A Maths Question

• kkhing

  22 Sep 12, 15:10

  Please help to me with these problwm:

  1) Show that 3^loga = a^log3. Hence solve 2a^log3(3^loga) - 5a^log3 = 3.

  2) Given that λ is the root of the equation 2x^2 = 3x-4, show that 4λ^3 = λ-12.

  3. Given that 2^x+1.12^x = 3^1-2x, evluate 6^3x without the use of a calculator.

  4a). In the expansion (1+3x)^n, where n is a positive constant, the ratio of the  coefficents  of x^2 to x is 15:2. Find the value of n.

    b). Using the value of n found in (a), find the coefficient of x^2 in the expansion of (1+3x+3x^2)^6.

  Thanks & Regards.

   

• wee_ws

  22 Sep 12, 19:52

  Hi,

  For Q1:
  
  Proof
  1. Let y = RHS expression, take log of both sides.
  2. Let y = LHS expression, take log of both sides.
  3. Conclude the result.
  
  Application
  1. Use the earlier result to convert the equation to a quadratic one involving either RHS or LHS expression.
  2. Factorise.

    Thanks.

  Cheers,
  Wen Shih

• wee_ws

  22 Sep 12, 20:00

  Hi,

  For Q2:

  1. Since λ is a root, we must have 2λ^2 = 3λ - 4.
  2. Multiply both sides by 2λ.
  3. Use 2λ^2 = 3λ - 4 on the RHS expression.

    Thanks.

  Cheers,
  Wen Shih

• wee_ws

  22 Sep 12, 20:09

  Hi,

  For Q3:

  1. Simplify LHS expression using the fact that 12^x = (4^x)(3^x) and obtain an expression involving bases 2 and 3.
  2. Make RHS a constant.
  3. Simplify LHS using the fact that 6^(3x) = 2^(3x).3^(3x).

    Thanks.

  Cheers,
  Wen Shih

• wee_ws

  22 Sep 12, 20:17

  Hi,

  For Q4:

  Part (a)
  1. Coefficient of x = 3n, coefficient of x^2 = 4.5n(n-1).
  2. Use the given ratio to form a quadratic equation involving n.
  3. Solve for n to obtain a positive root.

  Part (b)
  1. Substitute x with x + x^2.
  2. Consider those x^2 terms.

    Thanks.

  Cheers,
  Wen Shih