18. (This question is worth 12 points) As shown in Figure 4, the right triangular prism $\mathrm { ABC } - \mathrm { A } _ { 1 } \mathrm { B } _ { 1 } \mathrm { C } _ { 1 }$ has an equilateral triangle base with side length 2. $\mathrm { E }$ and $\mathrm { F }$ are the midpoints of $\mathrm { BC }$ and $\mathrm { CC } _ { 1 }$ respectively. (I) Prove that: plane $\mathrm { AEF } \perp$ plane $\mathrm { B } _ { 1 } \mathrm { BCC } _ { 1 }$ (II) If the angle between line $A _ { 1 } C$ and plane $A _ { 1 } A B B _ { 1 }$ is $45 ^ { \circ }$, find the volume of the triangular pyramid F-AEC. [Figure]

22. As shown in the figure, in the quadrangular pyramid $P - A B C D$, given that $P A \perp$ plane $A B C D$, and the quadrilateral $A B C D$ is a right trapezoid, $\angle A B C = \angle B A D = \frac { \pi } { 2 } , P A = A D = 2 , A B = B C = 1$
(1) Find the cosine of the dihedral angle between plane $P A B$ and plane $P C D$;
(2) Point Q is a moving point on segment BP. When the angle between line CQ and DP is minimized, find the length of segment BQ. [Figure]

17. (12 points) As shown in the figure, the rectangular prism $A B C D - A _ { 1 } B _ { 1 } C _ { 1 } D _ { 1 }$ has a square base $A B C D$. Point $E$ is on edge $A A _ { 1 }$, and $B E \perp E C _ { 1 }$. [Figure]
(1) Prove: $B E \perp$ plane $E B _ { 1 } C _ { 1 }$;
(2) If $A E = A _ { 1 } E , A B = 3$, find the volume of the quadrangular pyramid $E - B B _ { 1 } C _ { 1 } C$.

17. (12 points) As shown in the figure, the rectangular prism $A B C D - A _ { 1 } B _ { 1 } C _ { 1 } D _ { 1 }$ has a square base $A B C D$. Point $E$ is on edge $A A _ { 1 }$, and $B E \perp E C _ { 1 }$. [Figure]
(1) Prove that $B E \perp$ plane $E B _ { 1 } C _ { 1 }$;
(2) If $A E = A _ { 1 } E$, find the sine of the dihedral angle $B - E C - C _ { 1 }$.

Given a quadrangular pyramid with a square base of side length 4, and lateral edges of lengths $4, 4, 2\sqrt{2}, 2\sqrt{2}$ respectively, find the height of the quadrangular pyramid.

$A$ and $B$ are two like parallel forces. A couple of moment H lies in the plane of $A$ and $B$ and is contained with them. The resultant of $A$ and $B$ after combining is displaced through a distance
(1) $\frac{2\mathrm{H}}{\mathrm{A}-\mathrm{B}}$
(2) $\frac{H}{A+B}$
(3) $\frac{H}{2(A+B)}$
(4) $\frac{H}{A-B}$

A 'T' shaped object with dimensions shown in the figure, is lying on a smooth floor. A force $F$ is applied at the point $P$ parallel to $AB$, such that the object has only the translational motion without rotation. Find the location of $P$ with respect to $C$
(1) $\frac{2}{3}\ell$
(2) $\frac{3}{2}\ell$
(3) $\frac{4}{3}\ell$
(4) $\ell$

The magnitude of torque on a particle of mass 1 kg is 2.5 Nm about the origin. If the force acting on it is 1 N, and the distance of the particle from the origin is 5 m, the angle between the force and the position vector is (in radians):
(1) $\frac { \pi } { 6 }$
(2) $\frac { \pi } { 3 }$
(3) $\frac { \pi } { 8 }$
(4) $\frac { \pi } { 4 }$

A slab is subjected to two forces $\overrightarrow { \mathrm { F } } _ { 1 }$ and $\overrightarrow { \mathrm { F } } _ { 2 }$ of same magnitude $F$ as shown in the figure. Force $\overrightarrow { \mathrm { F } _ { 2 } }$ is in XY plane while force $\mathrm { F } _ { 1 }$ acts along $z$-axis at the point $( 2 \vec { i } + 3 \vec { j } )$. The moment of these forces about point O will be:
(1) $( 3 \hat { i } - 2 \hat { j } + 3 \hat { k } ) \mathrm { F }$
(2) $( 3 \hat { i } - 2 \hat { j } - 3 \hat { k } ) \mathrm { F }$
(3) $( 3 \hat { i } + 2 \hat { j } - 3 \hat { k } ) \mathrm { F }$
(4) $( 3 \hat { i } + 2 \hat { j } + 3 \hat { k } ) \mathrm { F }$

An $L$-shaped object, made of thin rods of uniform mass density, is suspended with a string as shown in figure. If $A B = B C$, and the angle made by $A B$ with downward vertical is $\theta$, then:
(1) $\tan \theta = \frac { 2 } { \sqrt { 3 } }$
(2) $\tan \theta = \frac { 1 } { 3 }$
(3) $\tan \theta = \frac { 1 } { 2 }$
(4) $\tan \theta = \frac { 1 } { 2 \sqrt { 3 } }$

A metre scale is balanced on a knife edge at its centre. When two coins, each of mass 10 g are put one on the top of the other at the 10.0 cm mark the scale is found to be balanced at 40.0 cm mark. The mass of the metre scale is found to be $x \times 10 ^ { - 2 } \mathrm {~kg}$. The value of $x$ is $\_\_\_\_$ .

A heavy iron bar, of weight $W$ is having its one end on the ground and the other on the shoulder of a person. The bar makes an angle $\theta$ with the horizontal. The weight experienced by the person is :
(1) $\mathrm { W } \cos \theta$
(2) $\frac { W } { 2 }$
(3) W
(4) $W \sin \theta$

The coordinates of a particle with respect to origin in a given reference frame is $(1,1,1)$ meters. If a force of $\overrightarrow{\mathrm{F}} = \hat{i} - \hat{j} + \hat{k}$ acts on the particle, then the magnitude of torque (with respect to origin) in the $z$-direction is \_\_\_\_ .