DP Mathematics: Analysis and Approaches Questionbank

SL 1.6—Simple proof
Description
[N/A]Directly related questions
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EXN.1.SL.TZ0.4a:
Show that u1=4.
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EXN.1.SL.TZ0.4b:
Prove that the sum of the first n terms of this arithmetic sequence is a square number.
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EXN.3.AHL.TZ0.2h:
Show that N(αβ)=N(α)N(β).
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21M.1.SL.TZ2.2:
Consider two consecutive positive integers, n and n+1.
Show that the difference of their squares is equal to the sum of the two integers.
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21N.1.SL.TZ0.6b:
Hence or otherwise, solve the equation 2 sin 2θ-3-6sin 2θ-1=0 for 0≤θ≤π, θ≠π4.
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21N.1.SL.TZ0.6a:
Show that 2x-3-6x-1=2x2-5x-3x-1, x∈ℝ, x≠1.
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21N.2.AHL.TZ0.6b:
The equation 2x2-5x+1=0 has two real roots, α and β.
Consider the equation x2+mx+n=0, where m, n∈ℤ and which has roots 1α3 and 1β3.
Without solving 2x2-5x+1=0, determine the values of m and n. -
21N.2.AHL.TZ0.6a:
Prove the identity (p+q)3-3pq(p+q)≡p3+q3.
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21N.3.AHL.TZ0.1b:
Show that (f(t))2+(g(t))2=f(2t).
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21N.3.AHL.TZ0.1c.i:
f(iu).
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21N.3.AHL.TZ0.1c.ii:
g(iu).
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21N.3.AHL.TZ0.1a:
Verify that u=f(t) satisfies the differential equation .
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21N.3.AHL.TZ0.1g:
The hyperbola with equation can be rotated to coincide with the curve defined by .
Find the possible values of .
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21N.3.AHL.TZ0.2b.ii:
By substituting , show that where is a constant.
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21N.3.AHL.TZ0.1e:
Show that .
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21N.3.AHL.TZ0.2a.i:
By solving the differential equation , show that where is a constant.
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21N.3.AHL.TZ0.2a.ii:
Show that .
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21N.3.AHL.TZ0.2b.iii:
Hence find as a function of .
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21N.3.AHL.TZ0.2c.i:
Show that .
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21N.3.AHL.TZ0.1d:
Hence find, and simplify, an expression for .
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21N.3.AHL.TZ0.1f:
Sketch the graph of , stating the coordinates of any axis intercepts and the equation of each asymptote.
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21N.3.AHL.TZ0.2b.iv:
Hence show that , where is a constant.
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21N.3.AHL.TZ0.2c.ii:
Find the two values for that satisfy .
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21N.3.AHL.TZ0.2c.iii:
Let the two values found in part (c)(ii) be and .
Verify that is a solution to the differential equation in (c)(i),where is a constant.
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21N.3.AHL.TZ0.2a.iii:
Solve the differential equation in part (a)(ii) to find as a function of .
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21N.3.AHL.TZ0.2b.i:
By differentiating with respect to , show that .
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22M.3.AHL.TZ1.1a.i:
For triangular numbers, verify that .
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22M.3.AHL.TZ1.1b.iii:
For , sketch a diagram clearly showing your answer to part (b)(ii).
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22M.3.AHL.TZ1.1c:
Show that is the square of an odd number for all .
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22M.3.AHL.TZ1.1b.i:
Show that .
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22M.3.AHL.TZ1.1b.ii:
State, in words, what the identity given in part (b)(i) shows for two consecutive triangular numbers.
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22M.3.AHL.TZ2.2a:
By expanding show that:
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22M.3.AHL.TZ2.2b.i:
Show that .
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22M.3.AHL.TZ2.2b.ii:
Hence show that .
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22M.1.SL.TZ2.3a:
Prove that the sum of these three integers is always divisible by .
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22M.1.SL.TZ2.3b:
Prove that the sum of the squares of these three integers is never divisible by .
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22M.1.SL.TZ1.9a.ii:
By using a suitable substitution for , show that .
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SPM.1.SL.TZ0.3a:
Show that , where .
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SPM.1.SL.TZ0.3b:
Hence, or otherwise, prove that the sum of the squares of any two consecutive odd integers is even.
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EXM.1.SL.TZ0.1a:
Explain why any integer can be written in the form or or or , where .
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EXM.1.SL.TZ0.1b:
Hence prove that the square of any integer can be written in the form or , where .
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18M.1.AHL.TZ2.H_10b.i:
Express in the form where A, B are constants.