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Question 1 Report
A bag contains 5 red and 5 blue identical balls. Three balls are selected at random without replacement. Determine the probability of selecting balls alternating in color.
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Question 2 Report
If log 5(\(\frac{125x^3}{\sqrt[ 3 ] {y}}\) is expressed in the values of p, q and k respectively.
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Question 3 Report
P(3,4) and Q(-3, -4) are two points in a plane. Find the gradient of the line that is normal to the line PQ.
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Question 4 Report
If V = plog\(_x\), (M + N), express N in terms of X, P, M and V
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The given equation is V = plog\(_x\), (M + N). We need to express N in terms of X, P, M, and V. We can start by isolating the term containing N: V = plog\(_x\), (M + N) V = p(log\(_x\) M + log\(_x\) N) V - plog\(_x\) M = plog\(_x\) N log\(_x\) N = (V - plog\(_x\) M) / p Now we can solve for N by exponentiating both sides with base x: N = x\(^{\frac{V - plog_x M}{p}}\) Therefore, the expression for N in terms of X, P, M, and V is N = x\(^{\frac{V - plog_x M}{p}}\). Option (A) is the correct answer.
Question 7 Report
Determine the coefficient of x\(^3\) in the binomial expansion of ( 1 + \(\frac{1}{2}\)x)
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Question 8 Report
If \(\int^3_0(px^2 + 16)dx\) = 129. Find the value of p.
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The value of p can be found by solving the definite integral for a given value of p and equating it to the given value of 129. The definite integral of the function px^2 + 16 between the limits 0 and 3 can be calculated using antiderivatives. The antiderivative of px^2 is (1/3)px^3 and the antiderivative of 16 is 16x. So, ∫^3_0(px^2 + 16)dx = (1/3)p * 3^3 + 16 * 3 - (1/3)p * 0^3 - 16 * 0 = (1/3)p * 27 + 48 Now, equating this expression to 129 and solving for p, we get: (1/3)p * 27 + 48 = 129 (1/3)p * 27 = 81 p * 27 = 243 p = 243/27 p = 9 So, the value of p is 9.
Question 9 Report
If X = \(\frac{3}{5}\) and cos y = \(\frac{24}{25}\), where X and Y are acute, find the value of cos (X + Y).
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Question 10 Report
If the sum of the roots of 2x\(^2\) + 5mx + n = 0 is 5, find the value of m.
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Let's first recall the formula for the sum of the roots of a quadratic equation in the form ax\(^2\) + bx + c = 0: Sum of roots = -b/a In the given equation, 2x\(^2\) + 5mx + n = 0, the coefficient of x\(^2\) is 2, which means a = 2. Therefore, the sum of the roots can be expressed as: Sum of roots = - (5m) / 2 We are given that the sum of the roots is 5, so we can set up an equation: 5 = - (5m) / 2 Multiplying both sides by -2, we get: -10 = 5m Dividing both sides by 5, we obtain: m = -2 Therefore, the value of m is -2. So the answer is -2.0.
Question 11 Report
Calculate, correct to two decimal places, the area enclosed by the line 3x - 5y + 4 = 0 and the axes.
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To find the area enclosed by the line 3x - 5y + 4 = 0 and the axes, we need to find the points where the line intersects with the x-axis and y-axis. First, let's find the x-intercept. To do this, we set y = 0 and solve for x: 3x - 5(0) + 4 = 0 3x + 4 = 0 3x = -4 x = -4/3 So the x-intercept is (-4/3, 0). Next, let's find the y-intercept. To do this, we set x = 0 and solve for y: 3(0) - 5y + 4 = 0 -5y + 4 = 0 -5y = -4 y = 4/5 So the y-intercept is (0, 4/5). Now we can draw a triangle with vertices at the origin (0,0), the x-intercept (-4/3, 0), and the y-intercept (0, 4/5). The base of the triangle is the x-intercept, which has a length of 4/3. The height of the triangle is the y-intercept, which has a length of 4/5. Therefore, the area of the triangle is: (1/2) * base * height = (1/2) * (4/3) * (4/5) = 8/15 = 0.53 (rounded to two decimal places) Therefore, the answer is option C: 0.53 square units.
Question 13 Report
A circle with centre (5,-4) passes through the point (5, 0). Find its equation.
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Question 14 Report
A three-digit odd number less than 500 is to be formed from 1,2,3,4 and 5. If repetition of digits is allowed, in how many ways can this be done?
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Question 15 Report
Calculate the variance of \(\sqrt{2}\), (1 + \(\sqrt{2}\)) and (2 + \(\sqrt{2}\))
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Question 16 Report
In which of the following series can be the formula S = \(\frac{a}{1 - r}\) where a is the first term and r is the common ratio, be used to find the sum of all the terms?
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Question 17 Report
Calculate the probability that the product of two numbers selected at random with replacement from the set {-5,-2,4, 8} is positive
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To calculate the probability that the product of two numbers selected at random with replacement from the set {-5,-2,4,8} is positive, we need to find the total number of ways to select two numbers from the set and the number of ways to select two numbers whose product is positive. There are four numbers in the set, so there are 4 x 4 = 16 possible ways to select two numbers with replacement. For example, we could select -5 and -2, or we could select 4 and 8. To find the number of ways to select two numbers whose product is positive, we need to consider the following cases: 1. Both numbers are positive: There are two positive numbers in the set, so there are 2 x 2 = 4 ways to select two positive numbers. 2. Both numbers are negative: There are two negative numbers in the set, so there are 2 x 2 = 4 ways to select two negative numbers. Therefore, there are a total of 4 + 4 = 8 ways to select two numbers whose product is positive. The probability of selecting two numbers whose product is positive is the number of ways to select two numbers whose product is positive divided by the total number of ways to select two numbers, which is 8/16 or 1/2. Therefore, the answer is, which is \(\frac{1}{2}\).
Question 18 Report
Find the angle between i + 5j and 5i - J
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To find the angle between the vectors i + 5j and 5i - j, we can use the dot product formula: a · b = |a||b| cos θ where a and b are the two vectors, |a| and |b| are their magnitudes, and θ is the angle between them. First, we need to find the dot product of the two vectors: (i + 5j) · (5i - j) = (1)(5) + (5)(-1) = 0 Next, we need to find the magnitudes of the two vectors: | i + 5j | = √(1² + 5²) = √26 | 5i - j | = √(5² + (-1)²) = √26 Now we can substitute the dot product and magnitudes into the dot product formula to solve for θ: 0 = (√26)(√26) cos θ cos θ = 0 θ = 90° Therefore, the angle between i + 5j and 5i - j is 90 degrees. Option (D) is the correct answer.
Question 19 Report
Given that X : R \(\to\) R is defined by x = \(\frac{y + 1}{5 - y}\) , y \(\in\) R, find the domain of x.
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Question 20 Report
The variables x and y are such that y =2x\(^3\) - 2x\(^2\) - 5x + 5. Calculate the corresponding change in y and x changes from 2.00 to 2.05.
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Question 21 Report
Given that F = 3i - 12j, R = 7i + 5j and N = pi + qj are forces acting on a body, if the body is in equilibrium. find the values of p and q.
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Question 22 Report
Find the inverse of \(\begin{pmatrix} 3 & 5 \\ 1 & 2 \end{pmatrix}\)
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To find the inverse of a matrix, we need to use the formula: \[A^{-1} = \frac{1}{\det(A)}\text{adj}(A)\] where \(\det(A)\) is the determinant of matrix \(A\) and \(\text{adj}(A)\) is the adjugate of matrix \(A\). In this case, we have: \[A = \begin{pmatrix} 3 & 5 \\ 1 & 2 \end{pmatrix}\] The determinant of \(A\) is: \[\det(A) = \begin{vmatrix} 3 & 5 \\ 1 & 2 \end{vmatrix} = (3\times2) - (5\times1) = 1\] The adjugate of \(A\) is: \[\text{adj}(A) = \begin{pmatrix} 2 & -5 \\ -1 & 3 \end{pmatrix}\] Therefore, the inverse of \(A\) is: \[A^{-1} = \frac{1}{\det(A)}\text{adj}(A) = \frac{1}{1}\begin{pmatrix} 2 & -5 \\ -1 & 3 \end{pmatrix} = \begin{pmatrix} 2 & -5 \\ -1 & 3 \end{pmatrix}\] So the answer is (B) \(\begin{pmatrix} 2 & -5 \\ -1 & 3 \end{pmatrix}\).
Question 23 Report
A force of 230N acts in its direction 065\(^o\). Find its horizontal component.
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When a force is acting at an angle to the horizontal, we can break it down into two components: the horizontal component and the vertical component. The horizontal component of the force is the part of the force that is acting in the horizontal direction, while the vertical component is the part of the force that is acting in the vertical direction. To find the horizontal component of the force of 230N acting at an angle of 65\(^o\), we use the formula: Horizontal component = Force × cos(angle) where angle is the angle between the force and the horizontal direction. Substituting the values given in the problem, we have: Horizontal component = 230N × cos(65\(^o\)) Using a calculator, we can find that cos(65\(^o\)) is approximately 0.4226. Therefore, the horizontal component of the force is: Horizontal component = 230N × 0.4226 ≈ 97.2N Therefore, the answer is approximately 97.2N.
Question 24 Report
A function f defined by f : x -> x\(^2\) + px + q is such that f(3) = 6 and f(3) = 0. Find the value of q.
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Question 26 Report
Given that 2x + 3y - 10 and 3x = 2y - 11, calculate the value of (x - y).
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Question 27 Report
A binary operation * is defined on the set of real number, R, by x*y = x\(^2\) - y\(^2\) + xy, where x, \(\in\) R. Evaluate (\(\sqrt{3}\))*(\(\sqrt{2}\))
\({\color{red}2x} \times 3\)
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Question 28 Report
Given that P = {x : 1 \(\geq\) x \(\geq\) 6} and Q = {x : 2 < x < 10}. Where x are integers, find n(p \(\cap\) Q)
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Question 29 Report
If \(\frac{6x + k}{2x^2 + 7x - 15}\) = \(\frac{4}{x + 5} - \frac{2}{2x - 3}\). Find the value of k.
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Question 30 Report
Find the median of the numbers 9,7, 5, 2, 12,9,9, 2, 10, 10, and 18.
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To find the median of a set of numbers, we need to arrange the numbers in order from least to greatest. Arranging the given set of numbers in order, we get: 2, 2, 5, 7, 9, 9, 9, 10, 10, 12, 18 There are 11 numbers in the set. Since 11 is an odd number, the median is the middle number when the set is arranged in order. The middle number is the sixth number, which is 9. Therefore, the median of the set of numbers 9, 7, 5, 2, 12, 9, 9, 2, 10, 10, and 18 is 9.
Question 31 Report
Find the nth term of the linear sequence (A.P) (5y + 1), ( 2y + 1), (1- y),...
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A linear sequence is a sequence in which each term differs from the previous term by a constant amount. In other words, if we subtract any term from the term that comes after it, we get the same value. To find the nth term of the linear sequence (5y + 1), (2y + 1), (1 - y), ..., we need to first find the common difference between consecutive terms. We can do this by subtracting the second term from the first term, and then subtracting the third term from the second term: (2y + 1) - (5y + 1) = -3y (1 - y) - (2y + 1) = -3y - 1 Since both subtractions give us the same value of -3y, we know that the common difference between consecutive terms is -3y. Now, we need to find the formula for the nth term of the sequence. We can use the general formula for an arithmetic progression: nth term = a\(_1\) + (n - 1)d where a\(_1\) is the first term, d is the common difference, and n is the term we want to find. In our case, the first term is (5y + 1), the common difference is -3y, and we want to find the nth term. Therefore, we have: nth term = (5y + 1) + (n - 1)(-3y) Simplifying this expression, we get: nth term = 5y + 1 - 3ny + 3y nth term = (8 - 3n)y + 1 Therefore, the nth term of the linear sequence (5y + 1), (2y + 1), (1 - y), ... is (8 - 3n)y + 1.
Question 32 Report
If \(\begin{pmatrix} p+q & 1\\ 0 & p-q \end {pmatrix}\) = \(\begin{pmatrix} 2 & 1 \\ 0 & 8 \end{pmatrix}\)
Find the values of p and q
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To solve for p and q, we can equate the corresponding elements of both matrices: p + q = 2 1 = 1 p - q = 8 From the first and third equations, we can obtain: p = 5 q = -3 Therefore, the correct answer is p = 5, q = -3.
Question 33 Report
In what interval is the function f : x -> 2x - x\(^2\) increasing?
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The function f(x) = 2x - x^2 is increasing on an interval if, as x increases, f(x) also increases. To determine this interval, we can first find the critical points of the function, which are the values of x where the derivative of the function is equal to 0 or undefined. The derivative of f(x) = 2x - x^2 is f'(x) = 2 - 2x. Setting f'(x) = 0, we get 2 - 2x = 0, so x = 1. This is the critical point of the function. To determine whether the function is increasing or decreasing at the critical point, we can use the second derivative test. The second derivative of the function f(x) = 2x - x^2 is f''(x) = -2, which is always negative. So, the first derivative (f'(x)) is decreasing at the critical point. Since the first derivative is decreasing at x = 1, the function f(x) = 2x - x^2 is decreasing to the left of x = 1 and increasing to the right of x = 1. So, the function is increasing on the interval x > 1.
Question 34 Report
Find the unit vector in the direction opposite to the resultant of forces. F\(_1\) = (-2i - 3j) and F\(_2\) = (5i - j)
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Question 35 Report
A stone was dropped from the top of a building 40m high. Find, correct to one decimal place, the time it took the stone to reach the ground. [Take g = 9.8ms\(^{-2}\)]
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Question 36 Report
The distance(s) in metres covered by a particle in motion at any time, t seconds, is given by S =120t - 16t\(^2\). Find in metres, the distance covered by the body before coming to rest.
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The given equation represents the distance covered by a particle in motion at any time t seconds. We are asked to find the distance covered by the particle before it comes to rest, which means the particle stops moving. To find the distance covered by the particle before coming to rest, we need to find the time when the particle stops moving. The particle stops moving when its velocity becomes zero. We can find the velocity of the particle at any time t seconds by differentiating the given equation with respect to time: v = dS/dt = 120 - 32t The particle stops moving when v = 0, so we can set the velocity equation to zero and solve for t: 120 - 32t = 0 t = 3.75 seconds Now we know that the particle stops moving after 3.75 seconds. To find the distance covered by the particle before it stops moving, we can substitute this value of t in the original equation: S = 120t - 16t^2 S = 120(3.75) - 16(3.75)^2 S = 225 meters Therefore, the distance covered by the particle before coming to rest is 225 meters. Option (D) is the correct answer.
Question 37 Report
Consider the statements:
x: Birds fly
y: The sky is blue
Which of the following statements can be represented as x \(\to\) y?
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Question 38 Report
If the binomial expansion of (1 + 3x)\(^6\) is used to evaluate (0.97)\(^6\), find the value of x.
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Question 39 Report
If cos x = -0.7133, find the values of x between 0\(^o\) and 360\(^o\)
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The values of x between 0° and 360° that satisfy cos x = -0.7133 are 135.5° and 224.5°. The cosine function has a range of values between -1 and 1, and for a given value of cosine, there are two angles in the interval [0°, 360°] that satisfy the equation. These angles are found by using the inverse cosine function, also known as arccos. The arccos of -0.7133 is equal to 135.5° and 224.5°, which are the two values of x that satisfy the equation cos x = -0.7133 in the interval [0°, 360°].
Question 40 Report
(a) Two functions p and q are defined on the set of real numbers, R, by p : y \(\to\) 2y +3 and q : y -> y - 2. Find QOP
(b) How many four digits odd numbers greater than 4000 can be formed from 1,7,3,8,2 if repetition is allowed?
(a) To find QOP: we need to apply the functions in the correct order. QOP means we apply Q first, then O, then P.
Q: y → y - 2, so QOP: y → 2(y - 2) + 3 = 2y - 1. Therefore, QOP: y → 2y - 1.
(b) To form a four-digit odd number greater than 4000: the leftmost digit must be 7. We can choose any of the remaining digits for the thousands place, giving us 4 choices. For the hundreds, tens, and ones places, we can choose any of the five digits (1, 3, 7, 8, 2) with repetition allowed, giving us 5 choices for each of those places.
Therefore, the total number of four-digit odd numbers greater than 4000 that can be formed with these digits is:
4 (choices for the thousands place) × 5 (choices for the hundreds place) × 5 (choices for the tens place) × 5 (choices for the ones place) = 500.
So, there are 500 four-digit odd numbers greater than 4000 that can be formed from 1, 7, 3, 8, 2 if repetition is allowed.
Answer Details
(a) To find QOP: we need to apply the functions in the correct order. QOP means we apply Q first, then O, then P.
Q: y → y - 2, so QOP: y → 2(y - 2) + 3 = 2y - 1. Therefore, QOP: y → 2y - 1.
(b) To form a four-digit odd number greater than 4000: the leftmost digit must be 7. We can choose any of the remaining digits for the thousands place, giving us 4 choices. For the hundreds, tens, and ones places, we can choose any of the five digits (1, 3, 7, 8, 2) with repetition allowed, giving us 5 choices for each of those places.
Therefore, the total number of four-digit odd numbers greater than 4000 that can be formed with these digits is:
4 (choices for the thousands place) × 5 (choices for the hundreds place) × 5 (choices for the tens place) × 5 (choices for the ones place) = 500.
So, there are 500 four-digit odd numbers greater than 4000 that can be formed from 1, 7, 3, 8, 2 if repetition is allowed.
Question 41 Report
Forces F\(_1\)(10N, 090°) and F\(_2\)(20N, 210\(^o\)) and (4N,330°) act on a particle, Find, correct to one decimal place, the magnitude of the resultant force.
To find the magnitude of the resultant force, we need to first calculate the x and y components of each force.
For F1:
For F2:
For F3:
To find the total x and y components, we can add up the individual x and y components:
To find the magnitude of the resultant force, we can use the Pythagorean theorem:
resultant force = sqrt((-13.8)^2 + (-2)^2) = 13.9 N
Therefore, the magnitude of the resultant force is 13.9 N.
Answer Details
To find the magnitude of the resultant force, we need to first calculate the x and y components of each force.
For F1:
For F2:
For F3:
To find the total x and y components, we can add up the individual x and y components:
To find the magnitude of the resultant force, we can use the Pythagorean theorem:
resultant force = sqrt((-13.8)^2 + (-2)^2) = 13.9 N
Therefore, the magnitude of the resultant force is 13.9 N.
Question 42 Report
(a) If (x + 2) is a factor of g(x) = 2x\(^3\) +11x\(^2\) - x - 30, find the zeros of g(x).
(b) Solve 3(2\(^x\)) +3\(^{y - 2}\) = 25 and 2x - 3\(^{y + 1}\) = -19 simultaneously.
(a) If (x + 2) is a factor of g(x), it means that (x + 2) is a root of the polynomial g(x). To find the zeros of g(x), we need to factorize the polynomial by using long division or synthetic division.
Using synthetic division, we get:
-2 | 2 11 -1 -30 |----------- | 2 -6 11 | 0
Therefore, g(x) can be factorized as g(x) = 2x^2 - 6x + 11. To find the zeros of g(x), we need to solve the equation 2x^2 - 6x + 11 = 0.
Using the quadratic formula, we get:
x = \(\frac{6 \pm \sqrt{6^2 - 4(2)(11)}}{4}\)
x = \(\frac{3}{2} \pm \frac{\sqrt{23}}{2}\)
Therefore, the zeros of g(x) are \(\frac{3}{2} + \frac{\sqrt{23}}{2}\), and \(\frac{3}{2} - \frac{\sqrt{23}}{2}\).
(b) To solve the simultaneous equations 3(2^x) + 3^(y - 2) = 25 and 2x - 3^(y + 1) = -19, we need to isolate one variable in terms of the other and substitute the expression into the other equation.
From the second equation, we have:
2x = 3^(y + 1) - 19
Substituting this expression into the first equation, we get:
3(2^x) + 3^(y - 2) = 25
3(2^x) + 3^(y - 2) = 3^2
Dividing both sides by 3^2, we get:
2^x + 3^(y - 2 - 2) = 1
2^x + 3^(y - 4) = 1
Substituting the expression for 2x from the second equation, we get:
2^x + 3^(y - 4) = 1 + 3^(y + 1) - 19
2^x + 3^(y - 4) = 3^(y + 1) - 18
Rearranging this equation, we get:
2^x = 3^(y + 1) - 3^(y - 4) - 18
Using logarithms, we can solve for x:
x = log2(3^(y + 1) - 3^(y - 4) - 18)
Substituting this expression for x into the second equation, we get:
2log2(3^(y + 1) - 3^(y - 4) - 18) - 3^(y + 1) = -19
Simplifying this
Answer Details
(a) If (x + 2) is a factor of g(x), it means that (x + 2) is a root of the polynomial g(x). To find the zeros of g(x), we need to factorize the polynomial by using long division or synthetic division.
Using synthetic division, we get:
-2 | 2 11 -1 -30 |----------- | 2 -6 11 | 0
Therefore, g(x) can be factorized as g(x) = 2x^2 - 6x + 11. To find the zeros of g(x), we need to solve the equation 2x^2 - 6x + 11 = 0.
Using the quadratic formula, we get:
x = \(\frac{6 \pm \sqrt{6^2 - 4(2)(11)}}{4}\)
x = \(\frac{3}{2} \pm \frac{\sqrt{23}}{2}\)
Therefore, the zeros of g(x) are \(\frac{3}{2} + \frac{\sqrt{23}}{2}\), and \(\frac{3}{2} - \frac{\sqrt{23}}{2}\).
(b) To solve the simultaneous equations 3(2^x) + 3^(y - 2) = 25 and 2x - 3^(y + 1) = -19, we need to isolate one variable in terms of the other and substitute the expression into the other equation.
From the second equation, we have:
2x = 3^(y + 1) - 19
Substituting this expression into the first equation, we get:
3(2^x) + 3^(y - 2) = 25
3(2^x) + 3^(y - 2) = 3^2
Dividing both sides by 3^2, we get:
2^x + 3^(y - 2 - 2) = 1
2^x + 3^(y - 4) = 1
Substituting the expression for 2x from the second equation, we get:
2^x + 3^(y - 4) = 1 + 3^(y + 1) - 19
2^x + 3^(y - 4) = 3^(y + 1) - 18
Rearranging this equation, we get:
2^x = 3^(y + 1) - 3^(y - 4) - 18
Using logarithms, we can solve for x:
x = log2(3^(y + 1) - 3^(y - 4) - 18)
Substituting this expression for x into the second equation, we get:
2log2(3^(y + 1) - 3^(y - 4) - 18) - 3^(y + 1) = -19
Simplifying this
Question 43 Report
Evaluate; \(\int^3_1(3x - 2)^5 dx\)
To evaluate the given definite integral, we can use the power rule of integration, which states that:
\(\int x^n dx = \frac{x^{n+1}}{n+1} + C\)
where C is the constant of integration.
Using this rule, we can integrate the integrand (3x - 2)^5 as follows:
\(\int (3x - 2)^5 dx = \frac{(3x - 2)^6}{18} + C\)
We have added the constant of integration C to the end of the result since integration is an inverse operation of differentiation and any constant added during differentiation will be lost during integration.
Now, to evaluate the definite integral from 1 to 3, we can substitute the limits of integration into the expression we just found and then subtract the value of the expression at the lower limit from the value at the upper limit, as follows:
\(\int^3_1 (3x - 2)^5 dx = [\frac{(3x - 2)^6}{18}]^3_1\)
\(= \frac{(3(3) - 2)^6}{18} - \frac{(3(1) - 2)^6}{18}\)
\(= \frac{145152}{18}\)
\(= 8064\)
Therefore, the value of the definite integral \(\int^3_1(3x - 2)^5 dx\) is 8064.
Answer Details
To evaluate the given definite integral, we can use the power rule of integration, which states that:
\(\int x^n dx = \frac{x^{n+1}}{n+1} + C\)
where C is the constant of integration.
Using this rule, we can integrate the integrand (3x - 2)^5 as follows:
\(\int (3x - 2)^5 dx = \frac{(3x - 2)^6}{18} + C\)
We have added the constant of integration C to the end of the result since integration is an inverse operation of differentiation and any constant added during differentiation will be lost during integration.
Now, to evaluate the definite integral from 1 to 3, we can substitute the limits of integration into the expression we just found and then subtract the value of the expression at the lower limit from the value at the upper limit, as follows:
\(\int^3_1 (3x - 2)^5 dx = [\frac{(3x - 2)^6}{18}]^3_1\)
\(= \frac{(3(3) - 2)^6}{18} - \frac{(3(1) - 2)^6}{18}\)
\(= \frac{145152}{18}\)
\(= 8064\)
Therefore, the value of the definite integral \(\int^3_1(3x - 2)^5 dx\) is 8064.
Question 44 Report
The diagram is that of a light inextensible string of length 4.2m, whose ends are attached to two fixed points X and Y, 3m apart, and on the same horizontal level. A body of mass 800g is hung on the string at a point O, 2.4m from Y. If the system is kept in equilibrium by a horizontal force P acting on the body and the tensions are equal, calculate:
(a) < XOY;
(b) the magnitude of the force P;
(c) the tension T in the string.
(a) To find the angle between the string and the horizontal, we can use the law of cosines:
c^2 = a^2 + b^2 - 2ab cos C
where c is the length of the string (4.2m), a is the distance from Y to the body (2.4m), and b is the distance from X to Y (3m).
We can rearrange the equation to solve for cos C:
cos C = (a^2 + b^2 - c^2) / 2ab
cos C = (2.4^2 + 3^2 - 4.2^2) / 2 * 2.4 * 3 = 0.8
Finally, we can find the angle using the inverse cosine function:
C = arccos 0.8 = 37.38°
So the angle between the string and the horizontal is 37.38°.
(b) To find the magnitude of the force P, we can use Newton's second law (F = ma), where F is the net force, m is the mass of the body (800g = 0.8kg), and a is the acceleration due to gravity (9.8m/s^2).
The net force is equal to the mass of the body multiplied by the acceleration due to gravity, so:
F = ma = 0.8 * 9.8 = 7.84N
So the magnitude of the force P is 7.84N.
(c) To find the tension T in the string, we can use the fact that the tension must be equal in both parts of the string to keep the system in equilibrium.
Let's call the tension in the part of the string from Y to O "T1" and the tension in the part of the string from O to X "T2".
T1 * cos 37.38° = T2 * cos (180° - 37.38°) = T2 * cos 142.62°
T1 = T2 * cos 142.62° / cos 37.38°
Since T1 = T2, we can substitute T2 for T1:
T2 = T2 * cos 142.62° / cos 37.38°
So, T2 / T2 = cos 142.62° / cos 37.38°
Since T2 is not equal to 0, we can divide both sides by T2:
1 = cos 142.62° / cos 37.38°
To find the tension T in the string, we can use either T1 or T2, since they are equal. Let's use T1:
T1 = T2 = 7.84 / (cos 142.62° / cos 37.38°) = 7.84 / 0.19 = 41.16 N
So the tension in the string is 41.16 N.
Answer Details
(a) To find the angle between the string and the horizontal, we can use the law of cosines:
c^2 = a^2 + b^2 - 2ab cos C
where c is the length of the string (4.2m), a is the distance from Y to the body (2.4m), and b is the distance from X to Y (3m).
We can rearrange the equation to solve for cos C:
cos C = (a^2 + b^2 - c^2) / 2ab
cos C = (2.4^2 + 3^2 - 4.2^2) / 2 * 2.4 * 3 = 0.8
Finally, we can find the angle using the inverse cosine function:
C = arccos 0.8 = 37.38°
So the angle between the string and the horizontal is 37.38°.
(b) To find the magnitude of the force P, we can use Newton's second law (F = ma), where F is the net force, m is the mass of the body (800g = 0.8kg), and a is the acceleration due to gravity (9.8m/s^2).
The net force is equal to the mass of the body multiplied by the acceleration due to gravity, so:
F = ma = 0.8 * 9.8 = 7.84N
So the magnitude of the force P is 7.84N.
(c) To find the tension T in the string, we can use the fact that the tension must be equal in both parts of the string to keep the system in equilibrium.
Let's call the tension in the part of the string from Y to O "T1" and the tension in the part of the string from O to X "T2".
T1 * cos 37.38° = T2 * cos (180° - 37.38°) = T2 * cos 142.62°
T1 = T2 * cos 142.62° / cos 37.38°
Since T1 = T2, we can substitute T2 for T1:
T2 = T2 * cos 142.62° / cos 37.38°
So, T2 / T2 = cos 142.62° / cos 37.38°
Since T2 is not equal to 0, we can divide both sides by T2:
1 = cos 142.62° / cos 37.38°
To find the tension T in the string, we can use either T1 or T2, since they are equal. Let's use T1:
T1 = T2 = 7.84 / (cos 142.62° / cos 37.38°) = 7.84 / 0.19 = 41.16 N
So the tension in the string is 41.16 N.
Question 45 Report
(a) Simplify; \(\frac{log_2 ^8 + log_2 ^{16} - 4 log_2 ^2}{log_4^{16}}\)
(b) The first, third, and seventh terms of an Arithmetic Progression (A.P) from three consecutive terms of a Geometric Progression (G.P). If the sum of the first two terms of the A.P is 6, find its:
(I) first term; (ii) common difference.
(a) To simplify the expression, we can use the logarithmic properties:
logam + logan = logam*n
logam - logan = logam/n
First, we simplify the numerator:
log28 + log216 - 4 log22 = log28*16 - 4 log222 = log2128 - 4 log24 = 7 - 4 * 2 = 3
Next, we simplify the denominator:
log416 = log224 = 4
Finally, we divide the numerator by the denominator:
3 / 4 = 0.75
So the simplified expression is 0.75.
(b) Given the first, third, and seventh terms of an Arithmetic Progression (A.P), we can find the common difference "d" and the first term "a" using the formula for the nth term of an A.P:
an = a + (n-1)d
Let's call the first term "a", the common difference "d", and the number of terms "n".
The third term is:
a + 2d
The seventh term is:
a + 6d
We know that the sum of the first two terms is 6, so we can write an equation using the formula for the nth term:
a + (a + d) = 6
2a + d = 6
2a = 6 - d
2a = 6 - d
a = (6 - d) / 2
We can substitute "a" in one of the other equations to find "d":
a + 2d = a + 2(6 - a) / 2 = 7
7 = a + 3 - a = 3
a = 4
So the first term is 4 and the common difference is (6 - a) / 2 = (6 - 4) / 2 = 1.
So the first term of the Arithmetic Progression is 4 and the common difference is 1.
Answer Details
(a) To simplify the expression, we can use the logarithmic properties:
logam + logan = logam*n
logam - logan = logam/n
First, we simplify the numerator:
log28 + log216 - 4 log22 = log28*16 - 4 log222 = log2128 - 4 log24 = 7 - 4 * 2 = 3
Next, we simplify the denominator:
log416 = log224 = 4
Finally, we divide the numerator by the denominator:
3 / 4 = 0.75
So the simplified expression is 0.75.
(b) Given the first, third, and seventh terms of an Arithmetic Progression (A.P), we can find the common difference "d" and the first term "a" using the formula for the nth term of an A.P:
an = a + (n-1)d
Let's call the first term "a", the common difference "d", and the number of terms "n".
The third term is:
a + 2d
The seventh term is:
a + 6d
We know that the sum of the first two terms is 6, so we can write an equation using the formula for the nth term:
a + (a + d) = 6
2a + d = 6
2a = 6 - d
2a = 6 - d
a = (6 - d) / 2
We can substitute "a" in one of the other equations to find "d":
a + 2d = a + 2(6 - a) / 2 = 7
7 = a + 3 - a = 3
a = 4
So the first term is 4 and the common difference is (6 - a) / 2 = (6 - 4) / 2 = 1.
So the first term of the Arithmetic Progression is 4 and the common difference is 1.
Question 46 Report
The essays of 10 candidates were ranked by three examiners as shown in the table.
| candidates | A | B | C | D | E | F | G | H | I | J |
| Examiner I | 1st | 3rd | 6th | 2nd | 10th | 9th | 7th | 4th | 8th | 5th |
| Examiner II | 2nd | 1st | 3rd | 9th | 7th | 4th | 8th | 10th | 5th | 6th |
| Examiner III | 3rd | 2nd | 1st | 6th | 9th | 8th | 7th | 5th | 4th | 10th |
a) Calculate the Spearman's rank correlation coefficient of the ranks assigned by:
(i) Examiners I and lI;
(ii) Examiners I and III
(iii) Examiners II and II.
(b) Using the results in (a), state which two examiners agree most.
The question presents a table showing how 10 candidates' essays were ranked by three examiners (I, II, and III). The question requires finding the Spearman's rank correlation coefficient for each pair of examiners to determine which examiners agree most.
(a) The Spearman's rank correlation coefficient measures the strength and direction of the relationship between two variables. It is given by the formula:
r = 1 - (6Σd²)/(n(n²-1))
where d is the difference between the ranks of the two examiners for each candidate, and n is the number of candidates.
(i) To find the correlation coefficient between examiners I and II, we calculate the differences between their ranks for each candidate and square them, then sum the squared differences. Using the formula above, we get r = 0.23, which indicates a weak positive correlation.
(ii) To find the correlation coefficient between examiners I and III, we repeat the same process as above and obtain r = -0.02, which indicates no correlation or a very weak negative correlation.
(iii) To find the correlation coefficient between examiners II and III, we repeat the same process as above and obtain r = 0.23, which indicates a weak positive correlation.
(b) Comparing the correlation coefficients, we can conclude that examiners I and II agree most since they have the highest correlation coefficient of 0.23. However, this correlation is still considered weak.
Answer Details
The question presents a table showing how 10 candidates' essays were ranked by three examiners (I, II, and III). The question requires finding the Spearman's rank correlation coefficient for each pair of examiners to determine which examiners agree most.
(a) The Spearman's rank correlation coefficient measures the strength and direction of the relationship between two variables. It is given by the formula:
r = 1 - (6Σd²)/(n(n²-1))
where d is the difference between the ranks of the two examiners for each candidate, and n is the number of candidates.
(i) To find the correlation coefficient between examiners I and II, we calculate the differences between their ranks for each candidate and square them, then sum the squared differences. Using the formula above, we get r = 0.23, which indicates a weak positive correlation.
(ii) To find the correlation coefficient between examiners I and III, we repeat the same process as above and obtain r = -0.02, which indicates no correlation or a very weak negative correlation.
(iii) To find the correlation coefficient between examiners II and III, we repeat the same process as above and obtain r = 0.23, which indicates a weak positive correlation.
(b) Comparing the correlation coefficients, we can conclude that examiners I and II agree most since they have the highest correlation coefficient of 0.23. However, this correlation is still considered weak.
Question 47 Report
(a) An association is made up of 6 farmers and 8 traders. If an executive body of 4 members is to be formed, find the probability that it will consist of at least two farmers. (b) The probability of an accident occurring in a given month in factories X, Y, and Z are \(\frac{1}{5}, \frac{1}{12} \) and \(\frac{1}{6}\) respectively.
Find the probability that the accident will occur in:
i) none of the factories;
(ii) all the factories;
(iii) at least one factory.
(a) To find the probability that the executive body will consist of at least two farmers, we need to calculate the probability of two farmers, three farmers, or four farmers being selected.
The total number of possible executive bodies of 4 members is given by:
n(S) = (6+8) choose 4 = 3003
The number of ways to select 2 farmers and 2 traders is:
n(A) = (6 choose 2) x (8 choose 2) = 420
The number of ways to select 3 farmers and 1 trader is:
n(B) = (6 choose 3) x (8 choose 1) = 160
The number of ways to select 4 farmers is:
n(C) = (6 choose 4) = 15
Therefore, the probability of selecting an executive body of 4 members that consists of at least two farmers is:
P(A or B or C) = (420 + 160 + 15) / 3003 = 0.297
So, the probability is 0.297 that the executive body will consist of at least two farmers.
(b) (i) The probability that an accident will occur in none of the factories is:
P(none) = (4/5) x (11/12) x (5/6) = 0.4583
(ii) The probability that an accident will occur in all three factories is:
P(all) = (1/5) x (1/12) x (1/6) = 0.0007
(iii) To find the probability that an accident will occur in at least one factory, we need to find the probability of the complement event (i.e., no accidents in any of the factories) and subtract it from 1:
P(at least one) = 1 - P(none) = 1 - 0.4583 = 0.5417
So, the probability is 0.4583 that no accidents will occur in any of the factories, the probability is 0.0007 that accidents will occur in all the factories, and the probability is 0.5417 that at least one accident will occur in any of the factories.
Answer Details
(a) To find the probability that the executive body will consist of at least two farmers, we need to calculate the probability of two farmers, three farmers, or four farmers being selected.
The total number of possible executive bodies of 4 members is given by:
n(S) = (6+8) choose 4 = 3003
The number of ways to select 2 farmers and 2 traders is:
n(A) = (6 choose 2) x (8 choose 2) = 420
The number of ways to select 3 farmers and 1 trader is:
n(B) = (6 choose 3) x (8 choose 1) = 160
The number of ways to select 4 farmers is:
n(C) = (6 choose 4) = 15
Therefore, the probability of selecting an executive body of 4 members that consists of at least two farmers is:
P(A or B or C) = (420 + 160 + 15) / 3003 = 0.297
So, the probability is 0.297 that the executive body will consist of at least two farmers.
(b) (i) The probability that an accident will occur in none of the factories is:
P(none) = (4/5) x (11/12) x (5/6) = 0.4583
(ii) The probability that an accident will occur in all three factories is:
P(all) = (1/5) x (1/12) x (1/6) = 0.0007
(iii) To find the probability that an accident will occur in at least one factory, we need to find the probability of the complement event (i.e., no accidents in any of the factories) and subtract it from 1:
P(at least one) = 1 - P(none) = 1 - 0.4583 = 0.5417
So, the probability is 0.4583 that no accidents will occur in any of the factories, the probability is 0.0007 that accidents will occur in all the factories, and the probability is 0.5417 that at least one accident will occur in any of the factories.
Question 48 Report
| Marks | 10 - 19 | 20 - 29 | 30 - 39 | 40 - 49 | 50 - 59 | 60 - 69 | 70 - 79 | 80 - 89 | 90 - 99 |
| Frequency | 2 | 2 | 2 | 8 | 13 | 11 | 12 | 10 | 4 |
The table shows the distribution of marks scored by 64 students in a test
(a) Draw a histogram for the distribution.
(b) Use the histogram to estimate the modal score.
a)
| Class intervals | F | Class boundaries |
| 10 - 19 | 2 | 9.5 - 19.5 |
| 20 - 29 | 2 | 19.5 - 29.5 |
| 30 - 39 | 2 | 29.5 - 39.5 |
| 40 - 49 | 8 | 39.5 - 49.5 |
| 50 - 59 | 13 | 49.5 - 59.5 |
| 60 - 69 | 11 | 59.5 - 69.5 |
| 70 - 79 | 12 | 69.5 - 79.5 |
| 80 - 89 | 10 | 79.5 - 89.5 |
| 90 - 99 | 4 | 89.5 - 99.5 |
b) The model score = 57
Answer Details
a)
| Class intervals | F | Class boundaries |
| 10 - 19 | 2 | 9.5 - 19.5 |
| 20 - 29 | 2 | 19.5 - 29.5 |
| 30 - 39 | 2 | 29.5 - 39.5 |
| 40 - 49 | 8 | 39.5 - 49.5 |
| 50 - 59 | 13 | 49.5 - 59.5 |
| 60 - 69 | 11 | 59.5 - 69.5 |
| 70 - 79 | 12 | 69.5 - 79.5 |
| 80 - 89 | 10 | 79.5 - 89.5 |
| 90 - 99 | 4 | 89.5 - 99.5 |
b) The model score = 57
Question 49 Report
(a) A bag contains 10 red and 8 green identical balls. Two balls are drawn at random from the bag, one after the other, without replacement. Find the probability that one is red and the other is green.
(b) There are 20% defective bulbs in a large box. If 12 bulbs are selected randomly from the box, calculate the probability that between two and five are defective.
(a) Probability of drawing one red and one green ball:
To find the probability that one ball is red and the other is green, we can use the formula:
P(red and green) = P(red) * P(green)
where P(red) is the probability of drawing a red ball on the first draw, and P(green) is the probability of drawing a green ball on the second draw, given that the first ball was red.
The probability of drawing a red ball on the first draw is 10/18, since there are 10 red balls out of a total of 18 balls in the bag. The probability of drawing a green ball on the second draw, given that the first ball was red, is 8/17, since there are now 8 green balls and 17 balls left in the bag.
Therefore, the probability of drawing one red and one green ball is:
P(red and green) = (10/18) * (8/17) = 0.2353 or approximately 23.53%
(b) Probability of between two and five defective bulbs:
To find the probability that between two and five bulbs are defective, we can use the binomial probability formula:
P(k successes) = (n choose k) * p^k * (1-p)^(n-k)
where n is the number of trials (bulbs selected), k is the number of successes (defective bulbs), p is the probability of success (defective bulb), and (n choose k) is the number of ways to choose k defective bulbs out of n total bulbs.
In this case, n = 12, p = 0.2 (since 20% of the bulbs are defective), and we want to find the probability of between two and five successes (defective bulbs). We can calculate this probability by adding the probabilities of getting exactly 2, 3, 4, or 5 defective bulbs:
P(2-5 defects) = P(2 defects) + P(3 defects) + P(4 defects) + P(5 defects)
P(2 defects) = (12 choose 2) * 0.2^2 * 0.8^10 = 0.301
P(3 defects) = (12 choose 3) * 0.2^3 * 0.8^9 = 0.318
P(4 defects) = (12 choose 4) * 0.2^4 * 0.8^8 = 0.185
P(5 defects) = (12 choose 5) * 0.2^5 * 0.8^7 = 0.058
Therefore, the probability of between two and five bulbs being defective is:
P(2-5 defects) = 0.301 + 0.318 + 0.185 + 0.058 = 0.862 or approximately 86.2%
Answer Details
(a) Probability of drawing one red and one green ball:
To find the probability that one ball is red and the other is green, we can use the formula:
P(red and green) = P(red) * P(green)
where P(red) is the probability of drawing a red ball on the first draw, and P(green) is the probability of drawing a green ball on the second draw, given that the first ball was red.
The probability of drawing a red ball on the first draw is 10/18, since there are 10 red balls out of a total of 18 balls in the bag. The probability of drawing a green ball on the second draw, given that the first ball was red, is 8/17, since there are now 8 green balls and 17 balls left in the bag.
Therefore, the probability of drawing one red and one green ball is:
P(red and green) = (10/18) * (8/17) = 0.2353 or approximately 23.53%
(b) Probability of between two and five defective bulbs:
To find the probability that between two and five bulbs are defective, we can use the binomial probability formula:
P(k successes) = (n choose k) * p^k * (1-p)^(n-k)
where n is the number of trials (bulbs selected), k is the number of successes (defective bulbs), p is the probability of success (defective bulb), and (n choose k) is the number of ways to choose k defective bulbs out of n total bulbs.
In this case, n = 12, p = 0.2 (since 20% of the bulbs are defective), and we want to find the probability of between two and five successes (defective bulbs). We can calculate this probability by adding the probabilities of getting exactly 2, 3, 4, or 5 defective bulbs:
P(2-5 defects) = P(2 defects) + P(3 defects) + P(4 defects) + P(5 defects)
P(2 defects) = (12 choose 2) * 0.2^2 * 0.8^10 = 0.301
P(3 defects) = (12 choose 3) * 0.2^3 * 0.8^9 = 0.318
P(4 defects) = (12 choose 4) * 0.2^4 * 0.8^8 = 0.185
P(5 defects) = (12 choose 5) * 0.2^5 * 0.8^7 = 0.058
Therefore, the probability of between two and five bulbs being defective is:
P(2-5 defects) = 0.301 + 0.318 + 0.185 + 0.058 = 0.862 or approximately 86.2%
Question 50 Report
(a) A car is moving with a velocity of 10ms\(^{-1}\) It then accelerates at 0.2ms\(^{-2}\) for 100m. Find, correct to two decimal places the time taken by the car to cover the distance.
(b) A particle moves along a straight line such that its distance S metres from a fixed point O is given by S = t\(^2\) - 5t + 6, where t is the time in seconds. Find its:
(i) initial velocity;
(ii) distance when it is momentarily at rest
(a) We can use the formula:
distance = initial velocity × time + 0.5 × acceleration × time\(^2\)
To solve for time, we can rearrange this equation to:
time = [(-initial velocity) ± sqrt((initial velocity)\(^2\) + 2 × acceleration × distance)] / acceleration
Plugging in the given values, we have:
distance = 100m
initial velocity = 10ms\(^{-1}\)
acceleration = 0.2ms\(^{-2}\)
So, time = [(-10) ± sqrt((10)\(^2\) + 2 × 0.2 × 100)] / 0.2
Simplifying, we get:
time = 155.8s or -45.8s
Since time can't be negative, we discard the negative solution and conclude that the time taken by the car to cover the distance is approximately 155.8 seconds.
(b) To find the initial velocity, we need to differentiate the equation for S with respect to time (t):
S = t\(^2\) - 5t + 6
dS/dt = 2t - 5
The initial velocity is the value of dS/dt when t = 0. So, plugging in t = 0, we get:
initial velocity = dS/dt (t = 0) = 2(0) - 5 = -5 m/s
To find the distance when the particle is momentarily at rest, we need to find the time when the velocity is zero.
Setting dS/dt = 0, we get:
2t - 5 = 0
t = 2.5 s
So, the distance when the particle is momentarily at rest is:
S (t = 2.5) = (2.5)\(^2\) - 5(2.5) + 6 = 1.25 m
Therefore, the initial velocity is -5 m/s and the distance when the particle is momentarily at rest is 1.25 m.
Answer Details
(a) We can use the formula:
distance = initial velocity × time + 0.5 × acceleration × time\(^2\)
To solve for time, we can rearrange this equation to:
time = [(-initial velocity) ± sqrt((initial velocity)\(^2\) + 2 × acceleration × distance)] / acceleration
Plugging in the given values, we have:
distance = 100m
initial velocity = 10ms\(^{-1}\)
acceleration = 0.2ms\(^{-2}\)
So, time = [(-10) ± sqrt((10)\(^2\) + 2 × 0.2 × 100)] / 0.2
Simplifying, we get:
time = 155.8s or -45.8s
Since time can't be negative, we discard the negative solution and conclude that the time taken by the car to cover the distance is approximately 155.8 seconds.
(b) To find the initial velocity, we need to differentiate the equation for S with respect to time (t):
S = t\(^2\) - 5t + 6
dS/dt = 2t - 5
The initial velocity is the value of dS/dt when t = 0. So, plugging in t = 0, we get:
initial velocity = dS/dt (t = 0) = 2(0) - 5 = -5 m/s
To find the distance when the particle is momentarily at rest, we need to find the time when the velocity is zero.
Setting dS/dt = 0, we get:
2t - 5 = 0
t = 2.5 s
So, the distance when the particle is momentarily at rest is:
S (t = 2.5) = (2.5)\(^2\) - 5(2.5) + 6 = 1.25 m
Therefore, the initial velocity is -5 m/s and the distance when the particle is momentarily at rest is 1.25 m.
Question 51 Report
If \(\frac{3x^2 + 3x - 2}{(x - 1)(x + 1)}\) = P + \(\frac{Q}{x - 1} + \frac{R}{x - 1}\)
Find the value of Q and R
To find the values of Q and R, we need to first simplify the right-hand side of the equation and then equate the numerators of both sides.
First, we need to find the common denominator of the fractions on the right-hand side:
P + \(\frac{Q}{x - 1}\) + \(\frac{R}{x + 1}\) = \(\frac{Px^2 - P + Q(x + 1) + R(x - 1)}{(x - 1)(x + 1)}\)
Next, we can equate the numerators of both sides:
3x^2 + 3x - 2 = Px^2 - P + Q(x + 1) + R(x - 1)
Now, we need to solve for Q and R by choosing appropriate values of x.
To find Q, we can choose x = 1:
3(1)^2 + 3(1) - 2 = P(1)^2 - P + Q(1 + 1) + R(1 - 1)
4 = P + 2Q
To find R, we can choose x = -1:
3(-1)^2 + 3(-1) - 2 = P(-1)^2 - P + Q(-1 + 1) + R(-1 - 1)
-2 = P - 2R
We now have two equations with two variables (P and Q), and we can solve for Q by substituting the expression we found for P in the previous step:
4 = (3 + 3 - 2P) + 2Q
2Q = 2P + 1
Q = P + \(\frac{1}{2}\)
To find R, we can substitute the expression we found for P and the expression we found for Q in the equation we found for R:
-2 = (3 + 3 - 2P) - 2(P + \(\frac{1}{2}\))
-2 = 6 - 4P
P = \(\frac{4}{3}\)
Substituting this value of P in the expression we found for Q, we get:
Q = \(\frac{4}{3}\) + \(\frac{1}{2}\) = \(\frac{11}{6}\)
Finally, substituting the values of P and Q in the expression we found for R, we get:
-2 = 6 - 4(\(\frac{4}{3}\)) - 2(\(\frac{11}{6}\))
R = -\(\frac{5}{6}\)
Therefore, the value of Q is \(\frac{11}{6}\) and the value of R is -\(\frac{5}{6}\).
Answer Details
To find the values of Q and R, we need to first simplify the right-hand side of the equation and then equate the numerators of both sides.
First, we need to find the common denominator of the fractions on the right-hand side:
P + \(\frac{Q}{x - 1}\) + \(\frac{R}{x + 1}\) = \(\frac{Px^2 - P + Q(x + 1) + R(x - 1)}{(x - 1)(x + 1)}\)
Next, we can equate the numerators of both sides:
3x^2 + 3x - 2 = Px^2 - P + Q(x + 1) + R(x - 1)
Now, we need to solve for Q and R by choosing appropriate values of x.
To find Q, we can choose x = 1:
3(1)^2 + 3(1) - 2 = P(1)^2 - P + Q(1 + 1) + R(1 - 1)
4 = P + 2Q
To find R, we can choose x = -1:
3(-1)^2 + 3(-1) - 2 = P(-1)^2 - P + Q(-1 + 1) + R(-1 - 1)
-2 = P - 2R
We now have two equations with two variables (P and Q), and we can solve for Q by substituting the expression we found for P in the previous step:
4 = (3 + 3 - 2P) + 2Q
2Q = 2P + 1
Q = P + \(\frac{1}{2}\)
To find R, we can substitute the expression we found for P and the expression we found for Q in the equation we found for R:
-2 = (3 + 3 - 2P) - 2(P + \(\frac{1}{2}\))
-2 = 6 - 4P
P = \(\frac{4}{3}\)
Substituting this value of P in the expression we found for Q, we get:
Q = \(\frac{4}{3}\) + \(\frac{1}{2}\) = \(\frac{11}{6}\)
Finally, substituting the values of P and Q in the expression we found for R, we get:
-2 = 6 - 4(\(\frac{4}{3}\)) - 2(\(\frac{11}{6}\))
R = -\(\frac{5}{6}\)
Therefore, the value of Q is \(\frac{11}{6}\) and the value of R is -\(\frac{5}{6}\).
Question 52 Report
A binary operation * is defined on the set of real numbers R, by p*q = p + q - \(\frac{pq}{2}\), where p, q \(\in\) R. Find the:
(a) inverse of -1 under * given that the identity clement is zero.
(b) truth set of m* 7 = m* 5,
(a) To find the inverse of -1 under *, we need to find a real number x such that -1 * x = x * (-1) = 0, where 0 is the identity element under *.
Substituting -1 and x into the given formula, we get:
-1 * x = -1 + x - \(\frac{(-1)x}{2}\) = 0
Simplifying this equation, we get:
x - \(\frac{x}{2}\) = 1
\(\frac{x}{2}\) = 1
x = 2
Therefore, the inverse of -1 under * is 2.
(b) To find the truth set of m * 7 = m * 5, we need to find all real numbers m that satisfy the equation.
Substituting m and 7 into the given formula, we get:
m * 7 = m + 7 - \(\frac{7m}{2}\)
Substituting m and 5 into the given formula, we get:
m * 5 = m + 5 - \(\frac{5m}{2}\)
Setting these two expressions equal to each other, we get:
m + 7 - \(\frac{7m}{2}\) = m + 5 - \(\frac{5m}{2}\)
Simplifying this equation, we get:
\(\frac{m}{2}\) = 2
m = 4
Therefore, the truth set of m * 7 = m * 5 is {4}, which means that the only real number that satisfies the equation is 4.
Answer Details
(a) To find the inverse of -1 under *, we need to find a real number x such that -1 * x = x * (-1) = 0, where 0 is the identity element under *.
Substituting -1 and x into the given formula, we get:
-1 * x = -1 + x - \(\frac{(-1)x}{2}\) = 0
Simplifying this equation, we get:
x - \(\frac{x}{2}\) = 1
\(\frac{x}{2}\) = 1
x = 2
Therefore, the inverse of -1 under * is 2.
(b) To find the truth set of m * 7 = m * 5, we need to find all real numbers m that satisfy the equation.
Substituting m and 7 into the given formula, we get:
m * 7 = m + 7 - \(\frac{7m}{2}\)
Substituting m and 5 into the given formula, we get:
m * 5 = m + 5 - \(\frac{5m}{2}\)
Setting these two expressions equal to each other, we get:
m + 7 - \(\frac{7m}{2}\) = m + 5 - \(\frac{5m}{2}\)
Simplifying this equation, we get:
\(\frac{m}{2}\) = 2
m = 4
Therefore, the truth set of m * 7 = m * 5 is {4}, which means that the only real number that satisfies the equation is 4.
Question 53 Report
(a) Find the derivative of y = x\(^2\) (1 + x)\(^{\frac{3}{2}}\) with respect to x.
(b) The centre of a circle lies on the line 2y - x = 3. If the circle passes through P(2,3) and Q(6,7), find its equation.
(a) To find the derivative of y = x\(^2\) (1 + x)\(^{\frac{3}{2}}\), we can use the product rule of differentiation, which states that if we have two functions u(x) and v(x), then the derivative of their product is given by: \((u\cdot v)' = u'v + uv'\) where u' and v' denote the derivatives of u and v with respect to x, respectively. Let u(x) = x\(^2\) and v(x) = (1 + x)\(^{\frac{3}{2}}\). Then, we have: u'(x) = 2x (by the power rule of differentiation) v'(x) = \(\frac{3}{2}\)(1 + x)\(\frac{1}{2}\) = \(\frac{3}{2}\)\(\sqrt{1 + x}\) (by the chain rule of differentiation) Using the product rule, we can now find the derivative of y as follows: y' = u'v + uv' = 2x(1 + x)\(^{\frac{3}{2}}\) + x\(^2\)\(\frac{3}{2}\)\(\sqrt{1 + x}\) = x(4 + 3x\(\sqrt{1 + x}\)). Therefore, the derivative of y = x\(^2\) (1 + x)\(^{\frac{3}{2}}\) with respect to x is x(4 + 3x\(\sqrt{1 + x}\)). (b) Since the centre of the circle lies on the line 2y - x = 3, we can express the centre as a point (h, k) such that 2k - h = 3. Since the circle passes through points P(2, 3) and Q(6, 7), we can use the distance formula to write two equations involving h and k, as follows: \((h-2)^2 + (k-3)^2 = r^2\) (from P) \((h-6)^2 + (k-7)^2 = r^2\) (from Q) where r is the radius of the circle. Expanding both equations and subtracting them, we get: \(8h + 8k = 32\) Substituting k = \(\frac{h+3}{2}\) (from the line equation 2k - h = 3) into the above equation, we get: \(h^2 - 14h + 25 = 0\) Solving this quadratic equation, we get h = 5 or h = 2.5. If h = 5, then from the line equation 2k - h = 3, we get k = 4. If h = 2.5, then k = 2. Thus, the centre of the circle is either (5, 4) or (2.5, 2). To find the radius of the circle, we can use either of the two equations we obtained earlier (from P or Q). Let's use the equation from P: \((h-2)^2 + (k-3)^2 = r^2\) Substituting (h, k) = (5, 4), we get: \((5-2)^2 + (4-3)^2 = r^2\) Simplifying, we get: \(r^2 = 10\) Taking the square root of both sides, we get: \(r = \sqrt{10}\) Therefore, the equation
Answer Details
(a) To find the derivative of y = x\(^2\) (1 + x)\(^{\frac{3}{2}}\), we can use the product rule of differentiation, which states that if we have two functions u(x) and v(x), then the derivative of their product is given by: \((u\cdot v)' = u'v + uv'\) where u' and v' denote the derivatives of u and v with respect to x, respectively. Let u(x) = x\(^2\) and v(x) = (1 + x)\(^{\frac{3}{2}}\). Then, we have: u'(x) = 2x (by the power rule of differentiation) v'(x) = \(\frac{3}{2}\)(1 + x)\(\frac{1}{2}\) = \(\frac{3}{2}\)\(\sqrt{1 + x}\) (by the chain rule of differentiation) Using the product rule, we can now find the derivative of y as follows: y' = u'v + uv' = 2x(1 + x)\(^{\frac{3}{2}}\) + x\(^2\)\(\frac{3}{2}\)\(\sqrt{1 + x}\) = x(4 + 3x\(\sqrt{1 + x}\)). Therefore, the derivative of y = x\(^2\) (1 + x)\(^{\frac{3}{2}}\) with respect to x is x(4 + 3x\(\sqrt{1 + x}\)). (b) Since the centre of the circle lies on the line 2y - x = 3, we can express the centre as a point (h, k) such that 2k - h = 3. Since the circle passes through points P(2, 3) and Q(6, 7), we can use the distance formula to write two equations involving h and k, as follows: \((h-2)^2 + (k-3)^2 = r^2\) (from P) \((h-6)^2 + (k-7)^2 = r^2\) (from Q) where r is the radius of the circle. Expanding both equations and subtracting them, we get: \(8h + 8k = 32\) Substituting k = \(\frac{h+3}{2}\) (from the line equation 2k - h = 3) into the above equation, we get: \(h^2 - 14h + 25 = 0\) Solving this quadratic equation, we get h = 5 or h = 2.5. If h = 5, then from the line equation 2k - h = 3, we get k = 4. If h = 2.5, then k = 2. Thus, the centre of the circle is either (5, 4) or (2.5, 2). To find the radius of the circle, we can use either of the two equations we obtained earlier (from P or Q). Let's use the equation from P: \((h-2)^2 + (k-3)^2 = r^2\) Substituting (h, k) = (5, 4), we get: \((5-2)^2 + (4-3)^2 = r^2\) Simplifying, we get: \(r^2 = 10\) Taking the square root of both sides, we get: \(r = \sqrt{10}\) Therefore, the equation
Question 54 Report
Given that w = 8i + 3j, x = 6i - 5j, y = 2i + 3j and |z| = 41. find z in the direction of w + x - 2y.
To find the vector "z" in the direction of the sum of vectors "w", "x", and "-2y", we can first find the sum of "w", "x", and "-2y", then find the unit vector in that direction and multiply it by the magnitude of "z".
First, let's find the sum of "w", "x", and "-2y":
w = 8i + 3j
x = 6i - 5j
y = 2i + 3j
-2y = -4i - 6j
w + x - 2y = (8i + 3j) + (6i - 5j) - (-4i - 6j) = 18i - 8j
Next, let's find the unit vector in the direction of the sum:
The unit vector in the direction of a vector can be found by dividing the vector by its magnitude.
The magnitude of the sum is:
√(18^2 + (-8)^2) = 20
So the unit vector in the direction of the sum is:
(18i - 8j) / 20 = 9/10i - 4/10j
Finally, let's multiply the unit vector by the magnitude of "z" to find "z":
z = 41 * (9/10i - 4/10j) = 36.9i - 16.4j
So the vector "z" in the direction of the sum of "w", "x", and "-2y" is approximately 36.9i - 16.4j.
Answer Details
To find the vector "z" in the direction of the sum of vectors "w", "x", and "-2y", we can first find the sum of "w", "x", and "-2y", then find the unit vector in that direction and multiply it by the magnitude of "z".
First, let's find the sum of "w", "x", and "-2y":
w = 8i + 3j
x = 6i - 5j
y = 2i + 3j
-2y = -4i - 6j
w + x - 2y = (8i + 3j) + (6i - 5j) - (-4i - 6j) = 18i - 8j
Next, let's find the unit vector in the direction of the sum:
The unit vector in the direction of a vector can be found by dividing the vector by its magnitude.
The magnitude of the sum is:
√(18^2 + (-8)^2) = 20
So the unit vector in the direction of the sum is:
(18i - 8j) / 20 = 9/10i - 4/10j
Finally, let's multiply the unit vector by the magnitude of "z" to find "z":
z = 41 * (9/10i - 4/10j) = 36.9i - 16.4j
So the vector "z" in the direction of the sum of "w", "x", and "-2y" is approximately 36.9i - 16.4j.
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