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Questions and Answers
What is the expression for $f3$ given the angles with the horizontal as shown?
What is the expression for $f3$ given the angles with the horizontal as shown?
C) $f3 = (F1 ext{ sin }35° - 2)i + (F1 ext{ cos }35° - 2)j$
At the circular table, how many ways can Abby, Betty, Cara, and their relatives sit so that each student sits adjacent to their relative?
At the circular table, how many ways can Abby, Betty, Cara, and their relatives sit so that each student sits adjacent to their relative?
B) $3! \times 2^3$
If $f(2) = 5$ and $f'(2) = -2$, what is the gradient of the tangent on the inverse function at $x = 5$?
If $f(2) = 5$ and $f'(2) = -2$, what is the gradient of the tangent on the inverse function at $x = 5$?
B) $-\frac{1}{2}$
What is the centre and radius of the circle represented by the parametric equations x = 4 + 3cos θ, y = 2 + 3sin θ?
What is the centre and radius of the circle represented by the parametric equations x = 4 + 3cos θ, y = 2 + 3sin θ?
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Which of the following polynomials has a root of multiplicity 2 at x = 1 in the equation f(x) = 0?
Which of the following polynomials has a root of multiplicity 2 at x = 1 in the equation f(x) = 0?
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In the domain 0 ≤ x ≤ π, which of the following is true?
In the domain 0 ≤ x ≤ π, which of the following is true?
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Identify the expression identical to 2 sin(3x) sin(5x).
Identify the expression identical to 2 sin(3x) sin(5x).
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Sketch the graph of y = x^2 - 4. Which option represents the correct sketch?
Sketch the graph of y = x^2 - 4. Which option represents the correct sketch?
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Find the integral of the function ∫(9 - 4x^2)dx from 3 to 3.
Find the integral of the function ∫(9 - 4x^2)dx from 3 to 3.
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What is the formula for the volume V of sand in the cone?
What is the formula for the volume V of sand in the cone?
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How many different teams can be created if at least two members must be women?
How many different teams can be created if at least two members must be women?
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What is the number of fish remaining after 6 months?
What is the number of fish remaining after 6 months?
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What is the solution to 3sin(2x) - 5cos(2x) = 5 for 0 ≤ x ≤ 2π?
What is the solution to 3sin(2x) - 5cos(2x) = 5 for 0 ≤ x ≤ 2π?
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Show the relationship between vectors a, b, and the angle θ using the dot product identity.
Show the relationship between vectors a, b, and the angle θ using the dot product identity.
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Calculate the value of h when the depth of sand is decreasing at a rate of 0.05 cm/s.
Calculate the value of h when the depth of sand is decreasing at a rate of 0.05 cm/s.
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Study Notes
Calculus and Functions
- Given ( f(2) = 5 ) and ( f'(2) = -2 ), the gradient of the tangent on the inverse function at ( x = 5 ) can be found using the formula: [ \text{slope of } f^{-1}(x) = \frac{1}{f'(f^{-1}(x))} ] Here, ( f^{-1}(5) = 2 ), thus slope becomes ( \frac{1}{-2} = -\frac{1}{2} ).
Geometry and Circles
- For the parametric equations ( x = 4 + 3\cos \theta ) and ( y = 2 + 3\sin \theta ):
- The center of the circle is ( (4, 2) ).
- The radius of the circle is 3.
Trigonometric Identities
- The expression identical to ( 2 \sin(3x) \sin(5x) ) can be obtained using the product-to-sum formulas: [ 2 \sin(A) \sin(B) = \cos(A - B) - \cos(A + B) ] Hence, it simplifies to: [ \cos(2x) - \cos(8x) ]
Polynomials and Roots
- A polynomial having a root of multiplicity 2 at ( x = 1 ) must be of the form: [ f(x) = (x - 1)^2 g(x) ] where ( g(x) ) is any polynomial.
Statistics and Combinatorics
- To create teams with at least two women, calculate combinations of men and women, ensuring that the team size condition is met.
Volumes and Geometric Shapes
- The formula for the volume ( V ) of sand in a cone (with height ( h ) and radius ( r )) is: [ V = \frac{1}{3}\pi r^2 h ]
Integrals and Areas
- The integral ( \int (9 - 4x^2)dx ) evaluated from 3 to 3 results in zero, as the limits are equal, implying no area under the curve.
Fish Population Dynamics
- The number of fish remaining after 6 months depends on the initial population and the rate of change. Use population models to determine remaining fish.
Trigonometric Equations
- The solution to the equation ( 3\sin(2x) - 5\cos(2x) = 5 ) for ( 0 \leq x \leq 2\pi ) involves using trigonometric identities and solving for ( x ).
Vectors
- The relationship among vectors ( a ) and ( b ) as well as the angle ( \theta ) is expressed via the dot product: [ a \cdot b = |a||b| \cos(\theta) ]
Depth of Sand
- To find the value of ( h ) when the depth is decreasing at a rate of ( 0.05 ) cm/s, apply related rates, considering the cone’s volume and the derivative with respect to time.
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Description
Test your knowledge on integration, functions, and polynomials with questions involving finding integrals, differentiation, and identifying polynomial roots. See if you can correctly solve the given math problems.
