1.02d Quadratic functions: graphs and discriminant conditions

In this question you must show detailed reasoning.

1. Show that the equation \(m \sec \theta + 3 \cos \theta = 4 \sin \theta\) can be expressed in the form $$m \tan^2 \theta - 4 \tan \theta + (m + 3) = 0.$$ [3]
2. It is given that there is only one value of \(\theta\), for \(0 < \theta < \pi\), satisfying the equation \(m \sec \theta + 3 \cos \theta = 4 \sin \theta\). Given also that \(m\) is a negative integer, find this value of \(\theta\), correct to 3 significant figures. [5]

A curve has equation $$y = 2x^2 + px + 1$$ A line has equation $$y = 5x - 2$$ Find the set of values of \(p\) for which the line intersects the curve at two distinct points. Give your answer in exact form. [5 marks]

The quadratic equation \(3x^2 + 4x + (2k - 1) = 0\) has real and distinct roots. Find the possible values of the constant \(k\) Fully justify your answer. [4 marks]

Find the values of \(k\) for which the equation \((2k - 3)x^2 - kx + (k - 1) = 0\) has equal roots. [4 marks]

A curve, C, has equation \(y = x^2 - 6x + k\), where \(k\) is a constant. The equation \(x^2 - 6x + k = 0\) has two distinct positive roots.

1. Sketch C on the axes below. [2 marks]
2. Find the range of possible values for \(k\). Fully justify your answer. [4 marks]

A curve \(C\) has equation \(y = x^2 - 4x + k\), where \(k\) is a constant. It crosses the \(x\)-axis at the points \((2 + \sqrt{5}, 0)\) and \((2 - \sqrt{5}, 0)\)

1. Find the value of \(k\). [2 marks]
2. Sketch the curve \(C\), labelling the exact values of all intersections with the axes. [3 marks]

The quadratic equation $$4x^2 + bx + 9 = 0$$ has one repeated real root. Find \(b\) Circle your answer. [1 mark] \(b = 0\) \quad \(b = \pm 12\) \quad \(b = \pm 13\) \quad \(b = \pm 36\)

The equation \(kx^2 + 4kx + 3 = 0\), where \(k\) is a constant, has no real roots. Prove that $$0 \leqslant k < \frac{3}{4}$$ [4]

Find the set of values of \(a\) for which the equation $$ax^2 + 8x + 2 = 0$$ has no real roots. [3]

Fig. 12 shows \(x\)- and \(y\)- coordinate axes with origin O and the trajectory of a particle projected from O with speed 28 m s\(^{-1}\) at an angle \(\alpha\) to the horizontal. After \(t\) seconds, the particle has horizontal and vertical displacements \(x\) m and \(y\) m. Air resistance should be neglected. \includegraphics{figure_12}

1. Show that the equation of the trajectory is given by $$\tan^2\alpha - \frac{160}{x}\tan\alpha + \frac{160y}{x^2} + 1 = 0.$$ (*) [5]
2. 1. Show that if (*) is treated as an equation with \(\tan\alpha\) as a variable and with \(x\) and \(y\) as constants, then (*) has two distinct real roots for \(\tan\alpha\) when \(y < 40 - \frac{x^2}{160}\). [3]
   2. Show the inequality in part (ii)(A) as a locus on the graph of \(y = 40 - \frac{x^2}{160}\) in the Printed Answer Booklet and label it R. [1]

S is the locus of points \((x, y)\) where (*) has one real root for \(\tan\alpha\). T is the locus of points \((x, y)\) where (*) has no real roots for \(\tan\alpha\).

1. Indicate S and T on the graph in the Printed Answer Booklet. [2]
2. State the significance of R, S and T for the possible trajectories of the particle. [3]

A machine can fire a tennis ball from ground level with a maximum speed of 28 m s\(^{-1}\).

1. State, with a reason, whether a tennis ball fired from the machine can achieve a range of 80 m. [1]

Find all the values of \(k\) for which the equation \(x^2 + 2kx + 9k = -4x\) has two distinct real roots. [7]

Find the range of values of \(k\) for which the quadratic equation \(x^2 + 2kx + 8k = 0\) has no real roots. [4]

The curve \(C\) has equation \(y = 2x^2 + 5x - 12\) and the line \(L\) has equation \(y = mx - 14\), where \(m\) is a real constant.

1. Given that \(L\) is a tangent to \(C\),
   1. show that \(m\) satisfies the equation $$m^2 - 10m + 9 = 0,$$ [5]
   2. find the coordinates of the two possible points of contact of \(C\) and \(L\). [6]
2. Given instead that \(L\) intersects \(C\) at two distinct points, find the range of values of \(m\). [2]

The quadratic equation \(4x^2 - 12x + m = 0\), where \(m\) is a positive constant, has two distinct real roots. Show that the quadratic equation \(3x^2 + mx + 7 = 0\) has no real roots. [7]

Show that, for any value of the real constant \(b\), the equation \(x^3 - (b + 1)x + b = 0\) has \(x = 1\) as a solution. Find all values of \(b\) for which this equation has exactly two real solutions [5]

The equation \(k(3x^2 + 8x + 9) = 2 - 6x\), where \(k\) is a real constant, has no real roots.

1. Show that \(k\) satisfies the inequality $$11k^2 - 30k - 9 > 0$$ [4]
2. Find the range of possible values for \(k\). [4]

Given that $$f(x) = x^2 - 4x + 5 \quad x \in \mathbb{R}$$

1. express \(f(x)\) in the form \((x + a)^2 + b\) where \(a\) and \(b\) are integers to be found. [2]

The curve with equation \(y = f(x)\) • meets the \(y\)-axis at the point \(P\) • has a minimum turning point at the point \(Q\)

1. Write down
   1. the coordinates of \(P\)
   2. the coordinates of \(Q\)
   [2]

The equation $$(k + 3)x^2 + 6x + k = 5$$, where \(k\) is a constant, has two distinct real solutions for \(x\).

1. Show that \(k\) satisfies $$k^2 - 2k - 24 < 0$$ [4]
2. Hence find the set of possible values of \(k\). [2]

A circle \(C\) with radius \(r\)

• lies only in the 1st quadrant
• touches the \(x\)-axis and touches the \(y\)-axis

The line \(l\) has equation \(2x + y = 12\)

1. Show that the \(x\) coordinates of the points of intersection of \(l\) with \(C\) satisfy $$5x^2 + (2r - 48)x + (r^2 - 24r + 144) = 0$$ [3]

Given also that \(l\) is a tangent to \(C\),

1. find the two possible values of \(r\), giving your answers as fully simplified surds. [4]

\includegraphics{figure_5} \includegraphics{figure_6} A suspension bridge cable \(PQR\) hangs between the tops of two vertical towers, \(AP\) and \(BR\), as shown in Figure 5. A walkway \(AOB\) runs between the bases of the towers, directly under the cable. The towers are 100 m apart and each tower is 24 m high. At the point \(O\), midway between the towers, the cable is 4 m above the walkway. The points \(P\), \(Q\), \(R\), \(A\), \(O\) and \(B\) are assumed to lie in the same vertical plane and \(AOB\) is assumed to be horizontal. Figure 6 shows a symmetric quadratic curve \(PQR\) used to model this cable. Given that \(O\) is the origin,

1. find an equation for the curve \(PQR\). [3] Lee can safely inspect the cable up to a height of 12 m above the walkway. A defect is reported on the cable at a location 19 m horizontally from one of the towers.
2. Determine whether, according to the model, Lee can safely inspect this defect. [2]

It is given that $$f(x) = x^2 - kx + (k+3),$$ where \(k\) is a constant. If the equation \(f(x) = 0\) has real roots find the range of the possible values of \(k\). [4]

In this question you must show detailed reasoning. A curve has equation $$y = 2x^2 + px + 1$$ A line has equation $$y = 5x - 2$$ Find the set of values of \(p\) for which the line intersects the curve at two distinct points. Give your answer in exact form using set notation. [6]