Welcome to **AMBiPi (Amans Maths Blogs)**. SAT (Scholastic Assessment Test) is a standard test, used for taking admission to undergraduate programs of universities or colleges in the United States. In this article, you will get * SAT 2022 Math Practice Problems Test 10 Grid Ins Questions with Answer Keys | SAT Online Classes AMBiPi*.

### SAT 2022 Math Practice Problems Test 10 Grid Ins Questions with Answer Keys

**SAT Math Practice Online Test Question No 1:**

What is the value of the complex number (1/4)i^{42} + i^{60}?

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**Correct Answer: 3/4 or .75 **

Difficulty: Hard

Category: Additional Topics in Math / Imaginary Numbers

Strategic Advice: To evaluate a high power of i, look for patterns and use the definition √-1 = i, which can be written in a more useful form as i^{2} = -1.

Getting to the Answer: Write out enough powers of i that allow you to see the pattern:

i^{1} = i

i^{2} = -1 (definition)

i^{3} = i x i^{2 }= i x -1 = -i

i^{4} = i^{2} x i^{2 }= -1 x -1 = 1

i^{5} = i^{4} x i^{ }= 1 x i = i

i^{6} = i^{4} x i^{2}^{ }= 1 x -1 = -1

i^{7} = i^{6} x i^{ }= -1 x i = -i i^{8} = i^{4} x i^{4}^{ }= 1 x 1 = 1

Notice that the pattern (i, -1, -i, 1, i, -1, -i, 1) repeats on a cycle of 4. To evaluate i^{42}, divide 42 by 4. The result is 10, remainder 2, which means 10 full cycles, and then back to i^{42}. This means i^{42} is equivalent to i^{2}, which is -1. Do the same for i^{60}: 60 ÷ 4 = 15, remainder 0, which means stop on the 4th cycle to find that i^{60} = 1. Make these substitutions in the original equation:

(1/4)i + i = 1/4(-1) + 1 = -1/4 + 1 = 3/4

Grid in the answer as 3/4 or .75.

**SAT Math Practice Online Test Question No 2:**

{x + 3y ≤ 18, 2x – 3y ≤ 9}

If (a, b) is a point in the solution region for the system of inequalities shown above and a = 6, what is the minimum possible value for b?

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**Correct Answer: 1 **

Difficulty: Medium

Category: Heart of Algebra / Inequalities

Strategic Advice: This question is extremely difficult to answer unless you draw a sketch. It doesn’t have to be perfect you just need to get an idea of where the solution region is. Don’t forget to flip the inequality symbol when you graph the second equation.

Getting to the Answer: Sketch the system.

**
**

If (a, b) is a solution to the system, then a is the x-coordinate of any point in the darkest shaded region and b is the corresponding y-coordinate. When a = 6, the minimum possible value for b lies on the lower boundary line, 2x – 3y ≤ 9. It looks like the y-coordinate is 1, but to be sure, substitute x = 6 into the equation and solve for y. You can use = in the equation, instead of the inequality symbol, because you are finding a point on the boundary line.

2x – 3y = 9

2(6) – 3y = 9

12 – 3y = 9

-3y = -3

y = 1

**SAT Math Practice Online Test Question No 3:**

If y = ax^{2} + bx + c passes through the points (-3, 10), (0, 1), and (2, 15), what is the value of a + b + c?

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**Correct Answer: 6 **

Difficulty: Hard

Category: Passport to Advanced Math / Quadratics

Strategic Advice: The highest power of x in the equation is 2, so the function is quadratic. Writing quadratic equations can be tricky and time-consuming. If you know the roots, you can use factors to write the equation. If you don’t know the roots, you need to create a system of equations to find the coefficients of the variable terms.

Getting to the Answer: You don’t know the roots of this equation, so start with the point that has the easiest values to work with, (0, 1), and substitute them into the equation y = ax^{2}+ bx + c.

1 = a(0)^{2}+ b(0) + c

1 = c

Now your equation looks like y = ax^{2} + bx + 1. Next, use the other two points to create a system of two equations in two variables. (-3, 10) → 10

= a(-3)^{2} + b(-3) + 1 → 9 = 9a – 3b

(2,15) → 15 = a(2)^{2} + b(2) + 1 → 14 = 4a + 2b

You now have a system of equations to solve. None of the variables has a coefficient of 1, so use elimination to solve the system. If you multiply the top equation by 2 and the bottom equation by 3, the b-terms will eliminate each other.

2[9a – 3b = 9] → 18a – 6b = 18

3[4a + 2b = 14] → 12a + 6b = 42

[18a – 6b = 18] + [12a + 6b = 42] = [30a = 60]

a = 2

Now, find b by substituting a = 2 into either of the original equations: Using the top equation, you get:

9(2) – 3b = 9

18 – 3b = 9

-3b = -9

b = 3

The value of a + b + c is 2 + 3 + 1 = 6.

**SAT Math Practice Online Test Question No 4:**

If the area of the shaded sector in circle C shown above is 6π square units, what is the diameter of the circle?

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**Correct Answer: 12 **

Difficulty: Medium

Category: Additional Topics in Math / Geometry

Strategic Advice: Use the relationship (area of sector/area of a circle) = central angle/360° to answer this question. To help you remember this relationship, just think (partial area/whole area) = (partial angle/whole angle).

Getting to the Answer: The unknown in this question is the diameter of the circle, which is twice the radius. You can find the radius of the circle by first finding the area of the whole circle, and then by using the area equation, A = πr2. You have everything you need to find the area of the circle. Because this is a no-calculator question, you can bet that numbers will simplify nicely.

(area of the sector)/(area of circle) = central angle/360°.

6π/A = 60/360

6π/A = 1/6

A = 36π

Now, solve for r using A = πr^{2}: 36π = πr^{2 }

36 = r^{2 }

±6 = r

The radius can’t be negative, so it must be 6, which means the diameter of the circle is twice that, or 12.

**SAT Math Practice Online Test Question No 5:**

In economics, the law of demand states that as the price of a commodity rises, the demand for that commodity goes down. A company determines that the monthly demand for a certain item that it sells can be modeled by the function q(p) = -2p + 34, where q represents the quantity sold in hundreds and p represents the selling price in dollars. It costs $7 to produce this item. How much more per month in profits can the company expect to earn by selling the item at $12 instead of $10? (Profit = sales – costs)

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**Correct Answer: 800 **

Difficulty: Hard

Domain: Passport to Advanced Math / Functions

Strategic Advice: You don’t need to take an economics class to understand this question. Think about it logically and in terms of function notation. You need to find the quantity sold (use the function for that), and then the profits (use logic for that). The calculations are fairly simple, but there are a lot of them, so try organizing them in a table.

Getting to the Answer: Find the quantity that the company can expect to sell at each price using the demand function. Don’t forget that the quantity is given in hundreds. Then, find the total sales, the total costs, and the total profits using simple multiplication.

The company will earn $5,000 – $4,200 = $800 more per month.

**SAT Math Practice Online Test Question No 6:**

What √x ⦁ x^{5/4} ⦁ x^{2}/ ∜x^{3}

If the expression above is combined into a single power of x with a positive exponent, what is that exponent?

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**Correct Answer: 3 **

Difficulty: Hard

Category: Passport to Advanced Math / Exponents

Strategic Advice: You need to use rules of exponents to simplify the expression. Before you can do that, you must rewrite the radicals as fraction exponents. Use the phrase “power over root” to help you convert the radicals:

and

Getting to the Answer: Write each factor in the expression in exponential form. Then use rules of exponents to simplify the expression. Add the exponents of the factors that are being multiplied and subtract the exponent of the factor that is being divided:

√x ⦁ x^{5/4} ⦁ x^{2}/∜x^{3}

= x^{1/2} ⦁ x^{5/4} ⦁ x^{3}x^{2/1}/x^{3/4}

= x^{1/2 + 5/4 + 2/1 – 3/4}

= x^{2/4 + 5/4 + 2/1 – 3/4}

= x^{12/4}

= x^{3}

= The exponent of the simplified expression is 3

**SAT Math Practice Online Test Question No 7:**

k(10x – 5) = 2(3 + x) – 7 If the equation above has infinitely many solutions and k is a constant, what is the value of k?

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**Correct Answer: 1/5 or .2 **

Difficulty: Medium

Category: Heart of Algebra / Linear Equations

Strategic Advice: There are two variables but only one equation, so you can’t actually solve the equation for k. Instead, recall that an equation has infinitely many solutions when the left side is identical to the right side. When this happens, everything cancels out and you get 0 = 0, which is always true.

Getting to the Answer: Start by simplifying the right-hand side of the equation. Don’t simplify the left side because k is already in a good position.

k(10x – 5) = 2(3 + x) – 7

k(10x – 5) = 6 + 2x – 7

k(10x – 5) = 2x – 1

Next, compare the left side of the equation to the right side. Rather than distributing the k, notice that 2x is a fifth of 10x and -1 is a fifth of -5, so if k were 1/2 (or 0.2), then both sides of the equation would equal 2x – 1, and it would therefore have infinitely many solutions. Thus, k is 1/5 or.2

**SAT Math Practice Online Test Question No 8:**

If the equation of the parabola shown in the graph is written in standard quadratic form, y = ax^{2} + bx + c, and a = -1, then what is the value of b?

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**Correct Answer: 6 **

Difficulty: Hard

Category: Passport to Advanced Math / Quadratics

Strategic Advice: When you are given the graph of a parabola, try to use what you know about intercepts, the vertex, and the axis of symmetry to answer the question. Here, you could try to use points from the graph to find its equation, but this is not necessary because the question only asks for the value of b. As a shortcut, recall that you can find the vertex of a parabola using the formula x = -b/2a (the quadratic formula without the radical part).

Getting to the Answer: You are given that a = -1. Now look at the graph-the vertex of the parabola is (3, 8), so substitute 3 for x, -1 for a, and solve for b.

3 = -b/2(-1)

3 = -(b/-2)

3 = b/2

3(2) = b

6 = b

As an alternate method, you could plug the value of a and the vertex (from the graph) into vertex form of a quadratic equation and simplify:

y = a(x – h)^{2}+ k =

= -1(x – 3)^{2}+ 8

= -1(x^{2} – 6x + 9)^{2}+ 8

= -x^{2} + 6x – 9 + 8

= -x^{2} + 6x – 1

The coefficient of x is b, so b = 6.

**SAT Math Practice Online Test Question No 9:**

If (3/4)x + (5/6)y = 12, what is the value of 9x + 10y?

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**Correct Answer: 144 **

Difficulty: Easy

Category: Heart of Algebra / Linear Equations

Strategic Advice: There is only one equation given and it has two variables. This means that you don’t have enough information to solve for either variable. Instead, look for the relationship between the left side of the equation and the other expression that you are trying to find.

Getting to the Answer: Start by clearing the fractions by multiplying both sides of the original equation by 12. This yields the expression that you are looking for, 9x + 10y, so no further work is required-just read the value on the right-hand side of the equation.

(3/4)x + (5/6)y = 12

12[(3/4)x + (5/6)y] = 12(12)

9x + 10y = 144

**SAT Math Practice Online Test Question No 10:**

How many degrees does the minute hand of an analogue clock rotate from 3:20 PM to 3:45 PM?

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**Correct Answer: 150 **

Difficulty: Medium

Category: Additional Topics in Math / Geometry

Strategic Advice: There are 360° in a circle. You need to figure out how many degrees each minute on the face of a clock represents.

Getting to the Answer: There are 60 minutes on the face of an analogue clock. This means that each minute represents 360 ÷ 60 = 6 degrees. Between 3:20 and 3:45, 25 minutes go by, so the minute hand rotates 25 × 6 = 150 degrees.