Explanation

Given x varies inversely to y:

x ∝ 1/y

x = k/y

When x = 3, y = 12:

3 = k/12

k = 36

So, the equation becomes:

x = 36/y

When x = 6:

6 = 36/y

y = 36/6

y = 6

Explanation

Given A = [a b; b a] and A² = [l m; ...], let's calculate A²:

A² = [a b; b a] * [a b; b a]

= [a² + b² 2ab; 2ab a² + b²]

Comparing A² = [l m; ...] with our calculation:

l = a² + b²

m = 2ab

The correct answer is l = a² + b², m = 2ab.

Explanation

If b² - 4ac < 0, the discriminant is negative, meaning the quadratic equation has complex (imaginary) roots.

These roots are conjugate pairs and unequal.

Explanation

Given A and B are square matrices of the same order:

(AB)^t = B^t * A^t

This property holds true for matrix transpose in multiplication.

The correct answer is (AB)^t = B^t * A^t.

Explanation

If AB = O (null matrix), it is possible that either A = O (null matrix) or B = O (null matrix), or both could be null matrices.

In matrix multiplication, if either matrix is a null matrix, the result will always be a null matrix.

Explanation

Given the equation:

log3 (4x + 3) = 3

We can rewrite this in exponential form:

3³ = 4x + 3

27 = 4x + 3

Subtracting 3 from both sides:

24 = 4x

Dividing both sides by 4:

x = 6

Explanation

Given the equations:

x + 3y = 53 ... (1)

4x - 2y = 2 ... (2)

We can solve these equations using substitution or elimination. Let's use elimination.

Multiplying equation (1) by 2 and equation (2) by 3 to eliminate y:

2x + 6y = 106

12x - 6y = 6

Adding both equations:

14x = 112

x = 8

Explanation

Let's solve the quadratic equation 6x² - 17x + 12 = 0.

Using the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

Here, a = 6, b = -17, and c = 12.

x = (17 ± √((-17)² - 4_6_12)) / (2*6)

x = (17 ± √(289 - 288)) / 12

x = (17 ± √1) / 12

x = (17 + 1) / 12 or x = (17 - 1) / 12

x = 18 / 12 or x = 16 / 12

x = 3/2 or x = 4/3

The solutions are x = 3/2 and x = 4/3.

Explanation

For a quadratic equation ax² + bx + c = 0, the sum of the roots is -b/a and the product of the roots is c/a.

Given the equation 2x² + kx + 6 = 0, we have:

Sum of the roots = -k/2

Product of the roots = 6/2 = 3

According to the problem, the sum of the roots is equal to four times the product of the roots:

-k/2 = 4 * 3

-k/2 = 12

k = -24

Explanation

Let's calculate what needs to be added:

(7 + 2a - a² + 4a³) - (2a³ + a² - 3a - 1)

= 7 + 2a - a² + 4a³ - 2a³ - a² + 3a + 1

= 2a³ - 2a² + 5a + 8