3D Geometry Questions and Answers

Which is NOT an undefined term in geometry A B D plane line point axis

8 Give an approximate measure for the angle

b Sketch a graph with a line of symmetry of at x 5 a range of 0 4 and x intercepts 2 units from the line of symmetry allste

Find an equation of the tangent plane to the surface z 2x 1y 2x 3y 3 at the point 5 1 59 Z

the curves y OF 3x and y 5 4 3 2 1 5 4 3 2 1 9 7 6 5 4 3 7 6 5 2 x 2 2 2 Find the area of the region between y y x 2 2 3x and

Find the tangent of ZZ Z 14 4 00 Y Write your answer in simplified rationalized form Do not rou tan Z X

X 8 W 89 5 Y Write your answer in simplified rationalized form Do not rour tan Y

7 Classify the pair of angles in the diagram shown at the right a complementary b adjacent 141 30

Find and graph the image of quadrilateral PQRS after a dilation centered at the origin with a scale factor of 1 What are the coordinates of Q P 0 4 Q 8 4 R 8 8 S 0 8 O Q 0 2 O Q 2 1 O Q 32 16 Q 2 2 YA X

Find the area of the triangle with vertices Q 5 3 3 R 7 2 5 S 4 6 5

Find an equation of the plane consisting of all points that are equidistant from 2 5 4 and 5 2 2 Note you have to enter the full equation

Find the angle of intersection of the plane 2x 4y 5z 4 with the plane 4x 4y 1z 5 Answer in radians and in degrees

1 point Find the point at which the line 2 3 2 t 5 4 3 intersects the plane 2x 5y 3z 10

is inside the sphere 2 x y x 9 outside the cone etc 2 z x y and above the xy plane a Draw the diagram using a mathematical application GeoGebra etc 3m b Draw the projection of the solid using a mathematical application GeoGebra 3 m c Hence find the volume of the solid using an appropriate coordinate system 9m

5 Let O be the centre of the circle x y r where r 2 equation of the line passing through P and Q is 2x 4y 5 If the centre of the circum OPQ lies on the line x 2y 4 then the value of r is Suppose PQ is a chord o

Each question has FOUR options A B C and D ONE OR MORE THAN ONE of these four option s is are correct For each question darken the bubble s corresponding to all the correct option s in the ORS For each question marks will be awarded in one of the following categories Full Marks 4 If only the bubble s corresponding to all the correct option s is are darkened For darkening a bubble corresponding to each correct option provided NO incorrect option is darkened If none of the bubbles is darkened In all other cases Zero Marks 0 Negative Marks 2 For example if A C and D are all the correct options for a question darkening all these three will result in 4 marks darkening only A and D will result in 2 marks and darkening A and B will result in 2 marks as a wrong option is also darkened Partial Marks Q 43 Let f x lim A f 1 C 2 0 1 n x n x 2 x n n x n x 4 n n x B D for all x 0 Then 3 2 2 f 2 f 3

Parameters a 0 b 9c 8 The equation ax1 bx2 cx3 0 is given Firstly Plug your parameters instead of a b c Then find the solution set Find a basis for the solution set Explain why the solution set is a subspace of R3

8 Find the horizontal asymptote of each of the following graphs 8x 7x 1 4x 6 a R x c R x x 3 2x 2x 5 9x6 4x 3 3x6 2x5 x4 7x 1 b R x

Problem 6 10 points Consider a triangle A that has vertices x y x2 y2 and x3 y3 Provide a short convincing explanation stating why or why not it is possible to find a 2D affine transformation that will map the vertices of triangle A onto the vertices of any new triangle B You do not need to write a detailed mathematical proof but your reasoning must be clear You do not need to consider degenerate cases such as all vertices being colinear

The radius of a cylindrical construction pipe is 2 ft and its length is 24 ft Answer the parts below Make sure that you use the correct units in your answers If necessary refer to the list of geometry formulas a Find the exact volume of the pipe Write your answer in terms of Exact volume 24 ft Approximate volume 2 ft b Using the ALEKS calculator approximate the volume of the pipe To do the approximation use your answer to part a and the button on the calculator Round your answer to the nearest hundredth 00 ft B JT ft

Triangle DEF has the dimensions shown 6 m 5m 15 m Which statement about the triangle is correct ADF 13 5 m since corresponding parts of similar triangles are proportional DF 19 5 m since a line drawn parallel to the base divides the other two sides proportionally DF 16 m since corresponding parts of similar triangles are proportional OF 24 m since line drawn parallel to the base dhades the other two sides proportionally

A cube has a surface area of 96 square yards If the length of each side is doubled what is the volume of the new cube A 512 cu yd B 384 cu yd C 128 cu yd D 64 cu yd

Daniel baked the cake shown He covered one half of the top of the cake with berries 9 in What is the area of the part of the cake he covered with berries rounded to the nearest hundredth Use 3 14 for m A 63 62 sq in B 254 47 in C 127 23 sq in 31 81 sq in

1 Give an example of an elliptic curve and show that it is supersingular

Which 3 D shaped fuel tank has more internal volume a fuel tank shaped like a cube with side 5 or a fuel tank shaped like a sphere with diameter 5 OO The Cube The volumes are equal The Sphere

What is the volume of a cube that measures 6 376 on a side 275 5 in3 259 2 in3

The solid below is made from 1 meter cubes Find the total surface area of the solid Be sure to include the correct unit in your answer 1 meter E X m m

Find all the local maxima local minima and saddle points of the function f x y 12x 5y 8x 15 1 4x

3 Each of these containers are filled with water through an opening at the top Graph the water level in the container as it is filled at a steady rate C A

Find the volume of each figure Round your answers to the nearest hundredth if necessary 1 2 3 4 6 in 16 in 5 in 6 ft 5 in 5 yd 3 yd 3 yd 4 yd

What does the notation Ry 3x 1 DAB mean Counter clockwise Reflection O Clockwise Over the x axis 180 270 Rotation Around the origin to the axis Over the y axis Over y 3x 1 90

Triangle XYZ is reflected across the y axis Then its image is rotated 90 degrees about the origin What are the coordinates of the final image of point X under the composition of transformations X 3 Z 2 0 3 0 3 2 3 2 O 2 1 O 2 3 2

Give a parametric description of the form r u v x u v y u v z u v for the following surface 5 The cap of the sphere x y z 25 for SZ 5 Select the correct choice below and fill in the answer boxes to complete your choice Type any angle measures in radians Use angle measures greater than or equal to 0 and less than 2x Type exact answers in terms of A r u v 5 sin u cos v 5 sin u sin v 5 cos u for OB r u v 25 cos u 25 sin u v for OC r u v 5 cos u sin v 5 sin u sin v 5 sin v for Sus r u y 5 cos u 5 sin u v for and Sus and sus and SVS sus SVS and SVS S SVS

Evaluate JJJE where x y z dV E is the solid in the first octant that lies under the sphere z x y 4

A surveyor on one side of a river wants to find the distance between points A and B on the opposite side of the river On her side she chooses points C and D which are 20 meters apart and measures the angles as shown in the figure below Find the distance between A and B in the most efficient way Show and label each step clearly A C B D mLACB 40 mzBCD 20 mLADC 45 mzADB 42 Call the intersection of the two diagonals point E This will make it easy to refer to the triangles

6 Complete the partial proof below Given AC L BD and AC bisects BD Prove AD AB 1 AC L BD and AC bisects BD 2 mLACB 90 mLACD 90 3 Statement 4 BC DC 5 6 AACD AACB B 1 Given 2 3 4 5 Reflexive Property 6 C Reason 7 D

6 You are given the line whose equation is 2x y 3 Answer the following questions a b a Write the given equation in slope intercept form 2 point b What is the slope of the line parallel to this line 1 point c What is the slope of the line perpendicular to this line 1 point d What is the equation of the line that passes through the point 5 4 and is perpendicular to the line whose equation is 2x y 3 in slope intercept form 2 point

1 Complete the two column proof 3 points Given Triangle ACD is isosceles with segment AC congruent to segment AD 1 is congruent to 3 Prove Segment AB Segment CD B 3 Statement Triangle ACD is isosceles with segment AC congruent to segment AD 1 is congruent to 3 1 is congruent to 4 Segment AB Segment CD Reason Isosceles triangle theorem Base angles of an isosceles triangle are congruent

2 Finish the two column proof 3 points Given 2 is congruent to 5 Segment AB is congruent to Segment DE Prove Segment BC is congruent to Segment EC B 3 C E Statement 2 is congruent to 5 Segment AB is congruent to Segment DE 3 and 4 are vertical angles 3 is congruent to 4 Triangle ABC is congruent to triangle DEC Segment BC is congruent to Segment EC Reason

2 A given statement and its annotated diagram are found below There are two mistakes Identify and correct the mistakes and state why you made those changes Focus on the given info and annotations Given AD bisects LBDC D is the midpoint of BC Mistake 1 D

The graphs of the functions y f2 x and y f3 r are given in Figures 5 and 6 respectively Let f x g x f3 x Figure 5 y f x a What is the domain of g x b Where is the function g r continuous I I I Figure 6 y f x

Find the perimeter of a parallelogram a 25 cm b 30 cm h 20 cm 130 cm 2 100 cm 2 110 cm 100 cm b a

The line has equation and r 4i 3j k i 2j 2k The plane p has equation 2x 3y z 4 4 marks a Find the acute angle between I and p b A second plane q is parallel to I perpendicular to p and contains the point with position vector 4j k Find the equation of q giving your answer in cartesian form 4 marks

Plot the functions et and 1 eversus log10 t from t 102 to t 10 Have two curves on the plot for each function one for t 1 and one for t 1

8 9 1 point Given the angle measure below what must be true O O O O A B 102 Both lines are parallel C D Neither Lines are parallel A B 78 50 102 78 1 point A diagnol brace strengthens the wire fenct and prevents it from sagging The brace makes a 50 angle with the wire as ashown What would the value of y be

14 Find the great circle containing the points P 0 1 2 3 2 and 3 5 0 4 5

Problem 3 Let MNPQ be a parallelogram Consider a point E inside MNPQ such that PE PN The points I and J are respectively the midpoints of the segments ME and PQ Show that NE L IJ

Find general and parametric equations of the plane containing the points A(3, 0, 0), B(0, 1, 0) and perpendicular to the XY-plane.

Which point lies on the line defined by (x,y,z) = (5, -1, 2) + t(1,2,-1)?

What is the cross section perpendicular to the base of a cylinder? circle triangle rectangle trapezoid