### Very hard questions for GCSE Maths

Do you need highly challenging questions for GCSE Maths exam? Here are some. Take the plunge and see whether you can crack them! They are from:

- Numbers
- Algebra
- Geometry
- Trigonometry
- Data handling
- Vectors
- Probability
- And many more...

**Answers are at the bottom of the page. ↓**

**Question 1**

If x² + y² = 13 and x + y = 5, find (x - y)², clearly showing the steps.

**Question 2**

If tan Θ = 3/2, find the equation of the line that goes through B and C. Leave your answer in the form of ax + by = c, where a, b and c are integers. Hence find the coordinates of B.

**Question 3**

**a)**
A group of girls in a class shared a bill for £22.09, equally between them. How many girls were there in the group? How much did it cost each?

**b)** u = v - ft/m is a formula in which v f, m and t are 18, 12, 6, 3, each correct to 1 significant figure. Find the lower and upper bound of u. Hence, provide a good approximation for u.

**Question 4**

**a)** The two circles intersect at D and E. Prove that the sides FH and GI are parallel.

**b)** The radii of the two circles are 8 cm and 6 cm respectively. If the centres of the large and small circles are A and C respectively, find the angle between AD and DC, if AC = 12 cm.

**Question 5**

A twelve-sided, fair die is rolled n times. Find the probability of getting 1 in two successive throws. How do you justify your answer? Hence find the probability of getting at least number 1 once, if the die
is thrown n times.

**Question 6**

**a)** A cyclist covers two sections of a road, 10 miles and 15 miles, at 5mph and 3mph respectively. Find his average speed.

**b)** In another cycle ride, he covers one half of a certain distance at 10 mph and the rest at 15 mph respectively. Find his average speed.

**Question 7**

**a)** Ted gave £1400 to share between his children, Berty, Clare and Damion in such a way that Berty gets twice as much as Clare and Damion gets half of what Clare gets. How much does Clare get?

**b)** If it takes 8 workers 6 days to build a wall 6 meters high, how many workers are needed to build a wall 9 meters high in 9 days?

**Question 8**

**a)** If x(x - 2) : x² - 6x + 8 = 7 : 3, find the value of x.

The ages of three siblings, A, B, and C, are in the ratio 3:4:7 respectively. The sum of their ages is 84 years.

**b)** Determine the age of each sibling.

**c)** If n years ago, C was twice as old as B was then, Find n.

**Question 9**

If 2x² + 4x + 1 ≡ a(x + b)² + c,

**a)** find a, b and c.

**b)** Find the line of symmetry of the curve.

**c)** Find the coordinates of the minimum point of the curve.

**d)** Find the y-intercept.

**e)** Find the values of x where the curve crosses the x-axis.

**f)** If the curve is brought down by p, the solutions of the equations become x = 0 and x = 2. Find p.

**Question 10**

The numerator of a fraction is 2 less than its denominator. If both numerator and denominator are increased by 2, the fraction becomes 7/9. Find the original fraction.

**Question 11**

Two trains remained stationary on the same track 90km apart, facing each other. They started moving towards each other at constant speeds, 40km/h and 20km/h, respectively. As soon as they started moving, a bird started flying from one train to the other and vice versa at
50 km/h until the two train just managed stop next to each other in time. Find the total distance that the bird had flown by then?

**Question 12**

Please use a pair of compasses and a ruler for the following constructions. You may use a protractor only for verifying the angles.

**a)** Construct an angle of 105°.

**b)** Construct a regular octagon inside a circle.

**c)** The three sides of the triangle, ABC, are 5cm, 12cm and 13 respectively. Construct the triangle. Hence construct a circle that touches the vertices, A, B and C. Find the area of the circle as well.

**Question 13**

The equation of a curve is f(x) = ax² + bx + c and f(-1), f(2) and f(3) are 10, 7 and 14 respectively. Find a, b and c.

**Question 14**

A courier driver has 72 parcels in his van. At a certain store, he drops 2 parcels and x were collected. At the next store, he drops a fifth
of the parcels, before collecting 4. If he has 64 parcels in his van at this stage, find x.

**Question 15**

**a)** Find the exact value of 202² - 198² without long multiplication.

**b)** Fully factorize x^{3} + 6x² - 4x - 24

**Question 16**

**a)** Three coins are tossed. Find the probability of getting at least 2 heads.

**b)** There are n balls in a container. Two of them are
black and the rest are white. A ball is taken out
at random, its colour noted and put back in.
Then a second ball is taken and the steps are
repeated. The probability of getting two balls of
different colours is 12/25. Show that 3n² - 25n +
50 = 0 and calculate the number balls in the
container.

**Question 17**

The nth term of a sequence of triangular numbers is
given by the formula, (n² + n)/2. Prove that the sum of
any two consecutive triangular numbers is a square
number.

**Question 18**

**a)** Find the angle BOC.

**b)** find the angle BCG.

**c)** If BE = EC, find the angle EBG.

**Question 19**

In the above diagram, FD and CA are parallel.

**a)** Show that angles DBF and DFB are equal.

**b)** Show that the triangles DFO and ECO are congruent.

**c)** Show that OC = ¼ BC.

**d)** Show that DFEC is a parallelogram.

**Question 20**

**a)** The mean of the numbers, 4, 5, 10, 11 , Y, is the same as the median of the numbers. Find Y.

**b)** The mean score for maths of 6 girls is 72 and that for 4 boys is 60. Find the mean score of the whole group. If two boys are added to the group with scores, 67 and 81, find the new mean of the group.

**Question 21**

PQRS is a trapezium with PQ being parallel to SR. The vectors, PQ = a, PS = b and SR = 3a. PA = PR/4.

**a)** Find the vectors, PR, PA, SA and SQ.

**b)** Show that the points, S, A and Q lie on the same straight line.

**Question 22**

**a)** If p : q = 4 : 5 and q : r = 3 : 2, find p : q : r.

**b)** If a : b = 3 : 5 and 3b : c = 4 : 7, find a : b : c.

**Question 23**

**a)** Solve x^{4} - x² = 6 and leave the answer in surd form.

**b)** Solve x - 7√x + 12 = 0 .

**Question 24**

**a)** Prove that the area of a kite is half of the product of the diagonals.

**b)** The coordinates of the points, A, B and C are (4 , 6), (8 , 4) and (5 , 3) respectively. Find the area of the triangle ABC.

**Question 25**

If the angle between the two extended sides is 36°, find the number of sides of the regular polygon. If the length of a side is 5cm, find its area.

**Question 26**

Find the area and the perimeter of EDC triangle, if the above composite shape consists of two identical triangles, ABC and ADE.

**Question 27**

Find the shaded area and the three inequalities that satisfy the region.

**Question 28**

**a) ** Find the value of 1/x, when x approaches infinity.

**b)** Hence, find the value of (n+1)/(n-1), when n gets really large.

**Question 29**

In the above diagram, AB = 7cm and AC = 24 cm, find the shaded area, if O is the centre of the circle.

**Question 30**

Prove that the sum of the squares of diagonals of a parallelogram is twice the sum of the squares of the two adjacent sides.

**Question 31**

The length of a wire is 64cm and it is divided into n equal parts. The sum of the reciprocals of the lengths is equal to 1. Find the value of n.

**Question 32**

**a)** If f(x) = 2x - 3 and g(x) = (x + 3)/2, show that gf(x) = fg(x).

**b)** If h(x) = x² - 2x, find h^{-1}(x).

**Question 33**

Two cars, Car-1 and Car-2, move as shown above. Both start from rest with a time interval 2 seconds. Both cars accelerate.

**a)** What is the significance of point B? Explain your answer.

**b)** Do they meet up at B? Justify your answer.

**c)** Find the acceleration of each car.

**Question 34**

Find the angle of AGC of the triangle CGA, without the use of the *cosine rule*. You may use basic trigonometry.

**Question 35**

**a)** In the above diagram, ED = 6.69cm, GD = 8.88cm, HD = 3.7cm. Find the length of FD.

**b)** If the radius of the circle is 3cm, find FH as well.

**Question 36**

The equation of a trigonometric curve is f(x) = a sin(x) + b, where a > b > 0. The maximum and minimum values of the functions are 7 and -3 respectively.

**a)** Sketch the graph, showing a and b clearly on the grid.

**b)** Find a and b.

**c)** Describe the transformation of f(x), from y = sin(x).

**Question 37**

**a)** Solve (2^{n})^{n} X (1/8)^{n} X 1/16 = 1, if n > 0.

**b)** Solve 32^{(x+2)/(x+7)} = 1/8

**Question 38**

**a)** Show that the curve y = x² - 2x and the line y = 4x - 9 intersect only at one point.

**b)** Find the coordinates of the above point, A.

**b)** Find the distance between the point A and the y-intercept.

**Question 39**

ABC triangle and ADE triangle are similar.

**a)** Find x.

**b)** Find y.

**Question 40**

The masses of two similar solid cylinders, made of the same substance, are 7kg and and 189kg respectively.

**a)** If the height of the shorter cylinder is 6cm, find the height of the taller cylinder.

**b)** Find the ratio of the surface areas of the cylinders.

**Question 41**

The composite object consists of two sections - a section with a triangular cross section and a rectangular cross section. The volume of the former is ¼ th of the latter. The lengths are in cm.

**a)** If the volume of the object is 10x^{3} + 720, find x.

**b)** If a liquid pours into the object at 33cm^{3}/min, how long will it take to fill it to a 1/6 the of its maximum height?

**Question 42**

**a)** When the sphere is dropped into the liquid in the cylinder, the height of the liquid goes up by 4/3 the of the radius of the cylinder. Prove that the radius of the sphere is the same as that of the cylinder.

**b)** If the height of the liquid in the cylinder is 4 times its radius, find the ratio of volume of the liquid to the volume of the sphere.

**Question 43**

The centres of the two circles and the square are the same.

**a)** Prove that the area of the larger circle is twice as big as the smaller circle.

**b)** Find the ratio of areas of red section to blue section.

**Question 44**

A large group of Year 11 students have been asked to learn the first 20 elements of the Periodic Table by heart. The above histogram shows frequency against the time taken by them.

There were 25 students who took between 10 to 20 seconds.

**a)** How do you justify the use of a histogram to represent the data?.

**b)** Find the total number of students.

**c)** What is the probability of a student taking between 60 to 70 seconds to remember the elements?

**Question 45**

Having mastered factorization for his GCSE, Ben showed the following to his friend, Piyal:

x = y

Multiply both sides by x,

x² = xy

Take away y²,

x² - y² = xy - y²

Factorizing both sides,

(x + y)(x - y) = y(x - y)

Dividing by (x - y)

x + y = y

Since x = y

2y = y

Dividing by y,

2 = 1! 😲

**a)** Piyal spotted the mistake. What was the mistake that Ben made?

**b)** Fully factorize 75x^{3} - 27xy².

**c)** Solve (2x - 3)² = 7, and leave the answer in surd form.

**Question 46**

Adrian thinks that n² + n + 41 is a prime number for any value of n. Jane does not agree with that.

**a)** How did Jane disprove Adrian's claim?

The firs four terms of a linear sequence are as follows:

5, 11, 17, 23

**b)** Find the nth term of the sequence.

**c)** Prove that the difference of squares of any two consecutive terms of the sequence is a multiple of 12.

**Question 47**

The two circles share the same centre, O and the radii are 2cm and 4cm respectively. Three longest lines drawn in the space between two circles are shown as above.

**a)** Show that EF, EI and FI are tangents to the smaller circle.

**b)** Find the total area between the larger circle and the triangle.

**Question 48**

Solve 3^{p} + 3^{p-1} + 3^{p-2} = 39 and find p.

**Question 49**

A hedgehog that wants to cross a road of width, d, sees an owl on top of a tree of height 8m, on the other side of the road. The hedgehog sees the owl at an angle of elevation of 42°.
Then, it reluctantly crosses the road and looks at the owl again, to note the angle of elevation being 53°.
**a)** How wide is the road?

**b)** Find the distance between the tree and the edge of the road, closer to the tree.

**Question 50**

**a)** Sketch f(x) = (x + 1)/x and clearly mark the asymptotes, if any.

**b)** Sketch f(x) = -1/x on the same grid.

**c)** Solve algebraically or otherwise (x+1)/x + 1/x = 0 and comment on the feasibility of the answers.

**Question 51**

ABCD is a parallelogram and the two diagonals intersect at E.

**a)** Find vector expressions for both diagonals.

**b)** Hence prove that the two diagonals bisect each other.

**Question 52**

Move the points with your mouse or finger and you will see the sum of angles B and C remains 180°, regardless of the individual angles. If O is the centre of the circle, prove this fact geometrically - and algebraically - without using any of the *circle theorems.*

**Question 53**

Sarah runs the above spinner by clicking, **Start Spin**, button for some experiments with probability.

**a)** Find the probability of getting a 2.

**b)** Find the probability of getting a 2 or a 5.

**c)** If she spins twice, find the probability of getting a 2, followed by a 4.

**d)** If she spins twice, find the probability of getting a 3 and a 4 in any order.

**e)** If she spins twice, find the probability of getting a 5, at least once.

**f)** If she runs the spinner until she gets a 4 or total spins are two. Find the probability.

**Question 54**

There are *n* number of chocolates with a number written on them in a container; three of them have different prime numbers written on them. Fionna is allowed to take two chocolates, one chocolate at a time. If a prime number is written on a chocolate, she is allowed to eat it. If the probability of Fionna eating at least one chocolate with a prime number on it, is 1/2, find the value of n.

**Question 55**

The numbers, 3, 4 and 7, are written on three cards. They are arranged next to each other in such a way that they form a number less than 700.

**a)** Find the probability of getting two numbers with the last digit being 7.

**b)** Find the probability of the difference between the last two digits being odd.

**Question 56**

The probability of raining tomorrow is 0.6. If it rains, the probability of identical twins, Mrs Dorcey as well as Mrs Bayles, catching a cold is 0.4. Due to low immunity, even if it will not rain tomorrow, the probability of both women catching a cold is 0.2.

**a)** Find the probability of Mrs Dorcey catching a cold tomorrow.

**b)** Find the probability of neither of them catching a cold tomorrow.

**Question 57**

A number that cannot be written as a fraction is an irrational number. Surds are considered to be irrational numbers.

**a)** Find an irrational number between 5 and 6.

**b)** Daniel says that the product of two irrational numbers is always irrational. Giving an example, show that Daniel is not correct.

**c) ** If a and b are integers, prove that (a + √b)(a - √b) is rational.

**d)** Show that 7√27/2√12 is rational.

**Question 58**

cos90 = cos60 x cos30 - sin30 x sin 60

The above is true for any three angles that follow the above pattern on both side of the equation.

**a)** Without using a calculator, find cos75.

**b)** Without using a calculator, find cos 63 x cos 27 - sin 27 sin 63.

**c) ** By using two appropriate angles, show that cos120 < 0.

**Question 59**

There are 3 boys and 2 girls in a group.

**a)** How many ways can two boys and a girl be chosen for educational purposes?.

**b)** How many ways can two boys and a girl be chosen for an expedition?

There are 12 boys and 8 girls in another group.

**c) ** How many ways can two boys and a girl be chosen for the same expedition from the above group?

**Question 60**

**a)** Solve 32^{2x - 3y} = 1/16 and 4^{x + y} = 1, to find x and y.

**b)** Hence, find the value of z, if 8^{x} x 4^{y} = 2^{z}.

**Question 61**

The radius of the circle is x and centre is O. CA and DA are two tangents to the circle. CA = x+7 and OA = x+8.
**a)** Find the value of x.

**b)** Find the angle between the two tangents.

**c)** Find the length of CD.

**Question 62**

The above shows a cross section of a container that is a semicircle. The container is placed horizontally and filled with a liquid to a height, 2cm. The length of the container is 10cm.

**a)** Find the volume of the liquid in the container.

**b)** If the density of the liquid is 5000kg/m^{3}, find the mass of the liquid in the container in grams.

**Question 63**

There are 5 star chocolates, 3 round chocolates and a square chocolate in a container. Maya takes three chocolates from it.

**a)** Find the probability of equal number of star chocolates and round chocolates left in the container.

**b)** Find the probability of more round chocolates than star chocolates left in the container.

**Question 64**

x_{n+1} = x_{n}(2 - 8x²_{n}) is an iterative formula.

**a)** By using the calculator, find x_{1} to x_{10}, if x_{0} = 0.02 .

**b)** Hence show that it is the exact value of 1/8.

**c)** Verify the answer to part, b), by rearranging the iterative formula.

**Question 65**

ABC is a triangle drawn inside a circle with AC being a diameter and O being the centre. BO is perpendicular to the diameter.

**a)** Show that cos 45° = 1/√2.

The point B is slightly moved so that the angle of BAC is equal to 53°.

**b)** If sin53 = 5/8, find cos37, without using a calculator.

**c)** Hence, find sin37.

**Question 66**

The distance between two towns, A and B, is 600km. David drives his car at a steady speed of x km/h in the hope of arriving at B on time. Having traveled 3/4 of the distance, his car broke down unexpectedly. Within half an hour, however, he rectified the problem and then, made it to town B on time, by just raising the speed by 15 km/h. Find the value of x.

**Question 67**

The perimeter of the inscribed circle is P and its radius is r. The area of the triangle is A. Prove that r = 2A/P.

**Question 68**

f(x) = x^{3} + 2x² - x - 2

**a)** Solve the above equation, f(x) = 0.

**b)** Sketch the curve, f(x).

**c)** Hence, sketch the curve f(2x) .

**Question 69**

x^{2} + (b/a)x + c/a = 0

**a)** Show that x = -b/2a ± √(b² - 4ac)/2a

**b)** Show that there will be no solution, if a = 2, c = 3 and b = 4

**c)** Hence, sketch the curve of the quadratic function and verify that there are no solutions.

**Question 70**

(x - p)(x - q) = x^{2} -4x - 5

**a)** Show that p + q = 4

**b)** Show that pq = -5

**c)** Find the value of 1/p + 1/q.

**Question 71**

The probability of getting a HEAD of a biased coin is ^{2}/_{3}. It is tossed up until a HEAD appears or the total number of tosses is equal to 3. Find the probability of getting at least 2 TAILs.

**Question 72**

The boys in a certain group indicate how they like two type of sports, cricket and football.

**a)** Find the value of x.

**b)** Hence, determine whether the events, 'like cricket' and 'like football', are independent.

**Question 73**

The lengths of the three sides of a triangles are shown above.

**a)** Find the value of x.

**b)** Hence, find the perimeter and area of the triangle.

**Question 74**

The product of the first and last of the four consecutive, positive integers is 154. Find the sum of the second and third integer.

**Question 75**

Factorize the following expression fully, stating in the form of linear expressions:

4x^{4} - 29x² + 25

**Answers:** ↓ please move the mouse over the following space:

Q1: 1

Q2: 2x + 3y = 23; (4,5)

Q3: 47 | 11.81

Q4: 117.7°

Q5: 1/144 | 1 - (11/12)^{n}

Q6: 3.6 | 12 8

Q7: 400 | 8

Q8: 7 | 8:18, 24, 42 | 6

Q9: 2, 1, -1 | x = -1 | (-1,-1) | 1 | -1.7, -0.3 | 1

Q10: 5/7

Q11: 75

Q13: 2, -3, 5

Q14: 5

Q15: 1600 | (x - 2)(x + 2)(x + 6)

Q16: 1/2 | 5
Q18: 144 | 84 | 30

Q20: 20 | 68.3

Q22: 12 : 15 : 10 | 12 : 20 : 105

Q23: ± √3 | ± 2, ± √3

Q24: 5

Q25: 5 | 43

Q26: 23.7 | 11.1

Q27: 3 | y ≥ 2x - 3; y ≥ 9 - x; y ≤ x + 3

Q28: 1

Q29: 406.87

Q31: 8

Q32: y = √(x+1) + 1

Q33: 1.5 | 3

Q34: 70.4°

Q35: 4.92 | 3.15

Q36: 5 | 2

Q37: -5

Q38: (3,3) | 12.4

Q39: x = 1.6 | y = 5.7

Q40: 18 | 1/9
Q41: 3 | 6

Q42: 3 : 1

Q43: 2(π-2)/(4-π)

Q44: 120 | 1/24

Q45: 3x(5x + 3y)(5x - 3y) | (3 ± √7)/2

Q46: 6n-1

Q47: 29.6

Q48: 3

Q49: 3, 6

Q50: -2

Q53: 1/8 | 3/8 | 5/128 | 5/64 | 7/16 | 135/256

Q54: 6

Q55: 1/2 | 1/2

Q56: 0.32 | 0.46

Q57: √31 | √2, √8

Q58: (√6 - √2)/4 | 0

Q59: 12 | 6 | 528

Q60: -4/25 | 4/25 | -4/25

Q61: 5 | 45.2 | 9.2

Q62: 112 | 560

Q63: 15/126 | 5/42

Q64: 60

Q65: 5/8 | √55/8

Q66: 60

Q68: -2 | 1 |-1

Q70: -0.8

Q71: 1/9

Q72: 0.06 | Not

Q73: 9 | 28 | 14√3

Q74: 25

Q75: (2x + 5)(x - 1)(x + 1)(2x - 5)

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