1.06g Equations with exponentials: solve a^x = b

2 Using the substitution \(u = 3 ^ { x }\), solve the equation \(3 ^ { x } + 3 ^ { 2 x } = 3 ^ { 3 x }\) giving your answer correct to 3 significant figures.

2

1. Solve the inequality \(| 3 x - 5 | < | x + 3 |\).
2. Hence find the greatest integer \(n\) satisfying the inequality \(\left| 3 ^ { 0.1 n + 1 } - 5 \right| < \left| 3 ^ { 0.1 n } + 3 \right|\).

2

1. Solve the equation \(| 4 + 2 x | = | 3 - 5 x |\).
2. Hence solve the equation \(\left| 4 + 2 e ^ { 3 y } \right| = \left| 3 - 5 e ^ { 3 y } \right|\), giving the answer correct to 3 significant figures.

3 It is given that \(k\) is a positive constant. Solve the equation \(2 \ln x = \ln ( 3 k + x ) + \ln ( 2 k - x )\), expressing \(x\) in terms of \(k\).

5 Given that \(\int _ { 0 } ^ { a } 6 \mathrm { e } ^ { 2 x + 1 } \mathrm {~d} x = 65\), find the value of \(a\) correct to 3 decimal places.

3

1. Express \(9 ^ { x }\) in terms of \(y\), where \(y = 3 ^ { x }\).
2. Hence solve the equation $$2 \left( 9 ^ { x } \right) - 7 \left( 3 ^ { x } \right) + 3 = 0 ,$$ expressing your answers for \(x\) in terms of logarithms where appropriate.

2 Solve the equation \(x ^ { 3.9 } = 11 x ^ { 3.2 }\), where \(x \neq 0\).

1 Solve the inequality \(( 0.8 ) ^ { x } < 0.5\).

2

1. Express \(4 ^ { x }\) in terms of \(y\), where \(y = 2 ^ { x }\).
2. Hence find the values of \(x\) that satisfy the equation $$3 \left( 4 ^ { x } \right) - 10 \left( 2 ^ { x } \right) + 3 = 0 ,$$ giving your answers correct to 2 decimal places.

2 Solve the equation \(\ln \left( 3 - x ^ { 2 } \right) = 2 \ln x\), giving your answer correct to 3 significant figures.

2 Use logarithms to solve the equation \(5 ^ { x } = 2 ^ { 2 x + 1 }\), giving your answer correct to 3 significant figures.

4 Solve the equation \(3 ^ { 2 x } - 7 \left( 3 ^ { x } \right) + 10 = 0\), giving your answers correct to 3 significant figures.

2 Use logarithms to solve the equation \(4 ^ { x + 1 } = 5 ^ { 2 x - 3 }\), giving your answer correct to 3 significant figures.

3 Solve the equation \(2 \ln ( x + 3 ) - \ln x = \ln ( 2 x - 2 )\).

2 Use logarithms to solve the equation \(5 ^ { x } = 3 ^ { 2 x - 1 }\), giving your answer correct to 3 significant figures.

5

1. Given that ( \(x + 2\) ) and ( \(x + 3\) ) are factors of $$5 x ^ { 3 } + a x ^ { 2 } + b$$ find the values of the constants \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, factorise $$5 x ^ { 3 } + a x ^ { 2 } + b$$ completely, and hence solve the equation $$5 ^ { 3 y + 1 } + a \times 5 ^ { 2 y } + b = 0$$ giving any answers correct to 3 significant figures.

4

1. Find the value of \(x\) satisfying the equation \(2 \ln ( x - 4 ) - \ln x = \ln 2\).
2. Use logarithms to find the smallest integer satisfying the inequality $$1.4 ^ { y } > 10 ^ { 10 }$$

1 Use logarithms to solve the equation $$5 ^ { x + 3 } = 7 ^ { x - 1 }$$ giving the answer correct to 3 significant figures.

1

1. Solve the equation \(| 3 x - 2 | = 5\).
2. Hence, using logarithms, solve the equation \(\left| 3 \times 5 ^ { y } - 2 \right| = 5\), giving the answer correct to 3 significant figures.

2

1. Solve the equation \(| 2 x + 3 | = | x + 8 |\).
2. Hence, using logarithms, solve the equation \(\left| 2 ^ { y + 1 } + 3 \right| = \left| 2 ^ { y } + 8 \right|\). Give the answer correct to 3 significant figures.

2

1. Given that \(\frac { 1 + 4 ^ { y } } { 3 + 2 ^ { y } } = 5\), find the value of \(2 ^ { y }\).
2. Use logarithms to find the value of \(y\) correct to 3 significant figures.

4 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 4 x ^ { 3 } + a x ^ { 2 } + a x + 4$$ where \(a\) is a constant.

1. Use the factor theorem to show that ( \(x + 1\) ) is a factor of \(\mathrm { p } ( x )\) for all values of \(a\).
2. Given that the remainder is - 42 when \(\mathrm { p } ( x )\) is divided by ( \(x - 2\) ), find the value of \(a\).
3. When \(a\) has the value found in part (ii), factorise \(\mathrm { p } \left( x ^ { 2 } \right)\) completely.

1 Candidates answer on the Question Paper.
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The total number of marks for this paper is 50. This document consists of 11 printed pages and 1 blank page. 1 Solve the equation \(\ln ( 3 x + 1 ) - \ln ( x + 2 ) = 1\), giving your answer in terms of e.

3 It is given that the variable \(x\) is such that $$1.3 ^ { 2 x } < 80 \quad \text { and } \quad | 3 x - 1 | > | 3 x - 10 | .$$ Find the set of possible values of \(x\), giving your answer in the form \(a < x < b\) where the constants \(a\) and \(b\) are correct to 3 significant figures.

4

1. Find \(\int \frac { 4 + \sin ^ { 2 } \theta } { 1 - \sin ^ { 2 } \theta } \mathrm {~d} \theta\).
2. Given that \(\int _ { 0 } ^ { a } \frac { 2 } { 3 x + 1 } \mathrm {~d} x = \ln 16\), find the value of the positive constant \(a\).