10 - Maths - Unit 3. Algebra

Explanation
If the system has a unique solution, it means that there is only one possible value for k that satisfies both equations. Therefore, k cannot be equal to 3, as this would result in a contradiction with the second equation. Hence, the correct answer is (2) k # 3.

Explanation
If the system of two linear equations is inconsistent, it means that the equations do not have a common solution. In other words, there is no point where the two graphs intersect. This is the same as saying that the graphs do not intersect at any point. Therefore, option (3) is the correct answer.

Explanation
The system of equations has infinitely many solutions because the two equations are equivalent. If we multiply the first equation by 3, we get 3x - 12y = 24, which is the same as the second equation. This means that the two equations represent the same line and every point on that line is a solution to the system. Therefore, there are infinitely many solutions.

Explanation
The question states that one zero of the polynomial p(x) is the reciprocal of the other zero. Let's assume that the zeros are a and 1/a. According to the sum of the zeros formula, a + 1/a = -13/k. Simplifying this equation, we get a^2 + 1 = -13a/k. Rearranging the equation, we get k = -13a/(a^2 + 1). Since a and 1/a are reciprocals, a^2 = 1. Substituting this into the equation, we get k = -13a/(1 + 1) = -13a/2. Therefore, k = 2.

Explanation
The sum of the two zeros of a polynomial is equal to the negative coefficient of the linear term divided by the leading coefficient. In this case, the linear term is (p + 3)x and the leading coefficient is 5. So, the sum of the zeros is -(p + 3)/5. Since it is given that the sum of the zeros is zero, we can set -(p + 3)/5 = 0 and solve for p. Simplifying the equation, we get p = -3. Therefore, the value of P is -3.

Explanation
When a number is divided by (x+4), the remainder will be one less than the divisor if the number is one more than a multiple of (x+4). In this case, the number is 31, which is one more than a multiple of (x+4). Therefore, the remainder when 31 is divided by (x+4) is 1.

Explanation
When a number is divided by (x-1), it means that the number is being divided by a factor of (x-1). In this case, the given number is x^2 + 1. To find the quotient, we divide x^2 + 1 by (x-1). The result is (x+1), which is option (2).

Explanation
The lowest form of a rational expression refers to the simplified form of the expression where the numerator and denominator have no common factors. In this case, the correct answer is (2) because it is the only option that represents the lowest form of the rational expression.

Explanation
The correct answer is (1) (x-5) (x-3). This can be determined by using the distributive property of multiplication. When diving by a binomial, each term in the numerator is divided by each term in the binomial. In this case, each term in the numerator (x) is divided by each term in the binomial (x-5) and (x-3). This results in the expression (x-5) (x-3). Therefore, the correct answer is (1) (x-5) (x-3).

Explanation
The square root of 49 is 7 because 7 multiplied by itself equals 49.

Explanation
The correct answer is (4) because the square root of a number is a value that, when multiplied by itself, gives the original number. In this case, the square root of 4 is 2, because 2 multiplied by itself equals 4.

Explanation
The correct answer is (4) because the square root of 121 is 11. When a number is squared, it is multiplied by itself. Therefore, when 11 is multiplied by itself, the result is 121.

Explanation
If a quadratic equation has equal roots, it means that the discriminant (b^2 - 4ac) is equal to zero. In this case, the value of c does not affect the discriminant and therefore can be any value. So, c can be equal to any other option (1), (3), or (4).

Explanation
If a quadratic equation has no real roots, it means that the discriminant (b^2 - 4ac) is negative. In option (3), the discriminant is represented as -b^2 - 4ac, which is negative. Therefore, option (3) is the correct answer.

Explanation
If α and β are the roots of the quadratic equation, then the equation can be written as (x-α)(x-β) = 0.
To find a quadratic equation with roots γ and δ, we can substitute α with γ and β with δ in the equation.
So, the equation becomes (x-γ)(x-δ) = 0, which is option (3).

Explanation
If a = c, then substituting c in place of a in the equation a = -c would result in a = -a. This is not possible because it would imply that a is equal to its negative value, which is only true for a = 0. Therefore, the correct answer is (1) a = c.