WMA11 Pure Mathematics 1 (P1): Oct/Nov 2019 Paper 01

Figure 1 shows a sector AOB of a circle with centre O and radius r cm.

The angle AOB is 1.25 radians.

Given that the area of the sector AOB is 15 cm²

(a)find the exact value of r,[2]

(b)find the exact length of the perimeter of the sector. Write your answer in simplest form.[3]

A tree was planted in the ground.

Exactly 2 years after it was planted, the height of the tree was 1.85 m.

Exactly 7 years after it was planted, the height of the tree was 3.45 m.

Given that the height, H metres, of the tree, t years after it was planted in the ground, can be modelled by the equation

where a and b are constants,

(a)find the value of a and the value of b.[4]

(b)State, according to the model, the height of the tree when it was planted.[1]

Solutions relying on calculator technology are not acceptable.

Figure 2 shows a sketch of the curve C with equation y = x²– 5x + 13

The point M is the minimum point of C.

The straight line l passes through the origin O and intersects C at the points M and N as shown.

Find, showing your working,

(a)the coordinates of M,[3]

(b)the coordinates of N.[5]

Figure 3 shows the curve C and the line l. The finite region R, shown shaded in Figure 3, is bounded by C, l and the y-axis.

(c)Use inequalities to define the region R.[2]

A parallelogram ABCD has area 40 cm²

Given that AB has length 10 cm, BC has length 6 cm and angle DAB is obtuse, find

(a)the size of angle DAB, in degrees, to 2 decimal places,[3]

(b)the length of diagonal BD, in cm, to one decimal place.[2]

A curve has equation

(a)Find dydx , giving the answer in simplest form.[3]

The point P ⎛⎝ 4 ,11⎞3 ⎠ lies on the curve.

(b)Find the equation of the normal to the curve at P. Write your answer in the form ax + by + c = 0, where a, b and c are integers to be found.[4]

The curve C has equation y = + k, where k is a positive constant.

(a)Sketch a graph of C, stating the equation of the horizontal asymptote and the coordinates of the point of intersection with the x-axis.[3]

The line with equation y = 10 – 2x is a tangent to C.

(b)Find the possible values for k.[5]

Figure 4 shows part of the curve with equation y = 2x²+ 5

The point P(2,13) lies on the curve.

(a)Find the gradient of the tangent to the curve at P.[2]

The point Q with x coordinate 2 + h also lies on the curve.

(b)Find, in terms of h, the gradient of the line PQ. Give your answer in simplest form.[3]

(c)Explain briefly the relationship between the answer to (b) and the answer to (a).[1]

Solve, using algebra, the equation

Fully simplify your answers, writing them in the form a + b c, where a, b and c are integers to be found.[5]

Figure 5 shows a sketch of part of the curve C with equation y = sin ⎛ x ⎞⎝ 12⎠ , where x is

measured in radians. The point M shown in Figure 5 is a minimum point on C.

(a)State the period of C.[1]

(b)State the coordinates of M.[1]

The smallest positive solution of the equation sin ⎛ x ⎞⎝ 12⎠ k, where k is a constant, is Į

Find, in terms of Į,

(c)(i)the negative solution of the equation sin ⎛ x ⎞⎝ 12⎠ k that is closest to zero,

(ii)the smallest positive solution of the equation cos ⎛ x ⎞⎝ 12⎠ k.

[2]

Figure 6 shows a sketch of part of the curve with equation y = f(x), where

(a)Deduce the values of x for which f(x) - 0[2]

The curve crosses the y-axis at the point P, as shown.

(b)Expand f(x) to the form

where a, b, c and d are integers to be found.[3]

(c)Hence, or otherwise, find

(i)the coordinates of P,

(ii)the gradient of the curve at P.[2]

The curve with equation y = f(x) is translated two units in the positive x direction to a curve with equation y = g(x).

(d)(i)Find g(x), giving your answer in a simplified factorised form.

(ii)Hence state the y intercept of the curve with equation y = g(x).[3]

A curve has equation y = f(x).

The point P ⎛⎝ 4 , 32⎞3 ⎠ lies on the curve.

Ɣ މމI x) = − 3

Ɣ މI x) = 5 at P

find

(a)the equation of the tangent to the curve at P, writing your answer in the form y = mx + c, where m and c are constants to be found,[2]

(b)f(x).[8]