WMA11 Pure Mathematics 1 (P1): January 2022 Paper 01

Find

giving each term in simplest form.[4]

(a)Express f(x) in the form

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

(b)Sketch the graph of the curve C with equation y = f(x), showing clearly the coordinates of the point where the curve crosses the y‑axis.[2]

(c)Write down the equation of the line of symmetry of C.[1]

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

(i)

Express f(x) in the form ax²+ bx 2 + c where a, b and c are integers to be found.[3]

(ii)Solve the equation

giving your answer in the form p + q 3 where p and q are simplified fractions to be found.[4]

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

Figure 1 shows a line l with equation x + y = 6 and a curve C with equation y = 6x – 2x²+ 1

The line l intersects the curve C at the points P and Q as shown in Figure 1.

(a)Find, using algebra, the coordinates of P and the coordinates of Q.[4]

The region R, shown shaded in Figure 1, is bounded by C, l and the x‑axis.

(b)Use inequalities to define the region R.[3]

Figure 2 shows a plan view of a semicircular garden ABCDEOA

The semicircle has

• centre O

• diameter AOE

• radius 3 m

The straight line BD is parallel to AE and angle BOA is 0.7 radians.

(a)Show that, to 4 significant figures, angle BOD is 1.742 radians.[1]

The flowerbed R, shown shaded in Figure 2, is bounded by BD and the arc BCD.

(b)Find the area of the flowerbed, giving your answer in square metres to one decimal place.[3]

(c)Find the perimeter of the flowerbed, giving your answer in metres to one decimal place.[3]

The curve C has equation y = f(x) where x > 0

Given that

• the point P(4, 20) lies on C

(a)(i)find the value of the gradient at P

(ii)Hence find the equation of the tangent to C at P, giving your answer in the form ax + by + c = 0 where a, b and c are integers to be found.[4]

(b)Find f(x), simplifying your answer.[7]

Figure 3 shows a sketch of part of the curve with equation y = f(x), where

Given that the curve with equation y = f(x) – p passes through the point with coordinates (0, 50)

(a)find the value of the constant p.[2]

Given that the curve with equation y = f(x + q) passes through the origin,

(b)write down the possible values of the constant q.[2]

(c)Find f′(x).[4]

(d)Hence find the range of values of x for which the gradient of the curve with equation y = f(x) is less than –18[3]

The line l₁ has equation

(a)Find the gradient of l₁[1]

Given that

• the point A has coordinates (6, –2)

• the line l₂ passes through A and is perpendicular to l₁

(b)find the equation of l₂ giving your answer in the form y = mx + c, where m and c are constants to be found.[3]

The lines l₁ and l₂ intersect at the point M.

(c)Using algebra and showing all your working, find the coordinates of M.

[3]

Given that the diagonals of a square ABCD meet at M,

(d)find the coordinates of the point C.[2]

Figure 4 shows part of the curve with equation

where A is a constant.

The point P is a minimum point on the curve and has coordinates (30, –3) as shown in Figure 4.

(a)Write down the value of A.[1]

The point Q is shown in Figure 4 and is a maximum point.

(b)Find the coordinates of Q.[3]

The curve C has equation

(a)Sketch the graph of C.

On your sketch

• show the coordinates of any points of intersection with the coordinate axes

• state clearly the equations of any asymptotes[4]

The curve D has equation y = kx² where k is a constant.

Given that C meets D at 4 distinct points,

(b)find the range of possible values for k.[5]