Questions and Answers of Complete Pure Mathematics

The equation px² + qx + r = 0, where p, q and r are constants, has roots −1/2 and 3/4. Find the smallest possible integer values of p, q and r.
Solve these simultaneous equations.y = x − 4y = 2 − x2
Use the formula to solve 7x² − 3x − 5 ≤ 0.Leave your answer in surd form.
The quadratic equation x + k + 9/x = 0 has equal roots. Find the two possible values of k.
Solve these simultaneous equations.x² + y = 3y + 3x = 5
Solve the equation 2x² + 5x + 1 = 0. Write your answer correct to 3 significant figures.
Prove that the quadratic equation (q − 5)x² + 5x − q = 0 has real roots for any value of q.
Express each of the quadratic equations in the form y = a (x + b)² + c. Sketch the curve, stating the coordinates of the vertex and whether there is a maximum or minimum value of y.y = 2x² − 5x
Solve these simultaneous equations.y = 2x + 5y = x² − 3x −1
Find the set of values for x for which 20 + 7x − 6x² ≥ 0.
Express each of the quadratic equations in the form y = a (x + b)² + c. Sketch the curve, stating the coordinates of the vertex and whether there is a maximum or minimum value of y.y = 5x² − 4x
Find the value of p for which the quadratic equation px² − 4px + 2 − p = 0 has equal roots.
Solve these simultaneous equations.y + 3 = 2xy = x² − 2x + 1
The first three terms, in ascending powers of x, in the expansion of (1 + a/x)n areFind the value of a and the value of n. 1 + 24 X + 252 +²
Find the set of values for x for which (x − 1)(x + 2) < 18.
a) Find the first four terms in the expansion of (1 − x/2)8 in ascending powers of x.b) Use these terms to find the value of (0.99)8 giving your answer correct to 4 decimal places.
a) Write down the expansion of (1 + x)5.b) By letting x = y + y², find the coefficient of y8 in the expansion of (1 + y + y²)5.
Find the coefficient of x² in the expansion of (2 − 3x + x²)(1 − x/2)10.
a) Write down the binomial expansion of (1 + x)5.b) Use your answer to (a) to express(1 −√2)5 in the form p + q√2 where p and q are integers.
Find the first four terms in the expansion ofin ascending power of x. (1 – X a
Find the coefficient of y5 in the expansion of (3 + 2y)8.
Find the term in y³ in the expansion of (2 − y)6.
a) Find the first four terms in the expansion ofin ascending powers of x.b) Hence determine the value of (1.05)10 correct to 2 decimal places. (1+ x 10
a) Write down the first three terms, in ascending powers of x, in the expansion of (1 + ax)n where a ≠ 0 and n > 2.b) The coefficient of the x² term is half thecoefficient of the term in x.Show
Use the binomial expansion to find the exact value of (1.001)4.
a) Expand (1 + 2x)³.b) Show that(1 + 2x)³ + (1 − 2x)³ = 2(1 + 12x²).c) Hence solve (1 + 2x)³ − (1 − 2x)³ = 8.
Find the 5th term in the expansion ofin ascending powers of x. X (4 - 3 9
a) Find the value of k, the value of a and the value of b.b) Use your values of k, a and b to findthe coefficient of the term in x³ in theexpansion of (1 − x)(1 + kx)8.(1 + kx)8 = 1 + 12x + ax² +
Find the 5th term in the expansion of (2 − 1/3x)11 in ascending powers of x.
Expand, in ascending powers of x, as far as the term in x².a) (5 − x)6b) (3 + 2x)9c)d)
Find the ratio of the coefficient of the x³ term to the coefficient of the x4 term in the expansion of (2x + 1/2)6.
Find the coefficient of x5 in the expansion of (2x − 1)15.
a) Find the first five terms in the expansion of (2 + x)6 in ascending powers of x.b) Using your expansion, show that(2.03)6 = 69.9804 correct to 4 decimal places.
Expand (2 + x + x²)4 in ascending powers of x up to and including the term in x².
Find (1 + x)4 − (1 − x)4. Write your answer inascending powers of x.
Find the coefficient of x4 in the expansion of (1 − x²)(2x + 1/x)6.
Find the 6th term in the expansion of (x/2 + 1)20 in ascending powers of x.
Find the coefficient of x³ in the expansion of (1 + x − 2x2)(x − 2/x)7.
The binomial expansion of (3 − x/4)8 is 6561 + px +qx² + ... Find the value of p and the value of q.
Find the coefficient of x4y² in the binomial expansion of (2x − y)6.
Find the first three terms in the expansion of (1 + ax)7 in ascending powers of x.
The 3rd term in ascending powers of x in the expansion of (1 + 1/x)n, n > 2, is 6/x2. Find the value of n.
a) Write down the first three terms in the expansion of (3 + x)7 in ascending powers of x.b) Use this result to write down the first three terms in the expansionof [(3 + (y + y²)]7 in ascending
Using the binomial expansion, expand each of these expressions.a) (3x + 4y)4b) (x − 2y)³c) d) (2 + 5x)4e)f) 1- x 2
Find the coefficient of the term in x4 in the expansion of (3 + 2x)8.
The coefficient of x² in the expansion of (1− x/2)5 − (a + x)4 is 2a where a > 0. Find a.
Find the coefficient of the 6th term in the expansion of (2 − x)10 in ascending powers of x.
Use Pascal's triangle to find the value of (1.03)4.
Use the binomial expansion to show that the value of (0.98)5 is 0.903921 correct to 6 decimal places.
Find the term independent of x in the expansion of (3 + x/2)(2 + 3/x)6.
The 2nd term in ascending powers of x in the expansion of (1 − 5x)" is − 25x.Find the value of n.
Find the term independent of x in the expansion of (2x² − 1/x)9.
Find the 3rd term in descending powers of x, in the expansion of (x + 3)(2x − 1)5.
Calculatea.b.c.d. 9 نیا
Find the coefficient of the term in x³ in the expansion of (1 − 6x)7.
Write down the last three terms in the expansion (4 + x)6 in ascending powers of x.
Find the 3rd term in the binomial expansion of (2 − 3x)10 in ascending powers of x.
i) Show that the equation 2 sin² x − cos x = 1 can be written as a quadratic equation in cos x.ii) Solve the equation 2 sin² x − cos x = 1 for 0° ≤ x ≤ 180°.iii) Hence solve the
Find, in ascending powers of x, the first three terms in the expansion of (1 − 2x)(1 + x/2)10.
Find the first four terms in the expansion of (3 + x)8 in ascending powers of x.
a) i) Given thatshow that tanθ = 3ii) Solve the equationfor 0° ≤ θ ≤ 360°.iii) Solve the equationfor 0° ≤ 0≤ 360°. b) i) Given that 2 sin² 2θ − cos2θ = 1, show that 2 cos²
Write down the first three terms in the expansion (2 − 10x)5 in ascending powers of x.
Write down the first three terms in the expansion of (1 − x/2)6 in ascending powers of x.
Find, in ascending powers of x, the first three terms of each of the following.a) (2 + x)(1 + x)8b) (1 − 2x) (3 + 2x)9
Find the first three terms in the expansion of (1 − x)10 in ascending powers of x.
Calculatea) 5!b) 10!c) 20!/17!d)8!/2!6!
Use Pascal's triangle to expanda) (2 + 3x)4b) (2x + 1)6c) (1 − 4x)5d) (x − 3)7.
The function f: x ↦1 + cos 2x is defined for the domain 0 ≤ x ≤ π/2.i) Solve the equation 2f(x) = 3.ii) Sketch the graph of y = f(x).iii) State the range of the function f.iv) State the domain
a) Sketch the curves y = 2 cos x and y = 3 sin x on the same axes for 0° ≤ x ≤ 360°.b) Find the range of values of x in theinterval 0° ≤ x ≤ 360° for which3sin x ≥ 2cos x.
i) Show that the equation 2 tan x = cos x can be written as a quadratic equation in sin x.ii) Solve the equation 2 tan x = cos x for 0° ≤ x ≤ 180°.iii) Hence, state the set of values of x
a) Use the identity sin² x + cos² x = 1 to prove thatb) Use the identity you proved in part (a) to find the minimum values of 1+ sinx cos.x cos²x = tan²x + tanx + 1.
a) Sketch, on the same diagram, the graphs of y = tanθ and y = tan(−θ) for −270° ≤ θ ≤ 270°.b) State the solutions of the equation tanθ = tan(−θ) for −270° ≤ θ ≤ 270°.
a) Sketch, on the same diagram, the graphs of y = 1/2 sinθ and sin1/2 θ for 0 ≤ θ ≤ 2π.b) State the number of solutions1of the equation sin 0 = 2 sin 1/2 θ for 0 ≤ θ ≤ 2π.
a) Sketch, on the same diagram, the graphs of y = 2cos x and y = cos² x for −360° ≤ x ≤ 360°.b) State the number of solutions of the equation 2cos x = cos² x for −360° ≤ x ≤ 360°.
a) Simplify b) Hence solve the equation 6 − 6 cos² θ = 3 sinθ for 0 ≤ θ ≤ 2π. 6-6cos²0 2 sin
Solve the following equation for 0° ≤ 0 ≤ 360°. a) 3 sin²θ + 5 cosθ − 1 = 0b) 2 sinθ tanθ = 3c) 7 cos²θ + sin²θ = 5cosθd) 3 tanθ = 5 sinθe) cos²θ − sin² θ = 0f) 1 +
a) Sketch the graph of y = sin² x for 0° ≤ x ≤ 360°.b) Prove that if sin² x > ³/ then cos² x < 1/4.c) Solve sin² x > 3/4 for 0° ≤ x ≤ 360°.
a) Express 3 sin² x − 4 cos² x in the form a + b cos² x, stating the values of a and b.b) Hence state the maximum and minimum values of 3 sin²x − 4 cos²x.c) Solve the equation 3 sin² x −
a) Solve the equation tan 1/2x = 3 for 0 ≤ x ≤ 4π.b) Solve the equation sinθ (3 cosθ − sinθ) = 0 for 0 ≤ θ≤ 2π.
Solve the equation tan³ θ − tan² θ − 2 tanθ = 0 for 0° ≤ θ ≤ 180°.
a) Given that 2 sin² θ − cos θ = 1, show that 2 cos² θ + cosθ − 1 = 0.b) Hence solve the equation 2 sin²(θ − 20°) − cos (θ − 20°) − 1 = 0for 0° ≤ θ ≤ 180°.
Sketch the graph of f(x) = sin(2x + π/4) for −π ≤ x ≤ π.
Solve the equation 2 cosθ − 5sinθ = 4 cosθ + 3 sinθ for 0 ≤ θ ≤ 2π.
Solve each of these equations, giving all solutions between 0 and 27 in radians correct to 3 significant figures.a) sin x = 0.4b) cos x = 0.4c) sin x = −0.8d) cos x= −0.21 e) tan x =
Solve each of these equations, giving all solutions between 0° and 360° to the nearest degree.a) sin x° = 0.9 b) cos x° = 0.9c) sin x° = −0.6d) cos x° = 0.33e) tan x° = 0.25f) tan x°
a) Sketch the graph of y = sin 4x for 0° ≤ x ≤ 180°.b) Find all the angles x in the interval 0° ≤ x ≤ 180° for whichi) sin 4x = √3/2ii) sin 4x = −1/2
Solve the equation sinθ + 5 cosθ = 3 sinθ for 0 ≤ θ ≤ 2π.
Prove the following identities.a.b.c.d. (2 − cos² θ)² − 4 sin² θ = cos4 θ. tan 0 = sin²0 1-sin¹0
Simplifya.b.c. sin²0-cos²0 sin 9-cos COS
Solve the equation 3 sin x = 2 cos x, giving all solutions between 0° and 360°.
a) Express 2 cos x − sin²x in the form (cos x + p)² + q, stating the values of p and q.b) Use your answer to part (a) to find the maximum and minimumvalues of 2 cos x − sin²x.
Find all the angles (to the nearest 0.1°) between 0° and 360° whose sine is −0.3636.
Show that (cos x − sin x)² + (cos x + sin x)² = 2.
Solve the equation for 0° ≤ 0 ≤ 360°.2 sinθ = 3 tanθ.
a) Express cos (3π + x) in terms of cos x.b) Express sin (x + 4π) in terms of sin x.c) Express sin (x − π) in terms of sin x.d) Express tan (x − π) in terms of tan x.
Here is the graph of f(x) = cos 2x.a) Which of the following domains ensures that the functionf(x) = cos 2x is one-to-one?b) What is the range of f(x) for this domain?c) Find an expression for a
Find all the angles (to the nearest 0.1°) between 0° and 360° whose cosine is 0.7660.
Solve the equation for 0° ≤ 0 ≤ 360°.3 cos 2θ = 4 sin 2θ.
a) Express sin (180° + x) in terms of sin x.b) Express cos (180° − x) in terms of cos x.c) Express tan (180° + x) in terms of tan x.d) Express tan (360° − x) in terms of tan x.
Solve 5+ 2 cos (3θ − π/4) = 6 for −π ≤ θ < π.

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