7/27/2019 Unit 1 - Practice 2 Revised

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Unit I Practice 2

1. Solve the equation 2 5 13. x − =  

2. Prove by induction that11 4 2 5 3 6 ( 3) ( 1)( 5)3

n n n n n× + × + × + ⋅ ⋅ ⋅ + + = + +  

for all integers 1.n ≥  

3. Fig. 1 shows the graph of  ( ) y f x= with domain 1 1. x− ≤ ≤  

Sketch the graphs of 

(a) ( ) y f x= −  

(b) 2 ( 1). y f x= −  

4. A circle with centre (2, 4) has equation 2 2 4 8 25. x y x y+ − − =  

(a) Show that the radius of the circle is 45 .  

(b) Prove that the point (8, 8) is outside of the circle.

(c) Find the equation of the line which is perpendicular to the line 2 8 y x+ = and which passes

through the centre of the circle.

(d) P and Q are the points where the line 2 8 y x+ = crosses the circle. Show that PQ is a

diameter of the circle and find the coordinates of P and Q.

5. Fig.2 shows the graph of  ( ) y f x= where2

3( ) , 0. f x ax x x= − ≥  

The curve crosses the x-axis at the point 1 , 08

 A

and has a turning point at B.

(a) Show that 2.a =  

(b) Find as an exact fraction the gradient of the curve at A. What happens to the gradient of thecurve near the origin?

(c) Find the exact coordinates of  B. State the range of the function ( ). f x  

(d) Calculate the area of the region enclosed by the curve and the x-axis.

Fig. 2

7/27/2019 Unit 1 - Practice 2 Revised

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2

6. Use the standard results for  2

1 1

and n n

r r 

r r = =

∑ ∑ to show that, for all positive integers n.

2 2

1

(6 2 1) (2 4 3).n

r 

r r n n n=

+ + = + +∑  

7. Express 12

3cos sin in the form cos( ), where 0 , Rθ θ θ α α π  + − < < stating the exact values of  R 

and  tan .α   

Hence, solve the equation

3cos sin 2,θ θ + =  

where 0 2 .θ π < <  

8.

The above diagram shows a sector OBC of a circle, centre O and radius 12 cm. The mid-points of OB

and OC are A and  D respectively. The length of  AD is 6cm.  AC is an arc of the circle, centre D and radius 6 cm. The shaded region is bounded by the line AB and the arcs AC and  BC .

(a) Show that 2

3angle . ADC  π =  

(b) Show that the perimeter of the shaded region is (8 6)π  + cm.

(c) Find the exact area of the shaded region.

9. (a) The quadratic equation 2 2 4 0 x x− + = has roots and .α β   

Write down the values of  and .α β αβ  +  

Show that 2 2 4.α β + = −  

Find a quadratic equation which has roots 2 2and .α β   

(b) The cubic equation 3 212 48 0 has roots , 2 and 3 . x x ax p p p− + − =  

Find the value of  p.

Hence find the value of a.

.

10. It is given that , and  α β γ  are three numbers such that

2 2 23, 19 and 1.α β γ α β γ αβγ  + + = + + = =  

Find 

(a) the value of  .αβ βγ γα  + +  

(b) a cubic equation with roots , and .α β γ   

(c) exact values of  , and .α β γ