3d planes practice · Mixed difficulty

Option D is correct because the locus of the centroid is found by using the equation of a plane and the distance from the origin.

Option C ((1, 0, 2)) is correct because the rotated plane passes through this point after finding the equation of the plane using the angle of rotation and the line of intersection.

Option C is correct because it accurately determines the equation of the plane containing the line of intersection and passing through the given point.

Option A ($a a^{\prime}+c c^{\prime}+1=0$) is correct because it represents the condition for perpendicularity of two lines in 3D space, derived from the dot product of their direction vectors.

Option B (1) is correct because equal intercepts on the axes imply that the coefficients of x, y, and z (after normalization) must be equal, leading to a = 1.

Option A ($\frac{x+3}{3}=\frac{y-5}{1}=\frac{z-2}{-5}$) is correct because it represents the image of the given line in the specified plane.

Option D ($\frac{3}{7} \sqrt{35}$) is correct because it represents the perpendicular distance from the point to the line, calculated using the formula involving direction ratios and coordinate differences.

Option B is correct. Planes ${P_1}$ and ${P_3}$ are parallel because their normal vectors are proportional, meaning they have the same direction ratios.

Option B (Assertion is correct, reason is correct; reason is not a correct explanation for assertion) is correct because both the assertion and the reason are correct statements, but the reason does not explain the assertion.

Option A (Statement -1 is true, Statement - 2 is true ; Statement 2 is not a correct explanation for Statement -1) is correct because both statements are true, but the second statement doesn't fully explain why A is the mirror image of B.

Option C (on the opposite sides of the plane) is correct because the perpendicular distances from the points to the plane have opposite signs, indicating they lie on opposite sides.

Option D ($2\sqrt {14}$) is correct because it is the distance between the points P and Q, where the given line intersects the two planes.

Option D ($100\,\frac{1}{2}$) is correct because it is the sum of the first 200 terms of the A.P. calculated using the given information.

Option D $\left(\frac{9}{5},-\frac{13}{5}, 4\right)$ is correct because it is the result of applying the image formula for a point in a plane.

Option D ($2 \sqrt{2}$) is correct because the distance of the plane x+z=4 from the origin (0,0,0) is calculated correctly using the point-to-plane distance formula.

Option A (5) is correct because solving the system of equations derived from the line and plane conditions yields the correct values for a, b, and c, and thus the correct answer.

Option B ($V \geq \frac{9}{2}$) is correct because using AM-GM inequality, the minimum volume of the tetrahedron is found to be $\frac{9}{2}$.

Option B ($\frac{{10}}{{\sqrt {83} }}$ units) is correct because it represents the perpendicular distance of the point from the plane, calculated using the formula for distance from a point to a plane.

Option B (2) is correct because it is the absolute value of the parameter 'm' that satisfies the given distance condition between the point and the plane, measured parallel to the given line.

Option B $\left( { - 2,\,0,\, - \frac{1}{2}} \right)$ is correct because it satisfies the equation of the acute angle bisector plane.

Option C (exactly two values) is correct because the coplanarity condition gives a quadratic equation in k, which has two solutions.

Option B ($\frac{{x-b-a}}{{a}}=\frac{{y-1}}{{1}}=\frac{{z-d-c}}{{c}}$) is correct because it represents the symmetrical form of the line of intersection of the given planes.

Option B (3/7) is correct because it is the perpendicular distance of the intersection point (3, 5, -2) from the plane 3x + 2y + 6z = 10, calculated using the point-to-plane distance formula.

Option A $2x + y - z = 1$ is correct because the foot of the perpendicular from the given point to the line lies on this plane.

Option B ($2 \sqrt{5}$) is correct because it calculates the distance AP where P is the point on the plane closest to the line AB.

Option A is correct because it represents the equation of a plane passing through the intersection of two planes and parallel to the line of intersection of another two planes, which is derived by setting the scalar triple product to zero.

Option A ($\frac{{10}}{{3 \sqrt{{3}}}}$) is correct because it represents the distance between the line and the plane, calculated using the formula for the distance between a point on the line and the plane, given that the line and plane are parallel.

Option C (13) is correct because by finding the intersection point of the line and the plane, the distance between that point and (-1, -5, -10) is calculated to be 13.

Option C (74) is correct because it involves finding the third vertex of a triangle given two vertices and the centroid, then finding the image of the third vertex in a plane, and finally calculating the required expression.

Option C ($-4$) is correct because after finding the values of a and b, and using the distance formula, the value of c is determined, leading to the correct ratio.