QuestionsQuestion 1Using binomial theorem, expand the following:(i)(x2−2x2)4\left( \dfrac{x}{2} - \dfrac{2}{x^2} \right)^4(2x−x22)4(ii)(2a−xa)7\left( 2a - \dfrac{x}{a} \right)^7(2a−ax)7(iii)(ax−xa)6\left( \sqrt{\dfrac{a}{x}} - \sqrt{\dfrac{x}{a}} \right)^6(xa−ax)6SolutionTheoryQuestion 2Calculate the following by means of binomial theorem:(i)(0.97)3(0.97)^3(0.97)3(ii)(2.02)4(2.02)^4(2.02)4(iii)(9.98)4(9.98)^4(9.98)4(iv)(2.1)5(2.1)^5(2.1)5SolutionTheoryQuestion 3Expand and simplify the following:(i)(a+2x)4+(a−2x)4(a + \sqrt{2}x)^4 + (a - \sqrt{2}x)^4(a+2x)4+(a−2x)4(ii)(2+3)5+(2−3)5(2 + \sqrt{3})^5 + (2 - \sqrt{3})^5(2+3)5+(2−3)5SolutionTheoryQuestion 4Expand the following in ascending power of xxx:(i)(2+x−x2)4(2 + x - x^2)^4(2+x−x2)4(ii)(1−x+x2)4(1 - x + x^2)^4(1−x+x2)4SolutionTheoryQuestion 5Find the term involving:(i)x4x^4x4 in the expansion of (3−2x)7(3 - 2x)^7(3−2x)7(ii)x−2x^{-2}x−2 in the expansion of (x−2x2)13\left( x - \dfrac{2}{x^2} \right)^{13}(x−x22)13(iii)a4a^4a4 in the expansion of (2x−a)9\left( \dfrac{2}{x} - a \right)^9(x2−a)9(iv)y3y^3y3 in the expansion of (x−y)11(x - \sqrt{y})^{11}(x−y)11SolutionTheoryQuestion 6Find the coefficient of:(i)x5x^5x5 in the expansion of (x2−32x)10\left( x^2 - \dfrac{3}{2x} \right)^{10}(x2−2x3)10(ii)xnx^nxn in the expansion of (x2−1x)2n\left( x^2 - \dfrac{1}{x} \right)^{2n}(x2−x1)2nSolutionTheoryQuestion 7Find 6th term in the expansion of (x2−32x)10\left( x^2 - \dfrac{3}{2x} \right)^{10}(x2−2x3)10.SolutionTheoryQuestion 8Find the term independent of xxx in the following expansions:(i)(x−2x)10\left( x - \dfrac{2}{x} \right)^{10}(x−x2)10(ii)(x+12x2)10\left( \sqrt{x} + \dfrac{1}{2x^2} \right)^{10}(x+2x21)10(iii)(1+x2)3(1+1x2)4(1 + x^2)^3 \left( 1 + \dfrac{1}{x^2} \right)^4(1+x2)3(1+x21)4SolutionTheoryQuestion 9Determine the middle term in the following expansions:(i)(1x−x22)12\left( \dfrac{1}{x} - \dfrac{x^2}{2} \right)^{12}(x1−2x2)12(ii)(32x−13x)11\left( \dfrac{3}{2}x - \dfrac{1}{3x} \right)^{11}(23x−3x1)11(iii)(2x−12x)2n+1\left( 2x - \dfrac{1}{2x} \right)^{2n+1}(2x−2x1)2n+1SolutionTheoryQuestion 10Show that: (n1)+(n2)+(n3)+⋯+(nn)=2n−1\dbinom{n}{1} + \dbinom{n}{2} + \dbinom{n}{3} + \cdots + \dbinom{n}{n} = 2^n - 1(1n)+(2n)+(3n)+⋯+(nn)=2n−1.SolutionTheory