DP Mathematics: Analysis and Approaches Questionbank

SL 2.7—Solutions of quadratic equations and inequalities, discriminant and nature of roots
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[N/A]Directly related questions
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20N.1.SL.TZ0.S_5:
Let f(x)=-x2+4x+5 and g(x)=-f(x)+k.
Find the values of k so that g(x)=0 has no real roots.
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EXN.1.AHL.TZ0.12d:
Hence find the exact value of cot23π8.
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EXN.2.SL.TZ0.5:
The quadratic equation (k-1)x2+2x+(2k-3)=0, where k∈ℝ, has real distinct roots.
Find the range of possible values for k.
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21N.1.AHL.TZ0.7b:
Consider the case when p=4. The roots of the equation can be expressed in the form x=a±√136, where a∈ℤ. Find the value of a.
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21N.1.AHL.TZ0.7a:
Find the possible values for p.
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21N.3.AHL.TZ0.2b.ii:
By substituting Y=dydt, show that Y=Be2t where B is a constant.
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21N.3.AHL.TZ0.2a.i:
By solving the differential equation dydt=y, show that y=Aet where A is a constant.
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21N.3.AHL.TZ0.2a.ii:
Show that dxdt-x=-Aet.
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21N.3.AHL.TZ0.2b.iii:
Hence find y as a function of t.
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21N.3.AHL.TZ0.2c.i:
Show that d2ydt2-2dydt-3y=0.
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21N.3.AHL.TZ0.2b.iv:
Hence show that x=-B2e2t+C, where C is a constant.
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21N.3.AHL.TZ0.2c.ii:
Find the two values for λ that satisfy d2ydt2-2dydt-3y=0.
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21N.3.AHL.TZ0.2c.iii:
Let the two values found in part (c)(ii) be λ1 and λ2.
Verify that y=Feλ1t+Geλ2t is a solution to the differential equation in (c)(i),where G is a constant.
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21N.3.AHL.TZ0.2a.iii:
Solve the differential equation in part (a)(ii) to find x as a function of t.
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21N.3.AHL.TZ0.2b.i:
By differentiating dydt=-x+y with respect to t, show that d2ydt2=2dydt.
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22M.3.AHL.TZ1.1a.ii:
The number 351 is a triangular number. Determine which one it is.
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22M.3.AHL.TZ1.2e:
Deduce from part (d)(i) that the complex roots of the equation (z-r)(z2-2az+a2+b2)=0 can be expressed as a±i√g'(a).
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22M.1.SL.TZ2.6b:
The third term in the expansion is the mean of the second term and the fourth term in the expansion.
Find the possible values of x.
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22M.1.SL.TZ2.8a:
Find the coordinates of B.
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22M.1.AHL.TZ2.11c:
Given that (h∘g)(a)=π4, find the value of a.
Give your answer in the form p+q2√r, where p, q, r∈ℤ+.
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22M.2.SL.TZ1.4a:
Show that 2k2-k+0.12=0.
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22M.2.SL.TZ1.4b:
Find the value of k, giving a reason for your answer.
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22M.2.SL.TZ1.4c:
Hence, find E(X).
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22M.2.AHL.TZ1.8a:
Write down an expression for the product of the roots, in terms of k.
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22M.2.AHL.TZ1.8b:
Hence or otherwise, determine the values of k such that the equation has one positive and one negative real root.
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22M.1.SL.TZ1.8b.iii:
The sum of the first n terms of the series is -3 ln x.
Find the value of n.
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22M.1.AHL.TZ1.10b.iii:
The sum of the first n terms of the series is ln(1x3).
Find the value of n.
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22M.1.AHL.TZ1.10b.ii:
Write down d in the form k ln x, where k∈ℚ.
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22M.1.AHL.TZ1.10b.i:
Show that p=23.
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22M.1.AHL.TZ2.8:
A continuous random variable X has the probability density function
f(x)={2(b-a)(c-a)(x-a),a≤x≤c2(b-a)(b-c)(b-x),c<x≤b0,otherwise.
The following diagram shows the graph of y=f(x) for a≤x≤b.
Given that c≥a+b2, find an expression for the median of X in terms of a, b and c.
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SPM.2.AHL.TZ0.12b:
Verify that x=tanθ and x=−cotθ satisfy the equation x2+(2cot2θ)x−1=0.
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SPM.2.AHL.TZ0.12a:
Show that cot2θ=1−tan2θ2tanθ.
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SPM.2.AHL.TZ0.12c:
Hence, or otherwise, show that the exact value of tanπ12=2−√3.
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SPM.2.AHL.TZ0.12d:
Using the results from parts (b) and (c) find the exact value of tanπ24−cotπ24.
Give your answer in the form a+b√3 where a, b∈Z.
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17M.1.SL.TZ1.S_9c:
The line y=kx−5 is a tangent to the curve of f. Find the values of k.
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16N.1.AHL.TZ0.H_5:
The quadratic equation x2−2kx+(k−1)=0 has roots α and β such that α2+β2=4. Without solving the equation, find the possible values of the real number k.
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17N.1.SL.TZ0.S_8d:
Find the area of the region enclosed by the graph of f and the line L.
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18M.2.SL.TZ1.S_1c:
Solve f '(x) = f "(x).
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17M.1.SL.TZ1.S_10a:
Show that cosθ=34.
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17M.1.SL.TZ1.S_10b:
Given that tanθ>0, find tanθ.
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17M.1.SL.TZ1.S_10c:
Let y=1cosx, for 0<x<π2. The graph of ybetween x=θ and x=π4 is rotated 360° about the x-axis. Find the volume of the solid formed.
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16N.2.SL.TZ0.S_4a:
Find the value of p and of q.
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16N.2.SL.TZ0.S_4b:
Hence, find the area of the region enclosed by the graphs of f and g.
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18N.1.SL.TZ0.S_5:
Consider the vectors a = (32p) and b = (p+18).
Find the possible values of p for which a and b are parallel.
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19N.1.SL.TZ0.S_5a:
Show that the discriminant of f′(x) is k2−16.
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19N.1.SL.TZ0.S_5b:
Given that f is an increasing function, find all possible values of k.
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17M.2.AHL.TZ2.H_4a:
Find the set of values of k that satisfy the inequality k2−k−12<0.
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17M.2.AHL.TZ2.H_4b:
The triangle ABC is shown in the following diagram. Given that cosB<14, find the range of possible values for AB.