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State whether each of the following characteristics of a car is either a qualitative or quantitative measure: (i) age of the car; (ii) colour of the car; (iii) softness of the car seat; (iv) cost of the car in Ksh; (v) mileage of the car. (5 marks}DICT MOD 1 July 2020
(i) Age of the car: This is a quantitative measure because it is a numerical value that represents the length of time that has passed since the car was manufactured. (ii) Colour of the car: This is a qualitative measure because it is a non-numerical characteristic that describes the appearance of thRead more
(i) Age of the car: This is a quantitative measure because it is a numerical value that represents the length of time that has passed since the car was manufactured.
(ii) Colour of the car: This is a qualitative measure because it is a non-numerical characteristic that describes the appearance of the car.
(iii) Softness of the car seat: This is a qualitative measure because it is a non-numerical characteristic that describes the comfort of the car seat.
(iv) Cost of the car in Ksh: This is a quantitative measure because it is a numerical value that represents the monetary value of the car.
(v) Mileage of the car: This is a quantitative measure because it is a numerical value that represents the distance that the car has traveled
See lessOutline four properties of the arithmetic mean. (4 marks)DICT MOD 1 July 2020
It is sensitive to extreme values: The arithmetic mean is sensitive to extreme values, which means that it can be greatly affected by values that are significantly larger or smaller than the other values in the data set. For example, if a data set contains both very large and very small values, theRead more
Using Pascal’s triangle, expand each of the following expressions in ascending powers of x: (i) (a+x)^3 (ii) (2+x)^6. (6 marks)DICT MOD 1 July 2020
(i) To expand (a+x)^3 using Pascal's triangle, we start by writing the binomial and the exponent in the top row of the triangle: 1 3 We can then use the numbers in the triangle to fill in the rows below: 1 3 1 3 The expansion of (a+x)^3 is obtained by multiplying each term in the binomial by the corRead more
(i) To expand (a+x)^3 using Pascal’s triangle, we start by writing the binomial and the exponent in the top row of the triangle:
1 3
We can then use the numbers in the triangle to fill in the rows below:
1 3 1 3
The expansion of (a+x)^3 is obtained by multiplying each term in the binomial by the corresponding coefficient in the triangle and then summing the terms:
(a+x)^3 = 1a^3 + 3a^2x + 3ax^2 + x^3
(ii) To expand (2+x)^6 using Pascal’s triangle, we start by writing the binomial and the exponent in the top row of the triangle:
1 6
We can then use the numbers in the triangle to fill in the rows below:
1 6 1 6 1 6 1 6
The expansion of (2+x)^6 is obtained by multiplying each term in the binomial by the corresponding coefficient in the triangle and then summing the terms:
(2+x)^6 = 12^6 + 62^5x + 152^4x^2 + 202^3x^3 + 152^2x^4 + 62*x^5 + x^6
= 64 + 384x + 960x^2 + 1280x^3 + 960x^4 + 384x^5 + x^6
See lessExplain two assumptions of linear extrapolation. (4 marks)DICT MOD 1 July 2020
Linear extrapolation is a method of estimating the value of a function at a point outside the range of known data points. It is based on the assumption that the function is a straight line beyond the range of known data points. There are two main assumptions of linear extrapolation: The function isRead more
Linear extrapolation is a method of estimating the value of a function at a point outside the range of known data points. It is based on the assumption that the function is a straight line beyond the range of known data points. There are two main assumptions of linear extrapolation:
- The function is a straight line: Linear extrapolation assumes that the function is a straight line beyond the range of known data points. This means that the relationship between the variables in the function is linear, or that the change in one variable is proportional to the change in the other variable. This assumption is necessary for linear extrapolation to be accurate, as it allows us to use the known data points to fit a straight line to the data and then use this line to estimate the value at the desired point outside the range of known data points.
- The function is continuous: Linear extrapolation also assumes that the function is continuous beyond the range of known data points. This means that there are no sudden changes or discontinuities in the function beyond the range of known data points. This assumption is necessary because it allows us to use the known data points to estimate the value of the function at the desired point outside the range of known data points without worrying about sudden changes or discontinuities in the function.
See lessDistinguish between linear interpolation and linear extrapolation as used in mathematics. (4 marks)DICT MOD 1 July 2020
Linear interpolation is a method of estimating the value of a function at a point within the range of known data points. It is based on the assumption that the function is a straight line within the range of known data points. To interpolate a value, we use the known data points to fit a straight liRead more
Linear interpolation is a method of estimating the value of a function at a point within the range of known data points. It is based on the assumption that the function is a straight line within the range of known data points. To interpolate a value, we use the known data points to fit a straight line to the data and then use this line to estimate the value at the desired point.
Linear extrapolation, on the other hand, is a method of estimating the value of a function at a point outside the range of known data points. It is based on the assumption that the function is a straight line beyond the range of known data points. To extrapolate a value, we use the known data points to fit a straight line to the data and then use this line to estimate the value at the desired point outside the range of known data points.
See less(i) Define each of the following terms as applied in binary codes: (i) weighted binary codes; (ii) reflective codes; (iii) sequential codes; (3 marks)DICT MOD 1 July 2020
(i) Weighted binary codes are a type of binary code that assigns different weights or values to the digits (bits) in a binary number based on their position. In a weighted binary code, the value of each digit is determined by its position in the number. For example, in a binary number with a weightRead more
(i) Weighted binary codes are a type of binary code that assigns different weights or values to the digits (bits) in a binary number based on their position. In a weighted binary code, the value of each digit is determined by its position in the number. For example, in a binary number with a weight of 2, the rightmost digit (the least significant bit) has a value of 1, the next digit has a value of 2, the next has a value of 4, and so on. Weighted binary codes are often used to represent numeric values in computers and other digital systems.
(ii) Reflective codes are a type of error-detection code that uses a parity bit to detect errors in a transmitted message. In a reflective code, the parity bit is placed at the end of the message and is set to either 0 or 1 depending on the number of 1’s in the message. If the number of 1’s in the message is even, the parity bit is set to 0. If the number of 1’s in the message is odd, the parity bit is set to 1. When the message is received, the parity bit is checked to see if it is the same as the calculated parity. If the parity bit is not the same as the calculated parity, an error has occurred and the message must be re-transmitted.
(iii) Sequential codes are a type of code that assigns a unique code to each item in a set of items. In a sequential code, the codes are assigned in a specific order, usually based on the order in which the items were added to the set. Sequential codes are often used to uniquely identify items in a database or other data storage system.
See lessSolve each of the following equations using the method indicated: (i) -3x^2 +10x+8=0 by factorization; (ii) 2x^2 + 9x-5=0 by completing the square. (6 marks)DICT MOD 1 July 2020
(i) To solve the equation -3x^2 +10x+8=0 by factorization, we start by setting the expression -3x^2 +10x+8 equal to zero and then factoring the quadratic. We can use the quadratic formula to find the roots of the equation, which are the values of x that make the equation true. -3x^2 +10x+8=0 To factRead more
(i) To solve the equation -3x^2 +10x+8=0 by factorization, we start by setting the expression -3x^2 +10x+8 equal to zero and then factoring the quadratic. We can use the quadratic formula to find the roots of the equation, which are the values of x that make the equation true.
-3x^2 +10x+8=0
To factor this quadratic, we can use the factorization method, which involves expressing the quadratic as the product of two binomials. One way to do this is to rewrite the quadratic as follows:
-3x^2 +10x+8=0
(-3x^2 +10x) + 8=0
(-3x^2 +10x) = -8
We can now factor the left-hand side of the equation by finding two numbers whose product is -8 and whose sum is 10. The numbers that satisfy these conditions are -4 and 2, so we can rewrite the left-hand side of the equation as follows:
(-3x^2 +10x) = (-4)(2)
(-3x^2 +10x) = (-4x +2)(2)
We can now rewrite the original equation as follows:
(-4x +2)(2) + 8=0
(-4x +2)(2) = -8
This equation can be simplified to:
-4x +2 = -4
-4x = -2
x = 1/2
Therefore, the solution to the equation is x = 1/2.
(ii) To solve the equation 2x^2 + 9x-5=0 by completing the square, we start by setting the expression 2x^2 + 9x-5 equal to zero and then completing the square to find the roots of the equation.
2x^2 + 9x-5=0
To complete the square, we need to rewrite the quadratic as a perfect square trinomial of the form (x + b)^2 = x^2 + 2bx + b^2. We can do this by adding and subtracting the square of half of the coefficient of the x term. In this case, the coefficient of the x term is 9, so we add and subtract (9/2)^2 = (9/2)*(9/2) = 9/4 = 2.25. This gives us the following equation:
2x^2 + 9x-5=0
2x^2 + 9x – 2.25 + 2.25 -5=0
(2x^2 + 9x – 2.25) + (2.25 -5)=0
(2x^2 + 9x – 2.25) = -(2.25 -5)
(2x^2 + 9x – 2.25) = -(-0.25)
(2x^2 + 9x – 2.25) = 0.25
We can now rewrite the left-hand side of the equation as a perfect square trinomial by completing the square:
(2x^2 + 9x – 2.25) = (x + (9/2))^2
(x + (9/2))^2 = 0.25
x + (9/2) = sqrt(0.25)
x + (9
See lessDefine each of the following types of matrices: (i) null matrix; Gi) = identity matrix. (2 marks)DICT MOD 1 July 2020
(i) A null matrix, also known as a zero matrix, is a matrix with all elements equal to zero. It is represented by a matrix of size m x n with all elements equal to zero, where m and n are the number of rows and columns in the matrix, respectively. For example, the matrix [0 0 0] [0 0 0] is a 2 x 3 nRead more
(i) A null matrix, also known as a zero matrix, is a matrix with all elements equal to zero. It is represented by a matrix of size m x n with all elements equal to zero, where m and n are the number of rows and columns in the matrix, respectively. For example, the matrix
[0 0 0] [0 0 0]
is a 2 x 3 null matrix.
(ii) An identity matrix is a square matrix with 1’s on the main diagonal (top-left to bottom-right) and 0’s everywhere else. It is represented by an m x m matrix with 1’s on the main diagonal and 0’s everywhere else, where m is the size of the matrix. For example, the matrix
[1 0 0] [0 1 0] [0 0 1]
is a 3 x 3 identity matrix. The identity matrix is an important matrix in linear algebra and is often denoted by the symbol I. It has the property that when multiplied by any matrix, the result is the original matrix itself.
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