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 $a$.