1)
a)
i) \( OS = \frac{5}{7} OQ \)
ii) \( PQ = p + q \)
iii) \( QR = \frac{3q}{4} \)
b)
Given that \( OT = hOR \) and \( PT = kPS \), we can rewrite this as:
\( OT = h(p - PR) \) and \( PT = k(PR) \)
From the information given, we know that \( PR = \frac{1}{4} PQ \) and \( PR = \frac{1}{3} QR \).
Therefore, substituting these values into the equations above, we get:
\( OT = h(p - \frac{1}{4} PQ) \) and \( PT = k(\frac{1}{4} PQ) \)
Solving for h and k:
\( h = \frac{1}{2} \) and \( k = 2 \)
2) Percentage error in calculating the area of the rectangle:
Actual area = Length x Width = 14.6 cm x 6.7 cm = 97.82 cm^2
Calculated area = 14.6 cm x 6.7 cm = 97.82 cm^2
Percentage error = \(\frac{(Calculated\ area - Actual\ area)}{Actual\ area} \times 100\%\)
Percentage error = \(\frac{97.82 - 97.82}{97.82} \times 100\% = 0\%\)
3) Solving the simultaneous equations:
4x - 2y = 6 ...(1)
\( x^2 - xy = -4 \)
\( x^2 = xy - 4 \)
Substitute x from (1) into the equation:
\( (4y + 6)^2 = y(4y + 6) - 4 \)
\( 16y^2 + 48y + 36 = 4y^2 + 6y - 4 \)
\( 12y^2 + 42y + 40 = 0 \)
Solve for y and then find x using (1).
4) Let r be the radius of the circle. Since MX is the perpendicular bisector of AB, then AM = MB = 10 cm.
From Pythagoras theorem:
\( r^2 = 10^2 + 3^2 \)
\( r^2 = 109 \)
\( r = \sqrt{109} \) cm
5)
a)
\( B^2 = \left(\begin{array}{cc} 0 & 3 \\ 1 & 0 \end{array}\right)\left(\begin{array}{cc} 0 & 3 \\ 1 & 0 \end{array}\right) = \left(\begin{array}{cc} 3 & 0 \\ 0 & 3 \end{array}\right) = 3\left(\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right) \)
Therefore, K = 3
b)
The image of PQR under the transformation whose matrix is B^2 is:
P' = B^2P = 3(3, 0) = (9, 0)
Q' = B^2Q = 3(2, 2) = (6, 6)
R' = B^2R = 3(0, 1) = (0, 3)
6) Given expression: \( \frac{\sqrt{3}-2\sqrt{2}}{3\sqrt{2}+\sqrt{3}} = c + a\sqrt{b} \)
Multiplying the numerator and denominator by the conjugate of the denominator:
\( \frac{(\sqrt{3}-2\sqrt{2})(3\sqrt{2}-\sqrt{3})}{(3\sqrt{2}+\sqrt{3})(3\sqrt{2}-\sqrt{3})} \)
Simplify to get c, a, and b.
7) Let the ratio in which brands x and y are mixed be \(x:y\).
Given that the total mixture is 35 kg made of x and y:
\(x + y = 35\)
The value of the mixture is given by:
\(68x + 53y = 35 \times 62\)
Solve these two equations to find x:y.
8) Calculate the position vector after each leg of the flight and use the cosine rule to find the distance from the starting point.
9) Use the compound interest formula:
\( A = P(1 + \frac{r}{n})^{nt} \)
Where:
A = final amount
P = principal amount (initial deposit)
r = annual interest rate (in decimal form)
n = number of times interest is compounded per year
t = number of years
Given:
P = 21000
r = 0.2
n = 2
t = 3
Calculate A and find the compound interest by subtracting the principal amount.
1)In a triangle OPQ,OP=p and OQ=q,R is a point on PQ such that PR:RQ=1:3 and 5OS:2OQ.PS intersects OR at T.
a)Express in terms of p and q;
i)OS
ii)PQ
iii)QR
b)Given that OT=hOR and PT=kPS.Determine the values of h and k
2)A rectangle has a length of 14.6cm and a width of 6.70cm.Find the percentage error in calculating the area of the rectangle
3)Solve the.simultaneous equation:
4x-2y=6
X^2-xy=-4
4)In a figure,a segment of a circle cut-off by a chord AB.Line MX is perpendicular bisector of chord AB.If AB is 20cm and MX is 6cm.Calculate the radius of the circle from which the chord was cut
5)If B=[(0 3,1 0)]
a)Find K if B^2=K[(1 0,0 1)]
b)Find the coordinates of the image of PQR given that P(3,0),Q(2,2) and R(0,1).Under the transformation whose matrix is B^2
6)If √3-2√2/3√2+√3=c+a√b,Find the values of a,b and c
7)A tea dealer mixes 2 brands of tea,x and y to obtain 35kg of the mixture worth ksh 62 per kg.If brand x is valued at ksh 68 per kg and brand y at ksh 53 per kg.Calculate the ratio, in its simplest form,in which the brands x and y are mixed
8)An aeroplane flies 125km in the direction 110°.It then changes course and flies for 170 km in the direction of 160°.Find by calculation its distance from the starting point
9)Find the compound interest on ksh 21000 in 3 years at a rate of 20%p.a compounded semi-annually
1 answer