IB Math AA SL Past Paper Questions - Quadratics PDF

## IB Math AA SL Questionbank - Quadratics ### Question 26 **[Maximum mark: 15]** Let *f(x) = a(x + 1)(x + 5)*, for *x ∈ R*, where *a ∈ Z*. The following diagram shows part of the graph of *f*. - The graph of *f* has *x*-intercepts at (*p*, 0) and (*q*, 0), and a *y*-intercept at (0, -10). **(a...

## IB Math AA SL Questionbank - Quadratics ### Question 26 **[Maximum mark: 15]** Let *f(x) = a(x + 1)(x + 5)*, for *x ∈ R*, where *a ∈ Z*. The following diagram shows part of the graph of *f*. - The graph of *f* has *x*-intercepts at (*p*, 0) and (*q*, 0), and a *y*-intercept at (0, -10). **(a)** * (i) Write down the value of *p* and the value of *q*. * (ii) Find the value of *a*. **(b)** Find the equation of the axis of symmetry. **(c)** Find the coordinates of the vertex. **(d)** The graph of a function *g* is obtained from the graph of *f* by a reflection in the *y*-axis, followed by a translation by the vector [ 2 ]. The point *P*(-2, 6) on the graph of *f* is mapped to point *Q* on the graph of *g*. Find the coordinates of *Q*. **SL Difficulty: Hard** **AA Formula Sheet** **Mark Scheme** * (a) Video Solution * (b) Video Solution * (c) Video Solution * (d) Video Solution ### Question 27 **[Maximum mark: 8]** The equation *kx² + 4kx - 1 = k* has two real, distinct roots. **(a)** Find the set of possible values for *k*. **(b)** Consider the case when *k = 3*. The roots of the equation can be expressed in the form p/√3, where *p, q ∈ Z*. Find the value of *p* and of *q*. **SL Difficulty: Hard** **AA Formula Sheet** **Mark Scheme** * (a) Video Solution * (b) Video Solution ### Question 28 **IB Math AA SL Exam Questionbank → Quadratics** ### Question 29 **[Maximum mark: 7]** Consider *f(x) = logk (8x - 2x²)*, for 0 < *x* < 4, where *k* > 0. The equation *f(x) = 3* has exactly one solution. Find the value of *k*. **SL Difficulty: Hard** **AA Formula Sheet** **Mark Scheme** **Video Solution** ### Question 30 **[Maximum mark: 15]** Let *f(x) = 16 - x²*, for *x ∈ R*. **(a)** Find the *x*-intercepts of the graph of *f*. - The following diagram shows part of the graph of *f*. - Rectangle *ABCD* is drawn with *A* & *B* on the *x*-axis and *C* & *D* on the graph of *f*. - Let *OA = a*. - Show that the area of *ABCD* is *32a - 2a³*. - Hence find the value of *a* > 0 such that the area of *ABCD* is a maximum. **(b)** Let *g(x) = (x - 4)² + k*, for *x ∈ R*, where *k* is a constant. - Show that when the graphs of *f* and *g* intersect, *2x² - 8x + k = 0*. - Given that the graphs of *f* and *g* intersect only once, find the value of *k*. **SL Difficulty: Hard** **AA Formula Sheet** **Mark Scheme** * (a) Video Solution * (b) Video Solution * (c) Video Solution * (d) Video Solution * (e) Video Solution ### Question 31 **[Maximum mark: 7]** Let *f(x) = 2(x - h₁)² + k₁* and *g(x) = (x - h₂)² + k₂*, where *h₁, h₂, k₁, k₂ ∈ R*. The vertex of the graph of *f* is at (*m*, -*m*²) and the vertex of the graph of *g* is at (-*m*, -*m*), where 0 < *m* < 1. The graphs of *f* and *g* intersect at exactly one point. Find the value of *m*. **SL Difficulty: Hard** **AA Formula Sheet** **Mark Scheme** **Video Solution** ### Question 32 **[Maximum mark: 14]** A quadratic function *f* can be written in the form *f(x) = p(x - 1)(x - q)*. The graph of *y = f(x)* has axis of symmetry *x = 2* and *y*-intercept at (0, -3). **(a)** Find the value of *q*. **(b)** Find the value of *p*. **(c)** The line *y = kx + 6* is a tangent to the graph of *y = f(x)*. Find the possible values of *k*. **SL Difficulty: Hard** **AA Formula Sheet** **Mark Scheme** * (a) Video Solution * (b) Video Solution * (c) Video Solution **End of Page**