1:2:1 Maths Tuition

Implicit Actions.....

Query:

Differentiate the implicit function....  x³ + y³ + 3xy² = 8, with respect to x.

Response:

 

 

Know Your Parameters.....

Query:

If x=sinθ and y=cos2θ, find d²y/dx².

Response:

 

 

A Stationary Event.....

Query:

Find the stationary points on the curve  y =  x³ - 12x + 5  and determine their nature.

Response:

 

A Turning Point.....

Query:

Find the stationary point on the curve  y = x² + 4x + 9  and determine its nature.

Response:

 

Polynomial Integral.....

Query:

Integrate with respect to x:  √x – 2/x³

Response:

 

Special Trig Equation.....

Query:  How do I set about solving  sinθ + 2 cosθ  = 1?

Response:

 

Just The Merest Touch.....

Query:

Two circles C1 and C2 are given by

(x-2)² + (y+4)²  =  100  &  x² + y² +14x – 16y + 88 = 0.

Show that they touch at the point (-4,4).

Response:

 

Tangent Needed.....

Query:

Find the equation of the tangent to the curve  y = 3x²- 4x + 1 at the point (2,5).

Response:

 

Don't Do It The Hard Way.....

Query:  Find the remainder when x³-2x²-3x+1 is divided by x+2.

Response:

 

Pure Hyperbolics.....

Query:

Find dy/dx if y = tan^-1(sinh x)

Response:

 

Gender Balancing Act.....

Query: A committee consists of 6 men and 4 women. A sub-committee of 4 members is to be formed at random. Find the probability that the sub-committee contains at least 1 member of each sex?

Response:

Very Complex!

Query: The complex numbers u, v and w are related

by the equation u^-1=v^-1+w^-1.

If v=1-3i, w=2+i, show that u=m(5-i), where m=5/13.

Response:

A Problem Of Progression.....

Query: An arithmetic progression has its first, second and fourth terms as consecutive terms of a geometric progression. Show that the first term of the arithmetic progression is equal to the common difference.

Response:

Just Touching.....

Query: The line y=2x+c is a tangent to the curve y²=8x. Evaluate 'c'.

Response:

Hint = This question is all about the roots of a quadratic equation.

A Power Series.....

Query: Expand (1+0.5x²)^1.5 up to the third term.

Response:

This is achieved using the power series expansion of (1+mx)^p and is valid for -1<mx<+1.

A Straight Line.....

Query: Find the equation of a straight line which passes through the coordinates (-3,-7) and (5,-1), expressing your answer in the form ax+by+c=0?

Response:

Another Trig. Equation To Solve.....

Query: If 8cos²x+2sinx=7, find all the values of x in the range 0 to 360 degrees.

Response:

Go Large.....

Query: Given that A=0.5r(3-2r) show that the maximum value of A as r varies is 9/16.

Response: Click on the graphic below to see the solution....

 

Where's The Median?

Query: How do I calculate the median of a continuous random variable, when all I'm given is the p.d.f.?

Response: Take a look at a worked example on this topic, by clicking the graphic below...

 

Continuous Random Variable...

Query: A continuous r.v. has probability density function 

f (x) = k(1 - x) for 0<x<1 and f (x) = 0 otherwise.

Evaluate k.

Response: 

Click on the graphic below...

Area By Integration

Query: See if you can solve the following problem. Click on the graphic to view the question only...

 

Solution: Click on the graphic below to view both the question and answer....

 

Watch those signs!

Query: Solve these simultaneous equations by an algebraic method.

x²+ y²= 25, 2x + y - 5 = 0.

Solution: Click on the graphic below to see the solution. This is an example of how important it is to remember that...

1. square roots are either positive or negative and
2. you should check to see whether your solution works for both equations.

A Composite Problem.....

Query: Why isn't it always possible to define a composite function completely from the domains of the functions involved.

Response: Consider the worked example below...

 

Indices Challenge.....

Query: I need to simplify the following algebraic terms to lowest terms...

Divide the fraction [16a²b³/3ac⁴ ] by the fraction [32ab⁴/25c²].

Response: Please click on the graphic below to view the solution.

Circular Measure...

Query: A sector of a circle radius r subtends an angle of theta at the centre. The perimeter of the sector is 3m and the area is A sq.m. Show that A=0.5r(3-2r).

Response: Please click on the graphic to view the solution.