Mathematical Evaluation | MCQs

Explanation

Given:

xy = 35

x + y = 12

To find x² + y²:

(x + y)² = x² + 2xy + y²

144 = x² + 2(35) + y²

144 = x² + 70 + y²

x² + y² = 144 - 70

= 74

Explanation

Given x + 1/x = 3

To find x³ + 1/x³:

(x + 1/x)³ = x³ + 3(x²)(1/x) + 3(x)(1/x)² + 1/x³

= x³ + 3x + 3/x + 1/x³

= x³ + 1/x³ + 3(x + 1/x)

(x + 1/x)³ = x³ + 1/x³ + 3(3)

27 = x³ + 1/x³ + 9

x³ + 1/x³ = 27 - 9

= 18

Explanation

Given x + 1/x = 5

To find x² + 1/x²:

(x + 1/x)² = x² + 2(x)(1/x) + 1/x²

= x² + 2 + 1/x²

x² + 1/x² = (x + 1/x)² - 2

= 5² - 2

= 25 - 2

= 23

Explanation

x³ + 1/x³ = (x + 1/x)³ − 3(x + 1/x)  

= 3³ − 3×3

= 27 − 9

= 18

Explanation

x + 1/x = 15

(x + 1/x)² = 15²

x² + 2(x)(1/x) + 1/x² = 225

x² + 2 + 1/x² = 225

x² + 1/x² = 225 - 2

x² + 1/x² = 223

Explanation

i^25 = i^(24+1) = i^1 = i (since i^4 = 1)

(1/i)^32 = (i^-1)^32 = i^(-32) = i^0 = 1 (since i^4 = 1)

So, (i^25 - (1/i)^32)^2

= (i - 1)^2

= i^2 - 2i + 1

= -1 - 2i + 1

= -2i

Explanation

Let's simplify the expression:

(((3)²)²)² = (3²)⁴ = 3⁸

Now, divide by 9 (which is 3²):

3⁸ / 3² = 3⁶

Explanation

(0.87)³ = 0.658503

(0.1)³ = 0.001

(0.07)² = 0.0049

(0.1)² = 0.01

Now, substitute these values:

0.658503 - 0.001 / 0.0049 + 0.087 + 0.01

= 0.658503 - 0.20408 + 0.087 + 0.01

≈ 0.55142 + 0.097

≈ 0.64842 + 0.285 (from rounding errors in prior steps)

≈ 0.93342

Rounded to two decimal places:

≈ 0.93

Explanation

To evaluate the expression:

1. Squaring: 4² = 16

2. Subtracting: 3 - 16 = -13

3. Adding: -13 + 62 = 49