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Beyond Advanced · Complex Numbers and Quadratic Equations

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1
Let NN be the number of ordered pairs (a,b)(a,b) of integers with 1≤a≤361\le a\le 36 and 1≤b≤301\le b\le 30 such that (3+i)a (1−i3)b(\sqrt3+i)^{a}\,(1-i\sqrt3)^{b} is a positive real number. Find NN.
Not attempted
Your answer: —
Correct answer: 90
2
The number of complex numbers zz satisfying z5=i zˉ 3z^{5}=i\,\bar z^{\,3} is
Not attempted
A
7
B
9
C
8
D
10
3
Let z,z1,z2z,z_1,z_2 be non-zero complex numbers. Which of the following statements are correct?
Not attempted
A
If z1z2z_1z_2 and z1+z2z_1+z_2 are both real, then z1z_1 and z2z_2 are both real.
B
If z+1zz+\dfrac1z is real, then zz is real or ∣z∣=1|z|=1.
C
If ∣z1∣=∣z2∣=1|z_1|=|z_2|=1 and z1z2≠−1z_1z_2\neq-1, then z1+z21+z1z2\dfrac{z_1+z_2}{1+z_1z_2} is real.
D
If Re(z1zˉ2)=0\text{Re}(z_1\bar z_2)=0, then ∣z1+z2∣2=∣z1∣2+∣z2∣2|z_1+z_2|^2=|z_1|^2+|z_2|^2.
4
Let S=∑k=12026(1+i)kS=\displaystyle\sum_{k=1}^{2026}(1+i)^{k}. Then ∣S∣2|S|^{2} equals
Not attempted
A
22027−22^{2027}-2
B
22026+22^{2026}+2
C
22026−22^{2026}-2
D
22027+22^{2027}+2
5
The value of ∑j=025(−1)⌊j/2⌋(25j)=(250)+(251)−(252)−(253)+(254)+⋯\displaystyle\sum_{j=0}^{25}(-1)^{\lfloor j/2\rfloor}\binom{25}{j}=\binom{25}{0}+\binom{25}{1}-\binom{25}{2}-\binom{25}{3}+\binom{25}{4}+\cdots is
Not attempted
A
8192
B
4096
C
16384
D
0
6
Consider the equation (1+ix1−ix)5=−1\left(\dfrac{1+ix}{1-ix}\right)^{5}=-1 for real xx. Which of the following are correct?
Not attempted
A
It has exactly 44 real solutions.
B
The sum of the squares of all its real solutions is 44.
C
The product of all its real solutions is 15\dfrac15.
D
x=tan⁡π2x=\tan\dfrac{\pi}{2} is not a solution only because it is undefined, so the equation has 55 real solutions.
7
The value of ∑k=12029i k(k+1)/2\displaystyle\sum_{k=1}^{2029}i^{\,k(k+1)/2} is
Not attempted
A
−2+i-2+i
B
−1−i-1-i
C
−2−i-2-i
D
−1+i-1+i
8
The principal argument of z=1+cos⁡8π7+isin⁡8π7z=1+\cos\dfrac{8\pi}{7}+i\sin\dfrac{8\pi}{7} is (the figure shows zz as the sum of the vectors 11 and cos⁡8π7+isin⁡8π7\cos\frac{8\pi}7+i\sin\frac{8\pi}7)
Not attempted
ReImO1z
A
4π7\dfrac{4\pi}{7}
B
−4π7-\dfrac{4\pi}{7}
C
−3π7-\dfrac{3\pi}{7}
D
3π7\dfrac{3\pi}{7}
9
Let z=(−1+i)2027 (3−i)−1011z=(-1+i)^{2027}\,(\sqrt3-i)^{-1011}. If θ∈(−180,180]\theta\in(-180,180] is the principal argument of zz in degrees, find θ\theta.
Not attempted
Your answer: —
Correct answer: 135
10
Here Arg\text{Arg} denotes the principal argument in (−π,π](-\pi,\pi]. Which of the following statements are true for the indicated non-zero complex numbers?
Not attempted
A
Arg(zˉ)=−Arg(z)\text{Arg}(\bar z)=-\text{Arg}(z) for every non-zero zz.
B
Arg(z1z2)=Arg z1+Arg z2\text{Arg}(z_1z_2)=\text{Arg}\,z_1+\text{Arg}\,z_2 whenever Re z1>0\text{Re}\,z_1>0 and Re z2>0\text{Re}\,z_2>0.
C
Arg(−z)=Arg(z)−π\text{Arg}(-z)=\text{Arg}(z)-\pi whenever Im z>0\text{Im}\,z>0.
D
Arg ⁣(1z)=−Arg(z)\text{Arg}\!\left(\dfrac1z\right)=-\text{Arg}(z) for every non-zero zz.
11
The value of ∑k=12026cos⁡ ⁣(kπ3+π4)\displaystyle\sum_{k=1}^{2026}\cos\!\left(\dfrac{k\pi}{3}+\dfrac{\pi}{4}\right) is
Not attempted
A
−2 (3+3)4-\dfrac{\sqrt2\,(3+\sqrt3)}{4}
B
2 (3−3)4\dfrac{\sqrt2\,(\sqrt3-3)}{4}
C
−2 (1+3)4-\dfrac{\sqrt2\,(1+\sqrt3)}{4}
D
2 (3+3)4\dfrac{\sqrt2\,(3+\sqrt3)}{4}
12
Let α=11π12\alpha=\dfrac{11\pi}{12} and z=1+itan⁡α1−itan⁡αz=\dfrac{1+i\tan\alpha}{1-i\tan\alpha}. The principal argument of zz is
Not attempted
A
11π6\dfrac{11\pi}{6}
B
π6\dfrac{\pi}{6}
C
−5π6-\dfrac{5\pi}{6}
D
−π6-\dfrac{\pi}{6}
13
A sequence of complex numbers is defined by z0=0z_0=0 and zn+1=(1+i) zn+1z_{n+1}=(1+i)\,z_n+1. Then ∣z14∣2|z_{14}|^{2} equals
Not attempted
A
16384
B
16385
C
32769
D
8193
14
List-IList-IIP)−1−i31)3π4Q)(1−i)52)−3π4R)1(1+i)33)−2π3S)sin⁡8π5+icos⁡8π54)9π105)π2\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} -1-i\sqrt3 & \text{1)} \dfrac{3\pi}{4} \\ \text{Q)} (1-i)^{5} & \text{2)} -\dfrac{3\pi}{4} \\ \text{R)} \dfrac{1}{(1+i)^{3}} & \text{3)} -\dfrac{2\pi}{3} \\ \text{S)} \sin\dfrac{8\pi}{5}+i\cos\dfrac{8\pi}{5} & \text{4)} \dfrac{9\pi}{10} \\ & \text{5)} \dfrac{\pi}{2} \end{array}
Not attempted
A
P-3, Q-1, R-2, S-4
B
P-3, Q-2, R-1, S-4
C
P-3, Q-1, R-2, S-5
D
P-2, Q-1, R-3, S-4
15
Complex numbers z≠0z\neq0 satisfy ∣z3−1z3∣=4\left|z^{3}-\dfrac{1}{z^{3}}\right|=4. Find the maximum possible value of ∣z∣|z|, correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 1.62
16
The quadrilateral ABCDABCD is labelled anticlockwise with A=0, B=6, C=7+4i, D=1+5iA=0,\ B=6,\ C=7+4i,\ D=1+5i. On each side a square is erected outwards (away from the interior), as in the figure. Let P,Q,R,SP,Q,R,S be the centres of the squares on AB,BC,CD,DAAB,BC,CD,DA, with complex numbers p,q,r,sp,q,r,s. The value of s−qr−p\dfrac{s-q}{r-p} is
Not attempted
ABCDPQRS
A
−i-i
B
11
C
ii
D
1+i2\dfrac{1+i}{\sqrt2}
17
Let z1,z2,z3z_1,z_2,z_3 be complex numbers with ∣z1∣=∣z2∣=∣z3∣=1|z_1|=|z_2|=|z_3|=1. Which of the following statements are correct?
Not attempted
A
If z1+z2+z3=0z_1+z_2+z_3=0, then z1,z2,z3z_1,z_2,z_3 are the vertices of an equilateral triangle.
B
If z1+z2+z3=1z_1+z_2+z_3=1, then at least one of z1,z2,z3z_1,z_2,z_3 equals 11.
C
If z1+z2+z3=1z_1+z_2+z_3=1, then z1z2z3=1z_1z_2z_3=1.
D
If z1+z2+z3=0z_1+z_2+z_3=0, then z12+z22+z32=0z_1^2+z_2^2+z_3^2=0.
18
For complex numbers zz with ∣z∣=1|z|=1, the maximum value of ∣z3−3z+2∣|z^{3}-3z+2| is MM. Find M2M^{2}.
Not attempted
Your answer: —
Correct answer: 27
19
z1,z2,z3z_1,z_2,z_3 are the vertices of an equilateral triangle in the Argand plane (in some order). Its centroid is G=1−iG=1-i and one vertex is z1=3+2iz_1=3+2i, as shown. Then z12+z22+z32z_1^2+z_2^2+z_3^2 equals
Not attempted
ReImOz₁G
A
6i6i
B
−6-6
C
15+36i15+36i
D
−6i-6i
20
The plane is first rotated anticlockwise through 120∘120^\circ about the point C1=1+2iC_1=1+2i, and the resulting plane is then rotated anticlockwise through 240∘240^\circ about C2=4−2iC_2=4-2i. The combined map is a translation of every point. Find the distance through which each point is translated, correct to two decimal places.
Not attempted
ReImOC₁C₂
Your answer: —
Correct answer: 8.66
21
Complex numbers z1,z2,z3z_1,z_2,z_3 satisfy ∣z1∣=1, ∣z2∣=2, ∣z3∣=3|z_1|=1,\ |z_2|=2,\ |z_3|=3 (one possible position is sketched). Let F=∣z1−z2∣2+∣z2−z3∣2+∣z3−z1∣2F=|z_1-z_2|^2+|z_2-z_3|^2+|z_3-z_1|^2, and let MM and mm be its maximum and minimum possible values. Find M⋅mM\cdot m.
Not attempted
ReImOz₁z₂z₃
Your answer: —
Correct answer: 252
22
Each of 3+4i\sqrt{3+4i} and 3−4i\sqrt{3-4i} may denote either of the two square roots of the number under the sign. The set of all possible values of (3+4i+3−4i)2\left(\sqrt{3+4i}+\sqrt{3-4i}\right)^{2} is
Not attempted
A
{16}\{16\}
B
{16, −4}\{16,\,-4\}
C
{−4}\{-4\}
D
{4, −4}\{4,\,-4\}
23
For a real number kk, the equation z2−(k+2i) z+(k+3+i)=0z^{2}-(k+2i)\,z+(k+3+i)=0 has a real root. For this kk, find the product of the moduli of the two roots, correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 3.64
24
A unique complex number zz satisfies z ∣z∣+2z+i=0z\,|z|+2z+i=0. The imaginary part of zz is
Not attempted
A
−1−2-1-\sqrt2
B
−12-\dfrac12
C
−13-\dfrac13
D
1−21-\sqrt2
25
Consider z2−(1+5i)z+(−6+2i)=0z^{2}-(1+5i)z+(-6+2i)=0 with roots z1,z2z_1,z_2. Which of the following are correct?
Not attempted
A
z1z_1 and z2z_2 are complex conjugates of each other.
B
The discriminant equals 2i2i and its square roots are ±(1+i)\pm(1+i).
C
z1−z2z_1-z_2 is a non-zero real number.
D
z1−z2=±(1+i)z_1-z_2=\pm(1+i).
26
The equation z2=2i zˉz^{2}=2i\,\bar z is satisfied by nn complex numbers zz. Which of the following are correct?
Not attempted
A
n=4n=4.
B
The non-zero solutions are the vertices of an equilateral triangle of side 232\sqrt3.
C
The sum of all solutions is 00.
D
Every solution is either real or purely imaginary.
27
The four roots of z4−6z2+25=0z^{4}-6z^{2}+25=0 are plotted in the Argand plane and joined to form a quadrilateral. Find its area.
Not attempted
Your answer: —
Correct answer: 8
28
List-IList-IIP)z2=zˉ1)1Q)z2=∣z∣2)2R)z3=zˉ3)3S)z2=5−12i4)45)5\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} z^{2}=\bar z & \text{1)} 1 \\ \text{Q)} z^{2}=|z| & \text{2)} 2 \\ \text{R)} z^{3}=\bar z & \text{3)} 3 \\ \text{S)} z^{2}=5-12i & \text{4)} 4 \\ & \text{5)} 5 \end{array}
Not attempted
A
P-4, Q-3, R-5, S-1
B
P-4, Q-3, R-5, S-2
C
P-3, Q-4, R-5, S-2
D
P-4, Q-2, R-5, S-3
29
Let z1,z2,z3,z4z_1,z_2,z_3,z_4 be the four roots of z4=−4z^{4}=-4. Find the value of ∑k=1412−zk\displaystyle\sum_{k=1}^{4}\frac{1}{2-z_k}.
Not attempted
Your answer: —
Correct answer: 1.6
30
Let θ1=20∘, θ2=140∘, θ3=260∘\theta_1=20^\circ,\ \theta_2=140^\circ,\ \theta_3=260^\circ. Which of the following are correct?
Not attempted
A
∑k=13cos⁡2θk=32\displaystyle\sum_{k=1}^{3}\cos^{2}\theta_k=\frac32
B
∑k=13sec⁡2θk=36\displaystyle\sum_{k=1}^{3}\sec^{2}\theta_k=36
C
∑k=13cos⁡4θk=32\displaystyle\sum_{k=1}^{3}\cos^{4}\theta_k=\frac32
D
∑k=13tan⁡2θk=36\displaystyle\sum_{k=1}^{3}\tan^{2}\theta_k=36
31
Let θ\theta be real. Which of the following statements are correct?
Not attempted
A
(sin⁡θ+icos⁡θ)n=sin⁡nθ+icos⁡nθ(\sin\theta+i\cos\theta)^{n}=\sin n\theta+i\cos n\theta for every positive integer nn.
B
(cos⁡θ+isin⁡θ)2/3(\cos\theta+i\sin\theta)^{2/3} takes exactly three distinct values, one of which is cos⁡2θ3+isin⁡2θ3\cos\frac{2\theta}{3}+i\sin\frac{2\theta}{3}.
C
(cos⁡θ+isin⁡θ)n=cos⁡nθ+isin⁡nθ(\cos\theta+i\sin\theta)^{n}=\cos n\theta+i\sin n\theta holds only for positive integers nn.
D
For every real θ\theta and positive integer nn, ∣1+cos⁡θ+isin⁡θ∣n=2ncos⁡nθ2\left|1+\cos\theta+i\sin\theta\right|^{n}=2^{n}\cos^{n}\frac{\theta}{2}.
32
The four dots in the figure are the roots z1,…,z4z_1,\dots,z_4 of z4=−16z^{4}=-16. The point PP moves on the circle ∣z∣=3|z|=3. Let Π(P)=∣P−z1∣ ∣P−z2∣ ∣P−z3∣ ∣P−z4∣\Pi(P)=|P-z_1|\,|P-z_2|\,|P-z_3|\,|P-z_4|. Find max⁡Πmin⁡Π\dfrac{\max\Pi}{\min\Pi}, correct to two decimal places.
Not attempted
ReImOP
Your answer: —
Correct answer: 1.49
33
For every real α\alpha, the sum ∑k=05sin⁡6 ⁣(α+kπ6)\displaystyle\sum_{k=0}^{5}\sin^{6}\!\left(\alpha+\frac{k\pi}{6}\right) equals the same constant. That constant is
Not attempted
A
54\dfrac{5}{4}
B
32\dfrac{3}{2}
C
158\dfrac{15}{8}
D
94\dfrac{9}{4}
34
Find the value of ∑k=011(11k)cos⁡2kπ3\displaystyle\sum_{k=0}^{11}\binom{11}{k}\cos\frac{2k\pi}{3}.
Not attempted
Your answer: —
Correct answer: 0.5
35
List-IList-IIP)(z−1)4=161)4Q)z6=642)33R)(z+2)3=−83)63S)z4=−44)85)12\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} (z-1)^{4}=16 & \text{1)} 4 \\ \text{Q)} z^{6}=64 & \text{2)} 3\sqrt3 \\ \text{R)} (z+2)^{3}=-8 & \text{3)} 6\sqrt3 \\ \text{S)} z^{4}=-4 & \text{4)} 8 \\ & \text{5)} 12 \end{array}
Not attempted
A
P-4, Q-3, R-2, S-1
B
P-4, Q-2, R-3, S-1
C
P-1, Q-3, R-2, S-4
D
P-4, Q-3, R-1, S-2
36
Find the number of ordered triples (a,b,c)(a,b,c) of integers with 1≤a,b,c≤101\le a,b,c\le10 for which ∣a+bω+cω2∣=1\left|a+b\omega+c\omega^{2}\right|=1, where ω\omega is a non-real cube root of unity.
Not attempted
Your answer: —
Correct answer: 54
37
For how many positive integers n≤100n\le100 is the polynomial xn+1+(x+1)nx^{n}+1+(x+1)^{n} divisible by x2+x+1x^{2}+x+1?
Not attempted
A
33
B
34
C
50
D
67
38
Let a=2+i, b=2−ia=2+i,\ b=2-i and x=a+b, y=aω+bω2, z=aω2+bωx=a+b,\ y=a\omega+b\omega^{2},\ z=a\omega^{2}+b\omega, where ω\omega is a non-real cube root of unity. Then x3+y3+z3x^{3}+y^{3}+z^{3} equals
Not attempted
A
4
B
36
C
0
D
12
39
Let ω\omega be a non-real cube root of unity. Which of the following are correct?
Not attempted
A
(1+ω−ω2)3=−8(1+\omega-\omega^{2})^{3}=-8
B
(1−ω+ω2)6=64(1-\omega+\omega^{2})^{6}=64
C
∑k=1100ωk2=33+67ω\displaystyle\sum_{k=1}^{100}\omega^{k^{2}}=33+67\omega
D
For real a,ba,b: (a+bω)(a+bω2)=a2+ab+b2(a+b\omega)(a+b\omega^{2})=a^{2}+ab+b^{2}
40
The points z1=1+iz_1=1+i, z2=3z_2=3 and z3z_3 form an equilateral triangle, and z1+ωz2+ω2z3=0z_1+\omega z_2+\omega^{2}z_3=0 where ω=−1+i32\omega=\frac{-1+i\sqrt3}{2}. Then z3z_3 equals
Not attempted
ReImOz₁z₂
A
32+2+i(12+3)\frac{\sqrt{3}}{2} + 2+i\left(\frac{1}{2} + \sqrt{3}\right)
B
2−32+i(12−3)2 - \frac{\sqrt{3}}{2}+i\left(\frac{1}{2} - \sqrt{3}\right)
C
−2−32+i(−3−12)-2 - \frac{\sqrt{3}}{2}+i\left(- \sqrt{3} - \frac{1}{2}\right)
D
32+2+i(12−23)\frac{\sqrt{3}}{2} + 2+i\left(\frac{1}{2} - 2 \sqrt{3}\right)
41
The value of (180)+(183)+(186)+⋯+(1818)\displaystyle\binom{18}{0}+\binom{18}{3}+\binom{18}{6}+\cdots+\binom{18}{18} is
Not attempted
A
87381
B
87383
C
87382
D
174762
42
Find ∣∑k=160k ωk∣\left|\displaystyle\sum_{k=1}^{60}k\,\omega^{k}\right|, correct to two decimal places, where ω\omega is a non-real cube root of unity. (The figure shows the first six terms kωkk\omega^k as vectors placed head to tail, labelled by kk.)
Not attempted
ReImO123456
Your answer: —
Correct answer: 34.64
43
The principal argument of z−1z+1\dfrac{z-1}{z+1} equals π3\dfrac{\pi}{3}. Which diagram shows the locus of zz? (In each diagram the dots mark −1-1 and 11; the end points themselves are not on the locus.)
Not attempted
A
Diagram II
ReImO−11
B
Diagram III
ReImO−11
C
Diagram I
ReImO−11
D
Diagram IV
ReImO−11
44
Complex numbers zz satisfy ∣z−1∣=2 ∣z−4∣|z-1|=2\,|z-4|. Find the maximum value of ∣z−3i∣|z-3i|, correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 7.83
45
Which of the following statements about loci in the Argand plane are correct?
Not attempted
A
The locus arg⁡z−1z+1=π2\arg\dfrac{z-1}{z+1}=\dfrac\pi2 is the whole circle ∣z∣=1|z|=1 with ±1\pm1 removed.
B
The locus ∣z−1∣+∣z+1∣=2|z-1|+|z+1|=2 is the line segment joining −1-1 and 11.
C
The locus ∣z−1∣−∣z+1∣=2|z-1|-|z+1|=2 is a ray starting at −1-1 and extending to the left, with −1-1 included.
D
The locus Rez−2z+2i=0\text{Re}\dfrac{z-2}{z+2i}=0 (for z≠−2iz\neq-2i) is the entire circle having 22 and −2i-2i as ends of a diameter.
46
Let w=4+2iw=4+2i. The complex number z1z_1 lies on the circle ∣z−w∣=1|z-w|=1, and z2z_2 satisfies ∣z2∣=∣z2−w∣|z_2|=|z_2-w| (so z2z_2 lies on the dashed line in the figure). The minimum possible value of ∣z1−z2∣|z_1-z_2| is
Not attempted
ReImO0w
A
5−1\sqrt5-1
B
5+1\sqrt5+1
C
5\sqrt5
D
52−1\dfrac{\sqrt5}{2}-1
47
Let RR be the shaded region in the figure: the set of points zz with ∣z−2∣≤2|z-2|\le2 and ∣z∣≥2|z|\ge2. Find the area of RR, correct to two decimal places.
Not attempted
ReImOAO
Your answer: —
Correct answer: 7.65
48
List-IList-IIP)arg⁡z−1z−i=01)the perpendicular bisector of the segment joining 1 and iQ)arg⁡z−1z−i=π2)the line through 1 and i with the closed segment between them removedR)∣z−1∣=∣z−i∣3)the open segment joining 1 and iS)Rez−1z−i=04)the circle with diameter joining 1 and i, with the point i removed5)the whole circle with diameter joining 1 and i\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} \arg\dfrac{z-1}{z-i}=0 & \text{1)} \text{the perpendicular bisector of the segment joining }1\text{ and }i \\ \text{Q)} \arg\dfrac{z-1}{z-i}=\pi & \text{2)} \text{the line through }1\text{ and }i\text{ with the closed segment between them removed} \\ \text{R)} |z-1|=|z-i| & \text{3)} \text{the open segment joining }1\text{ and }i \\ \text{S)} \text{Re}\dfrac{z-1}{z-i}=0 & \text{4)} \text{the circle with diameter joining }1\text{ and }i\text{, with the point }i\text{ removed} \\ & \text{5)} \text{the whole circle with diameter joining }1\text{ and }i \end{array}
Not attempted
A
P-3, Q-2, R-1, S-4
B
P-2, Q-3, R-1, S-5
C
P-2, Q-3, R-5, S-1
D
P-2, Q-3, R-1, S-4
49
A Gaussian integer is a complex number x+iyx+iy with x,yx,y integers. The number of Gaussian integers zz satisfying ∣z∣≥2 ∣z−3∣|z|\ge2\,|z-3| is
Not attempted
A
9
B
13
C
12
D
15
50
Let N=⌊(2+3)2026⌋N=\left\lfloor(2+\sqrt3)^{2026}\right\rfloor. Find the last (units) digit of NN.
Not attempted
Your answer: —
Correct answer: 3
51
Let α,β\alpha,\beta be the roots of x2−5x+3=0x^{2}-5x+3=0. The quadratic equation whose roots are α+1β\alpha+\dfrac1\beta and β+1α\beta+\dfrac1\alpha is
Not attempted
A
3x2−20x+16=03x^{2}-20x+16=0
B
3x2−20x+48=03x^{2}-20x+48=0
C
3x2−15x+16=03x^{2}-15x+16=0
D
x2−5x+16=0x^{2}-5x+16=0
52
Let a,b,ca,b,c be integers with a≠0a\neq0 and consider ax2+bx+c=0ax^{2}+bx+c=0. Which of the following statements are correct?
Not attempted
A
If a,b,ca,b,c are all odd, the equation has no rational root.
B
If a,b,ca,b,c are all odd, the equation must have non-real roots.
C
If a=n, b=n+1, c=n+2a=n,\ b=n+1,\ c=n+2 for a non-zero integer nn, the roots are non-real for every such nn.
D
If a+b+c=0a+b+c=0, both roots are rational.
53
If one root of x2+px+q=0x^{2}+px+q=0 is the cube of the other root, then pp and qq satisfy
Not attempted
A
p4=q[4p2+(q+1)2]p^{4}=q\left[4p^{2}+(q+1)^{2}\right]
B
p4=q[2p2+(q−1)2]p^{4}=q\left[2p^{2}+(q-1)^{2}\right]
C
p4=q[4p2+(q−1)2]p^{4}=q\left[4p^{2}+(q-1)^{2}\right]
D
p4=q[4p2−(q−1)2]p^{4}=q\left[4p^{2}-(q-1)^{2}\right]
54
For real kk, let α,β\alpha,\beta be the (real) roots of x2−(k−2)x+(k2+3k+5)=0x^{2}-(k-2)x+(k^{2}+3k+5)=0. As kk varies over all values for which the roots are real, α2+β2\alpha^{2}+\beta^{2} takes values between a minimum mm and a maximum MM. Find M⋅mM\cdot m.
Not attempted
Your answer: —
Correct answer: 100
55
Find the number of ordered pairs (p,q)(p,q) of positive integers such that the equation x2−px+q=0x^{2}-px+q=0 has real roots α,β\alpha,\beta with α2+β2=2026\alpha^{2}+\beta^{2}=2026.
Not attempted
Your answer: —
Correct answer: 9
56
The number of integers aa for which the equation x2−ax+2a=0x^{2}-ax+2a=0 has both roots integers (equal roots allowed) is
Not attempted
A
2
B
3
C
6
D
4
57
The values of the real number aa for which both roots of x2−2ax+a+2=0x^{2}-2ax+a+2=0 are real, distinct and lie strictly between 00 and 33 form an interval. Find its length. (The figure shows y=x2−2ax+a+2y=x^2-2ax+a+2 for the sample value a=2.5a=2.5, between the lines x=0x=0 and x=3x=3.)
Not attempted
xyO3
Your answer: —
Correct answer: 0.2
58
The graph of y=ax2+bx+cy=ax^{2}+bx+c is shown. It cuts the xx-axis at x=−1x=-1 and x=4x=4 and the yy-axis at y=1y=1. Which of the following are correct?
Not attempted
xyO−141
A
4a−2b+c<04a-2b+c<0
B
a+b+c>0a+b+c>0
C
abc<0abc<0
D
b2−4ac>0b^{2}-4ac>0 and the xx-coordinate of the vertex is 32\dfrac32
59
Find the number of integers aa for which x2−2(a−1)x+(a+7)>0x^{2}-2(a-1)x+(a+7)>0 holds for every real x>0x>0 (only the shaded half-plane x>0x>0 matters; the sketch shows the sample value a=−2a=-2).
Not attempted
xyO
Your answer: —
Correct answer: 12
60
For real aa, the equation x2−2ax+a2−1=0x^{2}-2ax+a^{2}-1=0 has exactly one root in the open interval (0,4)(0,4) (the other root is outside it, possibly equal to an end point). The set of all such aa is (the shaded strip is 0<x<40<x<4; the curves are y=x2−2ax+a2−1y=x^2-2ax+a^2-1 for three sample values of aa)
Not attempted
xyO
A
(−1,1)∪(3,5)(-1,1)\cup(3,5)
B
(−1,1]∪[3,5)(-1,1]\cup[3,5)
C
(−1,5)(-1,5)
D
[−1,1]∪[3,5][-1,1]\cup[3,5]
61
The real numbers a<ba<b and cc are such that ax2+bx+c≥0ax^{2}+bx+c\ge0 for every real xx. The minimum possible value of 4a+2b+cb−a\dfrac{4a+2b+c}{b-a} is
Not attempted
A
4
B
6
C
3
D
5
62
Which diagram shows the graph of y=x2−x−2x2−x−6y=\dfrac{x^{2}-x-2}{x^{2}-x-6} (dashed vertical lines mark vertical asymptotes)?
Not attempted
A
Graph II
xyO−4−3−2−112345
B
Graph I
xyO−4−3−2−112345
C
Graph III
xyO−4−3−2−112345
D
Graph IV
xyO−4−3−2−112345
63
List-IList-IIP)x2−4x+3(x−2)2<01)[1,3]Q)x2−4x+3(x−2)2≤02)(1,3)∖{2}R)(x2−4x+3)(x−2)2≤03)[1,3]∖{2}S)x2−4x+3x−2>04)(1,2)∪(3,∞)5)(−∞,1)∪(2,3)\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} \dfrac{x^{2}-4x+3}{(x-2)^{2}}<0 & \text{1)} [1,3] \\ \text{Q)} \dfrac{x^{2}-4x+3}{(x-2)^{2}}\le0 & \text{2)} (1,3)\setminus\{2\} \\ \text{R)} (x^{2}-4x+3)(x-2)^{2}\le0 & \text{3)} [1,3]\setminus\{2\} \\ \text{S)} \dfrac{x^{2}-4x+3}{x-2}>0 & \text{4)} (1,2)\cup(3,\infty) \\ & \text{5)} (-\infty,1)\cup(2,3) \end{array}
Not attempted
A
P-2, Q-3, R-3, S-4
B
P-3, Q-2, R-1, S-4
C
P-2, Q-3, R-1, S-4
D
P-2, Q-1, R-3, S-5
64
The number of real values of kk for which the equations x2+kx+6=0x^{2}+kx+6=0 and x2+6x+k=0x^{2}+6x+k=0 have at least one common root is
Not attempted
A
2
B
1
C
3
D
4
65
Consider x4−4x3+5x2−4x+1=0x^{4}-4x^{3}+5x^{2}-4x+1=0. Which of the following are correct?
Not attempted
A
All four roots are real.
B
The sum of the squares of all four roots (real and non-real) is 66.
C
The sum of the squares of the real roots is 66.
D
The non-real roots have modulus 3\sqrt3.
66
Which diagram shows the graph of y=x2−3x+2x2+x−2y=\dfrac{x^{2}-3x+2}{x^{2}+x-2}? (A small open circle marks a point that is missing from the graph; dashed lines are asymptotes.)
Not attempted
A
Graph II
xyO−5−4−3−2−112345
B
Graph I
xyO−5−4−3−2−112345
C
Graph III
xyO−5−4−3−2−112345
D
Graph IV
xyO−5−4−3−2−112345
67
Which of the following statements are correct?
Not attempted
A
x2−5∣x∣+6=0x^{2}-5|x|+6=0 has exactly four real roots.
B
x+5=x−1\sqrt{x+5}=x-1 has exactly two real roots.
C
22x−3⋅2x+2=02^{2x}-3\cdot2^{x}+2=0 has two real roots whose sum is 11.
D
(x+1)(x+2)(x+3)(x+4)=24(x+1)(x+2)(x+3)(x+4)=24 has exactly two real roots.
68
Let y=x2+x+1x2+2x+3y=\dfrac{x^{2}+x+1}{x^{2}+2x+3} for real xx. Which of the following are correct?
Not attempted
A
The range of yy is [3−34, 3+34]\left[\dfrac{3-\sqrt3}{4},\ \dfrac{3+\sqrt3}{4}\right].
B
The value y=1y=1 is attained, and only at x=−2x=-2.
C
The sum of the greatest and least values of yy is 32\dfrac32.
D
yy is not always greater than or equal to 13\dfrac13, and 11 is the only integer in its range.
69
The figure shows y=∣x2−2∣x∣−3∣y=\left|x^{2}-2|x|-3\right|. The equation ∣x2−2∣x∣−3∣=k\left|x^{2}-2|x|-3\right|=k, kk real, has exactly six distinct real solutions for
Not attempted
xyO12345−3−113
A
3≤k≤43\le k\le4
B
3<k≤43<k\le4
C
0<k<40<k<4
D
3<k<43<k<4
70
List-IList-IIP)x2−5∣x∣+6=01)1Q)4x−3⋅2x+2=02)2R)(x2−2x)2−2(x2−2x)−3=03)3S)x4−4x3+6x2−4x+1=04)45)0\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} x^{2}-5|x|+6=0 & \text{1)} 1 \\ \text{Q)} 4^{x}-3\cdot2^{x}+2=0 & \text{2)} 2 \\ \text{R)} (x^{2}-2x)^{2}-2(x^{2}-2x)-3=0 & \text{3)} 3 \\ \text{S)} x^{4}-4x^{3}+6x^{2}-4x+1=0 & \text{4)} 4 \\ & \text{5)} 0 \end{array}
Not attempted
A
P-4, Q-2, R-3, S-1
B
P-4, Q-2, R-1, S-3
C
P-2, Q-4, R-3, S-1
D
P-4, Q-5, R-3, S-1
Find the substitution for each equation and count the distinct real values of xx it produces.
List-II gives the number of distinct real roots: 1) 11 2) 22 3) 33 4) 44 5) 00. P: t=∣x∣≥0t=|x|\ge0, t2−5t+6=0t^2-5t+6=0, t=2,3t=2,3: x=±2,±3x=\pm2,\pm3. Four roots (entry 4). Q: t=2x>0t=2^x>0: t2−3t+2=0t^2-3t+2=0, t=1,2t=1,2, so x=0,1x=0,1. Two roots (entry 2). R: y=x2−2xy=x^2-2x: y2−2y−3=0y^2-2y-3=0, y=3y=3 or y=−1y=-1. x2−2x=3x^2-2x=3 gives x=3,−1x=3,-1; x2−2x=−1x^2-2x=-1 gives the double root x=1x=1. Three distinct roots (entry 3). S: (x−1)4=0(x-1)^4=0, a single distinct real root x=1x=1 (entry 1).