Find the exact value of the term independent of \(x\) in the expansion of

\(\left(2+\frac{3}{x^2}\right)^{10}(1-4x^2)^2.\)

In this question \(a\), \(b\) and \(n\) are constants. When \(5(2+ax)^n\) is written in ascending powers of \(x\), the first three terms are \(640+b^2x+30240x^2\).

Find the value of \(a\) and the possible values of \(b\).

(a) Write down the coefficient of \(x^r\) in the binomial expansion of \((2+x)^{59}\).

(b) For this expansion, find the value of \(r\) for which the coefficient of \(x^r\) is equal to the coefficient of \(x^{r+1}\).

(a) Using an appropriate quadratic factorisation, find the first three terms in the binomial expansion of \(\left(9 x^{2}+12 x+4\right)^{5}, \quad\) in ascending powers of \(x\). You must simplify your coefficients.

(b) Find the term independent of \(x\) in the expansion of \(\left(\frac{6}{x^{2}}+\frac{x^{4}}{2}\right)^{12}\).

The first three terms, in descending powers of \(x\), in the expansion of \(\left(3 x^{2}-a\right)^{n}\left(1+\frac{1}{x^{2}}\right)^{2}\) can be written as \(\quad 729 x^{12}+972 x^{10}+b x^{8}, \quad\) where \(a, b\) and \(n\) are constants.

Find the values of \(a, b\) and \(n\).

(a) Find the term independent of \(x\) in the expansion of \(\left(x^{2}-\frac{3}{x^{4}}\right)^{15}\). (b) In the expansion of \((1+a x)^{9}\) the coefficient of \(x^{3}\) is 7 times the coefficient of \(x^{2}\). Given that \(a\) is a positive constant, find the value of \(a\).

The expansion of \((ax-2)^4\left(1+\frac{b}{x}\right)^3\) is written in descending powers of \(x\).

The first 3 terms of this expansion are \(81x^4+999x^3+cx^2\).

It is given that \(a\), \(b\) and \(c\) are positive integers.

Find the values of \(a\), \(b\) and \(c\).

(a) In the expansion of \((1+k x)^{15}\), where \(k\) is a constant, the coefficient of \(x^{3}\) is -29120 . Find the value of \(k\). (b) Find the term independent of \(y\) in the expansion of \(\left(8 y^{2}-\frac{1}{2 y}\right)^{12}\).

(a) Find, in descending powers of \(x\), the first 3 terms in the expansion of \(\left(x+\frac{2}{x^{2}}\right)^{10}\). Simplify each term as far as possible. term as far as possible.

(b) Find the term independent of \(x\) in the expansion of \(\left(4 x^{2}+\frac{1}{2 x^{2}}\right)^{8}\).

(a) In the expansion of \(\left(x+x^{2}\right)^{8}\) in ascending powers of \(x\), the 3rd and 6th terms are equal. Find the value of \(x\).

(b) In the expansion of \(\left(x+\frac{2}{x}\right)^{n}\) in decreasing powers of \(x\), the 6th term is a constant. (i) Find the value of the positive integer \(n\).

(ii) Find the value of the 6th term.

The expansion of \(\left(a+\frac{x}{a}\right)^{n}\) in ascending powers of \(x\) begins \(b^{4}+48 b^{3} x\), where \(n, a\) and \(b\) are positive integers. (a) Show that \(a^{\frac{n}{2}-4}=\left(\frac{48}{n}\right)^{2}\). (b) Given also that the third term is \(1056 b^{2} x^{2}\), find the values of \(n, a\) and \(b\).

(a) The first three terms, in ascending powers of \(x\), in the expansion of \((3+p x)^{n}\) are \(243+810 x+q x^{2}, \quad\) where \(n, p\) and \(q\) are constants. Find the values of \(n, p\) and \(q\). (b) Find the term independent of \(y\) in the expansion of \(\left(2 y-\frac{1}{3 y^{2}}\right)^{6}\). Give your answer in exact form.

In the binomial expansion of \(\left(2+\frac{x}{2}\right)^{n}\), the first three terms in increasing powers of \(x\) are \(b+a b x+\frac{9}{8} a b x^{2}\). Find the values of the constants \(n, a\) and \(b\).

In this question \(a\) and \(b\) are integers.

Three terms in the expansion of \((2+ax)^5(1+bx)\) are

\(32+112x-240x^2.\)

Find the values of \(a\) and \(b\).

(a) Find and simplify the term independent of \(x\) in the expansion of \(\left(x^{2}-\frac{1}{2 x^{3}}\right)^{10}\). (b) DO NOT USE A CALCULATOR IN THIS PART OF THE QUESTION. (i) Use the binomial theorem to show that \((1+2 \sqrt{2})^{4}-(1-2 \sqrt{2})^{4}=k \sqrt{2}\), where \(k\) is an integer to be found. (ii) Hence write \(\frac{(1+2 \sqrt{2})^{4}-(1-2 \sqrt{2})^{4}}{1+\sqrt{2}}\) in the form \(a+b \sqrt{2}\), where \(a\) and \(b\) are integers.

Find the coefficient of \(x^8\) in the expansion of

\((1-x^2)\left(2x-\frac1x\right)^{10}.\)

(a) Find the first three terms in the expansion of

\(\left(x^2-\frac4{x^2}\right)^{10}\)

in descending powers of \(x\). Give each term in its simplest form.

(b) Hence find the coefficient of \(x^{16}\) in the expansion of

\(\left(x^2-\frac4{x^2}\right)^{10}\left(x^2+\frac2{x^2}\right)^2.\)

In the expansion of

\(\left(ax+\frac{b}{x^2}\right)^9,\)

where \(a\) and \(b\) are constants with \(a\gt 0\), the term independent of \(x\) is \(-145\,152\) and the coefficient of \(x^6\) is \(-6912\). Show that \(a^2b=-12\) and find the value of \(a\) and the value of \(b\).

(a)(i) Find the first three terms in the expansion of

\(\left(1+\frac{x}{7}\right)^5\)

in ascending powers of \(x\). Simplify the coefficient of each term.

(a)(ii) The expansion of

\(7(1+x)^n\left(1+\frac{x}{7}\right)^5,\)

where \(n\) is a positive integer, is written in ascending powers of \(x\). The first two terms in the expansion are \(7+89x\). Find the value of \(n\).

(b) In the expansion of \((k-2x)^8\), where \(k\) is a constant, the coefficient of \(x^4\) divided by the coefficient of \(x^2\) is \(\frac58\). The coefficient of \(x\) is positive. Form an equation and hence find the value of \(k\).

(a) It is given that the first four terms, in ascending powers of \(x\), in the expansion of \(\left(1-\frac{x}{2}\right)^n\) can be written in the form

\(1-8x+px^2+qx^3\),

where \(n\), \(p\) and \(q\) are integers. Find the values of \(n\), \(p\) and \(q\).

(b) Find the term independent of \(x\) in the expansion of \(\left(\frac{2}{x^2}+\frac{x}{3}\right)^6\), giving your answer as a rational number.

The first three terms, in descending powers of \(x\), in the expansion of

\(\displaystyle \left(2x^2-\frac1{4x}\right)^n\)

can be written in the form

\(256x^{16}+ax^{13}+bx^c\),

where \(n\), \(a\), \(b\) and \(c\) are integers. Find the values of \(n\), \(a\), \(b\) and \(c\).

(a)(i) Use the binomial theorem to expand \((1+3x)^7\) in ascending powers of \(x\), as far as the term in \(x^3\). Simplify each term.

(a)(ii) Show that your expansion from part (a)(i) gives the value of \(1.03^7\) as \(1.23\) to \(2\) decimal places.

(b) Find the term independent of \(x\) in the expansion of

\(\left(\frac{x^4}{2}+\frac{2}{x}\right)^{15}.\)

The first three terms, in ascending powers of \(x\), in the expansion of

\(\left(1+\frac{x}{6}\right)^{12}(2-3x)^3\)

can be written in the form \(8+px+qx^2\), where \(p\) and \(q\) are constants. Find the values of \(p\) and \(q\).

(a) Find the rational numbers \(a\), \(b\) and \(c\), such that the first three terms, in descending powers of \(x\), in the expansion of

\(\left(3x^2-\frac{1}{9x}\right)^5\)

can be written in the form \(ax^{10}+bx^7+cx^4\).

(b) Hence find the coefficient of \(x^4\) in the expansion of

\(\left(3x^2-\frac{1}{9x}\right)^5\left(1+\frac{1}{x^3}\right)^2.\)

(a) (i) Write down the first three terms, in ascending powers of \(x\), in the expansion of \((1+4x)^n\), where \(n\) is a positive integer.

(ii) In the expansion of \((1+4x)^n(1-4x)\), the coefficient of \(x^2\) is 6032. Find the value of \(n\).

(b) Find the term independent of \(x\) in the expansion of \(\left(\frac{x}{2}-\frac{8}{x^4}\right)^{10}\).

The first three terms, in ascending powers of \(x\), in the expansion of

\(\left(1-\frac{2x}{9}\right)^{18}(1+3x)^3\)

are written in the form

\(1+ax+bx^2,\)

where \(a\) and \(b\) are constants. Find the exact values of \(a\) and \(b\).

The first three terms, in descending powers of \(x\), of the expansion of

\(\left(ax+\frac25\right)^5\left(1-\frac{b}{x}\right)^2\)

can be written as

\(32x^5-160x^4+cx^3,\)

where \(a\), \(b\) and \(c\) are constants. Find the exact values of \(a\), \(b\) and \(c\).

The first four terms in the expansion of \((3+ax)^4\), in ascending powers of \(x\), are

\(81+bx+cx^2+\frac32x^3.\)

Find the values of \(a\), \(b\) and \(c\).

(a) In the expansion of

\(\left(2k-\frac{x}{k}\right)^5,\)

where \(k\) is a constant, the coefficient of \(x^2\) is \(160\). Find the value of \(k\).

(b)

(i) Find, in ascending powers of \(x\), the first 3 terms in the expansion of \((1+3x)^6\), simplifying the coefficient of each term.

(ii) When \((1+3x)^6(a+x)^2\) is written in ascending powers of \(x\), the first three terms are \(4+68x+bx^2\), where \(a\) and \(b\) are constants. Find the value of \(a\) and of \(b\).

The first 3 terms in the expansion of

\((a+x)^3\left(1-\frac{x}{3}\right)^5\)

in ascending powers of \(x\), can be written in the form

\(27+bx+cx^2,\)

where \(a\), \(b\) and \(c\) are integers. Find the values of \(a\), \(b\) and \(c\).

(a) Find the first three non-zero terms in the expansion of

\(\left(2-\frac{x^2}{4}\right)^6\)

in ascending powers of \(x\). Simplify each term.

(b) Hence find the term independent of \(x\) in the expansion of

\(\left(2-\frac{x^2}{4}\right)^6\left(3-\frac1{x^2}\right)^2.\)

Using the binomial theorem, expand

\((1+e^{2x})^4,\)

simplifying each term.

The first three terms, in ascending powers of \(x\), in the expansion of \((2+ax)^n\) can be written as \(64+bx+cx^2\), where \(n\), \(a\), \(b\) and \(c\) are constants.

(a) Find the value of \(n\).

(b) Show that \(5b^2=768c\).

(c) Given that \(b=12\), find the exact value of \(a\) and of \(c\).

(a) Find the first three terms, in ascending powers of \(x^2\), in the expansion of

\(\left(\frac12-\frac23x^2\right)^8.\)

Write your coefficients as rational numbers.

(b) Find the coefficient of \(x^2\) in the expansion of

\(\left(\frac12-\frac23x^2\right)^8\left(2x+\frac1x\right)^2.\)

(a) Expand \((2-3x)^4\), evaluating all of the coefficients.

(b) The sum of the first three terms in ascending powers of \(x\) in the expansion of

\((2-3x)^4\left(1+\frac ax\right)\)

is

\(\frac{32}{x}+b+cx,\)

where \(a\), \(b\) and \(c\) are integers. Find the values of \(a\), \(b\) and \(c\).

The first 3 terms in the expansion of \((3-ax)^5\), in ascending powers of \(x\), can be written in the form

\(b-81x+cx^2.\)

Find the value of each of \(a\), \(b\) and \(c\).

(a) Find the first three terms in the expansion of

\(\left(4-\frac{x}{16}\right)^6\)

in ascending powers of \(x\), giving each term in its simplest form.

(b) Hence find the term independent of \(x\) in the expansion of

\(\left(4-\frac{x}{16}\right)^6\left(x-\frac1x\right)^2.\)

(a) Expand \((2-x)^5\), simplifying each coefficient.

(b) Hence solve

\(\frac{e^{(2-x)^5}\times e^{80x}}{e^{10x^4+32}}=e^{-x^5}.\)

Do not use a calculator in this question.

(a) Find the term independent of \(x\) in the binomial expansion of

\(\left(3x-\frac1x\right)^6.\)

(b) In the expansion of

\(\left(1+\frac{x}{2}\right)^n,\)

the coefficient of \(x^4\) is half the coefficient of \(x^6\). Find the value of the positive constant \(n\).

Find the coefficient of \(x^2\) in the expansion of

\(\left(x-\frac{3}{x}\right)\left(x+\frac{2}{x}\right)^5.\)

Given that the coefficient of \(x^2\) in the expansion of

\((1+x)\left(1-\frac{x}{2}\right)^n\)

is \(\dfrac{25}{4}\), find the value of the positive integer \(n\).

The first three terms in the expansion of

\((a+bx)^5(1+x)\)

are

\(32-208x+cx^2.\)

Find the value of each of the integers \(a\), \(b\) and \(c\).

(i) Find the first three terms, in ascending powers of \(x\), in the expansion of \(\left(3-\dfrac{x}{9}\right)^6\).

(ii) Hence find the term independent of \(x\) in the expansion of \(\left(3-\dfrac{x}{9}\right)^6\left(x-\dfrac2x\right)^2\).

(i) The first 3 terms, in ascending powers of \(x\), in the expansion of \((2+bx)^8\) can be written as

\(a+256x+cx^2.\)

Find \(a\), \(b\) and \(c\).

(ii) Using the values found in part (i), find the term independent of \(x\) in the expansion of

\((2+bx)^8\left(2x-\frac3x\right)^2.\)

(a) In the binomial expansion of \(\left(a-\dfrac{x}{2}\right)^6\), the coefficient of \(x^3\) is 120 times the coefficient of \(x^5\). Find the possible values of the constant \(a\).

(b) (i) Expand \((1+2x)^{20}\) in ascending powers of \(x\), as far as the term in \(x^3\). Simplify each term.

(ii) Use your expansion to show that the value of \(0.98^{20}\) is \(0.67\) to 2 decimal places.

(a) (i) Given that \(\left(x^2-\dfrac1{px}\right)^8=x^{16}-4x^{13}+qx^{10}+rx^7+\cdots\), find the value of each of the constants \(p\), \(q\) and \(r\).

(ii) Explain why there is no term independent of \(x\) in the binomial expansion of \(\left(x^2-\dfrac1{px}\right)^8\).

(b) In the binomial expansion of \(\left(1-\dfrac{\sqrt{x}}2\right)^n\), where \(n\) is a positive integer, the coefficient of \(x\) is 30. Form an equation in \(n\) and hence find the value of \(n\).

The first three terms in the expansion of

\(\left(1-\frac{x}{7}\right)^{14}(1-2x)^4\)

can be written as \(1+ax+bx^2\). Find the value of each of the constants \(a\) and \(b\).

(i) Expand \((3+x)^4\), evaluating each coefficient.

In the expansion of

\(\left(x-\frac{p}{x}\right)(3+x)^4\)

the coefficient of \(x\) is zero.

(ii) Find the value of the constant \(p\).

(iii) Hence find the term independent of \(x\).

(iv) Show that the coefficient of \(x^2\) is \(90\).

The first four terms in the expansion of \((1+ax)^5(2+bx)\) are

\(2+32x+210x^2+cx^3,\)

where \(a\), \(b\) and \(c\) are integers. Show that

\(3a^2-16a+21=0\)

and hence find the values of \(a\), \(b\) and \(c\).

The first three terms, in ascending powers of \(x\), in the expansion of \((2+ax)^n\) are \(1024-1280x+bx^2\). Find the values of \(n\), \(a\), and \(b\).

Hence find the term independent of \(x\) in the expansion of \((2+ax)^n\left(x-\dfrac1x\right)^2\).

(i) Find the first \(3\) terms in the expansion of

\(\left(2x-\frac{1}{16x}\right)^8\)

in descending powers of \(x\).

(ii) Hence find the coefficient of \(x^4\) in the expansion of

\(\left(2x-\frac{1}{16x}\right)^8 \left(\frac{1}{x^2}+1\right)^2.\)

(i) The first three terms in the expansion of

\(\left(3-\frac1{9x}\right)^5\)

can be written as \(a+\dfrac bx+\dfrac c{x^2}\). Find the value of each of the constants \(a\), \(b\) and \(c\).

(ii) Use your values of \(a\), \(b\) and \(c\) to find the term independent of \(x\) in the expansion of

\(\left(3-\frac1{9x}\right)^5(2+9x)^2.\)

(i) Find the first \(3\) terms in the expansion of

\(\left(2x-\frac{1}{16x}\right)^8\)

in descending powers of \(x\).

(ii) Hence find the coefficient of \(x^4\) in the expansion of

\(\left(2x-\frac{1}{16x}\right)^8 \left(\frac{1}{x^2}+1\right)^2.\)

The coefficient of \(x^2\) in the expansion of \((2-x)(3+kx)^6\) is equal to \(972\). Find the possible values of the constant \(k\).

The 7th term in the expansion of \((a+bx)^{12}\) in ascending powers of \(x\) is \(924x^6\). It is given that \(a\) and \(b\) are positive constants.

(i) Show that \(b=\dfrac1a\).

The 6th term in the expansion of \((a+bx)^{12}\) in ascending powers of \(x\) is \(198x^5\).

(ii) Find the value of \(a\) and of \(b\).

(a) In the expansion of \((2+px)^5\), the coefficient of \(x^3\) is equal to \(-\dfrac{8}{25}\). Find the value of the constant \(p\).

(b) Find the term independent of \(x\) in the expansion of \(\left(2x^2+\dfrac{1}{4x^2}\right)^8\).

The first three terms in the expansion of

\(\left(3-\frac{x}{6}\right)^n\)

are \(81+ax+bx^2\). Find \(n\), \(a\), and \(b\).

(i) Given that \(a\) is a constant, expand \((2+ax)^4\), in ascending powers of \(x\), simplifying each term of your expansion.

Given also that the coefficient of \(x^2\) is equal to the coefficient of \(x^3\),

(ii) show that \(a=3\),

(iii) use your expansion to show that the value of \(1.97^4\) is \(15.1\) to 1 decimal place.

The first three terms of the binomial expansion of \((2-ax)^n\) are

\(64-16bx+100bx^2.\)

Find the value of each of the integers \(n\), \(a\) and \(b\).

(i) Find, in ascending powers of \(x\), the first 3 terms in the expansion of \(\left(2-\dfrac{x^2}{4}\right)^5\).

(ii) Hence find the term independent of \(x\) in the expansion of \(\left(2-\dfrac{x^2}{4}\right)^5\left(\dfrac1x-\dfrac3{x^2}\right)^2\).

(i) Find, in ascending powers of \(x\), the first 3 terms in the expansion of \(\left(2-\dfrac{x^2}{4}\right)^6\).

(ii) Hence find the coefficient of \(x^2\) in the expansion of \(\left(2-\dfrac{x^2}{4}\right)^6\left(\dfrac1x+x\right)^2\).

(i) Expand \((1+x)^4\), simplifying all coefficients.

(ii) Expand \((6-x)^4\), simplifying all coefficients.

(iii) Hence express \((6-x)^4-(1+x)^4=175\) in the form \(ax^3+bx^2+cx+d=0\), where \(a\), \(b\), \(c\), and \(d\) are integers.

(iv) Show that \(x=2\) is a solution of the equation in part (iii) and show that this equation has no other real roots.