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Differentiation

100

Differentiate 1/square root of x with respect to x

-1/2x to the power of -3/2

200

Find the value of dy/dx of y= (x+5)/x at the point (5,2)

-1/5

300

Use product rule to differentiate x²(x-1)³ with respect to x

x(x-1)²(5x-2)

400

Find d²y/dx² for y=(2x+5)/(3x-1)

y=102/(3x-1)³

500

Find the coordinates of the stationary points on the curve y=2x³-15x²+24x+6

(1,17) maximum point 

(4,-10) minimum point 

Indices and Surds

100

Simplify (x⁵)³

x¹⁵

200

Simplify (square root 8)²

8

300

Solve the equation 2^3x+1=8

x=2/3

400

Solve the simultaneous equation 

125^x/5^y=25

2^3x x (1/8)^1-y=32

x=7/6 y=3/2

500

Solve the equation 2y²-7y-4=0 then solve the equation 2(2^x)²-7(2^x)-4=0

2

Logarithm

100

Convert 2⁴=16 to logarithm form 

4=log₂16

200

Given that log_pX=5 and log_pY=2, find log_pX²

10

300

Given that log_pX=5 and log_pY=2, find log_xyp

1/7

400

Solve the simultaneous equation 

xy=64

log_xy=2

x=4 y=16

500

Solve the simultaneous equation 

2log₃y=log₅125+log₃x

2^y=4^x

x=6.75 y=13.5

Permutation and Combination

100

Find how many even numbers can be formed using the digits 1,3,5,6,7 and 9 if no number can be repeated. The first number must be a 3 and the last number must be a 6.

⁴P₂

200

John wants to put 5 books on the shelf out of the 8 books he has. Find the number of ways the books can be chosen if there are no restrictions. 

⁸C₅

300

A code consists of 3 letters selected from W,D,S,C,V,X followed by 2 digits selected from 1,3,2,4,6

⁶P₃x⁵P₂=2400

400

3 coats and 2 dresses are to be selected from 9 coats 7 dresses. Find the number of different selections that can be made. 

⁹C₃x⁷C₂=1764

500

John wants to put 5 books on the shelf out of the 8 books he has. There are 5 comics, 2 textbooks and 1 storybook. Find the number of ways the 5 books can be chosen if there must be at least one type of book.

20 (1 storybook, 1 textbook, 3 comics) or 10 (1 storybook, 2 textbooks, 2 comics)

Functions

100

G(x)=4/(w+1)

G(-6)=?

-4/5

200

f(x)=2x-3

g(x)=x/2

Find gf(x)

(2x-3)/2

300

f(x)=3x+2

g(x)=x²-1

fg(-3)=?

26

400

f(x)=x³+1

g(x)=2x-5

h(x)=f(x)-g(x)

Find h(-2)

-8

500

g(x)=(2x+3)/(x+1) for x>1

Find an expression for g^-1(x)

g^-1(x)=(x+3)/(x-2)