Algebra
The first three terms of an arithmetic sequence are 7, 9.5, and 12.
(a) What is the 41st term of the sequence?
(b) What is the sum of the first 101 terms of the sequence?
(a) u1 = 7, d = 2.5
u41 = u1 + (n – 1)d = 7 + (41 – 1)2.5= 107
(b) S101 = 2 n [2u1 + (n – 1)d]
= 2 101 [2(7) + (101 – 1)2.52]
= 2 101(264) = 13332
A family of functions is given by f (x) = x2 + 3x + k, where k ∈ {1, 2, 3, 4, 5, 6, 7}. Find the possible values of k if the curve of this function crosses the x-axis.
b 2 – 4ac = 9 – 4k
9 – 4k > 0
2.25 > k
In the triangle PQR, PR = 5 cm, QR = 4 cm, and PQ = 6 cm. Calculate (a) the size of PQR ; [4] (b) the area of triangle PQR.
(a) PQR=55.8
(B) Area=9.92 cm2
At a conference of 100 mathematicians, there were 72 men and 28 women. The men have a mean height of 1.79 m and the women have a mean height of 1.62 m. Find the mean height of the 100 mathematicians.
1.7424
Differentiate the following functions:
f(x)= (2x+5)3+(2x+5)2+2x+5
6(2x+5)2+8x+22
An arithmetic series has five terms. The first term is 2 and the last term is 32. Find the sum of the series.
S5 = 85
Express f (x) = x 2 – 6x + 14 in form
f (x) = (x – h) 2 + k, where h and k are to be determined.
f (x) = x2 – 6x + 14
f (x) = x2 – 6x + 9 – 9 + 14
f (x) = (x – 3)2 + 5
A triangle has sides of length 4, 5, and 7 units. Find, to the nearest tenth of a degree, the size of the largest angle.
A = 101.5°
For the events A and B, P(A )= 0.6, P(B )= 0.8, and P(A or B )= 1.
(a) Find P(A and B )
(b) Find P( A' or B')
(a) 0.4
(b)0.6
Consider the function f (x)=x3+3x2+3x (a) Find any stationary points and determine their nature.
x =-1
Find the term in x7 in the expansion of (1-X)10
−120x
log2 x +log2(x-7)=3
x = 8
Given that sin X= 1/3, where x is an acute angle, find the exact value of (a) cos x ; [4] (b) cos 2x.
a) 2(2)0.5/3
b) 7/9
Events E and F are independent, with P( E)=2/3 and
P( E and F)=1/3. Calculate (a) P(F ) ;
(b) P(E or F)
a) 1/2
b)5/6
(x-3+x-2+x-1+3) dx
(-1/2x2)-(1/x)+ln x+3x+c
Find the coefficient of a5 b7 in the expansion of
(a + b)12
792
$1000 is invested at 15% per annum interest, compounded monthly. Calculate the minimum number of months required for the value of the investment to exceed $3000.
15% per annum = 12 15 % = 1.25% per month
Total value of investment after n months, 1000(1.0125)n > 3000 => (1.0125)n > 3
n log (1.0125) > log (3) =>
so n = 89 months.
If A is an obtuse angle in a triangle and sin A=5/13 calculate the exact value of sin 2A .
120/169
Events A and B are independent such that
P(B)= 3P(A) and P(A or B) =0.68
. Find P(B)
0.6
Let f''(x)=12x2
a) find f'(x) , given that f'(0)=3
b) find f(x), given f(0)=2
f'(x)=4X3+3
f(x)=x4+3x+2
One of the terms of the expansion of (x + 2y)10 is
ax8 y2. Find the value of a.
a = 180
ln( x+2)=3
x + 2 = e3
x = e3-2
Solve cos 2x-3 cos x -3-cos2x=sin2x for 0<x<2pi
cos x=-1 x=pi
The random variable X follows the binomial distribution B(n,p). Given that E(X) = 10 and Var(X) = 6 find the values of n and p.
p=0.4
n=25
The function f is given by f(x)=2 sin (5x-3)
a) find f''(x)
b) write down integ of F(x) dx
a)f'(x)=2cos(5x-3).(5)=10cos(5x-3)
f''(x)= -(10sin (5x-3)).5=-50 sin(5x-3)
b)(2/5)(cos (5x-3))+c