Which real numbers are equal to their cubes?

Write 4*10-2 as a decimal.

Write 0.12*10-3 as a decimal.

Write 2 log3 x + log3 5 as a single logarithmic expression.

Factor the algebraic expression 6x2 - 21xy + 8xz - 28yz.

Factor the algebraic expression (x - 1)2 - (y - 2)2.

Factor the algebraic expression x2 - z4.

Evaluate the algebraic expression |-2x - y + 3| for x = 3 and y = 5

Simplify the algebraic expression -2(x - 3) + 4(-2x + 8)

Expand and simplify the algebraic expression (x + 3)(x - 3) - (-x - 9)

Which property is used to write a(x + y) = ax + ay

Simplify 8 x3 / 2 x-3

Simplify (-a2b3)2(c2)0

For what value of k is the point (-2, k) on the line with equation -3x + 3y = 4?

For what value of a will the system given below have no solutions?

2x + 6y = -2 -3x + ay = 4

Which equation best describes the relationship between x and y in this table?

x y

0 -4

4 -20

-4 12

8 -36

A. y = - x/4 - 4 B. y = - x/4 + 4 C. y = - 4x - 4

D. y = - 4x + 4

Which equation best represents the area of the rectangle below?

.

A. area = 2(x+1) + 2(x-1) B. area = 4(x+1)(x-1) C. area = 2x2 D. area = x2 - 1

Which line given by its equation below contains the points (1, -1) and (3, 5)?

A. -2y -6x = 0 B. 2y = 6x - 8 C. y = 3x + 4 D. y = -3x + 4

Solve the equation 2|3x - 2| - 3 = 7.

Solve for x the equation (1/2)x2 + mx - 2 = 0.

For what values of k the equation -x2 + 2kx - 4 = 0 has one real solution?

For what values of b the equation x2 - 4x + 4b = 0 has two real solutions?

Function f is described by the equation f (x) = -x2 + 7. What is the set of values of f(x) corresponding to the set for the independent variable x given by {1, 5, 7, 12}?

Find the length and width of a rectangle whose perimeter is equal to 160 cm and its length is equal to triple its width.

Simplify: |-x| + |3x| - |-2x| + 3|x|

If (x2 - y2) = 10 and (x + y) = 2, find x and y.

Geometry Problems with Answers and Solutions - Grade 10

Grade 10 geometry problems with answers are presented.

1. Each side of the square pyramid shown below measures 10 inches. The slant height, H, of this pyramid measures 12 inches.

.

a. What is the area, in square inches, of the base of the pyramid?

b. What is the total surface area, in square inches, of the pyramid?

c. What is h, the height, in inches, of the pyramid?

d. Using the height you determined in part (c), what is the volume, in cubic inches, of the pyramid?

2. The parallelogram shown in the figure below has a perimeter of 44 cm and an area of 64 cm2. Find angle T in degrees.

.

3. Find the area of the quadrilateral shown in the figure.(NOTE: figure not drawn to scale)

.

4. In the figure below triangle OAB has an area of 72 and triangle ODC has an area of 288. Find x and y.

.

5. Find the dimensions of the rectangle that has a length 3 meters more that its width and a perimeter equal in value to its area?

6. Find the circumference of a circular disk whose area is 100pi square centimeters.

7. The semicircle of area 1250 pi centimeters is inscribed inside a rectangle. The diameter of the semicircle coincides with the length of the rectangle. Find the area of the rectangle.

Answers to the Above Questions

1.a) 100 inches squared b) 100 + 4*(1/2)*12*10 = 340 inches squared c) h = sqrt(122 - 52) = sqrt(119) d) Volume = (1/3)*100*sqrt(119) = 363.6 inches cubed (approximated to 4 decimal digits)

2.

.

44 = 2(3x + 2) + 2(5x + 4) , solve for x x = 2

height = area / base = 64 / 14 = 6 cm

sin(T) = 6 / 8 , T = 48.6o

3.

.

ABD is a right triangle; hence BD2 = 152 + 152 = 450

Also BC2 + CD2 = 212 + 32 = 450

The above means that triangle BCD is also a right triangle and the total area of the

quadrilateral is the sum of the areas of the two

right triangles.

Area of quadrilateral = (1/2)*15*15 + (1/2)*21*3 = 144

4.

area of OAB = 72 = (1/2) sin (AOB) * OA * OB

solve the above for sin(AOB) to find sin(AOB) = 1/2

area of ODC = 288 = (1/2) sin (DOC) * OD * OD

Note that sin(DOC) = sin(AOB) = 1/2, OD = 18 + y and OC = 16 + x and substitute in the above to obtain the first equation in x and y

1152 = (18 + y)(16 + x)

We now use the theorem of the intersecting lines outside a circle to write a second equation in x and y

16 * (16 + x) = 14 * (14 + y)

Solve the two equations simultaneously to obtain

x = 20 and y = 14

5. Let L be the length and W be the width of the rectangle. L = W + 3

Perimeter = 2L + 2W = 2(W + 3) + 2W = 4W + 6

Area = L W = (W + 3) W = W2 + 3 W

Area and perimeter are equal in value; hence

W2 + 3 W = 4W + 6

Solve the above quadratic equation for W and substitute to find L

W = 3 and L + 6

6. Let r be the radius of the disk. Area is known and equal to 100Pi; hence

100Pi = Pi r2

Solve for r: r = 10

Circumference = 2 PI r = 20 Pi

7. Let r be the radius of the semicircle. Area of the semicircle is known; hence

1250Pi = (1/2) Pi r2 (note the 1/2 because of the semicircle)

Solve for r: r = 50

Length of rectangle = 2r = 100 (semicircle inscribed)

Width of rectangle = r = 50 (semicircle inscribed)

Area = 100 * 50 = 5000

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Simplify the following algebraic expressions.

A. -6x + 5 + 12x -6

B. 2(x - 9) + 6(-x + 2) + 4x C. 3x2 + 12 + 9x - 20 + 6x2 - x D. (x + 2)(x + 4) + (x + 5)(-x - 1) E. 1.2(x - 9) - 2.3(x + 4) F. (x2y)(xy2) G. (-x2y2)(xy2)

Simplify the expressions.

A. (a b2)(a3 b) / (a2 b3)

B. (21 x5) / (3 x4) C. (6 x4)(4 y2) / [ (3 x2)(16 y) ] D. (4x - 12) / 4 E. (-5x - 10) / (x + 2) F. (x2 - 4x - 12) / (x2 – 2x – 24)

Solve for x the following linear equations.

A. 2x = 6

B. 6x - 8 = 4x + 4 C. 4(x - 2) = 2(x + 3) + 7 D. 0.1 x - 1.6 = 0.2 x + 2.3 E. - x / 5 = 2 F. (x - 4) / (- 6) = 3 G. (-3x + 1) / (x - 2) = -3 H. x / 5 + (x - 1) / 3 = 1/5

Find any real solutions for the following quadratic equations.

A. 2 x2 - 8 = 0

B. x2 = -5 C. 2x2 + 5x - 7 = 0 D. (x - 2)(x + 3) = 0 E. (x + 7)(x - 1) = 9 F. 2x2 + 5x + 11 = 0 G. x(x - 6) = -9

Find any real solutions for the following equations.

A. x3 - 1728 = 0

B. x3 = - 64 C. sqrt(x) = -1 D. sqrt(x) = 5 E. sqrt(x/100) = 4 F. sqrt(200/x) = 2

Evaluate for the given values of a and b.

A. a2 + b2 , for a = 2 and b = 2

B. |2a - 3b| , for a = -3 and b = 5 C. 3a3 - 4b4 , for a = -1 and b = -2

Solve the following inequalities.

A. x + 3 < 0

B. x + 1 > -x + 5 C. 2(x - 2) < -(x + 7)

For what value of the constant k does the quadratic equation x2 +2x = - 2k have two distinct real solutions?

For what value of the constant b does the linear equation 2x + by = 2 have a slope equal to 2?

What is the y intercept of the line -4x + 6y = -12?

What is the x intercept of the line -3x + y = 3?

What is point of intersection of the lines x - y = 3 and -5x - 2y = -22?

For what value of the constant k does the line -4x + ky = 2 pass through the point (2,-3)?

What is the slope of the line with equation y - 4 = 10?

What is the slope of the line with equation 2x = -8?

Find the x and y intercepts of the line with equation x = - 3?

Find the x and y intercepts of the line with equation 3y - 6 = 3?

What is the slope of a line parallel to the x axis?

What is the slope of a line perpendicular to the x axis?