You're viewing 3 free preview questions for Maths AI HL — subscribe to unlock 219 more, plus the Full Question Bank, Mock Papers & Worksheets.Upgrade

Polynomials and Rational Functions · Paper 1 — Free Maths AI HL Practice Questions

1FoundationSAQ-SPolynomial functions and their graphs5 marksPaper 1~8 min

The volume,

V

cubic metres, of a rectangular box is given by

V(x) = x^3 + 6x^2 + 11x + 6, \quad x > 0

where

x

is a positive real number representing a scaled length.

(a)

Show that

(x+1)

is a factor of

V(x)

. [1 mark]

(b)

Hence write

V(x)

as a product of three linear factors. [2 marks]

(c)

The three linear factors of

V(x)

represent the three edge lengths of the box. Determine the value of

x

for which the box is a cube, and find the corresponding volume. [2 marks]

Solutions

2MasterySAQ-SPolynomial functions and their graphs5 marksPaper 1~8 min

A polynomial function is defined as

f(x) = 2x^3 + kx^2 - 8x - 12

, where

k

is a constant. It is given that

(x - 2)

is a factor of

f(x)

(a)

Show that

k = 3

. [2 marks]

(b)

Hence, write

f(x)

as a product of three linear factors. [2 marks]

(c)

Sketch the graph of

y = f(x)

for

-3 \leq x \leq 3

, labelling all

x

-intercepts and the

y

-intercept. [1 mark]

Solutions

3ChallengeSAQ-SPolynomial functions and their graphs5 marksPaper 1~8 min

A polynomial function

f

of degree 4 has a positive leading coefficient. The derivative of

f

f'(x) = 4(x+2)(x-1)(x-3).

(a)

Write down the

x

-coordinates of all local extrema of

f

, and state, with a reason, which is a local maximum and which is a local minimum. [2 marks]

(b)

Given that

f(0) = -36

, find

f(x)

in the form

f(x) = x^4 + bx^3 + cx^2 + dx + e

. [2 marks]

(c)

The tangent to the graph of

f

x = 3

passes through the point

(p,\, 0)

. Determine the value of

p

. [1 mark]

Solutions

Questions 4-222

Unlock these questions with Premium

That's every free question here — 3 of 222.

+219 more questions, plus Mock Papers & Worksheets, are waiting in Premium.

Upgrade