JEE Main Maths: Introduction to Three Dimensional Geometry questions with solutions

Q1. A line makes equal angles α, β and γ with the positive directions of the coordinate axes. If θ satisfies cos θ = (cos²α + cos²β + cos²γ) / (sin²α + sin²β + sin²γ), then what is the value of θ?

1. π/3
2. π/6
3. π/2
4. π/4

Answer: π/3

The equation given relates the cosines and sines of angles α, β, and γ, which are equal due to the line making equal angles with the axes. By substituting the equal angles into the equation, it simplifies to show that cos θ equals 1/2, which corresponds to θ being π/3.

Q2. A rectangular box is bounded by planes passing through the points (-1, 2, 5) and (1, -1, -1), each plane being parallel to one of the coordinate planes. What is the length of the space diagonal of this box?

1. 2
2. 3
3. 6
4. 7

Answer: 7

The opposite corners (-1,2,5) and (1,-1,-1) give edge lengths |1-(-1)|=2, |-1-2|=3, |-1-5|=6. The space diagonal is sqrt(2^2+3^2+6^2)=sqrt(49)=7.

Q3. If (2, 3, 5) is one end of a diameter of the sphere x² + y² + z² - 6x - 12y - 2z + 20 = 0, then the coordinates of the other end of the diameter are

1. (4, 3, 5)
2. (4, 3, -3)
3. (4, 9, -3)
4. (4, -3, 3)

Answer: (4, 9, -3)

The center of the sphere can be found by rewriting the equation in standard form, which reveals that the center is at (3, 6, 1). The coordinates of the other end of the diameter can be determined by using the midpoint formula; since one end is (2, 3, 5), the other end must be (4, 9, -3) to maintain the center at (3, 6, 1).

Q4. The line passing through the points (5, 1, a) and (3, b, 1) crosses the yz-plane at the point (0, 17/2, -13/2). Then

1. a = 2, b = 8
2. a = 4, b = 6
3. a = 6, b = 4
4. a = 8, b = 2

Answer: a = 6, b = 4

The correct option is right because the coordinates of the points on the line can be used to derive the equations of the line, and substituting the x-coordinate of the yz-plane (which is 0) into these equations yields the correct values for a and b, specifically a = 6 and b = 4, which match the intersection point given.

Q5. 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 -

1. 6
2. √53
3. 2√14
4. 2√21

Answer: 2√14

The point R lies on the line segment between P and Q, which means its coordinates can be expressed as a weighted average of P and Q. By calculating the coordinates of R, we find that R(4, -1, 7) results in a distance from the origin of 2√14, confirming that this is the correct option.

Q6. Q.65 Let PQR 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 ΔPQR from the point of intersection of the lines (x−2)/0 = y/2 = (z+3)/(−1) and (x−1)/1 = (y+3)/(−3) = (z+1)/1 is-

1. √69
2. 69
3. √99
4. 9

Answer: √69

The centroid of triangle PQR can be calculated using the coordinates of its vertices, and the distance from this centroid to the intersection point of the given lines can be determined using the distance formula. The calculations yield a distance of √69, confirming that this option is correct.

Q7. 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 AB 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) is equal to ____

1. 22
2. 21
3. 20
4. 19

Answer: 22

Direction AB=(12,4,6), |AB|=14, unit=(6/7,2/7,3/7). P=A+21*unit=(4+18,-6+6,-2+9)=(22,0,7), non-negative integers. PQ with Q(4,-12,3): sqrt(18^2+12^2+4^2)=sqrt(324+144+16)=sqrt484=22. The stored answer 21 is wrong.

Q8. 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

1. π/6
2. π/3
3. π/4
4. π/2

Answer: π/2

The line makes equal angles with the x-axis and y-axis, indicating it lies in the xy-plane. Since it does not extend in the z-direction, the angle with the positive z-axis is π/2, meaning it is perpendicular to the z-axis.

Q9. If (2, 3, 5) is one end of a diameter of the sphere x² + y² + z² − 6x − 12y − 2z + 20 = 0, then the coordinates of the other end of the diameter are

1. (4, 9, −3)
2. (4, −3, 3)
3. (4, 3, 5)
4. (4, 3, −3)

Answer: (4, 9, −3)

The sphere center is (3,6,1). The other end = 2(3,6,1) - (2,3,5) = (4,9,-3), not (4,-3,3).

Q10. If the projections of a line segment on the x, y and z-axes in 3-dimensional space are 2, 3 and 6 respectively, then the length of the line segment is:

1. 12
2. 7
3. 9
4. 6

Answer: 7

The length of the line segment can be calculated using the Pythagorean theorem in three dimensions, which states that the length is the square root of the sum of the squares of the projections on each axis. Therefore, the length is √(2² + 3² + 6²) = √(4 + 9 + 36) = √49 = 7.

Q11. Q88. Let ABC be a triangle with vertices at points A (2, 3, 5), B (−1, 3, 2) and C (λ, 5, μ) in three dimensional space. If the median through A is equally inclined with the axes, then (λ, μ) is equal to:

1. (10, 7)
2. (7, 5)
3. (7, 10)
4. (5, 7)

Answer: (7, 10)

The median through vertex A connects point A to the midpoint of side BC. For the median to be equally inclined with the axes, the direction ratios must be equal, which leads to the coordinates of point C being determined as (7, 10) to satisfy this condition.

Q12. A line segment joins A(-3, 2, 4) and B(0, 4, 7). Find the coordinates of the two points that trisect this line segment (divide it into three equal parts).

1. (-2, 8/3, 5) and (-1, 10/3, 6)
2. (-1, 8/3, 5) and (-2, 10/3, 6)
3. (-2, 10/3, 6) and (-1, 8/3, 5)
4. (-2, 8/3, 5) and (0, 4, 7)

Answer: (-2, 8/3, 5) and (-1, 10/3, 6)

Trisection points P1 and P2 divide AB internally in ratios 1:2 and 2:1 respectively from A. Applying the section formula gives P1=(-2, 8/3, 5) and P2=(-1, 10/3, 6).