Miscellaneous Problems




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If a problem is solved. It is not 'the' answer.
No attempt is made to search for the most elegant answer.
I highly recommend that you at least try to solve the problem before you read the solution.

Miscellaneous solved problems

About roots of a quadratic equation

Given is the equation
 
        (m2 - 5m + 6)x2 + (4 - m2)x + 20 = 0
Say the roots are x' and x", with x' < x" or x'=x".
Calculate the values of the parameter m such that x' < 1 < x".

Elimination

 
For which values of m the quadratic equation

       2x2 + mx - 12 = 0             (1)

has a common root with the quadratic equation

        x2 - (m+1)x + m = 0             (2)

Formula about the area of a triangle ABC

Given : the coordinates of A,B and C relative to an orthonormal coordinate system in a plane.
A(a,a') , B(b,b') , C(c,c')
Prove that the area of the triangle is the absolute value of
 
          1   |a        a'      1|
         --- .|b        b'      1|
          2   |c        c'      1|

Equation with integral

 
Calculate all positive x-values such that

        /x dt
        |  ---  =  2
        /1  t

Asymptotes, tangent line, integration

 
Let G be the graph of

                9x2 + mx + 4
        f(x) = ----------------  with m as real parameter
                2x - 7
  • Calculate m such that G and the x-axis have just one point in common.
  • Calculate m such that the line 9x - 2y + 3 = 0 is asymptote of G.
  • Calculate the slope of the tangent line in the intersection point of G and the y-axis.
  • Calculate
     
            /
            | f(x) dx
            /
    

Integral equation

 
Calculate all positive x-values such that

        /e    dt        /x    dt
        |     ---  =    |     --
        /1/e   t        /e     t

Trigonometry - formulas

 
Given : x and y in ]0, pi/2[
Show that :

   1 + 2cos(4y) - (sin(6x)/sin(2x)) = 16.sin(x-y)sin(x+y)cos(x-y)cos(x+y)

 
Solve
          _______________
         |      2  x
         | 2 cos  --- - 1  > 2 sin(x) - 3
        \|         2

 
a,b and c are the angles of a triangle.
Show that we have a right-angled triangle if

        sin(2 b) sin(2 c) = 2 - 2 sin2(b) cos2(c) - 2 cos2(b) sin2(c)

Arithmetic and geometric sequences

 
a, b, c form arithmetic sequence and x, y, z form a geometric sequence.
Prove that
                xb.yc.za = xc.ya.zb

Simplify a trigonometric expression

 
Simplify
        (1 + tan(a).tan(a/2))-1

Trigonometric equation

 
If m is a real parameter, then solve :

       m.cos2(x) + (2m2 - m + 1)sin(x) -3m + 1 = 0

Relative maxima and minima and trigonometry

 
Given:
        f(x) = (m - 1)cos2(x) -3m cos(x) + 2m

        with m a real parameter ( m is not 1).

Determine all m  such that there are four different x values in
[0, 2.pi[ so that the image is a relative maximum or minimum.

Relative maximum and minimum

Given : f(x) = sqrt(2x.x + k) - x/3 - 2
Calculate k such that the x value, corresponding with the relative maximum or minimum of f(x), is equal to that relative maximum or minimum.

Sequences and sums

Given :
Two constant values a and b.
A sequence {t(n)}. The sum of the first n terms is S(n)=a.n2 + b.n
With the sequence we construct a new sequence {t'(n)} such that t'(n) = t(2n).

Calculate the sum S'(n) of the first n terms of {t'(n)}.


Elimination and trigonometry

The numbers a,b,c are constant.
Eliminate x out of sin(a + x) = 2b ; sin(a - x) = 2c.

Factoring

 
Factor

    z = (x + y)5 - x5 - y5

Range or image of a function

 
Given : The function
                mx2 - 7x + 5
        f(x) = ---------------
                5x2 - 7x + m

The set of all the images is called the range or image of the function.

Determine m such that the range  of f(x) is the set of all real numbers.

Lines in space and linear systems

 
Given : Two lines in space with equations

        / 2x + my + z = 1       / 3x      + z = 2
        |                 and   |
        \ x - y + mz = 1        \ 2x + my + z = m - 1

  Determine the values of m such that the two lines have
  an intersection point.

Calculation about the range of a function

 
Calculate m such that the range of the function
                 m x2 + 3 x - 4
        f : x -> -----------------
                 m + 3 x  - 4 x2
 is equal to R.

Solving a system with exponential and trigonometric functions

 
Solve the system

        /
        | e2x  + cos2(y) = 1
        |
        |
        | e3x cos(y) = cos(3 y)
        \

Loci




Topics and Problems

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