-
Q1
5 marksa. Without using a calculator or four‑figure tables, simplify (3√5 − √2)/(2√5 + 3√2), leaving your answer with a rational denominator.
Add solution / teacher notes
-
Q2
6 marksa. Simplify (3√2 + 1)/(√3 + √6) without using four‑figure tables or a calculator.
Add solution / teacher notes
-
Q3
4 marksb. Without using a calculator or a four‑figure table, simplify (5 + √3)/(√7 + √5), leaving your answer with a rational denominator.
Add solution / teacher notes
-
Q4
4 marksa. Given that x = 9, y = 4 and z = 3, evaluate √((5y² + y⁰)/(x z⁴)).
Add solution / teacher notes
-
Q5
4 marksa. Simplify (√(n³) − √(9n))/√n in its simplest form.
Add solution / teacher notes
-
Q6
4 marksb. Without using a calculator or four‑figure tables, simplify √98 · (15√6) / (−√2) to its simplest form.
Add solution / teacher notes
-
Q7
3 marksa. Simplify (2 − √7)(2 + √7) without using a calculator or four‑figure tables.
Add solution / teacher notes
-
Q8
4 marksa. Simplify (√2 + √3)(√8 − √12) without using a calculator and four‑figure table.
Add solution / teacher notes
-
Q9
5 marksb. Express (√3 + 1)/(√3 − 1) with a rational denominator in its simplest form.
Add solution / teacher notes
-
Q10
4 marksb. Express 1/(3√2 − 3) with a rational denominator in its simplest form.
Add solution / teacher notes
Section 2 · Exponential and Logarithmic Functions
-
Q1
4 marksGiven that logac = 0.4475, evaluate loga(a²/c).
Add solution / teacher notes
-
Q2
4 marksWithout using four-figure tables or a calculator evaluate loga2 − loga6 + loga3.
Add solution / teacher notes
-
Q3
5 marksGiven that logy864 − logy6 = 2, find the value of y.
Add solution / teacher notes
-
Q4
7 marksSolve the equation 22a − 5(2a) + 4 = 0.
Add solution / teacher notes
-
Q5
3 marksGiven that log105 = 0.6990, without using a calculator or tables, evaluate log102.
Add solution / teacher notes
-
Q6
5 marksSolve the equation log7343 = 2x − 5.
Add solution / teacher notes
-
Q7
7 marksSimplify (4x × 8x−1) / 32x.
Add solution / teacher notes
-
Q8
4 marksEvaluate 2 logaa + 3 loga1 + log2√2.
Add solution / teacher notes
-
Q9
5 marksSolve for x if (2x)² − 9(2x) + 8 = 0.
Add solution / teacher notes
-
Q10
5 marksSolve the equation log9(27k) = k + 1.
Add solution / teacher notes
-
Q11
3 marksCalculate the value of x if 10x = 0.01.
Add solution / teacher notes
5. Progressions
- The second term of an arithmetic progression (AP) is 7 and the 10th term is 39. Calculate the common difference of the AP. (3 marks)
- Given that 1st term of an AP is -8 and the 11th term is 22, calculate the 7th term. (5 marks)
- The first term of an arithmetic progression is 5 and the last term is 43. If the sum of the terms is 480, calculate the number of terms. (3 marks)
- The last term in an arithmetic progression (AP) 2, 5, 8, … is 296. Calculate the number of terms. (4 marks)
- The second term of a geometric progression is -6 and the fourth term is -54. Calculate the common ratio, given that it is negative. (5 marks)
- The first term of a geometric progression is 81, and the common ratio is 1/3, calculate the fourth term. (4 marks)
- The sum of the first n terms of a geometric progression, GP, is 2(n+1) − 1. Calculate:
(i) the first term of the GP
(ii) the common ratio of the GP. (7 marks) - The ratio of the 2nd term to the 7th term of an arithmetic progression is 1:3 and their sum is 20. Calculate the sum of the first 10 terms of the progression. (10 marks)
- The sum of terms of an arithmetic progression is (5n2+3n)/2, calculate the first two terms of the arithmetic progression. (5 marks)
- Figure 7 shows the display of new types of bricks laid down on a table. On the 1st table there are 2 bricks, on the 2nd table there are 4 bricks, on the 3rd table there are 8 bricks and so on. If on the nth table there are 1024 bricks, calculate the value of n. (6 marks)
6. Variations
- The quantity q is partly constant and partly varies as p3. When q = 5, p = 1 and when q = 26, p = 2. Find q when p = -2. (8 marks)
- A quantity a is the sum of two parts, one of which is constant while the other varies inversely as the square of n. When n = 1, a = -1 and when n = 2, a = 2. Find the value of a when n = 4. (8 marks)
- The time, T, taken for students to finish discussing a topic in a group is partly constant and partly varies as the number of students in the group. A group of 5 students takes 120 minutes while a group of 9 students takes 180 minutes. Calculate the time a group of 12 students can take to finish discussing the topic. (7 marks)
- A variable y is partly constant and partly varies as x. When x = 5, y = 4 and when x = 9, y = 8. Find y when x = 1. (8 marks)
- x varies directly as y and inversely as the square of n. If x = 15, y = 24 and n = 4, calculate the value of n when x = 8 and y = 20. (7 marks)
- The bus fare per passenger (F) is partly constant and partly inversely proportional to the number (n) of passengers. The fare per passenger for 40 passengers is K240, and for 50 passengers is K200. Calculate the fare per passenger when there are 100 passengers. (9 marks)
- A quantity P varies directly as q and inversely as (q2 + 1). When P = 1, q = 2. Express P in terms of q only. (5 marks)
- The cost (C) for an international call from Malawi to Europe partly varies inversely as time (t), and partly as the square of the time. A one-minute call costs K120 and a two-minute call costs K200. Find the cost of a five-minute call. (8 marks)
- A quantity q is the difference between two parts. The first part is constant and the second part varies inversely as the square of p. If q = 1 when p = 2 and q = 6 when p = 3, find the positive value of p when q = 9. (10 marks)
- Given that y is partly constant and partly varies as x, and y = -3 when x = 3 and y = 22 when x = -2, calculate the value of y when x = 2. (5 marks)