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**Question 1**
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Simplify \(\frac{\tan 80° - \tan 20°}{1 + \tan 80° \tan 20°}\)

**Answer Details**

We can simplify this expression using the trigonometric identity: \[\tan(A-B) = \frac{\tan A - \tan B}{1+\tan A \tan B}\] In this case, we have A = 80° and B = 20°. Therefore: \[\tan(A-B) = \tan(80°-20°) = \tan 60° = \sqrt{3}\] Using the identity, we can rewrite the expression as: \[\frac{\tan 80° - \tan 20°}{1 + \tan 80° \tan 20°} = \tan(80°-20°) = \sqrt{3}\] Therefore, the simplified expression is \(\sqrt{3}\). So the correct option is (3) \(\sqrt{3}\).

**Question 2**
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The equation of a circle is given by \(x^{2} + y^{2} - 4x - 2y - 3\). Find the radius and the coordinates of its centre.

**Answer Details**

The equation of a circle in standard form is \((x-a)^2 + (y-b)^2 = r^2\), where \((a,b)\) are the coordinates of the center and \(r\) is the radius. We need to rewrite the given equation in this standard form by completing the square for both the \(x\) and \(y\) terms. \begin{align*} x^{2} + y^{2} - 4x - 2y - 3 &= 0\\ (x^{2} - 4x) + (y^{2} - 2y) &= 3\\ (x^{2} - 4x + 4) + (y^{2} - 2y + 1) &= 3 + 4 + 1\\ (x-2)^{2} + (y-1)^{2} &= 8 \end{align*} Comparing with the standard form, we see that the center of the circle is \((2,1)\) and the radius is \(\sqrt{8} = 2\sqrt{2}\). Therefore, the answer is: the radius of the circle is \(2\sqrt{2}\) and the coordinates of its center are \((2,1)\).

**Question 3**
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X and Y are two independent event. If \(P(X) = \frac{1}{5}\) and \(P(X \cap Y) = \frac{2}{15}\), find \(P(Y)\).

**Answer Details**

The formula to find the probability of the intersection of two independent events is given by: $$P(X \cap Y) = P(X) \times P(Y)$$ In this case, we know that: $$P(X) = \frac{1}{5}$$ $$P(X \cap Y) = \frac{2}{15}$$ Substituting the values into the formula, we have: $$\frac{2}{15} = \frac{1}{5} \times P(Y)$$ Solving for $P(Y)$, we get: $$P(Y) = \frac{\frac{2}{15}}{\frac{1}{5}} = \frac{2}{15} \times \frac{5}{1} = \frac{2}{3}$$ Therefore, the probability that event Y occurs is $\frac{2}{3}$. Answer: (a) \(\frac{2}{3}\).

**Question 4**
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Evaluate \(\lim \limits_{x \to 3} \frac{x^{2} - 2x - 3}{x - 3}\)

**Answer Details**

We can evaluate the given limit by plugging in the value 3 directly into the expression, but this results in a division by zero, which is undefined. Instead, we can use algebraic techniques to simplify the expression and evaluate the limit. We can factor the numerator of the expression as follows: \(x^{2} - 2x - 3 = (x - 3)(x + 1)\) Substituting this factorization into the original expression, we get: \(\frac{x^{2} - 2x - 3}{x - 3} = \frac{(x - 3)(x + 1)}{x - 3} = x + 1\) Now, we can take the limit of the simplified expression as x approaches 3: \(\lim \limits_{x \to 3} x + 1 = 3 + 1 = 4\) Therefore, the value of the given limit is 4, which is the correct answer.

**Question 5**
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The tangent to the curve \(y = 4x^{3} + kx^{2} - 6x + 4\) at the point P(1, m) is parallel to the x- axis, where k and m are constants. Determine the coordinates of P.

**Question 6**
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Given that \(\alpha\) and \(\beta\) are the roots of an equation such that \(\alpha + \beta = 3\) and \(\alpha \beta = 2\), find the equation.

**Answer Details**

We are given that \(\alpha\) and \(\beta\) are the roots of the equation, and we are also given two conditions based on these roots: 1. \(\alpha + \beta = 3\) 2. \(\alpha \beta = 2\) To find the equation, we can use the fact that if \(\alpha\) and \(\beta\) are roots of the equation, then the equation can be written in factored form as: $$(x - \alpha)(x - \beta) = 0$$ Expanding this equation, we get: $$x^2 - (\alpha + \beta)x + \alpha\beta = 0$$ Substituting the values of \(\alpha + \beta\) and \(\alpha \beta\) from the given conditions, we get: $$x^2 - 3x + 2 = 0$$ Therefore, the equation with roots \(\alpha\) and \(\beta\) is: $$x^2 - 3x + 2 = 0$$ Hence, the correct option is (a) \(x^{2} - 3x + 2 = 0\).

**Question 7**
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A particle is projected vertically upwards with a speed of 40 m/s. At what times will it be 35m above its point of projection? \(\text{Take g} = 10 ms^{-2}\)

**Question 8**
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Two functions f and g are defined on the set R of real numbers by \(f : x \to 2x - 1\) and \(g : x \to x^{2} + 1\). Find the value of \(f^{-1} \circ g(3)\).

**Answer Details**

We first need to find the composition \(f^{-1} \circ g(x)\), which means we need to find the inverse function of f, denoted by \(f^{-1}(x)\). To find \(f^{-1}(x)\), we solve the equation \(y = 2x - 1\) for x: \(y + 1 = 2x\) \(x = \frac{y + 1}{2}\) Thus, \(f^{-1}(x) = \frac{x + 1}{2}\). Now, we can find \(f^{-1} \circ g(3)\) by first computing g(3): \(g(3) = 3^{2} + 1 = 10\) Then, we can plug g(3) into the composition \(f^{-1} \circ g(x)\): \((f^{-1} \circ g)(3) = f^{-1}(g(3)) = f^{-1}(10) = \frac{10 + 1}{2} = \frac{11}{2}\) Therefore, the value of \(f^{-1} \circ g(3)\) is \(\frac{11}{2}\), which is the correct answer.

**Question 9**
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Which of the following is the same as \(\sin (270 + x)°\)?

**Answer Details**

We can use the identity \(\sin (a+b) = \sin a \cos b + \cos a \sin b\) to simplify the expression \(\sin (270 + x)°\): \begin{align*} \sin (270 + x)° &= \sin 270° \cos x + \cos 270° \sin x \\ &= (-1)(\cos x) + (0)(\sin x) \\ &= -\cos x \end{align*} Therefore, the answer is \(-\cos x\).

**Question 10**
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Given that \(R = (4, 180°)\) and \(S = (3, 300°)\), find the dot product.

**Question 11**
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Marks | 5 - 7 | 8 - 10 | 11 - 13 | 14 - 16 | 17 - 19 | 20 - 22 |

Frequency | 4 | 7 | 26 | 41 | 14 | 8 |

The table above shows the marks obtained by 100 pupils in a test. Find the probability that a student picked at random scored at least 14 marks.

**Answer Details**

To find the probability that a student picked at random scored at least 14 marks, we need to add the frequencies of the classes with marks 14-16, 17-19, and 20-22, since all of these classes correspond to scores of at least 14. Adding the frequencies for these classes, we get: 41 + 14 + 8 = 63. Therefore, out of the 100 students, 63 scored at least 14 marks. To find the probability, we divide the number of students who scored at least 14 by the total number of students: 63/100 = 0.63 Therefore, the probability that a student picked at random scored at least 14 marks is 0.63. Answer: 0.63

**Question 12**
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Simplify \(\frac{\sqrt{3}}{\sqrt{3} - 1} + \frac{\sqrt{3}}{\sqrt{3} +1}\)

**Question 13**
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Calculate in surd form, the value of \(\tan 15°\).

**Answer Details**

To calculate the value of \(\tan 15°\), we can use the half-angle formula for tangent: \[\tan \frac{\theta}{2} = \frac{\sin \theta}{1 + \cos \theta}\] Letting \(\theta = 30°\), we get: \[\tan 15° = \tan \frac{30°}{2} = \frac{\sin 30°}{1 + \cos 30°}\] We know that \(\sin 30° = \frac{1}{2}\) and \(\cos 30° = \frac{\sqrt{3}}{2}\). Substituting these values into the equation above, we get: \[\tan 15° = \frac{\frac{1}{2}}{1 + \frac{\sqrt{3}}{2}} = \frac{1}{2 + \sqrt{3}}\] Rationalizing the denominator, we get: \[\tan 15° = \frac{1}{2 + \sqrt{3}} \cdot \frac{2 - \sqrt{3}}{2 - \sqrt{3}} = \frac{2 - \sqrt{3}}{1} = \boxed{2 - \sqrt{3}}\] Therefore, the answer is (d) \(2 - \sqrt{3}\).

**Question 14**
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Simplify \(\frac{^{n}P_{3}}{^{n}C_{2}} + ^{n}P_{0}\)

**Question 15**
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Marks | 5 - 7 | 8 - 10 | 11 - 13 | 14 - 16 | 17 - 19 | 20 - 22 |

Frequency | 4 | 7 | 26 | 41 | 14 | 8 |

The table above shows the marks obtained by 100 pupils in a test. Find the upper class boundary of the class containing the third quartile.

**Question 16**
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Two bodies of masses 8 kg and 5 kg travelling in the same direction with speeds x m/s and 2 m/s respectively collide. If after collision, they move together with a speed of 3.85 m/s, find, correct to the nearest whole number, the value of x.

**Question 17**
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Three students are working independently on a Mathematics problem. Their respective probabilities of solving the problem are 0.6, 0.7 and 0.8. What is the probability that at least one of them solves the problem?

**Answer Details**

To find the probability that at least one of the three students solves the problem, we need to calculate the probability that none of them solve the problem and then subtract that from 1 (which represents the total probability). The probability that the first student does not solve the problem is 0.4 (since the probability of solving the problem is 0.6). Similarly, the probability that the second student does not solve the problem is 0.3 and the probability that the third student does not solve the problem is 0.2. To find the probability that none of them solve the problem, we multiply these probabilities together: 0.4 x 0.3 x 0.2 = 0.024. Therefore, the probability that at least one of the three students solves the problem is 1 - 0.024 = 0.976. So, the correct answer is: 0.976.

**Question 18**
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Find the area of the circle whose equation is given as \(x^{2} + y^{2} - 4x + 8y + 11 = 0\).

**Answer Details**

To find the area of the circle whose equation is given as \(x^{2} + y^{2} - 4x + 8y + 11 = 0\), we need to first complete the square for both \(x\) and \(y\). Rearranging the equation, we get: \begin{aligned} x^2 - 4x + y^2 + 8y &= -11 \\ x^2 - 4x + 4 + y^2 + 8y + 16 &= 9 \\ (x-2)^2 + (y+4)^2 &= 3^2 \end{aligned} Now we can see that the equation represents a circle with center \((2, -4)\) and radius \(3\). The formula for the area of a circle is \(A = \pi r^2\), where \(r\) is the radius. In this case, the radius is \(3\), so the area is: \begin{aligned} A &= \pi r^2 \\ &= \pi (3)^2 \\ &= \boxed{9\pi} \end{aligned} Therefore, the answer is \(\boxed{9\pi}\).

**Question 19**
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The sum of the first three terms of an Arithmetic Progression (A.P) is 18. If the first term is 4, find their product.

**Answer Details**

Let the common difference of the A.P be d. Then, the first three terms of the A.P can be written as 4, 4 + d, and 4 + 2d. Since the sum of the first three terms of the A.P is 18, we have: 4 + (4 + d) + (4 + 2d) = 18 Simplifying this equation, we get: 3d + 12 = 18 3d = 6 d = 2 Therefore, the first three terms of the A.P are 4, 6, and 8. The product of the first three terms of the A.P is: 4 × 6 × 8 = 192 Therefore, the answer is 192.

**Question 20**
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The gradient of the line passing through the points P(4, 5) and Q(x, 9) is \(\frac{1}{2}\). Find the value of x.

**Question 21**
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The tangent to the curve \(y = 4x^{3} + kx^{2} - 6x + 4\) at the point P(1, m) is parallel to the x- axis, where k and m are constants. Find the value of k.

**Question 22**
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Given that the straight lines \(kx - 5y + 6 = 0\) and \(mx + ny - 1 = 0\) are parallel, find a relationship connecting the constants m, n and k.

**Answer Details**

When two lines are parallel, their slopes are equal. In other words, the coefficient of x in one equation divided by the coefficient of y should be equal to the coefficient of x in the other equation divided by the coefficient of y. Therefore, we can write: k/(-5) = m/n Multiplying both sides by -5n, we get: kn = -5m This is the relationship connecting the constants m, n, and k. Therefore, the correct option is (2) kn + 5m = 0.

**Question 23**
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A force of 30 N acts at an angle of 60° on a body of mass 6 kg initially at rest on a smooth horizontal plane. Find the distance covered in 10 seconds.

**Answer Details**

To solve this problem, we need to use the equations of motion. Since the force is acting on the body, it will cause an acceleration in the direction of the force. The acceleration can be found using Newton's second law of motion: $$F = ma$$ where F is the force, m is the mass of the body, and a is the acceleration. In this case, the force is 30 N and the mass is 6 kg. Therefore, the acceleration can be found as: $$a = \frac{F}{m} = \frac{30}{6} = 5\text{ m/s}^2$$ Next, we can use the equations of motion to find the distance covered by the body in 10 seconds. Since the body starts from rest, we can use the equation: $$s = \frac{1}{2}at^2$$ where s is the distance covered, a is the acceleration, and t is the time. Substituting the values, we get: $$s = \frac{1}{2} \times 5 \times 10^2 = 125\text{ m}$$ However, the force is acting at an angle of 60 degrees to the horizontal. Therefore, only the horizontal component of the force will cause motion in the horizontal direction. The horizontal component of the force is: $$F_h = F \cos \theta = 30 \cos 60^\circ = 15\text{ N}$$ We can now find the acceleration in the horizontal direction as: $$a_h = \frac{F_h}{m} = \frac{15}{6} = 2.5\text{ m/s}^2$$ Using the equation of motion, the distance covered in 10 seconds in the horizontal direction is: $$s_h = \frac{1}{2} \times 2.5 \times 10^2 = 12.5\text{ m}$$ However, we also need to find the distance covered in the vertical direction. The vertical component of the force is: $$F_v = F \sin \theta = 30 \sin 60^\circ = 15\sqrt{3}\text{ N}$$ Since there is no force acting in the vertical direction, the body will move under the influence of gravity only. The acceleration due to gravity is 9.8 m/s^2. Therefore, the distance covered in the vertical direction in 10 seconds is: $$s_v = \frac{1}{2} \times 9.8 \times 10^2 = 490\text{ m}$$ The total distance covered by the body is the hypotenuse of the right-angled triangle formed by the horizontal and vertical distances: $$s_{total} = \sqrt{s_h^2 + s_v^2} = \sqrt{(12.5)^2 + (490)^2} \approx 490\text{ m}$$ Therefore, the answer is 125 m.

**Question 24**
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The binary operation * is defined on the set of R, of real numbers by \(x * y = 3x + 3y - xy, \forall x, y \in R\). Determine, in terms of x, the identity element of the operation.

**Answer Details**

To find the identity element of an operation, we need to find an element that when combined with any other element using the operation, results in that same element. Let's assume that the identity element is "a", so we need to find the value of "a" that satisfies the following equation for any real number "x": x * a = a * x = x Substituting the definition of the operation, we get: x * a = 3x + 3a - ax = x Simplifying this equation by subtracting 3a from both sides, we get: x - 3a = ax Dividing both sides by x, we get: 1 - 3a/x = a Now, we have an expression for "a" in terms of "x". Let's check each option to see which one satisfies the equation for any value of "x". : a = 2x/(x-3), x ≠ 3 Substituting this into our equation, we get: 1 - 3(2x/(x-3))/x = 2x/(x-3) Simplifying this expression, we get: -3x/(x-3) = 0 This equation is not true for all values of "x", since it is only true when x = 0. Therefore, is not the identity element. : a = 2x/(x+3), x ≠ -3 Substituting this into our equation, we get: 1 - 3(2x/(x+3))/x = 2x/(x+3) Simplifying this expression, we get: 3x/(x+3) - 1 = 0 This equation is true for all values of "x", except for x = -3. Therefore, is the identity element. : a = 3x/(x-3), x ≠ 3 Substituting this into our equation, we get: 1 - 3(3x/(x-3))/x = 3x/(x-3) Simplifying this expression, we get: -6x/(x-3) = 0 This equation is not true for all values of "x", since it is only true when x = 0. Therefore, is not the identity element. : a = 3x/(x+3), x ≠ -3 Substituting this into our equation, we get: 1 - 3(3x/(x+3))/x = 3x/(x+3) Simplifying this expression, we get: 6x/(x+3) - 1 = 0 This equation is true for all values of "x", except for x = -3. Therefore, is the identity element. Therefore, the identity element of the operation is: a = 2x/(x+3), x ≠ -3.

**Question 25**
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Given that \(P = \begin{pmatrix} 4 & 9 \end{pmatrix}\) and \(Q = \begin{pmatrix} -1 & -2 \\ 3 & 2 \end{pmatrix}\). Evaluate \(|Q|P\).

**Answer Details**

To evaluate \(|Q|P\), we first need to calculate the determinant of matrix \(Q\), denoted as \(|Q|\). $$ \begin{aligned} |Q| &= \begin{vmatrix} -1 & -2 \\ 3 & 2 \end{vmatrix} \\ &= (-1\times2) - (-2\times3) \\ &= -2 + 6 \\ &= 4 \end{aligned} $$ Next, we multiply the matrix \(P\) by the determinant of matrix \(Q\). $$ \begin{aligned} |Q|P &= 4\begin{pmatrix} 4 & 9 \end{pmatrix} \\ &= \begin{pmatrix} 16 & 36 \end{pmatrix} \end{aligned} $$ Therefore, the answer is \(\begin{pmatrix} 16 & 36 \end{pmatrix}\).

**Question 26**
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How many ways can 12 people be divided into three groups of 2, 7 and 3 in that order?

**Answer Details**

In this problem, we need to find the number of ways of dividing 12 people into three groups of 2, 7 and 3 in that order. Firstly, we can choose the first group of 2 people in \(\binom{12}{2}\) ways, which gives us 66 ways. Then, we can choose the second group of 7 people from the remaining 10 people in \(\binom{10}{7}\) ways, which gives us 120 ways. Finally, the last group of 3 people is determined by the previous two groups, so there is only one way to form this group. Therefore, the total number of ways of dividing 12 people into three groups of 2, 7 and 3 in that order is the product of the number of ways of forming each group, which is equal to \(66 \times 120 \times 1 = 7,920\). Thus, the answer is 7920.

**Question 27**
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A body is acted upon by two forces \(F_{1} = (5 N, 060°)\) and \(F_{2} = (10 N, 180°)\). Find the magnitude of the resultant force.

**Answer Details**

We can use vector addition to find the resultant force. Let's first convert the given forces into their x and y components. For \(F_1 = (5 N, 060°)\), the x component is \(F_{1x} = F_1 \cos(60°) = 5 \cos(60°) = 2.5 N\) and the y component is \(F_{1y} = F_1 \sin(60°) = 5 \sin(60°) = 4.33 N\). For \(F_2 = (10 N, 180°)\), the x component is \(F_{2x} = F_2 \cos(180°) = -10 N\) and the y component is \(F_{2y} = F_2 \sin(180°) = 0 N\). Now we can add the x and y components separately to get the resultant force: $$F_{Rx} = F_{1x} + F_{2x} = 2.5 N - 10 N = -7.5 N$$ $$F_{Ry} = F_{1y} + F_{2y} = 4.33 N + 0 N = 4.33 N$$ The magnitude of the resultant force is then given by: $$F_R = \sqrt{F_{Rx}^2 + F_{Ry}^2} = \sqrt{(-7.5)^2 + 4.33^2} \approx 8.66 N$$ Therefore, the answer is \boxed{8.66 N}.

**Question 28**
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Two vectors m and n are defined by \(m = 3i + 4j\) and \(n = 2i - j\). Find the angle between m and n.

**Question 29**
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A bag contains 2 red and 4 green sweets of the same size and shape. Two boys pick a sweet each from the box, one after the other, without replacement. What is the probability that at least a sweet with green wrapper is picked?

**Question 30**
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Calculate, correct to one decimal place, the standard deviation of the numbers: -1, 5, 0, 2 and 9.

**Answer Details**

To find the standard deviation of a set of numbers, we need to first calculate the mean of the numbers, then subtract the mean from each number to get the deviations, square each deviation, find the mean of the squared deviations (variance), and finally take the square root of the variance to get the standard deviation. So first, let's find the mean: mean = (-1 + 5 + 0 + 2 + 9) / 5 = 3 Next, we'll find the deviations by subtracting the mean from each number: -1 - 3 = -4 5 - 3 = 2 0 - 3 = -3 2 - 3 = -1 9 - 3 = 6 Now we'll square each deviation: (-4)^2 = 16 2^2 = 4 (-3)^2 = 9 (-1)^2 = 1 6^2 = 36 Next, we'll find the mean of the squared deviations: (16 + 4 + 9 + 1 + 36) / 5 = 13.2 Finally, we'll take the square root of the variance to get the standard deviation: sqrt(13.2) ≈ 3.6 So the answer is (c) 3.6, correct to one decimal place.

**Question 31**
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Find the value of p for which \(x^{2} - x + p\) becomes a perfect square.

**Answer Details**

To make the expression \(x^2-x+p\) a perfect square, we need to add and subtract a suitable constant term. To do this, we can start by finding the square of half the coefficient of the linear term. That is, we find the square of \(\frac{1}{2}\) which is \(\frac{1}{4}\). So, we have: $$x^2-x+\frac{1}{4}-\frac{1}{4}+p$$ Now, we can write this expression as a perfect square by factoring the first three terms: $$\left(x-\frac{1}{2}\right)^2+p-\frac{1}{4}$$ For this expression to be a perfect square, the last term must be zero, so we have: $$p-\frac{1}{4}=0$$ Solving for \(p\), we get: $$p=\frac{1}{4}$$ Therefore, the value of \(p\) that makes \(x^2-x+p\) a perfect square is \(\frac{1}{4}\).

**Question 32**
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If \(f(x) = mx^{2} - 6x - 3\) and \(f'(1) = 12\), find the value of the constant m.

**Answer Details**

The derivative of a quadratic function is obtained by differentiating the function term by term. Thus, differentiating the given function, we have: $$f'(x) = 2mx - 6$$ We are also given that $f'(1) = 12$. Substituting $x=1$ and equating to 12, we have: $$f'(1) = 2m(1) - 6 = 12$$ Simplifying this equation, we get: $$2m = 18$$ Therefore, $m=9$. Thus, the value of the constant $m$ is 9.

**Question 33**
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The polynomial \(2x^{3} + 3x^{2} + qx - 1\) has the same reminder when divided by \((x + 2)\) and \((x - 1)\). Find the value of the constant q.

**Question 34**
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Given that \(\log_{2} y^{\frac{1}{2}} = \log_{5} 125\), find the value of y.

**Answer Details**

We can simplify the left side of the equation as follows: $$\log_{2} y^{\frac{1}{2}} = \frac{1}{2}\log_{2} y$$ Similarly, we can simplify the right side of the equation as follows: $$\log_{5} 125 = \log_{5} 5^3 = 3\log_{5} 5 = 3$$ So we have: $$\frac{1}{2}\log_{2} y = 3$$ Multiplying both sides by 2, we get: $$\log_{2} y = 6$$ Using the definition of logarithms, we can write this as: $$2^6 = y$$ Simplifying, we get: $$y = 64$$ Therefore, the value of y is 64.

**Question 35**
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A group of 5 boys and 4 girls is to be chosen from a class of 8 boys and 6 girls. In how many ways can this be done?

**Answer Details**

To solve this problem, we need to use the concept of combinations. The number of ways to choose a group of r objects from a set of n distinct objects is given by the formula: C(n,r) = n! / (r! * (n-r)!) where n! denotes the factorial of n (i.e., the product of all positive integers up to and including n), and r! denotes the factorial of r. In this problem, we want to choose a group of 5 boys and 4 girls from a class of 8 boys and 6 girls. We can do this in two steps: Step 1: Choose the 5 boys from the 8 boys. This can be done in C(8,5) ways. Step 2: Choose the 4 girls from the 6 girls. This can be done in C(6,4) ways. The total number of ways to choose the group of 5 boys and 4 girls is the product of the number of ways in Step 1 and the number of ways in Step 2: C(8,5) * C(6,4) = (8! / (5! * 3!)) * (6! / (4! * 2!)) = (8*7*6 / (3*2*1)) * (6*5 / (2*1)) = 56 * 15 = 840 Therefore, the number of ways to choose a group of 5 boys and 4 girls from a class of 8 boys and 6 girls is 840.

**Question 36**
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Given that \(P = \begin{pmatrix} 4 & 9 \end{pmatrix}\) and \(Q = \begin{pmatrix} -1 & -2 \\ 3 & 2 \end{pmatrix}\). Which of the following operations is possible?

**Question 37**
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Given that \(p = 4i + 3j\), find the unit vector in the direction of p.

**Question 38**
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The equation of a curve is given by \(y = 2x^{2} - 5x + k\). If the curve has two intercepts on the x- axis, find the value(s) of constant k.

**Answer Details**

To find the x-intercepts of the curve, we need to set y = 0 and solve for x. Therefore, we have: $$ \begin{aligned} y &= 2x^{2} - 5x + k \\ 0 &= 2x^{2} - 5x + k \\ \end{aligned} $$ For the curve to have two x-intercepts, the discriminant of the quadratic equation must be greater than zero: $$ \begin{aligned} b^{2} - 4ac &> 0 \\ (-5)^{2} - 4(2)(k) &> 0 \\ 25 - 8k &> 0 \\ -8k &> -25 \\ k &< \frac{25}{8} \\ \end{aligned} $$ Therefore, the value of constant k must be less than \(\frac{25}{8}\). Thus, the answer is \(k < \frac{25}{8}\).

**Question 39**
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Find the fourth term of the binomial expansion of \((x - k)^{5}\) in descending powers of x.

**Question 40**
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Three forces \(F_{1} = (8 N, 300°), F_{2} = (6 N, 090°)\) and \(F_{3} = (4 N, 180°)\) act on a particle. Find the vertical component of the resultant force.

**Question 41**
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(a) Using the trapezium rule with seven ordinates, evaluate \(\int_{0}^{3} \frac{\mathrm d x}{x^{2} + 1}\), correct to two decimal places.

(b) Using matrix method, solve \(-2x + y = 3; - x + 4y = 1\).

None

**Answer Details**

None

**Question 42**
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Forces \(F_{1} = (10 N, 090°), F_{2} = (20 N, 210°)\) and \(F_{3} = (4 N, 330°)\) act on a body at rest on a smooth table. Find, correct to one decimal place, the magnitude of the resultant force.

None

**Answer Details**

None

**Question 43**
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Five digit numbers are formed from digits 4, 5, 6, 7 and 8.

(a)How many such numbers can be formed if repitition of digits is (i) allowed (ii) not allowed?

(b) How many of the numbers are odd, if repetition of digits is not allowed?

None

**Answer Details**

None

**Question 44**
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(a) Given that \(\begin{vmatrix} 5 & 2 & -3 \\ -1 & k & 6 \\ 3 & 9 & (k + 2) \end{vmatrix} = -207\), find the values of the constant k.

(b) The equation of a curve is \(x(y^{2} + 1) - y(x^{2} + 1) + 4 = 0\). Find the:

(i) gradient of the curve at any point (x, y).

(ii) equation of the tangent to the curve at the point (-1, -3).

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**Question 46**
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Four boys participated in a competition in which their respective chances of winning prizes are \(\frac{1}{5}, \frac{1}{4}, \frac{1}{3}\) and \(\frac{1}{2}\). What is the probability that at most two of them win prizes?

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**Question 47**
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(a) The roots of the equation \(x^{2} + mx + 11 = 0\) are \(\alpha\) and \(\beta\), where m is a constant. If \(\alpha^{2} + \beta^{2} = 27\), find the values of m.

(b) The line \(2x + 3y = 1\) intersects the circle \(2x^{2} + 2y^{2} + 4x + 9y - 9 = 0\) at points P and Q where Q lies in the fourth quadrant. Find the coordinates of P and Q.

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**Question 48**
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The table below shows the distribution of ages of workers in a company.

Age/ yr | 17 - 21 | 22 - 26 | 27 - 31 | 32 - 36 | 37 - 41 | 42 - 46 | 47 - 51 | 52 - 56 |

Workers | 12 | 24 | 30 | 37 | 45 | 25 | 10 | 7 |

(a) Using an assumed mean of 39, calculate the (i) mean (ii) standard deviation; of the distribution.

(b) If a worker is selected at random from the company for an award, what is the probability that he is at most 36 years old?

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**Question 49**
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The normal to the curve \(y = 2x^{2} + x - 3\) at the point (2, 7) meets the x- axis at the point P. Find the coordinates of P.

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**Question 50**
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An object is projected vertically upwards. Its height, h m, at time t seconds is given by \(h = 20t - \frac{3}{2}t^{2} - \frac{2}{3}t^{3}\). Find

(a) the time at which it is momentarily at rest (b) correct to two decimal places, the maximum height reached by the object.

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**Question 51**
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(a) Find, correct to one decimal place, the angle between \(p = \begin{pmatrix} 3 \\ -1 \end{pmatrix}\) and \(q = \begin{pmatrix} 3 \\ 4 \end{pmatrix}\).

(b) ABCD is a square with vertices at A(0, 0), B(2, 0), C(2, 2) and D(0, 2). Forces of magnitude 10 N, 15 N, 20 N and 5 N act along \(\overrightarrow{BA}, \overrightarrow{BC}, \overrightarrow{DC}\) and \(\overrightarrow{AD}\) respectively. Find the (i) magnitude (ii) direction; of the resultant.

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**Question 52**
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(a) If \(^{18}C_{r} = ^{18}C_{r + 2}\), find \(^{r}C_{5}\).

(b) In a community, 10% of the people tested positive to the HIV virus. If 6 persons from the community are selected at random, one after the other with replacement, calculate, correct to four decimal places, the probability that : (i) exactly 5 (ii) none (iii) at most 2; tested positive to the virus.

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**Question 53**
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The second term of a geometric progression is 3. If its sum to infinity is \(\frac{25}{2}\), find the value of its common ratio.

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**Question 54**
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(a) Solve the equation : \(\sqrt{4x - 3} - \sqrt{2x - 5} = 2\).

(b) Find the finite area enclosed by the curve \(y^{2} = 4x\) and the line \(y + x = 0\).

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**Question 55**
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(a) Write down the first four terms of the binomial expansion of \((2 - \frac{1}{2})^{5}\) in ascending powers of x.

(b) Use your expansion in (a) above to find, correct to two decimal places, the value of \((1.99)^{5}\).

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**Question 56**
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The table shows the marks obtained by some candidates in Physics (y) and Mathematics (x) tests.

Mathematics | 43 | 46 | 48 | 39 | 30 | 60 | 8 | 45 | 40 |

Physics | 54 | 53 | 63 | 30 | 44 | 75 | 20 | 33 | 49 |

(a)(i) Represent this information on a scatter diagram.

(ii) Find \(\bar{x}\) and \(\bar{y}\), the mean of x and y respectively.

(iii) Draw the line of best fit to pass through (x, y).

(b) Find the equation of the line in a(iii).

(c) Use your equation in (b) to find, correct to one decimal place, the mark in Physics for a candidate who scored 28 in Mathematics.

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