A Maths Question

kkhing SG

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 SG

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 SG

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 SG

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 SG

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