DP Mathematics: Analysis and Approaches Questionbank

AHL 1.15—Proof by induction, contradiction, counterexamples
Description
[N/A]Directly related questions
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20N.2.AHL.TZ0.H_6:
Use mathematical induction to prove that dndxn(xepx)=pn-1(px+n)epxdndxn(xepx)=pn−1(px+n)epx for n ∈ ℤ+, p ∈ ℚ.
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EXN.1.AHL.TZ0.9:
It is given that 2 cos A sin B≡sin(A+B)-sin(A-B). (Do not prove this identity.)
Using mathematical induction and the above identity, prove that nΣr=1cos(2r-1)θ=sin 2nθ2 sin θ for n∈ℤ+.
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EXN.2.AHL.TZ0.8:
Prove by contradiction that log2 5 is an irrational number.
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EXN.3.AHL.TZ0.2j:
Use proof by contradiction to prove that a prime number, p, that is not of the form a2+b2 is a Gaussian prime.
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21M.1.AHL.TZ1.12b:
Use mathematical induction to prove that f(n)(x)=(-14)n-1(2n-3)!(n-2)!(1+x)12-n for n∈ℤ, n≥2.
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21M.1.AHL.TZ2.12d:
Using mathematical induction and the result from part (b), prove that nΣr=1arctan(12r2)=arctan(nn+1) for n∈ℤ+.
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21N.1.AHL.TZ0.11a:
Prove by mathematical induction that dndxn(x2ex)=[x2+2nx+n(n-1)]ex for n∈ℤ+.
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21N.1.AHL.TZ0.11b:
Hence or otherwise, determine the Maclaurin series of f(x)=x2ex in ascending powers of x, up to and including the term in x4.
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21N.1.AHL.TZ0.11c:
Hence or otherwise, determine the value of limx→0[(x2ex-x2)3x9].
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22M.3.AHL.TZ1.1f:
A polygonal number, Pr(n), can be represented by the series
nΣm=1(1+(m-1)(r-2)) where r∈ℤ+, r≥3.
Use mathematical induction to prove that Pr(n)=(r-2)n2-(r-4)n2 where n∈ℤ+.
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22M.3.AHL.TZ2.2c:
Given that p2<3q, deduce that α, β and γ cannot all be real.
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22M.3.AHL.TZ2.2f.ii:
Hence state a condition in terms of p and q that would imply x4+px3+qx2+rx+s=0 has at least one complex root.
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22M.1.AHL.TZ1.8:
Consider integers a and b such that a2+b2 is exactly divisible by 4. Prove by contradiction that a and b cannot both be odd.
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22M.1.AHL.TZ2.9:
Prove by contradiction that the equation 2x3+6x+1=0 has no integer roots.
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17M.1.AHL.TZ1.H_8:
Use the method of mathematical induction to prove that 4n+15n−1 is divisible by 9 for n∈Z+.
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17N.1.AHL.TZ0.H_11a:
Determine whether fn is an odd or even function, justifying your answer.
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17N.1.AHL.TZ0.H_11b:
By using mathematical induction, prove that
fn(x)=sin2n+1x2nsin2x, x≠mπ2 where m∈Z.
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17N.1.AHL.TZ0.H_11c:
Hence or otherwise, find an expression for the derivative of fn(x) with respect to x.
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17N.1.AHL.TZ0.H_11d:
Show that, for n>1, the equation of the tangent to the curve y=fn(x) at x=π4 is 4x−2y−π=0.
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17M.1.AHL.TZ2.H_8:
Prove by mathematical induction that (22)+(32)+(42)+…+(n−12)=(n3), where n∈Z,n⩾3.
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19M.2.AHL.TZ1.H_8a:
Solve the inequality x2>2x+1.
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19M.2.AHL.TZ1.H_8b:
Use mathematical induction to prove that 2n+1>n2 for n∈Z, n⩾3.
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18M.1.AHL.TZ1.H_6:
Use the principle of mathematical induction to prove that
1+2(12)+3(12)2+4(12)3+…+n(12)n−1=4−n+22n−1, where n∈Z+.
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18M.2.AHL.TZ2.H_6:
Use mathematical induction to prove that (1−a)n>1−na for {n:n∈Z+,n⩾2} where 0<a<1.
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18N.1.AHL.TZ0.H_6:
Use mathematical induction to prove that n∑r=1r(r!)=(n+1)!−1, for n∈Z+.
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16N.1.AHL.TZ0.H_13a:
Find the value of sinπ4+sin3π4+sin5π4+sin7π4+sin9π4.
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16N.1.AHL.TZ0.H_13c:
Use the principle of mathematical induction to prove that
sinx+sin3x+…+sin(2n−1)x=1−cos2nx2sinx, n∈Z+, x≠kπ where k∈Z.
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16N.1.AHL.TZ0.H_13d:
Hence or otherwise solve the equation sinx+sin3x=cosx in the interval 0<x<π.
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19N.1.AHL.TZ0.H_6:
Consider the function f(x)=xe2x, where x∈R. The nth derivative of f(x) is denoted by f(n)(x).
Prove, by mathematical induction, that f(n)(x)=(2nx+n2n−1)e2x, n∈Z+.