Algebra, logarithmic and exponential functions — AS & A Level Mathematics (9709) questions by topic

Use logarithms to solve the inequality 4^x < 0.05. Give your answer in the form x < a, where the value of a is correct to 3 significant figures. [2]

Sketch, on a single diagram, the graphs of y = 3e^(−2x) and y = sec x for values of x such that 0 <= x < (1/2)π. [2]

The polynomial p(x) is defined by p(x) = ax^3 + bx^2 − ax − 24, where a and b are constants. It is given that (2x − 3) is a factor of p(x) and that the remainder is −15 when p(x)…

Sketch on the same diagram the graphs of y = 2x − 9 and y = 4x − 5.

The diagram shows parts of the curves with equations y = 4e^(−2x) and y = 1 + 0.5 sin 3x. Point P is a point of intersection of the curves, and the shaded region is bounded by the…

The polynomial p(x) is defined by p(x) = ax^4 + bx^3 + 13x^2 − 35x + 15, where a and b are constants. It is given that (2x − 1) and (x − 3) are factors of p(x). Find the values of…

A curve has equation (x^2 − 3) ln y + 6x = 14. Show that there is no point on the curve at which the y-coordinate is e^(−1).

Sketch on the same diagram the graphs of y = 2x − 9 and y = 4x − 5.

The diagram shows parts of the curves with equations y = 4e^(−2x) and y = 1 + 0.5 sin 3x. Point P is a point of intersection of the curves, and the shaded region is bounded by the…

The polynomial p(x) is defined by p(x) = ax^4 + bx^3 + 13x^2 − 35x + 15, where a and b are constants. It is given that (2x − 1) and (x − 3) are factors of p(x). Find the values of…

A curve has equation (x^2 − 3) ln y + 6x = 14. Show that there is no point on the curve at which the y-coordinate is e^(−1).

Sketch on the same diagram the graphs of y = 2x − 9 and y = 4x − 5.

The diagram shows parts of the curves with equations y = 4e^(−2x) and y = 1 + 0.5 sin 3x. Point P is a point of intersection of the curves, and the shaded region is bounded by the…

The polynomial p(x) is defined by p(x) = ax^4 + bx^3 + 13x^2 − 35x + 15, where a and b are constants. It is given that (2x − 1) and (x − 3) are factors of p(x). Find the values of…

Sketch the graph of y = 2x − 3 .

It is given that 2 ln p + ln(p − 1) − (1/2) ln(q + 1) = 3. Find q in terms of p.

The polynomial 3x³ + pax² + 7a²x + qa³ is denoted by f(x), where p, q and a are constants and a ≠ 0. When f(x) is divided by (x + 2a) the remainder is −22a³. When f(x) is divided…

Find the quotient and remainder when x³ + 5x² − 2x − 15 is divided by x² − 3.

Solve the equation (e^x + 2e^(-x)) / (e^x - 3) = 4. Give your answer correct to 3 decimal places. [5]

Find the quotient and remainder when x^2 is divided by 1 + 4x^2. [2]

Sketch the graph of y = 3x − 2a , where a is a positive constant.

Solve the equation 2 ln(2x + 3) − ln(2x + 5) = ln(3x).

Let f(x) = (3a − 5x) / ((3a + 2x)(2a − x)), where a is a positive constant. Express f(x) in partial fractions.

Solve the equation 3^(4−2x) = 5(6^(x−1)). Give your answer correct to 3 significant figures. [4]

Express (12x^2 + 55x − 2) / ((3x − 2)(x + 6)) in partial fractions. [5]

Sketch on the same diagram the graphs of y = 3x − 8 and y = 5 − x. [2]

The polynomial p(x) is defined by p(x) = 9x³ + 6x² + 12x + k, where k is a constant. Find the quotient when p(x) is divided by (3x + 2) and show that the remainder is (k − 8). [3]

Solve the inequality 5x + 7 2x − 3 .

Use logarithms to solve the equation 6^(2x−1) = 5e^(3x+2). Give your answer correct to 4 significant figures.

The polynomial p(x) is defined by p(x) = 9x³ + 18x² + 5x + 4. Find the quotient when p(x) is divided by (3x + 2), and show that the remainder is 6.

Solve the inequality 5x + 7 2x − 3 .

Use logarithms to solve the equation 6^(2x−1) = 5e^(3x+2). Give your answer correct to 4 significant figures.

The polynomial p(x) is defined by p(x) = 9x³ + 18x² + 5x + 4. Find the quotient when p(x) is divided by (3x + 2), and show that the remainder is 6.

Solve the equation ln(x − 5) = 7 − ln x. Give your answer correct to 2 decimal places.

The variables x and y satisfy the equation a^y = bx, where a and b are constants. The graph of y against ln x is a straight line passing through the points (0.336, 1.00) and…

Sketch the graph of y = x - 2a , where a is a positive constant. [1]

Express (6x^2 - 9x - 16) / (2x^2 - 5x - 12) in partial fractions. [5]

The variables x and y satisfy the equation a^(2y - 1) = b^(x - y), where a and b are constants. Show that the graph of y against x is a straight line. [3]

Solve the equation 8^(3 − 6x) = 4 × 5^(−2x). Give your answer correct to 3 decimal places.

The variables x and y satisfy the equation ky = e^(cx), where k and c are constants. The graph of ln y against x is a straight line passing through the points (2.80, 0.372) and…

Express (6x² − 2x + 2) / ((x − 1)(2x + 1)) in partial fractions.

Let f(x) = 8x³ + 54x² − 17x − 21. Show that x + 7 is a factor of f(x).

The variables x and y satisfy the equation a^(2y) = e^(3x+k), where a and k are constants. The graph of y against x is a straight line. Use logarithms to show that the gradient of…

Solve the inequality x - 7 4x + 3. [4]

The polynomial p(x) is defined by p(x) = ax³ − ax² − 15x + 18, where a is a constant. It is given that (x + 2) is a factor of p(x). Find the value of a. [2]

Use logarithms to show that the equation 5^(8y) = 6^(7x) can be expressed in the form y = kx. Give the value of the constant k correct to 3 significant figures.

Sketch the graphs of y = 1 + e^(2x) and y = x − 4 on the same diagram.

The polynomial p(x) is defined by p(x) = ax³ + bx² − ax + 8, where a and b are constants. It is given that (x + 2) is a factor of p(x), and that the remainder is 24 when p(x) is…

The variables x and y satisfy the equation a^(2y) = e^(3x+k), where a and k are constants. The graph of y against x is a straight line. Use logarithms to show that the gradient of…

Solve the inequality x − 7 4x + 3.

The polynomial p(x) is defined by p(x) = ax³ − ax² − 15x + 18, where a is a constant. It is given that (x + 2) is a factor of p(x). Find the value of a.

The polynomial 4x³ + ax² + 5x + b, where a and b are constants, is denoted by p(x). It is given that (2x + 1) is a factor of p(x). When p(x) is divided by (x − 4) the remainder is…

Let f(x) = (5x² + 8x + 5) / ((1 + 2x)(2 + x²)). Express f(x) in partial fractions.

Solve the equation 5^x = 5^(x+2) − 10. Give your answer correct to 3 decimal places.

The variables x and y satisfy the equation ay = b^x, where a and b are constants. The graph of ln y against x is a straight line passing through the points (0.50, 2.24) and (3.40,…

The number of bacteria in a population, P, at time t hours is modelled by the equation P = ae^{kt}, where a and k are constants. The graph of ln P against t, shown in the diagram,…

Let f(x) = 7a² / ((a − 2x)(3a + x)), where a is a positive constant. Express f(x) in partial fractions.

Find the quotient and remainder when x⁴ + 16 is divided by x² + 4.

Use logarithms to solve the equation 12^x = 3^(2x+1). Give your answer correct to 3 significant figures. [4]

The polynomial p(x) is defined by p(x) = 2x^3 + 3x^2 + kx − 30, where k is a constant. It is given that (x − 3) is a factor of p(x). Find the value of k. [2]

The variables x and y satisfy the equation y = Ae^((A − B)x), where A and B are constants. The graph of ln y against x is a straight line passing through the points (0.4, 3.6) and…

The variables x and y satisfy the equation y = Ae^((A−B)x), where A and B are constants. The graph of ln y against x is a straight line passing through the points (0.4, 3.6) and…

Solve the equation 3e^(2x) − 4e^(−2x) = 5. Give the answer correct to 3 decimal places.

Sketch the graph of y = 2x + 3 .

Let f(x) = (3 − 3x²) / ((2x + 1)(x + 2)²). Express f(x) in partial fractions.

The polynomial x³ + 5x² + 31x + 75 is denoted by p(x). Show that (x + 3) is a factor of p(x).

Solve the inequality 5x − 3 < 2 3x − 7 .

Solve the equation ln(2x² − 3) = 2 ln x − ln 2, giving your answer in an exact form.

Let f(x) = (2x² + 17x − 17) / ((1 + 2x)(2 − x)²). Express f(x) in partial fractions.

Solve the equation ln(x + 5) = 5 + ln x. Give your answer correct to 3 decimal places.

Find the quotient and remainder when 2x⁴ − 27 is divided by x² + x + 3.

Let f(x) = (21 − 8x − 2x²) / ((1 + 2x)(3 − x)²). Express f(x) in partial fractions.

Verify the identity (2x − 1)(4x² + 2x − 1) ≡ 8x³ − 4x + 1.

Sketch, on the same diagram, the graphs of y = 3x − 5 and y = 2x + 7. [2]

The polynomial p(x) is defined by p(x) = 6x^3 + ax^2 + bx − 20, where a and b are constants. It is given that (x + 2) is a factor of p(x) and that the remainder is −11 when p(x)…

When the polynomial ax^3 + 4ax^2 − 7x − 5 is divided by (x + 2), the remainder is 33. Find the value of the constant a.

Sketch, on the same diagram, the graphs of y = 3 − x and y = 9 − 2x.

Find the quotient when 6x^3 − 5x^2 − 24x − 4 is divided by (2x + 1), and show that the remainder is 6.

Sketch, on the same diagram, the graphs of y = 3x − 5 and y = 2x + 7.

The polynomial p(x) is defined by p(x) = 6x³ + ax² + bx − 20, where a and b are constants. It is given that (x + 2) is a factor of p(x) and that the remainder is −11 when p(x) is…

The variables x and y are related by the equation y = ab^x, where a and b are constants. The diagram shows the result of plotting ln y against x for two pairs of values of x and…

Let f(x) = (24x + 13) / ((1 − 2x)(2 + x)²). Express f(x) in partial fractions.

Sketch the graph of y = 4x − 2 .

The polynomial 2x³ + ax² − 11x + b is denoted by p(x). It is given that p(x) is divisible by (2x − 1) and that when p(x) is divided by (x + 1) the remainder is 12. Find the values…

Find the set of values of x satisfying the inequality 2^(x+1) − 2 < 0.5, giving your answer to 3 significant figures.

The polynomial 2x³ + ax² + bx + 6, where a and b are constants, is denoted by p(x). When p(x) is divided by (x + 2) the remainder is −38 and when p(x) is divided by (2x − 1) the…

Let f(x) = (17x² − 7x + 16) / ((2 + 3x²)(2 − x)). Express f(x) in partial fractions.