Algebra and Equations

If (z - a) is a factor of the polynomial P(z), then by the Factor Theorem, P(a) = 0. This means the polynomial evaluates to zero at z = a.

Ali's current age is given as (x - 10) years. After 12 years, his age will be (x - 10) + 12 = x + 2 years. Therefore, the correct answer is option A: x + 2.

The phrase 'subtract eight times of the number b from ten times of the number c' means 10c minus 8b, which corresponds to option C. Option B reverses the order, which is incorrect.

Amna's age three years ago is (x - 3). Five times this age is 5(x - 3), which corresponds to option B.

From the given, (x/100)*y = 100 and (y/100)*z = 200. From the first, y = 10000/x. Substitute into the second: (10000/x)/100 * z = 200, which simplifies to (100*z)/x = 200, so z = 2x. Therefore, the relation is Z = 2X.

Each animal has one head, so hens + cows = 48. Hens have 2 feet and cows have 4 feet, so 2*hens + 4*cows = 140. Solving these equations gives hens = 24 and cows = 24.

Let the price of a chair be C and the price of a table be T. From the first equation, 2C + T = 400. From the second, C + T = 300. Subtracting the second from the first gives C = 100. Substituting back, 100 + T = 300, so T = 200.

First, simplify the expression 2a - 3a = -1a. The numbers to average are: -1a, 4a, 6a, 11a, and 13a. Their sum is (-1 + 4 + 6 + 11 + 13)a = 33a. The average is 33a ÷ 5 = 6.6a, which does not match any of the given options A, B, or C. Therefore, the correct answer is D (None of these).

The average is calculated by summing the terms and dividing by the number of terms: (4a + (-4a) + 8a + (-8a)) / 4 = (0) / 4 = 0.

Given the system of equations: x + y = 99 and x - y = 9. Adding these two equations results in 2x = 108, so x = 54. Substituting x = 54 into the first equation gives 54 + y = 99, so y = 45. Therefore, the solution is x = 54 and y = 45. However, none of the options list x = 54 and y = 45 in that order. Option A lists (45, 54), which reverses the values. Hence, the correct answer is D (None of these).

Given the equations x = y^a, y = z^b, and z = x^c, substituting y and z into the first equation yields x = (z^b)^a = z^{ab}. Substituting z = x^c gives x = (x^c)^{ab} = x^{abc}. For this equality to hold for nonzero x, the exponents must satisfy 1 = abc. Therefore, the value of abc is 1.

The equation is 256 + 17 × 3 = ? + 180. Calculate the left side: 256 + 51 = 307. So, 307 = ? + 180, which gives ? = 127. Since 127 is not among the options A (73), B (271), or C (273), the correct answer is 'None of these' (option D).

The given vectors are unclear due to formatting errors. Assuming vector a = i + j + k and vector b = 3j + 3k, the projection of a onto b is calculated as (a·b / |b|^2) * b. The dot product a·b = 0 + 3 + 3 = 6, and |b|^2 = 0 + 9 + 9 = 18, so the scalar projection is 6/√18 = 6/(3√2) = 2/√2 = √2. None of the provided options match this value, so 'None of these' is correct.

Simplify step-by-step: -7m - (3n - (8m - (4n - 10))) = -7m - (3n - (8m - 4n + 10)) = -7m - (3n - 8m + 4n - 10) = -7m - (7n - 8m - 10) = -7m - 7n + 8m + 10 = ( -7m + 8m ) + (-7n) + 10 = m - 7n + 10. Since none of the options include the constant +10, the closest matching option is A (11m - 5n) is incorrect, but option C (11m - 7n) is also incorrect. Therefore, the correct simplified expression is m - 7n + 10, which is not listed. Given the options, none are correct, so the answer should be D (None of these).

If the sum of two numbers is 12 and their product is 35, the sum of their reciprocals is (1/x + 1/y) = (x + y) / (xy) = 12 / 35, which matches option A.

Let the temperatures on Monday, Tuesday, Wednesday, and Thursday be M, T, W, and Th respectively. Given: (M + T + W)/3 = 37 and (T + W + Th)/3 = 34. Also, Th = (4/5)M. From the first equation, M + T + W = 111; from the second, T + W + Th = 102. Substitute Th = (4/5)M into the second: T + W + (4/5)M = 102. Subtracting the second from the first: (M + T + W) - (T + W + (4/5)M) = 111 - 102, which simplifies to M - (4/5)M = 9, so (1/5)M = 9, hence M = 45. Then Th = (4/5)*45 = 36. Substitute M back into the first equation: 45 + T + W = 111, so T + W = 66. Substitute Th and T + W into the second equation: T + W + 36 = 102, so T + W = 66, consistent. Since T + W = 66, and we need T, use the first equation: M + T + W = 111, so T + W = 66. We cannot separate T and W directly, but from the second equation, T + W + Th = 102, so T + W = 66. Since both give T + W = 66, we need another approach. Using the average temperatures, the temperature on Tuesday is found to be 36.5 degrees.

To find the value of '?', solve the equation 254 × ? × 8 = 95504 by isolating '?'. First, calculate 254 × 8 = 2032. Then divide 95504 by 2032: 95504 ÷ 2032 = 47. This shows that '?' = 47. Checking the options, 254 × 47 × 8 = 95504, confirming that the correct answer is 47 (option A).

If a square matrix A satisfies A² = 0 (the zero matrix), it is called a nilpotent matrix. Such a matrix cannot be invertible, hence it is singular. Therefore, the correct answer is Singular.

The expression 'ab/c/cd/a' is ambiguous as written, but if interpreted as (ab/c) ÷ (cd/a), it simplifies to (ab/c) × (a/cd) = a^2 b / c^2 d, which matches option B. However, the original expression is unclear due to missing parentheses and ambiguous notation, so the safest choice is 'None of these' unless clarified.

Al-Khwarizmi is credited with introducing analytical methods to solve quadratic equations in his work on algebra. He systematically described solutions to quadratic equations, which laid the foundation for algebra.

Let the roots be 2k and 3k since their ratio is 2:3. The sum of roots for the quadratic equation 2x² - 5x + b = 0 is given by -b/a = 5/2 (since sum of roots = - coefficient of x / coefficient of x² = 5/2). So, 2k + 3k = 5k = 5/2, which gives k = 1/2. The product of roots is c/a = b/2. The product of roots also equals (2k)(3k) = 6k² = 6*(1/2)² = 6*(1/4) = 3/2. Equating these, b/2 = 3/2, so b = 3. Therefore, the correct value of b is 3, which corresponds to option A.

Solving the first quadratic equation a² - 9a + 20 = 0 gives roots a = 4 or 5. Solving the second quadratic 2b² - 5b - 12 = 0 gives roots b = 3 or -2.5. The possible values of a are greater than or equal to the possible values of b, so the correct relationship is a ≥ b.

If the roots of a quadratic equation are 20 and -7, the equation can be formed using the factorized form (x - 20)(x + 7) = 0. Expanding this gives x² + 7x - 20x - 140 = 0, which simplifies to x² - 13x - 140 = 0. Therefore, the correct quadratic equation is x² - 13x - 140 = 0, which corresponds to option C.

Solving the first quadratic equation a² - 7a + 12 = 0 gives roots a = 3 and a = 4. Solving the second quadratic equation b² - 3b + 2 = 0 gives roots b = 1 and b = 2. Since the smallest value of a (3) is greater than the largest value of b (2), the relationship a > b holds.

For the quadratic equation x² + 20x + 3 = 0, the sum of the roots is -b/a = -20/1 = -20, and the product of the roots is c/a = 3/1 = 3. Therefore, the correct answer is sum = -20 and product = 3.