# Grade 10 Math Questions and Solutions

The Smarter Balanced Assessment Consortium (SBAC) is a standardized test that includes a variety of new technology-enhanced questions.

Some of them are Multiple choice-single correct responses, Multiple choice-multiple correct responses, Matching Tables, Drag and Drop, Hot text, Table Fill in, Graphing, Equation/numeric, Extended constructed response, Short answer, and many more.

This page contains several sample questions along with practice test links for Grade 10 Math that gives you an idea of questions that your students are likely to see on the test. After each sample question, an answer explanation follows. The explanation includes essential aspects of the task that you may need to consider for the skills, processes, and information your students need to know.

## Domain: Grade 10 >> Number and Quantity – The Real Number System

Sample Question: Evaluate 9 150/300

1. 18
2. 9
3. 3
4. 81

Answer Explanation: 9150/300 = 91/2 = square root of 9 = 3. In a problem with a rational exponent, the numerator tells you the power, and the denominator the root. However, in this problem the exponent can be reduced, so we should reduce that first. The exponent 150/300 = 1/2. So the problem becomes 9 to the 1/2 power. The denominator is 2 so we take the square root of 9 which is 3. The numerator is 1 so we raise 3 to the 1st power and the answer is 3.

Standards: HSN.RN.A.1

Click here to practice: Number and Quantity – The Real Number System Questions on Grade 10 Math

## Domain: Grade 10 >> Number and Quantity – Quantities

Sample Question: Rewrite x1/2 in radical form.

1. √x
2. √x2
3. 1/√x
4. -√x

Answer Explanation: In a problem with a rational exponent, the numerator tells you the power, and the denominator the root. Since the problem is, x1/2, the denominator is 2 indicating we should take a square root and the numerator is 1 so we would raise that to the first power or there will be no exponent since an exponent of 1 is rarely used. That makes the answer the square root of x, written as √x.

Standards: HSN.RN.A.1

## Domain: Grade 10 >> Number and Quantity – The Complex Number System

Sample Question: Simplify completely i(7−i)

1. 7i−i2
2. 1+7i
3. 6i
4. −1+7i

Start by using the distributive method. Now simplify −i2=1 by definition. Now rearrange and put the real part first and the imaginary part last so that it looks like this a+bi.

Standards: HSN.CN.A.2

## Domain: Grade 10 >> Number and Quantity – Vector & Matrix Quantities

Sample Question: A vector in standard form has components <3, 10>. What is the initial point?

1. (0, 0)
2. (3, 10)
3. (6, 20)
4. Not enough information given

Answer Explanation: Since the vector is in standard position, we know that the initial point is (0, 0) or the origin.

Standards: HSN.VM.A.2

Click here to practice: Number and Quantity – Vector & Matrix Quantities Questions for Grade 10 Math

## Domain: Grade 10 >> Algebra – Seeing Structure in Expressions

Sample Question: Which expression is equivalent to 9x2 – 16y2?

1. (3x – 4y) (3x – 4y)
2. (3x + 4y) (3x + 4y)
3. (3x + 4y) (3x – 4y)
4. (3x – 4y)2

Answer Explanation: Student must recognize the expression is the difference of two perfect squares

Standards: HSA.SSE.A.2

## Domain: Grade 10 >> Algebra – Arithmetic with Polynomials & Rational Expressions

Sample Question: Evaluate f(x)=−a3+6a−7 at a = – 1 and state the remainder.

1. -14
2. -12
3. 14
4. 12

Answer Explanation: student must substitute – 1 into the function as follows −(−1)3+6(−1)−7=−12 and find the value to get the remainder

Standards: HSA.APR.B.2

Click here to practice: Algebra – Arithmetic with Polynomials & Rational Expressions Questions for Grade 10 Math

## Domain: Grade 10 >> Algebra – Creating Equations

Sample Question: The ratio of staff to guests at the gala was 3 to 5. There were a total of 576 people in the ballroom. How many guests were at the gala?

1. 276
2. 300
3. 360
4. 216

Answer Explanation: Setup a proportion of guests to the total number of people, 8/5 = x/576. Solve by cross multiplying. 8x = 2880. Divide both sides by 8. So x=360.

Standards: HSA.CED.A.3

## Domain: Grade 10 >> Algebra – Reasoning with Equations & Inequalities

Sample Question: Solve the quadratic x2+10x=−25.

1. -10
2. 10
3. 5
4. -5

Answer Explanation: This problem can be easily solved by rearranging the equation so that it is solved for zero and then factoring as shown:

x2+10x=−25

x2+10x+25=0

(x+5)(x+5)=0

Since both factors are exactly the same, you will only have one solution to this problem.

x+5=0

x=−5

Standards: HSA.REI.B.4

## Domain: Grade 10 >> Functions – Interpreting Functions

Sample Question: Which graph could represent the graph of f(x)=sin(x)?

1. 2. 3. 4. Answer Explanation: The sin function always graphs to look like a wave. The only one that could be the sin function is D.

Standards: HSF.IF.C.7

## Domain: Grade 10 >> Functions – Building Functions

Sample Question: Describe how the graph of g(x)=x3 – 5 can be obtained by shifting f(x) = x3 + 2.

1. Shift right 7 units
2. Shift left 7 units
3. Shift up 7 units
4. Shift down 7 units

Answer Explanation: The only thing that changed in the two equations is the y-intercept which controls the vertical shift (up or down). To get the graph of g(x) by shifting the graph of f(x), you would shift f(x) down 7 units to change from +2 to -5.

Standards: HSF.BF.B.3

## Domain: Grade 10 >> Functions – Interpreting Functions

Sample Question: Solve 3x=12 using logarithmic form.

1. x = ln12/ln3
2. x = ln(4)
3. x = ln(9)
4. None of these

Solve using logs as follows
3x=12
x=log(base 3) 12
x=ln12/ln3

Standards: HSF.LE.A.4

## Domain: Grade 10 >> Functions – Trigonometric Functions

Sample Question: In the unit circle, one can see that tan(5π/4)=1 . What is the value of cos(5π/4)?

1. −√2/2
2. undefined
3. √2/2
4. -1

Answer Explanation: The trigonometric ratio of cosine is the ratio of the length of the adjacent side divided by the length of the hypotenuse. The length of the adjacent side is the x−value in a point on the unit circle. The hypotenuse is the radius of the unit circle, so the hypotenuse is 1. Thus, the value of the cosine ratio of any angle in the unit circle is the x−value of the point on the unit circle that corresponds to that angle. The trigonometric ratio of tangent is the length of the opposite side divided by the length of the adjacent side. The length of the opposite side is the y−value in a point on the unit circle and the length of the adjacent side is the x−value in a point on the unit circle. The hypotenuse is the radius of the unit circle, so the hypotenuse is 1. Thus, the value of the tangent ratio of any angle in the unit circle is the ratio yx from the point on the unit circle that corresponds to that angle. In this question, tan(5π/4)=1. This ratio is taken from the point (−2/√2,−2/√2) that corresponds to the angle with a measure of 5π/4 radians. Thus, using the information above, the value of cos(5π4) is the same as the x−value in the point (−2/√2,−2/√2).Therefore, the value of cos(5π/4)=−2/√2.

Standards: HSF.TF.A.2

## Domain: Grade 10 >> Geometry – Congruence

Sample Question: What would be the coordinates of point S after applying the following rule: (x+3, y -2)? 1. (1, -4)
2. (-2, -2)
3. (2, -2)
4. (3, -2)

Explanation: The transformation rule that is give is to translate the point 3 units to the right and 2 units down as shown by the following diagram: Standards:

## Domain: Grade 10 >> Geometry – Similarity, Right Triangles, & Trigonometry

Sample Question: By what property can the angles BAX and TSX be found to be congruent? 1. Corresponding angles
2. Vertical angles
3. Alternate interior angles
4. Congruent angles

Although they are congruent angles, the question is asking for the property. Since they are in corresponding locations with the transversal (AX) the correct answer is A

Standards: HSG.SRT.A.3

## Domain: Grade 10 >> Geometry – Circles

Sample Question: What is the translation rule and the scale factor of the dilation as Circle F→Circle F′ ? 1. (x,y)→1/4(x,y+10)
2. (x,y)→4(x,y+10)
3. (x,y)→1/4(x+10,y)
4. (x,y)→1/4(x,y−10)

Answer Explanation: The original circle F has its center at the point (−5,−6) with a radius of 4 units. The translated/dilated circle F’ has its center at the point (−5,4) with a radius of 1 units. This means the center was translated up 10 units. As a transformation, this translation is written as (x,y)→(x,y+10). Circle F was also dilated by a factor of 1/4 because the radius was reduced from 4 units to 1 units. As a transformation, this dilation is written as (x,y)→1/4(x,y). Putting the translation and dilation together, the rule is (x,y)→1/4(x,y+10).

Standards:

## Domain: Grade 10 >> Geometry – Expressing Geometric Properties with Equations

Sample Question: What value on the number line in the figure below divides segment EF into two parts having a ratio of their lengths of 3:1? 1. -5
2. -3
3. -2
4. -1

Answer Explanation: Point E is at -7 on the number line in the figure, and pointF is at 1. Thus, the length of segment EF is 8. To divide the segment into two parts with a ratio of their lengths of 3:1, change the ratio to 3x:1x to allow variation in the location on the number line. Next, set the sum of the two parts equal to 8 and solve for x. 3x+1x=8;4x=8;x=2.Now, that you know that x=2, find 3x, which equals 6. Find the value on the number line by adding 6 to the position of point E. −7+6=−1.The value on the number line that divides segment EF in a ratio of 3:1 is -1.

Standards: HSG.GPE.B.6

## Domain: Grade 10 >> Geometry – Geometric Measurement & Dimension

Sample Question: What is the volume of the prism shown below? 1. 1350 cm3
2. 1350 cm
3. 675 cm3
4. 675 cm

Answer Explanation: Use the formula for volume of a pyramid:

V=1/2.a.c.h

In this case the length is 15cm, the base is 10 cm in length, and the height is 9 cm. Therefore :

V=1/2.15.10.9=675cm3

Standards: HSG.GMD.A.3

## Domain: Grade 10 >> Geometry – Modeling with Geometry

Sample Question: A company ships spherical paperweights in cubic boxes. The circumference of the paperweight is 9π cm. If the box fits the sphere exactly with the sides of the sphere touching the box, what is the volume of the smallest box the company can use for shipping.

1. 81 cm3
2. 81 π cm3
3. 729 cm3
4. 1009 π cm3 Notice that the diameter of the sphere will be the same as the side of the cubic box. Using the value of the circumference the diameter of the paperweight can be determined.
C = πd9π
cm = πd9
cm = d
Since the diameter is equal in measure to the sides{\dots}
V=s3
V=(9 cm)3
V=729 cm3

Standards: HSG.MG.A.3

## Domain: Grade 10 >> Statistics & Probability – Interpreting Categorical & Quantitative Data

Sample Question: Given the scatter plot below, what type of function expresses the correlation between the two variables? 1. Linear
2. Exponential
4. Polar

Answer Explanation: Notice that the trend of the graph (in red) between the data points forms a line.

Standards: HSS.ID.A.4

Click here to practice: Statistics & Probability – Interpreting Categorical & Quantitative Data Questions for Grade 10 Math

## Domain: Grade 10 >> Statistics & Probability – Making Inferences & Justifying Conclusions

Sample Question: In a research project about pet behavior, a random sample of 400 cats was chosen. The study showed that 60% of the cats preferred to sleep inside the house. Chicken was the favorite food for 35% of those cats. The study also showed that 85% of the cats that preferred to sleep outside the house had a different favorite dish. How many cats in the sample liked chicken the best and preferred to sleep inside?

1. 84
2. 56
3. 160
4. 156

Answer Explanation: If the sample has 400 cats and 60% of the cats preferred to sleep inside, then 400.0.60=240 cats preferred to sleep inside and 160 cats preferred to sleep outside. Next, if the favorite dish of 35% of those cats that preferred to sleep inside was chicken, then, 240.0.35=84 cats in the sample preferred to sleep inside and had chicken as their favorite dish.

Standards: HSS.IC.B.6

Click here to practice: Statistics & Probability – Making Inferences & Justifying Conclusions Questions for Grade 10 Math

## Domain: Grade 10 >> Statistics & Probability – Conditional Probability & the Rules of Probability

Sample Question: A student council has one upcoming vacancy. The school is holding an election and has eight equally likely candidates. The AP Statistics class wants to simulate the results of the election, so the students have to choose an appropriate simulation method. They intend to do trials with the simulation. Which of these methods would be the most appropriate?

1. Spin a wheel with eight equal spaces
2. Toss a coin eight times for each election
3. Throw a dice
4. Throw four die

Answer Explanation: The question states that there are eight equally likely candidates. This means that each candidate has the same chance of winning the election. Only the spinning wheel with eight equal spaces could simulation this situation because the wheel has an equal chance of landing on each space.

Standards: HSS.IC.A.1

Click here to practice: Statistics & Probability – Conditional Probability & the Rules of Probability Questions for Grade 10 Math

## Domain: Grade 10 >> Statistics & Probability – Using Probability to Make Decisions

Sample Question: Using just the Venn diagram above, find P(C or E).

1. 1/3
2. 7/24
3. 5/24
4. None of these

Answer Explanation: Simply count the data points in circles C and E. There are 8 of them out of 24 total data points and by reducing we get 8/24=1/3.

Standards: HSS.CP.B.7

Click here to practice: Statistics & Probability – Using Probability to Make Decisions Questions for Grade 10 Math

## Domain: Grade 10 >> Statistics & Probability – Using Probability to Make Decisions

Sample Question: A statistician is working for Sweet Shop USA and has been given the task to find out what the probability is that the fudge machine malfunctions messing up a whole batch of fudge in the process. Each malfunction of the machine costs the company \$250. The statistician calculates the probability is 1 in 20 batches of fudge will be lost due to machine malfunction. What is the expected value of this loss for one month if the company produces 20 batches of fudge each day?

1. \$3750
2. \$150,000
3. \$7500
4. \$375

Answer Explanation: Since most months have 30 days we will assume 30 days in a month. We can use E(x)=x1p1+x2p2+…+xipi or simply calculate as follows
E(X)=.05*250*30=\$375

Standards: HSS.MD.A.4

Click here to practice: Statistics & Probability – Using Probability to Make Decisions Questions for Grade 10 Math