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 #1
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+1

We can analyze the system of quadratic equations to determine the number of real solutions for different values of c. Here's how:

 

Analyzing the Discriminant:

 

The discriminant of a quadratic equation determines the nature of its roots (solutions). It is denoted by the symbol b2−4ac. In this case, considering the first equation (y = 6x^2 - 9x + c):

 

a = 6

 

b = -9

 

c (variable)

 

The discriminant (d) for the first equation is:

 

d = (-9)^2 - 4 * 6 * c

 

The number of real solutions depends on the value of the discriminant:

 

d > 0: Two real and distinct solutions (roots)

 

d = 0: One repeated real solution (root)

 

d < 0: No real solutions (complex roots)

 

Relating Discriminant to c:

 

We want to find the values of c that correspond to each case.

 

(a) Exactly one real solution:

 

For exactly one real solution (repeated root), the discriminant needs to be zero.

 

Therefore, we need to solve:

 

0 = (-9)^2 - 4 * 6 * c

 

This simplifies to:

 

c = \frac{81}{24} = \dfrac{7}{2}

 

(b) More than one real solution:

 

For more than one real solution (distinct roots), the discriminant needs to be positive.

 

Therefore, we need to solve:

 

0 < (-9)^2 - 4 * 6 * c

 

This simplifies to:

 

c < \dfrac{81}{24} = \dfrac{7}{2}

 

(c) No real solutions:

 

For no real solutions (complex roots), the discriminant needs to be negative.

 

Therefore, we need to solve:

 

0 > (-9)^2 - 4 * 6 * c

 

This simplifies to:

 

c > \dfrac{81}{24} = \dfrac{7}{2}

 

Summary:

 

(a) Exactly one real solution: c = dfrac{7}{2}

 

(b) More than one real solution: c < dfrac{7}{2}

 

(c) No real solutions: c > dfrac{7}{2}

Apr 15, 2024
 #1
avatar+179 
0

We can solve for a and b by utilizing the given information about A and its effect on x and y, along with the property of matrix multiplication.

 

Here's how we proceed:

 

Analyze the equation (A^5)x = ax + by:

 

This equation states that applying matrix A to vector x five times consecutively (A raised to the power of 5) results in a linear combination of x and y, with coefficients a and b.

 

Utilize the information about A:

 

We are given that Ax = y and Ay = x + 2y. These equations define how A transforms x and y.

 

Express (A^5)x in terms of x and y:

 

We can't directly expand (A^5) as it's a high power. However, we can use the given information about A iteratively.

 

Start with the first equation: (A^2)x = A(Ax) = A(y).

 

Substitute from the first given equation: (A^2)x = Ay.

 

Use the second given equation: (A^2)x = x + 2y.

 

Similarly, we can continue:

 

(A^3)x = A((A^2)x) = A(x + 2y) = Ax + 2Ay (using the definition of matrix multiplication).

 

Substitute from the first given equation: (A^3)x = y + 2(x + 2y) = 3x + 4y.

 

We can repeat this process further, but the pattern should be clear.

 

Find an expression for (A^4)x and (A^5)x:

 

Following the established pattern, we can see that:

 

(A^4)x = 3(A^3)x = 3(3x + 4y) = 9x + 12y.

 

(A^5)x = 3(A^4)x = 3(9x + 12y) = 27x + 36y.

 

Substitute (A^5)x in the original equation:

 

The original equation is: (A^5)x = ax + by.

 

Substitute the expression we found for (A^5)x: 27x + 36y = ax + by.

 

Solve for a and b:

 

We want to isolate a and b. Since x and y are not multiples of each other, we can treat them as independent variables.

 

If we set y = 0, the equation becomes 27x = ax, which implies a = 27.

 

If we set x = 0, the equation becomes 36y = by, which implies b = 36.

 

Therefore, in the equation (A^5)x = ax + by, the values of a and b are:

 

a = 27

 

b = 36

Apr 15, 2024
 #1
avatar+179 
+1

Analyzing the function f(x) = floor((2 - 3x) / (x + 3)):

 

The denominator (x + 3) is 0 when x = -3. This means f(x) is undefined at x = -3.

 

We will need to consider different cases based on the sign of the denominator (x + 3) and the relative values of 2 - 3x compared to 0.

 

Cases for f(x):

 

x < -3: In this case, both denominator (x + 3) and numerator (2 - 3x) are negative. Dividing two negative numbers results in a positive value.

 

Since we take the floor (greatest integer less than or equal to), f(x) will be -1.

 

-3 < x < 2/3: Here, the denominator (x + 3) is positive, but the numerator (2 - 3x) is negative.

 

Dividing a positive by a negative results in a negative number.

 

Taking the floor of a negative number keeps it negative, so f(x) will be -2.

 

x = 2/3: At this specific point, the numerator becomes 0, and the result of the division is 0. The floor of 0 is 0, so f(x) = 0.

 

x > 2/3: In this case, both the numerator (2 - 3x) and denominator (x + 3) are positive. Dividing two positive numbers results in a positive value.

 

Taking the floor doesn't change the positive sign, so f(x) will be 1.

 

Evaluating the sum:

 

The key to evaluating the sum efficiently is to recognize that for a large range of x values (between -3 and 2/3), f(x) will be -2.

 

We can exploit this by calculating the number of terms that fall into this range and summing the contributions from the remaining terms separately.

 

Number of terms where f(x) = -2:

 

We know x = -3 falls outside this range (f(x) is undefined).

 

The range ends when x = 2/3, which is between terms 1000 and 1001 (1000th term is x = 999 and 1001st term is x = 1000).

 

Therefore, there are 1000 - (-3) + 1 = 1004 terms where f(x) = -2.

 

Contribution from terms where f(x) = -2:

 

Each term contributes -2 to the sum.

 

Total contribution = -2 * (number of terms) = -2 * 1004 = -2008

 

Remaining terms:

 

We need to consider terms for x < -3 (f(x) = -1), x = 2/3 (f(x) = 0), and x > 2/3 (f(x) = 1).

 

There are very few terms less than -3 (all negative x values), and they can be ignored for a large sum like this (their contribution will be negligible).

 

There's only one term for x = 2/3, contributing f(2/3) = 0.

 

The remaining terms from x slightly greater than 2/3 to x = 1000 will all have f(x) = 1. The exact number of these terms depends on the specific values, but there will be significantly fewer compared to the 1004 terms with f(x) = -2.

 

Overall Sum:

 

Sum from terms with f(x) = -2: -2008

 

Contribution from f(2/3) (x = 2/3): 0

 

Contribution from remaining terms with f(x) = 1 (positive but less than those with -2): + (positive value)

 

Since the number of terms with f(x) = 1 is significantly less than those with -2, and there's a negligible contribution from terms less than -3, the positive value from the remaining terms will be much smaller than 2008.

 

Therefore, the overall sum f(1) + f(2) + ... + f(999) + f(1000) is equal to -2008.

Mar 24, 2024