3D Geometry

153 questions · Mathematics · JEE Main

3D Geometry

153 questions · Mathematics · JEE Main

Let the vertices Q and R of the triangle PQR lie on the line $\frac{x + 3}{5} = \frac{y - 1}{2} = \frac{z + 4}{3}, QR = 5$ and the coordinates of the point $P$ be $(0, 2, 3)$ . If the area of the triangle $P QR$ is $\frac{m}{n}$ then :2025 · 2 Apr · Shift 1 · Q27 · MCQ
Let $A B C D$ be a tetrahedron such that the edges $A B, A C$ and $A D$ are mutually perpendicular. Let the areas of the triangles $ABC, ACD$ and ADB be 5,6 and 7 square units respectively. Then the area (in square units)…2025 · 2 Apr · Shift 1 · Q33 · MCQ
If the image of the point $P (1, 0, 3)$ in the line joining the points $A (4, 7, 1)$ and $B (3, 5, 3)$ is $Q (α, β, γ)$ , then $α + β + γ$ is equal to :2025 · 2 Apr · Shift 2 · Q33 · MCQ
The line $L_{1}$ is parallel to the vector $a = - 3 \hat{i} + 2 \hat{j} + 4 \hat{k}$ and passes through the point $(7, 6, 2)$ and the line $L_{2}$ is parallel to the vector $b = 2 \hat{i} + \hat{j} + 3 \hat{k}$ …2025 · 2 Apr · Shift 2 · Q38 · MCQ
Let a line passing through the point $(4, 1, 0)$ intersect the line $L_{1} : \frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}$ at the point $A (α, β, γ)$ and the line $L_{2} : x - 6 = y = - z + 4$ at the point $B (a, b, c)$ . Then $1 α a 0 β b 1 γ c$ …2025 · 3 Apr · Shift 1 · Q35 · MCQ
Line $L_{1}$ passes through the point $(1, 2, 3)$ and is parallel to $z$ -axis. Line $L_{2}$ passes through the point $(λ, 5, 6)$ and is parallel to $y$ -axis. Let for $λ = λ_{1}, λ_{2}, λ_{2} < λ_{1}$ , the shortest…2025 · 3 Apr · Shift 1 · Q38 · MCQ
Each of the angles $β$ and $γ$ that a given line makes with the positive $y$ - and $z$ -axes, respectively, is half of the angle that this line makes with the positive $x$ -axes. Then the sum of all possible values of the angle $β$ …2025 · 3 Apr · Shift 2 · Q28 · MCQ
The distance of the point $(7, 10, 11)$ from the line $\frac{x - 4}{1} = \frac{y - 4}{0} = \frac{z - 2}{3}$ along the line $\frac{x - 9}{2} = \frac{y - 13}{3} = \frac{z - 17}{6}$ is2025 · 3 Apr · Shift 2 · Q39 · MCQ
Let the shortest distance between the lines $\frac{x - 3}{3} = \frac{y - α}{- 1} = \frac{z - 3}{1}$ and $\frac{x + 3}{- 3} = \frac{y + 7}{2} = \frac{z - β}{4}$ be $330$ . Then the positive value of $5 α + β$ is2025 · 4 Apr · Shift 1 · Q27 · MCQ
Let $A$ and $B$ be two distinct points on the line $L : \frac{x - 6}{3} = \frac{y - 7}{2} = \frac{z - 7}{- 2}$ . Both $A$ and $B$ are at a distance $217$ from the foot of perpendicular drawn from the point $(1, 2, 3)$ on the line $L$ . If $O$ is…2025 · 4 Apr · Shift 1 · Q40 · MCQ
Let A be the point of intersection of the lines $L_{1} : \frac{x - 7}{1} = \frac{y - 5}{0} = \frac{z - 3}{- 1}$ and $L_{2} : \frac{x - 1}{3} = \frac{y + 3}{4} = \frac{z + 7}{5}$ . Let B and C be the points on the lines $L_{1}$ and $L_{2}$ …2025 · 4 Apr · Shift 2 · Q39 · MCQ
Let the values of p , for which the shortest distance between the lines $\frac{x + 1}{3} = \frac{y}{4} = \frac{z}{5}$ and $r = (p \hat{i} + 2 \hat{j} + \hat{k}) + λ (2 \hat{i} + 3 \hat{j} + 4 \hat{k})$ is $\frac{1}{6}$ …2025 · 4 Apr · Shift 2 · Q44 · MCQ
Let the line L pass through $(1, 1, 1)$ and intersect the lines $\frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 1}{4}$ and $\frac{x - 3}{1} = \frac{y - 4}{2} = \frac{z}{1}$ . Then, which of the following points lies on the line $L$ ?2025 · 7 Apr · Shift 1 · Q40 · MCQ
If the shortest distance between the lines $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}$ and $\frac{x}{1} = \frac{y}{α} = \frac{z - 5}{1}$ is $\frac{5}{6}$ , then the sum of all possible values of $α$ is2025 · 7 Apr · Shift 1 · Q42 · MCQ
If the equation of the line passing through the point $(0, - \frac{1}{2}, 0)$ and perpendicular to the lines $r = λ (\hat{i} + a \hat{j} + b \hat{k})$ and $r = (\hat{i} - \hat{j} - 6 \hat{k}) + μ (- b \hat{i} + a \hat{j} + 5 \hat{k})$ …2025 · 7 Apr · Shift 2 · Q35 · MCQ
Consider the lines L1: x - 1 = y - 2 = z and L2: x - 2 = y = z - 1. Let the feet of the perpendiculars from the point P(5, 1, -3) on the lines L1 and L2 be Q and R respectively. If the area of the triangle PQR is A, then 4A2 is equal to :2025 · 7 Apr · Shift 2 · Q39 · MCQ
Let the values of $λ$ for which the shortest distance between the lines $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}$ and $\frac{x - λ}{3} = \frac{y - 4}{4} = \frac{z - 5}{5}$ is $\frac{1}{6}$ be $λ_{1}$ and $λ_{2}$ .…2025 · 8 Apr · Shift 2 · Q26 · MCQ
Let the area of the triangle formed by the lines $x + 2 = y - 1 = z, \frac{x - 3}{5} = \frac{y}{- 1} = \frac{z - 1}{1}$ and $\frac{x}{- 3} = \frac{y - 3}{3} = \frac{z - 2}{1}$ be $A$ . Then $A^{2}$ is equal to .2025 · 8 Apr · Shift 2 · Q49 · Numerical
Let $L_{1} : \frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}$ and $L_{2} : \frac{x - 2}{3} = \frac{y - 4}{4} = \frac{z - 5}{5}$ be two lines. Then which of the following points lies on the line of the shortest distance between $L_{1}$ …2025 · 22 Jan · Shift 1 · Q28 · MCQ
Let $L_{1} : \frac{x - 1}{3} = \frac{y - 1}{- 1} = \frac{z + 1}{0}$ and $L_{2} : \frac{x - 2}{2} = \frac{y}{0} = \frac{z + 4}{α}, α \in R$ , be two lines, which intersect at the point $B$ . If $P$ is the foot of perpendicular…2025 · 22 Jan · Shift 1 · Q48 · Numerical
Let a line pass through two distinct points $P (- 2, - 1, 3)$ and $Q$ , and be parallel to the vector $3 \hat{i} + 2 \hat{j} + 2 \hat{k}$ . If the distance of the point Q from the point $R (1, 3, 3)$ is 5 , then the square of the area of $△ P QR$ …2025 · 22 Jan · Shift 2 · Q27 · MCQ
The perpendicular distance, of the line $\frac{x - 1}{2} = \frac{y + 2}{- 1} = \frac{z + 3}{2}$ from the point $P (2, - 10, 1)$ , is :2025 · 22 Jan · Shift 2 · Q31 · MCQ
Let P be the foot of the perpendicular from the point $Q (10, - 3, - 1)$ on the line $\frac{x - 3}{7} = \frac{y - 2}{- 1} = \frac{z + 1}{- 2}$ . Then the area of the right angled triangle $P QR$ , where $R$ is the point $(3, - 2, 1)$ , is2025 · 23 Jan · Shift 1 · Q27 · MCQ
If the square of the shortest distance between the lines $\frac{x - 2}{1} = \frac{y - 1}{2} = \frac{z + 3}{- 3}$ and $\frac{x + 1}{2} = \frac{y + 3}{4} = \frac{z + 5}{- 5}$ is $\frac{m}{n}$ , where $m$ , $n$ are coprime numbers, then $m + n$ is equal to :2025 · 23 Jan · Shift 2 · Q31 · MCQ
The distance of the line $\frac{x - 2}{2} = \frac{y - 6}{3} = \frac{z - 3}{4}$ from the point $(1, 4, 0)$ along the line $\frac{x}{1} = \frac{y - 2}{2} = \frac{z + 3}{3}$ is :2025 · 23 Jan · Shift 2 · Q36 · MCQ
Let in a $△ A B C$ , the length of the side $A C$ be 6 , the vertex $B$ be $(1, 2, 3)$ and the vertices $A, C$ lie on the line $\frac{x - 6}{3} = \frac{y - 7}{2} = \frac{z - 7}{- 2}$ . Then the area (in sq. units) of $△ A B C$ is:2025 · 24 Jan · Shift 1 · Q35 · MCQ
Let the line passing through the points $(- 1, 2, 1)$ and parallel to the line $\frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z}{4}$ intersect the line $\frac{x + 2}{3} = \frac{y - 3}{2} = \frac{z - 4}{1}$ at the point $P$ . Then the distance of $P$ from the point $Q (4, - 5, 1)$ …2025 · 24 Jan · Shift 1 · Q36 · MCQ
Let P be the image of the point $Q (7, - 2, 5)$ in the line $L : \frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z}{4}$ and $R (5, p, q)$ be a point on $L$ . Then the square of the area of $△ P QR$ is …2025 · 24 Jan · Shift 2 · Q46 · Numerical
Let $A (x, y, z)$ be a point in $x y$ -plane, which is equidistant from three points $(0, 3, 2), (2, 0, 3)$ and $(0, 0, 1)$ . Let $B = (1, 4, - 1)$ and $C = (2, 0, - 2)$ . Then among the statements (S1) : $△ ABC$ is…2025 · 28 Jan · Shift 1 · Q26 · MCQ
If the image of the point $(4, 4, 3)$ in the line $\frac{x - 1}{2} = \frac{y - 2}{1} = \frac{z - 1}{3}$ is $(α, β, γ)$ , then $α + β + γ$ is equal to2025 · 28 Jan · Shift 1 · Q38 · MCQ
The square of the distance of the point $(\frac{15}{7}, \frac{32}{7}, 7)$ from the line $\frac{x + 1}{3} = \frac{y + 3}{5} = \frac{z + 5}{7}$ in the direction of the vector $\hat{i} + 4 \hat{j} + 7 \hat{k}$ is:2025 · 28 Jan · Shift 2 · Q30 · MCQ
Let $L_{1} : \frac{x - 1}{1} = \frac{y - 2}{- 1} = \frac{z - 1}{2}$ and $L_{2} : \frac{x + 1}{- 1} = \frac{y - 2}{2} = \frac{z}{1}$ be two lines. Let $L_{3}$ be a line passing through the point $(α, β, γ)$ and be perpendicular to both $L_{1}$ …2025 · 29 Jan · Shift 1 · Q26 · MCQ
Let a straight line $L$ pass through the point $P (2, - 1, 3)$ and be perpendicular to the lines $\frac{x - 1}{2} = \frac{y + 1}{1} = \frac{z - 3}{- 2}$ and $\frac{x - 3}{1} = \frac{y - 2}{3} = \frac{z + 2}{4}$ . If the line $L$ intersects the…2025 · 29 Jan · Shift 2 · Q28 · MCQ
Let P be the foot of the perpendicular from the point $(1, 2, 2)$ on the line $L : \frac{x - 1}{1} = \frac{y + 1}{- 1} = \frac{z - 2}{2}$ . Let the line $r = (- \hat{i} + \hat{j} - 2 \hat{k}) + λ (\hat{i} - \hat{j} + \hat{k}), λ \in R$ …2025 · 29 Jan · Shift 2 · Q43 · MCQ
If the shortest distance between the lines $\frac{x - λ}{- 2} = \frac{y - 2}{1} = \frac{z - 1}{1}$ and $\frac{x - 3}{1} = \frac{y - 1}{- 2} = \frac{z - 2}{1}$ is 1 , then the sum of all possible values of $λ$ is :2024 · 1 Feb · Shift 1 · Q50 · MCQ
Let the line of the shortest distance between the lines $L_{1} : r = (\hat{i} + 2 \hat{j} + 3 \hat{k}) + λ (\hat{i} - \hat{j} + \hat{k}) and L_{2} : r = (4 \hat{i} + 5 \hat{j} + 6 \hat{k}) + μ (\hat{i} + \hat{j} - \hat{k})$ …2024 · 1 Feb · Shift 1 · Q59 · Numerical
Consider a $△ A B C$ where $A (1, 3, 2), B (- 2, 8, 0)$ and $C (3, 6, 7)$ . If the angle bisector of $∠ B A C$ meets the line $B C$ at $D$ , then the length of the projection of the vector $A D$ on the vector $A C$ …2024 · 1 Feb · Shift 2 · Q33 · MCQ
Let $P$ and $Q$ be the points on the line $\frac{x + 3}{8} = \frac{y - 4}{2} = \frac{z + 1}{2}$ which are at a distance of 6 units from the point $R (1, 2, 3)$ . If the centroid of the triangle PQR is $(\alpha, \beta,…2024 · 1 Feb · Shift 2 · Q45 · MCQ
If the mirror image of the point $P (3, 4, 9)$ in the line $\frac{x - 1}{3} = \frac{y + 1}{2} = \frac{z - 2}{1}$ is $(α, β, γ)$ , then 14 $(α + β + γ)$ is :2024 · 1 Feb · Shift 2 · Q47 · MCQ
Let the point, on the line passing through the points $P (1, - 2, 3)$ and $Q (5, - 4, 7)$ , farther from the origin and at a distance of 9 units from the point $P$ , be $(α, β, γ)$ . Then $α^{2} + β^{2} + γ^{2}$ is equal to :2024 · 4 Apr · Shift 1 · Q40 · MCQ
Let $P$ be the point of intersection of the lines $\frac{x - 2}{1} = \frac{y - 4}{5} = \frac{z - 2}{1}$ and $\frac{x - 3}{2} = \frac{y - 2}{3} = \frac{z - 3}{2}$ . Then, the shortest distance of $P$ from the line $4 x = 2 y = z$ is2024 · 4 Apr · Shift 2 · Q35 · MCQ
Consider a line $L$ passing through the points $P (1, 2, 1)$ and $Q (2, 1, - 1)$ . If the mirror image of the point $A (2, 2, 2)$ in the line $L$ is $(α, β, γ)$ , then $α + β + 6 γ$ …2024 · 4 Apr · Shift 2 · Q59 · Numerical
If the line $\frac{2 - x}{3} = \frac{3 y - 2}{4 λ + 1} = 4 - z$ makes a right angle with the line $\frac{x + 3}{3 μ} = \frac{1 - 2 y}{6} = \frac{5 - z}{7}$ , then $4 λ + 9 μ$ is equal to :2024 · 5 Apr · Shift 1 · Q34 · MCQ
Let $d$ be the distance of the point of intersection of the lines $\frac{x + 6}{3} = \frac{y}{2} = \frac{z + 1}{1}$ and $\frac{x - 7}{4} = \frac{y - 9}{3} = \frac{z - 4}{2}$ from the point $(7, 8, 9)$ . Then $d^{2} + 6$ is equal to :2024 · 5 Apr · Shift 1 · Q43 · MCQ
Let $(α, β, γ)$ be the image of the point $(8, 5, 7)$ in the line $\frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 2}{5}$ . Then $α + β + γ$ is equal to :2024 · 5 Apr · Shift 2 · Q34 · MCQ
Let the point $(- 1, α, β)$ lie on the line of the shortest distance between the lines $\frac{x + 2}{- 3} = \frac{y - 2}{4} = \frac{z - 5}{2}$ and $\frac{x + 2}{- 1} = \frac{y + 6}{2} = \frac{z - 1}{0}$ . Then $(α - β)^{2}$ is equal to …2024 · 5 Apr · Shift 2 · Q54 · Numerical
If $A (3, 1, - 1), B (\frac{5}{3}, \frac{7}{3}, \frac{1}{3}), C (2, 2, 1)$ and $D (\frac{10}{3}, \frac{2}{3}, \frac{- 1}{3})$ are the vertices of a quadrilateral $A B C D$ , then its area is2024 · 6 Apr · Shift 1 · Q35 · MCQ
The shortest distance between the lines $\frac{x - 3}{2} = \frac{y + 15}{- 7} = \frac{z - 9}{5}$ and $\frac{x + 1}{2} = \frac{y - 1}{1} = \frac{z - 9}{- 3}$ is2024 · 6 Apr · Shift 1 · Q39 · MCQ
Let $P$ be the point $(10, - 2, - 1)$ and $Q$ be the foot of the perpendicular drawn from the point $R (1, 7, 6)$ on the line passing through the points $(2, - 5, 11)$ and $(- 6, 7, - 5)$ . Then the length of the line segment $P Q$ is equal to …2024 · 6 Apr · Shift 1 · Q55 · Numerical
Let $P (α, β, γ)$ be the image of the point $Q (3, - 3, 1)$ in the line $\frac{x - 0}{1} = \frac{y - 3}{1} = \frac{z - 1}{- 1}$ and $R$ be the point $(2, 5, - 1)$ . If the area of the triangle $PQR$ is $λ$ …2024 · 6 Apr · Shift 2 · Q40 · MCQ
If the shortest distance between the lines $\frac{x - λ}{3} = \frac{y - 2}{- 1} = \frac{z - 1}{1}$ and $\frac{x + 2}{- 3} = \frac{y + 5}{2} = \frac{z - 4}{4}$ is $\frac{44}{30}$ , then the largest possible value of $∣ λ ∣$ is equal to …2024 · 6 Apr · Shift 2 · Q54 · Numerical
Let $P (x, y, z)$ be a point in the first octant, whose projection in the $x y$ -plane is the point $Q$ . Let $O P = γ$ ; the angle between $O Q$ and the positive $x$ -axis be $θ$ ; and the angle between $O P$ and the positive $z$ -axis…2024 · 8 Apr · Shift 1 · Q42 · MCQ
If the shortest distance between the lines $L_{1} : r = (2 + λ) \hat{i} + (1 - 3 λ) \hat{j} + (3 + 4 λ) \hat{k}, L_{2} : r = 2 (1 + μ) \hat{i} + 3 (1 + μ) \hat{j} + (5 + μ) \hat{k}, λ \in R μ \in R$ …2024 · 8 Apr · Shift 1 · Q49 · MCQ
If the shortest distance between the lines $\frac{x - λ}{2} = \frac{y - 4}{3} = \frac{z - 3}{4}$ and $\frac{x - 2}{4} = \frac{y - 4}{6} = \frac{z - 7}{8}$ is $\frac{13}{29}$ , then a value of $λ$ is :2024 · 8 Apr · Shift 2 · Q32 · MCQ
Let $P (α, β, γ)$ be the image of the point $Q (1, 6, 4)$ in the line $\frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}$ . Then $2 α + β + γ$ is equal to 2024 · 8 Apr · Shift 2 · Q55 · Numerical
The shortest distance between the lines $\frac{x - 3}{4} = \frac{y + 7}{- 11} = \frac{z - 1}{5}$ and $\frac{x - 5}{3} = \frac{y - 9}{- 6} = \frac{z + 2}{1}$ is:2024 · 9 Apr · Shift 1 · Q44 · MCQ
Let the line $L$ intersect the lines $x - 2 = - y = z - 1, 2 (x + 1) = 2 (y - 1) = z + 1$ and be parallel to the line $\frac{x - 2}{3} = \frac{y - 1}{1} = \frac{z - 2}{2}$ . Then which of the following points lies on $L$ ?2024 · 9 Apr · Shift 1 · Q49 · MCQ
Consider the line $L$ passing through the points $(1, 2, 3)$ and $(2, 3, 5)$ . The distance of the point $(\frac{11}{3}, \frac{11}{3}, \frac{19}{3})$ from the line $L$ along the line $\frac{3 x - 11}{2} = \frac{3 y - 11}{1} = \frac{3 z - 19}{2}$ …2024 · 9 Apr · Shift 2 · Q46 · MCQ
The square of the distance of the image of the point $(6, 1, 5)$ in the line $\frac{x - 1}{3} = \frac{y}{2} = \frac{z - 2}{4}$ , from the origin is .2024 · 9 Apr · Shift 2 · Q56 · Numerical
The distance, of the point $(7, - 2, 11)$ from the line $\frac{x - 6}{1} = \frac{y - 4}{0} = \frac{z - 8}{3}$ along the line $\frac{x - 5}{2} = \frac{y - 1}{- 3} = \frac{z - 5}{6}$ , is :2024 · 27 Jan · Shift 1 · Q37 · MCQ
If the shortest distance between the lines $\frac{x - 4}{1} = \frac{y + 1}{2} = \frac{z}{- 3}$ and $\frac{x - λ}{2} = \frac{y + 1}{4} = \frac{z - 2}{- 5}$ is $\frac{6}{5}$ , then the sum of all possible values of $λ$ is :2024 · 27 Jan · Shift 1 · Q43 · MCQ
Let the image of the point $(1, 0, 7)$ in the line $\frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}$ be the point $(α, β, γ)$ . Then which one of the following points lies on the line passing through $(α, β, γ)$ and making…2024 · 27 Jan · Shift 2 · Q35 · MCQ
The lines $\frac{x - 2}{2} = \frac{y}{- 2} = \frac{z - 7}{16}$ and $\frac{x + 3}{4} = \frac{y + 2}{3} = \frac{z + 2}{1}$ intersect at the point $P$ . If the distance of $P$ from the line $\frac{x + 1}{2} = \frac{y - 1}{3} = \frac{z - 1}{1}$ is $l$ , then $14 l^{2}$ …2024 · 27 Jan · Shift 2 · Q53 · Numerical
Let $O$ be the origin and the position vectors of $A$ and $B$ be $2 \hat{i} + 2 \hat{j} + \hat{k}$ and $2 \hat{i} + 4 \hat{j} + 4 \hat{k}$ respectively. If the internal bisector of $∠ AOB$ meets the line $AB$ at…2024 · 29 Jan · Shift 1 · Q41 · MCQ
Let $P QR$ be a triangle with $R (- 1, 4, 2)$ . Suppose $M (2, 1, 2)$ is the mid point of $PQ$ . The distance of the centroid of $△ PQR$ from the point of intersection of the lines $\frac{x - 2}{0} = \frac{y}{2} = \frac{z + 3}{- 1}$ …2024 · 29 Jan · Shift 1 · Q47 · MCQ
A line with direction ratios $2, 1, 2$ meets the lines $x = y + 2 = z$ and $x + 2 = 2 y = 2 z$ respectively at the points $P$ and $Q$ . If the length of the perpendicular from the point $(1, 2, 12)$ to the line $PQ$ is $l$ , then $l^{2}$ …2024 · 29 Jan · Shift 1 · Q54 · Numerical
Let $P (3, 2, 3), Q (4, 6, 2)$ and $R (7, 3, 2)$ be the vertices of $△ PQR$ . Then, the angle $∠ QPR$ is2024 · 29 Jan · Shift 2 · Q39 · MCQ
Let O be the origin, and M and $N$ be the points on the lines $\frac{x - 5}{4} = \frac{y - 4}{1} = \frac{z - 5}{3}$ and $\frac{x + 8}{12} = \frac{y + 2}{5} = \frac{z + 11}{9}$ respectively such that $MN$ is the shortest distance between the…2024 · 29 Jan · Shift 2 · Q51 · Numerical
Let $(α, β, γ)$ be the foot of perpendicular from the point $(1, 2, 3)$ on the line $\frac{x + 3}{5} = \frac{y - 1}{2} = \frac{z + 4}{3}$ . Then $19 (α + β + γ)$ is equal to :2024 · 30 Jan · Shift 1 · Q37 · MCQ
Let $A (2, 3, 5)$ and $C (- 3, 4, - 2)$ be opposite vertices of a parallelogram $A B C D$ . If the diagonal $BD = \hat{i} + 2 \hat{j} + 3 \hat{k}$ , then the area of the parallelogram is equal to :2024 · 30 Jan · Shift 1 · Q44 · MCQ
If $d_{1}$ is the shortest distance between the lines $x + 1 = 2 y = - 12 z, x = y + 2 = 6 z - 6$ and $d_{2}$ is the shortest distance between the lines $\frac{x - 1}{2} = \frac{y + 8}{- 7} = \frac{z - 4}{5}, \frac{x - 1}{2} = \frac{y - 2}{1} = \frac{z - 6}{- 3}$ …2024 · 30 Jan · Shift 1 · Q57 · Numerical
Let $L_{1} : r = (\hat{i} - \hat{j} + 2 \hat{k}) + λ (\hat{i} - \hat{j} + 2 \hat{k}), λ \in R$ , $L_{2} : r = (\hat{j} - \hat{k}) + μ (3 \hat{i} + \hat{j} + p \hat{k}), μ \in R, and L_{3} : r = δ (ℓ \hat{i} + m \hat{j} + n \hat{k}), δ \in R$ …2024 · 30 Jan · Shift 2 · Q41 · MCQ
Let a line passing through the point $(- 1, 2, 3)$ intersect the lines $L_{1} : \frac{x - 1}{3} = \frac{y - 2}{2} = \frac{z + 1}{- 2}$ at $M (α, β, γ)$ and $L_{2} : \frac{x + 2}{- 3} = \frac{y - 2}{- 2} = \frac{z - 1}{4}$ at $N (a, b, c)$ . Then, the value of $\frac{( α + β + γ ) ^{2}}{( a + b + c ) ^{2}}$ …2024 · 30 Jan · Shift 2 · Q55 · Numerical
Let $Q$ and $R$ be the feet of perpendiculars from the point $P (a, a, a)$ on the lines $x = y, z = 1$ and $x = - y, z = - 1$ respectively. If $∠ QPR$ is a right angle, then $12 a^{2}$ is equal to …2024 · 31 Jan · Shift 1 · Q56 · Numerical
Let $(α, β, γ)$ be the mirror image of the point $(2, 3, 5)$ in the line $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}$ . Then, $2 α + 3 β + 4 γ$ is equal to2024 · 31 Jan · Shift 2 · Q34 · MCQ
The shortest distance, between lines $L_{1}$ and $L_{2}$ , where $L_{1} : \frac{x - 1}{2} = \frac{y + 1}{- 3} = \frac{z + 4}{2}$ and $L_{2}$ is the line, passing through the points $A (- 4, 4, 3), B (- 1, 6, 3)$ and perpendicular to the line $\frac{x - 3}{- 2} = \frac{y}{3} = \frac{z - 1}{1}$ …2024 · 31 Jan · Shift 2 · Q38 · MCQ
A line passes through $A (4, - 6, - 2)$ and $B (16, - 2, 4)$ . The point $P (a, b, c)$ , where $a, b, c$ are non-negative integers, on the line $A B$ lies at a distance of 21 units, from the point $A$ . The distance between the points $P (a, b, c)$ and $Q (4, - 12, 3)$ …2024 · 31 Jan · Shift 2 · Q60 · Numerical
The shortest distance between the lines $\frac{x - 5}{1} = \frac{y - 2}{2} = \frac{z - 4}{- 3}$ and $\frac{x + 3}{1} = \frac{y + 5}{4} = \frac{z - 1}{- 5}$ is :2023 · 1 Feb · Shift 1 · Q25 · MCQ
One vertex of a rectangular parallelopiped is at the origin $O$ and the lengths of its edges along $x, y$ and $z$ axes are $3, 4$ and $5$ units respectively. Let $P$ be the vertex $(3, 4, 5)$ . Then the shortest distance…2023 · 6 Apr · Shift 1 · Q26 · MCQ
If the lines $\frac{x - 1}{2} = \frac{2 - y}{- 3} = \frac{z - 3}{α}$ and $\frac{x - 4}{5} = \frac{y - 1}{2} = \frac{z}{β}$ intersect, then the magnitude of the minimum value of $8 α β$ is .2023 · 6 Apr · Shift 2 · Q42 · Numerical
The shortest distance between the lines $\frac{x - 4}{4} = \frac{y + 2}{5} = \frac{z + 3}{3}$ and $\frac{x - 1}{3} = \frac{y - 3}{4} = \frac{z - 4}{2}$ is :2023 · 8 Apr · Shift 1 · Q28 · MCQ
The shortest distance between the lines $\frac{x + 2}{1} = \frac{y}{- 2} = \frac{z - 5}{2}$ and $\frac{x - 4}{1} = \frac{y - 1}{2} = \frac{z + 3}{0}$ is :2023 · 10 Apr · Shift 1 · Q25 · MCQ
Let a line $l$ pass through the origin and be perpendicular to the lines $l_{1} : r = (\overset{}{^} - 11 \overset{}{^} - 7 \hat{k}) + λ (\hat{i} + 2 \overset{}{^} + 3 \hat{k}), λ \in R$ and $l_{2} : r = (- \overset{}{^} + \hat{k}) + μ (2 \overset{}{^} + 2 \overset{}{^} + \hat{k}), μ \in R$ …2023 · 11 Apr · Shift 1 · Q45 · Numerical
Let the lines $l_{1} : \frac{x + 5}{3} = \frac{y + 4}{1} = \frac{z - α}{- 2}$ and $l_{2} : 3 x + 2 y + z - 2 = 0 = x - 3 y + 2 z - 13$ be coplanar. If the point $P (a, b, c)$ on $l_{1}$ is nearest to the point $Q (- 4, - 3, 2)$ , then $∣ a ∣ + ∣ b ∣ + ∣ c ∣$ …2023 · 12 Apr · Shift 1 · Q26 · MCQ
The line, that is coplanar to the line $\frac{x + 3}{- 3} = \frac{y - 1}{1} = \frac{z - 5}{5}$ , is :2023 · 13 Apr · Shift 2 · Q26 · MCQ
Let $S$ be the set of all values of $λ$ , for which the shortest distance between the lines $\frac{x - λ}{0} = \frac{y - 3}{4} = \frac{z + 6}{1}$ and $\frac{x + λ}{3} = \frac{y}{- 4} = \frac{z - 6}{0}$ is 13. Then $8 λ \in S \sum λ$ …2023 · 15 Apr · Shift 1 · Q36 · MCQ
The shortest distance between the lines $\frac{x - 2}{3} = \frac{y + 1}{2} = \frac{z - 6}{2}$ and $\frac{x - 6}{3} = \frac{1 - y}{2} = \frac{z + 8}{0}$ is equal to 2023 · 24 Jan · Shift 1 · Q38 · Numerical
If the shortest between the lines $\frac{x + 6}{2} = \frac{y - 6}{3} = \frac{z - 6}{4}$ and $\frac{x - λ}{3} = \frac{y - 2 6}{4} = \frac{z + 2 6}{5}$ is 6, then the square of sum of…2023 · 24 Jan · Shift 2 · Q38 · Numerical
The distance of the point P(4, 6, $-$ 2) from the line passing through the point ( $-$ 3, 2, 3) and parallel to a line with direction ratios 3, 3, $-$ 1 is equal to :2023 · 25 Jan · Shift 1 · Q31 · MCQ
Consider the lines $L_{1}$ and $L_{2}$ given by $L_{1} : \frac{x - 1}{2} = \frac{y - 3}{1} = \frac{z - 2}{2} L_{2} : \frac{x - 2}{1} = \frac{y - 2}{2} = \frac{z - 3}{3}$ . A line $L_{3}$ having direction ratios 1, $-$ 1, $-$ 2,…2023 · 25 Jan · Shift 1 · Q33 · MCQ
The foot of perpendicular of the point (2, 0, 5) on the line $\frac{x + 1}{2} = \frac{y - 1}{5} = \frac{z + 1}{- 1}$ is ( $α, β, γ$ ). Then, which of the following is NOT correct?2023 · 25 Jan · Shift 2 · Q23 · MCQ
The shortest distance between the lines $x + 1 = 2 y = - 12 z$ and $x = y + 2 = 6 z - 6$ is :2023 · 25 Jan · Shift 2 · Q28 · MCQ
If the shortest distance between the line joining the points (1, 2, 3) and (2, 3, 4), and the line $\frac{x - 1}{2} = \frac{y + 1}{- 1} = \frac{z - 2}{0}$ is $α$ , then 28 $α^{2}$ is equal to .2023 · 25 Jan · Shift 2 · Q45 · Numerical
Let the co-ordinates of one vertex of $Δ A B C$ be $A (0, 2, α)$ and the other two vertices lie on the line $\frac{x + α}{5} = \frac{y - 1}{2} = \frac{z + 4}{3}$ . For $α \in Z$ , if the area of $Δ A B C$ …2023 · 29 Jan · Shift 1 · Q48 · Numerical
The shortest distance between the lines $\frac{x - 1}{2} = \frac{y + 8}{- 7} = \frac{z - 4}{5}$ and $\frac{x - 1}{2} = \frac{y - 2}{1} = \frac{z - 6}{- 3}$ is :2023 · 29 Jan · Shift 2 · Q37 · MCQ
Let a line $L$ pass through the point $P (2, 3, 1)$ and be parallel to the line $x + 3 y - 2 z - 2 = 0 = x - y + 2 z$ . If the distance of $L$ from the point $(5, 3, 8)$ is $α$ , then $3 α^{2}$ is equal to :2023 · 30 Jan · Shift 2 · Q41 · Numerical
Let the shortest distance between the lines $L : \frac{x - 5}{- 2} = \frac{y - λ}{0} = \frac{z + λ}{1}, λ \geq 0$ and $L_{1} : x + 1 = y - 1 = 4 - z$ be $26$ . If $(α, β, γ)$ lies on $L$ , then which of the following is…2023 · 31 Jan · Shift 1 · Q34 · MCQ
Let a line having direction ratios, 1, $-$ 4, 2 intersect the lines $\frac{x - 7}{3} = \frac{y - 1}{- 1} = \frac{z + 2}{1}$ and $\frac{x}{2} = \frac{y - 7}{3} = \frac{z}{1}$ at the points A and B. Then (AB)2 is equal to …2022 · 24 Jun · Shift 1 · Q38 · Numerical
If the shortest distance between the lines $r = (- i + 3 k) + λ (i - a j)$ and $r = (- j + 2 k) + μ (i - j + k)$ …2022 · 24 Jun · Shift 1 · Q43 · Numerical
If the shortest distance between the lines $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{λ}$ and $\frac{x - 2}{1} = \frac{y - 4}{4} = \frac{z - 5}{5}$ is $\frac{1}{3}$ , then the sum of all possible value…2022 · 24 Jun · Shift 2 · Q33 · MCQ
The shortest distance between the lines $\frac{x + 7}{- 6} = \frac{y - 6}{7} = z$ and $\frac{7 - x}{2} = y - 2 = z - 6$ is :2022 · 25 Jul · Shift 2 · Q33 · MCQ
Let l1 be the line in xy-plane with x and y intercepts $\frac{1}{8}$ and $\frac{1}{4 2}$ respectively, and l2 be the line in zx-plane with x and z intercepts $- \frac{1}{8}$ and $- \frac{1}{6 3}$ respectively. If d is the…2022 · 25 Jun · Shift 2 · Q45 · Numerical
The length of the perpendicular from the point $(1, - 2, 5)$ on the line passing through $(1, 2, 4)$ and parallel to the line $x + y - z = 0 = x - 2 y + 3 z - 5$ is :2022 · 26 Jul · Shift 1 · Q35 · MCQ
Let $Q$ and $R$ be two points on the line $\frac{x + 1}{2} = \frac{y + 2}{3} = \frac{z - 1}{2}$ at a distance $26$ from the point $P (4, 2, 7)$ . Then the square of the area of the triangle $P QR$ is .2022 · 26 Jul · Shift 1 · Q45 · Numerical
If the two lines $l_{1} : \frac{x - 2}{3} = \frac{y + 1}{- 2}, z = 2$ and $l_{2} : \frac{x - 1}{1} = \frac{2 y + 3}{α} = \frac{z + 5}{2}$ are perpendicular, then an angle between the lines l2 and $l_{3} : \frac{1 - x}{3} = \frac{2 y - 1}{- 4} = \frac{z}{4}$ …2022 · 26 Jun · Shift 1 · Q30 · MCQ
Let $a = i + j + 2 k$ , $b = 2 i - 3 j + k$ and $c = i - j + k$ be three given vectors. Let $v$ …2022 · 26 Jun · Shift 2 · Q35 · MCQ
If the length of the perpendicular drawn from the point $P (a, 4, 2)$ , a $> 0$ on the line $\frac{x + 1}{2} = \frac{y - 3}{3} = \frac{z - 1}{- 1}$ is $26$ units and $Q (α_{1}, α_{2}, α_{3})$ is the image of the…2022 · 27 Jul · Shift 2 · Q32 · MCQ
If two straight lines whose direction cosines are given by the relations $l + m - n = 0$ , $3 l^{2} + m^{2} + c n l = 0$ are parallel, then the positive value of c is :2022 · 27 Jun · Shift 1 · Q31 · MCQ
The shortest distance between the lines $\frac{x - 3}{2} = \frac{y - 2}{3} = \frac{z - 1}{- 1}$ and $\frac{x + 3}{2} = \frac{y - 6}{1} = \frac{z - 5}{3}$ , is :2022 · 27 Jun · Shift 2 · Q31 · MCQ
Let $P (- 2, - 1, 1)$ and $Q (\frac{56}{17}, \frac{43}{17}, \frac{111}{17})$ be the vertices of the rhombus PRQS. If the direction ratios of the diagonal RS are $α, - 1, β$ , where both $α$ and $β$ …2022 · 28 Jul · Shift 1 · Q39 · Numerical
Let the image of the point P(1, 2, 3) in the line $L : \frac{x - 6}{3} = \frac{y - 1}{2} = \frac{z - 2}{3}$ be Q. Let R ( $α$ , $β$ , $γ$ ) be a point that divides internally the line segment PQ in the ratio 1 : 3. Then the…2022 · 28 Jun · Shift 2 · Q41 · Numerical
Consider a triangle ABC whose vertices are A(0, $α$ , $α$ ), B( $α$ , 0, $α$ ) and C( $α$ , $α$ , 0), $α$ > 0. Let D be a point moving on the line x + z $-$ 3 = 0 = y and G be the centroid of $Δ$ ABC. If…2022 · 30 Jun · Shift 1 · Q40 · Numerical
The distance of line $3 y - 2 z - 1 = 0 = 3 x - z + 4$ from the point (2, $-$ 1, 6) is :2021 · 1 Sep · Shift 2 · Q32 · MCQ
Let the position vectors of two points P and Q be 3 $i - j$ + 2 $k$ and $i$ + 2 $j -$ 4 $k$ , respectively. Let R and S be two points such that the direction ratios of lines PR and QS are…2021 · 16 Mar · Shift 1 · Q25 · MCQ
If the foot of the perpendicular from point (4, 3, 8) on the line $L_{1} : \frac{x - a}{l} = \frac{y - 2}{3} = \frac{z - b}{4}$ , l $e$ 0 is (3, 5, 7), then the shortest distance between the line L1 and line $L_{2} : \frac{x - 2}{3} = \frac{y - 4}{4} = \frac{z - 5}{5}$ …2021 · 16 Mar · Shift 2 · Q27 · MCQ
The lines x = ay $-$ 1 = z $-$ 2 and x = 3y $-$ 2 = bz $-$ 2, (ab $e$ 0) are coplanar, if :2021 · 20 Jul · Shift 2 · Q26 · MCQ
If the shortest distance between the straight lines $3 (x - 1) = 6 (y - 2) = 2 (z - 1)$ and $4 (x - 2) = 2 (y - λ) = (z - 3), λ \in R$ is $\frac{1}{38}$ , then the integral value of $λ$ is equal to :2021 · 22 Jul · Shift 2 · Q30 · MCQ
Let a, b $\in$ R. If the mirror image of the point P(a, 6, 9) with respect to the line $\frac{x - 3}{7} = \frac{y - 2}{5} = \frac{z - 1}{- 9}$ is (20, b, $-$ a $-$ 9), then | a + b |, is equal to :2021 · 24 Feb · Shift 2 · Q25 · MCQ
Let $λ$ be an integer. If the shortest distance between the lines x $- λ$ = 2y $-$ 1 = $-$ 2z and x = y + 2 $λ$ = z $- λ$ is $\frac{7}{2 2}$ , then the value of | $λ$ | is .2021 · 24 Feb · Shift 2 · Q36 · Numerical
The equation of the line through the point (0, 1, 2) and perpendicular to the line $\frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 1}{- 2}$ is :2021 · 25 Feb · Shift 1 · Q25 · MCQ
Let $α$ be the angle between the lines whose direction cosines satisfy the equations l + m $-$ n = 0 and l2 + m2 $-$ n2 = 0. Then the value of sin4 $α$ + cos4 $α$ is :2021 · 25 Feb · Shift 1 · Q33 · MCQ
A line 'l' passing through origin is perpendicular to the lines $l_{1} : r = (3 + t) i + (- 1 + 2 t) j + (4 + 2 t) k l_{2} : r = (3 + 2 s) i + (3 + 2 s) j + (2 + s) k$ …2021 · 25 Feb · Shift 2 · Q44 · Numerical
The angle between the straight lines, whose direction cosines are given by the equations 2l + 2m $-$ n = 0 and mn + nl + lm = 0, is :2021 · 27 Aug · Shift 2 · Q23 · MCQ
If (a, b, c) is the image of the point (1, 2, -3) in the line $\frac{x + 1}{2} = \frac{y - 3}{- 2} = \frac{z}{- 1}$ , then a + b + c is :2020 · 5 Sep · Shift 1 · Q36 · MCQ
If the foot of the perpendicular drawn from the point (1, 0, 3) on a line passing through ( $α$ , 7, 1) is $(\frac{5}{3}, \frac{7}{3}, \frac{17}{3})$ , then $α$ is equal to .2020 · 7 Jan · Shift 2 · Q29 · Numerical
The shortest distance between the lines $\frac{x - 3}{3} = \frac{y - 8}{- 1} = \frac{z - 3}{1}$ and $\frac{x + 3}{- 3} = \frac{y + 7}{2} = \frac{z - 6}{4}$ is :2020 · 8 Jan · Shift 1 · Q24 · MCQ
The projection of the line segment joining the points (1, –1, 3) and (2, –4, 11) on the line joining the points (–1, 2, 3) and (3, –2, 10) is .2020 · 9 Jan · Shift 1 · Q30 · Numerical
The length of the perpendicular from the point (2, –1, 4) on the straight line, $\frac{x + 3}{10}$ = $\frac{y - 2}{- 7}$ = $\frac{z}{1}$ is :2019 · 8 Apr · Shift 1 · Q35 · MCQ
If a point R(4, y, z) lies on the line segment joining the points P(2, –3, 4) and Q(8, 0, 10), then the distance of R from the origin is :2019 · 8 Apr · Shift 2 · Q31 · MCQ
The vertices B and C of a $Δ$ ABC lie on the line, $\frac{x + 2}{3} = \frac{y - 1}{0} = \frac{z}{4}$ such that BC = 5 units. Then the area (in sq. units) of this triangle, given that the point A(1, –1, 2), is :2019 · 9 Apr · Shift 2 · Q38 · MCQ
If the lines x = ay + b, z = cy + d and x = a'z + b', y = c'z + d' are perpendicular, then :2019 · 9 Jan · Shift 2 · Q27 · MCQ
If the length of the perpendicular from the point ( $β$ , 0, $β$ ) ( $β e$ 0) to the line, $\frac{x}{1} = \frac{y - 1}{0} = \frac{z + 1}{- 1}$ is $\frac{3}{2}$ , then $β$ is equal to :2019 · 10 Apr · Shift 1 · Q25 · MCQ
An angle between the lines whose direction cosines are gien by the equations, $l$ + 3m + 5n = 0 and 5 $l$ m $-$ 2mn + 6n $l$ = 0, is :2018 · 15 Apr · Shift 2 · Q33 · MCQ
If the angle between the lines, $\frac{x}{2} = \frac{y}{2} = \frac{z}{1}$ and $\frac{5 - x}{- 2} = \frac{7 y - 14}{p} = \frac{z - 3}{4}$ is $cos^{- 1} (\frac{2}{3}),$ then p is equal to :2018 · 16 Apr · Shift 1 · Q44 · MCQ
The shortest distance between the lines $\frac{x}{2} = \frac{y}{2} = \frac{z}{1}$ and $\frac{x + 2}{- 1} = \frac{y - 4}{8} = \frac{z - 5}{4}$ lies in the interval :2016 · 9 Apr · Shift 1 · Q36 · MCQ
ABC is a triangle in a plane with vertices A(2, 3, 5), B(−1, 3, 2) and C( $λ$ , 5, $μ$ ). If the median through A is equally inclined to the coordinate axes, then the value of ( $λ$ 3 + $μ$ 3 + 5) is :2016 · 10 Apr · Shift 1 · Q33 · MCQ
The angle between the lines whose direction cosines satisfy the equations $l + m + n = 0$ and $l^{2} = m^{2} + n^{2}$ is :2014 · Shift 0 · Q44 · MCQ
If the lines $\frac{x - 2}{1} = \frac{y - 3}{1} = \frac{z - 4}{- k}$ and $\frac{x - 1}{k} = \frac{y - 4}{2} = \frac{z - 5}{1}$ are coplanar, then $k$ can have :2013 · Shift 0 · Q47 · MCQ
If the line $\frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 1}{4}$ and $\frac{x - 3}{1} = \frac{y - k}{2} = \frac{z}{1}$ intersect, then $k$ is equal to :2012 · Shift 0 · Q35 · MCQ
Statement - 1 : The point $A (1, 0, 7)$ is the mirror image of the point $B (1, 6, 3)$ in the line : $\frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}$ Statement - 2 : The line $\frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}$ bisects…2011 · Shift 0 · Q50 · MCQ
A line $A B$ in three-dimensional space makes angles $4 5^{\circ}$ and $12 0^{\circ}$ with the positive $x$ -axis and the positive $y$ -axis respectively. If $A B$ makes an acute angle $θ$ with the positive $z$ -axis, then $θ$ …2010 · Shift 0 · Q48 · MCQ
The projections of a vector on the three coordinate axis are $6, - 3, 2$ respectively. The direction cosines of the vector are :2009 · Shift 0 · Q46 · MCQ
If the straight lines $\frac{x - 1}{k} = \frac{y - 2}{2} = \frac{z - 3}{3}$ and $\frac{x - 2}{3} = \frac{y - 3}{k} = \frac{z - 1}{2}$ intersects at a point, then the integer…2008 · Shift 0 · Q55 · MCQ
The line passing through the points $(5, 1, a)$ and $(3, b, 1)$ crosses the $y z$ -plane at the point $(0, \frac{17}{2}, - \frac{- 13}{2})$ . Then2008 · Shift 0 · Q56 · MCQ
If a line makes an angle of $π /4$ with the positive directions of each of $x$ -axis and $y$ -axis, then the angle that the line makes with the positive direction of the $z$ -axis is :2007 · Shift 0 · Q44 · MCQ
Let $L$ be the line of intersection of the planes $2 x + 3 y + z = 1$ and $x + 3 y + 2 z = 2.$ If $L$ makes an angle $α$ with the positive $x$ -axis, then cos $α$ equals2007 · Shift 0 · Q62 · MCQ
The two lines $x = a y + b, z = cy + d;$ and $x = a^{'} y + b^{'}, z = c^{'} y + d^{'}$ are perpendicular to each other if :2006 · Shift 0 · Q66 · MCQ
The angle between the lines $2 x = 3 y = - z$ and $6 x = - y = - 4 z$ is :2005 · Shift 0 · Q92 · MCQ
If the straight lines $x = 1 + s, y = - 3 - λ s, z = 1 + λ s$ and $x = \frac{t}{2}, y = 1 + t, z = 2 - t,$ with parameters $s$ and $t$ respectively, are co-planar, then $λ$ equals :2004 · Shift 0 · Q114 · MCQ
A line with direction cosines proportional to $2, 1, 2$ meets each of the lines $x = y + a = z$ and $x + a = 2 y = 2 z$ . The co-ordinates of each of the points of intersection are given by :2004 · Shift 0 · Q93 · MCQ
A line makes the same angle $θ$ , with each of the $x$ and $z$ axis. If the angle $β$ , which it makes with y-axis, is such that $sin^{2} β = 3 sin^{2} θ,$ then $cos^{2} θ$ equals :2004 · Shift 0 · Q96 · MCQ
The two lines $x = a y + b, z = cy + d$ and $x = a^{'} y + b^{'}, z = c^{'} y + d^{'}$ will be perpendicular, if and only if :2003 · Shift 0 · Q92 · MCQ
The lines $\frac{x - 2}{1} = \frac{y - 3}{1} = \frac{z - 4}{- k}$ and $\frac{x - 1}{k} = \frac{y - 4}{2} = \frac{z - 5}{1}$ are coplanar if :2003 · Shift 0 · Q93 · MCQ