DP Mathematics SL Questionbank

• Syllabus

Topic 2 - Functions and equations

Sub sections and their related questions

2.1

• 12M.1.sl.TZ2.2a: Find \({f^{ - 1}}(x)\) . 
• 12M.1.sl.TZ2.2b: Find \((f \circ g)(1)\) . 
• 08N.1.sl.TZ0.4b: Write down the range of \({f^{ - 1}}\) .
• 08N.1.sl.TZ0.4c: Find \({f^{ - 1}}(x)\) .
• 08M.1.sl.TZ1.7a: Find \({f^{ - 1}}(x)\) .
• 08M.1.sl.TZ1.7b: Let \(g(x) = {{\rm{e}}^x}\) . Find \((g \circ f)(x)\) , giving your answer in the form...
• 08M.1.sl.TZ2.5b(i) and (ii): Let \(g(x) = f(x + 3)\) . (i)     Find \(g( - 3)\) . (ii)    Describe fully the transformation...
• 12M.1.sl.TZ1.9a: Find p .
• 12M.1.sl.TZ1.9b: The graph of f is reflected in the line \(y = x\) to give the graph of g . (i)     Write down...
• 12M.1.sl.TZ1.9c: The graph of \(f\) is reflected in the line \(y = x\) to give the graph of \(g\) . Find \(g(x)\) .
• 10N.1.sl.TZ0.9a: Find \((f \circ g)(x)\) .
• 10N.1.sl.TZ0.9b: The vector \(\left( {\begin{array}{*{20}{c}}3\\{ - 1}\end{array}} \right)\) translates the graph...
• 10N.1.sl.TZ0.9c: The vector \(\left( {\begin{array}{*{20}{c}}3\\{ - 1}\end{array}} \right)\) translates the graph...
• 10N.1.sl.TZ0.9d: The vector \(\left( {\begin{array}{*{20}{c}}3\\{ - 1}\end{array}} \right)\) translates the graph...
• 10M.1.sl.TZ1.7a: Show that \({f^{ - 1}}(x) = {3^{2x}}\) .
• 10M.1.sl.TZ1.7b: Write down the range of \({f^{ - 1}}\) .
• 10M.1.sl.TZ1.7c: Let \(g(x) = {\log _3}x\) , for \(x > 0\) . Find the value of \(({f^{ - 1}} \circ g)(2)\) ,...
• 10M.1.sl.TZ2.4a: Find \(f\left( {\frac{\pi }{2}} \right)\) .
• 10M.1.sl.TZ2.4b: Find \((g \circ f)\left( {\frac{\pi }{2}} \right)\) .
• 10M.1.sl.TZ2.4c: Given that \((g \circ f)(x)\) can be written as \(\cos (kx)\) , find the value of k,...
• 09N.1.sl.TZ0.1a: (i)     Find \(g(0)\) . (ii)    Find \((f \circ g)(0)\) .
• 09N.1.sl.TZ0.1b: Find \({f^{ - 1}}(x)\) .
• 09N.1.sl.TZ0.7a: Given that \({f^{ - 1}}(1) = 8\) , find the value of \(k\) .
• 09N.1.sl.TZ0.7b: Find \({f^{ - 1}}\left( {\frac{2}{3}} \right)\) .
• 09M.1.sl.TZ1.6a: (i)     Show that \({f^{ - 1}}(x) = \ln x - 3\) . (ii)    Write down the domain of \({f^{ - 1}}\) .
• 09M.2.sl.TZ1.3a: Find an expression for \(h(x)\) .
• 09M.2.sl.TZ1.10e: Write down the range of values for the gradient of \(f\) .
• 09M.1.sl.TZ2.1a: Find \({g^{ - 1}}(x)\) .
• 09M.1.sl.TZ2.1b: Find \((f \circ g)(4)\) .
• SPNone.2.sl.TZ0.9a: Find \({h^{ - 1}}(x)\) .
• SPNone.2.sl.TZ0.9b(i), (ii) and (iii): (i)     Sketch the graph of h for \( - 4 \le x \le 4\) and \( - 5 \le y \le 8\) , including any...
• SPNone.2.sl.TZ0.10a(i), (ii) and (iii): Jane thinks that the function \(f(t) = - 0.25{t^3} - 2.32{t^2} + 1.93t + 106\) is a suitable...
• 11N.2.sl.TZ0.1a: Find \({f^{ - 1}}(x)\) .
• 11N.2.sl.TZ0.1c: Find \((f \circ g)(3.5)\) .
• 11M.1.sl.TZ1.1a: Find \((g \circ f)(x)\) .
• 11M.1.sl.TZ1.1b: Write down \({g^{ - 1}}(x)\) .
• 11M.1.sl.TZ1.1c: Find \((f \circ {g^{ - 1}})(5)\) .
• 11M.2.sl.TZ2.1a: Find \(h(x)\) .
• 11M.2.sl.TZ2.1b: Find \({h^{ - 1}}(x)\) .
• 13M.1.sl.TZ1.5a: Find \({f^{ - 1}}(2)\) .
• 13M.1.sl.TZ1.5b: Let \(g\) be a function such that \({g^{ - 1}}\) exists for all real numbers. Given that...
• 13M.1.sl.TZ1.10a: Find the value of \(f(0)\) .
• 13M.2.sl.TZ1.9a: Write down \(f(0)\) .
• 13M.2.sl.TZ1.9c: Find the range of \(f\) .
• 13M.1.sl.TZ2.1a: Find \({f^{ - 1}}(x)\) .
• 13M.1.sl.TZ2.1b: Find \((f \circ g)(1)\) .
• 13M.1.sl.TZ2.4a.i: Write down the value of \(f(2)\).
• 13M.1.sl.TZ2.10a: Write down the value of \(g(3)\) , of \(f'(3)\) , and of \(h''(2)\) .
• 13M.1.sl.TZ2.10c: find the \(y\)-coordinate of P.
• 11N.2.sl.TZ0.1b: Find \((f \circ g)(x)\) .
• 13M.1.sl.TZ2.4a.ii: Write down the value of \({f^{ - 1}}( - 1)\) .
• 14M.1.sl.TZ2.3a(ii): Write down the value of  \({f^{ - 1}}(1)\).
• 14M.1.sl.TZ2.3a(i): Write down the value of \(f( - 3)\).
• 14M.1.sl.TZ2.3b: Find the domain of \({f^{ - 1}}\).
• 13N.1.sl.TZ0.8a: Find \({f^{ - 1}}(x)\).
• 13N.1.sl.TZ0.8b: Show that \(\left( {g \circ {f^{ - 1}}} \right)(x) = \frac{5}{{x + 2}}\).
• 13N.1.sl.TZ0.8e: Given that \({h^{ - 1}}(a) = 3\), find the value of \(a\).
• 14N.2.sl.TZ0.1a: Find \((f \circ g)(x)\).
• 15M.1.sl.TZ1.4a: Find \({f^{ - 1}}( - 1)\).
• 15M.1.sl.TZ1.4b: Find \((f \circ f)( - 1)\).
• 15M.2.sl.TZ1.10c: Let \(g(x) = \ln \left( {f(x)} \right)\) and \(f(2) = 3\). Find \(g'(2)\).
• 15N.1.sl.TZ0.5a: Find \({f^{ - 1}}(x)\).
• 15N.1.sl.TZ0.5b: Let \(g\) be a function so that \((f \circ g)(x) = 8{x^6}\). Find \(g(x)\).
• 16M.1.sl.TZ1.1a: Write down \(g(2)\).
• 16M.1.sl.TZ1.1b: Find \((f \circ g)(x)\).
• 16M.1.sl.TZ1.1c: Find \({f^{ - 1}}(x)\).
• 16M.1.sl.TZ2.6a: Write \(h(x)\) in the form \(a\sin (bx)\), where \(a,{\text{ }}b \in \mathbb{Z}\).
• 16M.1.sl.TZ2.6b: Hence find the range of \(h\).
• 16N.2.sl.TZ0.1a: Find \(f(8)\).
• 16N.2.sl.TZ0.1b: Find \((g \circ f)(x)\).
• 16N.2.sl.TZ0.1c: Solve \((g \circ f)(x) = 0\).
• 17M.1.sl.TZ1.2a: Find \({f^{ - 1}}(x)\).
• 17M.1.sl.TZ1.2b: Find \((f \circ g)(7)\).
• 17M.2.sl.TZ1.10b.i: Find \(\int_{0.111}^{3.31} {\left( {h(x) - x} \right){\text{d}}x} \).
• 17M.2.sl.TZ1.10b.ii: Hence, find the area of the region enclosed by the graphs of \(h\) and \({h^{ - 1}}\).
• 17M.2.sl.TZ1.10c: Let \(d\) be the vertical distance from a point on the graph of \(h\) to the line \(y = x\)....
• 17M.2.sl.TZ2.3a: Write down the range of \(f\).
• 17M.2.sl.TZ2.3b: On the grid above, sketch the graph of \(g\).
• 17M.2.sl.TZ2.3c: Write down the domain of \(g\).
• 17M.2.sl.TZ2.6a: Show that \((f \circ g)(x) = {x^4} - 4{x^2} + 3\).
• 17M.2.sl.TZ2.6b: On the following grid, sketch the graph of \((f \circ g)(x)\), for...
• 17M.2.sl.TZ2.6c: The equation \((f \circ g)(x) = k\) has exactly two solutions, for...
• 17N.1.sl.TZ0.3a: Write down the range of \(f\).
• 17N.1.sl.TZ0.3b.i: Write down \(f(2)\);
• 17N.1.sl.TZ0.3b.ii: Write down \({f^{ - 1}}(2)\).
• 17N.1.sl.TZ0.3c: On the grid, sketch the graph of \({f^{ - 1}}\).
• 17N.1.sl.TZ0.5a: Find \((g \circ f)(x)\).
• 17N.1.sl.TZ0.5b: Given that \(\mathop {\lim }\limits_{x \to  + \infty } (g \circ f)(x) =  - 3\), find the value of...
• 18M.1.sl.TZ1.1a: Write down f (14).
• 18M.1.sl.TZ1.1b: Find \(\left( {g \circ f} \right)\) (14).
• 18M.1.sl.TZ1.1c: Find g−1(x).
• 18M.1.sl.TZ1.3a.i: Write down the value of f (0).
• 18M.1.sl.TZ1.3a.ii: Write down the value of f −1 (1).
• 18M.1.sl.TZ1.3b: Write down the range of f −1.
• 18M.1.sl.TZ1.3c: On the grid above, sketch the graph of f −1.
• 18M.1.sl.TZ2.10a.i: Write down \(f'\left( 2 \right)\).
• 18M.1.sl.TZ2.10a.ii: Find \(f\left( 2 \right)\).
• 18M.1.sl.TZ2.10b: Show that the graph of g has a gradient of 6 at P.
• 18M.1.sl.TZ2.10c: Let L2 be the tangent to the graph of g at P. L1 intersects L2 at the point Q. Find the...

2.2

• 12N.2.sl.TZ0.3a: Write down the x-coordinate of P.
• 12N.2.sl.TZ0.3b: Write down the gradient of the curve at P.
• 12N.2.sl.TZ0.3c: Find the equation of the normal to the curve at P, giving your equation in the form \(y = ax + b\) .
• 12N.2.sl.TZ0.5a(i) and (ii): Write down the value of (i)     \(a\) ; (ii)    \(c\) .
• 12N.2.sl.TZ0.5b: Find the value of b .
• 12N.2.sl.TZ0.5c: Find the x-coordinate of R.
• 12N.2.sl.TZ0.7a: On the grid below, sketch the graph of \(s\) .
• 12N.2.sl.TZ0.7b: Find the maximum velocity of the particle.
• 12N.2.sl.TZ0.9a: Sketch the graph of f , for \( - 1 \le x \le 5\) .
• 12N.2.sl.TZ0.9b: This function can also be written as \(f(x) = {(x - p)^2} - 3\) . Write down the value of p .
• 12N.2.sl.TZ0.9c: The graph of g is obtained by reflecting the graph of f in the x-axis, followed by a translation...
• 12N.2.sl.TZ0.9d: The graph of g is obtained by reflecting the graph of f in the x-axis, followed by a translation...
• 12N.2.sl.TZ0.9e: The graph of \(g\) is obtained by reflecting the graph of \(f\) in the x-axis, followed by a...
• 12M.1.sl.TZ2.10a: Use the quotient rule to show that \(f'(x) = \frac{{2{x^2} - 2}}{{{{( - 2{x^2} + 5x - 2)}^2}}}\) .
• 12M.1.sl.TZ2.10b: Hence find the coordinates of B.
• 12M.1.sl.TZ2.10c: Given that the line \(y = k\) does not meet the graph of f , find the possible values of k .
• 12M.2.sl.TZ2.2a: Find \(f'(x)\) .
• 12M.2.sl.TZ2.2b: On the grid below, sketch the graph of \(f'(x)\) .
• 12M.2.sl.TZ2.9a: Show that \(8a + 4b + c = 9\) .
• 12M.2.sl.TZ2.9b: The graph of f has a local minimum at \((1{\text{, }}4)\) . Find two other equations in a , b...
• 12M.2.sl.TZ2.10a: Use the cosine rule to show that \({\rm{PQ}} = 2r\sin \theta \) .
• 12M.2.sl.TZ2.9c: Find the value of a , of b and of c .
• 12M.2.sl.TZ2.10b: Let l be the length of the arc PRQ . Given that \(1.3{\rm{PQ}} - l = 0\) , find the value of...
• 12M.2.sl.TZ2.10c(i) and (ii): Consider the function \(f(\theta ) = 2.6\sin \theta  - 2\theta \) , for...
• 12M.2.sl.TZ2.10d: Use the graph of f to find the values of \(\theta \) for which \(l < 1.3{\rm{PQ}}\) .
• 08N.1.sl.TZ0.4a: On the same diagram, sketch the graph of \({f^{ - 1}}\) .
• 08N.2.sl.TZ0.4a: Sketch the graph of f on the following set of axes.
• 08N.2.sl.TZ0.4b: The graph of f intersects the x-axis when \(x = a\) , \(a \ne 0\) . Write down the value of a.
• 08M.2.sl.TZ1.4a: On the grid below, sketch the graph of \(y = f(x)\) .
• 08M.2.sl.TZ2.9b: Write down the equation of the horizontal asymptote.
• 12M.1.sl.TZ1.9a: Find p .
• 12M.1.sl.TZ1.9b: The graph of f is reflected in the line \(y = x\) to give the graph of g . (i)     Write down...
• 12M.1.sl.TZ1.9c: The graph of \(f\) is reflected in the line \(y = x\) to give the graph of \(g\) . Find \(g(x)\) .
• 12M.2.sl.TZ1.4a: Find k .
• 12M.2.sl.TZ1.4b: The shaded region is rotated \(360^\circ \) about the x-axis. Let V be the volume of the solid...
• 12M.2.sl.TZ1.4c: The shaded region is rotated \(360^\circ \) about the x-axis. Let V be the volume of the solid...
• 12M.2.sl.TZ1.10a(i) and (ii): Find the distance between the ships (i)     at 13:00; (ii)    at 14:00.
• 12M.2.sl.TZ1.10b: Let \(s(t)\) be the distance between the ships t hours after noon, for \(0 \le t \le 4\) . Show...
• 12M.2.sl.TZ1.10c: Sketch the graph of \(s(t)\) .
• 12M.2.sl.TZ1.10d: Due to poor weather, the captain of ship A can only see another ship if they are less than 8 km...
• 10M.1.sl.TZ2.10a(i) and (ii): Solve for \(0 \le x < 2\pi \) (i)     \(6 + 6\sin x = 6\) ; (ii)    \(6 + 6\sin x = 0\) .
• 10M.1.sl.TZ2.10b: Write down the exact value of the x-intercept of f , for \(0 \le x < 2\pi \) .
• 10M.1.sl.TZ2.10c: The area of the shaded region is k . Find the value of k , giving your answer in terms of \(\pi \) .
• 10M.1.sl.TZ2.10d: Let \(g(x) = 6 + 6\sin \left( {x - \frac{\pi }{2}} \right)\) . The graph of f is transformed to...
• 10M.1.sl.TZ2.10e: Let \(g(x) = 6 + 6\sin \left( {x - \frac{\pi }{2}} \right)\) . The graph of f is transformed to...
• 09N.1.sl.TZ0.9a: (i)     Find the coordinates of A. (ii)    Show that \(f'(x) = 0\) at A.
• 09N.1.sl.TZ0.9d: Write down the range of \(f\) .
• 09N.1.sl.TZ0.10b: Point A is the x-intercept of L . Find the x-coordinate of A.
• 09N.2.sl.TZ0.7: The fencing used for side AB costs \(\$ 11\) per metre. The fencing for the other three sides...
• 09N.2.sl.TZ0.9a: On the same diagram, sketch the graphs of f and g .
• 09M.2.sl.TZ2.10a: Sketch the graph of f .
• 09M.2.sl.TZ2.10b: Write down (i)     the amplitude; (ii)    the period; (iii)   the x-intercept that lies...
• 10N.2.sl.TZ0.2a: On the grid below, sketch the graph of v , clearly indicating the maximum point.
• 10N.2.sl.TZ0.2b(i) and (ii): (i)     Write down an expression for d . (ii)    Hence, write down the value of d .
• 10N.2.sl.TZ0.7a: There are two points of inflexion on the graph of f . Write down the x-coordinates of these points.
• 10N.2.sl.TZ0.7b: Let \(g(x) = f''(x)\) . Explain why the graph of g has no points of inflexion.
• 10N.2.sl.TZ0.8a: Find the value of a and of b .
• 10N.2.sl.TZ0.8b: The graph of f has a maximum value when \(x = c\) . Find the value of c .
• 10N.2.sl.TZ0.8c: The region under the graph of f from \(x = 0\) to \(x = c\) is rotated \({360^ \circ }\) about...
• 10N.2.sl.TZ0.8d: Let R be the region enclosed by the curve, the x-axis and the line \(x = c\) , between \(x = a\)...
• 10M.2.sl.TZ1.3a: Find \(f'(x)\) .
• 10M.2.sl.TZ1.3b: On the grid below, sketch the graph of \(y = f'(x)\) .
• 10M.2.sl.TZ1.5a(i), (ii) and (iii): Find the value of (i)     p ; (ii)    q ; (iii)   r.
• 10M.2.sl.TZ2.5a: On the diagram above, sketch the graph of g.
• 10M.2.sl.TZ2.5b: Solve \(f(x) = g(x)\) .
• 10M.2.sl.TZ2.5c: Write down the set of values of x such that \(f(x) > g(x)\) .
• 10M.2.sl.TZ2.6a: Write down the x-coordinate of A.
• 10M.2.sl.TZ2.6b(i) and (ii): Find the value of (i)     p ; (ii)    q .
• 10M.2.sl.TZ2.6c: Find \(\int_p^q {f(x){\rm{d}}x} \) . Explain why this is not the area of the shaded region.
• 10M.2.sl.TZ2.10a(i) and (ii): Let P and Q be points on the curve of f where the tangent to the graph of f is parallel to the...
• 10M.2.sl.TZ2.10b: Let \(g(x) = {x^3}\ln (4 - {x^2})\) , for \( - 2 < x < 2\) . Show that...
• 10M.2.sl.TZ2.10c: Let \(g(x) = {x^3}\ln (4 - {x^2})\) , for \( - 2 < x < 2\) . Sketch the graph of \(g'\) .
• 10M.2.sl.TZ2.10d: Let \(g(x) = {x^3}\ln (4 - {x^2})\) , for \( - 2 < x < 2\) . Consider \(g'(x) = w\) ....
• SPNone.2.sl.TZ0.2a: Find the x-intercepts of the graph of f .
• SPNone.2.sl.TZ0.2b: On the grid below, sketch the graph of f .
• SPNone.2.sl.TZ0.9a: Find \({h^{ - 1}}(x)\) .
• SPNone.2.sl.TZ0.9b(i), (ii) and (iii): (i)     Sketch the graph of h for \( - 4 \le x \le 4\) and \( - 5 \le y \le 8\) , including any...
• SPNone.2.sl.TZ0.10c(i) and (ii): (i)     On graph paper, using a scale of 1 cm to 1 second, and 1 cm to 10 m, plot the data given...
• 11N.2.sl.TZ0.10a: Sketch the graph of f .
• 11N.2.sl.TZ0.10b(i) and (ii): (i)     Write down the x-coordinate of the maximum point on the graph of f . (ii)    Write down...
• 11N.2.sl.TZ0.10c: Show that \(f'(x) = \frac{{20 - 6x}}{{{{\rm{e}}^{0.3x}}}}\) .
• 11N.2.sl.TZ0.10d: Find the interval where the rate of change of f is increasing.
• 11M.1.sl.TZ1.7a: Find the value of k .
• 11M.1.sl.TZ1.7b: The line \(y = p\) intersects the graph of f . Find all possible values of p .
• 11M.1.sl.TZ1.10a: Write down the velocity of the particle when \(t = 0\) .
• 11M.1.sl.TZ1.10b(i) and (ii): When \(t = k\) , the acceleration is zero. (i)     Show that \(k = \frac{\pi }{4}\) . (ii)   ...
• 11M.1.sl.TZ1.10c: When \(t < \frac{\pi }{4}\) , \(\frac{{{\rm{d}}v}}{{{\rm{d}}t}} > 0\) and when...
• 11M.1.sl.TZ1.10d(i) and (ii): Let d be the distance travelled by the particle for \(0 \le t \le 1\) . (i)     Write down an...
• 11M.2.sl.TZ1.10a: Show that \(f(x) = {\log _3}2x\) .
• 11M.2.sl.TZ1.10b: Find the value of \(f(0.5)\) and of \(f(4.5)\) .
• 11M.2.sl.TZ1.10c(i), (ii) and (iii): The function f can also be written in the form \(f(x) = \frac{{\ln ax}}{{\ln b}}\) . (i)    ...
• 11M.2.sl.TZ1.10d: Write down the value of \({f^{ - 1}}(0)\) .
• 11M.2.sl.TZ1.10e: The point A lies on the graph of f . At A, \(x = 4.5\) . On your diagram, sketch the graph of...
• 11M.2.sl.TZ2.2a: Sketch the graph of g on the following set of axes.
• 11M.2.sl.TZ2.2b: Hence find the value of x for which \(g(x) = - 1\) .
• 11M.2.sl.TZ2.3a: Write down the number of terms in this expansion.
• 11M.2.sl.TZ2.3b: Find the term containing \({x^2}\) .
• 13M.1.sl.TZ1.10d: There is a point of inflexion on the graph of \(f\) at \(x = \sqrt[4]{3}\)...
• 13M.2.sl.TZ1.5a: On the grid below, sketch the graph of \(v\) .
• 13M.2.sl.TZ1.5b.i: Find the total distance travelled by the particle in the first five seconds.
• 13M.1.sl.TZ2.4a.i: Write down the value of \(f(2)\).
• 13M.1.sl.TZ2.4b: Sketch the graph of \({f^{ - 1}}\) on the grid below.
• 13M.2.sl.TZ2.10a: (i)     Sketch the graphs of \(f\) and \(g\) on the same axes. (ii)     Find the area of \(R\) .
• 13M.1.sl.TZ2.4a.ii: Write down the value of \({f^{ - 1}}( - 1)\) .
• 14M.2.sl.TZ1.9a: Sketch the graph of \(f\).
• 14M.2.sl.TZ1.9b: Find the values of \(x\) where the function is decreasing.
• 14M.1.sl.TZ2.3c: On the grid above, sketch the graph of \({f^{ - 1}}\).
• 14M.2.sl.TZ2.2a: Find the \(x\)-coordinate of \({\text{A}}\) and of \({\text{B}}\).
• 13N.1.sl.TZ0.8c(i): Find the \(y\)-intercept of the graph of \(h\).
• 13N.1.sl.TZ0.8d(i): For the graph of \({h^{ - 1}}\), write down the \(x\)-intercept;
• 13N.1.sl.TZ0.8d(ii): For the graph of \({h^{ - 1}}\), write down the equation of the vertical asymptote.
• 13N.2.sl.TZ0.5a: On the grid below, sketch the graph of \(v\), for \(0 \leqslant t \leqslant 4\).
• 15M.2.sl.TZ1.4a: For the graph of \(f\) (i)     find the \(x\)-intercept; (ii)     write down the equation of...
• 15M.2.sl.TZ1.5a: On the following grid, sketch the graph of \(G\).
• 15M.2.sl.TZ1.5b: Robin and Pat are planning a wedding banquet. The cost per guest, \(G\) dollars, is modelled by...
• 15M.2.sl.TZ1.7a: Find the coordinates of the local minimum point.
• 15M.2.sl.TZ1.7b: The graph of \(f\) is translated to the graph of \(g\) by the vector...
• 15M.2.sl.TZ1.10a: The graph of \(f\) has a local maximum point when \(x = p\). State the value of \(p\), and...
• 15M.2.sl.TZ2.5a: On the following grid, sketch the graph of \(f\).
• 15M.2.sl.TZ2.7: Let \(f(x) = k{x^2} + kx\) and \(g(x) = x - 0.8\). The graphs of \(f\) and \(g\) intersect at two...
• 15N.2.sl.TZ0.3a: Find the equation of the vertical asymptote to the graph of \(f\).
• 15N.2.sl.TZ0.3b: Find the \(x\)-intercept of the graph of \(f\).
• 16M.1.sl.TZ1.3a: (i)     Write down the amplitude of \(f\). (ii)     Find the period of \(f\).
• 16M.1.sl.TZ1.3b: On the following grid sketch the graph of \(f\).
• 16M.1.sl.TZ1.5a: Show that the two zeros are 3 and \( - 6\).
• 16M.2.sl.TZ1.9a: Find the displacement of P from O after 5 seconds.
• 16M.2.sl.TZ1.9b: Find when P is first at rest.
• 16M.2.sl.TZ1.9c: Write down the number of times P changes direction.
• 16M.2.sl.TZ1.9d: Find the acceleration of P after 3 seconds.
• 16M.2.sl.TZ1.9e: Find the maximum speed of P.
• 16M.1.sl.TZ2.1a: Write down the value of \(h\) and of \(k\).
• 16M.1.sl.TZ2.1b: Write down the value of \(a\) and of \(b\).
• 16M.1.sl.TZ2.1c: Find the \(y\)-intercept.
• 16M.2.sl.TZ2.9a: Write down the equation of the horizontal asymptote of the graph of \(f\).
• 16M.2.sl.TZ2.9b: Find \(f'(x)\).
• 16M.2.sl.TZ2.9c: Write down the value of \(b\).
• 16M.2.sl.TZ2.9d: Given that \(g'(1) =  - e\), find the value of \(a\).
• 16M.2.sl.TZ2.9e: There is a value of \(x\), for \(1 < x < 4\), for which the graphs of \(f\) and \(g\) have...
• 16N.2.sl.TZ0.2a: Find the coordinates of A.
• 16N.2.sl.TZ0.2b: (i)     sketch the graph of \(f\), clearly indicating the point A; (ii)    sketch the tangent to...
• 17M.1.sl.TZ1.9a: Find the value of \(p\).
• 17M.1.sl.TZ1.9b: Find the value of \(a\).
• 17M.1.sl.TZ1.9c: The line \(y = kx - 5\) is a tangent to the curve of \(f\). Find the values of \(k\).
• 17M.1.sl.TZ2.10a.i: Write down \(f'(x)\).
• 17M.1.sl.TZ2.10a.ii: Find the gradient of \(L\).
• 17M.1.sl.TZ2.10b: Show that the \(x\)-coordinate of B is \( - \frac{k}{2}\).
• 17M.1.sl.TZ2.10c: Find the area of triangle ABC, giving your answer in terms of \(k\).
• 17M.1.sl.TZ2.10d: Given that the area of triangle ABC is \(p\) times the area of \(R\), find the value of \(p\).
• 17M.2.sl.TZ1.3a: Find the \(y\)-intercept.
• 17M.2.sl.TZ1.3b: Find the equation of the vertical asymptote.
• 17M.2.sl.TZ1.3c: Find the minimum value of \(f(x)\) for \(x > 2\).
• 17M.2.sl.TZ2.6a: Show that \((f \circ g)(x) = {x^4} - 4{x^2} + 3\).
• 17M.2.sl.TZ2.6b: On the following grid, sketch the graph of \((f \circ g)(x)\), for...
• 17M.2.sl.TZ2.6c: The equation \((f \circ g)(x) = k\) has exactly two solutions, for...
• 17M.2.sl.TZ2.8a: Find the value of \(p\).
• 17M.2.sl.TZ2.8b.i: Write down the coordinates of A.
• 17M.2.sl.TZ2.8b.ii: Write down the rate of change of \(f\) at A.
• 17M.2.sl.TZ2.8c.i: Find the coordinates of B.
• 17M.2.sl.TZ2.8d: Let \(R\) be the region enclosed by the graph of \(f\) , the \(x\)-axis, the line \(x = b\) and...
• 17N.1.sl.TZ0.3a: Write down the range of \(f\).
• 17N.1.sl.TZ0.3b.i: Write down \(f(2)\);
• 17N.1.sl.TZ0.3b.ii: Write down \({f^{ - 1}}(2)\).
• 17N.1.sl.TZ0.3c: On the grid, sketch the graph of \({f^{ - 1}}\).
• 17N.2.sl.TZ0.2a: Find the \(x\)-intercept of the graph of \(f\).
• 17N.2.sl.TZ0.2b: The graph of \(f\) has a maximum at the point A. Write down the coordinates of A.
• 17N.2.sl.TZ0.2c: On the following grid, sketch the graph of \(f\).
• 18M.1.sl.TZ1.3a.i: Write down the value of f (0).
• 18M.1.sl.TZ1.3a.ii: Write down the value of f −1 (1).
• 18M.1.sl.TZ1.3b: Write down the range of f −1.
• 18M.1.sl.TZ1.3c: On the grid above, sketch the graph of f −1.
• 18M.2.sl.TZ1.4a: Write down the coordinates of the vertex of the graph of g.
• 18M.2.sl.TZ1.4b: On the grid above, sketch the graph of g for −2 ≤ x ≤ 4.
• 18M.2.sl.TZ1.4c: Find the area of the region enclosed by the graphs of f and g.
• 18M.1.sl.TZ2.5a: On the same axes, sketch the graph of \(f\left( { - x} \right)\).
• 18M.1.sl.TZ2.5b: Another function, \(g\), can be written in the form...
• 18M.1.sl.TZ2.10a.i: Write down \(f'\left( 2 \right)\).
• 18M.1.sl.TZ2.10a.ii: Find \(f\left( 2 \right)\).
• 18M.1.sl.TZ2.10b: Show that the graph of g has a gradient of 6 at P.
• 18M.1.sl.TZ2.10c: Let L2 be the tangent to the graph of g at P. L1 intersects L2 at the point Q. Find the...
• 18M.2.sl.TZ2.3a: Find the x-intercept of the graph of \(f\).
• 18M.2.sl.TZ2.3b: The region enclosed by the graph of \(f\), the y-axis and the x-axis is rotated 360° about the...
• 18M.2.sl.TZ2.7a: The line x = 3 is a vertical asymptote to the graph of f. Find the value of c.
• 18M.2.sl.TZ2.7b: Write down the equation of the horizontal asymptote to the graph of f.
• 18M.2.sl.TZ2.7c: The line y = k, where \(k \in \mathbb{R}\) intersects the graph...

2.3

• 12N.2.sl.TZ0.9a: Sketch the graph of f , for \( - 1 \le x \le 5\) .
• 12N.2.sl.TZ0.9b: This function can also be written as \(f(x) = {(x - p)^2} - 3\) . Write down the value of p .
• 12N.2.sl.TZ0.9c: The graph of g is obtained by reflecting the graph of f in the x-axis, followed by a translation...
• 12N.2.sl.TZ0.9d: The graph of g is obtained by reflecting the graph of f in the x-axis, followed by a translation...
• 12N.2.sl.TZ0.9e: The graph of \(g\) is obtained by reflecting the graph of \(f\) in the x-axis, followed by a...
• 12M.1.sl.TZ2.5a: Sketch the graph of \(f( - x)\) on the grid below.
• 12M.1.sl.TZ2.5b: The graph of f is transformed to obtain the graph of g . The graph of g is shown below. The...
• 08N.1.sl.TZ0.10b: The transformation P is given by a horizontal stretch of a scale factor of \(\frac{1}{2}\) ,...
• 08N.1.sl.TZ0.10c: The graph of g is the image of the graph of f under P. Find \(g(t)\) in the form...
• 08N.1.sl.TZ0.10d: The graph of g is the image of the graph of f under P. Give a full geometric description of the...
• 08M.1.sl.TZ1.9d: Let \(g(x) = {x^2}\) . The graph of f may be obtained from the graph of g by the two...
• 08M.1.sl.TZ2.5a: On the same diagram sketch the graph of \(y = - f(x)\) .
• 08M.1.sl.TZ2.5b(i) and (ii): Let \(g(x) = f(x + 3)\) . (i)     Find \(g( - 3)\) . (ii)    Describe fully the transformation...
• 10N.1.sl.TZ0.9a: Find \((f \circ g)(x)\) .
• 10N.1.sl.TZ0.9b: The vector \(\left( {\begin{array}{*{20}{c}}3\\{ - 1}\end{array}} \right)\) translates the graph...
• 10N.1.sl.TZ0.9c: The vector \(\left( {\begin{array}{*{20}{c}}3\\{ - 1}\end{array}} \right)\) translates the graph...
• 10N.1.sl.TZ0.9d: The vector \(\left( {\begin{array}{*{20}{c}}3\\{ - 1}\end{array}} \right)\) translates the graph...
• 10M.1.sl.TZ1.8a: Find the coordinates of A.
• 10M.1.sl.TZ1.8b(i), (ii) and (iii): Write down the coordinates of (i)     the image of B after reflection in the y-axis; (ii)   ...
• 09N.1.sl.TZ0.4a: Let \(h(x) = f( - x)\) . Sketch the graph of \(h\) on the grid below.
• 09N.1.sl.TZ0.4b: Let \(g(x) = \frac{1}{2}f(x - 1)\) . The point \({\text{A}}(3{\text{, }}2)\) on the graph of...
• 09M.1.sl.TZ1.5a: The graph of g can be obtained from the graph of f using two transformations. Give a full...
• 09M.1.sl.TZ1.5b: The graph of g is translated by the...
• 09M.1.sl.TZ1.10a: Show that \(f( - x) = - f(x)\) .
• 09M.1.sl.TZ1.10c: It is given that \(\int {f(x){\rm{d}}x = \frac{a}{2}} \ln ({x^2} + 1) + C\) . (i)     Find the...
• 09M.2.sl.TZ2.2a: On the same grid sketch the graph of \(y = f( - x)\) .
• 09M.2.sl.TZ2.2b: Complete the following table.
• 09M.2.sl.TZ2.2c: Give a full geometric description of the transformation that gives the image in Diagram A.
• SPNone.1.sl.TZ0.8d: Let \(g(x) = {x^2}\) . The graph of f may be obtained from the graph of g by the following two...
• 11N.2.sl.TZ0.7a: Find \(v(t)\) , giving your answer in the form \(a{(t - b)^2} + c\) .
• 11N.2.sl.TZ0.7b: A particle moves along a straight line so that its velocity in ms−1 , at time t seconds, is given...
• 11M.2.sl.TZ1.2a: Write down the coordinates of the vertex of the graph of g .
• 11M.2.sl.TZ1.2b: Express g in the form \(g(x) = 3{(x - p)^2} + q\) .
• 11M.2.sl.TZ1.2c: The graph of h is the reflection of the graph of g in the x-axis. Write down the coordinates of...
• 11M.1.sl.TZ2.5a: Express \(g(x)\) in the form \(f(x) + \ln a\) , where \(a \in {{\mathbb{Z}}^ + }\) .
• 11M.1.sl.TZ2.5b: The graph of g is a transformation of the graph of f . Give a full geometric description of this...
• 14M.1.sl.TZ2.8e: The graph of \(f\) has its vertex on the \(x\)-axis. The graph of a function \(g\) is obtained...
• 13M.2.sl.TZ1.6a: The graph of \(f\) is mapped to the graph of \(g\) under the following transformations: vertical...
• 13M.2.sl.TZ1.6b: Let \(h(x) = - g(3x)\) . The point A(\(6\), \(5\)) on the graph of \(g\) is mapped to the point...
• 15M.1.sl.TZ1.4c: On the same diagram, sketch the graph of \(y = f( - x)\).
• 15M.2.sl.TZ1.7b: The graph of \(f\) is translated to the graph of \(g\) by the vector...
• 15M.2.sl.TZ2.5b: The graph of \(f\) is translated by the vector...
• 15N.1.sl.TZ0.8c: Let \(g(x) =  - {(x - 3)^2} + 1\). The graph of \(g\) is obtained by a reflection of the graph of...
• 16M.1.sl.TZ1.9a: Find the \(x\)-coordinate of P.
• 16M.1.sl.TZ1.9b: Find \(f(x)\), expressing your answer as a single logarithm.
• 16M.1.sl.TZ1.9c: The graph of \(f\) is transformed by a vertical stretch with scale factor \(\frac{1}{{\ln 3}}\)....
• 16N.2.sl.TZ0.4a: Find the value of \(p\) and of \(q\).
• 17M.2.sl.TZ1.10a.i: Write down the value of \(q\);
• 17M.2.sl.TZ1.10a.ii: Write down the value of \(h\);
• 17M.2.sl.TZ1.10a.iii: Write down the value of \(k\).
• 17M.2.sl.TZ1.10b.i: Find \(\int_{0.111}^{3.31} {\left( {h(x) - x} \right){\text{d}}x} \).
• 17M.2.sl.TZ1.10b.ii: Hence, find the area of the region enclosed by the graphs of \(h\) and \({h^{ - 1}}\).
• 17M.2.sl.TZ1.10c: Let \(d\) be the vertical distance from a point on the graph of \(h\) to the line \(y = x\)....
• 17M.2.sl.TZ2.3a: Write down the range of \(f\).
• 17M.2.sl.TZ2.3b: On the grid above, sketch the graph of \(g\).
• 17M.2.sl.TZ2.3c: Write down the domain of \(g\).
• 18M.1.sl.TZ2.5a: On the same axes, sketch the graph of \(f\left( { - x} \right)\).
• 18M.1.sl.TZ2.5b: Another function, \(g\), can be written in the form...

2.4

• 12M.1.sl.TZ2.8a(i) and (ii): The function can be written in the form \(f(x) = a{(x - h)^2} + k\) . (i)     Write down the...
• 12M.1.sl.TZ2.8b: Find \(f(x)\)  , giving your answer in the form \(A{x^2} + Bx + C\) .
• 12M.1.sl.TZ2.8c: Calculate the area enclosed by the graph of f , the x-axis, and the lines \(x = 2\) and \(x = 4\) .
• 08N.2.sl.TZ0.1a: Express \(f(x)\) in the form \(f(x) = 2{(x - h)^2} + k\) .
• 08N.2.sl.TZ0.1b: Write down the equation of the axis of symmetry of the graph of f .
• 08N.2.sl.TZ0.1c: Express \(f(x)\) in the form \(f(x) = 2(x - p)(x - q)\) .
• 08M.1.sl.TZ1.9b(i), (ii), (iii) and (iv): For the graph of f (i)     write down the coordinates of the vertex; (ii)    write down the...
• 08M.1.sl.TZ1.9c: Hence sketch the graph of f .
• 08M.1.sl.TZ2.2a: Find both x-intercepts.
• 08M.1.sl.TZ2.2b: Find the x-coordinate of the vertex.
• 12M.2.sl.TZ1.2a(i) and (ii): (i)     Write down the coordinates of the vertex. (ii)    Hence or otherwise, express the...
• 12M.2.sl.TZ1.2b: Solve the equation \(f(x) = 0\) .
• 10N.1.sl.TZ0.9a: Find \((f \circ g)(x)\) .
• 10N.1.sl.TZ0.9b: The vector \(\left( {\begin{array}{*{20}{c}}3\\{ - 1}\end{array}} \right)\) translates the graph...
• 10N.1.sl.TZ0.9c: The vector \(\left( {\begin{array}{*{20}{c}}3\\{ - 1}\end{array}} \right)\) translates the graph...
• 10N.1.sl.TZ0.9d: The vector \(\left( {\begin{array}{*{20}{c}}3\\{ - 1}\end{array}} \right)\) translates the graph...
• 10M.1.sl.TZ1.1a: Find the x-intercepts of the graph.
• 10M.1.sl.TZ1.1b(i) and (ii): (i)     Write down the equation of the axis of symmetry. (ii)    Find the y-coordinate of the...
• 10M.1.sl.TZ2.1a: Write down the value of q and of r.
• 10M.1.sl.TZ2.1b: Write down the equation of the axis of symmetry.
• 10M.1.sl.TZ2.1c: Find the value of p.
• 09N.2.sl.TZ0.9c: Consider the graph of g . Write down (i)     the two x-intercepts; (ii)    the equation of the...
• 09M.2.sl.TZ1.10e: Write down the range of values for the gradient of \(f\) .
• SPNone.1.sl.TZ0.8a: Show that \(f(x) = 3{x^2} + 6x - 9\) .
• SPNone.1.sl.TZ0.8b(i), (ii) and (iii): For the graph of f (i)     write down the coordinates of the vertex; (ii)    write down the...
• SPNone.1.sl.TZ0.8c: Hence sketch the graph of f .
• 11N.1.sl.TZ0.1a: Write down the equation of the axis of symmetry.
• 11N.1.sl.TZ0.1b: The function f can be written in the form \(f(x) = a{(x - h)^2} + k\) . Write down the value of...
• 11N.1.sl.TZ0.1c: The function f can be written in the form \(f(x) = a{(x - h)^2} + k\) . Find a .
• 11M.2.sl.TZ1.2a: Write down the coordinates of the vertex of the graph of g .
• 11M.2.sl.TZ1.2b: Express g in the form \(g(x) = 3{(x - p)^2} + q\) .
• 11M.2.sl.TZ1.2c: The graph of h is the reflection of the graph of g in the x-axis. Write down the coordinates of...
• 11M.1.sl.TZ2.9a: Write down \(f(x)\) in the form \(f(x) = - 10(x - p)(x - q)\) .
• 11M.1.sl.TZ2.9b: Find another expression for \(f(x)\) in the form \(f(x) = - 10{(x - h)^2} + k\) .
• 11M.1.sl.TZ2.9c: Show that \(f(x)\) can also be written in the form \(f(x) = 240 + 20x - 10{x^2}\) .
• 11M.1.sl.TZ2.9d(i) and (ii): A particle moves along a straight line so that its velocity, \(v{\text{ m}}{{\text{s}}^{ - 1}}\)...
• 13M.1.sl.TZ2.9b: Find \(g(4)\) .
• 13M.1.sl.TZ2.9c: (i)     Write down the value of \(h\) . (ii)     Find the value of \(a\) .
• 14M.1.sl.TZ1.1a: Write down the value of \(h\) and of \(k\).  
• 14M.1.sl.TZ1.1b: Find the value of \(a\).
• 14M.1.sl.TZ2.8b: The graph of \(f\)has its vertex on the \(x\)-axis. Find the coordinates of the vertex of the...
• 14M.1.sl.TZ2.8d(i): The graph of \(f\) has its vertex on the \(x\)-axis. The function can be written in the form...
• 14M.1.sl.TZ2.8d(ii): The graph of \(f\) has its vertex on the \(x\)-axis. The function can be written in the form...
• 14M.1.sl.TZ2.8d(iii): The graph of \(f\) has its vertex on the \(x\)-axis. The function can be written in the form...
• 14M.2.sl.TZ2.2a: Find the \(x\)-coordinate of \({\text{A}}\) and of \({\text{B}}\).
• 13N.2.sl.TZ0.2a: Find the \(x\)-intercepts of the graph of \(f\).
• 13M.1.sl.TZ1.2a: Write down the \(x\)-intercepts of the graph of \(f\) .
• 13M.1.sl.TZ1.2b: Find the coordinates of the vertex of the graph of \(f\) .
• 14N.1.sl.TZ0.1a: Write down the \(y\)-intercept of the graph of \(f\).
• 16M.1.sl.TZ1.5b: Find the value of \(q\) and of \(r\).
• 16M.1.sl.TZ2.1a: Write down the value of \(h\) and of \(k\).
• 16M.1.sl.TZ2.1b: Write down the value of \(a\) and of \(b\).
• 16M.1.sl.TZ2.1c: Find the \(y\)-intercept.
• 16N.1.sl.TZ0.1a: Find the equation of the axis of symmetry of the graph of \(f\).
• 16N.1.sl.TZ0.1b: (i)     Write down the value of \(h\). (ii)     Find the value of \(k\).
• 16N.2.sl.TZ0.6a: Use the model to find the volume of the barrel.
• 18M.1.sl.TZ1.4a.i: The equation of the axis of symmetry is x = p. Find p.
• 18M.1.sl.TZ1.4a.ii: Hence, show that a = 2.
• 18M.1.sl.TZ1.4b: The equation of L is y = 5 . Find the value of c.
• 18M.1.sl.TZ2.6: Let \(f\left( x \right) = p{x^2} + qx - 4p\), where p ≠ 0. Find Find the number of roots for the...

2.5

• 14M.2.sl.TZ1.10a: Write down the equations of the vertical and horizontal asymptotes of the graph of \(f\).
• 14M.2.sl.TZ1.10b: The vertical and horizontal asymptotes to the graph of \(f\) intersect at the point...
• 13N.1.sl.TZ0.8c(ii): Hence, sketch the graph of \(h\).
• 14N.1.sl.TZ0.5a: Write down the value of \(q\).
• 14N.1.sl.TZ0.5b: The graph of \(f\) has a \(y\)-intercept at \((0,{\text{ }}4)\). Find the value of \(p\).
• 14N.1.sl.TZ0.5c: The graph of \(f\) has a \(y\)-intercept at \((0,{\text{ }}4)\). Write down the equation of the...
• 16N.1.sl.TZ0.7: Let \(f(x) = m - \frac{1}{x}\), for \(x \ne 0\). The line \(y = x - m\) intersects the graph of...
• 18M.2.sl.TZ2.7a: The line x = 3 is a vertical asymptote to the graph of f. Find the value of c.
• 18M.2.sl.TZ2.7b: Write down the equation of the horizontal asymptote to the graph of f.
• 18M.2.sl.TZ2.7c: The line y = k, where \(k \in \mathbb{R}\) intersects the graph...

2.6

• 08M.2.sl.TZ2.10b(i) and (ii): At the end of 2000 there were \(25600\) people in the city who used taxis. After n years the...
• 08M.2.sl.TZ2.10c(i) and (ii): Let R be the ratio of the number of people using taxis in the city to the number of taxis. The...
• 12M.1.sl.TZ1.9a: Find p .
• 12M.1.sl.TZ1.9b: The graph of f is reflected in the line \(y = x\) to give the graph of g . (i)     Write down...
• 12M.1.sl.TZ1.9c: The graph of \(f\) is reflected in the line \(y = x\) to give the graph of \(g\) . Find \(g(x)\) .
• 09M.1.sl.TZ1.6a: (i)     Show that \({f^{ - 1}}(x) = \ln x - 3\) . (ii)    Write down the domain of \({f^{ - 1}}\) .
• 10M.2.sl.TZ1.9a: Show that \(A = 10\) .
• 10M.2.sl.TZ1.9b: Given that \(f(15) = 3.49\) (correct to 3 significant figures), find the value of k.
• 10M.2.sl.TZ1.9d: Let \(g(x) = - {x^2} + 12x - 24\) . Find the area enclosed by the graphs of f and g .
• 10M.2.sl.TZ2.7a: Find the value of n when \(t = 0\) .
• 10M.2.sl.TZ2.7b: Find the rate at which n is increasing when \(t = 15\) .
• 10M.2.sl.TZ2.7c: After k minutes, the rate of increase in n is greater than \(10000\) bacteria per minute. Find...
• 11N.2.sl.TZ0.6a: Write down \(A(0)\) .
• 11N.2.sl.TZ0.6b: Find the concentration of medication left in his bloodstream after 50 minutes.
• 11N.2.sl.TZ0.6c: At 13:00, when there is no medication in Jose’s bloodstream, he takes his first dose of...
• 11M.2.sl.TZ1.10a: Show that \(f(x) = {\log _3}2x\) .
• 11M.2.sl.TZ1.10b: Find the value of \(f(0.5)\) and of \(f(4.5)\) .
• 11M.2.sl.TZ1.10c(i), (ii) and (iii): The function f can also be written in the form \(f(x) = \frac{{\ln ax}}{{\ln b}}\) . (i)    ...
• 11M.2.sl.TZ1.10d: Write down the value of \({f^{ - 1}}(0)\) .
• 11M.2.sl.TZ1.10e: The point A lies on the graph of f . At A, \(x = 4.5\) . On your diagram, sketch the graph of...
• 13M.1.sl.TZ1.7b: Find the value of \({8^{{{\log }_2}5}}\) .
• 13M.2.sl.TZ2.10a: (i)     Sketch the graphs of \(f\) and \(g\) on the same axes. (ii)     Find the area of \(R\) .
• 14M.2.sl.TZ2.8a: Find the number of bacteria in colony \({\text{A}}\) after four hours.
• 14M.2.sl.TZ2.8b: Find the number of bacteria in colony \({\text{A}}\) after four hours.
• 14M.2.sl.TZ2.8c: How long does it take for the number of bacteria in colony \({\text{A}}\) to reach \(400\)?
• 14M.2.sl.TZ2.8d: The number of bacteria in colony \({\text{B}}\) after \(t\) hours is modelled by the function...
• 14M.2.sl.TZ2.8e: The number of bacteria in colony \({\text{B}}\) after \(t\) hours is modelled by the function...
• 15N.2.sl.TZ0.3a: Find the equation of the vertical asymptote to the graph of \(f\).
• 15N.2.sl.TZ0.3b: Find the \(x\)-intercept of the graph of \(f\).
• 16M.2.sl.TZ1.7a: (i)     Find the value of \(k\). (ii)     Interpret the meaning of the value of \(k\).
• 16M.2.sl.TZ1.7b: Find the least number of whole years for which \(\frac{{{P_t}}}{{{P_0}}} < 0.75\).
• 16M.2.sl.TZ2.8a: (i)     Find the correlation coefficient. (ii)     Write down the value of \(a\) and of \(b\).
• 16M.2.sl.TZ2.8b: Use the regression equation to estimate the price of Lina’s car, giving your answer to the...
• 16M.2.sl.TZ2.8c: Calculate the price of Lina’s car after 6 years.
• 16M.2.sl.TZ2.8d: Find the year when Lina sells her car.
• 17M.2.sl.TZ1.10a.i: Write down the value of \(q\);
• 17M.2.sl.TZ1.10a.ii: Write down the value of \(h\);
• 17M.2.sl.TZ1.10a.iii: Write down the value of \(k\).
• 17M.2.sl.TZ1.10b.i: Find \(\int_{0.111}^{3.31} {\left( {h(x) - x} \right){\text{d}}x} \).
• 17M.2.sl.TZ1.10b.ii: Hence, find the area of the region enclosed by the graphs of \(h\) and \({h^{ - 1}}\).
• 17M.2.sl.TZ1.10c: Let \(d\) be the vertical distance from a point on the graph of \(h\) to the line \(y = x\)....

2.7

• 12N.1.sl.TZ0.7: The equation \({x^2} - 3x + {k^2} = 4\) has two distinct real roots. Find the possible values of k .
• 12N.2.sl.TZ0.9a: Sketch the graph of f , for \( - 1 \le x \le 5\) .
• 12N.2.sl.TZ0.9b: This function can also be written as \(f(x) = {(x - p)^2} - 3\) . Write down the value of p .
• 12N.2.sl.TZ0.9c: The graph of g is obtained by reflecting the graph of f in the x-axis, followed by a translation...
• 12N.2.sl.TZ0.9d: The graph of g is obtained by reflecting the graph of f in the x-axis, followed by a translation...
• 12M.1.sl.TZ2.6: Consider the equation \({x^2} + (k - 1)x + 1 = 0\) , where k is a real number. Find the values...
• 08M.2.sl.TZ1.4b: Solve the equation \(f(x) = 1\) .
• 08M.2.sl.TZ1.10c: There are two values of x for which the gradient of f is equal to the gradient of g. Find both...
• 08M.2.sl.TZ2.10a(i) and (ii): (i)     Find the number of taxis in the city at the end of 2005. (ii)    Find the year in...
• 08M.2.sl.TZ2.10b(i) and (ii): At the end of 2000 there were \(25600\) people in the city who used taxis. After n years the...
• 08M.2.sl.TZ2.10c(i) and (ii): Let R be the ratio of the number of people using taxis in the city to the number of taxis. The...
• 12M.2.sl.TZ1.2a(i) and (ii): (i)     Write down the coordinates of the vertex. (ii)    Hence or otherwise, express the...
• 12M.2.sl.TZ1.2b: Solve the equation \(f(x) = 0\) .
• 10M.1.sl.TZ2.6: Solve \({\log _2}x + {\log _2}(x - 2) = 3\) , for \(x > 2\) .
• 09M.2.sl.TZ1.6b: Find the smallest value of n for which \({S_n} > 40\) .
• 09M.2.sl.TZ2.3: Solve the equation \({{\rm{e}}^x} = 4\sin x\) , for \(0 \le x \le 2\pi \) .
• 09M.2.sl.TZ2.7a: Find the possible values of k.
• 09M.2.sl.TZ2.7b: Write down the values of k for which \({x^2} + (k - 3)x + k = 0\) has two equal real roots.
• 09M.2.sl.TZ2.10f: Let \(g(x) = \ln (x + 1)\) , for \(0 \le x \le \pi \) . There is a value of x, between \(0\) and...
• 10M.2.sl.TZ1.8a: Use the cosine rule to show that \({\rm{AC}} = \sqrt {41 - 40\cos x} \) .
• 10M.2.sl.TZ1.8b: Use the sine rule in triangle ABC to find another expression for AC.
• 10M.2.sl.TZ1.8c: (i)     Hence, find x, giving your answer to two decimal places. (ii)    Find AC .
• 10M.2.sl.TZ1.8d(i) and (ii): (i)     Find y. (ii)    Hence, or otherwise, find the area of triangle ACD.
• 10M.2.sl.TZ1.9c(i), (ii) and (iii): (i)     Using your value of k , find \(f'(x)\) . (ii)    Hence, explain why f is a decreasing...
• 10M.2.sl.TZ2.5a: On the diagram above, sketch the graph of g.
• 10M.2.sl.TZ2.5b: Solve \(f(x) = g(x)\) .
• 10M.2.sl.TZ2.5c: Write down the set of values of x such that \(f(x) > g(x)\) .
• 10M.2.sl.TZ2.7a: Find the value of n when \(t = 0\) .
• 10M.2.sl.TZ2.7b: Find the rate at which n is increasing when \(t = 15\) .
• 10M.2.sl.TZ2.7c: After k minutes, the rate of increase in n is greater than \(10000\) bacteria per minute. Find...
• SPNone.2.sl.TZ0.10a(i), (ii) and (iii): Jane thinks that the function \(f(t) = - 0.25{t^3} - 2.32{t^2} + 1.93t + 106\) is a suitable...
• SPNone.2.sl.TZ0.10b: Kevin thinks that the function \(g(t) = - 5.2{t^2} + 9.5t + 100\) is a better model for the data....
• 11N.1.sl.TZ0.7a: Find the values of k such that \(f(x) = 0\) has two equal roots.
• 11N.1.sl.TZ0.7b: Each value of k is equally likely for \( - 5 \le k \le 5\) . Find the probability that...
• 11N.2.sl.TZ0.8a(i) and (ii): Consider an infinite geometric sequence with \({u_1} = 40\) and \(r = \frac{1}{2}\) . (i)    ...
• 11N.2.sl.TZ0.8b(i) and (ii): Consider an arithmetic sequence with n terms, with first term (\( - 36\)) and eighth term...
• 11N.2.sl.TZ0.8c: The sum of the infinite geometric sequence is equal to twice the sum of the arithmetic sequence....
• 11M.1.sl.TZ1.7a: Find the value of k .
• 11M.1.sl.TZ1.7b: The line \(y = p\) intersects the graph of f . Find all possible values of p .
• 11M.2.sl.TZ1.6: Let \(f(x) = \cos ({x^2})\) and \(g(x) = {{\rm{e}}^x}\) , for \( - 1.5 \le x \le 0.5\) . Find...
• 11M.2.sl.TZ1.9a: Identify the two points of inflexion.
• 11M.2.sl.TZ1.9b(i) and (ii): (i)     Find \(f'(x)\) . (ii)    Show that \(f''(x) = (4{x^2} - 2){{\rm{e}}^{ - {x^2}}}\) .
• 11M.2.sl.TZ1.9c: Find the x-coordinate of each point of inflexion.
• 11M.2.sl.TZ1.9d: Use the second derivative to show that one of these points is a point of inflexion.
• 13M.2.sl.TZ1.9b: Solve \(f(x) = 95\) .
• 13M.2.sl.TZ2.7b: The area of the shaded region is \(25\) cm2 . Find the value of \(r\) .
• 13M.2.sl.TZ2.10b: Consider all values of \(m\) such that the graphs of \(f\) and \(g\) intersect. Find the value of...
• 14M.1.sl.TZ1.4b: Hence, solve \({\log _3}27 + {\log _8}\frac{1}{8} - {\log _{16}}4 = {\log _4}x\).
• 14M.1.sl.TZ1.7b: Given that \(f'(x) \geqslant 0\), show that \({p^2} \leqslant 3pq\).
• 14M.1.sl.TZ2.8a(i): Write down the value of the discriminant.
• 14M.1.sl.TZ2.8a(ii): Hence, show that \(p = 3\).
• 14M.1.sl.TZ2.8c: The graph of \(f\) has its vertex on the \(x\)-axis. Write down the solution of \(f(x) = 0\).
• 14M.2.sl.TZ2.8c: How long does it take for the number of bacteria in colony \({\text{A}}\) to reach \(400\)?
• 14M.2.sl.TZ2.8d: The number of bacteria in colony \({\text{B}}\) after \(t\) hours is modelled by the function...
• 14M.2.sl.TZ2.8e: The number of bacteria in colony \({\text{B}}\) after \(t\) hours is modelled by the function...
• 13N.1.sl.TZ0.7: The equation \({x^2} + (k + 2)x + 2k = 0\) has two distinct real roots. Find the possible values...
• 13N.1.sl.TZ0.10d: The graph of \(g\) intersects the graph of \(f'\) when \(x = q\). Find the value of \(q\).
• 14N.1.sl.TZ0.1b: Solve \(f(x) = 0\).
• 14N.1.sl.TZ0.4b: Hence or otherwise, solve \(3\ln 2 - \ln 4 =  - \ln x\).
• 14N.2.sl.TZ0.4b: Solve \(f(x) = 0\).
• 14N.2.sl.TZ0.9b: Another sequence \({v_n}\) is defined by \({v_n} = a{n^k}\), where...
• 14N.2.sl.TZ0.9c: Find the smallest value of \(n\) for which \({v_n} > {u_n}\).
• 15M.1.sl.TZ1.3b: Hence or otherwise solve \({8^{2x + 1}} = {16^{2x - 3}}\).
• 15M.1.sl.TZ1.6a: Show that the discriminant of \(f(x)\) is \(100 - 4{p^2}\).
• 15M.1.sl.TZ1.6b: Find the values of \(p\) so that \(f(x) = 0\) has two equal roots.
• 15M.2.sl.TZ1.7b: The graph of \(f\) is translated to the graph of \(g\) by the vector...
• 15M.2.sl.TZ2.7: Let \(f(x) = k{x^2} + kx\) and \(g(x) = x - 0.8\). The graphs of \(f\) and \(g\) intersect at two...
• 15N.2.sl.TZ0.4c: Find the least value of \(n\) such that \({S_n} > 75\,000\).
• 15N.2.sl.TZ0.9e: During which year were the number of coyotes the same as the number of foxes?
• 16M.1.sl.TZ1.7: Let \(f(x) = 3{\tan ^4}x + 2k\) and \(g(x) =  - {\tan ^4}x + 8k{\tan ^2}x + k\), for...
• 16M.2.sl.TZ1.2a: Solve \(f(x) = g(x)\).
• 16M.2.sl.TZ1.2b: Find the area of the region enclosed by the graphs of \(f\) and \(g\).
• 16M.2.sl.TZ1.7a: (i)     Find the value of \(k\). (ii)     Interpret the meaning of the value of \(k\).
• 16M.2.sl.TZ1.7b: Find the least number of whole years for which \(\frac{{{P_t}}}{{{P_0}}} < 0.75\).
• 16M.2.sl.TZ1.9a: Find the displacement of P from O after 5 seconds.
• 16M.2.sl.TZ1.9b: Find when P is first at rest.
• 16M.2.sl.TZ1.9c: Write down the number of times P changes direction.
• 16M.2.sl.TZ1.9d: Find the acceleration of P after 3 seconds.
• 16M.2.sl.TZ1.9e: Find the maximum speed of P.
• 16M.2.sl.TZ1.10a: Find \(\overrightarrow {{\text{AB}}} \).
• 16M.2.sl.TZ1.10b: Show that...
• 16M.2.sl.TZ1.10c: Find \(\theta \).
• 16M.2.sl.TZ1.10d: (i)     Show that...
• 16M.1.sl.TZ2.4: Three consecutive terms of a geometric sequence are \(x - 3\), 6 and \(x + 2\). Find the...
• 16M.2.sl.TZ2.7: A particle moves in a straight line. Its velocity \(v{\text{ m}}\,{{\text{s}}^{ - 1}}\) after...
• 16M.2.sl.TZ2.8a: (i)     Find the correlation coefficient. (ii)     Write down the value of \(a\) and of \(b\).
• 16M.2.sl.TZ2.8b: Use the regression equation to estimate the price of Lina’s car, giving your answer to the...
• 16M.2.sl.TZ2.8c: Calculate the price of Lina’s car after 6 years.
• 16M.2.sl.TZ2.8d: Find the year when Lina sells her car.
• 16M.2.sl.TZ2.9a: Write down the equation of the horizontal asymptote of the graph of \(f\).
• 16M.2.sl.TZ2.9b: Find \(f'(x)\).
• 16M.2.sl.TZ2.9c: Write down the value of \(b\).
• 16M.2.sl.TZ2.9d: Given that \(g'(1) =  - e\), find the value of \(a\).
• 16M.2.sl.TZ2.9e: There is a value of \(x\), for \(1 < x < 4\), for which the graphs of \(f\) and \(g\) have...
• 16N.1.sl.TZ0.7: Let \(f(x) = m - \frac{1}{x}\), for \(x \ne 0\). The line \(y = x - m\) intersects the graph of...
• 16N.2.sl.TZ0.1c: Solve \((g \circ f)(x) = 0\).
• 16N.2.sl.TZ0.4b: Hence, find the area of the region enclosed by the graphs of \(f\) and \(g\).
• 17M.1.sl.TZ1.9a: Find the value of \(p\).
• 17M.1.sl.TZ1.9b: Find the value of \(a\).
• 17M.1.sl.TZ1.9c: The line \(y = kx - 5\) is a tangent to the curve of \(f\). Find the values of \(k\).
• 17M.1.sl.TZ1.10a: Show that \(\cos \theta  = \frac{3}{4}\).
• 17M.2.sl.TZ1.10a.i: Write down the value of \(q\);
• 17M.2.sl.TZ1.10a.ii: Write down the value of \(h\);
• 17M.2.sl.TZ1.10a.iii: Write down the value of \(k\).
• 17M.2.sl.TZ1.10b.i: Find \(\int_{0.111}^{3.31} {\left( {h(x) - x} \right){\text{d}}x} \).
• 17M.2.sl.TZ1.10b.ii: Hence, find the area of the region enclosed by the graphs of \(h\) and \({h^{ - 1}}\).
• 17M.2.sl.TZ1.10c: Let \(d\) be the vertical distance from a point on the graph of \(h\) to the line \(y = x\)....
• 17M.2.sl.TZ2.5: Consider a geometric sequence where the first term is 768 and the second term is 576. Find the...
• 17M.2.sl.TZ2.6a: Show that \((f \circ g)(x) = {x^4} - 4{x^2} + 3\).
• 17M.2.sl.TZ2.6b: On the following grid, sketch the graph of \((f \circ g)(x)\), for...
• 17M.2.sl.TZ2.6c: The equation \((f \circ g)(x) = k\) has exactly two solutions, for...
• 17M.2.sl.TZ2.10a.i: Find \(q\).
• 17M.2.sl.TZ2.10a.ii: Find \(p\).
• 17M.2.sl.TZ2.10b.i: Write down the probability of drawing three blue marbles.
• 17M.2.sl.TZ2.10b.ii: Explain why the probability of drawing three white marbles is \(\frac{1}{6}\).
• 17M.2.sl.TZ2.10b.iii: The bag contains a total of ten marbles of which \(w\) are white. Find \(w\).
• 17M.2.sl.TZ2.B10c: Jill plays the game nine times. Find the probability that she wins exactly two prizes.
• 17M.2.sl.TZ2.10d: Grant plays the game until he wins two prizes. Find the probability that he wins his second prize...
• 17N.1.sl.TZ0.8a: Show that \(f’(1) = 1\).
• 17N.1.sl.TZ0.8b: Find the equation of \(L\) in the form \(y = ax + b\).
• 17N.1.sl.TZ0.8d: Find the area of the region enclosed by the graph of \(f\) and the line \(L\).
• 17N.1.sl.TZ0.8c: Find the \(x\)-coordinate of Q.
• 17N.2.sl.TZ0.5a: Find the value of \(p\).
• 17N.2.sl.TZ0.5b: The following diagram shows part of the graph of \(f\). The region enclosed by the graph of...
• 18M.2.sl.TZ1.1a: Find f '(x).
• 18M.2.sl.TZ1.1b: Find f "(x).
• 18M.2.sl.TZ1.1c: Solve f '(x) = f "(x).
• 18M.2.sl.TZ1.7a: Given that xk + 1 = xk + a, find a.
• 18M.2.sl.TZ1.7b: Hence find the value of n such that \(\sum\limits_{k = 1}^n {{x_k} = 861} \).
• 18M.1.sl.TZ2.6: Let \(f\left( x \right) = p{x^2} + qx - 4p\), where p ≠ 0. Find Find the number of roots for the...

2.8

• 08M.2.sl.TZ2.10a(i) and (ii): (i)     Find the number of taxis in the city at the end of 2005. (ii)    Find the year in...
• 08M.2.sl.TZ2.10b(i) and (ii): At the end of 2000 there were \(25600\) people in the city who used taxis. After n years the...
• 08M.2.sl.TZ2.10c(i) and (ii): Let R be the ratio of the number of people using taxis in the city to the number of taxis. The...
• 12M.2.sl.TZ1.10a(i) and (ii): Find the distance between the ships (i)     at 13:00; (ii)    at 14:00.
• 12M.2.sl.TZ1.10b: Let \(s(t)\) be the distance between the ships t hours after noon, for \(0 \le t \le 4\) . Show...
• 12M.2.sl.TZ1.10c: Sketch the graph of \(s(t)\) .
• 12M.2.sl.TZ1.10d: Due to poor weather, the captain of ship A can only see another ship if they are less than 8 km...
• 09N.2.sl.TZ0.7: The fencing used for side AB costs \(\$ 11\) per metre. The fencing for the other three sides...
• 10M.2.sl.TZ2.7a: Find the value of n when \(t = 0\) .
• 10M.2.sl.TZ2.7b: Find the rate at which n is increasing when \(t = 15\) .
• 10M.2.sl.TZ2.7c: After k minutes, the rate of increase in n is greater than \(10000\) bacteria per minute. Find...
• SPNone.2.sl.TZ0.10a(i), (ii) and (iii): Jane thinks that the function \(f(t) = - 0.25{t^3} - 2.32{t^2} + 1.93t + 106\) is a suitable...
• SPNone.2.sl.TZ0.10b: Kevin thinks that the function \(g(t) = - 5.2{t^2} + 9.5t + 100\) is a better model for the data....
• SPNone.2.sl.TZ0.10c(i) and (ii): (i)     On graph paper, using a scale of 1 cm to 1 second, and 1 cm to 10 m, plot the data given...
• 11N.2.sl.TZ0.6a: Write down \(A(0)\) .
• 11N.2.sl.TZ0.6b: Find the concentration of medication left in his bloodstream after 50 minutes.
• 11N.2.sl.TZ0.6c: At 13:00, when there is no medication in Jose’s bloodstream, he takes his first dose of...
• 14M.2.sl.TZ2.8c: How long does it take for the number of bacteria in colony \({\text{A}}\) to reach \(400\)?
• 14M.2.sl.TZ2.8d: The number of bacteria in colony \({\text{B}}\) after \(t\) hours is modelled by the function...
• 14M.2.sl.TZ2.8e: The number of bacteria in colony \({\text{B}}\) after \(t\) hours is modelled by the function...
• 15M.2.sl.TZ1.5b: Robin and Pat are planning a wedding banquet. The cost per guest, \(G\) dollars, is modelled by...
• 15M.2.sl.TZ2.10b: When \(\theta  = \alpha \), the area of the square \(ABCD\) is equal to the area of the sector...
• 15N.2.sl.TZ0.9c: Let \(f\) be the number of foxes in the reserve after \(t\) years. The number of foxes can be...
• 15N.2.sl.TZ0.9e: During which year were the number of coyotes the same as the number of foxes?
• 17M.2.sl.TZ1.10a.i: Write down the value of \(q\);
• 17M.2.sl.TZ1.10a.ii: Write down the value of \(h\);
• 17M.2.sl.TZ1.10a.iii: Write down the value of \(k\).
• 17M.2.sl.TZ1.10b.i: Find \(\int_{0.111}^{3.31} {\left( {h(x) - x} \right){\text{d}}x} \).
• 17M.2.sl.TZ1.10b.ii: Hence, find the area of the region enclosed by the graphs of \(h\) and \({h^{ - 1}}\).
• 17M.2.sl.TZ1.10c: Let \(d\) be the vertical distance from a point on the graph of \(h\) to the line \(y = x\)....