1 (i) Are there values of \( k \) such that the curve \( y=(k+3) x^{2}-3 x \) lies completely above the line \( y=x+k \) ? Justify your answer. (ii) Explain the relationship between the curve and the line in part (i).

To determine if there are values of \( k \) such that the curve \( y = (k + 3)x^2 - 3x \) lies completely above the line \( y = x + k \), we need to analyze the relationship between these two equations. ### Step 1: Set up the inequality We want to find when the curve is above the line, which means we need to solve the inequality: \[ (k + 3)x^2 - 3x > x + k \] ### Step 2: Rearrange the inequality Rearranging the inequality gives us: \[ (k + 3)x^2 - 3x - x - k > 0 \] This simplifies to: \[ (k + 3)x^2 - 4x - k > 0 \] ### Step 3: Analyze the quadratic inequality This is a quadratic inequality in the form \( ax^2 + bx + c > 0 \), where: - \( a = k + 3 \) - \( b = -4 \) - \( c = -k \) For the quadratic to be positive for all \( x \), the following conditions must be satisfied: 1. The coefficient \( a \) must be positive: \( k + 3 > 0 \) or \( k > -3 \). 2. The discriminant must be negative: \( b^2 - 4ac < 0 \). ### Step 4: Calculate the discriminant The discriminant \( D \) is given by: \[ D = b^2 - 4ac = (-4)^2 - 4(k + 3)(-k) \] Calculating this gives: \[ D = 16 + 4(k + 3)k = 16 + 4k^2 + 12k \] We need to set this less than zero: \[ 4k^2 + 12k + 16 < 0 \] ### Step 5: Solve the quadratic inequality To solve \( 4k^2 + 12k + 16 < 0 \), we can first find the roots of the equation \( 4k^2 + 12k + 16 = 0 \) using the quadratic formula: \[ k = \frac{-b \pm \sqrt{D}}{2a} = \frac{-12 \pm \sqrt{(12)^2 - 4 \cdot 4 \cdot 16}}{2 \cdot 4} \] Calculating the discriminant for this quadratic: \[ D = 144 - 256 = -112 \] Since the discriminant is negative, the quadratic \( 4k^2 + 12k + 16 \) does not cross the k-axis and is always positive. Therefore, there are no values of \( k \) such that \( 4k^2 + 12k + 16 < 0 \). ### Conclusion for part (i) Thus, there are no values of \( k \) such that the curve \( y = (k + 3)x^2 - 3x \) lies completely above the line \( y = x + k \). ### Part (ii): Relationship between the curve and the line The relationship between the curve and the line is that the curve will either intersect the line or lie below it for all values of \( k \). Since we found that the quadratic does not have any values of \( k \) for which it is entirely above the line, it indicates that the curve either touches or crosses the line at some points, depending on the specific value of \( k \). In summary, the curve and the line will always have points of intersection or will be below the line for all values of \( k \).