Mathematics : Three Dimensional Geometry

Direction ratios of the line which is perpendicular to the lines with direction ratios - 1, 2, 2 and 0, 2, 1 are

• 1, 1, 2

• 2, - 1, 2

• - 2, 1, 2

• 2, 1, - 2

If the angle between the planes $\mathrm{r} . \left(\mathrm{m}\hat{\mathrm{i}} - \hat{\mathrm{j}} + 2\hat{\mathrm{k}}\right)$ + 3 = 0 and $\mathrm{r} . \left(2\hat{\mathrm{i}} - \mathrm{m}\hat{\mathrm{j}} - \hat{\mathrm{k}}\right) - 5$ = 0 is $\frac{\mathrm{\pi }}{3}$, then m =

• 2

• $\pm 3$

• 3

• - 2

If the origin and the points P(2, 3, 4 ), Q(1, 2, 3) and R(x, y, z) are coplanar, then

• x - 2y - z = 0

• x + 2y + z = 0

• x - 2y + z = 0

• 2x - 2y + z = 0

If lines represented by equation px² - qy² = 0 are distinct, then

• pq > 0

• pq < 0

• pq = 0

• p + q = 0

The equation of the plane through (- 1, 1, 2) whose normal makes equal acute angles with coordinate axes is

• $\mathrm{r}. \left(\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}\right) = 2$

• $\mathrm{r}. \left(\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}\right) = 6$

• $\mathrm{r}. \left(3\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + 3\hat{\mathrm{k}}\right) = 2$

• $\mathrm{r}. \left(\hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}\right) = 3$

If distance of points $2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + \mathrm{\lambda }\hat{\mathrm{k}}$ from the plane $\mathrm{r} . \left(3\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 6\hat{\mathrm{k}}\right)$= 13 is 5 units, then $\mathrm{\lambda }$ = n

• $6, - \frac{17}{3}$

• $6, \frac{17}{3}$

• $- 6, - \frac{17}{3}$

• $- 6, \frac{17}{3}$

$∆$ABC has vertices at A = (2, 3, 5), B = (-1, 3, 2) and C = $\left(\mathrm{\lambda }, 5, \mathrm{\mu }\right)$. If the median through A is equally inclined to the axes, then the values of $\mathrm{\lambda } \mathrm{and} \mathrm{\mu }$ respectively are

• 10, 7

• 9, 10

• 7, 9

• 7, 10

A plane is flying horizontally at a height of 1 km from ground. Angle of elevation of the plane at a certain instant is 60°. After 20 s, angle of elevation is found 30°. The speed of plane is

• $100\sqrt{3} \mathrm{m}/\mathrm{s}$

• $200\sqrt{3} \mathrm{m}/\mathrm{s}$

• $\frac{100}{\sqrt{3}} \mathrm{m}/\mathrm{s}$

• $\frac{200}{\sqrt{3}} \mathrm{m}/\mathrm{s}$

The maximum horizontal range of a ball projected with a velocity of 40 m/s is (take g = 9.8m/s²)

• 157 m

• 127 m

• 163 m

• 153 m

A body of 6 kg rests in limiting equilibrium on an inclined plane whose slope is 30°. If the plane is raised to slope of 60°, then force (in kg-wt) along the plane required to support it is

• 3

• $2\sqrt{3}$

• $\sqrt{3}$

• $3\sqrt{3}$