Questions and Answers of Complete Pure Mathematics

Given that y = 3x³ − ax² and d2y/dx2 = 54 when x = 2 work out the value of a.
Find the equation of the tangent and the normal to the curve y = 2x³ − xat the point where x = 2.
By considering the gradient of the chord joining (x, f(x)) and ((x + h), f(x + h)) for the following functions f(x), deduce what f´(x) will be.f(x) = x2 + 3x
The curve y = 2x³ + 6x − 5 crosses the y-axis at the point P. Find theequation of the tangent to the curve at the point P.
Find dy/dx in each case. a.b. y = (x² + 4)5c)d)e)f)g)h)i)j) y= y = √√√4x-1
Find the gradient of the tangent to the curve y = x³ − 8x² − 7 where x = −2.
Find dy/dx in each case.a) y = 2x1/2 b) y = −5/xc) y = √x/3d) y = 12x7e) y = 8/x3f) y = 3/√xg) y = 1/3√x h) y = 2/9xi) y = x × x5j) y = 12x8 ÷ 3xk) y = 4x × 5x² l) y
Find the derived function in each case.a) f(x) = x³b) f(x) = x9c) f(x) = 4x²d) f(x) = −15xe) f(x) = 7x−4f) f(x) = 3x−1g) f(x) = −1/2xh) f(x) = 4/5 x10i) f(x) = √xj) f(x) =
By considering the gradient of the chord joining (x, f(x)) and ((x + h), f(x + h)) for the following functions f(x), deduce what f´(x) will be.f(x) = 2x²
a) Using a scale of 1 cm to 1 unit, plot the graph of y = x² + x − 2 for −3 ≤ x ≤ 3.b) Estimate the gradient of the tangent to the curve at the point (2, 4).c) Estimate the gradient of the
Find the equation of the tangent to the curvea) y = x³ − 6x + 3 at the point (2, −1)b) y = 2x4 + 9x³ + x when x = −2c) y = (x − 7)(x + 4) at the point when x = 1d) y= x4 – 3x³/x at the
Find the gradient of the tangent to the curve y = 5x³ + 3x² − x + 4at the point (1, 11).
Find the equation of the normal to the curve y = 3x² at the point (t, 3t²).
Find f´´(x) in each case.a)b)c)d)e)f)g)h)i) f(x) = 8x²-x 2x³
Find the gradient of the tangent to the curve y = 2x² + 3x + 1 at each of the points where the curve meets the line y = 4(x + 1).
Find the equation y = √x (√x − 2)of the normal to the curveat the point where x = 9.
Finda)b)c)d)e)f)g)h)i)j)k)l) d dx (x-²)
By considering the gradient of the chord joining (x, f(x)) and ((x + h), f(x + h)) for the following functions f(x), deduce what f´(x) will be.f(x) = x4
Given that y = 8x³ − 3x + 4/x, find the value of d2y/dx2 when x = −2.
Find the coordinates of the points on the curve with equation y = 2x³ + 3x² − 12x where the gradient is 24.
Find the gradient of the tangent to the curve y = 5 − 3x + x³ at the point (1, 3).
Find dy/dx in each case.a)b)c)d)e) y = 2x³(1 − 4x − 8x²)f) y=(√x + 3)(2√x + 5)g)h)i)j) y=x¹ + ¹ X I 3 √x
By considering the gradient of the chord joining (x, f(x)) and ((x + h), f(x + h)) for the following functions f(x), deduce what f´(x) will be.f(x) = 8x
Find the equation of the normal to the curvea) y = 7x² − 8x + 9 when x = 2b) y = 5x³ + x² −2 at the point (−1, −6)c) y = (3x + 10)² at the point (−3, 1)d) y = 2x²(6 − x) when x =
Find dy/dx whena) y = x(x + 3)²b)c)d) 3 录 y = 65 +
x + 5, x and x − 4 are three consecutive terms of a geometric progression. Find the value of x.
All the terms of a geometric progression are positive. The sum of the first two terms is 100 and the sum to infinity is 180. Find the first term and the common ratio.
A geometric progression has a common ratio 3/4 and a sum to infinity of 92. Find the 1st term.
The nth term of a geometric progression is (−1/3)n. Find the 1st term and the common ratio, r.
Show that the sum of the integers from 1 to n is 1/2n(n + 1).
Write the following series using the sigma notation.a) 2 + 5 + 8 + 11 + ...b) 1 + 4 + 9 + 16 + ... + 121c) 7 + 3 − 1 − 5 + ...d) x² − x4 + x6 − x8 + x10 − x12 e) 1 + 2+ 3 + 4 + 5 + 6
Find the first three terms of the sequence defined by U n1+1 = 1 - 3 1-2un where u, = 1.
Find the sum of all the odd numbers between 20 and 100.
Nita invests $50 at the beginning of each year for 10 years at 4% compound interest. Calculate the total value of the investment at the end of the 10 years.
A geometric sequence is 1 − 2x + 4x² − 8x³ + ... Find an expression for the nth term.
The sum of the first nine terms of a geometric sequence is 684 and the common ratio is −2.Find the 1st and the 8th terms.
The 3rd and 4th terms of a geometric progression are 5 and −20, respectively.Find the 1st term, the common ratio, r, and the 9th term.
A geometric progression has a sum to infinity of −30. The 1st term of the geometric progression is −9. Find the common ratio, r.
In an arithmetic progression the 8th term is 39 and the 4th term is 19. Find the 1st term and the sum of the first 12 terms.
An expression for the nth triangular number is n(n + 1)/2. Write down the 20th triangular number.
The 1st and 2nd terms of a geometric progression are 4 and 12, respectively. Find the sum of the first ten terms.
A geometric progression has a 1st term of 80. The sum of the first two terms is 40 Find the common ratio, r, and the sum to infinity of the geometric progression.
The 3rd term of an arithmetic progression is 1 and the 6th term is 10. Find the 4th term.
Work out the 15th term in the series Σ(2r + 1). r=1
Find which of the following series are geometric progressions. If they are, write down the common ratio.a) 5 + 15 + 45 + 135 + ...b) −60 + 30 − 15 + 7.6 − 3.8 + ...c) 1/2 + 1/3 +1/4 + 1/5 +
The 10th and the 20th terms of an arithmetic progression are 18 and 88, respectively. Find the 1st term and the sum of the first 16 terms.
The 9th term of an arithmetic progression is 8 and the 4th term is 18. Find the 1st term and the common difference, d.
The first four terms of a geometric progression are 3, −1, 1/3, −1/9. Find the sum to infinity.
Write down the first three terms, the last term (if the series is finite) and the nth term of the following series.a)b)c)d) Σ(3r−2) r=1
i) Prove the identityii) Solve the equation 5 sin cos 0 = (cosθ − sinθ) (cosθ + sinθ) for 0° ≤ θ ≤ 180°. sin cos tan 0 (cos-sin)(cos+sin 0) (1-tan²0)
i) Prove the identityii) Hence, solve the equation. 1 cos - tane 1-sin0 1+ sin 0
Find the value of a and the value of b.(1 + x)4(1 + ax)7 = 1 + 74x² + bx² + ...
Find the value of a, the value of b and the value of c.(2 − ax)8 = 2b − 5120x + cx² + ...
Find the value of p and the value of a.(2 − ax)6 = 64 + px + 135x² + ... with p > 0 and a > 0.
The line PQ has equation 3x − 2y = 12. P is the point with coordinates (6, 3) and Q is the point with coordinates (−2, k).a) Find the value of k.b) Find an equation of the line through P that is
P(2, 1) is a point on the circumference of the circle x² + y² − 10x + 2y + 13 = 0.PQ is a diameter of the circle. Find the equation of the line through P and Q.
Find the equation of the line perpendicular to the line y = 5x and passing through the point (2, −1).
Show that f(x) is a self-inverse function, that is f(x) = f−1(x)f(x)= x/ x −1, x ∈ R, x ≠ 1.
Find the domain and range for which this function is defined. f: x ↦ 1/1− x
A curve has equation y = 2x² + x + 11.i) Find the set of values of x for which y ≤ 17.
The sum of the first n terms of a series is given by Sn = 2n(n + 3). Show that the terms of the series form an arithmetic progression.
The sum of the first six terms of an arithmetic progression is 51 and the sum of the first 12 terms of the same progression is 282. Find the 1st term and the common difference, d.
Find the sum to 11 terms of the geometric progression 1/2 − 1/4 + 1/8 − 1/16 + ...
The nth term of a geometric progression is (−1)n (2)n.a) Find the first three terms.b) Explain why this geometric progression does not have a sum to infinity.
In an arithmetic progression the 8th term is twice the 4th term and the 20th term is 40. Find the common difference, d, and the sum of the first 10 terms.
Find the sum of the first nine terms of a geometric sequence that has a 3rd term of 45 and a 6th term of 1215.
The 1st term of a geometric progression is 1 and the 5th term is 0.0016. Find the value of r and the sum of the first eight terms.
A geometric progression has the sum to infinity equal to twice the 1st term. Find the common ratio, r.
In an arithmetic progression the sum of the 1st term and the 5th term is 18. The 5th term is 6 more than the 3rd term. Find the sum of the first 12 terms.
The sum of the first three terms of an arithmetic progression is 15. The product of the 1st and 3rd terms is −24.Find two possible values for the 1st term.
The first three terms of an arithmetic progression are 1/2, x and 25. The first three terms of a geometric progression are x + 1/4,and y, where x and y are positive numbers. Find the value of x and
The 4th term of a geometric series is 6. The 7th term is 162.a) Find the 1st term and the common ratio, r. b) Show that the sum of the first n terms can be written as S = 71 3"-1 9
The sum of the first n terms of an arithmetic progression is n² − 3n.Write down the 10th term.
The sum of the first two terms of a geometric progression is 10. The sum to infinity of the geometric progression is 18. Work out the two possible values of the common ratio, r.
The 4th term of a geometric progression is 9 times the 6th term. The 5th term is 3 and r > 0. Find the sum of the first six terms.
Find the sum of the first 30 terms of the arithmetic progression 2, −5, −12, −19, ...
The 1st and 3rd term of a geometric progression are 18 and 2, respectively. Find two possible values of the common ratio, r, and the sum to infinity for each.
A geometric progression has a common ratio of 2/5. The sum of the 13 first four terms is 2 13/25.
The first four terms of a geometric progression areFind the sum to infinity of this geometric progression. 1, X x+1' 2 x³ (x+1)²(x+1)³*
Find the 13th term of the geometric progression 1.1, 1.21, 1.331, ...
The 1st term of a geometric progression is 5. The sum to infinity of the geometric progression is 20. Find the common ratio, r.
The nth term of an arithmetic progression is 1/2(7 − n). Write down the firstthree terms and the 20th term.
By expressing the recurring decimal 0.3̇6̇ as the sum of a geometric progression, write 0.3̇6̇ as a fraction in its simplest form.
The 4th and 6th terms of a geometric progression are 200 and 800, respectively. Find two possible values for the common ratio, r, and two possible values for the 1st term.
The 1st term and the last term of an arithmetic progression are 15 and −20,respectively. The sum of all the terms is −40.Work out how many terms there are in thearithmetic progression.
Given that work out the value of n. 4 11. Σ6r=2 25r, r=1 r=1
The sum of the terms of an arithmetic progression is 38.5. The 1st term is 1 and the last term is 6. Work out how many terms there are in the arithmetic progression.
Calculate the sum to infinity of the series 125 + 75 + 45 + 27 + ...
The nth term of the sequence 2, 0, −2, −4, −6, ... is 4 times the nth term of thesequence −22, −20, −18, −16, −14, ... Work out the value of n.
Write down the nth term of each sequence.a) 7, 13, 19, 25, 31, ... b) 7, 4, 1, −2, −5, ...c) 1/2, 1/3, 1/4, 1/5, 1/6, ...d) −15, −4, 7, 18, 29, ...
The 20th term of an arithmetic progression is −41 and the sum of the first20 terms is −440. Work out the sum of the first 10 terms.
A geometric progression has its 1st term and 3rd term as −3/7 and −12/175, respectively. The common ratio, r, is positive. Find r and the sum to infinity of this geometric progression.
Mo has seven pieces of wood. Each piece of wood has a different length. The lengths are in arithmetic progression. The length of the largest piece of wood is 5 times the length of the smallest piece
Work out the number of positive terms in the arithmetic progression 200 + 194 + 188 + 182 + ...
The sum to n terms of a geometric progression is 3n − 1. Find the first four terms and the common ratio, r.
Megan borrows some money from her aunt. She gives her aunt $10 the first month, $12 the second month, $14 the third month. The payments continue to rise by $2 each month. The final payment she makes
The nth term of a geometric progression is (−1)n + 1(3)n (−x)n. Find the common ratio, r.
The 1st term of a finite arithmetic progression is 18 and the common progression difference, d, is The sum of the terms is −12. Find the number of terms. 두다
A clock has a pendulum. The time it takes for the pendulum's first swing is 5 seconds. The time for each successive swing is 3/5 of the time of the previous swing. Find how long it takes the pendulum
The first three terms of an arithmetic progression are (m − 3), (m + 1) and (5m + 5).a) Work out the value of m.b) Work out the sum of the first 10 terms.

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