WMA11 Pure Mathematics 1 (P1): January 2020 Paper 01

Find, in simplest form,

[4]

Given y = 3^x, express each of the following in terms of y. Write each expression in its simplest form.

(a)3^3x[1]

(b)_x−2 3[2]

(c)_2−3x

[2]

Figure 1 shows part of the curve with equation y = x²+ 3x – 2

The point P(3,16) lies on the curve.

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

The point Q with x coordinate 3 + h also lies on the curve.

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

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

Figure 2 shows the plan view of a house ABCD and a lawn APCDA.

ABCD is a rectangle with AB = 16 m.

APCOA is a sector of a circle centre O with radius 12 m.

The point O lies on the line DC, as shown in Figure 2.

(a)Show that the size of angle AOD is 1.231 radians to 3 decimal places.[2]

The lawn APCDA is shown shaded in Figure 2.

(b)Find the area of the lawn, in m², to one decimal place.[4]

(c)Find the perimeter of the lawn, in metres, to one decimal place.[3]

(a)Find, using algebra, all solutions of

[3]

(b)Hence find all real solutions of

[4]

The line l₁ has equation 3x – 4y + 20 = 0

The line l₂ cuts the x-axis at R(8,0) and is parallel to l₁

(a)Find the equation of l₂, writing your answer in the form ax + by + c = 0, where a, b and c are integers to be found.[3]

The line l₁ cuts the x-axis at P and the y-axis at Q.

Given that PQRS is a parallelogram, find

(b)the area of PQRS,[3]

(c)the coordinates of S.[2]

Figure 3 shows part of the curve C₁ with equation y = 3sin x, where x is measured in degrees.

The point P and the point Q lie on C₁ and are shown in Figure 3.

(a)State

(i)the coordinates of P,

(ii)the coordinates of Q.[3]

A different curve C₂ has equation y = 3sin x + k, where k is a constant.

The curve C₂ has a maximum y value of 10

The point R is the minimum point on C₂ with the smallest positive x coordinate.

(b)State the coordinates of R.[2]

The straight line l has equation y = k(2x – 1), where k is a constant.

The curve C has equation y = x²+ 2x + 11

Find the set of values of k for which l does not cross or touch C.[6]

A curve has equation

Find the x coordinate of the point on the curve at which = 0

[6]

The curve C₁ has equation y = f(x), where

(a)Sketch C₁ showing the coordinates of any point where the curve touches or crosses the coordinate axes.[3]

(b)Hence or otherwise

(i)find the values of x for which f  1 4 x 0

(ii)find the value of the constant p such that the curve with equation y = f(x) + p passes through the origin.[2]

A second curve C₂ has equation y = g(x), where g(x) = f(x + 1)

(c)(i)Find, in simplest form, g(x). You may leave your answer in a factorised form.

(ii)Hence, or otherwise, find the y intercept of curve C₂[3]

A curve has equation y = f(x), where

The point P(4, –50) lies on the curve.

Given that fʹ(x) = –4 at P,

(a)find the equation of the normal at P, writing your answer in the form y = mx + c, where m and c are constants,[3]

(b)find f(x).[8]