Find the dimensions of rectangle ABCD of maximum area.

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1 QURTIS (hapter 1) 47 7 Infinitel man rectangles ma e inscried within the right angled triangle shown alongside. One of them is illustrated. a Let = cm and = cm. Use similar triangles to find in terms of. Find the dimensions of rectangle of maimum area. 6 cm 8 cm 8 The points P 1 (a 1, 1 ), P(a, ), P 3 (a 3, 3 ),..., Mn P n (a n, n ) are eperimental data. The points are approimatel linear through the origin O(0, 0). P3 M4 Pn P1 To find the equation of the line of est fit through the M M3 P4 origin, we decide to minimise (P 1 M 1 ) +(P M ) +(P 3 M 3 ) + :::: +(P n M n ) where M1 P [P i M i ] is the vertical line segment connecting each point P i with the corresponding point M i on the line with the same -coordinate a i. Find the gradient of the line of est fit in terms of a i and i, i =1,, 3, 4,..., n. 9 Write =( a )( a + )( + a )( + a + ) in epanded form and hence determine the least value of. ssume that a and are real constants. 10 considering the function =(a 1 1 ) +(a ), use quadratic theor to prove the auch-schwarz inequalit: ja a j 6 p a1 + a p , c 1,, and c are real numers such that 1 =(c 1 + c ). Show that at least one of the equations c 1 =0, + + c =0 has two real roots. REVIEW SET 1 NON-LULTOR 1 onsider the quadratic function = ( + )( 1). a State the -intercepts. State the equation of the ais of smmetr. c Find the -intercept. d Find the coordinates of the verte. e Sketch the function. Solve the following equations, giving eact answers: a 3 1 = = 0 c 11 =60 3 Solve using the quadratic formula: a +5 +3= =0 4 Solve completing the square : +7 4=0 5 Use the verte, ais of smmetr, and -intercept to graph: a =( ) 4 = 1 ( +4) +6 6 Find, in the form = a + + c, the equation of the quadratic whose graph: a touches the -ais at 4 and passes through (, 1) has verte ( 4, 1) and passes through (1, 11).

2 48 QURTIS (hapter 1) 7 Find the maimum or minimum value of the relation = and the value of at which this occurs. 8 The roots of 3 =4 are and. Find the simplest quadratic equation which has roots 1 and. 1 9 Solve the following equations: a +10=7 + 1 =7 c 7 +3=0 10 Find the points of intersection of = 3 and = For what values of k does the graph of = +5 + k not cut the -ais? 1 Find the values of m for which 3 + m =0 has: a a repeated root two distinct real roots c no real roots. 13 The sum of a numer and its reciprocal is Find the numer. 14 Show that no line with a -intercept of (0, 10) will ever e tangential to the curve with equation = One of the roots of k +(1 3k) +(k 6) = 0 is the negative reciprocal of the other root. Find k and the two roots. REVIEW SET 1 1 onsider the quadratic function = a onvert it to the form = a( h) + k. State the coordinates of the verte. c Find the -intercept. d Sketch the graph of the function. LULTOR Solve: a ( )( +1)=3 4 1 =5 3 raw the graph of = +. 4 onsider the quadratic function = Find the equation of the ais of smmetr, and the coordinates of the verte. 5 Using the discriminant onl, determine the nature of the solutions of: a 5 7= =0 6 a For what values of c do the lines with equations =3 + c intersect the paraola = + 5 in two distinct points? hoose one such value of c from part a and find the points of intersection in this case. 7 Suppose [] has the same length as [], [] is cm shorter than [], and [E] is 7 cm in length. Find the length of []. E

3 QURTIS (hapter 1) m of chicken wire is availale to construct a rectangular chicken enclosure against an eisting wall. a If = m, show that the area of rectangle is given = (30 1 ) m. Find the dimensions of the enclosure which will maimise the area enclosed. m eisting wall 9 onsider the quadratic function = a State the ais of smmetr. Find the coordinates of the verte. c Find the aes intercepts. d Hence sketch the function. 10 n open square-ased container is made cutting 4 cm square pieces out of a piece of tinplate. If the volume of the container is 10 cm 3, find the size of the original piece of tinplate. 11 onsider = 5 +3 and = a Solve for : 5 +3= Hence, or otherwise, determine the values of for which > Find the maimum or minimum value of the following quadratics, and the corresponding value of : a = = m of fencing is used to construct 6 rectangular animal pens as shown. a Show that the area of each pen is ³ = m. 9 c Find the dimensions of each pen so that it has the maimum possile area. What is the area of each pen in this case? m m 14 Two different quadratic functions of the form =9 k +4 each touch the -ais. a Find the two values of k. Find the point of intersection of the two quadratic functions. REVIEW SET 1 1 onsider the quadratic function = 1 ( ) 4. a State the equation of the ais of smmetr. Find the coordinates of the verte. c Find the -intercept. d Sketch the function. Solve the following equations: a 5 3=0 7 3=0

4 50 QURTIS (hapter 1) 3 Solve the following using the quadratic formula: a 7 +3=0 5 +4=0 4 Find the equation of the quadratic function with graph: a c (, -0) =4-3 5 Use the discriminant onl to find the relationship etween the graph and the -ais for: a = +3 7 = etermine whether the following quadratic functions are positive definite, negative definite, or neither: a = +3 + = Find the equation of the quadratic function shown: (, 5) 1 8 Find the -intercept of the line with gradient 3 that is tangential to the paraola = For what values of k would the graph of = + k cut the -ais twice? 10 Find the quadratic function which cuts the -ais at 3 and and which has -intercept 4. Give our answer in the form = a + + c. 11 For what values of m are the lines = m 10 tangents to the paraola =3 +7+? 1 a +(3 a) 4=0 has roots which are real and positive. What values can a have? 13 a etermine the equation of: i the quadratic function ii the straight line. For what values of is the straight line aove the curve? Show that the lines with equations = 5 + k are tangents to the paraola = 3 + c if and onl if c k = =0 has roots p, q. Find all quadratic equations with roots p 3 and q 3.

5 hapter Functions Sllaus reference:.1,.,.4,.7 ontents: E F G H I J K L Relations and functions Function notation omain and range omposite functions Even and odd functions Sign diagrams Inequalities (inequations) The modulus function Rational functions Inverse functions Graphing functions Finding where graphs meet

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