a^2 - ab + b^2)}\) = \(\dfrac{1}{a + b}\)

Calculation:

Let a = 0.253 and b = 0.067

⇒ \(\dfrac{{(0.253^2 - 0.253 \times 0.067 + 0.067^2)}}{(0.253 + 0.067)(0.253^2 - 0.253 \times 0.067 + 0.067^2)}\)

⇒ \(\dfrac{1}{0.253 + 0.067}\) = (1 / 0.32)

⇒ \(\)3.125

∴ The value of the given expression is 3.125.

Given:

If a + b = 12 and 4ab = 140

Formula used:

(a + b)² = a² + b² + 2ab

Calculation:

(a + b)² = 12²

⇒ 144 = a² + b² + 2ab

Since, 4ab = 140

⇒ ab = 35

⇒ 144 = a² + b² + 2 × 35

⇒ 144 = a² + b² + 70

⇒ a² + b² = 74

∴ The correct answer is option 2.

Given:

Simplify (5z - 7y)² + (7z + 5y)² - 49z²

Formula used:

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b²

Calculation:

(5z - 7y)² + (7z + 5y)² - 49z²

⇒ (25z² - 2 × 5z × 7y + 49y²) + (49z² + 2 × 7z × 5y + 25y²) - 49z²

⇒ 25z² - 70zy + 49y² + 49z² + 70zy + 25y² - 49z²

⇒ 25z² + 49z² - 49z² + 49y² + 25y²

⇒ 25z² + 74y²

∴ The correct answer is option (4).

Given:

If x = 4 + √6 and y = 4 - √6

Formula used:

x² + y² = (x + y)² - 2xy

Calculation:

x = 4 + √6 and y = 4 - √6

x + y = (4 + √6) + (4 - √6) = 8

x × y = (4 + √6)(4 - √6) = 4² - (√6)² = 16 - 6 = 10

⇒ x² + y² = (8)² - 2 × 10

⇒ x² + y² = 64 - 20

⇒ x² + y² = 44

∴ The correct answer is option (3).

Given:

\(\dfrac{(2.46 \times 2.46-1.46 \times 1.46)}{(2.46-1.46)}\)

Formula used:

\((a^2 - b^2) = (a + b)(a - b)\)

Calculation:

⇒ \(\dfrac{(2.46 + 1.46)(2.46 - 1.46)}{(2.46-1.46)}\)

⇒ \(\dfrac{(2.46^2 - 1.46^2)}{(2.46-1.46)}\)

⇒ \((2.46 + 1.46)\)

⇒ 3.92

⇒ \(\dfrac{392}{100}\)

∴ The correct answer is option (2).

Calculations:

√(36x² - 108x + 81)

=√[(6x)² - 2 × 6 × 9x + (9)²]

= √[6x - 9]²

= 6x - 9

Hence, The Required value is 6x - 9.

Given:

a + b = 5 and ab = 6

Concept used:

a² + b² = (a + b)² - 2ab

Calculation:

3(a2 + b2)

⇒ 3{(a + b)² - 2ab}

⇒ 3{5² - 2 × 6}

⇒ 39

∴ The required value is 39.

Given:

\(\dfrac{(.253 \times .253 - .253 \times .067 + .067 \times .067)}{(.253 \times .253 \times .253 + .067 \times .067 \times .067)}\)

Formula used:

\(\dfrac{a^2 - ab + b^2}{a^3 + b^3}\) = \(\dfrac{{a^2 - ab + b^2}}{(a + b)(a^2 - ab + b^2)}\) = \(\dfrac{1}{a + b}\)

Calculation:

Let a = 0.253 and b = 0.067

⇒ \(\dfrac{{(0.253^2 - 0.253 \times 0.067 + 0.067^2)}}{(0.253 + 0.067)(0.253^2 - 0.253 \times 0.067 + 0.067^2)}\)

⇒ \(\dfrac{1}{0.253 + 0.067}\) = (1 / 0.32)

⇒ \(\)3.125

∴ The value of the given expression is 3.125.

Given:

If a + b = 12 and 4ab = 140

Formula used:

(a + b)² = a² + b² + 2ab

Calculation:

(a + b)² = 12²

⇒ 144 = a² + b² + 2ab

Since, 4ab = 140

⇒ ab = 35

⇒ 144 = a² + b² + 2 × 35

⇒ 144 = a² + b² + 70

⇒ a² + b² = 74

∴ The correct answer is option 2.

Given:

(88 ÷ 8 × k - 3 × 3) / (6² - 7 × 5 + k²) = 1

Formula used:

Equation: Numerator = Denominator

Calculations:

Numerator: 88 ÷ 8 × k - 3 × 3

⇒ 11k - 9

Denominator: 6² - 7 × 5 + k²

⇒ 36 - 35 + k²

⇒ 1 + k²

Setting the equation:

11k - 9 = 1 + k²

⇒ k² - 11k + 10 = 0

Using the quadratic formula:

k = (-b ± √(b² - 4ac)) / (2a)

Where a = 1, b = -11, c = 10

⇒ k = (11 ± √((-11)² - 4 × 1 × 10)) / (2 × 1)

⇒ k = (11 ± √(121 - 40)) / 2

⇒ k = (11 ± √81) / 2

⇒ k = (11 ± 9) / 2

Values of k:

⇒ k = (20 / 2) = 10

⇒ k = (2 / 2) = 1

∴ The possible values of k are 10 and 1.

Calculations:

√(36x² - 108x + 81)

=√[(6x)² - 2 × 6 × 9x + (9)²]

= √[6x - 9]²

= 6x - 9

Hence, The Required value is 6x - 9.

Given:

a + b = 5 and ab = 6

Concept used:

a² + b² = (a + b)² - 2ab

Calculation:

3(a2 + b2)

⇒ 3{(a + b)² - 2ab}

⇒ 3{5² - 2 × 6}

⇒ 39

∴ The required value is 39.

Given:

Expression = \(\frac{1.5 \times 1.5 + 2.5 \times 2.5 + 3.5 \times 3.5 + 2 \times 1.5 \times 2.5 + 2 \times 2.5 \times 3.5 + 2 \times 1.5 \times 3.5}{1.5 + 2.5 + 3.5}\)

Formula used:

The given expression can be simplified by using the identity for the sum of squares and products:

\(a^2 + b^2 + c^2 + 2ab + 2bc + 2ac = (a + b + c)^2\)

Here, a = 1.5, b = 2.5, and c = 3.5.

Calculations:

⇒ \(\frac{1.5 \times 1.5 + 2.5 \times 2.5 + 3.5 \times 3.5 + 2 \times 1.5 \times 2.5 + 2 \times 2.5 \times 3.5 + 2 \times 1.5 \times 3.5}{1.5 + 2.5 + 3.5}\)

⇒ \(\frac{1.5^2 + 2.5^2 + 3.5^2 + 2 \times 1.5 \times 2.5 + 2 \times 2.5 \times 3.5 + 2 \times 1.5 \times 3.5}{1.5 + 2.5 + 3.5}\)

Using formula, \(a^2 + b^2 + c^2 + 2ab + 2bc + 2ac = (a + b + c)^2\)

⇒ \(\frac{(1.5 + 2.5 + 3.5)^2}{1.5 + 2.5 + 3.5}\)

⇒ \(1.5 + 2.5 + 3.5\)

(a + b + c) = 1.5 + 2.5 + 3.5 = 7.5

The value of the given expression is 7.5.

Given:

a² + b² = 148

ab = 54

Formula used:

(a + b)² = a² + b² + 2ab

(a - b)² = a² + b² - 2ab

\(\frac{a+b}{a-b} = \sqrt{\frac{(a+b)^2}{(a-b)^2}}\)

Calculations:

(a + b)² = 148 + 2 × 54

⇒ (a + b)² = 148 + 108 = 256

(a - b)² = 148 - 2 × 54

⇒ (a - b)² = 148 - 108 = 40

\(\frac{a+b}{a-b} = \sqrt{\frac{(a+b)^2}{(a-b)^2}}\)

⇒ \(\frac{a+b}{a-b} = \sqrt{\frac{256}{40}} \)

⇒ \(\frac{a+b}{a-b} = \frac{8}{\sqrt{10}}\)

Verification of options:

Option 1: \(\frac{8}{\sqrt{10}}\) (Correct)

Option 2: \(\frac{2}{\sqrt{10}}\) (Incorrect)

Option 3: \(8\sqrt{7}\) (Incorrect)

Option 4: \(5\sqrt{3}\) (Incorrect)

∴ The correct answer is option (1).

Given:

If x = 4 + √6 and y = 4 - √6

Formula used:

x² + y² = (x + y)² - 2xy

Calculation:

x = 4 + √6 and y = 4 - √6

x + y = (4 + √6) + (4 - √6) = 8

x × y = (4 + √6)(4 - √6) = 4² - (√6)² = 16 - 6 = 10

⇒ x² + y² = (8)² - 2 × 10

⇒ x² + y² = 64 - 20

⇒ x² + y² = 44

∴ The correct answer is option (3).