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**Question 1**
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Given that \(\begin{pmatrix} 1 & -3 \\ 1 & 4 \end{pmatrix} \begin{pmatrix} -6 \\ P \end{pmatrix} = \begin{pmatrix} 3 \\ -26 \end{pmatrix}\), find the value of P.

**Answer Details**

We have the equation \(\begin{pmatrix} 1 & -3 \\ 1 & 4 \end{pmatrix} \begin{pmatrix} -6 \\ P \end{pmatrix} = \begin{pmatrix} 3 \\ -26 \end{pmatrix}\). We can solve for P by performing matrix multiplication and solving the resulting system of linear equations: \(\begin{pmatrix} 1 & -3 \\ 1 & 4 \end{pmatrix} \begin{pmatrix} -6 \\ P \end{pmatrix} = \begin{pmatrix} -6 - 3P \\ P + 4P \end{pmatrix} = \begin{pmatrix} 3 \\ -26 \end{pmatrix}\) This gives us two equations: \(-6 - 3P = 3\) \(5P = -26\) Solving for P, we get: \(-6 - 3P = 3 \Rightarrow -3P = 9 \Rightarrow P = -3\) \(5P = -26 \Rightarrow P = \frac{-26}{5}\) Therefore, the value of P is -3. Note that the system of linear equations is consistent, meaning that there is a unique solution.

**Question 2**
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Find the coordinates of the centre of the circle \(4x^{2} + 4y^{2} - 5x + 3y - 2 = 0\).

**Answer Details**

To find the center of the given circle, we need to write the equation in standard form, which is: \[(x - h)^2 + (y - k)^2 = r^2\] Where (h, k) is the center of the circle and r is the radius. We start by completing the square for both x and y terms, by moving the constant term to the right side and grouping the x and y terms together: \[4x^2 - 5x + 4y^2 + 3y = 2\] To complete the square for the x terms, we need to add and subtract \(\frac{5}{4}\) times the square of the coefficient of x (which is 2), inside the first bracket: \[4\left(x^2 - \frac{5}{4}x + \left(\frac{5}{4}\right)^2\right) + 4y^2 + 3y = 2 + 4\left(\frac{5}{4}\right)^2\] We do the same for the y terms, by adding and subtracting \(\frac{3}{8}\) times the square of the coefficient of y (which is \(\frac{3}{2}\)), inside the second bracket: \[4\left(x^2 - \frac{5}{4}x + \left(\frac{5}{4}\right)^2\right) + 4\left(y^2 + \frac{3}{8}y + \left(\frac{3}{16}\right)^2\right) = 2 + 4\left(\frac{5}{4}\right)^2 + 4\left(\frac{3}{16}\right)^2\] Now we have the equation in standard form, and we can read off the center and radius: Center = \(\left(\frac{5}{4}, -\frac{3}{8}\right)\) Radius = \(\sqrt{\frac{129}{16}} = \frac{3\sqrt{43}}{4}\) Therefore, the correct answer is option (C) \((\frac{5}{8}, -\frac{3}{8})\).

**Question 4**
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Age(in years) | 1 - 5 | 6 - 10 | 11 - 15 |

Frequency | 3 | 5 | 2 |

Calculate the standard deviation of the distribution.

**Question 5**
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If \(f(x) = \frac{1}{2 - x}, x \neq 2\), find \(f^{-1}(-\frac{1}{2})\).

**Answer Details**

To find \(f^{-1}(-\frac{1}{2})\), we need to find the value of x such that \(f(x) = -\frac{1}{2}\). We know that \(f(x) = \frac{1}{2 - x}\), so we set \(\frac{1}{2 - x} = -\frac{1}{2}\) and solve for x: \[\frac{1}{2 - x} = -\frac{1}{2}\] \[2 - x = -2\] \[x = 4\] Therefore, \(f^{-1}(-\frac{1}{2}) = 4\). Note that we were able to solve for \(f^{-1}(y)\) by first setting \(y = f(x)\), then solving for x. This is because the inverse function of f, denoted by \(f^{-1}\), is defined such that \(f^{-1}(f(x)) = x\) for all x in the domain of f.

**Question 6**
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Which of the following is a singular matrix?

**Question 7**
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Find the coordinates of the point on the curve \(y = x^{2} + 4x - 2\), where the gradient is zero.

**Question 8**
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Simplify \(\frac{^{n}P_{4}}{^{n}C_{4}}\)

**Answer Details**

The expression \(\frac{^{n}P_{4}}{^{n}C_{4}}\) represents the number of ways to choose and arrange 4 objects out of a total of n objects, divided by the number of ways to choose 4 objects out of n objects without arranging them. The number of ways to choose and arrange 4 objects out of n objects is given by the formula for permutations, which is \(^{n}P_{4} = \frac{n!}{(n-4)!}\). The number of ways to choose 4 objects out of n objects without arranging them is given by the formula for combinations, which is \(^{n}C_{4} = \frac{n!}{4!(n-4)!}\). So, \(\frac{^{n}P_{4}}{^{n}C_{4}} = \frac{\frac{n!}{(n-4)!}}{\frac{n!}{4!(n-4)!}} = \frac{4!}{1!} = 24\). Therefore, the answer is 24.

**Question 9**
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The parallelogram PQRS has vertices P(-2, 3), Q(1, 4), R(2, 6) and S(-1,5). Find the coordinates of the point of intersection of the diagonals.

**Answer Details**

To find the point of intersection of the diagonals of a parallelogram, we need to find the midpoint of the segment that connects the two diagonals. This midpoint is the point of intersection of the diagonals. Let's start by finding the equation of the line containing the diagonal PR. The slope of PR is: $$m_{PR}=\frac{y_R-y_P}{x_R-x_P}=\frac{6-3}{2-(-2)}=\frac{3}{4}$$ Using point-slope form, the equation of line PR is: $$y-3=\frac{3}{4}(x+2)$$ Simplifying, we get: $$y=\frac{3}{4}x+\frac{15}{4}$$ Now let's find the equation of the line containing the diagonal QS. The slope of QS is: $$m_{QS}=\frac{y_S-y_Q}{x_S-x_Q}=\frac{5-4}{-1-1}=\frac{-1}{2}$$ Using point-slope form, the equation of line QS is: $$y-4=\frac{-1}{2}(x-1)$$ Simplifying, we get: $$y=-\frac{1}{2}x+\frac{9}{2}$$ To find the point of intersection of the diagonals, we need to find the midpoint of the segment that connects the two diagonals. Let's call this midpoint M. The midpoint M is the average of the coordinates of the endpoints of the segment, which are the intersection points of the diagonals with their respective sides. To find the intersection point of PR with side QS, we solve the system of equations formed by the equations of lines PR and QS: $$\begin{cases}y=\frac{3}{4}x+\frac{15}{4}\\y=-\frac{1}{2}x+\frac{9}{2}\end{cases}$$ Solving for x and y, we get: $$\begin{cases}x=\frac{6}{5}\\y=\frac{39}{10}\end{cases}$$ So the intersection point of PR with side QS is (\(\frac{6}{5}, \frac{39}{10}\)). To find the intersection point of QS with side PR, we solve the system of equations formed by the equations of lines QS and PR: $$\begin{cases}y=-\frac{1}{2}x+\frac{9}{2}\\y=\frac{3}{4}x+\frac{15}{4}\end{cases}$$ Solving for x and y, we get: $$\begin{cases}x=-\frac{6}{5}\\y=\frac{21}{10}\end{cases}$$ So the intersection point of QS with side PR is (-\(\frac{6}{5}\), \(\frac{21}{10}\)). The midpoint M is the average of these two points, so we have: $$M=(\frac{\frac{6}{5}-\frac{6}{5}}{2}, \frac{\frac{39}{10}+\frac{21}{10}}{2})=(0, \frac{3}{2})$$ Therefore, the coordinates of the point of intersection of the diagonals is (0, \(\frac{3}{2}\)). So the answer is option (C), \((0, 4\frac{1}{2})\).

**Question 10**
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Solve the inequality \(2x^{2} + 5x - 3 \geq 0\).

**Answer Details**

To solve this inequality, we can start by finding the roots of the quadratic equation \(2x^2+5x-3=0\). We can do this by factoring the quadratic or by using the quadratic formula. Factoring the quadratic, we get: \begin{align*} 2x^2+5x-3&=0\\ (2x-1)(x+3)&=0 \end{align*} So the roots are \(x=\frac{1}{2}\) and \(x=-3\). These are the values of \(x\) that make the left-hand side of the inequality equal to zero. Now, we can use these roots to determine the sign of the expression \(2x^2+5x-3\) for different values of \(x\). We can do this by testing values of \(x\) in the intervals between the roots and outside the roots. For example, if we pick a value of \(x\) less than \(-3\), say \(x=-4\), then we have: \begin{align*} 2x^2+5x-3&=2(-4)^2+5(-4)-3\\ &=32-20-3\\ &=9 \end{align*} Since \(9\) is positive, this means that any value of \(x\) less than \(-3\) makes the expression \(2x^2+5x-3\) positive. Similarly, if we pick a value of \(x\) between \(-3\) and \(\frac{1}{2}\), say \(x=0\), then we have: \begin{align*} 2x^2+5x-3&=2(0)^2+5(0)-3\\ &=-3 \end{align*} Since \(-3\) is negative, this means that any value of \(x\) between \(-3\) and \(\frac{1}{2}\) makes the expression \(2x^2+5x-3\) negative. Finally, if we pick a value of \(x\) greater than \(\frac{1}{2}\), say \(x=1\), then we have: \begin{align*} 2x^2+5x-3&=2(1)^2+5(1)-3\\ &=4+5-3\\ &=6 \end{align*} Since \(6\) is positive, this means that any value of \(x\) greater than \(\frac{1}{2}\) makes the expression \(2x^2+5x-3\) positive. Putting all of this together, we can see that the inequality \(2x^2+5x-3\geq 0\) is true when \(x\leq -3\) or \(x\geq \frac{1}{2}\). Therefore, the correct option is: - \(x \leq -3\) or \(x \geq \frac{1}{2}\)

**Question 11**
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A lift moving upwards with a uniform acceleration of 5\(ms^{-2}\) carries a body of mass p kg. If the reaction on the floor is 480 N, find the value of p. [Take g = \(10 ms^{-2}\)].

**Question 12**
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If \((x - 3)\) is a factor of \(2x^{3} + 3x^{2} - 17x - 30\), find the remaining factors.

**Answer Details**

If \((x - 3)\) is a factor of \(2x^{3} + 3x^{2} - 17x - 30\), then we can use long division or synthetic division to divide the polynomial by \((x - 3)\) and find the remaining factors.

Using long division, we have:

2x^2 + 9x + 10 ------------------------ x - 3 | 2x^3 + 3x^2 - 17x - 30 - (2x^3 - 6x^2) -------------- 9x^2 - 17x - (9x^2 - 27x) ----------- 10x - 30 - (10x - 30) --------- 0

The result of the division is \(2x^{2} + 9x + 10\), which is a quadratic polynomial. Therefore, the remaining factors are given by factoring this quadratic polynomial. We can factor it as follows:

\[2x^{2} + 9x + 10 = (2x + 5)(x + 2)\]Therefore, the complete factorization of \(2x^{3} + 3x^{2} - 17x - 30\) is:

\[2x^{3} + 3x^{2} - 17x - 30 = (x - 3)(2x + 5)(x + 2)\]So, the correct option is **(2x + 5)(x + 2)**.

**Question 13**
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If \(a = \begin{pmatrix} 3 \\ 2 \end{pmatrix}\) and \(b = \begin{pmatrix} -3 \\ 5 \end{pmatrix}\), find a vector c such that \(4a + 3c = b\).

**Answer Details**

To solve this problem, we want to find a vector c that satisfies the equation 4a + 3c = b. We can rearrange this equation to get: 3c = b - 4a Now we just need to solve for c. We can do this by dividing both sides by 3: c = (b - 4a) / 3 Substituting the values of a and b given in the problem, we get: c = (\begin{pmatrix} -3 \\ 5 \end{pmatrix} - 4\begin{pmatrix} 3 \\ 2 \end{pmatrix}) / 3 c = (\begin{pmatrix} -3 \\ 5 \end{pmatrix} - \begin{pmatrix} 12 \\ 8 \end{pmatrix}) / 3 c = \begin{pmatrix} -15/3 \\ -3/3 \end{pmatrix} c = \begin{pmatrix} -5 \\ -1 \end{pmatrix} Therefore, the answer is option B, \(\begin{pmatrix} -5 \\ -1 \end{pmatrix}\).

**Question 14**
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A force of 32 N is applied to an object of mass m kg which is at rest on a smooth horizontal surface. If the acceleration produced is 8\(ms^{-2}\), find the value of m.

**Answer Details**

We can use Newton's second law of motion, which states that the force acting on an object is equal to its mass multiplied by its acceleration. The formula can be written as F = ma, where F is the force, m is the mass, and a is the acceleration. In this case, the force acting on the object is 32 N, and the acceleration produced is 8 \(ms^{-2}\). Since the object is at rest, its initial velocity is zero. So, we can use the formula F = ma to find the value of m. Rearranging the formula, we get m = F/a. Substituting the given values, we get m = 32 N / 8 \(ms^{-2}\) = 4 kg. Therefore, the value of m is 4 kg.

**Question 15**
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Simplify \(8^{n} \times 2^{2n} \div 4^{3n}\)

**Answer Details**

We can simplify the expression by first rewriting each of the bases as powers of 2: $$8^{n} \times 2^{2n} \div 4^{3n} = (2^{3})^{n} \times 2^{2n} \div (2^{2})^{3n}.$$ Using the laws of exponents, we can simplify this to: $$(2^{3n}) \times 2^{2n} \div 2^{6n} = 2^{3n+2n-6n} = 2^{-n}.$$ Therefore, the simplified expression is \(2^{-n}\).

**Question 16**
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Find the equation of the line passing through (0, -1) and parallel to the y- axis.

**Answer Details**

The equation of a line can be represented in the form y = mx + b, where m is the slope of the line and b is the y-intercept (the point where the line intersects the y-axis). Since the line is parallel to the y-axis, its slope is undefined (division by zero is not defined). Therefore, the equation of the line cannot be represented in the form y = mx + b. However, we know that the line passes through the point (0, -1). This means that the x-coordinate of every point on the line is 0 (since the line is parallel to the y-axis and does not move horizontally). Therefore, the equation of the line is x = 0. This equation means that every point on the line has an x-coordinate of 0, which is the y-axis. So, the line is simply the y-axis itself. Thus, the answer is x = 0.

**Question 17**
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Find the least value of the function \(f(x) = 3x^{2} + 18x + 32\).

**Question 18**
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Calculate, correct to one decimal place, the angle between 5i + 12j and -2i + 3j.

**Answer Details**

To find the angle between two vectors, we can use the dot product formula: cos θ = (A dot B) / (|A| |B|) where A and B are the two vectors, A dot B is their dot product, and |A| and |B| are their magnitudes. Let's apply this formula to the given vectors: A = 5i + 12j B = -2i + 3j A dot B = (5*-2) + (12*3) = -10 + 36 = 26 |A| = sqrt(5^2 + 12^2) = 13 |B| = sqrt((-2)^2 + 3^2) = sqrt(13) cos θ = (26) / (13 * sqrt(13)) Using a calculator, we can find that cos θ ≈ 0.801 To find the angle θ itself, we can take the inverse cosine (cos^-1) of this value: θ ≈ cos^-1(0.801) ≈ 37.8° However, this is only half of the actual angle between the two vectors. To get the full angle, we need to take into account the direction of the vectors. If we draw the two vectors on a coordinate plane, we can see that they are in different quadrants: A is in the second quadrant and B is in the fourth quadrant. The angle between them will be the difference between their angles relative to the positive x-axis. Using the inverse tangent (tan^-1) function, we can find the angles of each vector: Angle of A = tan^-1(12/5) ≈ 67.4° Angle of B = tan^-1(-3/2) ≈ -56.3° (Note that we use a negative angle for B because it is in the fourth quadrant, where angles are negative.) The angle between the vectors will be the difference between these two angles: θ = |67.4° - (-56.3°)| = 123.7° However, this is the full angle between the vectors, and the question only asks for the acute angle (the smaller of the two possible angles). To find the acute angle, we subtract the full angle from 180°: Acute angle = 180° - 123.7° ≈ 56.3° Therefore, the answer is option B: 56.3°.

**Question 19**
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Two functions f and g are defined by \(f : x \to 3x - 1\) and \(g : x \to 2x^{3}\), evaluate \(fg(-2)\).

**Question 20**
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ABCD is a square. Forces of magnitude 14N, 4N, 2N and \(2\sqrt{2} N\) act along the sides AB, BC, CD and DA respectively. Find in Newtons, the magnitude of the resultant of the forces.

**Question 21**
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The mean age of n men in a club is 50 years. Two men aged 55 and 63 years left the club, and the mean age reduced by 1 year. Find the value of n.

**Answer Details**

Let's denote the sum of ages of the n men in the club by S. Then we know that: Mean age of n men = S/n = 50 Multiplying both sides by n, we get: S = 50n After two men aged 55 and 63 years leave the club, the sum of ages of the remaining men becomes: S' = S - 55 - 63 = 50n - 118 The new mean age is reduced by 1 year, so we have: S'/(n-2) = 50 - 1 Simplifying this equation and substituting S', we get: (50n - 118)/(n-2) = 49 Multiplying both sides by n-2, we get: 50n - 118 = 49(n-2) Simplifying this equation, we get: n = 20 Therefore, there were originally 20 men in the club.

**Question 22**
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The mean and median of integers x, y, z and t are 5 and z respectively. If x < y < z < t and y = 4, find (x + t).

**Question 23**
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A particle of mass 2.5 kg is moving at a speed of 12 m/s. If a force of magnitude 10 N acts against it, find how long it takes to come to rest.

**Answer Details**

To find the time it takes for the particle to come to rest, we can use the formula: time = (initial velocity) / (deceleration) where the deceleration is the rate at which the particle's velocity decreases due to the force acting against it. The deceleration can be calculated using Newton's Second Law, which states that force is equal to mass times acceleration: force = mass x acceleration Rearranging this formula, we get: acceleration = force / mass Substituting the given values, we get: acceleration = 10 N / 2.5 kg = 4 m/s^2 Therefore, the time it takes for the particle to come to rest is: time = 12 m/s / 4 m/s^2 = 3.0 s So, the answer is 3.0 s.

**Question 24**
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Two forces, each of magnitude 16 N, are inclined to each other at an angle of 60°. Calculate the magnitude of their resultant.

**Question 25**
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The area of a sector of a circle is 3\(cm^{2}\). If the sector subtends an angle of 1.5 radians at the centre, calculate the radius of the circle.

**Answer Details**

To solve this problem, we need to use the formula for the area of a sector of a circle, which is: Area of sector = (θ/2) × r² where θ is the central angle subtended by the sector, r is the radius of the circle, and the angle is measured in radians. In this problem, we are given the area of the sector and the central angle, so we can plug in these values and solve for the radius: 3 = (1.5/2) × r² Multiplying both sides by 2/1.5, we get: 4 = r² Taking the square root of both sides, we get: r = 2 Therefore, the radius of the circle is 2 cm. We can check our answer by plugging it back into the formula for the area of the sector: Area of sector = (1.5/2) × 2² = 1.5 × 2 = 3 This confirms that our answer is correct.

**Question 28**
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Evaluate \(\lim \limits_{x \to 1} \frac{1 - x}{x^{2} - 3x + 2}\)

**Answer Details**

To evaluate the limit \(\lim \limits_{x \to 1} \frac{1 - x}{x^{2} - 3x + 2}\), we can start by substituting \(x = 1\) directly into the expression. However, this results in an expression of the form \(\frac{0}{0}\), which is undefined. To evaluate such a limit, we can use algebraic techniques to simplify the expression before taking the limit. In this case, we can factor the denominator to obtain: \[\frac{1 - x}{x^{2} - 3x + 2} = \frac{1 - x}{(x - 1)(x - 2)} = -\frac{1}{x - 2}\] Now, we can take the limit as \(x\) approaches 1 by substituting \(x = 1\) directly into the simplified expression. This gives: \[\lim \limits_{x \to 1} \frac{1 - x}{x^{2} - 3x + 2} = \lim \limits_{x \to 1} -\frac{1}{x - 2} = -\frac{1}{-1} = 1\] Therefore, the limit of the expression as \(x\) approaches 1 is equal to 1. So, the correct option is 1.

**Question 29**
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Given that \((\sqrt{3} - 5\sqrt{2})(\sqrt{3} + \sqrt{2}) = p + q\sqrt{6}\), find q.

**Question 30**
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A particle is projected vertically upwards from a height 45 metres above the ground with a velocity of 40 m/s. How long does it take it to hit the ground? [Take g = \(10 ms^{-2}\)].

**Answer Details**

We can use the kinematic equation: h = ut + 1/2gt^2 where h is the initial height (45m), u is the initial velocity (40 m/s), g is the acceleration due to gravity (10 m/s^2), and t is the time taken for the particle to hit the ground. At the point where the particle hits the ground, h = 0, so we can solve for t: 0 = 40t - 1/2(10)t^2 Simplifying this equation gives: 0 = 5t(8 - t) This equation has two solutions: t = 0 and t = 8. We can ignore the t = 0 solution because it corresponds to the initial moment when the particle is projected upward. So, the particle will hit the ground after 8 seconds. Therefore, the answer is: 9s (since the particle was projected from a height of 45m, it takes 8 seconds to fall and hit the ground. However, the total time from the projection until it hits the ground would be 9 seconds).

**Question 31**
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A committee of 4 is to be selected from a group of 5 men and 3 women. In how many ways can this be done if the chairman of the committee must be a man?

**Answer Details**

To form a committee of 4 with a chairman, we first need to select a man to be the chairman from the 5 men available. This can be done in 5 ways. Then, we need to select 3 people from the remaining 7 people (4 men and 3 women) to complete the committee. This can be done in 7 choose 3 ways, which is equal to 35. Therefore, the total number of ways to form a committee of 4 with a chairman from a group of 5 men and 3 women is: 5 * 35 = 175 So the correct answer is option (D) 175.

**Question 32**
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The roots of the equation \(2x^{2} + kx + 5 = 0\) are \(\alpha\) and \(\beta\), where k is a constant. If \(\alpha^{2} + \beta^{2} = -1\), find the values of k.

**Answer Details**

The sum of the roots of the quadratic equation \(ax^{2} + bx + c = 0\) is given by the formula \(-\frac{b}{a}\), and the product of the roots is given by the formula \(\frac{c}{a}\). Therefore, for the quadratic equation \(2x^{2} + kx + 5 = 0\), we have: \[\alpha + \beta = -\frac{k}{2}\] \[\alpha \beta = \frac{5}{2}\] Squaring the first equation, we get: \[(\alpha + \beta)^{2} = \alpha^{2} + 2\alpha\beta + \beta^{2} = \frac{k^{2}}{4}\] Substituting the given value of \(\alpha^{2} + \beta^{2} = -1\), we have: \[-1 + 2\alpha\beta = \frac{k^{2}}{4}\] Substituting the value of \(\alpha \beta = \frac{5}{2}\), we have: \[-1 + 2\left(\frac{5}{2}\right) = \frac{k^{2}}{4}\] Simplifying the left-hand side, we get: \[4 = \frac{k^{2}}{4}\] Multiplying both sides by 4, we get: \[16 = k^{2}\] Taking the square root of both sides, we get: \[k = \pm 4\] Therefore, the values of k are \(\pm 4\).

**Question 33**
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Find the sum of the exponential series \(96 + 24 + 6 +...\)

**Answer Details**

The given series is 96 + 24 + 6 + ... , and we need to find its sum. First, we can observe that each term in the series is obtained by dividing the previous term by 4. So, the common ratio between consecutive terms is 1/4. Let S be the sum of the series. Then we have: S = 96 + 24 + 6 + ... Dividing both sides by 4, we get: S/4 = 24 + 6 + 1.5 + ... Now, if we subtract the second equation from the first, we get: S - S/4 = 96 Simplifying, we get: 3S/4 = 96 Multiplying both sides by 4/3, we get: S = 128 Therefore, the sum of the given series is 128. So, the correct option is (B) 128.

**Question 34**
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\(P = {x : 1 \leq x \leq 6}\) and \(Q = {x : 2 < x < 9}\) where \(x \in R\), find \(P \cap Q\).

**Answer Details**

\(P = {x : 1 \leq x \leq 6}\) means that P is the set of all real numbers between 1 and 6 (including 1 and 6). Similarly, \(Q = {x : 2 < x < 9}\) means that Q is the set of all real numbers between 2 and 9 (excluding 2 and 9). To find \(P \cap Q\), we need to find the set of all elements that are common to both P and Q. From the definitions of P and Q, we can see that the range of values that satisfy both P and Q is from 2 to 6 (including 2 and 6). Therefore, \(P \cap Q\) is the set of all real numbers between 2 and 6 (including 2 and 6). Hence, the correct option is (d) \({x : 2 < x \leq 6}\).

**Question 35**
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Evaluate \(\log_{0.25} 8\)

**Answer Details**

We can use the change of base formula to evaluate \(\log_{0.25} 8\). The change of base formula states that for any positive numbers \(a\), \(b\), and \(c\) with \(a \neq 1\), we have: \(\log_{a} b = \frac{\log_{c} b}{\log_{c} a}\) In this case, we can choose any base we like, but it is convenient to use base 2 because 8 and 0.25 are powers of 2. So, we can rewrite the expression as: \(\log_{0.25} 8 = \frac{\log_{2} 8}{\log_{2} 0.25}\) Now, we can evaluate the logarithms on the right-hand side using the rules of logarithms. We know that \(2^{3} = 8\) and \(2^{-2} = 0.25\), so we have: \(\log_{2} 8 = 3\) and \(\log_{2} 0.25 = -2\) Substituting these values into the equation above, we get: \(\log_{0.25} 8 = \frac{3}{-2} = -\frac{3}{2}\) Therefore, the value of \(\log_{0.25} 8\) is \(-\frac{3}{2}\). Hence, the answer is \(-\frac{3}{2}\).

**Question 36**
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The equation of the line of best fit for variables x and y is \(y = 19.33 + 0.42x\), where x is the independent variable. Estimate the value of y when x = 15.

**Answer Details**

The equation given is in the form y = mx + b, where m is the slope and b is the y-intercept of the line of best fit. In this case, the slope is 0.42 and the y-intercept is 19.33. To estimate the value of y when x = 15, we simply substitute x = 15 into the equation and solve for y: y = 19.33 + 0.42(15) y = 19.33 + 6.3 y = 25.63 Therefore, the estimated value of y when x = 15 is 25.63.

**Question 37**
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Find the coefficient of \(x^{4}\) in the binomial expansion of \((1 - 2x)^{6}\).

**Answer Details**

We can use the binomial theorem to find the coefficient of \(x^{4}\) in the expansion of \((1 - 2x)^{6}\). The binomial theorem states that the \(n\)th power of a binomial \((a + b)^{n}\) can be expressed as the sum of the terms of the form \(a^{n-r}b^{r}\), where \(r\) ranges from 0 to \(n\). In this case, the binomial is \((1 - 2x)\), so we can expand it using the binomial theorem as follows: \[(1 - 2x)^{6} = \sum_{r=0}^{6} {6 \choose r} (1)^{6-r} (-2x)^{r}\] where \({6 \choose r}\) is the binomial coefficient. To find the coefficient of \(x^{4}\), we need to find the term where \(r\) is 2. So, we can substitute \(r = 2\) into the formula above to get: \[{6 \choose 2} (1)^{4} (-2x)^{2} = 15 \times 4x^{2} = 60x^{2}\] However, we are looking for the coefficient of \(x^{4}\), not \(x^{2}\). Since the exponent of \(x\) is 2 in the term we just calculated, we need to multiply it by another power of \(x^{2}\) to get \(x^{4}\). This means we need to find the term where \(r\) is 4. Substituting \(r = 4\) into the formula above, we get: \[{6 \choose 4} (1)^{2} (-2x)^{4} = 15 \times 16x^{4} = 240x^{4}\] Therefore, the coefficient of \(x^{4}\) in the binomial expansion of \((1 - 2x)^{6}\) is 240. Hence, the answer is 240.

**Question 38**
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In a firing contest, the probabilities that Kojo and Kwame hit the target are \(\frac{2}{5}\) and \(\frac{1}{3}\) respectively. What is the probability that none of them hit the target?

**Answer Details**

The probability that Kojo hits the target is \(\frac{2}{5}\), which means that the probability that he misses the target is \(1-\frac{2}{5}=\frac{3}{5}\). Similarly, the probability that Kwame misses the target is \(1-\frac{1}{3}=\frac{2}{3}\). Now, we want to find the probability that none of them hit the target. Since they are shooting independently, we can multiply their probabilities of missing the target: Probability that both miss the target = Probability that Kojo misses the target AND Probability that Kwame misses the target = \(\frac{3}{5} \times \frac{2}{3} = \frac{6}{15} = \frac{2}{5}\) Therefore, the probability that none of them hit the target is \(\frac{2}{5}\).

**Question 39**
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A and B are two independent events such that \(P(A) = \frac{2}{5}\) and \(P(A \cap B) = \frac{1}{15}\). Find \(P(B)\).

**Question 40**
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A binary operation ♦ is defined on the set R, of real numbers by \(a ♦ b = \frac{ab}{4}\). Find the value of \(\sqrt{2} ♦ \sqrt{6}\).

**Answer Details**

The binary operation ♦ is defined as \(a ♦ b = \frac{ab}{4}\). So, to find the value of \(\sqrt{2} ♦ \sqrt{6}\), we just need to substitute \(\sqrt{2}\) for \(a\) and \(\sqrt{6}\) for \(b\), and then simplify the expression. \[\sqrt{2} ♦ \sqrt{6} = \frac{\sqrt{2}\cdot\sqrt{6}}{4} = \frac{\sqrt{12}}{4} = \frac{\sqrt{4}\cdot\sqrt{3}}{4} = \frac{2\sqrt{3}}{4} = \frac{\sqrt{3}}{2}\] Therefore, the value of \(\sqrt{2} ♦ \sqrt{6}\) is \(\frac{\sqrt{3}}{2}\).

**Question 41**
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(a) A ball P moving with velocity \(2u ms^{-1}\), collides with a similar ball Q, of different mass, which is at rest. After collision, Q moves with \(u ms^{-1}\) and P with velocity \(\frac{1}{2} u ms^{-2}\), in the opposite direction. Find the ratio of the masses of P and Q.

(b) Two forces of magnitudes 3 N and 7 N have a resultant of magnitude 5 N. Calculate, correct to one decimal place, the angle between the two forces.

(c) \(AB = \begin{pmatrix} -4 \\ 6 \end{pmatrix}\) and \(CB = \begin{pmatrix} 2 \\ -3 \end{pmatrix}\) are two vectors in the XY- plane. If V is the midpoint of AB, find CV.

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**Question 42**
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The table shows the distribution of hours spent at work by the employees of a factory in a week.

Time (in hours) | 20 - 29 | 30 - 39 | 40 - 49 | 50 - 59 | 60 - 69 | 70 - 79 |

No of persons | 8 | 11 | 23 | 25 | 8 | 5 |

(a) Draw an ogive for the distribution.

(b) Using your graph, estimate the (i) lower quartile (ii) median (iii) 40th percentile (iv) number of employees that spent at least 50 hours 30 minutes at work.

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**Question 43**
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Find the equation of the tangent to the curve \(y = \frac{x - 1}{2x + 1}, x \neq -\frac{1}{2}\) at the point (1, 0).

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**Question 44**
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(a) Using the substitution \(u = 5 - x^{2}\), evaluate \(\int_{1}^{2} \frac{x}{\sqrt{5 - x^{2}}} \mathrm {d} x\).

(b) If \(y = px^{2} + qx; \frac{\mathrm d y}{\mathrm d x} = 6x + 7\) and \(\frac{\mathrm d^{2} y}{\mathrm d x^{2}} = 6\), find the values of p and q.

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**Question 45**
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(a) A body of mass 5 kg is placed on a smooth plane inclined at an angle 30° to the horizontal. Find the magnitude of the force: (i) acting parallel to the plane (ii) required to keep the body in equilibrium. [Take g = \(10 ms^{-2}\)].

(b) A uniform plank PQ of length 10m and mass m kg rests on two supports A and B, where |PA| = |BQ| = 1m. A load of mass 8 kg is placed on the plank at point C such that |AC| = 3.5 m. If the reaction at B is 100 N, calculate the (i) value of m (ii) reaction at A. [Take g = \(10 ms^{-2}\)].

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**Question 46**
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(a) A body of mass 15 kg is suspended at a point P by two light inextensible strings \(\overrightarrow{XP}\) and \(\overrightarrow{YP}\). The strings are inclined at 60° and 40° respectively to the downward vertical. Find, correct to two decimal places, the tensionsin the strings. [Take g = \(10 ms^{-2}\)].

(b) The height h metres, of a ball thrown into the air is \(2 + 20t + kt^{2}\), after t seconds. If it takes 2 seconds for the ball to reach its highest point, find

(i) the value of k (ii) its highest point from the point of throw.

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**Answer Details**

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**Question 47**
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A body of mass 5 kg resting on a smooth horizontal plane, is acted upon by force 6i + 2j, 5i + 4j and 4i - j. Calculate the:

(a) velocity of the body

(b) Magnitude of its velocity after 4s.

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**Question 48**
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If \(2^{2x - 3y} = 32\) and \(\log_{y} x = 2\), find the values of x and y.

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**Question 49**
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(a) There are 6 points in a plane. How many triangles can be formed with the points?

(b) A family of 6 is to be seated in a row . In how many ways can this be done if the father and mother are not to be seated together?

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**Answer Details**

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**Question 50**
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(a) Simplify \(\frac{\sqrt{75} - 3}{\sqrt{3} + 1}\), leaving your answer in the form \(a + b\sqrt{c}\); where a, b and c are rational numbers.

(b) The points (7, 3), (2, 8) and (-3, 3) lie on a circle. Find the (i) equation and (ii) radius of the circle.

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**Question 51**
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(a) Two pupils are chosen at random from a group of 4 boys and 5 girls. Find the probability that the two pupils chosen would be boys.

(b) Twenty percent of the total production of transistors produced by a machine are below standard. If a random sample of 6 transistors produced by the machine is taken, what is the probability of getting (i) exactly 2 standard transistors (ii) exactly 1 standard transistor (iii) at least 2 standard transistors (iv) at most 2 standard transistors?

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**Question 52**
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The sum of the 2nd and 5th terms of an arithmetic progression (AP) is 42. If the difference between the 6th and 3rd term is 12, find the

(i) Common difference

(ii) first term

(iii) 20th term.

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**Question 53**
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The table shows the distribution of the ages of a group of people in a village.

Ages (in years) | 15 - 18 | 19 - 22 | 23 - 26 | 27 - 30 | 31 - 34 | 35 - 38 |

Frequency | 40 | 33 | 25 | 10 | 8 | 4 |

Using an assumed mean of 24.5, calculate the mean of the distribution.

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**Question 54**
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If (x + 2) and (x - 1) are factors of \(f(x) = 6x^{4} + mx^{3} - 13x^{2} + nx + 14\), find the

(a) values of m and n.

(b) remainder when f(x) is divided be (x + 1).

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**Question 55**
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An object is projected vertically upwards with a velocity of 80 m/s. Find the :

(a) Maximum height reached

(b) Time taken to return to the point of projection. [Take g = \(10 ms^{-2}\)].

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**Question 56**
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(a) A man P has 5 red, 3 blue and 2 white buses. Another man Q has 3 red, 2 blue and 4 white buses. A bus owned by P is involved in an accident with a bus belonging to Q. Calculate the probability that the two buses are not of the same color.

(b) A man travels from Nigeria to Ghana by air and from Ghana to Liberia by ship. He returns by the same means. He has 6 airlines and 4 shipping lines to choose from. In how many ways can he make his journey without using the same airline or shipping line twice?

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**Question 57**
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(a) The gradient of the tangent to the curve \(y = 4x^{3}\) at points P and Q is 108. Find the coordinates of P and Q.

(b) Given that \(A = 45°, B = 30°, \sin (A + B) = \sin A \cos B + \sin B \cos A\) and \(\cos (A + B) = \cos A \cos B - \sin A \sin B\)

(i) Show that \(\sin 15° = \frac{\sqrt{6} - \sqrt{2}}{4}\) and \(\cos 15° = \frac{\sqrt{6} + \sqrt{2}}{4}\)

(ii) hence find \(\tan 15°\).

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