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

1FoundationSAQ-SPolynomial functions and their graphs5 marksPaper 1~8 min
The graph of f(x)=a(x−p)(x−q)(x−r)f(x) = a(x - p)(x - q)(x - r) is shown below.
(a)
State the number of real zeros of f(x)f(x). [1 mark]
(b)
State the degree of f(x)f(x). [1 mark]
(c)
Determine the value of aa, the leading coefficient of f(x)f(x). [3 marks]
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Solutions

2FoundationSAQ-SPolynomial functions and their graphs5 marksPaper 1~8 min
The graph of a polynomial function f(x)f(x) is shown below.
(a)
State the number of distinct real zeros of f(x)f(x). [1 mark]
(b)
State the degree of f(x)f(x). [1 mark]
(c)
Given that f(x)=a(x+1)(x−2)2(x−4)f(x) = a(x+1)(x-2)^{2}(x-4), find the value of the leading coefficient aa. [3 marks]
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Solutions

3MasterySAQ-SPolynomial functions and their graphs5 marksPaper 1~8 min
A ball is thrown vertically upwards from a platform. Its height, hh metres, above the ground after tt seconds is given by h(t)=−4.9t2+19.6t+2.5,t≥0.h(t) = -4.9t^2 + 19.6t + 2.5, \quad t \geq 0.
(a)
Write down the height of the platform above the ground. [1 mark]
(b)
Find the time at which the ball reaches its maximum height. [2 marks]
(c)
The ball is considered "in safe range" if it remains above 1010 metres. Determine the total time, in seconds, that the ball is in safe range. Give your answer correct to 3 significant figures. [2 marks]
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Solutions

4ChallengeSAQ-LPolynomial functions and their graphs8 marksPaper 1~12 min

Data

$$
A manufacturing company produces cylindrical containers. Each container has radius r cmr\,\text{cm}, height h cmh\,\text{cm}, and includes a top and a bottom. The total surface area of each container is fixed at 600π cm2600\pi\,\text{cm}^2.
(a)
Show that the volume of the container can be expressed as $$V(r) = 300\pi r - \pi r^3. \quad [3 marks]
(b)
State the values of rr for which V(r)>0V(r) > 0, giving a reason based on the physical constraints of the container. [2 marks]
(c)
Calculate the value of rr that maximises V(r)V(r), correct to 3 significant figures. [2 marks]
(d)
A competitor claims their container holds more than 6400 cm36400\,\text{cm}^3 while satisfying the same surface area constraint. Determine whether this claim is possible, justifying your answer. [1 mark]
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Solutions

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