StudyDojo

Question 1

SLPaper 1

A renewable energy firm is conducting research on the efficiency of solar panels. They found that the output of a specific panel can be modeled by the equation $3^{x + 1} = 81$ .

1.

Find the value of x

[4]

Question 2

HLPaper 1

1.

Consider a three-digit code xyz, where each of x, y and z is assigned one of the values 1, 2, 3, 4 or 5. Find the total number of possible codes assuming that each value can be repeated (for example, 121 or 444).

[1]

2.

Find the total number of possible codes assuming that no value is repeated.

[4]

3.

Hence show that the only way to assign the values is p = 4, q = 5 and r = 2.

[1]

4.

Express Q(x) as a product of linear factors.

[1]

5.

Hence or otherwise, sketch the graph of y = Q(x), clearly showing the coordinates of any intercepts with the axes.

[12]

6.

Let Q(x) = x^3 + px^2 + qx + r, where each of p, q and r is assigned one of the values 1, 2, 3, 4 or 5. Assume that no value is repeated. Consider the case where Q(x) has a factor of (x^2 + 3x + 2). Find an expression for q in terms of p.

[1]

Question 3

SLPaper 1

In a study of energy consumption, researchers need to solve the equation $5^{2x-1} = 125$ to find the value of $x$ that represents a specific energy threshold.

1.

Solve for $x$ .

[3]

Question 4

SLPaper 1

A technology company is analyzing user data to improve their application. They have found that the number of active users, represented by ( x + 3 ), can be modeled using logarithmic functions. They discovered that the logarithm of the number of active users to the base 2 equals 4.

1.

Solve for x in the equation $\log_2(x + 3) = 4$

[3]

Question 5

HLPaper 1

A polynomial p(x) with real coefficients has roots 2+i and 3.

1.

State another root of p(x).

[1]

2.

Write the polynomial p(x) in its factorized form, assuming it is of degree 3.

[3]

Question 6

SLPaper 1

Consider the double angle identities of $\cos$ and $\sin$

1.

Prove that $\tan(2x)=\frac{2\tan(x)}{1-\tan^2(x)}$

[4]

2.

Hence, find $\int(\tan(2x)-\tan(2x)\tan^2(x))\,dx$ in the form $a\ln x+b$ where $a>0$

[6]

Question 7

HLPaper 1

Consider a group of 10 people who are planning to form a committee.

1.

How many different committees of 4 people can be formed from this group?

[3]

2.

If two specific people, Alice and Bob, must be in the committee, how many different committees of 4 people can be formed?

[3]

Question 8

SLPaper 1

In a research project, a student is combining logarithmic expressions to analyze data trends. They need to express the following logarithmic expression as a single logarithm: $\log _3(9)+2 \log _3(x)$

1.

Express the given logarithmic expression as a single logarithm.

[3]

Question 9

SLPaper 1

A renewable energy company is analyzing the efficiency of solar panels. They have determined that the output of a solar panel can be represented by the expression: $\frac{\left(3^4 \cdot 3^{-2}\right)^3}{\left(3^{-1} \cdot 3^3\right)^2}$

1.

Simplify the expression to find the overall output of the solar panel.

[3]

Question 10

SLPaper 1

A researcher is studying the effects of a new agricultural technique on crop yield. The yield can be represented by the expression $(16 x^4 y^{-2})^{\frac{1}{4}}$ .

1.

Simplify this expression to understand the yield per unit area.

[3]

Number and Algebra

Number and Algebra