## SAT MAth Practice questions – all topics

**Advanced Math**Weightage: 35% Questions: 13-15- Equivalent expressions
- Nonlinear equations in one variable and systems of equations in two variables
- Nonlinear functions

## SAT MAth and English – full syllabus practice tests

**[Calc]**** ****Question*** *** ****medium**

The graph models the mass \(y\), in nanograms, of cobalt-60 (Co-60) remaining in a sample after \(x\) halflives. The half-life of Co-60 is 5.27 years.

What is the mass, in nanograms, of Co-60 remaining in the sample after 10.54 years?

A. 0.47

B. 1.25

C. 2.00

D. 2.64

**▶️Answer/Explanation**

Ans:B

The relationship between the remaining mass of cobalt-60 (Co-60) and the number of half-lives that have passed. The half-life of Co-60 is 5.27 years, and the mass of Co-60 decreases by half every 5.27 years.

Initial mass \( y_0 = 5 \) nanograms

Half-life \( t_{1/2} = 5.27 \) years

We need to find the mass remaining after 10.54 years. First, we determine how many half-lives have passed in 10.54 years:

\[ \text{Number of half-lives} = \frac{10.54 \text{ years}}{5.27 \text{ years/half-life}} = 2 \]

Since 10.54 years corresponds to 2 half-lives, the mass of Co-60 remaining after 2 half-lives can be calculated as follows:

\[ y = y_0 \times \left(\frac{1}{2}\right)^{\text{Number of half-lives}} \]

Substitute the values into the equation:

\[ y = 5 \times \left(\frac{1}{2}\right)^2 \]

\[ y = 5 \times \frac{1}{4} \]

\[ y = 5 \times 0.25 \]

\[ y = 1.25 \]

So, the mass of Co-60 remaining in the sample after 10.54 years is \( 1.25 \) nanograms.

**Alternative method:**

**Which is only in option B (because left mass is greater then 1 and less then 2)**

**[No calc]**** ****Question**** **medium

The graph of y =f(x)-1 is shown. Which equation could define function f ?

A.\(f(x)=2^{x}\)

B. \(f(x)=2^{x}-1\)

c. \(f(x)=2^{x}+1\)

D. \(f(x)=2^{x}+2\)

**▶️Answer/Explanation**

Ans: D

For $x= 0$ , the value of $f(x)-1$ is 2 .

equating , $2=f(0)-1\Rightarrow f(0)=3$

If \(f(x) – 1\) equals \(2\) when \(x = 0\), then \(f(0) = 3\)

Given the options, let’s evaluate each one for \(x = 0\):

**A.** \(f(x) = 2^x\)

\(f(0) = 2^0 = 1\)

**B.** \(f(x) = 2^x – 1\)

\(f(0) = 2^0 – 1 = 1 – 1 = 0\)

**C.** \(f(x) = 2^x + 1\)

\(f(0) = 2^0 + 1 = 1 + 1 = 2\)

**D.** \(f(x) = 2^x + 2\)

\(f(0) = 2^0 + 2 = 1 + 2 = 3\)

So, among the options, only option D yields \(f(0) = 3\), consistent with what we’ve determined. Hence, the correct answer is option D) \(f(x) = 2^x + 2\).

**[Calc]**** ****Question*** *** Medium**

Which table could represent values of x and their corresponding values of f(x) for a decreasing exponential function f ?

**▶️Answer/Explanation**

Ans: B

An exponential function is characterized by its growth or decay rate. For a decreasing exponential function, as \(x\) increases, \(f(x)\) decreases. This means that as the input \(x\) increases, the output \(f(x)\) decreases at an increasing rate.

Looking at the tables:

A) The values of \(f(x)\) increase as \(x\) increases. This does not represent a decreasing exponential function.

B) The values of \(f(x)\) decrease as \(x\) increases. This represents a decreasing exponential function.

C) The values of \(f(x)\) increase as \(x\) increases. This does not represent a decreasing exponential function.

D) The values of \(f(x)\) decrease as \(x\) increases. This represents a decreasing exponential function.

So, the table that could represent values of \(x\) and their corresponding values of \(f(x)\) for a decreasing exponential function is:B

**[Calc]**** ****Question** ** ** **Medium**

The graph of the exponential function \(\mathrm{f}\) is shown. For what value of \(x\) is \(f(x)=0\) ?

A) -4

B) -3

C) -2

D) -1

**▶️Answer/Explanation**

Ans:C

The value of y at x=0 is -3.

The value of x at y=0 is -2.

**[Calc]**** ****Question** ** **medium

The graph of y = f(x) is shown. What is the graph of y = f(x) − 2 ?

**▶️Answer/Explanation**

**C**

To find the graph of $y = f(x) – 2$, We need to shift the original graph $y = f(x)$ downwards by 2 units.

The graph provided shows a curved increasing function passing through the points (2, 0), and continuing to increase.

To get y = f(x) – 2, I will shift this entire curve downwards by 2 units on the y-axis:

The new curve will pass through the point (0, 0) instead of (2, 0). Which is option- C

**[Calc]**** ****Question** ** ** **Medium**

The half-life of the radioactive isotope iodine-131 is approximately 8 days, which means that at the end of each 8 -day time interval only half of the mass of the isotope that was present at the beginning of the time interval remains. Which of the following best describes how the amount of iodine-131 changes over time?

A) It increases linearly.

B) It decreases linearly.

C) It increases exponentially.

D) It decreases exponentially.

**▶️Answer/Explanation**

Ans:D

The half-life of iodine-131 indicates that the amount of iodine-131 halves every 8 days. This kind of change over time is best described by an exponential decay.

The correct description is:

\[

\boxed{\text{It decreases exponentially.}}

\]

The equation of the circle is:

\[

\boxed{x^2 + y^2 = 4}

\]

**[Calc]**** ***Questions ***Medium**

An advertising agency guarantees that its services will increase website traffic by 3.5% compared to each previous week. Which type of function best models the weekly guaranteed website traffic as the number of weeks increases?

A) Increasing exponential

B) Decreasing exponential

C) Increasing linear

D) Decreasing linear

**▶️Answer/Explanation**

Ans: A

The type of function that best models the weekly guaranteed website traffic as the number of weeks increases is an \(\mathbf{increasing\ exponential}\). This is because the guaranteed increase in website traffic each week (3.5%) indicates exponential growth over time. So, the correct answer is \(\mathbf{A}\).

**[Calc]**** ****Question**** **** Medium**

What is the $y$-intercept of the graph of $y=4^x$ in the $x y$-plane?

A) $(1,4)$

B) $(1,0)$

C) $(0,1)$

D) $(4,1)$

**▶️Answer/Explanation**

C

*Question*

Which expression is equivalent to \(g\)^{4/5 }\(h\)^{2/5} , where \(g\) and \(h\) are positive? 2.5

- \(\sqrt[4]{g^{5}h^{10}}\)
- \(\sqrt[5]{g^{4}h^{2}}\)
- \(\frac{1}{\sqrt[4]{g^{5}h^{10}}}\)
- \(\frac{1}{\sqrt[5]{g^{4}h^{2}}}\)

**▶️Answer/Explanation**

B

*Question*

\(S(t)=10(0.5)^{\frac{t}{29}}\)

A sample of strontium-90 will radioactively decay to half the original quantity in approximately 29 years. The function \(S\) above models the amount of strontium-90, in grams, that remains \(t\) years after a 10-gram sample starts to decay. Which of the following is the best interpretation of the number 0.5 in the function? 2.14

- The approximate number of years it would take for 5 grams of strontium-90 to remain in the sample
- The proportion of strontium-90 that remains after approximately 29 years
- The number of grams of strontium 90 in the sample that will decay approximately every 29 years
- The number of grams of strontium 90 by which the sample will be reduced each year over approximately 29 years

**▶️Answer/Explanation**

B

*Question*

What is the graph of the equation ? 2.11

**▶️Answer/Explanation**

**A**

*Question*

The graph of the function is shown. what is the value of for f(\(x\))=0? 2.9

- -6
- -3
- 0
- 3

**▶️Answer/Explanation**

D

*Questions *

In the $x y$-plane, the graph of a linear equation of the form $y=m x+b$ and the graph of an exponential equation of the form $y=a b^x$ both contain points $(1,3)$ and $(2,4)$. If the point $(r, s)$ is on the graph of the linear equation and the point $(r, t)$ is on the graph of the exponential equation, where $0<r<4$ and $s>t$, which of the following must be true?

A. $0<r<1$

B. $1<\mathrm{r}<2$

C. $2<r<3$

D. $3<r<4$

**▶️Answer/Explanation**

Ans: B

*Questions *

From 2005 through 2014, the number music CDs sold in the United States declined each year by approximately $15 \%$ of the number sold the preceding year. In 2005, approximately 600 million CDs were sold in the United States. Of the following, which best models $C$, the number of millions of CDs sold in the United States, $t$ years after 2005?

A. $C=600(0.15)^t$

B. $C=600(0.85)^t$

C. $C=600(1.15)^t$

D. $C=600(1.85)^t$

**▶️Answer/Explanation**

Ans: B

*Questions *

$P(t)=60(3)^{\frac{t}{2}}$

The number of microscopic organisms in a petri dish grows exponentially with time. The function $P$ above models the number of organisms after growing for $t$ days in the petri dish. Based on the function, which of the following statements is true?

A. The predicted number of organisms in the dish triples every two days.

B. The predicted number of organisms in the dish doubles every three days.

C. The predicted number of organisms in the dish triples every day.

D. The predicted number of organisms in the dish doubles every day.

**▶️Answer/Explanation**

Ans: A

*Questions *

$P=215(1.005)^{\frac{t}{3}}$

The equation above can be used to model the population, in thousands, of a certain city $t$ years after 2000. According to the model, the population is predicted to increase by $0.5 \%$ every $\mathrm{n}$ months. What is the value of $\mathrm{n}$ ?

A. 3

B. 4

C. 12

D. 36

**▶️Answer/Explanation**

Ans: D