IB Mathematics Higher Level Sample Questions

Mathematics HL – Paper One Style Questions

ALGEBRA Question 1 An arithmetic sequence is defined in such a way that the 15th term, u 15 = 92 , and the 3rd term, u 3 = 56 . (a) Determine (i) the common difference. (ii) the first term. (b) Calculate S 12 .

Question 2 u A geometric sequence is such that its 4th term, u 4 = 135 and the ratio ----9- = 3 5 . u4 (a) Find (i) the common ratio. (ii) the first term. (b) If S 10 = a ( b – 1 ) , determine the real constants a and b.

Question 3 Consider the sequence defined by the equation u n = 1 – 9n, n = 1, 2, 3, … , where u n represents the nth term of the sequence. (a)

Write down the value of u 1, u 2 and u 3 . 10

(b)

Calculate

∑ ( 1 – 9n ) .

n=1

Question 4 Solve for x in each of the following equations (a)

log 2x 4 = log 16 .

(b)

log x27 = 3 .

(c)

log 432 = x .

(d)

log 3( 1 – x ) – log 3x = 2 .

2

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Question 5 The nth term of a sequence is given by 2 u n = --- × 3 n, n = 1, 2, 3, … . 3 20

If

∑  --3- × 3 n 2

= a ( 3 20 – b ) , determine the values of a and b.

n=1

Question 6 Solve the following equations for x. (a) (b) (c)

x – 4 = 4. x – 4 = 4. log 2( x – 4 ) = 4 .

(d) (e)

x2 – 4 = 4 . log 2( x – 2 ) + log 2( x + 2 ) = 2 .

Question 7 Solve the following equations for x. (a)

(b)

2x – 1 = 14 . 3–x (ii) ----------- = x . 2 ( x – 2)( x + 1) = 1 . (i)

Question 8 Factorise the quadratic expression x 2 – 6x + 5 .

(a)

(i)

(b)

(ii) Solve the inequality for x, where x 2 – 6x + 5 > 0 . For what values of x is log 10( 6 – x ) defined? (i) (ii)

(c)

log 10( x – 4 ) defined?

Solve for x, the equation log 10( x 2 – 6x + 5 ) = log 10( 6 – x ) + log 10( x – 4 ) .

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Question 9 (a)

(b)

(i)

Expand ( a + 1 ) ( a 2 + 2a – 1 ) .

(ii)

Solve for a the equation a 3 + 3a 2 + a – 1 = 0 .

Solve for x where ( ln x ) 3 + 3 ( ln x ) 2 + ln x = 1 .

Question 10 (a)

(b)

(i)

Expand ( x + a ) ( x + 1 ) .

(ii)

Factorise x 2 + 3x + 2 .

Solve each of the following equations for x, x 2 – ( e + 1 )x + e = 0 . (i) (ii)

e 2x – ( e + 1 )e x + e = 0 .

Question 11 The first three consecutive terms of an arithmetic sequence are k – 2, 2k +1 and 4k +2. (a)

Find the value of k.

(b)

Find (i)

u 10 .

(ii)

S 10 .

Question 12 The first three terms of a geometric sequence are given by (a)

Find x.

(b)

Find S ∞ .

3 + 1, x and

3 – 1 , where x > 0.

Question 13 1 Given that --- + 1 + 2 + 2 2 + … + 2 10 = a × 2 b + c , find the values a, b and c. 2

Question 14 A geometric sequence is such that each term is equal to the sum of the two terms directly following it. Find the positive common ratio.

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Question 15 An arithmetic sequence is such that u p = q and u q = p . (a)

Find, in terms of p and q, u 1 .

(b)

Find S p + q .

Question 16 Solve each of the following equations for x. (a)

x2 + x – 2 = 0 .

(b)

1 ( ln ( x ) ) 2 – ln  --- – 2 = 0 .  x

(c)

2 –2×2

x

–x

+ 1 = 0.

Question 17 (a)

Expand ( 2a + 1 ) ( 3 – a ) .

(b)

Solve

(i)

log 10( 2x – 5 ) + log 10( x ) = log 103 .

(ii)

2×3

2x – 1

–5×3

x–1

– 1 = 0.

Question 18 Find k if

(i)

log 2( 2k – a ) = 3 .

(ii)

log 10k = 1 + log 105 .

(iii) 3 ln 2 – 2 ln 3 = – ln k .

Question 19 (a)

If log ( a – b ) = log a – log b , find an expression for a in terms of b, stating any restrictions on b.

(b)

If a 2 + b 2 – 2ab = 4 and ab = 4 , where a > 0, b > 0, find the value of 2 log 20( a + b ) .

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Question 20 Solve the simultaneous equations

9 x – y = 27 2 x + y = 32

.

Question 21 If x = log an and y = log cn where a, c > 0 and n ≠ 1 , prove that log bc – log ba x–y ------------ = --------------------------------, b > 0 . log bc – log ba x+y

Question 22 A sequence of numbers have the property that x, 12, y, where x > 0, y > 0 form a geometric sequence while 12, x, 3y form an arithmetic sequence. (a)

If xy = k, find k.

(b)

Find the value of x and y.

(c)

For the sequence x, 12, y, find

(i) (ii)

the common ratio. the sum to infinity.

Question 23 The numbers α and β are such that their arithmetic mean is 17 while their geometric mean is 8. Find α and β .

Question 24 n

Given that the sequence with general term u i , has its sum defined by

∑ ui

i=1 10

(a)

∑ ui .

i=1

(b)

(c)

(i)

the first term, u 1 .

(ii)

the second term, u 2 .

(i) (ii)

the rth term. the 20th term.

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= n ( n + 2 ) , find


Question 24 n

(a)

Given that

1

∑ ------------------k(k + 1)

k=1 99

(b)

Find

∑

k = 10

b = a + ------------ , find a and b. n+1

m 1 -------------------- , giving your answer in the form ---- , where m and n have no common n k(k + 1)

factors.

Question 25 (a)

Show that log ( a n )( x n ) = log ax , a > 0, a ≠ 1, x > 0, n ≠ 0.

(b)

Hence find log 55 .

Question 26 One solution to the equation 5 x + 1 = 3 x and the values a and b.

2

–1

is given by x = a + log 3b . Find the other solution

Question 27 (a)

5 2n – 1 – 5 2n – 3 -. Simplify ---------------------------------5 2n + 5 2n – 2

(b)

If 2 x = 5 and 2 = 5 y , find xy .

Question 28 Solve the equation log 4( 3 × 2 x + 1 – 8 ) = x .

Question 29 Prove by mathematical induction, that 8 n – 3 n where n ∈

Question 30 Find all values of x such that 2 2x – 2 x + 1 < 3 .

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, is divisible by 5.


Question 31 1 2n Find the constant term in the expansion  2x + --- .  x Question 32 n n  n  Three consecutive coefficients in the expansion of ( 1 + x ) n are   ,  and  and  r   r + 1  r + 2 are in the ratio 6 : 3 : 1. (a)

Show that 2n – 3r = 1 and 3n – 4r = 5 .

(b)

Which terms are they?

Question 33 Given that ( 3 – i ) is a zero of the polynomial p ( x ) = x 3 – 5x 2 + 4x + 10 , find the other zeros.

Question 34 Given that ( 3 + i ) 8 = – a 7 ( b + ci ) , find the values of a, b and c.

Question 35 1+i 3 Express ------------------ in the form z = re iθ . 1–i 3

Question 36 1 1 If -----------------2- + -----------------2- = a + bi , a, b ∈ , find a and b. (1 + i) (1 – i)

Question 37 z Given that w = ---------- , where z = a + bi , a, b ∈ , show that if w is a real number then z is a z–i pure imaginary number.

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Question 38 i ( 4 – 3i ) If m + ni = -------------------, ( 1 – i )2 (a)

find m and n.

(b)

i ( 4 – 3i ) express -------------------in the form z = re iθ . 2 (1 – i)

Question 39 Given that w = – 1 + 3i and z = 1 – i , find w , Arg ( w ) . z , Arg ( z ) . zw , Arg ( zw ) .

(a)

(i) (ii) (iii)

(b)

(i)

Find wz.

(ii)

5π Hence find the exact value of sin  ------ .  12 

Question 40 a b If z = ----------- and w = -------------- where a, b ∈ , find a and b given that z + w = 1. 1+i 1 + 2i

Question 41 When the polynomial p ( x ) = x 4 + ax + 2 is divided by x 2 + 1 the remainder is 2x + 3 . Find the value of a.

Question 42 (a)

(i)

π 4 If z 1 = 2cis  --- find z 1 .  4

(ii)

Draw z 1 on an Argand diagram. 4

(b)

How many solutions to z = – 16 are there if z ∈ ?

(c)

Solve z 5 + 16z = 0 , z ∈ .

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Question 43 The two square roots of 5 – 12i take on the form x + iy , x, y ∈ . Find both values of z for which z 2 = 5 – 12i , giving your answer in the form x + iy , x, y ∈ .

Question 44 Find the four fourth roots of the complex number 1 + 3i , giving your answer in the form re iθ, – π < θ ≤ π .

Question 45 Find a and b, where a, b ∈ (a)

a + bi = i .

(b)

( a + bi ) 2 = i .

(c)

( a + bi ) 3 = i .

in each of the following cases

Question 46 If ( a + bi ) n = x + iy, n ∈ Expresss k in terms of n.

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, show that x 2 + y 2 = ( a 2 + b 2 ) k where k is a function of n.

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FUNCTIONS AND EQUATIONS Question 1 1 Consider the function g ( x ) = ----------------, x ∈ X . x–3 (a)

g(4) . Find (i) (ii) g ( 12 ) . (iii) g ( 9 ) .

(b)

Find the largest set X for which g ( x ) is defined.

(c)

Sketch the graph of g ( x ) using its maximal domain X.

Question 2 1 Consider the functions f ( x ) = ----------- and g ( x ) = x 3 . x–1 (a)

Determine the maximal domain of

(b)

(i) (ii)

(i) (ii)

f ( x) . g( x) .

Justify the existence of ( g o f ) ( x ) . Find ( g o f ) ( x ) .

Question 3 Consider the function f ( x ) =

7 – x.

f (3) . f (7) .

(a)

Find (i) (ii)

(b)

What is the maximal domain of f ( x ) .

(c)

Find (i) (ii)

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1 g ( x ) = ----------- and determine the implicit domain of g ( x ) . f ( x) h ( x ) = f –1 ( x ) and determine the implicit domain of h ( x ) .

1


Question 4 1 -, x ∈ Given the function f ( x ) = ---------------1 + e 2x a, b ∈ , find the values of a and b.

and the fact that f ( x ) + f ( – x ) = ax + b , where

Question 5

Consider the function f ( x ) = 3x 2 – 6x + 7 . (a)

Express f ( x ) in the form a ( x – h ) 2 + k , where a, b, c ∈

(b)

For the graph of f ( x ) write down the (i) coordinates of its vertex. (ii) equation of the axis of symmetry.

(c)

(i) (ii)

.

Find the coordinates of the y-intercept. Sketch the graph of f ( x ) , clearly labelling the vertex, y-intercept and axis of symmetry.

Question 6 (a)

Determine the maximal domain of f ( x ) = ln ( x + 1 ) .

(b)

If g ( x ) = e x , (i) justify the existence of ( f o g ) ( x ) . (ii) find ( f o g ) ( x ) . (iii) state the maximal domain of ( f o g ) ( x ) .

Question 7 The graph of the function g ( x ) = 2x 2 – 12x + 7 is shown below. The coordinates of B are  a + b---, 0 where a, b, c ∈ . Find the values of a, b and c.  c  y (0,7) A

B

x=3

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x


Question 8 1 Consider the function g ( x ) = x + ---, x > 0 . x (a)

(b)

(i) (ii)

Find g ( 2 ) . Find the x-value for which g ( x ) = 2 .

f ( x) , x > 0 . Find f ( x ) . The composite function ( g o g ) ( x ) = ---------------------x( 1 + x2 )

Question 9 (a)

Find the ( f o g ) ( x ) where g ( x ) = e x + 1, x ∈

and f ( x ) = ( x – 1 ) ln ( x – 1 ), x > 1 .

(b)

Find { x : ( f o g ) ( x ) = 2x } .

(c)

What would the solution to ( f o g ) ( x ) = 2x be if g ( x ) = e x + 1, x > 0 ?

Question 10 (a)

1 Find the inverse function, f –1 , of f ( x ) = -----------, x > 2 . x–2

(b)

On the same set of axes, sketch the graphs of f ( x ) and f –1 ( x ) .

(c)

Find the coordinates of the point of intersection of f ( x ) and f –1 ( x ) .

Question 11 e –kx Consider the function f k ( x ) = ---------------,x∈ . 2 + e–x (a)

(b)

Find (i)

{ x : f 2( x) = 1} .

(ii)

 1  x : f 1 ( x ) = --2-  .  

 1 For what values of a will  x : f 1 ( x ) = ---  = a 

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.


Question 12 (a)

On the same set of axes, sketch the graphs of f ( x ) = x 2 – 4 and g ( x ) =

(b)

What is the domain of ( f o g ) ( x ) ?

(c)

Find (i) (ii)

x – 2.

( f og)( x) . the range of ( f o g ) ( x ) .

Question 13 The graph of f ( x ) = – 1 + ln ( x – 1 ), x > a is shown. (a)

y

y = f ( x)

Find the values of a and b. (b,0)

f –1 ( x ) .

(b)

Find

(c)

Sketch the graph of f –1 ( x ) . x=a

Question 14 1 Find the maximal domain of h ( x ) = ------------------ . ln ( ln x )

Question 15 (a)

On the same set of axes sketch the graphs of f ( x ) = 5x + 2 and g ( x ) = 3x 2 .

(b)

Find (i) (ii)

{ x : f ( x) = g( x)} . { x : 3x 2 ≤ 5x + 2 } .

Question 16 Consider the function f : A → , where f ( x ) = log 3( 3x + 2 ) . (a)

Find the largest set A for which f is defined.

(b)

(i)

Define fully, the inverse, f –1 .

(ii)

Sketch the graph of f –1 ( x ) .

(iii) If g ( x ) = 3 x , express f –1 ( x ) in terms of g ( x ) .

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x


Question 17 For the graph shown below, sketch, on different sets of axes, the graphs of y (4,4) 4 (a) y = f ( x + 1 ) . (b) (c)

1 --- y = f ( x ) . 2 y = f ( x) – 2 .

3 y = f ( x)

2 1

1

2

3

4

x

Question 18 1 Let f ( x ) = --- , x ≠ 0 . The graph of g ( x ) is a translation of the graph of f ( x ) defined by the x 3 matrix   .  2 (a)

Find an expression for g ( x ) (i) in terms of f ( x ) . (ii) in terms of x.

(b)

On the same set of axes, sketch the graphs of f ( x) . (i) (ii) g ( x ) , for x > 3.

Question 19 The function g ( x ) = ax – b passes through the points with coordinates ( 1, 9 ) and ( – 3, 1 ) . Find a and b.

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Question 20 Let f ( x ) = x 2, x ∈ (a)

.

Sketch the graph of f .

–1 The graph of f is transformed to the graph of g by a translation   .  2 (b)

Find an expression for g ( x ) in terms of (i) f ( x) . (ii) x.

(c)

(i) Find { x :g ( x ) = 0 } . (ii) Find g ( 0 ) . (iii) Sketch the graph of g ( x ) .

Question 21 x–1 Given that f ( x ) = ------------ , find x+1

(a)

f (1) .

(b)

f –1 ( – 1 ) .

Question 22 a The function f ( x ) undergoes a transformation defined by the matrix   to produce the new  – b function g ( x ) . (a)

Express g ( x ) in terms of f ( x ) .

(b)

If f ( x ) = x 2 – 2x + 4 and g ( x ) = x 2 – 2x + 8 , find a and b.

Question 23 Let f ( x ) = 2e – x + 3, x ∈ , find (a)

f ( –1 ) .

(b)

f –1 ( 4 ) .

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Question 24 Let f ( x ) = x 3 and g ( x ) = x – 1 . Find (a)

( f og)(1) .

(b)

(go f )(2) .

Question 25  x – 2, if x < 0 Consider the function f ( x ) =  . 2  ( x – 1 ) – 3k, if x ≥ 0 (a)

Find the value of k for which f ( x ) is continuous for x ∈

(b)

Using the value of k in (a), sketch the graph of f ( x ) , clearly labelling all intercepts with the axes.

.

Question 26 Find all real values of k so that the graph of the function h ( x ) = 2x 2 – kx + k , x ∈ x-axis at two distinct points.

cuts the

Question 27 (a)

Sketch the graph of f ( x ) = x 2 – 4x + 5 , clearly showing and labelling the coordinates of its vertex.

(b)

Find

(i) (ii)

(c)

4 lim -------------------------2 x → ∞ x – 4x + 5 4 lim -------------------------2 x → – ∞ x – 4x + 5

4 Given that g ( x ) = ----------- , f ( x) (i) state the maximal domain of g ( x ) . (ii) sketch the graph of g ( x ) . (iii) find the range of g ( x ) .

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Question 28 (a)

On the same set of axes sketch the curves of y = x and y = ln ( 1 – x ) .

The graph of the function f ( x ) = x ln ( 1 – x ), x < a is shown below. y y = f ( x)

a

O

(b)

Find the value of a.

(c)

Sketch the graph of

(d)

Sketch the graph of g ( x ) = x ln ( 2 – x ) .

x

y = f ( x) . y = f( x ). 1 (iii) y = ----------- . f ( x) (i) (ii)

Question 29 (a)

(i) (ii)

Sketch the graph of f ( x ) = 2x + 1 . Solve 2x + 1 = 3 .

(iii) Hence solve for x if g ( x ) = 0 where g ( x ) = ( 2x + 1 ) 2 – 2x + 1 – 6 . (b)

For what values of x will ( 2x + 1 ) 2 – 2x + 1 > 6 ?

Question 30

(a)

(b)

(i)

 f ( x ), x ≥ 3 x–3 . Find f ( x ) and g ( x ) . -------------- =  2x  g ( x ), x < 3

(ii)

x–3 Sketch the curve of y = -------------- . 2x

x–3 Find all real values of x that satisfy the inequality -------------- < 1 . 2x

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Question 31 Sketch the curve y = ( x + 2 ) x + 2 . Hence find all real values of x for which ( x + 2 ) x + 2 < 9 .

(a)

(i) (ii)

(b)

Find { x ( x + 2 ) x – 2 < 3 } .

Question 32 The polynomial p ( x ) = ax 3 – x 2 + bx + 6 is divisible by ( x + 2 ) and has a remainder of 10 when divided by ( x + 1 ) . (a)

Find the values of a and b.

(b)

Find { x p ( x ) < 6 – 7x } .

Question 33 x Consider the curve with equation y = -------------. 2 x +4 (a)

Show that yx 2 – x + 4y = 0 .

(b)

Hence, given that – a ≤ y ≤ a for all real values of x, find a.

Question 34  x + 7, x ≤ 3 Find the value of a which makes the function defined by f ( x ) =  continuous for  a × 2 x, x > 3 all real values of x.

Question 35 (a)

Sketch the graph of the polynomial p ( x ) = x 3 – x 2 – 5x – 3 .

(b)

Find all real values of x for which x 3 – x 2 – 5x – 3 < 4 ( x – 3 ) . (i) (ii)

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x 3 – x 2 – 5x – 3 > ( x + 1 ) ( x – 3 ) .

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Question 36  ax + 1, x < π --2  Given that a + b = 2 , find the values of a and b so that the function f ( x ) =   sin x + b, x ≥ π -- 2 is continuous for all values of x.

Question 37 Given that h ( x ) = 9 x + 9 and g ( x ) = 10 × 3 x , find { x h ( x ) < g ( x ) } .

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CIRCULAR FUNCTIONS AND TRIGONOMETRY Question 1 π 3 Given that 0 ≤ θ ≤ --- and tan θ = --- , find 2 4 (a)

cos θ .

(b)

sin 2θ .

(c)

π tan  --- – θ . 2 

Question 2 π 1 Given that 0 < θ ≤ --- , arrange, in increasing order, sin θ, -----------, sin 2 θ . 2 sin θ

Question 3 (a)

1 If 0 < θ < 90° and cos θ = --- a , find sin θ . 3

(b)

π Express --- in degrees. 5

(c)

Evaluate cos 300° cos 30°

(d)

sin ( π – θ ) 3π Express in terms of tan θ , --------------------------- ⋅ tan  ------ – θ .   2 π cos  --- + θ 2 

Question 4 (a)

1 1 Express --------------------- – --------------------- in terms of sin θ . cos θ – 1 cos θ + 1

(b)

1 1 Solve --------------------- – --------------------- = – 8, 0° < θ < 360° cos θ – 1 cos θ + 1

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Question 5 (a)

1 Given that cos θ sin θ = --- , evaluate ( cos θ – sin θ ) 2 . 2

(b)

Find all values of θ such that

(i)

sin 2 θ – sin θ = 0, 0 ≤ θ ≤ 2π .

(ii)

sin 2 θ – sin θ = 2, 0 ≤ θ ≤ 2π .

Question 6 x The figure below shows the graph of f ( x ) = 2 sin  --- .  2 y

A(a,b)

B(c,0)

x

O y = f ( x) (a)

Find a, b and c.

(b)

Solve for x, where f ( x ) =

3, 0 ≤ x ≤ c .

Question 7 Consider the graph of the function h ( x ) = acos ( bx ) + c : y y = h( x)

5

2 O

π

π --2

Find the values a, b and c.

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Question 8 (a)

State the range of 2 sin θ .

(b)

Find (i) (ii)

1 -. the smallest value of -------------------3 + 4 sin θ 1 -. the largest value of -------------------3 + 4 sin θ

Question 9 (a)

Find all values of x such that 2 cos ( x ) + sin ( 2x ) = 0, x ∈ [ 0, 2π ] .

(b)

How many solutions to 2 cos ( x ) + sin ( 2x ) = cos x are there in the interval [0, 2π].

Question 10 Part of a particular tile pattern, made up by joining hexagonal shaped tiles, is shown below.

A

B

F

C

θ

E

D

The side lengths of every hexagon is x cm. (a) (b)

Find in terms of x and θ the length of [AE]. 1 If the area of the shaded region in any one of the hexagons is 9 cm 2 and sin θ = --- , find 8 the value of x.

Question 11 For the diagram shown alongside, the value 1 of sin ( α – β ) = ------ , where k ∈ . k

a β a

Find the value of k.

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α a


Question 12 The following diagram shows continually decreasing quarter circles, where each successive quarter circle has a radius half that of the previous one.

r cm 1 --- r 4 1 --- r 2

Let A 1 equal the area of the quarter circle of radius r, A 2 equal the area of the quarter circle of 1 1 radius --- r , A 3 equal the area of the quarter circle of radius --- r and so on. 2 4 (a)

Find (i)

A 1 in terms of r.

(ii)

A 2 in terms of r.

(iii) A 3 in terms of r. (b)

525 If A 1 + A 2 + A 3 = --------- π , find r. 64 ∞

(c)

If quarter circles are drawn and shaded in indefinitely, find r if

∑ Ai

= 27π .

i=1

Question 13 A pole of length b metres resting against a wall makes an angle of 60˚ with the ground. The end of the pole making contact with the ground starts to slip away from the wall until it comes to rest, 1 m from its initial position, where it now makes an angle of 30˚ with the ground. Find the value of b.

Question 14 The figure below shows a square ABCD. If ∠XBA = 30° and ∠CXA = θ° , find tan θ° . 30˚

B

C

A

D

θ° X

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Question 15 3 In the diagram below, cos B = --- , AB = 6 and BC = 2, AC = 8

b , find b.

A 6

C

2

B

Question 16 3 The area of the triangle shown is 27 sq. units. Given that sin θ = --- , find x. 4 2x θ 4x Question 17 A circular badge of radius 2 cm has the following design: (a)

State the value of α .

(b)

Find (i) (ii)

(c)

Find the area of the shaded region shown.

If 0° ≤ x ≤ 90° , solve each of the following (a)

cos x = sin 36° .

(b)

cos x = sin x .

(c)

cos 2x = sin x .

5

30˚

O

the length of the minor BC. the area of the minor sector OBC.

Question 18

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A

α° B

C


Question 19 (a)

4 sin θ If tan θ = a , express -------------------------------- in terms of a. 5 cos θ – sin θ

(b)

π 4 sin θ Hence find the value of θ if -------------------------------- = 1 , 0 < θ < --- . 2 5 cos θ – sin θ

Question 20 B

Consider the triangle ABC:

2x cm

60°

x cm

(a)

Find x.

(b)

Find the area of ∆ ABC

A

C

48 cm

Question 21 The segments [CA] and [CB] are tangents to the circle at the points A and B respectively. If the circle has a radius of 4 cm 2π and ∠ AOB = ------ . 3 (a)

Find the length of [OC].

(b)

Find the area of the shaded region.

A

O

2π -----3

C

B

Question 22 The graph of f ( x ) = acos ( bx ) + c, 0 ≤ x ≤ π is shown below. Find the values a, b and c. y 10

y = f ( x)

2 O

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3π

3π -----2

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x


Question 23 (a)

Part of the graph of the function defined by y = f ( t ) is shown below. y 3 y = f (t)

1 1 O

0.5

t

–1 (i) (ii)

What is the period of f ( t ) ? Given that f ( t ) = acos ( kt ) + c , find a, k and c.

(b)

What is the least positive value of x for which cos ( 2x ) =

(c)

Let g ( x ) = mcos ( x ) + n , n, m ∈ g ( x ) < 0 for all real values of x.

(d)

Sketch the graph of h : ]–π,π[

3 sin ( 2x ) ?

and m > 0. Write an expression for n in terms of m if

x , where h ( x ) = tan  --- + 1 , clearly determining and  2

labelling the x-intercepts. (e)

1 x Find the sum of the solutions to cos  --- = --- 3 , x ∈ [ 0, 4π ] .  2 2

Question 24 Destructive interference generated by two out of tune violins results in the production of a sound intensity, I ( t ) , given by the equation 4π I ( t ) = 30 + 4 cos  ------t dB (decibels) 3  where t is the time in seconds after the violins begin to sound. (a)

What is the intensity after 1 second?

(b)

What is the least intensity generated?

(c)

When does the intensity first reach 32 dB?

(d)

What time difference exists between successive measures of maximum intensity?

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Question 25 The rabbit population, N ( t ) , over a ten year cycle in a small region of South Australia fluctuates according to the equation N ( t ) = 950 cos ( 36t° ) + 3000, 0 ≤ t ≤ 10 where t is measured in years. (a)

Find the rabbit population after 2 and a half years.

(b)

What is the minimum number of rabbits in this region that is predicted by this model?

(c)

Sketch the graph of y = N ( t ) , 0 ≤ t ≤ 10 .

(d)

For how long, over a 10 year cycle, will the rabbit population number at most 3475?

Question 26 Solve the equation for x, where sin 2x =

3 cos x, 0 ≤ x ≤ π .

Question 27 π 2 cos  θ – --- .  4

(a)

Expand

(b)

Solve cos 2 θ + sin θ cos θ = – ( cos θ + sin θ ) , 0 ≤ θ ≤ π .

Question 28 2 3 If sin A = ------- and cos B = ------- where both A and B are acute angles, find 4 4 (a)

(i) (ii)

sin2A. cos2B.

(c)

sin(A + B).

Question 29 (a)

(b)

(i)

If T ( x ) = 6 – ( x – 1 ) 2 , what is the maximum value of T ( x ) ?

(ii)

Express sin 2 x + 2 cos x + 6 in the form a + b ( cos x – 1 ) 2 .

What is the maximum value of sin 2 x + 2 cos x + 6 ?

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Question 30 (a)

Let s = sin θ . Show that the equation 15 sin θ + cos 2 θ = 8 + sin 2 θ can be expressed in the form 2s 2 – 15s + 7 = 0 .

(b)

Hence solve 15 sin θ + cos 2 θ = 8 + sin 2 θ for θ ∈ [ 0, 2π ] .

Question 31 Find { x 4sin 3 x + 4sin 2 x – sin x – 1 = 0, 0 ≤ x ≤ 2π }

Question 32 asinθ If sin A = 2 sin ( θ – A ) , find the values of a and b such that tan A = -----------------------1 + b cos θ

Question 33 cos 3θ = 4cos 3 θ – 3 cos θ . sin θ° = cos ( 90° – θ° ) .

(a)

Show that (i) (ii)

(b)

Noting that 2 × 18 = 36 and 3 × 18 = 54 = 90 – 36 , find the exact value of sin 18° .

Question 34 1 + sin θ Given that u = -------------------- , prove that cos θ (a)

u2 – 1 sin θ = -------------. u2 + 1

(b)

2u cos θ = -------------u2 + 1

Question 35 3 3 Given that sin x sin y = ------- and cos x cos y = ------- , find all real x and y that satisfy these 4 4 equations simultaneously, where 0 ≤ x ≤ π and 0 ≤ y ≤ π.

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Question 36 Given that asecθ = 1 + tan θ and b sec θ = 1 – tan θ , show that a 2 + b 2 = k and hence state the value of k.

Question 37 A sector OAB of radius 4 cm is shown below, where the point C is the midpoint of [OB]. A

30˚

C

B

O

(a)

Find the area of the shaded region.

(b)

Where, along [OB], should the point C be placed so that [AC] divides the area of the sector OAB in two equal parts.

Question 38 C In the figure shown, DC = a cm, BC = b cm, ∠ ABC = β and ∠ ADC = α . If AD = x cos α + y sin β , find x and y in terms of a and b.

B

β α

A

D

Question 39 Point A is x metres due south of a vertical pole OP and is such that the angle of elevation of P from A is 60˚. Point B is due east of the pole OP and makes an angle of elevation to P of 30˚. The points O, A and B all lie on the same horizontal plane. A metal wire is attached from A to P and another wire is attached from B to P. (a)

Find an expression in terms of x for

(b)

a b The sine of the angle between the two wires is ---------- where a, b, c ∈ c

© IBID Press

(i) AP. (ii) BP. (iii) AB.

10

. Find a, b and c.


Question 40 π Consider the function f ( x ) = 2 sin  x – --- + 1, 0 ≤ x ≤ π .  2 (a)

(i) (ii)

Find the range of f. Sketch the graph of f.

(b)

Find f –1 .

(c)

Sketch the graph of f –1 .

Question 41 (a)

(b)

Sketch on the same set of axes the graphs of f ( x ) = Arccos ( x ) and g ( x ) = Arcsin ( x ) . (ii) Hence sketch the graph of h ( x ) = f ( x ) + g ( x ) . (iii) Deduce the value of Arccos ( x ) + Arcsin ( x ) for – 1 ≤ x ≤ 1 . (i)

4 4 Find sin  Arccos  --- + Arctan  – ---    5   5

© IBID Press

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MATRICES Question 1 Let the matrix A = – 2 1 define a 2 × 2 matrix. 3 0 (a)

(b)

Find (i)

A2 .

(ii)

A –1 .

If B = 4 3 , find the matrix X given that 2X + B = A . 1 –2

Question 2 Given that A = 2 4 and B = 1 – 2 3 –1 3 2 (a)

find (i) (ii)

AB. A2 B – A .

(b)

If xA – yB = 5 6 where x, y ∈ , find x and y. 90

Question 3 Consider the matrices A = 2 5 , B = – 1 and C = – 1 0 . Find, where possible, 3 2 4 (a)

AB.

(b)

A + B.

(c)

BC.

(d)

CB.

(e)

CA.

© IBID Press

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Question 4

Find x given that

(i)

x2 7 = 9 4 4 6

(ii)

x 2 + 6 7 = 5x 4 x+2 3

7 . x+3 7 . 3

Question 5

(a)

(b)

0 –1 1 2 0 1 Find 1 5 – 4 2 1 – 1 . 1 2 –2 3 1 –1

(i)

(ii)

2x + z = 5 Set up the system of simultaneous equations 2x + y – z = 4 in the form AX = B , 3x + y – z = 3 x where X = y z Hence solve for x, y and z.

Question 6 If 2x + 8 4 = 3x – y 4 , find x and y. 0.5x y 5 y

Question 7

(a)

Given that A = – 1 3 , find A –1 . 2 4

(b)

Hence find the coordinates ( a, b ) of point A in the diagram shown below. y – x + 3y = 3

x + 2y = 4 A ( a, b )

x

© IBID Press

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Question 8 2 2 If A = cos θ – cos θ and B = sin θ cos θ , find sin θ sin 2 θ – sin θ cos 2 θ

(a)

(i) A + B. (ii) ( A + B ) –1 . (iii) A – B.

(b)

For what values of θ where 0 < θ < π , does A – B =

0

2 . 2 0

Question 9 Consider the function T ( x ) = 3x 2 + 4x + I , where I = 1 0 . 01

(a)

If A = 1 1 , find T ( A ) . 01

(b)

(i)

Find ( 3 A + I ) ( A + I ) .

(ii)

Hence find 3 A 2 – 4 A + I .

Question 10 Given that A = x 2

1 , for what values of x will A be singular? x–1

Question 11

(a)

(b)

Given that A = 3 2 , find 74 (i)

A .

(ii)

A –1 .

Hence solve the system of simultaneous equations

© IBID Press

3

7x + 4y = 4 . 3x + 2y = – 2


Question 12

If A =

0 3 , find scalars α, β and γ , where α ≠ 0, β ≠ 0, γ ≠ 0 , for which –1 1 α A 2 + βA + γI = 0 0 . 00

Question 13

(a)

(b)

(i)

–1 1 1 011 Let A = 1 – 1 1 and B = 1 0 1 , find AB . 1 1 –1 110

(ii)

Hence find A –1 .

–x+y+z = 3 Hence solve the system of simultaneous equations x – y + z = 4 . x+y–z = 5

Question 14

Find x if A = 0 and A =

x–1 1 2 2 x–3 1 . –1 1 x–2

Question 15 3 5 1

1 2 . 2 3 1 (a – 2)

(a)

Find, in terms of a, ∆ =

(b)

3x + y + 2z = 4 Consider the system of equations 5x + 2y + 3z = 6 . x + y + ( a – 2 )z = 0 (i) (ii)

© IBID Press

Solve the system of equations when a = 3. For what values(s) of a will the system of equations have exactly one solution.

4


Question 16 3x + 2y – z = 1 Consider the system of equations x + y + z = 2 . kx + 2y – z = 1 (a)

Find the value of k for which the system of equations has more than one solution.

(b)

Solve the system of equations for the value of k in part (a).

Question 17

(a)

If ∆ = (i) (ii)

(b)

(c)

1 3 0

1 – 13 p+3

1 p , find –1

an expression for ∆ in terms of p. the value(s) of p for which ∆ = 0 .

x+y+z = 2 For what values of p will the system of equations 3x – 13y + pz = p + 1 have a unique ( p + 3 )y – z = 0 solution? For each value of p for which a unique solution does not exist, determine whether a solution exists and if it does, find all solutions.

Question 18 1 Consider the matrices A = a 2

a 1 –2

2 5 x 1 , B = 1 and X = y , a ∈ a+2 8 z

.

(a)

Show that the equation AX = B has a unique solution when a is neither 0 nor –1.

(b)

(i) (ii)

© IBID Press

If a = 0, find the particular solution for which x + y + z = 0 . If a = –1, show that the system of equations has no solution.

5


VECTORS Question 1 The points A, B and C are shown in the diagram below y i

(b)

θ

A

C

1 (a)

j

B

9 8 7 6 5 4 3 2 1

2

3 4

5 6

7

8 9 10

Find in terms of the unit vectors i and j (i)

AB .

(ii)

AC .

(i)

Find AB • AC .

(ii)

Hence find cos 2 θ .

x

Question 2 ABCD is a quadrilateral where P, Q and R are the midpoints of [AB], [BC] and [CD] respectively. If AD + BC = kPR , where k is a positive integer, find k.

Question 3 2 Given that a =   , b =  – 1 and c =  5 , find  1  3  3 (a)

a + 2b + c.

(b)

a • b.

(c)

1 a – --- ( c – b ) . 3

(d)

c .

© IBID Press

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Question 4  2    Consider the vector r =  2  .    –1  (a)

(b)

Find (i) (ii)

r • r. rˆ .

 2  x     If a =  1  and b =  3  , find x and y given that r = – 4a + 2b .      1  y

Question 5 If a = 2i – 6 j + 3k and b = – i + 2 j – 2k , find (a)

(i)

a

(ii)

(b)

b–a .

(c)

(a – b) • (a + b) .

b .

Question 6  3  1    The vector a =  2  is a linear combination of the vectors x =  2     6  1

  1   , y =  2     0

   and  

 1   z =  0  . That is, a = αx + βy + γz . Find α, β and γ .    2

Question 7  cos θ   sin θ    The vector a =  sin θ  makes an angle of 60˚ with the vector b =  cos θ     0.5   0.5 of sin2θ. © IBID Press

2

   . Find the value  


Question 8  x   2    If vectors a =  3  and b =  x 2     –1   1

   are perpendicular, find x.  

Question 9 Given that a = 4 , b = 3 and that a is perpendicular to b, find (a)

(a – b) • (a + b) .

(b)

(a + b) • (a + b) .

(c)

(a + b) • (a + b) – (a – b) • (a – b) .

Question 10 (a)

The straight line, l 1 , passes through the points (2, 3, –1) and (5, 6, 3). Write down the vector equation, r 1 , of this straight line.

(b)

 1   2    A second line is defined by the vector equation r 2 =  – 1  + µ  – 1     3   1

   , where µ is a real  

number. If the angle between the two lines is θ , find the value of cos θ .

Question 11  1  4     The vector equation of line l 1 is given by r 1 =  – 3  + t  3     1  2  (a)

  ,t∈  

.

If the point P on r 1 corresponds to when t = 1 , find (i) the position vector of P. (ii) how far P is from the origin O.

A second line, l 2 , is defined by the vector equation r 2 = i + 2 j – 3k + s ( 2i – j + 3k ) , s ∈ . (b)

If l 2 intersects l 1 when t = 1, find the value of s.

© IBID Press

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IB Mathematics Higher Level Sample Questions

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