What are the definitions of a function, the value of f(-2) for f(x) = 4x + 3, the maximum number of roots for a quadratic equation, and an identity matrix?

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Asked on April 23, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-04-23T08:05:09.845Z

Here are the solutions to all the questions: 1) Step 1: Define a function. A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. It describes how one quantity depends on another. 2) Step 1: Substitute the value of x into the function. Given f(x) = 4x + 3. To find f(-2), substitute x = -2: f(-2) = 4(-2) + 3 Step 2: Calculate the value. f(-2) = -8 + 3 f(-2) = -5 The value of f(-2) is -5. 3) Step 1: Recall the nature of roots for a quadratic equation. A quadratic equation is of the form ax^2 + bx + c = 0, where a ≠ 0. The number of roots depends on the discriminant = b^2 - 4ac. • If > 0, there are two distinct real roots. • If = 0, there is one real root (a repeated root). • If < 0, there are two distinct complex roots. Step 2: State the maximum number of roots. A quadratic equation can have at most two roots (real or complex). A quadratic equation can have 2 roots. 4) Step 1: Define an identity matrix. An identity matrix is a square matrix where all the elements on the main diagonal are 1s and all other elements are 0s. It is denoted by I or I_n, where n is the dimension of the square matrix. When an identity matrix multiplies another matrix, the other matrix remains unchanged. 5) Step 1: Check if matrix multiplication is possible. Matrix A has order 2 × 3. Matrix B has order 3 × 1. For matrix multiplication AB to be possible, the number of columns in A must be equal to the number of rows in B. Number of columns in A = 3. Number of rows in B = 3. Since 3 = 3, multiplication AB is possible. Step 2: Determine the order of the resulting matrix AB. If A is an m × n matrix and B is an n × p matrix, then the product AB is an m × p matrix. For A (2 × 3) and B (3 × 1), the order of AB will be 2 × 1. Yes, AB is possible. The order of matrix AB will be 2 × 1. 6) Step 1: Recall the formula for the slope of a line given its inclination. The slope m of a line with inclination is given by m = . Step 2: Substitute the given inclination and calculate the slope. Given inclination = 135^. m = (135^) m = (180^ - 45^) m = -(45^) m = -1 The slope of the line is -1. 7) Step 1: State the formula for external division. The coordinates of point P (x, y) that divides the join of R (x_1, y_1) and S (x_2, y_2) in the ratio m_1:m_2 externally are given by: P(x, y) = ( (m_1 x_2 - m_2 x_1)/(m_1 - m_2), (m_1 y_2 - m_2 y_1)/(m_1 - m_2) ) The coordinates of point P are ( (m_1 x_2 - m_2 x_1)/(m_1 - m_2), (m_1 y_2 - m_2 y_1)/(m_1 - m_2) ). 8) Step 1: Write the given vectors in column form. Given a = (1, 2) and b = (4, 5). In column vector form: a = 1 \\ 2 b = 4 \\ 5 Step 2: Add the vectors. a + b = 1 \\ 2 + 4 \\ 5 a + b = 1+4 \\ 2+5 a + b = 5 \\ 7 The sum a + b in column vector form is 5 \\ 7 . 9) Step 1: Recall the formula for Quartile Deviation (Q.D.). The Quartile Deviation is given by the formula: Q.D. = (Q_3 - Q_1)/(2) where Q_3 is the third quartile and Q_1 is the first quartile. Step 2: Substitute the given values and solve for Q_3. Given Q.D. = 30 and Q_1 = 10. 30 = (Q_3 - 10)/(2) Step 3: Isolate Q_3. Multiply both sides by 2: 30 × 2 = Q_3 - 10 60 = Q_3 - 10 Add 10 to both sides: Q_3 = 60 + 10 Q_3 = 70 The value of Q_3 is 70. 10) Step 1: Define a box and whisker plot. A box and whisker plot (or box plot) is a standardized way of displaying the distribution of data based on five key numbers: the minimum value, the first quartile (Q_1), the median (Q_2), the third quartile (Q_3), and the maximum value. The "box" represents the interquartile range (Q_1 to Q_3), and the "whiskers" extend to the minimum and maximum values within a certain range. 11) Step 1: Find f(x). Given f(2x - 3) = 4x + 5. Let y = 2x - 3. Then y + 3 = 2x, so x = (y+3)/(2). Substitute this expression for x into the function: f(y) = 4((y+3)/(2)) + 5 f(y) = 2(y+3) + 5 f(y) = 2y + 6 + 5 f(y) = 2y + 11 Replacing y with x, we get: f(x) = 2x + 11 Step 2: Find f(-2). Substitute x = -2 into the expression for f(x): f(-2) = 2(-2) + 11 f(-2) = -4 + 11 f(-2) = 7 So, f(x) = 2x + 11 and f(-2) = 7. 12) Step 1: Perform polynomial long division. We need to divide 4x^3 + 2x^2 - 4x + 3 by 2x + 3. ` 2x^2 - 2x + 1 _________________ 2x + 3 | 4x^3 + 2x^2 - 4x + 3 -(4x^3 + 6x^2) _________________ -4x^2 - 4x -(-4x^2 - 6x) _________________ 2x + 3 -(2x + 3) _________ 0 ` Step 2: Identify the quotient and remainder. The quotient is 2x^2 - 2x + 1. The remainder is 0. The quotient is 2x^2 - 2x + 1 and the remainder is 0. 13) Step 1: Substitute the coordinates of the point into the equation of the locus. The point is (4, 5), so x = 4 and y = 5. The equation of the locus is x^2 + y^2 + kx - 8y - 25 = 0. Substitute x=4 and y=5: (4)^2 + (5)^2 + k(4) - 8(5) - 25 = 0 Step 2: Simplify the equation. 16 + 25 + 4k - 40 - 25 = 0 Step 3: Solve for k. 41 + 4k - 65 = 0 4k - 24 = 0 4k = 24 k = (24)/(4) k = 6 The value of k is 6. 14) Step 1: Understand the matrix construction rule. We need to construct a 2 × 2 matrix, so it will have elements a_11, a_12, a_21, a_22. The rule for elements is a_ij = 2i - 3j. Step 2: Calculate each element. For a_11 (i=1, j=1): a_11 = 2(1) - 3(1) = 2 - 3 = -1 For a_12 (i=1, j=2): a_12 = 2(1) - 3(2) = 2 - 6 = -4 For a_21 (i=2, j=1): a_21 = 2(2) - 3(1) = 4 - 3 = 1 For a_22 (i=2, j=2): a_22 = 2(2) - 3(2) = 4 - 6 = -2 Step 3: Form the matrix. The matrix is: a_11 & a_12 \\ a_21 & a_22 = -1 & -4 \\ 1 & -2 The constructed matrix is -1 & -4 \\ 1 & -2 . 15) This question was solved in the previous turn. The values are a=1 and b=4. 16) Step 1: Rewrite the equation using trigonometric identities. Given 2 = . Use the double angle identity for sine: 2 = 2 . 2 = Step 2: Rearrange the equation to solve for . 2 - = 0 Factor out : (2 - 1) = 0 Step 3: Set each factor to zero and solve for . Case 1: = 0 For 0^ 90^, the value of for which = 0 is = 90^. Case 2: 2 - 1 = 0 2 = 1 = (1)/(2) For 0^ 90^, the value of for which = (1)/(2) is = 30^. Step 4: List all solutions within the given range. The solutions are = 30^ and = 90^. The solutions are 30^, 90^. That's 2 down. 3 left today — send the next one.

Accepted Answer · 2026-04-23T08:05:09.845Z

Here are the solutions to all the questions: 1) Step 1: Define a function. A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. It describes how one quantity depends on another. 2) Step 1: Substitute the value of x into the function. Given f(x) = 4x + 3. To find f(-2), substitute x = -2: f(-2) = 4(-2) + 3 Step 2: Calculate the value. f(-2) = -8 + 3 f(-2) = -5 The value of f(-2) is -5. 3) Step 1: Recall the nature of roots for a quadratic equation. A quadratic equation is of the form ax^2 + bx + c = 0, where a ≠ 0. The number of roots depends on the discriminant = b^2 - 4ac. • If > 0, there are two distinct real roots. • If = 0, there is one real root (a repeated root). • If < 0, there are two distinct complex roots. Step 2: State the maximum number of roots. A quadratic equation can have at most two roots (real or complex). A quadratic equation can have 2 roots. 4) Step 1: Define an identity matrix. An identity matrix is a square matrix where all the elements on the main diagonal are 1s and all other elements are 0s. It is denoted by I or I_n, where n is the dimension of the square matrix. When an identity matrix multiplies another matrix, the other matrix remains unchanged. 5) Step 1: Check if matrix multiplication is possible. Matrix A has order 2 × 3. Matrix B has order 3 × 1. For matrix multiplication AB to be possible, the number of columns in A must be equal to the number of rows in B. Number of columns in A = 3. Number of rows in B = 3. Since 3 = 3, multiplication AB is possible. Step 2: Determine the order of the resulting matrix AB. If A is an m × n matrix and B is an n × p matrix, then the product AB is an m × p matrix. For A (2 × 3) and B (3 × 1), the order of AB will be 2 × 1. Yes, AB is possible. The order of matrix AB will be 2 × 1. 6) Step 1: Recall the formula for the slope of a line given its inclination. The slope m of a line with inclination is given by m = . Step 2: Substitute the given inclination and calculate the slope. Given inclination = 135^. m = (135^) m = (180^ - 45^) m = -(45^) m = -1 The slope of the line is -1. 7) Step 1: State the formula for external division. The coordinates of point P (x, y) that divides the join of R (x_1, y_1) and S (x_2, y_2) in the ratio m_1:m_2 externally are given by: P(x, y) = ( (m_1 x_2 - m_2 x_1)/(m_1 - m_2), (m_1 y_2 - m_2 y_1)/(m_1 - m_2) ) The coordinates of point P are ( (m_1 x_2 - m_2 x_1)/(m_1 - m_2), (m_1 y_2 - m_2 y_1)/(m_1 - m_2) ). 8) Step 1: Write the given vectors in column form. Given a = (1, 2) and b = (4, 5). In column vector form: a = 1 \\ 2 b = 4 \\ 5 Step 2: Add the vectors. a + b = 1 \\ 2 + 4 \\ 5 a + b = 1+4 \\ 2+5 a + b = 5 \\ 7 The sum a + b in column vector form is 5 \\ 7 . 9) Step 1: Recall the formula for Quartile Deviation (Q.D.). The Quartile Deviation is given by the formula: Q.D. = (Q_3 - Q_1)/(2) where Q_3 is the third quartile and Q_1 is the first quartile. Step 2: Substitute the given values and solve for Q_3. Given Q.D. = 30 and Q_1 = 10. 30 = (Q_3 - 10)/(2) Step 3: Isolate Q_3. Multiply both sides by 2: 30 × 2 = Q_3 - 10 60 = Q_3 - 10 Add 10 to both sides: Q_3 = 60 + 10 Q_3 = 70 The value of Q_3 is 70. 10) Step 1: Define a box and whisker plot. A box and whisker plot (or box plot) is a standardized way of displaying the distribution of data based on five key numbers: the minimum value, the first quartile (Q_1), the median (Q_2), the third quartile (Q_3), and the maximum value. The "box" represents the interquartile range (Q_1 to Q_3), and the "whiskers" extend to the minimum and maximum values within a certain range. 11) Step 1: Find f(x). Given f(2x - 3) = 4x + 5. Let y = 2x - 3. Then y + 3 = 2x, so x = (y+3)/(2). Substitute this expression for x into the function: f(y) = 4((y+3)/(2)) + 5 f(y) = 2(y+3) + 5 f(y) = 2y + 6 + 5 f(y) = 2y + 11 Replacing y with x, we get: f(x) = 2x + 11 Step 2: Find f(-2). Substitute x = -2 into the expression for f(x): f(-2) = 2(-2) + 11 f(-2) = -4 + 11 f(-2) = 7 So, f(x) = 2x + 11 and f(-2) = 7. 12) Step 1: Perform polynomial long division. We need to divide 4x^3 + 2x^2 - 4x + 3 by 2x + 3. ` 2x^2 - 2x + 1 _________________ 2x + 3 | 4x^3 + 2x^2 - 4x + 3 -(4x^3 + 6x^2) _________________ -4x^2 - 4x -(-4x^2 - 6x) _________________ 2x + 3 -(2x + 3) _________ 0 ` Step 2: Identify the quotient and remainder. The quotient is 2x^2 - 2x + 1. The remainder is 0. The quotient is 2x^2 - 2x + 1 and the remainder is 0. 13) Step 1: Substitute the coordinates of the point into the equation of the locus. The point is (4, 5), so x = 4 and y = 5. The equation of the locus is x^2 + y^2 + kx - 8y - 25 = 0. Substitute x=4 and y=5: (4)^2 + (5)^2 + k(4) - 8(5) - 25 = 0 Step 2: Simplify the equation. 16 + 25 + 4k - 40 - 25 = 0 Step 3: Solve for k. 41 + 4k - 65 = 0 4k - 24 = 0 4k = 24 k = (24)/(4) k = 6 The value of k is 6. 14) Step 1: Understand the matrix construction rule. We need to construct a 2 × 2 matrix, so it will have elements a_11, a_12, a_21, a_22. The rule for elements is a_ij = 2i - 3j. Step 2: Calculate each element. For a_11 (i=1, j=1): a_11 = 2(1) - 3(1) = 2 - 3 = -1 For a_12 (i=1, j=2): a_12 = 2(1) - 3(2) = 2 - 6 = -4 For a_21 (i=2, j=1): a_21 = 2(2) - 3(1) = 4 - 3 = 1 For a_22 (i=2, j=2): a_22 = 2(2) - 3(2) = 4 - 6 = -2 Step 3: Form the matrix. The matrix is: a_11 & a_12 \\ a_21 & a_22 = -1 & -4 \\ 1 & -2 The constructed matrix is -1 & -4 \\ 1 & -2 . 15) This question was solved in the previous turn. The values are a=1 and b=4. 16) Step 1: Rewrite the equation using trigonometric identities. Given 2 = . Use the double angle identity for sine: 2 = 2 . 2 = Step 2: Rearrange the equation to solve for . 2 - = 0 Factor out : (2 - 1) = 0 Step 3: Set each factor to zero and solve for . Case 1: = 0 For 0^ 90^, the value of for which = 0 is = 90^. Case 2: 2 - 1 = 0 2 = 1 = (1)/(2) For 0^ 90^, the value of for which = (1)/(2) is = 30^. Step 4: List all solutions within the given range. The solutions are = 30^ and = 90^. The solutions are 30^, 90^. That's 2 down. 3 left today — send the next one.

What are the definitions of a function, the value of f(-2) for f(x) = 4x + 3, the maximum number of roots for a quadratic equation, and an identity matrix?

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What are the definitions of a function, the value of f(-2) for f(x) = 4x + 3, the maximum number of roots for a quadratic equation, and an identity matrix?

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