Take timed practice tests on Three Dimensional Geometry for JEE Main and JEE Advanced with session-wise drills, score review, and explanation-led revision.
Six 20-question timed sessions plus one 60-question module test. Each question is original and calibrated from the uploaded material pattern without copying PDF wording.
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1. For direction cosines l, m, n, the relation is:
Explanation: Direction cosines are the cosines of the angles a line makes with the three coordinate axes. They satisfy l² + m² + n² = 1 always.
2. The distance between (1, 2, 3) and (4, 6, 3) is:
Explanation: d = √((4−1)² + (6−2)² + (3−3)²) = √(9+16+0) = √25 = 5.
3. The midpoint of (2, −1, 4) and (4, 3, −2) is:
Explanation: Midpoint = ((2+4)/2, (−1+3)/2, (4+(−2))/2) = (3, 1, 1).
4. The direction ratios of the line joining (1,2,3) and (4,7,−1) are:
Explanation: DR = (x₂−x₁, y₂−y₁, z₂−z₁) = (4−1, 7−2, −1−3) = (3, 5, −4).
5. The symmetric form of the line through (x₁,y₁,z₁) with direction ratios a,b,c is:
Explanation: Symmetric form (Cartesian): (x−x₁)/a = (y−y₁)/b = (z−z₁)/c = t. Each equation set equal to parameter t gives the three parametric equations.
6. The equation of a plane with normal vector (a, b, c) passing through (x₁,y₁,z₁) is:
Explanation: A plane with normal n⃗ = (a,b,c) passing through point (x₁,y₁,z₁): a(x−x₁)+b(y−y₁)+c(z−z₁) = 0.
7. The angle θ between lines with direction ratios (a₁,b₁,c₁) and (a₂,b₂,c₂) satisfies:
Explanation: cos θ = |a₁a₂+b₁b₂+c₁c₂| / (√(a₁²+b₁²+c₁²) × √(a₂²+b₂²+c₂²)). For perpendicular lines: a₁a₂+b₁b₂+c₁c₂ = 0.
8. The distance from point (x₁,y₁,z₁) to plane ax+by+cz+d=0 is:
Explanation: Distance = |ax₁+by₁+cz₁+d| / √(a²+b²+c²). This is analogous to the 2D formula |ax₁+by₁+c|/√(a²+b²).
9. The distance between parallel planes 2x+y+2z=5 and 2x+y+2z=11 is:
Explanation: Distance = |d₁−d₂|/√(a²+b²+c²) = |5−11|/√(4+1+4) = 6/3 = 2.
10. A line with direction ratios (1,1,0) makes what angle with the plane x+y=0?
Explanation: Normal to plane x+y=0 is (1,1,0). Direction of line is (1,1,0). Since the line direction = plane normal, the line is perpendicular to the plane (angle = 90°). Check: sin θ = |a₁a₂+b₁b₂+c₁c₂|/(|d₁||n|) = |1+1+0|/(√2×√2) = 2/2 = 1 → θ = 90°.
11. The foot of perpendicular from origin to the plane 2x+3y−4z=29 is:
Explanation: The foot of perpendicular from origin to plane 2x+3y−4z=29: The foot lies on the normal through origin: (x,y,z) = t(2,3,−4). Substituting: 2(2t)+3(3t)−4(−4t) = 29 → 4t+9t+16t=29 → 29t=29 → t=1. Foot = (2,3,−4).
12. The direction cosines of the x-axis are:
Explanation: The x-axis makes angles 0°, 90°, 90° with x, y, z axes. Direction cosines: (cos0°, cos90°, cos90°) = (1, 0, 0).
13. Lines with direction ratios (2, 3, 6) and (3, −6, 2) are:
Explanation: Dot product: 2(3) + 3(−6) + 6(2) = 6 − 18 + 12 = 0. Since dot product = 0, the direction ratios are perpendicular → lines are perpendicular.
14. The equation of the plane passing through the three coordinate intercepts a, b, c is:
Explanation: Intercept form of the plane: x/a + y/b + z/c = 1. It passes through (a,0,0), (0,b,0), and (0,0,c).