3d planes practice · Mixed difficulty

Option D ($90^{\circ}$) is correct because the dot product of the direction vectors is zero, indicating perpendicular lines.

Option A ((-2,0,1)) is correct because the plane equation derived from the given points and parallel line condition satisfies this point.

Option C ($x+3 y+6 z=7$) is correct because it represents the equation of the plane containing the line and parallel to the given plane.

Option C ($\frac{x-3}{1}=\frac{y-2}{0}=\frac{z-5}{4}$) is correct because it represents the equation of the reflected ray derived from the image of point P and point Q.

The correct answer is D ($\sin ^{-1}(\frac{4}{21})$) because it is calculated using the formula for the angle between a line and a plane.

Option B ($4 x+3 y+5 z=50$) is correct because the plane passes through the intersection of the two lines and is perpendicular to the line connecting the origin and the intersection point, maximizing the distance from the origin.

Option A (620) is correct because it is the square of the distance calculated using the formula for the distance of a point from a plane.

Option A (-23) is correct because it represents the sum of the coefficients $\alpha$, $\beta$, and $\gamma$ in the equation of the plane that passes through the given point and contains the line of intersection of the two planes.

Option D (7/2) is correct because substituting the direction cosines into the angle formula and solving for P yields P = 7/2.

Option A (5) is correct because after finding the foot of the perpendicular P, its distance from the plane is calculated to be 5.

Option C ((4, 2, 2)) is correct because it does not satisfy the equation of the plane formed by the intersection of the two given planes and passing through the point (1, 2, 3).

Option C ($\frac{12}{\sqrt{29}}$) is correct because it is the length of the perpendicular from the origin to the plane, calculated using the formula involving intercepts.

The correct answer is B (1) because it represents the distance of the point from the plane, measured parallel to the given line, calculated by finding the intersection point and then the distance.

Option A ($x+y+z=0$) is correct because it represents the equation of the plane that contains the given line and the point (0,7,-7).

Option D (-9) is correct because by substituting the parametric form of the line into the plane equation, we can solve for α and β, and then find their sum.

The equation of the rotated plane is found by taking a linear combination of the original plane and the plane of intersection, then using the perpendicularity condition, resulting in $5 x+y+4 z=20$.

Option A ((-3,2,1)) is correct because it satisfies the equation of the plane that bisects the line segment at right angles.

Option B ($3: 10$) is correct because it represents the ratio in which the plane divides the line segment, calculated by substituting the section formula into the plane equation.

Option D (13) is correct because it is the distance between the given point and the intersection point of the line and plane.

Option C ($2x - y + 2z = 0$) is correct because it represents the equation of a plane that is perpendicular to the given line and passes through the origin.

The correct answer is $\frac{5}{3}$ because it is the value of $\lambda$ that satisfies the given condition $\sin \theta = \frac{1}{3}$ using the formula for the angle between a line and a plane.

Option C ($\cos^{{-1}}(\frac{{4}}{{9}})$) is correct because it is the angle between the pair of planes represented by the given equation.

Option A ((-2,5,0)) is correct because after finding the equation of the plane, substituting the coordinates of the point (-2,5,0) satisfies the equation.

Option C ($\frac{9}{2}$) is correct because it is the value of k for which the two lines intersect, found by equating the coordinates of a general point on each line and solving the resulting system of equations.

Option D (2) is correct because it is the value of $l^2 + m^2$ calculated from the conditions that the line lies in the plane.

Option D ($\sin^{-1} \sqrt{\frac{3}{17}}$) is correct because the angle between the plane and the line of intersection is found using the dot product of the normal vector of the plane and the direction vector of the line.

Option C ($3:1$) is correct because the y-axis divides the line segment in the ratio $\lambda:1$ where the x-coordinate of the point of division is 0.

Option D (4) is correct because by finding the normal vector to the plane and using the point (-1, 0, -2), we can determine the equation of the plane and then calculate a + b + c.

Option A (0) is correct because the points A, O, and G are collinear, resulting in a triangle with zero area.

Option B ($3 \sqrt{{6}}$) is correct because after finding λ, the distance from the point to the plane P2 is calculated using the distance formula.