DP Further Mathematics HL Questionbank

Description

The aim of this section is to develop students’ geometric intuition, visualization and deductive reasoning.

Directly related questions

• 18M.2.hl.TZ0.7c: A parallelogram is positioned inside a circle such that all four vertices lie on the circle....
• 18M.2.hl.TZ0.7b.ii: Using two diagrams, explain why there are two values of (DE)2.
• 18M.2.hl.TZ0.7b.i: Find in terms of R, the two values of (DE)2 such that the area of the shaded region is twice the...
• 18M.2.hl.TZ0.7a.iv: Hence show that the area of the triangle ABC is \(\frac{{abc}}{{4R}}\).
• 18M.2.hl.TZ0.7a.iii: Given that R denotes the radius of the circumscribed circle prove that...
• 18M.2.hl.TZ0.7a.ii: Prove that the area of the triangle ABC is \(\frac{1}{2}ab\,{\text{sin}}\,{\text{C}}\).
• 18M.2.hl.TZ0.7a.i: In a triangle ABC,...
• 18M.2.hl.TZ0.5c: The centre of another ellipse is now given as the point (2, 1). The minor and major axes are of...
• 18M.2.hl.TZ0.5b.iv: Find the equations of the directrices.
• 18M.2.hl.TZ0.5b.iii: Find the coordinates of the foci.
• 18M.2.hl.TZ0.5b.ii: Hence find the eccentricity.
• 18M.2.hl.TZ0.5b.i: Determine which coordinate axis the major axis of the ellipse lies along.
• 18M.2.hl.TZ0.5a: Show that the area enclosed by the ellipse is \(\pi ab\).
• 18M.1.hl.TZ0.15: Given that the tangents at the points P and Q on the parabola \({y^2} = 4ax\) are perpendicular,...
• 16M.2.hl.TZ0.3c: Find the coordinates of the second point on \(C\) on the chord through \((1,{\text{ }}2)\)...
• 16M.2.hl.TZ0.3b: Write down the equation of \(C\).
• 16M.2.hl.TZ0.3a: Find the coordinates of the centre of \(C\) and its radius.
• 16M.1.hl.TZ0.5: Consider the curve C given by \(y = {x^3}\). The tangent at a point P on \(C\) meets the curve...
• 16M.1.hl.TZ0.11: The points P, Q and R, lie on the sides [AB], [AC] and [BC], respectively, of the triangle ABC....
• 16M.1.hl.TZ0.8b: Let M be any point on \(C\) and N be the \(x\)-intercept of \(C\) between A and B. Prove that...
• 16M.1.hl.TZ0.8a: (i)     the value of \(k\); (ii)     the radius of \(C\); (iii)     the \(x\)-intercepts of \(C\).
• 17M.2.hl.TZ0.9b.iv: Hence find the coordinates of the foci of the hyperbola prior to rotation.
• 17M.2.hl.TZ0.9b.iii: Determine the equation of the rotated hyperbola in this case, giving your answer in the...
• 17M.2.hl.TZ0.9b.ii: Verify that the coefficient of \(XY\) in the equation is zero when \(\tan \alpha  = \frac{1}{2}\).
• 17M.2.hl.TZ0.6b.iii: By writing down and using two further similar expressions, show that the points D, E, F are...
• 17M.2.hl.TZ0.6b.ii: By considering a pair of similar triangles, show...
• 17M.2.hl.TZ0.6b.i: Show that...
• 17M.2.hl.TZ0.6a.i: Show that \({\text{P}}{{\text{Q}}^2} = {\text{PR}} \times {\text{PS}}\).
• 17M.1.hl.TZ0.8b: Given that \({\rm{S\hat OT}}\) is a right-angle, where O is the origin, determine the possible...
• 17M.1.hl.TZ0.8a: Show that \({t^2} + st + 2 = 0\).
• 17M.1.hl.TZ0.12b.iii: Explain briefly why this result shows that the three altitudes of a triangle are concurrent.
• 17M.1.hl.TZ0.12b.ii: Show, by expanding these equations, that (a – f ) \( \bullet \) (c – b) = 0.
• 17M.1.hl.TZ0.12b.i: Show that the position vector f of F satisfies the equations (b – f ) \( \bullet \) (c – a) =...
• 17M.1.hl.TZ0.12a.ii: Explain briefly why this result shows that the three medians of a triangle are concurrent.
• 17M.1.hl.TZ0.12a.i: Show that the position vector of E is \(\frac{1}{3}\) (a + b + c).
• 15M.2.hl.TZ0.4f: Find the equations of the directrices of the ellipse.
• 15M.2.hl.TZ0.4e: Find the coordinates of the foci of the ellipse.
• 15M.2.hl.TZ0.4d: State the coordinates of the point where (PQ) touches the ellipse.
• 15M.2.hl.TZ0.4c: Hence show that (PQ) touches the ellipse.
• 15M.2.hl.TZ0.4b: Given that the tangent crosses the \(x\)-axis at P and the normal crosses the \(y\)-axis at Q,...
• 15M.2.hl.TZ0.4a: (i)     Find the equation of the tangent to the ellipse at the point...
• 15M.1.hl.TZ0.16: A circle \({x^2} + {y^2} + dx + ey + c = 0\) and a straight line \(lx + my + n = 0\) intersect....
• 15M.1.hl.TZ0.13b: Explain why the result also holds if the line segments meet externally at \(\rm{Y}\).
• 15M.1.hl.TZ0.13a: Two line segments [\(\rm{AB}\)] and [\(\rm{CD}\)] meet internally at the point \(\rm{Y}\). Given...
• 15M.1.hl.TZ0.10c: Find the equation of the tangent to the curve when \(\theta  = \frac{\pi }{3}\).
• 15M.1.hl.TZ0.10b: As the wheel rolls, the point A traces out a curve. Show that the gradient of this curve is...
• 15M.1.hl.TZ0.10a: Find the coordinates of \({\text{A}}\) in terms of \(r\) and \(\theta \).
• 07M.1.hl.TZ0.1b: Show that this equation represents a circle and state its radius and the coordinates of its centre.
• 11M.1.hl.TZ0.5: The rectangle ABCD is inscribed in a circle. Sides [AD] and [AB] have lengths \(3\) cm and (\9\)...
• 11M.2.hl.TZ0.2b: ABCD is a quadrilateral. (AD) and (BC) intersect at F and (AB) and (CD) intersect at H. (DB) and...
• 10M.1.hl.TZ0.6a: Show that \(\frac{{{\rm{OR}}}}{{{\rm{OT}}}} = \frac{{{\rm{OT}}}}{{{\rm{OS}}}}\) .
• 10M.1.hl.TZ0.6b: (i)     Show that \({\rm{PR}} - {\rm{RQ}} = 2{\rm{OR}}\) . (ii)     Show...
• 10M.2.hl.TZ0.2A.a: By considering (BE) as a transversal to the triangle ACD, show...
• 10M.2.hl.TZ0.2A.b: Determine the ratios   (i)     \(\frac{{{\rm{AF}}}}{{{\rm{FB}}}}\) ;   (ii)    ...
• 10M.2.hl.TZ0.2B: The diagram shows a hexagon ABCDEF inscribed in a circle. All the sides of the hexagon are...
• 09M.1.hl.TZ0.3b: AP : PD
• 09M.1.hl.TZ0.3a: AF : FB .
• 09M.2.hl.TZ0.5A: Prove that the interior bisectors of two of the angles of a non-isosceles triangle and the...
• 09M.2.hl.TZ0.5B.a: An equilateral triangle QRT is inscribed in a circle. If S is any point on the arc QR of the...
• 09M.2.hl.TZ0.5B.b: Perpendiculars are drawn from a point P on the circumcircle of triangle LMN to the three sides....
• 13M.1.hl.TZ0.3a: A triangle \(T\) has sides of length \(3\), \(4\) and \(5\).   (i)     Find the radius of the...
• 13M.1.hl.TZ0.3b: A triangle \(U\) has sides of length \(4\), \(5\) and \(7\).   (i)     Show that the...
• 13M.2.hl.TZ0.6A.a: Show that the opposite angles of a cyclic quadrilateral add up to \({180^ \circ }\) .
• 13M.2.hl.TZ0.6A.b: A quadrilateral ABCD is inscribed in a circle \(S\) . The four tangents to \(S\) at the vertices...
• 13M.2.hl.TZ0.6B.a: Show that the locus of a point \({\rm{P'}}\) , which satisfies...
• 13M.2.hl.TZ0.6B.b: Show that the two tangents to \(C\) from Q are also tangents to \({\rm{C'}}\) .
• 08M.1.hl.TZ0.4a: The triangle ABC is isosceles and AB = BC = 5. D is the midpoint of AC and BD = 4. Find the...
• 08M.2.hl.TZ0.3b: A straight line meets the sides (PQ), (QR), (RS), (SP) of a quadrilateral PQRS at the points U,...
• 07M.1.hl.TZ0.1a: Find the equation of the locus of P.
• 08M.2.hl.TZ0.3a: The diagram shows the line \(l\) meeting the sides of the triangle ABC at the points D, E and F....
• 07M.2.hl.TZ0.3a: The diagram shows triangle ABC together with its inscribed circle. Show that [AD], [BE] and [CF]...
• 07M.2.hl.TZ0.3b: PQRS is a parallelogram and T is a point inside the parallelogram such that the sum of...
• 12M.1.hl.TZ0.4: The diagram below shows a quadrilateral ABCD and a straight line which intersects (AB), (BC),...
• 12M.2.hl.TZ0.1A.a: the circumscribed circle.
• 12M.2.hl.TZ0.1A.b: the inscribed circle.
• 12M.2.hl.TZ0.1B.a: When \(k = 1\) , show that the locus of P is a straight line.
• 12M.2.hl.TZ0.1B.b: When \(k \ne 1\) , the locus of P is a circle.   (i)     Find, in terms of \(k\) , the...
• SPNone.1.hl.TZ0.5a: The point \({\rm{T}}(a{t^2},2at)\) lies on the parabola \({y^2} = 4ax\) . Show that the tangent...
• SPNone.1.hl.TZ0.5b: The distinct points \({\rm{P}}(a{p^2}, 2ap)\) and \(Q(a{q^2}, 2aq)\) , where \(p\), \(q \ne 0\) ,...
• SPNone.1.hl.TZ0.15a: Prove the internal angle bisector theorem, namely that the internal bisector of an angle of a...
• SPNone.1.hl.TZ0.15b: The bisector of the exterior angle \(\widehat A\) of the triangle ABC meets (BC) at P. The...
• SPNone.2.hl.TZ0.3a: (i)     Show that CEFB is a cyclic quadrilateral. (ii)     Show that \({\rm{HE}} = {\rm{EP}}\) .
• SPNone.2.hl.TZ0.3b: The line (AH) meets [BC] at D. (i)     By considering cyclic quadrilaterals show that...
• SPNone.2.hl.TZ0.8c: The ellipse \(E\) has equation \({{\boldsymbol{X}}^T}{\boldsymbol{AX}} = 24\) where...
• 14M.1.hl.TZ0.9: \({\text{ABCDEF}}\) is a hexagon. A circle lies inside the hexagon and touches each of the six...
• 14M.1.hl.TZ0.6: The parabola \(P\) has equation \({y^2} = 4ax\). The distinct points...
• 14M.1.hl.TZ0.14: (a)     The function \(g\) is defined by \(g(x,{\text{ }}y) = {x^2} + {y^2} + dx + ey + f\) and...
• 14M.2.hl.TZ0.7: (a)   (i)     By first noting that \({\text{OP}} = x\sec \alpha \), show that...
• 14M.2.hl.TZ0.9: (a)     Show that, with the usual convention for the signs of lengths in a triangle,...