DP Mathematics HL Questionbank

• Syllabus

2.6

Sub sections and their related questions

Solving quadratic equations using the quadratic formula.

• 12N.2.hl.TZ0.2: Show that the quadratic equation \({x^2} - (5 - k)x - (k + 2) = 0\) has two distinct real...
• 09M.1.hl.TZ1.10: The diagram below shows a solid with volume V , obtained from a cube with edge \(a > 1\) when...
• 13M.2.hl.TZ1.9a: Prove that the equation \(3{x^2} + 2kx + k - 1 = 0\) has two distinct real roots for all values...
• 13M.2.hl.TZ1.9b: Find the value of k for which the two roots of the equation are closest together.
• 13M.1.hl.TZ2.7b: Find the minimum value of \(\left| {{z_1} + \alpha{z_2}} \right|\), where \(\alpha \in \mathbb{R}\).
• 11M.2.hl.TZ1.5a: Write down the quadratic expression \(2{x^2} + x - 3\) as the product of two linear factors.
• 13N.2.hl.TZ0.7b: find the smallest possible positive value of \(\theta \).
• 15M.2.hl.TZ1.12d: Hence express \(\sin 72^\circ \) in the form \(\frac{{\sqrt {a + b\sqrt c } }}{d}\) where...

Use of the discriminant \(\Delta = {b^2} - 4ac\) to determine the nature of the roots.

• 12M.2.hl.TZ1.1: Given that the graph of \(y = {x^3} - 6{x^2} + kx - 4\) has exactly one point at which...
• 12N.2.hl.TZ0.2: Show that the quadratic equation \({x^2} - (5 - k)x - (k + 2) = 0\) has two distinct real...
• 11N.1.hl.TZ0.9a: Find the set of values of y for which this equation has real roots.
• 11M.2.hl.TZ1.4b: Show that for this value of \(a\) there is a unique real solution to the equation \(f (x) = 0\) .
• 13N.2.hl.TZ0.8b: Find the value of \(\theta \) for which the shaded area is equal to half that of the unshaded...
• 14N.2.hl.TZ0.6: Consider \(p(x) = 3{x^3} + ax + 5a,\;\;\;a \in \mathbb{R}\). The polynomial \(p(x)\) leaves a...
• 17N.2.hl.TZ0.7: In the quadratic equation \(7{x^2} - 8x + p = 0,{\text{ }}(p \in \mathbb{Q})\), one root is three...

Solving polynomial equations both graphically and algebraically.

• 12M.1.hl.TZ2.12B.c: Find the two complex roots of the equation \(f(x) = 0\) in Cartesian form.
• 10M.2.hl.TZ2.11: The function f is defined...
• 09N.2.hl.TZ0.1: Find the values of \(k\) such that the equation \({x^3} + {x^2} - x + 2 = k\) has three distinct...
• 13N.2.hl.TZ0.7b: find the smallest possible positive value of \(\theta \).
• 16M.2.hl.TZ1.11a: Find the solutions of \(f(x) > 0\).
• 16M.2.hl.TZ2.12b: (i)     By forming a quadratic equation in \({{\text{e}}^x}\), solve the equation \(h(x) = k\),...
• 16N.1.hl.TZ0.5: The quadratic equation \({x^2} - 2kx + (k - 1) = 0\) has roots \(\alpha \) and \(\beta \) such...
• 16N.2.hl.TZ0.11b: Find the values of the constants \(a\) and \(b\).
• 16N.2.hl.TZ0.11c: Deduce that...
• 17N.2.hl.TZ0.7: In the quadratic equation \(7{x^2} - 8x + p = 0,{\text{ }}(p \in \mathbb{Q})\), one root is three...
• 18M.1.hl.TZ2.2b: Solve the equation \(\frac{x}{2} + 1 = \left| {x - 2} \right|\).

Sum and product of the roots of polynomial equations.

• SPNone.1.hl.TZ0.2a: Write down the numerical value of the sum and of the product of the roots of this equation.
• 14M.1.hl.TZ1.4: The equation \(5{x^3} + 48{x^2} + 100x + 2 = a\) has roots \({r_1}\), \({r_2}\) and...
• 14M.1.hl.TZ2.4: The roots of a quadratic equation \(2{x^2} + 4x - 1 = 0\) are \(\alpha \) and \(\beta...
• 14N.1.hl.TZ0.2a: Without solving the equation, find the value of (i)     \(\alpha  + \beta \); (ii)    ...
• 14N.1.hl.TZ0.2b: Another quadratic equation \({x^2} + px + q = 0,{\text{ }}p,{\text{ }}q \in \mathbb{Z}\) has...
• 15M.1.hl.TZ1.7a: For the polynomial equation \(p(x) = 0\), state (i)     the sum of the roots; (ii)     the...
• 15M.1.hl.TZ1.7b: A new polynomial is defined by \(q(x) = p(x + 4)\). Find the sum of the roots of the equation...
• 15M.1.hl.TZ2.12a: (i)     \(p =  - (\alpha  + \beta  + \gamma )\); (ii)    ...
• 15M.1.hl.TZ2.12b: It is now given that \(p =  - 6\) and \(q = 18\) for parts (b) and (c) below. (i)     In the...
• 15M.1.hl.TZ2.12c: In another case the three roots \(\alpha ,{\text{ }}\beta ,{\text{ }}\gamma \) form a geometric...
• 15N.1.hl.TZ0.10a: the degree of the polynomial;
• 15N.1.hl.TZ0.10b: the value of \({a_0}\).
• 16M.2.hl.TZ1.11a: Find the solutions of \(f(x) > 0\).
• 16M.2.hl.TZ2.12b: (i)     By forming a quadratic equation in \({{\text{e}}^x}\), solve the equation \(h(x) = k\),...
• 16N.1.hl.TZ0.5: The quadratic equation \({x^2} - 2kx + (k - 1) = 0\) has roots \(\alpha \) and \(\beta \) such...
• 16N.2.hl.TZ0.11b: Find the values of the constants \(a\) and \(b\).
• 16N.2.hl.TZ0.11c: Deduce that...
• 17N.2.hl.TZ0.7: In the quadratic equation \(7{x^2} - 8x + p = 0,{\text{ }}(p \in \mathbb{Q})\), one root is three...
• 18M.2.hl.TZ1.2: The equation \({x^2} - 5x - 7 = 0\) has roots \(\alpha \) and \(\beta \). The equation...
• 18M.2.hl.TZ2.10d.i: Find \(\alpha \) and β in terms of \(k\).
• 18M.2.hl.TZ2.10d.ii: Show that \(\alpha \) + β < −2.

Solution of \({a^x} = b\) using logarithms.

• 13N.1.hl.TZ0.9: Solve the following equations: (a)     \({\log _2}(x - 2) = {\log _4}({x^2} - 6x + 12)\); (b)  ...

Use of technology to solve a variety of equations, including those where there is no appropriate analytic approach.

• 12M.2.hl.TZ1.11a: Write down the coordinates of the minimum point on the graph of f .
• 12M.2.hl.TZ1.11c: Find the coordinates of the point, on \(y = f(x)\) , where the gradient of the graph is 3.
• 08N.2.hl.TZ0.6: (a)     Sketch the curve \(y = \left| {\ln x} \right| - \left| {\cos x} \right| - 0.1\) ,...
• 11M.2.hl.TZ2.7a: Find the coordinates of P.
• 11M.2.hl.TZ2.7b: Calculate the size of the acute angle between the tangents to the two graphs at the point P.
• 13M.2.hl.TZ2.4b: Determine the value of m if \(\int_0^m {x{{\sec }^2}x{\text{d}}x = 0.5} \), where m > 0.
• 13M.2.hl.TZ2.11b: A number of distinct points are marked on the circumference of a circle, forming a polygon....
• 11N.2.hl.TZ0.8b: find the value of x such that \(f(x) = {f^{ - 1}}(x)\).
• 14M.2.hl.TZ2.3a: Find the value of \({x_{\text{A}}}\) and the value of \({x_{\text{B}}}\).
• 13N.2.hl.TZ0.8b: Find the value of \(\theta \) for which the shaded area is equal to half that of the unshaded...
• 14N.2.hl.TZ0.13c: Once empty, water is pumped back into the container at a rate of...
• 15M.2.hl.TZ2.3b: Hence, or otherwise, solve the equation \({(x - 5)^2} - 2\left| {x - 5} \right| - 9 = 0\).
• 15M.2.hl.TZ2.8c: Find the value of \(a\).