Activities
P.o.t.W. #25
■ No GDC ■ for HL students Given \(a + b = 1 and \({a^4} + {b^4} = 7 for \(a,b \in \mathbb{R}, find the value of \({a^2} + {b^2} and the value of \({a^3} + {b^3}. Hint: Begin by letting \(P = {a^2}...
Activities
P.o.t.W. #24
■ GDC allowed ■ for SL and HL students The length of the perimeter of a right triangle is exactly 90 cm and the length of the altitude perpendicular to the hypotenuse is exactly 11.25 cm. The three...
Activities
P.o.t.W. #23
■ GDC allowed ■ for HL students Note: All working is performed with angles in radian measure. (a) Consider the statement: If \(\tan \left( A \right) = \tan \left( B \right), then \(A = B. (i) ...
Activities
P.o.t.W. #22
■ No GDC ■ for SL and HL students A point of inflexion is a point on a graph where: (i) the concavity of the graph changes, and (ii) a line is tangent to the graph. Also, the graph crosses its tangent...
Activities
P.o.t.W. #21
■ No GDC ■ for SL and HL students The first three terms of a geometric sequence are \(\dfrac{{x\left( {x + 1} \right)}}{{x - 2}}, \(\dfrac{{x{{\left( {x + 1} \right)}^2}}}{{{{\left( {x - 2} \right)}^2}}} and...
Activities
P.o.t.W. #20
■ No GDC ■ for SL and HL students Two real numbers, \(a and \(b, are chosen at random such that \(0 < a < 1 and \(0 < b < 1. Given that \(c = 1, find the exact probability that the three numbers \(a, \(b...
Activities
P.o.t.W. #19
■ No GDC ■ for SL and HL students The figure shows an outer regular pentagon and a smaller inner regular pentagon that is shaded. Find the ratio of the area of the outer pentagon to the area of the...
Activities
P.o.t.W. #18
■ No GDC ■ for SL and HL students Alicia collects autographs of well-known people. She keeps the autographs in three different boxes. One-fifth of her autographs are in box #1, several sevenths are...
Activities
P.o.t.W. #17
■ No GDC ■ for HL students (a) Using the substitution \(x = \sin {\textrm{\theta }}, show that \(\displaystyle\int {\sqrt {1 - {x^2}} } \,{\textrm{d}}x = \frac{{\arcsin x + x\sqrt {1 - {x^2}} }}{2}....
Activities
P.o.t.W. #16
■ GDC is allowed ■ for SL and HL students Anika and Bella are equally strong chess players. Hence, when they play a chess match it is equally likely that either of them will win. Determine whether...