Higher Maths · SQA past paper

2023 Paper 1

Non-Calculator · 13 questions

Question 1

Differentiate or evaluate derivative: polynomial
3 Marks
2023 P1 Q1
Given that
y=x53−10x4\displaystyle y=x^{\frac{5}{3}}-\frac{10}{x^{4}} where x≠0\displaystyle x\ne0
find dydx\displaystyle \frac{dy}{dx}.
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53x23+40x−5\displaystyle \frac{5}{3}x^{\frac{2}{3}}+40x^{-5}

Question 2

Altitudes, medians, perpendicular bisectors
4 Marks
2023 P1 Q2

P and Q are the points (-2, 6) and (10, 0).
Find the equation of the perpendicular bisector of PQ.

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y=2x−5\displaystyle y=2x-5

Question 3

Solving equations containing a logarithm
3 Marks
2023 P1 Q3
Solve
log⁡5x−log⁡53=2\displaystyle \log_{5}x-\log_{5}3=2
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x=75\displaystyle x=75

Question 4

Apply compound angle formula to simplify or evaluate
(1, 1, 3)5 Marks
2023 P1 Q4

The diagram shows two right-angled triangles with angles p and q as marked.

Two right-angled triangles with angles p and q

(a)  Determine the value of:
      (i) cos⁡p\displaystyle \cos p
      (ii) cos⁡q\displaystyle \cos q
(b)  Hence determine the value of cos⁡(p+q)\displaystyle \cos(p+q).

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(a) (i) 35\displaystyle \frac{3}{5} (ii) 15\displaystyle \frac{1}{\sqrt{5}} or 345\displaystyle \frac{3}{\sqrt{45}}
(b) −15\displaystyle -\frac{1}{\sqrt{5}} or −345\displaystyle -\frac{3}{\sqrt{45}}

Question 5

Discriminant and Quadratics
3 Marks
2023 P1 Q5

The equation 2x2+(3p−2)x+p=0\displaystyle 2x^{2}+(3p-2)x+p=0 has equal roots.
Determine the possible values of p.

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p=29,2\displaystyle p=\frac{2}{9}, 2

Question 6

Integrate (definite or indefinite): polynomial
4 Marks
2023 P1 Q6
Find
∫(2x5−6x)dx,x≥0\displaystyle \int(2x^{5}-6\sqrt{x})dx, x\ge0
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13x6−4x32+c\displaystyle \frac{1}{3}x^{6}-4x^{\frac{3}{2}}+c

Question 7

Simplify numerical expression involving logs/exponentialsGraphs of logarithmic or exponential functions
(2, 1)3 Marks
2023 P1 Q7

(a)  Evaluate
log⁡25+log⁡2140\displaystyle \log_{2}5+\log_{2}\frac{1}{40}
(b)  Given that a∈R\displaystyle a\in\mathbb{R} and that log⁡8a\displaystyle \log_{8}a is negative, state the range of possible values of a.

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(a) -3
(b) 0<a<1\displaystyle 0 \lt a \lt 1

Question 8

Find stationary points and determine nature
6 Marks
2023 P1 Q8

A function, f, is defined on R\displaystyle \mathbb{R}, the set of real numbers, by
f(x)=x3+3x2−9x+5\displaystyle f(x)=x^{3}+3x^{2}-9x+5
Find the coordinates of the stationary points of f and determine their nature.

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Max at (-3, 32), Min at (1, 0)

Question 9

Identifying/sketching graphs of related functions (non-trigonometric)Inverse functions
3 Marks
2023 P1 Q9

The diagram shows the graph of the function f(x)=log⁡3x\displaystyle f(x)=\log_{3}x where x>0\displaystyle x \gt 0.

Graph of f(x)=log3x

The inverse function, f−1\displaystyle f^{-1}, exists.
On the diagram in your answer booklet, sketch the graph of y=f−1(x)−1\displaystyle y=f^{-1}(x)-1.

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Sketch showing concave up curve passing through (0,0) and (1,2), approaching y=−1\displaystyle y=-1

Question 10

Cubic/quartic expressions/equations: factorise or solveDiscriminant and Quadratics
(2, 5)7 Marks
2023 P1 Q10

(a)  Show that (x+5)\displaystyle (x+5) is a factor of
x4+3x3−7x2+9x−30\displaystyle x^{4}+3x^{3}-7x^{2}+9x-30
(b)  Hence, or otherwise, solve
x4+3x3−7x2+9x−30=0\displaystyle x^{4}+3x^{3}-7x^{2}+9x-30=0
x∈R.\displaystyle x\in\mathbb{R}.

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(a) Proof
(b) x=−5,x=2\displaystyle x=-5, x=2

Question 11

Integrate (definite or indefinite): trigonometric expressionAreas using integration
(3, 1)4 Marks
2023 P1 Q11

(a)  Evaluate
∫π2π(5sin⁡x−3cos⁡x)dx\displaystyle \int_{\frac{\pi}{2}}^{\pi}(5\sin x-3\cos x)dx
The diagram in your answer booklet shows the graphs with equations y=5sin⁡x\displaystyle y=5\sin x and y=3cos⁡x\displaystyle y=3\cos x, 0≤x≤2π\displaystyle 0\le x\le2\pi.

Graphs of y = 5 sin x and y = 3 cos x for x from 0 to 2 pi

(b)  On the diagram in your answer booklet, shade the area represented by the integral in (a).

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(a) 8
(b) Area shaded between the curves y=5sin⁡x\displaystyle y=5\sin x and y=3cos⁡x\displaystyle y=3\cos x, from x=π2\displaystyle x=\frac{\pi}{2} to x=π\displaystyle x=\pi

Question 12

Completing the square
3 Marks
2023 P1 Q12
Express −2x2−12x+7\displaystyle -2x^{2}-12x+7 in the form a(x+b)2+c.\displaystyle a(x+b)^{2}+c.
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−2(x+3)2+25\displaystyle -2(x+3)^{2}+25

Question 13

Composite functionsSolving a trigonometric equation using formula for sin(2x)
(1, 2, 1, 3)7 Marks
2023 P1 Q13

Functions f and g are defined by:
f(x)=2sin⁡x\displaystyle f(x)=2\sin x where 0<x<π2\displaystyle 0 \lt x \lt \frac{\pi}{2}
g(x)=2x\displaystyle g(x)=2x where 0<x<π4\displaystyle 0 \lt x \lt \frac{\pi}{4}
(a)  (i) Evaluate f(g(π6)).\displaystyle f(g(\frac{\pi}{6})).
(ii) Determine an expression for f(g(x)).\displaystyle f(g(x)).
(b)  (i) Given that f(p)=13\displaystyle f(p)=\frac{1}{3}, determine the exact value of sin⁡p.\displaystyle \sin p.
(ii) Hence, determine the exact value of f(g(p)).\displaystyle f(g(p)).

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(a) (i) 3\displaystyle \sqrt{3} (ii) 2sin⁡2x\displaystyle 2\sin 2x
(b) (i) 16\displaystyle \frac{1}{6} (ii) 359\displaystyle \frac{\sqrt{35}}{9}