Pegasys Publishing. CfE Higher Mathematics. Expressions and Functions Practice Assessment A

Transcription

1 Pegasys Publishing CfE Higher Mathematics Epressions and Functions Practice ssessment otes:. Read the question fully before answering it.. lways show your working.. Check your paper at the end if you have time.

2 FORMULE SHEET Scalar Product: a.b = a b cos θ, where θ is the angle between a and b. a or a.b = a b + ab + ab where a = a a b and b = b. b Trigonometric formulae: sin ( ± B) = sin cos B± cos sin B cos ( ± B) = cos cos Bm sin sin B sin = sin cos cos = cos = cos sin = sin

3 nswer all the questions E&F ssessment Standard.. (a) Simplify the epression log 4+log. (b) Hence solve log 4+log = () (#. ). Given the equation log log = 4log find the value of n.. Solve =0 correct to decimal places. E&F ssessment Standard. 4. Epress 4cos 5sin in the form cos+ given that >0 and 0 <60. (4) 5. kite is shown in the diagram below. cm y cm 4 cm Find the eact value of sin+. (4) 6. Given that cos= show that the eact value of sin is. (#. ) E&F ssessment Standard. 7. function is given as =cos for 0 π. (a) State the maimum and minimum turning points of the function. (b) Find the point where the graph intersects the y-ais. ()

4 8. The diagram shows the graph of a cubic equation, =. (, 4) y 4 O (, ) Sketch the graph of = showing the new coordinates of the stationary points. () 9. Sketch the graph of =sin4+ for 0 () 0. The graph below shows a function in the form =log. y O (4, 0) (7, ) 7 Find the values of a and b.. Two functions are defined as: f = 6 ( ) g( ) =, 0. (a) Find an epression for. (b) Find an epression for. (c) Which epression has the same domain as? Justify your answer. function is given by ( ) = 5 + Find the inverse function f ( ). f. () (#. )

5 E&F ssessment Standard.4. On orienteering course is being laid out in a forest. Relative to suitable aes, the first three control points can be represented by the points (,, 0), B(,, 9) and C(4,, 7). B C (,, 0) (,, 9) (4,, 7) Show that the three points are collinear. () 4. canine agility test features a see saw with a pivot point that is slightly off-centre. The diagram below represents the see saw, relative to a suitable aes. P(,, 7) Q(5, 6, 45) R(0, 4, 0) The points P, Q and R lie in a straight line, as shown. Q should divide PR in the ratio :5. Is Q in the correct position? Justify your answer. () (#. ) 5. BCDEF is a triangular prism as shown. E The vectors, B D and F are given by: F B = 4i + 8j + 4k D = 0i + 4j + k F = i 4j + k D C Epress FB in component form. B () Turn over for question 5

6 6. The points S(, 0, ), T(7,, -5) and U(4,, -) are shown in the diagram below. U(4,, -) T(7,, -5) S(, 0, ) Find the size of the shaded angle. (5) End of question paper

7 Epressions and Functions, Practice ssessment Qu. Points of Epected Response Illustrative Scheme E&F Outcome.: Logs and Eponentials (a) pply log +log =log log (b) #. Strategy: Substitute for LHS Start to solve Find m #. log = =6 OR log =log 6 = pply log log =log OR log =log Find n Take natural logarithm of both sides Find (round to d.p.) E&F Outcome.: Trigonometry 4 Epand cos+ Match coefficients Find k Find a 5 Find missing sides Epand sin+ and begin subs Complete substitution Process log = 5=ln0 =0 46 OR 7log log 7 + #. coscos sinsin stated eplicitly cos=4 and sin=5 stated eplicitly = 4 =5 5 and cos+cossin + or 6 Method Square both sides #. Strategy: Use sin = cos Simplify Method #. Strategy: Start to find sin State sin Square both sides Method cos = #. sin = Method #. Find opposite side in right-angled triangle, 7 sin= sin = 0 + #.

8 Qu. Points of Epected Response Illustrative Scheme E&F Outcome.: Trigonometry 7(a) Maimum turning point, Minimum turning point, 7(b) y-intercept or equivalent (= ) 8 Correct horizontal translation Correct vertical translation Shape and stationary points coordinates -coordinates correct y-coordinates correct Correct shape and annotation y (-, ) O - -5 (, -5) 9 Correct amplitude Correct period Shape and correct vertical translation 4 + y O π 0 Find a Find b (a) (b) Start composite process Complete process Start composite process Complete process (c) 5 State epression #. Eplain solution a = b = 4 5 = =6 =6 = #. The denominator of a fraction cannot be zero. Start inverse process State inverse function =5 = 8 + #.

9 Qu. Points of Epected Response Illustrative Scheme E&F Outcome.4: Vectors Find vector between two points e.g. = B Find second vector and interpret multiple e.g.bc = 4=B Complete proof of collinearity =B BC hence the vectors are parallel. B is a common point. Therefore the points, B and C are collinear. 4 Method Ratio to find Q Method 8 Find PR PR = Use correct ratio PQ = 40 Find point Q #. Justify answer Method Section Formula to find Q Begin substitution Complete substitution Find point Q #. Justify answer Method Component Vectors Find PQ Find QR Interpret multiples #. Justify answer 5 Find pathway for FB Identify F Complete calculation of FB 6 Find component vectors 5 Use scalar product Find scalar product Find magnitudes Find angle 4 Q = (5, 6, 9) #. Q is in correct position as it divides PR in the ratio :5 Method = = Q = (5, 6, 9) #. Q is in correct position as it divides PR in the ratio :5 Method PQ = = QR PQ =QR or equivalent #. Q is in correct position as it divides PR in the ratio :5 FB = F +B OR 9 Do not award for (,, 9) TU = and TS = cos=. and 66 5 θ = 5 6 or 0 6 (radians) 4 + #.