3d planes practice · Easy

Option D ($(0,4,5)$) is correct because when the value of $\lambda$ is found and the equation of the plane is determined, this point does not satisfy the plane's equation.

Option D ($P$ and $R$ are parallel) is correct because the coefficients of x, y, and z in the equations of planes P and R are proportional.

Option A (88) is correct because solving the equations derived from the midpoint and direction ratios gives |a + b| = 88.

Option B ($(3 a, 2 a, 3 a),(a, a, a)$) is correct because these coordinates are obtained by solving the equations of the given lines and the direction ratios.

Option B ($10\sqrt 3$) is correct because it is the distance calculated by finding the intersection of the line and plane and then calculating the distance between the points.

Option D ($a^2+b^2+c^2+2 a b c=1$) is correct because this condition ensures that the determinant of the coefficients of the planes is zero, indicating that the planes pass through one line.

Option A (4) is correct because the least value of $p^2 + q^2 + r^2$ is found using Cauchy-Schwarz inequality, given the plane equation.

Option B (Both are true) is correct because both statements are fundamental properties of combinations.

Option B ((-6,7)) is correct because these are the values of $\alpha$ and $\beta$ for which the given line lies in the given plane, found by ensuring the direction ratios are perpendicular to the normal and the point lies on the plane.

Option A (Statement I is true, Statement II is true; Statement II is the correct explanation of Statement I) is correct because the point A is indeed the mirror image of B in the given plane, and the plane bisects the line segment joining A and B.

Option B (11) is correct because it involves solving a system of linear equations derived from the matrix equation and then substituting the values into the given expression.

Option A is the correct answer because it represents the equation of a line passing through the given point and perpendicular to the line of intersection of the given planes.

Option C (-6) is correct because it is found by using the formula for the mirror image of a point in a plane and then substituting the values into the expression 2a+b+2c.

Option D ($\mathrm{k}=0$ or -3) is correct because these are the values of k that make the determinant of the matrix formed by the direction ratios and difference of points equal to zero, indicating coplanarity.

Option D (-5) is correct because it is the integer value of k that makes the two given lines intersect.

Option D (5151) is correct because it's the result of calculating $\frac{N(N+1)}{2}$ where N is derived from the distance between the two planes.

Option A ($\frac{{p^5}}{{2 h k l}}$) is correct because it represents the area of the triangle ABC, derived from the equation of the plane and the intercepts on the coordinate axes.

Option C (c=0) is correct because for the plane to be perpendicular to the xy-plane, its normal vector must be orthogonal to the normal vector of the xy-plane, which is $\hat{k}$.

Option C ($3 x^2-10 x+7=0$) is correct because it is the equation satisfied by $\lambda$ when the points are equidistant from the plane.

Option B (r. $(4 \hat{\mathbf{i}}-12 \hat{\mathbf{j}}-3 \hat{\mathbf{k}})=32$) is correct because it represents a plane parallel to the given plane and passing through the specified point.

Option C ((0,-3,1)) is correct because it satisfies the equation of the plane containing the given point and line.

Option A ($2 x-y-z=1$) is correct because it represents the plane formed by the intersection of the two given spheres.

Option A ($\left(\frac{8}{3}, \frac{4}{3},-\frac{7}{3}\right)$) is correct because it is the foot of the perpendicular from P to the plane, calculated using the distance formula and the equation of the line perpendicular to the plane.

Option B (0) is correct because the point (1,2,-3) lies on the plane, making the shortest distance 0.

Option D ($4x - y - 5z + 2 = 0\,\,$) is the correct equation of the plane because it satisfies both the intersection condition and the distance from the origin.

Option B (2, -10, -2) is correct because by finding the value of $\alpha$ for which the lines are coplanar and substituting it into the equation of line $L_2$, the point (2, -10, -2) satisfies the equation.

Option D (1) is correct because the distance is calculated by finding the intersection point of the line and plane, then finding the distance between the given point and the intersection point.

The correct answer is B. The shortest distance between the Z-axis and the given line is found by finding a plane containing the line that is parallel to the Z-axis, and then calculating the distance from a point on the Z-axis to that plane, which results in 2.

Option B ($\frac{2 a}{\sqrt{6}}$) is correct because it represents the shortest distance between the specified opposite edges of the tetrahedron.

Option B ($3x+2z=6$) is correct because it represents a plane parallel to the y-axis with the given intercepts on the x and z axes.