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JEE • MATHEMATICS • DCS AND DRS

Dcs and drs practice · Easy

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Practice format and chapter quiz

These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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Showing 1-30 of 45 questions
1 Easy
1 of 45
Application +4 | -1
If the line 2−x3=3y−24λ+1=4−z\frac{2-x}{3}=\frac{3 y-2}{4 \lambda+1}=4-z makes a right angle with the line x+33μ=1−2y6=5−z7,\frac{x+3}{3 \mu}=\frac{1-2 y}{6}=\frac{5-z}{7}, then 4λ+9μ4 \lambda+9 \mu is equal to:
View answer and explanation
Correct option: D
6
The correct answer is D (6). The condition for perpendicular lines in 3D space is used to find the relationship between λ\lambda and μ\mu.
2 Easy
2 of 45
Application +4 | -1
The shortest distance between the lines x2=y2=z1\frac{x}{2}=\frac{y}{2}=\frac{z}{1} and x+2−1=y−48=z−54\frac{x+2}{-1}=\frac{y-4}{8}=\frac{z-5}{4} lies in the interval :
View answer and explanation
Correct option: B
(2,3](2,3]
Option B ((2,3]) is correct because the shortest distance between the given lines is approximately 2.68, which lies in the interval (2,3].
3 Easy
3 of 45
Application +4 | -1
A uniform rod, 4 m4 {~m} in length and weight 2 kg2 {~kg} turns freely about a point distant one metre from one end and from that end a weight of 10 kg10 {~kg} is suspended. The weight that must be placed at the other end to produce equilibrium is
View answer and explanation
Correct option: A
223 kg2 \frac{2}{3} {~kg}
Option A (223 kg2 \frac{2}{3} {~kg}) is correct because it is the weight required to balance the moments about the pivot point, considering the weight of the rod and the suspended weight.
4 Easy
4 of 45
Application +4 | -1
A line makes angles of 45∘45^{\circ} and 60∘60^{\circ} with the positive axes of XX and YY respectively. The angle made by the same line with the positive axis of ZZ, is.
View answer and explanation
Correct option: D
60∘60^{\circ} or 120∘120^{\circ}
Option D (60∘60^{\circ} or 120∘120^{\circ}) is correct because the sum of the squares of the cosines of the angles with the axes must equal 1.
5 Easy
5 of 45
Application +4 | -1
The direction cosines of a line segment AB\mathrm{AB} are −2/17-2 / \sqrt{17}, 3/17,−2/173 / \sqrt{17},-2 / \sqrt{17} If AB=17\mathrm{AB}=\sqrt{17} and the co-ordinates of A\mathrm{A} are (3,−6,10)(3,-6,10) then the co-ordinates of B\mathrm{B} are
View answer and explanation
Correct option: D
(1,−3,8)(1,-3,8)
Option D ((1,−3,8)(1,-3,8)) is correct because it is one of the two possible coordinates calculated using the given direction cosines, distance, and coordinates of point A.
6 Easy
6 of 45
Application +4 | -1
A line ABA B in three-dimensional space makes angles 45∘45^{\circ} and 120∘120^{\circ} with the positive XX-axis and the positive YY-axis, respectively. If ABA B makes an acute angle θ\theta with the positive ZZ-axis, then θ\theta equals
View answer and explanation
Correct option: C
60∘60^{\circ}
Option C (60∘60^{\circ}) is correct because it is the acute angle that the line makes with the positive Z-axis, calculated using the relationship between direction cosines.
7 Easy
7 of 45
Analyzing +4 | -1
Direction ratios of two lines are a,b,ca, b, c and 1bc,1ca,1ab\frac{1}{b c}, \frac{1}{c a}, \frac{1}{a b}, The lines are
View answer and explanation
Correct option: B
Parallel
Option B (Parallel) is correct because the ratios of the corresponding direction ratios are equal, indicating parallel lines.
8 Easy
8 of 45
Analyzing +4 | -1
The line x+65=y+103=z+148\frac{x+6}{5}=\frac{y+10}{3}=\frac{z+14}{8} is the hypotenuse of an isosceles right angled triangle whose opposite vertex is (7,2,4)(7,2,4). Then which of the following is not the side of the triangle ?
View answer and explanation
Correct option: B
x−73=y−26=z−42\frac{x-7}{3}=\frac{y-2}{6}=\frac{z-4}{2}
Option B (x−73=y−26=z−42\frac{x-7}{3}=\frac{y-2}{6}=\frac{z-4}{2}) is correct because this line is not a side of the isosceles right-angled triangle.
9 Easy
9 of 45
Application +4 | -1
There is a point P(a,a,a){P}({a}, {a}, {a}) on the line passing through the origin and equally inclined with axes. The equation of the plane perpendicular to OPO P and passing through PP cuts the intercepts on axes, the sum of whose reciprocals is-
View answer and explanation
Correct option: D
1/a1/a
Option D (1/a1/a) is correct because the sum of the reciprocals of the intercepts is equal to the reciprocal of the distance from the origin to the plane.
10 Easy
10 of 45
Application +4 | -1
The projections of the segment PQP Q on the co-ordinate axes are −9,12,−8-9,12,-8 respectively. The direction woiues of the liue PQ\mathrm{PQ} as
View answer and explanation
Correct option: D
⟨−917,1217,−817⟩\left\langle-\frac{9}{17}, \frac{12}{17}, \frac{-8}{17}\right\rangle
The correct answer is ⟨−917,1217,−817⟩\left\langle-\frac{9}{17}, \frac{12}{17}, \frac{-8}{17}\right\rangle because direction cosines are the direction ratios divided by the magnitude of the vector.
11 Easy
11 of 45
Application +4 | -1
A line passes through the points (6,−7,−1)(6,-7,-1) and (2,−3,1)(2,-3,1). The direction cosines of the line so directed that the angle made by it with the positive direction of xx-axis is acute, are
View answer and explanation
Correct option: A
23,−23,−13\frac{2}{3},-\frac{2}{3},-\frac{1}{3}
Option A is correct because the direction cosines are calculated from the direction ratios, ensuring the x-axis angle is acute.
12 Easy
12 of 45
Application +4 | -1
A line makes the same angle θ\theta with XX-axis and ZZ-axis. If the angle β\beta, which it makes with YY-axis, is such that sin⁡2β=3sin⁡2θ\sin ^2 \beta=3 \sin ^2 \theta, then the value of cos⁡2θ\cos ^2 \theta is
View answer and explanation
Correct option: C
35\frac{3}{5}
Option C (35\frac{3}{5}) is correct because it satisfies the direction cosine relationship and the given condition sin⁡2β=3sin⁡2θ\sin ^2 \beta=3 \sin ^2 \theta.
13 Easy
13 of 45
Application +4 | -1
The angle between the lines whose direction cosines satisfy the equations l+m+n=0l+m+n=0 and l2=m2+n2l^2=m^2+n^2 is
View answer and explanation
Correct option: C
π3\frac{\pi}{3}
Option C (π3\frac{\pi}{3}) is correct because it is the angle between the two lines whose direction cosines satisfy the given equations, found by solving for the direction cosines and using the dot product formula.
14 Easy
14 of 45
Application +4 | -1
The projection of line joining (3,4,5)(3,4,5) and (4,6,3)(4,6,3) on the line joining (−1,2,4)(-1,2,4) and (1,0,5)(1,0,5) is -
View answer and explanation
Correct option: A
43\frac{4}{3}
Option A (43\frac{4}{3}) is correct because it represents the magnitude of the projection of the line segment AB onto the line PQ, calculated using the dot product of their direction ratios.
15 Easy
15 of 45
Application +4 | -1
A line makes an angle θ\theta both with XX and YY-axes. A possible range of θ\theta is
View answer and explanation
Correct option: C
[π4,π2]\left[\frac{\pi}{4}, \frac{\pi}{2}\right]
Option C ([π4,π2\frac{\pi}{4}, \frac{\pi}{2}]) is correct because it represents the possible range of the angle θ\theta that a line makes with both the X and Y axes, derived from the direction cosines relationship.

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16 Easy
16 of 45
Application +4 | -1
The direction cosines of the line joining the points ( 4 , 3,−5)3,-5) and (−2,1,−8)(-2,1,-8) are
View answer and explanation
Correct option: A
(67,27,37)\left(\frac{6}{7}, \frac{2}{7}, \frac{3}{7}\right)
Option A ((67,27,37)\left(\frac{6}{7}, \frac{2}{7}, \frac{3}{7}\right)) is correct because the direction cosines are found by dividing the difference in coordinates by the distance between the points.
17 Easy
17 of 45
Analyzing +4 | -1
The direction cosines l,m,nl, m, n of two lines are connected by the relations l+m+n=0,lm=0l+m+n=0, l m=0, then the angle between them is:
View answer and explanation
Correct option: A
π/3\pi / 3
Option A (π/3\pi / 3) is correct because solving the given relations for direction cosines leads to an angle of 60 degrees between the lines.
18 Easy
18 of 45
Application +4 | -1
The equation of the line through the point (0, 1, 2) and perpendicular to the line x−12=y+13=z−1−2\frac{{x - 1}}{2} = \frac{{y + 1}}{3} = \frac{{z - 1}}{{ - 2}} is
View answer and explanation
Correct option: D
x−3=y−14=z−23\frac{x}{{ - 3}} = \frac{{y - 1}}{4} = \frac{{z - 2}}{3}
Option D is correct. The equation of the line is found by using the perpendicularity condition and the given point.
19 Easy
19 of 45
Application +4 | -1
If the directions cosines of a line are k,k,kk, k, k, then
View answer and explanation
Correct option: D
k=13k=\frac{1}{\sqrt{3}} or −13-\frac{1}{\sqrt{3}}
Option D (k=13k=\frac{1}{\sqrt{3}} or −13-\frac{1}{\sqrt{3}}) is correct because the sum of the squares of direction cosines must equal 1.
20 Easy
20 of 45
Analyzing +4 | -1
If the direction ratios of two lines are given by a+b+c=0a+b+c=0 and 2ab+2ac−bc=02 a b+2 a c-b c=0, then the angle between the lines is
View answer and explanation
Correct option: B
2π3\frac{2 \pi}{3}
The correct answer is B (2π3\frac{2 \pi}{3}) because it is the angle between the two lines whose direction ratios satisfy the given conditions.
21 Easy
21 of 45
Application +4 | -1
If PP is a point in space such that OPO P is inclined to OXO X at 45∘45^{\circ} and OYO Y to 60∘60^{\circ} then OPO P is inclined to OZO Z at
View answer and explanation
Correct option: B
60∘60^{\circ} and 120∘120^{\circ}
Option B is correct because the angle to OZ is found using the relationship between direction cosines and the given angles to OX and OY.
22 Easy
22 of 45
Understanding +4 | -1
Two lines x−xil1=y−yim1=z−zin1(i=1,2)\frac{x-x_i}{l_1}=\frac{y-y_i}{m_1}=\frac{z-z_i}{n_1}(i=1,2) are perpendicular to each other if their direction ratios satisfy
View answer and explanation
Correct option: B
l1l2+m1m2+n1r2=0l_1 l_2+m_1 m_2+n_1 r_2=0
Option B (l1l2+m1m2+n1r2=0l_1 l_2+m_1 m_2+n_1 r_2=0) is correct because it represents the condition for two lines to be perpendicular in 3D space.
23 Easy
23 of 45
Application +4 | -1
If (12,13,n)\left(\frac{1}{2}, \frac{1}{3}, n\right) are the direction cosines of a line, then the value of nn is
View answer and explanation
Correct option: A
236\frac{\sqrt{23}}{6}
Option A (236\frac{\sqrt{23}}{6}) is correct because the sum of the squares of direction cosines must equal 1.
24 Easy
24 of 45
Application +4 | -1
The angle between the lines ABA B and CDC D, where A=(0,0,0),B=(1,1,1),C=(−1,−1,−1)A=(0,0,0), B=(1,1,1), C=(-1,-1,-1) and D=(0,1,0)D=(0,1,0) is given by
View answer and explanation
Correct option: B
cos⁡θ=432\cos \theta=\frac{4}{3 \sqrt{2}}
Option B (cos⁡θ=432\cos \theta=\frac{4}{3 \sqrt{2}}) is correct because it is calculated by finding the direction ratios of the lines and using the formula for the angle between two lines.
25 Easy
25 of 45
Analyzing +4 | -1
The angle between the lines whose direction cosines are given by the equations l2+m2−n2=0,l+m+n=0l^2+m^2-n^2=0, l+m+n=0 is
View answer and explanation
Correct option: C
π3\frac{\pi}{3}
The angle between the lines is found using the direction cosines and the formula for the angle between two lines, resulting in π3\frac{\pi}{3}.
26 Easy
26 of 45
Analyzing +4 | -1
If the direction cosines of two lines are such that l+m+n=0,l2+m2−n2=0l+m+n=0, l^2+m^2-n^2=0, then the angle between them is
View answer and explanation
Correct option: B
π3\frac{\pi}{3}
Option B (π3\frac{\pi}{3}) is correct because solving the given equations for direction cosines and then finding the angle between the lines results in π3\frac{\pi}{3}.
27 Easy
27 of 45
Application +4 | -1
If a line makes an angle of π4\frac{\pi}{4} with the positive directions of each of XX-axis and YY-axis, then the angle that the line makes with the positive direction of the ZZ-axis is
View answer and explanation
Correct option: D
π/2\pi / 2
Option D (π/2\pi / 2) is correct because it is the angle that the line makes with the positive direction of the Z-axis, calculated using the relationship between direction cosines.
28 Easy
28 of 45
Understanding +4 | -1
Let Axy,Ayz,AzxA_{x y}, A_{y z}, A_{z x} be the area of the projection of a plane area AA on the xy,yz,zxx y, y z, z x plane respectively Then A2=A^2=
View answer and explanation
Correct option: A
Axy2+Ayz2+Azx2A_{x y}^2+A_{y z}^2+A_{z x}^2
The correct answer is A (Axy2+Ayz2+Azx2A_{xy}^2+A_{yz}^2+A_{zx}^2) because it represents the relationship between the area of a plane and its projections onto the coordinate planes, using direction cosines.
29 Easy
29 of 45
Application +4 | -1
The angle between a line with direction ratios proportional to 2,2,12,2,1 and a line joining (3,1,4)(3,1,4) to (7,2,12)(7,2,12), is
View answer and explanation
Correct option: A
cos⁡−1(23)\cos ^{-1}\left(\frac{2}{3}\right)
Option A (cos⁡−1(23)\cos ^{-1}\left(\frac{2}{3}\right)) is correct because it's the angle calculated using the dot product of the direction vectors and their magnitudes.
30 Easy
30 of 45
Analyzing +4 | -1
If two straight lines whose direction cosines are given by the relations l+m−n=0, 3l2+m2+cnl=0l + m - n = 0,\,3{l^2} + {m^2} + cnl = 0 are parallel, then the positive value of cc is:
View answer and explanation
Correct option: A
6
Option A (6) is correct because the condition for parallel lines implies that the discriminant of the quadratic equation in l/m must be zero.
Revision notes and key takeaways

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Key Takeaways

  • ✓ This set contains 45 single-correct-answer MCQs on DCS AND DRS.
  • ✓ Try each question before opening its answer. Read the explanation where one is available.
  • ✓ Sign in to save submitted answers. Revisit incorrect answers alongside your textbook or class notes.
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