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JEE • MATHEMATICS • 3D GEOMETRY SYSTEM

3d geometry system practice · Mixed difficulty

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Showing 31-60 of 140 questions

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31 Easy
31 of 140
Application +4 | -1
The point PP is equidistant from A(1,  3),  B(−3,  5)A\left( {1,\;3} \right),\;B\left( { - 3,\;5} \right) and C(5,  −1)C\left( {5,\; - 1} \right), then PAPA is equal to
View answer and explanation
Correct option: D
5105\sqrt {10}
Option D (5105\sqrt {10}) is correct because it is the distance PA calculated after finding the coordinates of point P, which is equidistant from A, B, and C.
32 Easy
32 of 140
Analyzing +4 | -1
Let A be a fixed point (0, 6) and BB be a moving point (2t, 0). Let MM be the mid-point of ABAB and the perpendicular bisectors of ABAB meets the yy-axis at CC.
The locus of the mid-point PP of MCMC is
View answer and explanation
Correct option: C
2x2+3y−9=02{x^2} + 3y - 9 = 0
Option C (2x2+3y−9=02{x^2} + 3y - 9 = 0) is correct because it represents the locus of the midpoint P of MC, derived by finding the coordinates of M and C and then eliminating the parameter t.
33 Medium
33 of 140
Application +4 | -1
The shortest distance of the point (1,2,3)(1,2,3) from X,YX, Y and ZZ-axes, respectively are
View answer and explanation
Correct option: D
13,10,5\sqrt{13}, \sqrt{10}, \sqrt{5}
Option D (13,10,5\sqrt{13}, \sqrt{10}, \sqrt{5}) is correct because the shortest distance from the X, Y, and Z axes are calculated using the formulas y2+z2\sqrt{y^2+z^2}, x2+z2\sqrt{x^2+z^2}, and x2+y2\sqrt{x^2+y^2} respectively.
34 Easy
34 of 140
Application +4 | -1
The equation of the plane which contains the origin and the line of intersection of the planes r⋅a=p{r} \cdot {a}=p and r⋅b={r} \cdot {b}= qq is
View answer and explanation
Correct option: D
r⋅(qa−pb)=0{r} \cdot(q {a}-p {b})=0
Option D (r⋅(qa−pb)=0{r} \cdot(q {a}-p {b})=0) is correct because it represents the equation of a plane passing through the origin and the line of intersection of the given planes.
35 Easy
35 of 140
Application +4 | -1
If the line through the points (4,1,2)(4,1,2) and (5,k,0)(5, k, 0) is parallel to the line through the points (2,1,1)(2,1,1) and (3(3, 3,−1)3,-1), then kk is equal to
View answer and explanation
Correct option: A
3
Option A (3) is correct because the lines are parallel, so their direction ratios are proportional.
36 Medium
36 of 140
Application +4 | -1
The sum of magnitudes of two forces acting at a point is 18 and the magnitude of their resultant is 12 . If the resultant is at 90∘90^{\circ} with the force of smaller magnitude, then their magnitudes are
View answer and explanation
Correct option: C
5,13
Option C (5,13) is correct because these magnitudes satisfy the given conditions: their sum is 18, their resultant is 12, and the resultant is at 90 degrees with the smaller force.
37 Easy
37 of 140
Application +4 | -1
The point moves such that the area of the triangle formed by it with the points (1, 5) and (3,-7) is 21 sq unit. The locus of the point is
View answer and explanation
Correct option: A
6x+y−32=06x + y - 32=0
Option A (6x+y−32=06x + y - 32=0) is correct because it represents the locus of the point derived from the area of the triangle formed by the given points.
38 Medium
38 of 140
Application +4 | -1
Let A(2,3,5)A(2,3,5) and C(−3,4,−2)C(-3,4,-2) be opposite vertices of a parallelogram ABCDA B C D. If the diagonal BD→=i^+2j^+3k^\overrightarrow{B D}=\hat{i}+2 \hat{j}+3 \hat{k}, then the area of the parallelogram is equal to:
View answer and explanation
Correct option: B
12474\frac{1}{2} \sqrt{474}
The correct answer is B (12474\frac{1}{2} \sqrt{474}) because it is the area of the parallelogram calculated using the cross product of its diagonals.
39 Easy
39 of 140
Application +4 | -1
If P(3,2,−4),Q(5,4,−6)P(3,2,-4), Q(5,4,-6) and R(9,8,−10)R(9,8,-10) are collinear, then RR divides PQP Q in the ratio
View answer and explanation
Correct option: B
3:23: 2 externally
Option B is correct because R divides PQ externally in the ratio found using the section formula and equating the coordinates.
40 Medium
40 of 140
Analyzing +4 | -1
Let O\mathrm{O} be origin and point P1(1,1,1)P_1(1,1,1) lie on L1=x=y=zL_1=x=y=z. Projection of OP1O P_1 on x-axis be OM1O M_1. Projection of OM1O M_1 on L1L_1 be OP2O P_2 and so on. Then the value of ∣P1M1∣+∣P2M2∣……∞\left|P_1 M_1\right|+\left|P_2 M_2\right| \ldots \ldots \infty
View answer and explanation
Correct option: B
32\frac{3}{\sqrt{2}}
Option B (32\frac{3}{\sqrt{2}}) is correct because it represents the sum of an infinite geometric series formed by the projections.
41 Medium
41 of 140
Application +4 | -1
A(3,2,0),B(5,3,2),C(−9,6,−3){A}(3,2,0), {B}(5,3,2), {C}(-9,6,-3) are three points forming a triangle. If AD{AD}, the bisector of ∠BAC\angle {BAC} meets BC{BC} in D{D}, then coordinates of D{D} are
View answer and explanation
Correct option: D
(198,5716,1716)\left(\frac{19}{8}, \frac{57}{16}, \frac{17}{16}\right)
Option D ( (19/8, 57/16, 17/16) ) is correct because it represents the coordinates of point D, which divides BC in the ratio of AB:AC, calculated using the section formula.
42 Easy
42 of 140
Application +4 | -1
The direction ratios of the diagonals of a cube which joins the origin to the opposite corner are (when the three concurrent edges of the cube are coordinate axes)
View answer and explanation
Correct option: B
1,1,11,1,1
Option B (1,1,11,1,1) is correct because the direction ratios of the diagonal of a cube joining the origin to the opposite corner are equal to the coordinates of that corner.
43 Medium
43 of 140
Application +4 | -1
What is the length of foot of perpendicular drawn from the point P(3,4,5)P(3,4,5) on YY-axis?
View answer and explanation
Correct option: B
34\sqrt{34} units
Option B (34\sqrt{34} units) is correct because it represents the distance between the point P(3,4,5) and its projection on the Y-axis, which is (0,4,0).
44 Easy
44 of 140
Application +4 | -1
If orthocentre and circumcentre of triangle are respectively (1, 1) and (3, 2), then the coordinates of its centroid are
View answer and explanation
Correct option: A
(73,53)\left( {\frac{7}{3},\frac{5}{3}} \right)
Option A, (73,53)\left( {\frac{7}{3},\frac{5}{3}} \right), is correct because the centroid divides the line segment joining the orthocenter and circumcenter in the ratio 2:1.
45 Easy
45 of 140
Understanding +4 | -1
If r{r} be position vector of any point on a sphere and a and b{b} are respectively position vectors of the extremities of a diameter, then
View answer and explanation
Correct option: D
(r−a)⋅(r−b)=0(\mathbf{r}-\mathbf{a}) \cdot(\mathbf{r}-\mathbf{b})=0
Option D ((\mathbf{r}-\mathbf{a}) ·(\mathbf{r}-\mathbf{b})=0) is correct because the angle in a semicircle is a right angle, implying the dot product of the vectors representing the sides is zero.

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46 Easy
46 of 140
Application +4 | -1
Let AA be the image of (2,−1)(2,-1) with respect to YY-axis. Without transforming the origin, coordinate axis are turned at an angle 45∘45^{\circ} in the clockwise direction. Then, the coordiates of AA in the new system are
View answer and explanation
Correct option: A
(−12,−32)\left(-\frac{1}{\sqrt{2}},-\frac{3}{\sqrt{2}}\right)
Option A (−12,−32)\left(-\frac{1}{\sqrt{2}},-\frac{3}{\sqrt{2}}\right) is correct because it represents the coordinates of the image of the point after rotation.
47 Medium
47 of 140
Application +4 | -1
The point (0, 3) is nearest to the curve x2=2y{x^2} = 2y at
View answer and explanation
Correct option: C
(2, 2)
Option C (2, 2) is correct because it represents the point on the curve x2=2y{x^2} = 2y that minimizes the distance to the point (0, 3).
48 Easy
48 of 140
Understanding +4 | -1
If a line makes α,β,γ\alpha, \beta, \gamma angles with the positive directions of the axes, then sin⁡2α+sin⁡2β+sin⁡2γ\sin ^2 \alpha+\sin ^2 \beta+\sin ^2 \gamma is equal to
View answer and explanation
Correct option: B
2
Option B (2) is correct because sin⁡2α+sin⁡2β+sin⁡2γ=2\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 2 given that α,β,γ\alpha, \beta, \gamma are the angles a line makes with the positive directions of the axes.
49 Easy
49 of 140
Understanding +4 | -1
If AA and BB are two fixed points, then the locus of a point which moves in such a way that the angle, APBAPB is a right angle is
View answer and explanation
Correct option: A
A circle
Option A is correct because the locus of a point P such that angle APB is a right angle, where A and B are fixed points, is a circle with AB as the diameter.
50 Medium
50 of 140
Application +4 | -1
The locus of a point PP which moves such that 2PA=3PB2PA = 3PB, where A(0,  0)A\left( {0,\;0} \right) and B(4,  −3)B\left( {4,\; - 3} \right) are points, is
View answer and explanation
Correct option: B
5x2+5y2−72x+54y+225=05{x^2} + 5{y^2} - 72x + 54y + 225 = 0
Option B (5x2+5y2−72x+54y+225=05{x^2} + 5{y^2} - 72x + 54y + 225 = 0) is correct because it represents the locus of point P satisfying the condition 2PA=3PB2PA = 3PB, derived by squaring both sides and simplifying the equation.
51 Medium
51 of 140
Application +4 | -1
If the coordinates of orthocentre O′O' and centroid GG of a Δ  ABC\Delta \;ABC are (0,1)\left( {0,1} \right) and (2,3)\left( {2,3} \right) respectively, then the coordinates of the circumcentre are
View answer and explanation
Correct option: D
(3,  4)\left( {3,\;4} \right)
Option D (3,  4)\left( {3,\;4} \right) is correct because the circumcentre coordinates are found using the section formula with the centroid dividing the line joining orthocentre and circumcentre in the ratio 1:2.
52 Medium
52 of 140
Application +4 | -1
The point on Y-axis, which is at a distance of 10\sqrt{10} units from the point (1,2,3)(1,2,3) is
View answer and explanation
Correct option: B
(0,2,0)(0,2,0)
Option B ((0,2,0)) is correct because the distance between (1,2,3) and (0,y,0) is (1−0)2+(2−y)2+(3−0)2=10\sqrt{{(1-0)^2 + (2-y)^2 + (3-0)^2}} = \sqrt{{10}}, which gives y=2y=2.
53 Easy
53 of 140
Understanding +4 | -1
If the vertices P,  Q,  R  P,\;Q,\;R\; of a Δ  PQR\Delta \;PQR are rational points, which of the following points of the Δ  PQR\Delta \;PQR is (are) always rational points?

(A rational point is a point both of whose coordinates are rational numbers)
View answer and explanation
Correct option: A
Centroid
Option A (Centroid) is correct because the centroid's coordinates are the average of the vertices' coordinates, and the average of rational numbers is always rational.
54 Easy
54 of 140
Application +4 | -1
A straight line which makes an angle of 60∘60^{\circ} with each of yy and zz-axis, inclines with xx-axis at an angle
View answer and explanation
Correct option: D
45∘45^{\circ}
Option D (45∘45^{\circ}) is correct because it satisfies the direction cosine relationship: cos⁡2α+cos⁡2β+cos⁡2γ=1\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1.
55 Medium
55 of 140
Application +4 | -1
The angle between the lines whose direction cosines are given by the equations 3l+m+5n=0,6nm−2nl+5lm=03 l+{m}+5 {n}=0,6 {nm}-2 {n} l+5 l {m}=0 is
View answer and explanation
Correct option: B
cos⁡−1(−16)\cos ^{-1}\left(-\frac{1}{6}\right)
Option B, cos⁡−1(−16)\cos ^{-1}\left(-\frac{1}{6}\right), is the correct answer because it is the angle calculated using the direction cosines derived from the given equations.
56 Easy
56 of 140
Application +4 | -1
The equation of the line passing through the point with position vector 2i^−3j^−5k^2 \hat{i}-3 \hat{j}-5 \hat{k} and perpendicular to the plane r⋅(6i^−3j^−5k^)+2=0{r} \cdot(6 \hat{i}-3 \hat{j}-5 \hat{k})+2=0 is
View answer and explanation
Correct option: A
r=2i^−3j^−5k^+λ(−6i^+3j^+5k^){r}=2 \hat{i}-3 \hat{j}-5 \hat{k}+\lambda(-6 \hat{i}+3 \hat{j}+5 \hat{k})
Option A (r=2i^−3j^−5k^+λ(−6i^+3j^+5k^){r}=2 \hat{i}-3 \hat{j}-5 \hat{k}+\lambda(-6 \hat{i}+3 \hat{j}+5 \hat{k})) is correct because it represents the equation of the line passing through the given point and perpendicular to the given plane.
57 Easy
57 of 140
Application +4 | -1
The equation of straight line passing through the points (a,b,c)(a, b, c) and (a−b,b−c,c−a)(a-b, b-c, c-a), is
View answer and explanation
Correct option: B
x−ab=y−bc=z−ca\frac{x-a}{b}=\frac{y-b}{c}=\frac{z-c}{a}
Option B (x−ab=y−bc=z−ca\frac{x-a}{b}=\frac{y-b}{c}=\frac{z-c}{a}) is correct because it represents the equation of a line passing through two points (a, b, c) and (a-b, b-c, c-a) in 3D space.
58 Medium
58 of 140
Application +4 | -1
The equation of the locus of a point equidistant from the points (a1,  b1)\left( {{a_1},\;{b_1}} \right) and (a2,  b2)\left( {{a_2},\;{b_2}} \right) is (a1−a2)x+(b1−b2)y+c=0\left( {{a_1} - {a_2}} \right)x + \left( {{b_1} - {b_2}} \right)y + c = 0, then the value of cc is
View answer and explanation
Correct option: C
12(a22+b22−a12−b12)\frac{1}{2}\left( {a_2^2 + b_2^2 - a_1^2 - b_1^2} \right)
Option C (12(a22+b22−a12−b12)\frac{1}{2}\left( {a_2^2 + b_2^2 - a_1^2 - b_1^2} \right)) is correct because it represents the constant term in the equation of the locus of a point equidistant from two given points.
59 Medium
59 of 140
Application +4 | -1
If A(6,3,2),B(5,1,4),C(3,−4,7),D(0,2,5)A(6,3,2), B(5,1,4), C(3,-4,7), D(0,2,5) are four points then length of projection of CDC D on ABA B is
View answer and explanation
Correct option: A
133\frac{13}{3}
Option A (133\frac{13}{3}) is correct because it is the length of the projection of vector CD onto vector AB, calculated using the dot product and magnitude.
60 Easy
60 of 140
Application +4 | -1
The coordinates of the centroid of a triangle having its circumcentre and orthocentre at (7/2,  5/2)\left( {7/2,\;5/2} \right) and (2,1)\left( {2,1} \right) respectively, are
View answer and explanation
Correct option: A
(3,  2)\left( {3,\;2} \right)
Option A (3,  2)\left( {3,\;2} \right) is correct because the centroid divides the line joining the orthocentre and circumcentre in the ratio 2:1.
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