Practice gce model questions vol1 mathematics 2019 Past Questions

Question 1 View Details

The sum of the first \(n\) terms of an arithmetic sequence is given by \(S_n = 3n^{2}+4n\) for all positive integers \(n\). Find the 10th term of the sequence.

A. 55

Correct B. 61

C. 60

D. 62

Correct Answer: B

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 2 View Details

In triangle \(ABC\) the sides satisfy \(AB = 13\text{ cm}\) and \(AC = 15\text{ cm}\). The internal angle bisector of \(\angle A\) meets side \(BC\) at \(D\). The incircle of the triangle touches \(BC\) at \(D\) and has radius \(r\). Given that \(BD = 6\text{ cm}\), find the radius \(r\) of the incircle (in centimetres).

A. 6

B. 3

C. 5

Correct D. 4

Correct Answer: D

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 3 View Details

The time \(T\) (in minutes) required for a reaction varies directly as the square root of the concentration \(C\) (in mol\,L\(^{-1}\)) and inversely as the cube of the temperature difference \(\Delta\theta\) (in \(^{\circ}\)C) from a reference temperature. When \(C = 0.25\) mol L\(^{-1}\) and \(\Delta\theta = 5^{\circ}\)C, the reaction takes \(12\) minutes. Find the time required when \(C = 0.36\) mol L\(^{-1}\) and \(\Delta\theta = 3^{\circ}\)C. Give your answer as a fraction in simplest form.

A. 40/3

Correct B. 200/3

C. 400/3

D. 200/9

Correct Answer: B

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 4 View Details

Simplify \(\displaystyle \log_{3}\bigl(\sqrt{27}\bigr) + \log_{3}\bigl(9^{-\frac12}\bigr)\).

Correct A. 1/2

B. 1/4

C. 3/2

D. 2/3

Correct Answer: A

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 5 View Details

Solve for \(x\): \(\displaystyle \frac{|2x-5|}{x-3} < 2\).

A. x>3

Correct B. x<3

C. x<5

D. x≤3

Correct Answer: B

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 6 View Details

In a school of 250 students, let \(A\) be the set of students who play football, \(B\) those who play basketball and \(C\) those who play cricket. The numbers are \(|A| = 110\), \(|B| = 90\) and \(|C| = 70\). Exactly 10 students play all three sports and exactly 25 students play exactly two of the sports. It is also known that \(|A \cap C| = 25\). Find \(|A \cap B|\).

A. 15

Correct B. 20

C. 30

D. 25

Correct Answer: B

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 7 View Details

The quadratic equation \(x^{2}-(k+3)x+2k=0\) has two distinct positive integer roots, the larger exceeding the smaller by 3. Determine the value of \(k\) and the two roots.

A. k = 1, roots 1 and 2

Correct B. k = 2, roots 1 and 4

C. k = 2, roots 2 and 5

D. k = 3, roots 1 and 5

Correct Answer: B

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 8 View Details

The diagram shows triangle \(ABC\) with vertices \(A(0,0)\), \(B(6,0)\) and \(C(6,8)\). Let \(M\) be the midpoint of \(AB\). A line through \(M\) is drawn parallel to \(AC\) and meets side \(BC\) at \(D\). Find the coordinates of \(D\) and the area of triangle \(ABD\).

A. (6,3), 9

B. (5,4), 10

Correct C. (6,4), 12

D. (6,5), 15

Correct Answer: C

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 9 View Details

When the integer 23 487 is divided by 9 it leaves a remainder \(r\). Find \(r\) and state the smallest positive integer that must be added to 23 487 to obtain a multiple of 9.

A. 5, 4

B. 7, 2

Correct C. 6, 3

D. 3, 6

Correct Answer: C

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 10 View Details

The velocity-time graph of a particle moving in a straight line passes through the points \((0,0)\) and \((6,18)\) and is a straight line between them. Assuming the motion starts from rest, determine (a) the constant acceleration of the particle and (b) the total distance travelled during the first 6 seconds.

A. 3 m/s^2, 48 m

Correct B. 3 m/s^2, 54 m

C. 2 m/s^2, 36 m

D. 2.5 m/s^2, 45 m

Correct Answer: B

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 11 View Details

The system of equations \(3x + 2y = 7\) and \(kx - y = 4\) has a solution \((x,y)\) with \(x>y\). Find the positive integer value of \(k\) for which \(x\) is an integer, and then give that integer value of \(x\).

Correct A. 3

B. 2

C. 5

D. 4

Correct Answer: A

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 12 View Details

The expression \(P(x)= (x^{2}-6x+k)(x-2)+8\) is known to be divisible by \((x-4)\). (a) Find the value of \(k\). (b) Using this value, evaluate \(P(1)\).

Correct A. 9

B. 10

C. 7

D. 8

Correct Answer: A

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 13 View Details

The time \(t\) (in minutes) for a reaction varies directly as the square of the concentration \(c\) (in mol\,L\(^{-1}\)) and inversely as the temperature \(T\) (in K). When \(c=0.2\) mol L\(^{-1}\) and \(T=300\) K, the reaction takes \(5\) minutes. (a) Determine the constant of variation \(k\). (b) If the concentration is increased to \(0.3\) mol L\(^{-1}\) and we wish the reaction to still take \(5\) minutes, what temperature must be used? Give your answer in kelvin.

A. 600

B. 450

Correct C. 675

D. 900

Correct Answer: C

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 14 View Details

A square‑based pyramid has base side length \(6\) cm and height \(9\) cm. A cube is placed inside the pyramid so that one face of the cube lies on the base and the top vertices of the cube touch the sloping faces of the pyramid. Find the volume of the cube in cubic centimetres, expressed as an exact fraction.

Correct A. 5832/125

B. 5832/150

C. 5832/100

D. 5400/125

Correct Answer: A

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 15 View Details

In an arithmetic sequence the first term is \(a\) and the common difference is \(d\). The sum of the first five terms is \(40\) and the sum of the next five terms (terms 6 to 10) is \(70\). Find the eighth term of the sequence.

Correct A. 14

B. 15

C. 12

D. 13

Correct Answer: A

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 16 View Details

In a school there are 108 students. Let set \(A\) be the students who study Mathematics and set \(B\) be those who study Physics. It is known that \(|A| = 84\), \(|B| = 70\) and that 10 students study neither subject. Moreover, the number of students who study both subjects is twice the number who study Mathematics only. How many students study both Mathematics and Physics?

A. 84

B. 66

Correct C. 56

D. 46

Correct Answer: C

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 17 View Details

The diagram shows three overlapping sets \(A\), \(B\) and \(C\) representing students who study Chemistry, Biology and Physics respectively. The numbers shown in the diagram are: only \(A\): 30, only \(B\): 25, only \(C\): 20, \(A\cap B\) only: 15, \(B\cap C\) only: 10, \(A\cap C\) only: 5, and none of the three: 40. The number of students who study all three subjects is not shown. The total number of students surveyed is 200. How many students study all three subjects?

A. 65

B. 75

C. 45

Correct D. 55

Correct Answer: D

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 18 View Details

A quadratic equation \(x^{2} - Sx + P = 0\) has two distinct integer roots that differ by 2 and whose sum is 10. Find the value of \(P - S\).

A. 16

Correct B. 14

C. 20

D. 12

Correct Answer: B

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 19 View Details

Simplify \(\displaystyle \frac{2^{3}\,\times\,5^{2}}{10^{2}}\times\sqrt{50}\) and give your answer in simplest surd form.

A. 10\sqrt{5}

B. 5\sqrt{2}

Correct C. 10\sqrt{2}

D. 20\sqrt{2}

Correct Answer: C

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 20 View Details

In a class of 20 students, the mean score for Exam 1 is 68 with a standard deviation of 5, and the mean score for Exam 2 is 72 with a standard deviation of 6. The correlation coefficient between the two exam scores is 0.8. (a) Determine the slope of the least‑squares regression line that predicts the Exam 2 score from the Exam 1 score. (b) Using this regression line, estimate the Exam 2 score of a student who obtained 75 in Exam 1. Give your answer to two decimal places.

A. Slope = 0.67; Predicted Exam 2 score = 76.69

B. Slope = 1.20; Predicted Exam 2 score = 80.40

C. Slope = 0.96; Predicted Exam 2 score = 72.00

Correct D. Slope = 0.96; Predicted Exam 2 score = 78.72

Correct Answer: D

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 21 View Details

In the plane, points A\(2, -1\) and B\(8, k\) are given. A line L is perpendicular to AB and passes through the point P\(1, 2\). It is also given that the distance from the point Q\(4, 0\) to line AB equals the distance from Q to line L. Determine the value of k and give the equation of line L in the form \(ax + by = c\).

A. k = 3; the equation of L is \(x - y = 1\).

B. k = 7; the equation of L is \(2x + y = 5\).

Correct C. k = 5; the equation of L is \(x + y = 3\).

D. k = 4; the equation of L is \(3x - 2y = 6\).

Correct Answer: C

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 22 View Details

A factory produces two types of widgets. Type A accounts for 60 % of production and has a defect probability of 2 %; type B accounts for 40 % and has a defect probability of 5 %. From a large batch, exactly three defective widgets are found. What is the probability that exactly two of these defective widgets are of type A? Give your answer as a decimal correct to three significant figures.

Correct A. 0.264

B. 0.236

C. 0.224

D. 0.284

Correct Answer: A

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 23 View Details

A solid consists of a right circular cylinder of radius \(5\) cm and height \(h\) cm, surmounted by a hemisphere of the same radius. A right circular cone of height \(8\) cm and base radius \(5\) cm is removed from the top of the hemisphere (forming a conical cavity). The total surface area of the solid, including the interior surface of the cavity but excluding the base of the cylinder, is \(1500\) cm\(^2\). Find the value of \(h\) (in cm) correct to two decimal places.

A. 34.12 cm

B. 33.95 cm

Correct C. 35.53 cm

D. 36.78 cm

Correct Answer: C

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 24 View Details

The diagram shows the graph of a quadratic function \(y = ax^2 + bx + c\). The vertex of the parabola is at \((2, -3)\) and the graph passes through the point \((0, 1)\). Using the graph, determine the \(x\)-coordinate at which the tangent to the curve has gradient \(4\).

A. 2

B. 6

C. 5

Correct D. 4

Correct Answer: D

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

Question 25 View Details

The lines \(L_1\) and \(L_2\) are given by \((k+1)x - 2y = 4\) and \(3x + (k-2)y = 7\) respectively. Find the value of \(k\) for which the two lines are perpendicular, and then determine the coordinates of their point of intersection for that value of \(k\).

Correct A. k = -7; intersection point \((-11/30, -9/10)\)

B. k = 4; intersection point \((1\/2, -7\/4)\)

C. k = 0; intersection point \((2, -1)\)

D. k = -5; intersection point \((-3\/5, -2\/3)\)

Correct Answer: A

Want to see the full step-by-step solution? Unlock AI Explanation & Analytics

gce model questions vol1 2019 mathematics | Objective

Master the Exam!

Help others prepare! Share this practice hub: