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Understanding Basic Math – Test 3

Monday, May 13th, 2013

 

Euclidean Geometry

This is the third in a series of tests of understanding basic math. You need to be able to answer these questions accurately and confidently if you intend to take college-level math courses in Calculus, or Statistics. In answering the questions, please provide typical examples.

Angles and Triangles

  1. Define the term “angle.” When are two angles equal?
  2. Define the terms “straight angle”, “right angle”, and “acute angle.”
  3. Explain how the size of an angle is specified, in terms of degrees. (How much is 360° ?)
  4. Define the term “triangle.” Also define “right triangle”, “isosceles triangle”, and “equilateral triangle.”
  5. State the theorem about the sum of the interior angles of a triangle. Give the proof.
  6. Find the size of each angle in (a) an equiangular triangle, and (b) an isosceles right triangle.
  7. State Pythagoras’s Theorem and give the proof.
  8. Define “congruence” for triangles (and other geometric figures).
  9. One set of sufficient conditions for two triangles to be congruent is that the three sides of the first triangle are equal to the three sides of the second triangle. (a) Explain why this is true. (b) Is the analogous statement true for quadrilaterals?
  10. Give two other sets of sufficient conditions for congruence of triangles.
  11. Prove that the base angles of an isosceles triangle are equal. What is the converse?
  12. Prove that an equilateral triangle is also equiangular. State and prove the converse.

 

Circles, Arcs, and Sectors

  1. Define radian measure for angles. Give examples.
  2. 1° is how many radians? 1 radian is how many degrees?
  3. Define the terms “circle”, “arc”, and “sector.”
  4. State the formulas for the circumference of a circle, and for the area inside a circle. What do the formulas have to do with scaling?
  5. State and derive the formula for arc length.
  6. Ditto for the area inside a sector of a circle.
  7. IF ABC is a triangle inscribed in a circle (i.e., the vertices A, B and C all lie on the circle), with AB being a diameter of the circle, prove that ABC is a right triangle. Suggeston: draw the line OC from the center of the circle O to point C, and look for isosceles triangles.

Solving Right Triangles

  1. Given two sides of a right triangle, show how to find the third side.
  2. Define sin θ, cos θ, and tan θ for positive acute angles θ.
  3. Calculate these functions for θ = 30°, 45°, and 60°. Use geometry, not a calculator.
  4. Given two sides of a right triangle, show how to find the angles using trigonometry (and a calculator).
  5. Ditto, given one side and one angle of the triangle.

 

Understanding Basic Math – Test 2

Saturday, May 4th, 2013

 

Algebra of Polynomials

This is the second in a series of tests of understanding basic math. You need to be able to answer these questions accurately and confidently if you intend to take college-level math courses in Calculus, or Statistics. In answering the questions, please provide typical examples.

Exponents

  1. Explain how the expression xn is defined, if n is a positive whole number (n = 1,2,3,…) and x is any real number.
  2. Prove that xn xm  = xn+m.
  3. Prove that (xy)n  = xn yn and that (xn)m  = xnm.
  4. Explain why x0 is defined as x0  = 1 (for x not equal to 0).
  5. How is xn defined if n is a negative integer?
  6. Prove that xn / xm  = xn-m.

Polynomials

  1. Define “polynomial.” Also define “degree” of a polynomial.
  2. Show how to add and multiply two polynomials.
  3. Expand (x + a)2 and explain.
  4. Explain division of polynomials with remainder. Illustrate with an example.
  5. Show how to solve a linear equation ax + b = 0.

Quadratic Polynomials

  1. What is a quadratic polynomial?
  2. Explain how to factor a quadratic polynomial by trial and error.
  3. State the quadratic formula, with examples.
  4. Explain the method called “completing the square.”
  5. Use completing the square to prove the quadratic formula.
  6. Define the discriminant of a quadratic polynomial.

The Binomial Theorem

  1. Expand (a + b)3.
  2. Describe Pascal’s triangle.
  3. What are the binomial coefficients C(n,k)? How are they related to Pascal’s triangle?
  4. State the Binomial Theorem, using ∑ notation.

Roots of Polynomials

  1. Define the terms “root of a polynomial” and “factor of a polynomial.”
  2. Explain why any polynomial of odd degree must have at least one real root.
  3. State and prove the remainder theorem.
  4. State and prove the factor theorem.