3d planes practice · Mixed difficulty

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

Option B (3) is correct because by substituting the plane equation into the expression and simplifying using the given condition, the expression reduces to 3.

The angle between the lines is $90^{\circ}$ because the dot product of their direction ratios is zero, indicating perpendicularity.

Option D ($\sqrt{\frac{474}{35}}$) is correct because it is the length of the projection calculated using direction ratios and the angle between the line and the plane.

Option B ($(3,-2,1)$) is correct because substituting the parametric form of the line into the plane equation yields r=1, resulting in the point (3,-2,1).

Option A ($\frac{q}{\text { i.n }}$) is correct because it represents the x-coordinate where the plane intersects the x-axis, derived from the plane equation.

Option D ($\frac{2}{3}$) is correct because it is the value of $\lambda$ that satisfies the given condition for the angle between the line and the plane, found by using the formula relating the angle to the direction ratios of the line and the normal to the plane.

Option B ($(1,-2,7)$) is correct because it is found by determining the ratio in which the line intersects the plane and then finding the point of intersection.

Option A ($\sqrt{\frac{2}{7}}$) is correct because it represents the distance of the plane from the origin, calculated using the formula for the distance of a point from a plane.

Option C ($4 x+3 y+5 z=50$) is correct because it represents the equation of the plane passing through the intersection of the given lines and having the largest distance from the origin.

Option C ($x + 3y + 6z = 7$) is correct because the plane equation is found by combining the given lines and ensuring parallelism with the specified plane.

Option D ($\sqrt{14}$) is correct because it involves finding the equation of the plane, using the fact that it's parallel to the line, and then calculating the distance between the plane and the line.

Option C (A pair of planes at right angles) is correct because the equation xy = 0 represents the union of the x = 0 plane and the y = 0 plane, which are perpendicular.

Option C ($2 \sqrt{2}$) is correct because it is the distance calculated from the point to the plane, where the plane is perpendicular to the given planes.

Option C (7) is correct because there are 4 planes with one point on one side and the other three on the other side, and 3 planes with two points on each side.

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 C ($\sqrt{{2}}$) is correct because the projection of PQ on the plane is equal to the length of PQ since PQ is parallel to the plane.

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 C (5) is correct because substituting the direction ratios of the line into the plane equation and solving for k yields k=5.

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 D ($\frac{{4 \sqrt{{7}}}}{{3}}$) is correct because it represents the minimum distance calculated by projecting the vector connecting the point to the plane onto a vector perpendicular to the given line.

Option A (5) is correct because it is the value of $\alpha$ that satisfies the equation of the plane containing the given lines and passing through the point ($\alpha$, -3, 5).

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.