LOCI, LINEAR INEQUALITIES MATHS NOTES

LOCI,LINEAR INEQUALITIES

CALCULUS: CURVE SKETCHING,AREA APPROXIMATIONS

1. The equation of the curve is y ³– 2x²+ 3x + 5.
2. (i) Determine the stationary points of the curve. (3 marks)

 

(ii) For each point in a (i) above determine the nature of the points hence sketch the curve.                                                                                     (4 marks)

 

2. Find the equation of the tangent to the curve at x = 2. (3 marks)

 

2018 Mocks PP1/87

2.a)      i)         Find the coordinates of the stationary points on the curve y=x³-3x+2. (3 marks)

1. ii) For each stationary point determine whether  it is a minimum or a maximum.

(4 marks)

2. b) In the space provided, sketch the graph of the function y=x³-3x+2. (3 marks)

2018 Mocks PP1/72

3.The equation of a curve is y = -2x² + x + 1

 

1. Find

 

• The gradient of the curve at P(5, -44)                                   (3mks)

 

• The y intercept (1mk)

 

(b)       (i)        Determine the stationary point of the curve              (3mks)

 

(ii)       Sketch the curve                                                         (3mks)

 

2018 Mocks PP1/48

 

4.Using a ruler and pair of compas only construct :

(a)        An equilateral triangle ABC of side 6cm

(b)        The focus of a point P inside the triangle such that AP < PB

(c)        The locus of a point Q such that AQ > 4cm

(d)        Mark and label the region x inside the triangle which satisfy the two loci.                                  (4mks)

 

2018/31 PP2

5.Above line AB = 10cm drawn below, construct and label in a single diagram, using a pair of a compasses and ruler only;

2. The locus of a point X such that the area of a triangle ABX is 15cm².
3. The locus of a point Y such that angle AYB = 90^o.
4. Locate points P and Q where loci X and Y intersect. Measure PQ.
5. Show by shading and labeling the region R which satisfies the conditions below simultaneously:
6. Angle ARB ≥ 90^o
7. Area of triangle ABR ≥ 15cm²
8. Calculate the area of the shaded region R in (d) above. (Take = 3.142)                        (10 marks)

2018/24/PP2

 

6.(a)Construct  rectangle ABCD with sideAB= 6.4cm and diagonal AC = 8cm.            (3mks)

(b)Locus,L_1, is a set of points equidistant from A and B and locus, L_2,is a set of points equidistant fromBC and BA.If L₁and L₂meets at N inside the rectangle, locate point N.(3mks)

(c)A point x is to be located inside the rectangle such that it is nearer B than Aand also nearer

ABthanBC. If its not greater than 3cm from Nshade the region where the points could be located.                                                                                                                              (4mks)

2018/21/PP2

7.Using a ruler and a pair of compasses only construct

1. Triangle ABC, such that AB = 9cm, AC = 7cm  and  < CAB = 60°      (2mks)

 

1. The locus of P , such that AP ≤  BP                                                             (2mks)

 

iii.           The locus of Q such that  CQ ≤  3.5cm

 

1. Locus of R such that angle ACR ≤ angle BCR (2mks)

2018/11/PP2

 

8.The line segment BC = 7.5 cm long is one side of triangle ABC.

1. a) Use a ruler and compasses only to complete the construction of triangle ABC in which

ÐABC = 45^o, AC = 5.6 cm and angle BAC is obtuse.                                                             {3 marks}

1. b) Draw the locus of a point P such that P is equidistant from a point O and passes through the vertices of triangle ABC.                                                                                                           {3 marks}
2. c) Locate point D on the locus of P equidistant from lines BC and BO. Q lies in the region enclosed by lines BD, BO extended and the locus of P. Shade the locus of Q.                                            {4 marks}