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JEE • MATHEMATICS • DCS AND DRS

Dcs and drs practice · Mixed difficulty

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1 Easy
1 of 61
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 61
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 61
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 61
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 61
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 61
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 61
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 Medium
8 of 61
Application +4 | -1
The reflection of the point P(1,0,0)P(1,0,0) in the line x−12=y+1−3=z+108\frac{x-1}{2}=\frac{y+1}{-3}=\frac{z+10}{8} is
View answer and explanation
Correct option: B
(5,−8,−4)(5,-8,-4)
Option B ((5,−8,−4)(5,-8,-4)) is correct because it represents the reflection of the point P in the given line, found by determining the foot of the perpendicular and then extending to the reflection point.
9 Easy
9 of 61
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.
10 Easy
10 of 61
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.
11 Medium
11 of 61
Application +4 | -1
Let d{d} be the distance of the point of intersection of the lines x+63=y2=z+11\frac{x+6}{3}=\frac{y}{2}=\frac{z+1}{1} and x−74=y−93=z−42\frac{x-7}{4}=\frac{y-9}{3}=\frac{z-4}{2} from the point (7,8,9).(7,8,9). Then d2+6d^2+6 is equal to:
View answer and explanation
Correct option: C
75
The correct answer is C (75) because it's the value of d2 + 6, where d is the distance between the intersection point of two lines and the point (7,8,9).
12 Easy
12 of 61
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.
13 Medium
13 of 61
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 k3\frac{k}{3} then 'kk' equals.
View answer and explanation
Correct option: A
4
Option A (4) is correct because the projection is calculated using the dot product of the direction vectors divided by the magnitude of the line on which the projection is being found, resulting in 4/3, thus k=4.
14 Medium
14 of 61
Application +4 | -1
A line passing through A(1,2,3){A}(1,2,3) and having direction ratios (3,4,5)(3,4,5) meets a plane x+2y−3z=5x+2 y-3 z=5 at BB, then distance ABA B is equal to-
View answer and explanation
Correct option: D
None of these
Option D (None of these) is correct because the distance AB calculated using the parametric form of the line and the plane equation does not match any of the given options.
15 Medium
15 of 61
Application +4 | -1
If the direction cosines of a straight line are <k,k,k>< k, k, k >, then
View answer and explanation
Correct option: D
k=13k=\frac{1}{\sqrt{3}} or −13-\frac{1}{\sqrt{3}}
Option D (30°) is correct. Since the direction ratios of the line and the normal to the plane are proportional, the line is perpendicular to the plane, making the angle between them 90°. Therefore, the angle between the line and the plane is 90-60 = 30°.

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16 Medium
16 of 61
Analyzing +4 | -1
Let PP be the plane containing the straight line x−39=y+4−1=z−7−5\frac{x-3}{9}=\frac{y+4}{-1}=\frac{z-7}{-5} and perpendicular to the plane containing the straight lines x2=y3=z5\frac{x}{2}=\frac{y}{3}=\frac{z}{5} and x3=y7=z8\frac{x}{3}=\frac{y}{7}=\frac{z}{8}. If dd is the distance PP from the point (2,−5,11)(2,-5,11), then d2d^2 is equal to :
View answer and explanation
Correct option: C
323\frac{32}{3}
Option C is correct. The equation of the plane is found using the cross product of the direction vectors and then the distance formula is applied.
17 Medium
17 of 61
Application +4 | -1
Angle between the rays with d.r.'s 4,−3,5 & 3,4,54,-3,5 \,\&\, 3, 4,5 is-
View answer and explanation
Correct option: C
cos⁡−1(12)\cos ^{-1}\left(\frac{1}{2}\right)
Option C (cos⁡−1(12)\cos ^{-1}(\frac{1}{2})) is correct because it's the angle between the two rays, calculated using the dot product formula.
18 Medium
18 of 61
Application +4 | -1
If the d.r's of the two perpendicular lines are (2,−3,5)(2,-3,5) and (λ,1+λ,2+λ)(\lambda, 1+\lambda, 2+\lambda) then λ=\lambda=
View answer and explanation
Correct option: C
−74-\frac{7}{4}
Option C (-7/4) is correct because for two perpendicular lines, the dot product of their direction ratios is zero. Solving the equation 2λ - 3(1+λ) + 5(2+λ) = 0 gives λ = -7/4.
19 Easy
19 of 61
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.
20 Medium
20 of 61
Analyzing +4 | -1
Let L1=x−31=y−22=z−13L_{1}=\frac{x-3}{1}=\frac{y-2}{2}=\frac{z-1}{3} and L2=x−11=y−22=z−33L_{2}=\frac{x-1}{1}=\frac{y-2}{2}=\frac{z-3}{3} and direction ratios of line L3L_{3} are <1,−1,3>.P<1,-1,3>. P and QQ are points of intersection of L1L_{1} and L3L_{3} and L2&L3L_{2} \& L_{3}, respectively. Then, distance between PP and QQ is:
View answer and explanation
Correct option: C
4311\frac{4}{3} \sqrt{11}
Option C (4311\frac{4}{3} \sqrt{11}) is correct because it represents the distance between points P and Q, calculated using the given line equations and direction ratios.
21 Easy
21 of 61
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.
22 Easy
22 of 61
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.
23 Easy
23 of 61
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.
24 Easy
24 of 61
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.
25 Easy
25 of 61
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.
26 Easy
26 of 61
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.
27 Easy
27 of 61
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.
28 Easy
28 of 61
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.
29 Easy
29 of 61
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.
30 Medium
30 of 61
Application +4 | -1
A vector r\mathbf{r} is equally inclined with the co-ordinate axes. If the tip of r\mathbf{r} is in the positive octant and ∣r∣=6|\mathbf{r}|=6, then r\mathbf{r} is
View answer and explanation
Correct option: D
23(i+j+k)2 \sqrt{3}(\mathbf{i}+\mathbf{j}+\mathbf{k})
Option D, 23(i+j+k)2 \sqrt{3}(\mathbf{i}+\mathbf{j}+\mathbf{k}), is correct because it represents a vector with magnitude 6, equally inclined to the coordinate axes, and lying in the positive octant.
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Key Takeaways

  • ✓ This set contains 61 single-correct-answer MCQs on DCS AND DRS.
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