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igcse model questions vol1 2025 mathematics | Objective

Prepare for your exams with igcse model questions vol1 questions? Reviewing past/model questions is one of the most effective ways to guarantee a high score. This practice hub features authentic 2025 mathematics (Objective) questions designed to simulate the real exam environment.

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Question 1 View Details

A point M has coordinates (2, 3). It is first reflected in an unknown straight line l, producing point M₁, and then the image M₁ is rotated 90° clockwise about the fixed point O(5, 1) to give the final point M₂(7, ‑2). Determine the equation of the line l.

Correct A. x = 5

B. x = 7

C. x = 2

D. x = 3

Correct Answer: A

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Question 2 View Details

A class of 40 students sat a mathematics test. Their scores (in whole marks) are grouped into the intervals 0-9, 10-19, 20-29 and 30-39. Let the frequencies in these intervals be f₁, f₂, f₃ and f₄ respectively. It is known that: • the overall mean score is 22.5; • the median score lies in the 20-29 interval; • the frequency in the 0-9 interval is half the frequency in the 30-39 interval. Find the number of students whose scores are in the 10-19 interval.

A. 10

Correct B. 8

C. 6

D. 12

Correct Answer: B

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Question 3 View Details

The graph of a straight line passes through the point A(1, 2) and meets the x‑axis at a point C that lies between x = 1 and x = 4. The line also passes through a second point B(4, m). The region bounded by the line, the x‑axis and the vertical lines x = 1 and x = 4 has an area of 9 square units. Determine the value of m.

A. 3 - \sqrt{17}

B. -3 - \sqrt{15}

C. -3 + \sqrt{17}

Correct D. -3 - \sqrt{17}

Correct Answer: D

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Question 4 View Details

A garden is shaped as a rectangle with a semicircle attached to one of its longer sides. The rectangle's length is twice its width. The total perimeter of the whole figure is 50 m. Find the area of the garden, giving your answer to one decimal place.

Correct A. 175.0

B. 150.0

C. 165.0

D. 185.0

Correct Answer: A

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Question 5 View Details

A function f is linear and can be written as \(f(x)=ax+b\) with \(a\neq0\). Its inverse is of the form \(f^{-1}(x)=mx+n\) for constants m and n. It is also known that the function \(h\) defined by \(h(x)=f(2x-1)+f^{-1}(x)\) satisfies \(h(x)=5x+7\) for every real x. Determine the values of a and b.

A. a = (5±√17)\/2, b = a(7+a)\/(a‑1)

Correct B. a = (5±√17)/4, b = a(7+a)/(a‑1)

C. a = (5±√17)\/4, b = a(7‑a)\/(a‑1)

D. a = (5±√13)\/4, b = a(7+a)\/(a‑1)

Correct Answer: B

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Question 6 View Details

The quadratic equation \(x^{2}+p x+q=0\) has two distinct real roots. When the entire equation is multiplied by 3 and then the constant term is increased by 12, the resulting quadratic has a double root. Moreover, the sum of the roots of the original equation is 6. Find the ordered pair \((p,q)\).

A. (-6, 4)

Correct B. (-6, 5)

C. (-5, 5)

D. (-6, 6)

Correct Answer: B

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Question 7 View Details

The diagram shows the curve \(f(x)=x^{2}-4x+5\) and the line \(g(x)=x+1\). They intersect at points A and B. Using the diagram, determine the exact area enclosed between the curve and the line.

A. \frac{7}{2}

B. \frac{8}{2}

Correct C. \frac{9}{2}

D. \frac{9}{3}

Correct Answer: C

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Question 8 View Details

Determine all integer values of \(k\) for which the system of inequalities \(3x - 2 < k\) and \(x^{2}-(k-3)x+(k-2)\le 0\) has at least one real solution for \(x\).

A. k \le 1 \text{ or } k \ge 9

Correct B. k \le 2 \text{ or } k \ge 8

C. k < 2 \text{ or } k > 8

D. k \le 2 \text{ or } k > 8

Correct Answer: B

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Question 9 View Details

A bag contains red, blue and green balls. The total number of balls is 30. The probability of drawing a green ball is 0.2, and the probability of drawing a red ball is twice that of drawing a blue ball. Two balls are drawn successively with replacement. Given that at least one green ball is drawn, find the probability that exactly one red ball is drawn.

A. \frac{16}{9}

Correct B. \frac{16}{27}

C. \frac{8}{27}

D. \frac{4}{9}

Correct Answer: B

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Question 10 View Details

Find all real values of \(k\) such that the solution set of \(|2x-5|<k\) is entirely contained within the solution set of \(x^{2}-4x+3\le 0\).

Correct A. 0 < k \le 1

B. 0 < k < 1

C. 0 < k \le 2

D. k \le 0

Correct Answer: A

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Question 11 View Details

The line passes through the points \( (2, a) \) and \( (b, 8) \). It also passes through the point \( (5, 5) \). If \(a\) and \(b\) are integers between 0 and 10 inclusive, find the value of \(a + b\).

Correct A. 10

B. 14

C. 12

D. 8

Correct Answer: A

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Question 12 View Details

Two vectors \(\mathbf{u}\) and \(\mathbf{v}\) lie in a plane. Their magnitudes are \(|\mathbf{u}| = 5\) and \(|\mathbf{v}| = 7\). The vector \(\mathbf{w} = 2\mathbf{u} - \mathbf{v}\) has magnitude 6. Find the angle between \(\mathbf{u}\) and \(\mathbf{v}\) in degrees, correct to one decimal place.

A. 60.1

B. 45.5

Correct C. 36.2

D. 30.0

Correct Answer: C

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Question 13 View Details

The variable \(x\) satisfies the quadratic equation \(2x^{2} - 5x + 3 = 0\). Find the sum of the reciprocals of the two roots of this equation, expressed as a fraction in lowest terms.

A. 3/5

B. 10/3

C. 5/2

Correct D. 5/3

Correct Answer: D

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Question 14 View Details

In a school of 150 students, 80 study Mathematics, 70 study Physics and 60 study Chemistry. The number of students who study exactly two of these subjects is twice the number who study exactly one subject. Twenty students study all three subjects. How many students study none of the three subjects?

A. 20

Correct B. 40

C. 30

D. 50

Correct Answer: B

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Question 15 View Details

Consider the system of equations \(2x + 3y = k\) \(5x - y = 7\) Find the smallest integer value of \(k\) for which the system has a solution in positive integers \(x\) and \(y\) satisfying \(x < y\).

A. 14

Correct B. 13

C. 12

D. 15

Correct Answer: B

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Question 16 View Details

A positive integer leaves a remainder of 3 when divided by \(7\) and a remainder of 2 when divided by \(5\). What is the smallest such integer?

A. 24

B. 31

Correct C. 17

D. 10

Correct Answer: C

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Question 17 View Details

In a bag there are red and blue balls. The probability that two balls drawn at random without replacement are both red is \(\frac{3}{10}\). After these two red balls are removed, a third ball is drawn. What is the probability that this third ball is red?

A. 3/8

B. 2/5

C. 1/2

Correct D. 1/3

Correct Answer: D

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Question 18 View Details

Simplify the expression \(\displaystyle \frac{2^{3}\sqrt{50}}{5^{2}\sqrt{2}}\) and give your answer in simplest fractional form.

A. 6/5

B. 9/5

Correct C. 8/5

D. 8/7

Correct Answer: C

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Question 19 View Details

Vectors \(\mathbf{a}\) and \(\mathbf{b}\) have magnitudes \(|\mathbf{a}| = 5\) cm and \(|\mathbf{b}| = 7\) cm. The magnitude of their sum \(\mathbf{a}+\mathbf{b}\) is \(9\) cm. (i) Find the angle between \(\mathbf{a}\) and \(\mathbf{b}\) to the nearest degree. (ii) Find the magnitude of the vector \(2\mathbf{a}-\mathbf{b}\) (to one decimal place).

Correct A. 84°, 11.6

B. 84°, 13.5

C. 78°, 12.3

D. 90°, 10.2

Correct Answer: A

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Question 20 View Details

A product costs \(N\) naira. Its price is first increased by 20% and then decreased by 15% of the new price. What is the overall percentage change in the price of the product?

A. 2% decrease

B. 3% increase

Correct C. 2% increase

D. 5% increase

Correct Answer: C

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Question 21 View Details

Seven test scores are 50, 60, 68, 71, 77, 78 and 84. An eighth score \(x\) is unknown. The mean of all eight scores is 70 and the median of the eight scores is 71.5. Find the value of \(x\).

A. 68

B. 80

C. 74

Correct D. 72

Correct Answer: D

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Question 22 View Details

The \(n\)th term of a sequence is given by \(t_n = n^{2}+kn\). The sum of the first four terms of the sequence is 100. Find the eighth term \(t_{8}\).

A. 124

Correct B. 120

C. 130

D. 112

Correct Answer: B

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Question 23 View Details

Real numbers \(a\) and \(b\) satisfy \(a+b=6\) and \(a^{2}+b^{2}=20\). Find the value of \(a^{3}+b^{3}\).

Correct A. 72

B. 80

C. 68

D. 64

Correct Answer: A

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Question 24 View Details

A circle has centre \(C(2,-1)\) and radius \(5\). A line passes through the point \(P(4,2)\) and is perpendicular to the line joining \(C\) to \(Q(7,5)\). Find the coordinates of the point where this line meets the circle in the first quadrant.

A. (125+5√409)/34 , (77-3√409)/34

B. (131-5√409)/34 , (71+3√409)/34

C. (131+5√409)/34 , (71-2√409)/34

Correct D. (131+5√409)/34 , (71-3√409)/34

Correct Answer: D

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Question 25 View Details

In the diagram, points \(A(2,3)\), \(B(8,3)\) and \(C(5,7)\) are plotted on a coordinate plane. Point \(D(5,0)\) is also shown. A line through \(D\) is drawn parallel to \(AB\) and meets \(AC\) at point \(E\). Find the coordinates of \(E\).

A. (5/2, 0)

B. (5/4, 3)

C. (-1/4, 0)

Correct D. (5/4, 0)

Correct Answer: D

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