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(a)Solve the following inequality for x ∈ ℝ:
|x − 3| ≤ 12
(a)
(b)Multiply out and simplify:
(4x − 10√x)(2x + 5√x − 7)
(b)
(c)(2x + 3) is a factor of 4x3 − 12x2 − 7x + 30. Use this information to find the three solutions to the following equation in x:
4x3 − 12x2 − 7x + 30 = 0
(c)
(a)The function g(x) is defined for x ∈ ℝ by g(x) = 5x2 + 20x − 12. Write g(x) in the following form (by completing the square), where a, h, k ∈ ℤ are constants:
g(x) = a(x + h)2 + k
(a)
(b)p is a positive constant. Use the laws of logs to write the expression ln[ (e3p)5 ] in the form c + d ln p, where c, d ∈ ℤ are constants.
(b)
(c)Below is a pair of simultaneous equations in x and y, where n ∈ ℝ is a constant. One of the solutions to this pair of equations is on the y-axis. Use this information to find the value of n.
2x − y = 7
x2 + y + 2y2 = n
x2 + y + 2y2 = n
(c)
(b)h(x) is the following function of x, where m and r are positive constants and x ∈ ℝ:
h(x) = 6m x2 − 4r x + 54m
(i)The equation h(x) = 0 has exactly one solution. Use this to show that r = 9m. (Hint: the discriminant is 0.)
(b)(i)
(ii)Use the fact that r = 9m to find the value of this solution.
(b)(ii)
(a)Solve the following equation for n ∈ ℕ: n − 3 = √(3n + 1)
(a)
(b)Write the following expression as a single fraction in terms of t: 42t + 1 − 712t
(b)
(c)Solve the following simultaneous equations for x, y, w ∈ ℤ: x + 2y = 143
y + 3w = −74
4x + 5w = 4
y + 3w = −74
4x + 5w = 4
(c)
(a)h(x) = x2 + bx − 12, where x ∈ ℝ and b is a constant. Find the value of b for which x − 4 is a factor of h(x).
(a)
A sphere has a radius of R units. The part of the sphere that is cut off by a flat surface is called a "cap", and has a volume of C = πk23(3R − k). Here, C is the volume of the cap and k is the height of the cap, with 0 < k ≤ R.
(a)Find the value of C, the volume of the cap, when R = 13 and k = 4. Give your answer in terms of π.
(a)
(b)A different sphere has a radius of 8 units and a cap with a height of y units, with 0 < y ≤ 8. The volume of this cap is (36πy) units3.
(i)Use this, and the expression for C above, to show that: y3(24 − y) = 36
(b)(i)
(ii)Multiply out and solve the equation y3(24 − y) = 36 to find the height of this cap.
(b)(ii)
(c)A hemisphere has a diameter of x cm. V(x), the volume of this hemisphere in cm3, is given by V(x) = π12 x3. Find the value of x when the volume of the hemisphere is 3 litres. Give your answer correct to 1 decimal place.
(c)
(e)A cone has a radius of r cm and a height of h cm. The curved surface area of the cone, S, can be written as S = πr√(r2 + h2). Rearrange this to write h in terms of S, r, and π. Give your answer in the form √(S2 − arn)br, where a, b, and n are constants.
(e)
(d)In this part, p and r are constants, with p, r ∈ ℝ and 0 < r < 0·9p. The company sells bags of plant food. The usual price of one bag is €p. In a sale, the customer can choose to pay using either Option 1 or Option 2, as follows:
• Option 1: the usual price reduced by 10%, and then reduced by a further €r.
• Option 2: the usual price reduced by €r, and then this new price reduced by 10%.
Which option (1 or 2), if either, is cheaper? Write the price for each option (1 and 2) in terms of p and r, to support your answer.
• Option 1: the usual price reduced by 10%, and then reduced by a further €r.
• Option 2: the usual price reduced by €r, and then this new price reduced by 10%.
Which option (1 or 2), if either, is cheaper? Write the price for each option (1 and 2) in terms of p and r, to support your answer.
(d)
(a)Find the two values of m ∈ ℝ for which |5 + 3m| = 11.
(a)
(b)For the real numbers h, j, and k: 1h = kj + k Express k in terms of h and j.
(b)
(c)x2 − px + 1 is a factor of x3 − 2x − 3r, where p, r ∈ ℝ and p < 0. Find the value of p and the value of r.
(c)
(a)Prove that √2 is not a rational number.
(a)
(b)t is a positive real number, with: log3 t + log9 t + log27 t + log81 t = 10 Find the value of t. Give your answer in the form 3r, where r ∈ ℚ. Hint: use the formula loga b = logc blogc a.
(b)
(c)(i) Explain what log6 m means, where m is a positive real number.
(c)(i)
(ii)m is a real number, and m > 6. What information does this give about the value of log6 m?
(c)(ii)
Ava is investigating factors of different numbers.
(a)First, she looks at numbers that can be written as powers of a prime number.
(i)List the 5 different factors of 24. You can write each one as a power of 2.
(a)(i)
(ii)Work out how many different factors 37 has.
(a)(ii)
(iii)Work out how many different factors 210 × 312 has.
(a)(iii)
(a)Find the two values of m ∈ ℤ for which the following equation in x has exactly one solution: 3x2 − mx + 3 = 0
(a)
(b)Explain why the following equation in x has no real solutions: (2x + 3)2 + 7 = 0
(b)
(c)(i) Show that x = −1 is not a solution of 3x2 + 2x + 5 = 0.
(c)(i)
(ii)Find the remainder when 3x2 + 2x + 5 is divided by x + 1. That is, find the value of c when 3x2 + 2x + 5 = (x + 1)(ax + b) + c, where a, b, c ∈ ℤ.
(c)(ii)
(b)f(x) = 2x3 − 21x2 + 40x + 63, where x ∈ ℝ.
(i)x + 1 is a factor of f(x). Find the three values of x for which f(x) = 0.
(b)(i)
(a)Given that x = −3 is a solution to |x + p| = 5, find the two values of p, where p ∈ ℤ.
(a)
(b)(x + 4) is a factor of f(x) = x3 + qx2 − 22x + 56, where x ∈ ℝ and q ∈ ℤ. Show that q = −5, and find the three roots of f(x).
(b)
The diagram shows a cuboid with dimensions x, y and z cm. The areas (cm2) of three of its faces are shown: 2√2, 4√3, and 8√6.
(a)Find the volume of the cuboid in the form a√b cm3, where a, b ∈ ℕ.
(a)
(b)(i) Given that f(x) = 3x2 + 8x − 35, where x ∈ ℝ, find the two roots of f(x) = 0.
(b)(i)
(ii)Hence or otherwise, solve the equation 32m+1 = 35 − 8(3m), where m ∈ ℝ. Give your answer in the form m = log3 p − q, where p, q ∈ ℕ.
(b)(ii)
(a)f(x) = x2 + 5x + p where x ∈ ℝ, −3 ≤ p ≤ 8, and p ∈ ℤ.
(i)Find the value of p for which x + 3 is a factor of f(x).
(a)(i)
(ii)Find the value of p for which f(x) has roots which differ by 3.
(a)(ii)
(iii)Find the two values of p for which the graph of f(x) will not cross the x-axis.
(a)(iii)
(b)Find the range of values of x for which |2x + 5| − 1 ≤ 0, where x ∈ ℝ.
(b)
The real variables y and x are related by y = 5x2.
(i)The equation y = 5x2 can be rewritten in the form log5 y = a + b log5 x. Find the value of a and the value of b.
(b)(i)
(ii)Hence, or otherwise, find the real values of y for which log5 y = 2 + log5 (12625 x − 1).
(b)(ii)
(a)In the expansion of (2x + 1)(x2 + px + 4), where p ∈ ℕ, the coefficient of x is twice the coefficient of x2. Find the value of p.
(a)
(b)Solve the equation 32x + 1 + 25 = 23x − 1 where x ≠ −12, 13, and x ∈ ℝ.
(b)
(a)Factorise fully: 3xy − 9x + 4y − 12.
(a)
(b)g(x) = 3x ln x − 9x + 4 ln x − 12. Using your answer to part (a) or otherwise, solve g(x) = 0.
(b)
(a)(i)Given that x − √32 = √128 − 5x, find the value of x, where x ∈ ℝ. Give your answer in the form a√2, where a ∈ ℕ (working with surds).
(a)(i)
(ii)A = {√(32k2), √(50k2), √(128k2), √(98k2)}, where k ∈ ℕ. Show that the mean of set A is equal to the median of set A.
(a)(ii)
(b)Prove, using contradiction, that √2 is not a rational number.
(b)
(a)Solve the simultaneous equations: 2x + 3y − z = −4; 3x + 2y + 2z = 14; x − 3z = −13.
(a)
(b)Solve the inequality 2x − 3x + 2 ≥ 3, where x ∈ ℝ and x ≠ −2.
(b)
(a)Write the function f(x) = 2x2 − 7x − 10, where x ∈ ℝ, in the form a(x + h)2 + k, where a, h, k ∈ ℚ (completing the square).
(a)
(b)Hence, write the minimum point of f.
(b)
(c)(i)Explain why f must have two real roots.
(c)(i)
(ii)Write the roots of f(x) = 0 in the form p ± √q, where p, q ∈ ℚ.
(c)(ii)
The function f is such that f(x) = 2x3 + 5x2 − 4x − 3, where x ∈ ℝ.
(a)Show that x = −3 is a root of f(x) and find the other two roots (using the factor theorem).
(a)
(a)Find the range of values of x for which |x − 4| ≥ 2, where x ∈ ℝ (a modulus inequality).
(a)
(b)Solve the simultaneous equations: x2 + xy + 2y2 = 4; 2x + 3y = −1.
(b)
(b)Given loga 2 = p and loga 3 = q, where a > 0, write each of the following in terms of p and q (using the laws of logs):
(i)loga 83
(b)(i)
(ii)loga 9a216.
(b)(ii)
(a)(i)The lengths of the sides of a right-angled triangle are x − 1, 4x, and 5x − 9, as shown. Find the value of x (using Pythagoras to form a quadratic).
(a)(i)
(ii)Verify, with this value of x, that the lengths of the sides form a Pythagorean triple.
(a)(ii)
(a)Solve the equation x3 − 3x2 − 9x + 11 = 0. Write any irrational solution in the form a + b√c, where a, b, c ∈ ℤ.
(a)
(a)Solve the equation x = √(x + 6), x ∈ ℝ (a surd equation).
(a)
The graph of a cubic function f(x) cuts the x-axis at x = −3, x = −1 and x = 2, and the y-axis at (0, −6), as shown.
(a)Verify that f(x) can be written as f(x) = x3 + 2x2 − 5x − 6.
(a)
(b)(i)The graph of the function g(x) = −2x − 6 intersects the graph of f(x) above. Let f(x) = g(x) and solve the resulting equation to find the co-ordinates of the points where the graphs of f(x) and g(x) intersect.
(b)(i)
(ii)Draw the graph of the function g(x) = −2x − 6 on the diagram above.
(b)(ii)
Three natural numbers a, b and c, such that a2 + b2 = c2, are called a Pythagorean triple.
(i)Let a = 2n + 1, b = 2n2 + 2n and c = 2n2 + 2n + 1. Pick one natural number n and verify that the corresponding values of a, b and c form a Pythagorean triple.
(i)
(ii)Prove that a = 2n + 1, b = 2n2 + 2n and c = 2n2 + 2n + 1, where n ∈ ℕ, will always form a Pythagorean triple.
(ii)
Questions reproduced from State Examinations Commission Leaving Certificate examination papers,
© State Examinations Commission (examinations.ie). Gathered here for study use.
© State Examinations Commission (examinations.ie). Gathered here for study use.