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Beyond Advanced · Three Dimensional Geometry

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1
A directed line has unit direction vector uu, perpendicular to (1,1,1)(1,1,1). The absolute projections of (1,−1,0)(1,-1,0) and (1,1,−2)(1,1,-2) on it are in the ratio 1:21:2. Let TT be the set of distinct squared projections of (2,1,0)(2,1,0) on all such lines. The product of the elements of TT is
Not attempted
A
81/19681/196
B
27/19627/196
C
81/4981/49
D
9/149/14
2
Let A=(−2,5,7)A=(-2,5,7) and B=(2,−1,3)B=(2,-1,3). Points P,QP,Q divide ABAB internally and externally, respectively, in the same ratio t:1t:1, where t>1t>1. If OP⊥OQOP\perp OQ, then PQ2PQ^2 equals
Not attempted
A
4641/2564641/256
B
4641/644641/64
C
663/16663/16
D
4641/494641/49
3
The vertices A=(1,2,0)A=(1,2,0), B=(3,0,2)B=(3,0,2), C=(0,1,4)C=(0,1,4) and DD form a tetrahedron. The point DD is equidistant from A,B,CA,B,C, and the tetrahedron centroid lies in x+y+z=11/2x+y+z=11/2. If its volume is VV, then V2V^2 is
Not attempted
Oblique view; lengths and angles not to scaleABCD
A
99
B
144144
C
3636
D
324324
4
Let P=(2,1,3)P=(2,1,3) and let FF be its perpendicular foot on a variable line through OO contained in x+y+z=0x+y+z=0. The smaller angle of that line with direction (0,−1,1)(0,-1,1) is at most 45∘45^\circ. Put X=F⋅(0,−1,1)/2X=F\cdot(0,-1,1)/\sqrt2 and Y=F⋅(−2,1,1)/6Y=F\cdot(-2,1,1)/\sqrt6. Which diagram and caption give exactly the locus of (X,Y)(X,Y)? All displayed boundaries are included.
Not attempted
A
X2+Y2=2X, Y≤0X^2+Y^2=\sqrt2X,\ Y\le0
XYBoundaries included; equal scale
B
X2+Y2=2X, X≤1/2X^2+Y^2=\sqrt2X,\ X\le1/\sqrt2
XYBoundaries included; equal scale
C
X2+Y2=2X, Y≥0X^2+Y^2=\sqrt2X,\ Y\ge0
XYBoundaries included; equal scale
D
X2+Y2=2X, X≥1/2X^2+Y^2=\sqrt2X,\ X\ge1/\sqrt2
XYBoundaries included; equal scale
5
A directed line has direction cosines l≥m≥nl\ge m\ge n satisfying l+m+n=1l+m+n=1. Which statements hold for every such direction, or give the stated exact extremum?
Not attempted
A
2/3≤l≤12/3\le l\le1
B
lm+mn+nl=0lm+mn+nl=0
C
The maximum of 27lmn27lmn is 44.
D
n<0n<0 for every admissible direction.
6
The segment from A=(1,2,−3)A=(1,2,-3) to B=(−3,−2,5)B=(-3,-2,5) meets the yz,zx,xyyz,zx,xy planes at X,Y,ZX,Y,Z, respectively. Ignore the crossing points themselves when counting octants. Which statements are correct?
Not attempted
Oblique view; lengths and angles not to scaleABXZY
A
The centroid of X,Y,ZX,Y,Z lies in xyxy.
B
The segment passes through four octants.
C
AZ:ZB=3:5AZ:ZB=3:5
D
The projection of XYXY on direction (1,1,1)(1,1,1) has length 6\sqrt6.
7
For a unit vector u=(l,m,n)u=(l,m,n), let cic_i be its signed projections on the unit directions proportional to (1,1,1),(1,−1,−1),(−1,1,−1),(−1,−1,1)(1,1,1),(1,-1,-1),(-1,1,-1),(-1,-1,1). Put S=∑i=14ci4S=\sum_{i=1}^4c_i^4. Which statements are correct?
Not attempted
A
max⁡S=28/27\max S=28/27
B
min⁡S=4/9\min S=4/9
C
SS is maximal exactly along body diagonals of a cube with coordinate-parallel edges.
D
∑i=14ci3=0\sum_{i=1}^4c_i^3=0 for every uu.
8
A first-octant point PP has distances from the x,y,zx,y,z axes in the ratio 5:10:13\sqrt5:\sqrt{10}:\sqrt{13} and lies on 2x−y+z=152x-y+z=15. The line OPOP meets z=−2z=-2 at QQ. Find PQ2PQ^2.
Not attempted
Your answer: —
Correct answer: 350
9
A point P=(x,y,z)P=(x,y,z) satisfies x+y+z=6x+y+z=6 and xy+yz+zx=9xy+yz+zx=9. For Q=(5,−1,2)Q=(5,-1,2) let MM and mm be the largest and smallest values of PQ2PQ^2. Find (M−m)2(M-m)^2.
Not attempted
Your answer: —
Correct answer: 1728
10
Exactly two unit vectors u1,u2u_1,u_2 satisfy l+m+n=0l+m+n=0 and l+2m+3n=1l+2m+3n=1, where (l,m,n)(l,m,n) are their components. Put pi=ui⋅(1,3,2)p_i=u_i\cdot(1,3,2). Find 16(p14+p24)16(p_1^4+p_2^4).
Not attempted
Your answer: —
Correct answer: 56
11
Let A=(3,0,0)A=(3,0,0), B=(0,6,0)B=(0,6,0), C=(0,0,9)C=(0,0,9). A point PP lies in x+y+z=0x+y+z=0 and is at distance 22 from the line r=t(1,1,−2)r=t(1,1,-2). If MM is the minimum of PA2+PB2+PC2PA^2+PB^2+PC^2, find (267−2M)2(267-2M)^2.
Not attempted
Your answer: —
Correct answer: 288
12
Let u=(l,m,n)u=(l,m,n) range over all unit directions perpendicular to (1,2,2)(1,2,2). Match each extremum in List-I to its value in List-II. List-IList-IIP) max⁡l21) 8/9Q) max⁡m22) 5/9R) max⁡(l+m)23) 1S) min⁡(l2+m2)4) 4/95) 0\begin{array}{ll}\text{List-I}&\text{List-II}\\\text{P)}\ \max l^2&\text{1)}\ 8/9\\\text{Q)}\ \max m^2&\text{2)}\ 5/9\\\text{R)}\ \max(l+m)^2&\text{3)}\ 1\\\text{S)}\ \min(l^2+m^2)&\text{4)}\ 4/9\\&\text{5)}\ 0\end{array}
Not attempted
A
P-1, Q-2, R-3, S-4
B
P-2, Q-3, R-4, S-1
C
P-3, Q-4, R-1, S-2
D
P-4, Q-1, R-2, S-3
13
The line LL is (2x−2)/4=(3−y)/2=(z+1)(2x-2)/4=(3-y)/2=(z+1). A point PP lies in x+y+z=6x+y+z=6, and its image under a half-turn about LL lies in x+y+z=0x+y+z=0. The least possible value of OP2OP^2 is
Not attempted
A
1212
B
45926\frac{459}{26}
C
25/925/9
D
459104\frac{459}{104}
14
For each t∈Rt\in\mathbb R, reflect P(t)=(2,−1,3)+t(2,−1,0)P(t)=(2,-1,3)+t(2,-1,0) by a half-turn about L:r=(1,1,1)+s(1,2,2)L:r=(1,1,1)+s(1,2,2). The original trajectory and its image trajectory are distinct parallel lines. The unique plane containing both is
Not attempted
Oblique view; lengths and angles not to scaleABCDF
A
x+2y+2z=9x+2y+2z=9
B
x+2y+2z=0x+2y+2z=0
C
x+2y+2z=6x+2y+2z=6
D
2x−y=52x-y=5
15
For real aa, let La:r=(a,1,0)+t(1,a,0)L_a:r=(a,1,0)+t(1,a,0). The image of OO under a half-turn about LaL_a is required to lie on x+y=2x+y=2. The sum of the squares of all admissible values of aa is
Not attempted
A
22
B
44
C
88
D
66
16
Let P(t)=(t,1,2t)P(t)=(t,1,2t) for all real tt. First apply a half-turn about L:r=(1,0,0)+s(1,1,0)L:r=(1,0,0)+s(1,1,0), then orthogonally project the resulting points onto y+z=0y+z=0. Which diagram and equation describe the final locus?
Not attempted
A
x=2, y+z=0x=2,\ y+z=0
Candidate line; O marks the originOUV
B
x=1, y+z=0x=1,\ y+z=0
Candidate line; O marks the originOUV
C
x=2, y−z=0x=2,\ y-z=0
Candidate line; O marks the originOUV
D
x=1, y−z=0x=1,\ y-z=0
Candidate line; O marks the originOUV
17
For each PP in x+z=0x+z=0, let FF and II be its foot and half-turn image with respect to L:r=(1,2,3)+t(1,0,1)L:r=(1,2,3)+t(1,0,1). Which statements are correct?
Not attempted
A
The locus of FF consists of one point.
B
The locus of II is the entire plane x+z=0x+z=0.
C
min⁡PI2=0\min PI^2=0
D
I=−PI=-P for every admissible PP.
18
A line through A=(−1,0,1)A=(-1,0,1) varies over all directions perpendicular to (1,1,0)(1,1,0). For P=(1,2,3)P=(1,2,3), let FF and II be its foot and half-turn image in that line. Which statements are correct?
Not attempted
Oblique view; lengths and angles not to scaleAPUV
A
The foot locus is a circle of radius 11 centred at (−1,0,2)(-1,0,2).
B
min⁡PI2=32\min PI^2=32
C
max⁡PF2=12\max PF^2=12
D
The image locus lies in x+y=−5x+y=-5.
19
For every finite real tt, let LtL_t join A=(0,1,0)A=(0,1,0) to B=(t,0,1)B=(t,0,1), and let I(t)I(t) be the half-turn image of OO about LtL_t. Which statements are correct?
Not attempted
A
The image locus is a complete circle.
B
yI+zI=2y_I+z_I=2 for every tt.
C
The point (0,2,0)(0,2,0) occurs for some real tt.
D
The maximum of OI2OI^2 is 44.
20
For P(t)=(t,1−t,2+t)P(t)=(t,1-t,2+t), let I(t)I(t) be its half-turn image about r=(1,−2,0)+s(1,2,−1)r=(1,-2,0)+s(1,2,-1). If M=min⁡t∈ROI(t)2M=\min_{t\in\mathbb R}OI(t)^2, find MM.
Not attempted
Your answer: —
Correct answer: 24
21
A line LL is parallel to the plane x−y=0x-y=0 and is contained in x−y=1x-y=1. The perpendicular feet on LL of (1,0,0),(0,1,0),(0,0,1)(1,0,0),(0,1,0),(0,0,1) are F1,F2,F3F_1,F_2,F_3, with F3=(F1+F2)/2F_3=(F_1+F_2)/2. Also d(O,L)=2d(O,L)=2. If H1,H2H_1,H_2 are the two possible feet of OO and their zz-coordinates are z1,z2z_1,z_2, find 3(z12+z22)3(z_1^2+z_2^2).
Not attempted
Your answer: —
Correct answer: 14
22
A variable line through A=(−1,0,−1)A=(-1,0,-1) has perpendicular foot FF from P=(1,2,3)P=(1,2,3) lying in x+y+z=0x+y+z=0. Let M,mM,m be the greatest and least possible values of PF2PF^2. Find 9(M−m)29(M-m)^2.
Not attempted
Your answer: —
Correct answer: 448
23
Let P(t)=(t,t2,1)P(t)=(t,t^2,1), and let I(t)I(t) be its half-turn image about the line r=s(1,1,1)r=s(1,1,1). Let TT consist of all real tt for which I(t)I(t) lies in x−y+z=1x-y+z=1. Find 625∑t∈TOI(t)2625\sum_{t\in T}OI(t)^2.
Not attempted
Your answer: —
Correct answer: 3156
24
Fix P=(2,1,−1)P=(2,1,-1). For each pair (a,b)(a,b) in List-I, let II be the half-turn image of PP about r=a+tbr=a+tb, and JJ its mirror image in the plane (r−a)⋅b=0(r-a)\cdot b=0. Match IJ2IJ^2 to List-II. List-IList-IIP) a=(1,0,0), b=(0,1,1)1) 12Q) a=(0,1,0), b=(1,0,1)2) 20R) a=(0,0,1), b=(1,1,0)3) 36S) a=(−1,1,1), b=(1,−1,2)4) 525) 0\begin{array}{ll}\text{List-I}&\text{List-II}\\\text{P)}\ a=(1,0,0),\ b=(0,1,1)&\text{1)}\ 12\\\text{Q)}\ a=(0,1,0),\ b=(1,0,1)&\text{2)}\ 20\\\text{R)}\ a=(0,0,1),\ b=(1,1,0)&\text{3)}\ 36\\\text{S)}\ a=(-1,1,1),\ b=(1,-1,2)&\text{4)}\ 52\\&\text{5)}\ 0\end{array}
Not attempted
A
P-2, Q-3, R-4, S-1
B
P-1, Q-2, R-3, S-4
C
P-3, Q-4, R-1, S-2
D
P-4, Q-1, R-2, S-3
25
Let L:r=t(1,0,0)L:r=t(1,0,0) and Ma:r=(0,1,a)+s(1,a,1)M_a:r=(0,1,a)+s(1,a,1), with a∈Ra\in\mathbb R. Let TT be the set of all aa for which the shortest distance between LL and MaM_a equals 11. Then ∑a∈Ta2\sum_{a\in T}a^2 is
Not attempted
A
99
B
33
C
66
D
00
26
For real aa, let L:r=t(1,0,0)L:r=t(1,0,0) and Ma:r=(0,1,2)+s(1,a,a2)M_a:r=(0,1,2)+s(1,a,a^2). For which values of aa does their shortest distance attain its global maximum?
Not attempted
A
{1/2}\{1/2\}
B
{−1/2}\{-1/2\}
C
{0,2}\{0,2\}
D
{0,−1/2}\{0,-1/2\}
27
Consider L:r=(1,0,0)+t(1,1,0)L:r=(1,0,0)+t(1,1,0) and M:r=(0,1,k)+s(0,1,1)M:r=(0,1,k)+s(0,1,1). The midpoint of their shortest joining segment lies in x+y+z=3x+y+z=3. The squared shortest distance is
Not attempted
A
643\frac{64}{3}
B
163\frac{16}{3}
C
6464
D
99
28
Let P=(t,0,0)P=(t,0,0) with 0≤t≤10\le t\le1, and Q=(3−2s,1+s,2−s)Q=(3-2s,1+s,2-s) with 0≤s≤10\le s\le1. The minimum of PQ2PQ^2 is
Not attempted
Oblique view; lengths and angles not to scaleABCD
A
9/29/2
B
29/629/6
C
99
D
55
29
Three lines are Lx:r=(t,0,0)L_x:r=(t,0,0), Ly:r=(0,s,1)L_y:r=(0,s,1) and Lz:r=(1,1,u)L_z:r=(1,1,u). Let U,V,WU,V,W be the midpoints of the three pairwise shortest joining segments. Which statements are correct?
Not attempted
Oblique view; lengths and angles not to scaleOABCDE
A
Every pair of the three lines has shortest distance 11.
B
Triangle UVWUVW is equilateral.
C
The square of its area is 27/6427/64.
D
The three common-perpendicular lines are concurrent.
30
For L:r=(t,0,0)L:r=(t,0,0) and M:r=(s,1+s,1)M:r=(s,1+s,1), let Π\Pi be the plane containing LL and parallel to the direction of MM. Which statements are correct?
Not attempted
A
d(L,M)=d(M,Π)=1d(L,M)=d(M,\Pi)=1
B
The shortest-segment midpoint is (0,0,1/2)(0,0,1/2).
C
Every connector perpendicular to LL is a shortest connector.
D
The projections of L,ML,M onto Π\Pi intersect.
31
Let L:r=t(1,1,0)L:r=t(1,1,0) and Ma:r=(0,1,1)+s(1+a,1+a,a)M_a:r=(0,1,1)+s(1+a,1+a,a) for all real aa. Which statements are correct?
Not attempted
A
d(L,Ma)2=1/2d(L,M_a)^2=1/2 for every a≠0a\ne0.
B
d(L,M0)2=3/2d(L,M_0)^2=3/2
C
The shortest distance is continuous at a=0a=0.
D
The common-perpendicular feet for a≠0a\ne0 remain bounded as a→0a\to0.
32
Let P=(t,0,0)P=(t,0,0) and Q=(s,1+s,1)Q=(s,1+s,1), with unrestricted real t,st,s. Require the midpoint of PQPQ to lie in x+y=2x+y=2. If MM is the minimum of PQ2PQ^2 under this constraint, find 5M5M.
Not attempted
Your answer: —
Correct answer: 23
33
Points P=(t,0,0)P=(t,0,0) and Q=(0,s,3)Q=(0,s,3) have t,s>0t,s>0 and ts=4ts=4. Their midpoint is at distance 11 from the line r=(0,0,3/2)+v(1,1,0)r=(0,0,3/2)+v(1,1,0). Find PQ2PQ^2.
Not attempted
Your answer: —
Correct answer: 25
34
The lines L:r=(t,0,0)L:r=(t,0,0) and M:r=(0,s,1)M:r=(0,s,1) are skew. For 0<c<10<c<1, the line through C=(1,1,c)C=(1,1,c) meeting both LL and MM cuts them at A,BA,B. Find the minimum possible AB2AB^2.
Not attempted
Your answer: —
Correct answer: 9
35
For every real aa, let La:r=t(1,a,0)L_a:r=t(1,a,0) and Ma:r=(1,0,1)+s(0,1,a)M_a:r=(1,0,1)+s(0,1,a). Let DD be the largest possible squared shortest distance between them. Find 3D3D.
Not attempted
Your answer: —
Correct answer: 4
36
Fix L:r=t(1,0,0)L:r=t(1,0,0). For each family Mk:r=a(k)+sb(k)M_k:r=a(k)+s b(k) in List-I, match the number of real kk for which MkM_k intersects LL in exactly one point. Coincident or distinct parallel lines do not count. List-IList-IIP) a=(0,k,1), b=(1,1,k)1) 0Q) a=(0,1,k2), b=(1,k,k2)2) 1R) a=(0,k,1), b=(1,−1,k)3) 2S) a=(0,k(k−1)(k+1),1), b=(1,0,1)4) 35) 4\begin{array}{ll}\text{List-I}&\text{List-II}\\\text{P)}\ a=(0,k,1),\ b=(1,1,k)&\text{1)}\ 0\\\text{Q)}\ a=(0,1,k^2),\ b=(1,k,k^2)&\text{2)}\ 1\\\text{R)}\ a=(0,k,1),\ b=(1,-1,k)&\text{3)}\ 2\\\text{S)}\ a=(0,k(k-1)(k+1),1),\ b=(1,0,1)&\text{4)}\ 3\\&\text{5)}\ 4\end{array}
Not attempted
A
P-1, Q-4, R-3, S-2
B
P-2, Q-1, R-4, S-3
C
P-3, Q-2, R-1, S-4
D
P-4, Q-3, R-2, S-1
37
Among all planes containing the line x+y+z=1, x−y=0x+y+z=1,\ x-y=0, let Π\Pi have the largest possible angle in [0,90∘][0,90^\circ] with the plane 2x+z=02x+z=0. Which equation gives Π\Pi?
Not attempted
A
x+y−2z=−2x+y-2z=-2
B
x+5y−2z=−2x+5y-2z=-2
C
2x+z=12x+z=1
D
x−5y−2z=−2x-5y-2z=-2
38
Let P\mathcal P be all distinct planes through A=(1,1,2)A=(1,1,2) and B=(2,0,2)B=(2,0,2) whose distance from OO is 22. For Q=(1,2,4)Q=(1,2,4), the sum ∑Π∈Pd(Q,Π)2\sum_{\Pi\in\mathcal P}d(Q,\Pi)^2 equals
Not attempted
Oblique view; lengths and angles not to scaleABQO
A
52/952/9
B
16/916/9
C
40/940/9
D
20/320/3
39
For real tt, let A=(1,0,0)A=(1,0,0), B=(0,1,0)B=(0,1,0) and C=(t,t,1−t2)C=(t,t,1-t^2). Let TT consist of parameters for which the unique plane through A,B,CA,B,C is perpendicular to x+y+z=0x+y+z=0. Find ∑t∈Tt2\sum_{t\in T}t^2.
Not attempted
A
7/27/2
B
44
C
33
D
11
40
Consider all planes containing r=(1,0,1)+t(1,1,0)r=(1,0,1)+t(1,1,0) and making equal smaller angles with the planes x=0x=0 and y+z=0y+z=0. If their squared distances from OO are d12,d22d_1^2,d_2^2, find d12d22d_1^2d_2^2.
Not attempted
A
16/1716/17
B
4/94/9
C
4/174/17
D
2/52/5
41
A plane cuts the axes at finite nonzero intercepts a,b,ca,b,c, of arbitrary signs. Let GG be the centroid of the intercept triangle and HH the perpendicular foot from OO. Which statements are correct?
Not attempted
A
OH2≤(a2+b2+c2)/9OH^2\le(a^2+b^2+c^2)/9
B
Equality in that bound holds exactly when ∣a∣=∣b∣=∣c∣|a|=|b|=|c|.
C
Equality is impossible if abc<0abc<0.
D
Fixing a2+b2+c2a^2+b^2+c^2 forces a positive lower bound for OHOH.
42
For t∈Rt\in\mathbb R, consider the planes containing OO, (1,t,0)(1,t,0) and (t,1,0)(t,1,0) and perpendicular to x+y+z=0x+y+z=0. Which statements are correct?
Not attempted
A
Exactly two real parameter values admit such a plane.
B
At t=1t=1, the unique required plane is x−y=0x-y=0.
C
At t=−1t=-1, the unique required plane is x+y−2z=0x+y-2z=0.
D
At t=0t=0, z=0z=0 is a required plane.
43
For finite real λ\lambda, let Πλ:x+λy+(1−λ)z=1\Pi_\lambda:x+\lambda y+(1-\lambda)z=1, and let FλF_\lambda be the foot of OO. Which statements are correct?
Not attempted
A
The maximum of OFλ2OF_\lambda^2 is 2/32/3.
B
The minimum of OFλ2OF_\lambda^2 is 00.
C
The locus of FλF_\lambda is a complete circle.
D
Among positive intercepts, the minimum intercept tetrahedron volume is 1/61/6.
44
A plane through P=(1,1,1)P=(1,1,1) cuts the positive axes at A,B,CA,B,C. Its foot from OO lies on the ray r=t(1,2,2)r=t(1,2,2), t>0t>0. If KK is the area of triangle ABCABC, find 64K264K^2.
Not attempted
Oblique view; lengths and angles not to scaleOABCP
Your answer: —
Correct answer: 5625
45
A plane cuts the positive axes at intercepts a,b,ca,b,c in arithmetic progression. It passes through (1,2,3)(1,2,3) and is parallel to direction (1,−2,1)(1,-2,1). Find the volume of its intercept tetrahedron with OO.
Not attempted
Your answer: —
Correct answer: 36
46
Let E\mathcal E be the set of all points in space at equal perpendicular distances from the four planes x+y+z=6x+y+z=6, x−y−z=0x-y-z=0, −x+y−z=0-x+y-z=0, −x−y+z=0-x-y+z=0. Find 4∑P∈EOP24\sum_{P\in\mathcal E}OP^2.
Not attempted
Your answer: —
Correct answer: 243
47
There are exactly two distinct planes containing r=(1,2,0)+t(1,0,1)r=(1,2,0)+t(1,0,1) and at perpendicular distance 11 from OO. If their smaller angle is θ\theta, find 81sin⁡2θ81\sin^2\theta.
Not attempted
Your answer: —
Correct answer: 56
48
Every plane in this question contains the line x+y=1,z=0x+y=1,z=0. Match each additional condition in List-I to the resulting plane in List-II. List-IList-IIP) largest distance from O1) x+y=1Q) perpendicular to x+y+z=02) x+y−2z=1R) passes through (0,0,1)3) x+y+z=1S) passes through O4) z=05) x+y−z=1\begin{array}{ll}\text{List-I}&\text{List-II}\\\text{P)}\ \text{largest distance from }O&\text{1)}\ x+y=1\\\text{Q)}\ \text{perpendicular to }x+y+z=0&\text{2)}\ x+y-2z=1\\\text{R)}\ \text{passes through }(0,0,1)&\text{3)}\ x+y+z=1\\\text{S)}\ \text{passes through }O&\text{4)}\ z=0\\&\text{5)}\ x+y-z=1\end{array}
Not attempted
A
P-4, Q-1, R-2, S-3
B
P-2, Q-3, R-4, S-1
C
P-3, Q-4, R-1, S-2
D
P-1, Q-2, R-3, S-4
49
Let A=(1,0,2)A=(1,0,2) and B=(3,0,4)B=(3,0,4). A point PP varies on the line x+y=1,z=0x+y=1,z=0. The square of the minimum possible value of AP+PBAP+PB is
Not attempted
A
24+12224+12\sqrt2
B
24+8224+8\sqrt2
C
4444
D
20+12220+12\sqrt2
50
A line L:r=(1,0,0)+t(1,1,1)L:r=(1,0,0)+t(1,1,1) is first reflected in 2x−y+2z=32x-y+2z=3 and then in −4x+2y−4z=12-4x+2y-4z=12, giving L1L_1. Find the squared shortest distance between LL and L1L_1.
Not attempted
A
3636
B
2424
C
66
D
9696
51
A variable line makes equal smaller angles with the planes x=0x=0 and x+y+z=0x+y+z=0. If ϕ\phi is its angle with the xyxy-plane, find the greatest possible sin⁡2ϕ\sin^2\phi.
Not attempted
A
(3+3)/6(3+\sqrt3)/6
B
(9−3)/12(9-\sqrt3)/12
C
(9+3)/12(9+\sqrt3)/12
D
3/43/4
52
Let P(t)=(t,1,0)P(t)=(t,1,0) range over a line. Reflect it first in x−y=0x-y=0, then in y−z=0y-z=0, and finally project orthogonally onto x+y+z=0x+y+z=0. Which diagram and equations give the final locus?
Not attempted
A
x+y+z=0, y−z=−1x+y+z=0,\ y-z=-1
Candidate line in x + y + z = 0OUV
B
x+y+z=0, x−y=−1x+y+z=0,\ x-y=-1
Candidate line in x + y + z = 0OUV
C
x+y+z=0, y−z=1x+y+z=0,\ y-z=1
Candidate line in x + y + z = 0OUV
D
x+y+z=0, x−y=1x+y+z=0,\ x-y=1
Candidate line in x + y + z = 0OUV
53
Consider Π1:2x−y+2z=3\Pi_1:2x-y+2z=3 and Π2:x+2y+2z=−3\Pi_2:x+2y+2z=-3. Their two equal-distance bisector planes are B1,B2B_1,B_2. Let F1,F2F_1,F_2 be the perpendicular feet from OO on Π1,Π2\Pi_1,\Pi_2. Which statements are correct?
Not attempted
Oblique view; lengths and angles not to scaleOFG
A
One bisector is 3x+y+4z=03x+y+4z=0.
B
B1⊥B2B_1\perp B_2
C
Both bisectors pass through OO.
D
The bisector through OO is perpendicular to F1F2F_1F_2.
54
Let L:r=(t,0,t)L:r=(t,0,t) and M:r=(0,s,1+s)M:r=(0,s,1+s). Orthogonally project both lines onto x+y+z=0x+y+z=0. Which statements are correct?
Not attempted
Oblique view; lengths and angles not to scaleABCDPQ
A
The projected lines intersect at (1/3,−2/3,1/3)(1/3,-2/3,1/3).
B
The original lines have shortest distance 00.
C
The connector joining points with that common projection is the common perpendicular.
D
d(L,M)2=1/3d(L,M)^2=1/3
55
A line LL makes an angle 0<ϕ<90∘0<\phi<90^\circ with a plane Π\Pi. Let L1L_1 be its mirror image in Π\Pi, and let α∈[0,90∘]\alpha\in[0,90^\circ] be the smaller angle between L,L1L,L_1. Which statements are correct?
Not attempted
A
L,L1L,L_1 are coplanar.
B
α=min⁡(2ϕ,180∘−2ϕ)\alpha=\min(2\phi,180^\circ-2\phi)
C
L⊥L1L\perp L_1 exactly when ϕ=45∘\phi=45^\circ.
D
The plane containing L,L1L,L_1 is perpendicular to Π\Pi.
56
Let A=(1,2,3)A=(1,2,3) and B=(4,5,6)B=(4,5,6). Points PP and QQ vary on x=0x=0 and y=0y=0, respectively, in the prescribed path order A→P→Q→BA\to P\to Q\to B. Find the square of the minimum path length. Zero-length legs are allowed.
Not attempted
One admissible path; P and Q varyABPQ
Your answer: —
Correct answer: 83
57
Let A=(5,1,0)A=(5,1,0) and B=(2,4,3)B=(2,4,3). Points P∈{x=0}P\in\{x=0\} and Q∈{y=0}Q\in\{y=0\} occur in the order A→P→Q→BA\to P\to Q\to B. Zero-length legs are allowed. If MM is the square of the minimum total length, find (M−55)2(M-55)^2.
Not attempted
Your answer: —
Correct answer: 2080
58
Let TT mean reflection first in x+y=0x+y=0 and then in y+z=0y+z=0. Starting with P=(1,2,3)P=(1,2,3), put A=PA=P, B=T(P)B=T(P) and C=T(T(P))C=T(T(P)). If KK is the area of triangle ABCABC, find 4K24K^2.
Not attempted
Your answer: —
Correct answer: 1083
59
Fix P=(1,2,3)P=(1,2,3). A plane Π\Pi varies over all planes containing r=t(1,1,1)r=t(1,1,1). Let FF and II be the perpendicular foot and mirror image of PP in Π\Pi. Match each quantity to its value. List-IList-IIP) squared radius of the F locus1) 1/2Q) squared radius of the I locus2) 2R) min⁡OF23) 12S) max⁡OF24) 145) 0\begin{array}{ll}\text{List-I}&\text{List-II}\\\text{P)}\ \text{squared radius of the }F\text{ locus}&\text{1)}\ 1/2\\\text{Q)}\ \text{squared radius of the }I\text{ locus}&\text{2)}\ 2\\\text{R)}\ \min OF^2&\text{3)}\ 12\\\text{S)}\ \max OF^2&\text{4)}\ 14\\&\text{5)}\ 0\end{array}
Not attempted
A
P-1, Q-2, R-3, S-4
B
P-2, Q-3, R-4, S-1
C
P-3, Q-4, R-1, S-2
D
P-4, Q-1, R-2, S-3
60
Let L:r=(1,0,1)+t(1,1,0)L:r=(1,0,1)+t(1,1,0) and Ma:r=(0,1,a)+s(1,a,1)M_a:r=(0,1,a)+s(1,a,1). For the two real values of aa making these lines coplanar, let Π1,Π2\Pi_1,\Pi_2 be their containing planes. Find d(O,Π1)2d(O,Π2)2d(O,\Pi_1)^2d(O,\Pi_2)^2.
Not attempted
A
1/41/4
B
1/161/16
C
9/169/16
D
11
61
The parallel lines L:r=(1,2,0)+t(1,−1,1)L:r=(1,2,0)+t(1,-1,1) and M:r=(3,1,2)+s(2,−2,2)M:r=(3,1,2)+s(2,-2,2) determine a plane Π\Pi. Reflect P=(0,1,0)P=(0,1,0) in Π\Pi to obtain II. Find d(I,L)2d(I,L)^2.
Not attempted
Oblique view; lengths and angles not to scaleABCDP
A
33
B
11
C
22
D
66
62
The set of points XX satisfying (X−A)⋅(X−B)=0(X-A)\cdot(X-B)=0, with A=(1,−2,0)A=(1,-2,0) and B=(5,2,4)B=(5,2,4), is cut by planes containing r=(1,0,0)+t(1,1,0)r=(1,0,0)+t(1,1,0). Exactly two such planes have a circular section of radius 33. If their distances from OO are d1,d2d_1,d_2, find d12d22d_1^2d_2^2.
Not attempted
Oblique view; lengths and angles not to scaleABUV
A
1/91/9
B
1/361/36
C
1/161/16
D
1/1441/144
63
For real λ\lambda, the surface SλS_\lambda is x2+y2+z2−2λx−4y+2z+4λ−4=0x^2+y^2+z^2-2\lambda x-4y+2z+4\lambda-4=0. A unique member has exactly one point in common with x=0x=0. Its section by y=0y=0 is a circle of squared radius
Not attempted
A
17/1617/16
B
81/1681/16
C
65/1665/16
D
5/165/16
64
For real kk, consider the real locus (k−1)(x2+y2+z2)+2x+2ky+2(k+1)z+k=0(k-1)(x^2+y^2+z^2)+2x+2ky+2(k+1)z+k=0. Which statements are correct?
Not attempted
A
Every k≠1k\ne1 gives a nondegenerate sphere.
B
At k=−1k=-1, the locus is (1/2,−1/2,0)(1/2,-1/2,0).
C
At k=1k=1, the locus is a plane.
D
For −2<k<−1-2<k<-1, the real locus is empty.
65
Let PP vary over the common points of x2+y2+z2=25x^2+y^2+z^2=25 and x2+y2+(z−6)2=25x^2+y^2+(z-6)^2=25. For each PP, take the plane through PP perpendicular to the radius from each centre. Which statements are correct?
Not attempted
A
The common locus is a circle of radius 44.
B
The cosine of the smaller angle between the two radius-normal planes is 7/257/25.
C
The intersection of those two planes lies in z=3z=3.
D
Those two planes are perpendicular for every PP.
66
For real aa, let La:r=(a,0,0)+t(1,a,0)L_a:r=(a,0,0)+t(1,a,0) and let S:x2+y2+z2=1S:x^2+y^2+z^2=1. Put τ=(1+5)/2\tau=(1+\sqrt5)/2. Which statements are correct?
Not attempted
A
At a=0a=0, the line cuts a diameter of SS.
B
The line has exactly one common point with SS iff a=±τa=\pm\sqrt\tau.
C
Exactly four real values of aa give a single common point.
D
The sum of squared real tangent parameters is 1+51+\sqrt5.
67
A surface consists of points at a fixed distance from an unknown centre and passes through O,(4,0,0),(0,6,0),(0,0,8)O,(4,0,0),(0,6,0),(0,0,8). On its section by x+y+z=3x+y+z=3, let MM be the maximum value of x−yx-y. Find (M+1)2(M+1)^2.
Not attempted
Oblique view; lengths and angles not to scaleOABDC
Your answer: —
Correct answer: 34
68
A closed ball of the smallest possible radius contains all four points (2,0,0),(−2,0,0),(0,3,3),(0,−3,3)(2,0,0),(-2,0,0),(0,3,3),(0,-3,3). Its centre is unrestricted. If its squared radius is R2R^2, find 9R29R^2.
Not attempted
Oblique view; lengths and angles not to scaleABCD
Your answer: —
Correct answer: 85
69
There are two planes containing the line r=(3,0,0)+t(0,1,0)r=(3,0,0)+t(0,1,0) and each meeting x2+y2+z2=4x^2+y^2+z^2=4 in exactly one point. Let those contact points be A,BA,B. Find 9AB29AB^2.
Not attempted
Section y = 0; the given line passes through EOExzThe given line is perpendicular to this section
Your answer: —
Correct answer: 80
70
The surface (X−A)⋅(X−B)=0(X-A)\cdot(X-B)=0 has A=(−1,0,1)A=(-1,0,1) and B=(3,4,5)B=(3,4,5). Match each section plane in List-I to its complete intersection description in List-II. List-IList-IIP) x+y+z=61) circle with squared radius 12Q) 2x−y+2z=122) circle with squared radius 8R) x+y+z=123) one pointS) 2x−y+2z=184) no real points5) circle with squared radius 4\begin{array}{ll}\text{List-I}&\text{List-II}\\\text{P)}\ x+y+z=6&\text{1)}\ \text{circle with squared radius }12\\\text{Q)}\ 2x-y+2z=12&\text{2)}\ \text{circle with squared radius }8\\\text{R)}\ x+y+z=12&\text{3)}\ \text{one point}\\\text{S)}\ 2x-y+2z=18&\text{4)}\ \text{no real points}\\&\text{5)}\ \text{circle with squared radius }4\end{array}
Not attempted
A
P-2, Q-3, R-4, S-1
B
P-1, Q-2, R-3, S-4
C
P-3, Q-4, R-1, S-2
D
P-4, Q-1, R-2, S-3
The diameter condition fixes both the centre and the radius; compare each plane distance with that radius before subtracting squares.
The centre is (1,2,3)(1,2,3) and the squared radius is ∣B−A∣2/4=12|B-A|^2/4=12. For x+y+z=6x+y+z=6, the centre lies in the plane, so the section has squared radius 1212. For 2x−y+2z=122x-y+2z=12, the distance is 22, giving squared section radius 88. For x+y+z=12x+y+z=12, the distance is 6/3=126/\sqrt3=\sqrt{12}, giving exactly one point, not a zero-radius circle with multiple points. For 2x−y+2z=182x-y+2z=18, the distance is 4>124>\sqrt{12}, so the intersection is empty. Negative section-radius squares describe no real locus.