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Higher Maths · SQA past paper

2022 Paper 1

Non-Calculator · 14 questions
1

Question 1

Perpendicular and parallel lines
3 Marks
2022 P1 Q1
Determine the equation of the line perpendicular to 5x+2y=7\displaystyle 5x+2y=7 passing through (−1,6)\displaystyle (-1,6).
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5y=2x+32\displaystyle 5y=2x+32 or 2x−5y+32=0\displaystyle 2x-5y+32=0
2

Question 2

Simplify numerical expression involving logs/exponentials
3 Marks
2022 P1 Q2
Evaluate
2log⁡36−log⁡34\displaystyle 2 \log_{3} 6-\log_{3} 4
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2
3

Question 3

Inverse functions
3 Marks
2022 P1 Q3

A function, h\displaystyle h, is defined by
h(x)=4+13x\displaystyle h(x)=4+\frac{1}{3}x
where x∈R\displaystyle x\in\mathbb{R}.
Find the inverse function, h−1(x)\displaystyle h^{-1}(x).

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h−1(x)=3(x−4)\displaystyle h^{-1}(x)=3(x-4)
4

Question 4

Differentiate or evaluate derivative: polynomial
3 Marks
2022 P1 Q4
Differentiate
y=x3−2x−1\displaystyle y=\sqrt{x^{3}}-2x^{-1}
where x>0\displaystyle x \gt 0.
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32x12+2x−2\displaystyle \frac{3}{2}x^{\frac{1}{2}}+2x^{-2}
5

Question 5

Angles and straight lines: m = tanθ
3 Marks
2022 P1 Q5

A line makes an angle of π3\displaystyle \frac{\pi}{3} radians with the y-axis, and passes through the point (−2,0)\displaystyle (-2,0) as shown below.

Graph showing a line passing through (-2,0) making an angle of pi/3 with the positive y-axis

Determine the equation of the line.

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y=13x+23\displaystyle y=\frac{1}{\sqrt{3}}x+\frac{2}{\sqrt{3}} or 3y=x+2\displaystyle \sqrt{3}y=x+2
6

Question 6

Integrate (definite or indefinite): (px + q)^n
4 Marks
2022 P1 Q6
Evaluate
∫−52(10−3x)−12dx\displaystyle \int_{-5}^{2}(10-3x)^{-\frac{1}{2}}dx
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2
7

Question 7

Apply compound angle formula to simplify or evaluate
(1, 1, 3)5 Marks
2022 P1 Q7

Triangles ABC and ADE are both right angled. Angle BAC=q\displaystyle BAC=q and angle DAE=r\displaystyle DAE=r as shown in the diagram.

Diagram showing right-angled triangles ABC and ADE with angles q and r

(a)  Determine the value of:
      (i) sin⁡r\displaystyle \sin r
      (ii) sin⁡q\displaystyle \sin q
(b)  Hence determine the value of sin⁡(q−r)\displaystyle \sin(q-r).

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(a)(i) 110\displaystyle \frac{1}{\sqrt{10}} (ii) 313\displaystyle \frac{3}{\sqrt{13}}
(b) 7130\displaystyle \frac{7}{\sqrt{130}}
8

Question 8

Solving equations containing a logarithm
4 Marks
2022 P1 Q8
Solve
log⁡6x+log⁡6(x+5)=2\displaystyle \log_{6}x+\log_{6}(x+5)=2
where x>0\displaystyle x \gt 0.
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x=4\displaystyle x=4
9

Question 9

Solving a trigonometric equation using formula for cos(2x)
5 Marks
2022 P1 Q9
Solve the equation
cos⁡2x∘=5cos⁡x∘−3\displaystyle \cos 2x^{\circ}=5 \cos x^{\circ}-3
for 0≤x<360\displaystyle 0\le x \lt 360.
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x=60,300\displaystyle x=60, 300
10

Question 10

Identifying/sketching graphs of related functions (non-trigonometric)
(3, 1)4 Marks
2022 P1 Q10

The diagram shows the graph of a cubic function with equation y=f(x)\displaystyle y=f(x). The curve has stationary points at (0,3)\displaystyle (0,3) and (4,0)\displaystyle (4,0).

Graph of y=f(x) with stationary points at (0,3) and (4,0)

(a)  Sketch the graph of y=2f(x)+1\displaystyle y=2f(x)+1.
(b)  State the coordinates of the stationary points on the graph of y=f(12x)\displaystyle y=f(\frac{1}{2}x).

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(a) Sketch with max turning point at (0,7)\displaystyle (0,7) and min turning point at (4,1)\displaystyle (4,1).
(b) (0,3)\displaystyle (0,3) and (8,0)\displaystyle (8,0).
11

Question 11

Completing the square
3 Marks
2022 P1 Q11
Express
2x2+12x+23\displaystyle 2x^{2}+12x+23
in the form p(x+q)2+r\displaystyle p(x+q)^{2}+r.
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2(x+3)2+5\displaystyle 2(x+3)^{2}+5
12

Question 12

Differentiate or evaluate derivative: trigonometric expression
3 Marks
2022 P1 Q12
Given that f(x)=4sin⁡(3x−π3)\displaystyle f(x)=4 \sin(3x-\frac{\pi}{3})
evaluate f′(π6)\displaystyle f^{\prime}(\frac{\pi}{6}).
Show answer
63\displaystyle 6\sqrt{3}
13

Question 13

Cubic/quartic expressions/equations: factorise or solveIdentifying/sketching graphs of related functions (non-trigonometric)
(2, 3, 1)6 Marks
2022 P1 Q13

(a)  (i) Show that (x+2)\displaystyle (x+2) is a factor of
f(x)=x3−2x2−20x−24\displaystyle f(x)=x^{3}-2x^{2}-20x-24
      (ii) Hence, or otherwise, solve f(x)=0\displaystyle f(x)=0.

The diagram shows the graph of y=f(x)\displaystyle y=f(x).

Graph of y=f(x)

(b)  The graph of y=f(x−k)\displaystyle y=f(x-k), k>0\displaystyle k \gt 0 has a stationary point at (1,0)\displaystyle (1,0). State the value of k\displaystyle k.

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(a)(ii) x=−2,6\displaystyle x=-2, 6
(b) k=3\displaystyle k=3
14

Question 14

Circle equation from radius/centre or vice versaIntersections of two circles
(2, 2, 2)6 Marks
2022 P1 Q14

C1\displaystyle C_{1} is the circle with equation
(x−7)2+(y+5)2=100\displaystyle (x-7)^{2}+(y+5)^{2}=100
(a)  (i) State the centre and radius of C1\displaystyle C_{1}.
      (ii) Hence, or otherwise, show that the point P(−2,7)\displaystyle P(-2, 7) lies outside C1\displaystyle C_{1}.

C2\displaystyle C_{2} is a circle with centre P and radius r\displaystyle r.
(b)  Determine the value(s) of r\displaystyle r for which circles C1\displaystyle C_{1} and C2\displaystyle C_{2} have exactly one point of intersection.

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(a)(i) Centre (7,−5)\displaystyle (7, -5), Radius 10. (ii) Distance PC = 15, which is greater than radius 10.
(b) r=5\displaystyle r=5 or r=25\displaystyle r=25