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Master College-Level Mathematics: 40 Challenging Exam
Questions with In-depth Explanations
PS: Please note that some solutions might not be unique, and some can
represent a family of solutions
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What is the value of the determinant of the following matrix?
A = [[3, 2], [4, 5]]
Answer: 1

Explanation:
The determinant of a 2x2 matrix can be found by multiplying the entries on the main diagonal
(top-left to bottom-right) and subtracting the product of the entries on the other diagonal
(top-right to bottom-left)
...


Q:
2
...

Answer: x^4/4 + x^3/3 + C (C is a constant)

Explanation:
To integrate x^3 + x^2 with respect to x, we can use the power rule for integration, which
states that the integral of x^n with respect to x is (x^(n+1))/(n+1) + C
...
Adding these two integrals together, we
get (x^4)/4 + (x^3)/3 + C
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Solve the following system of equations:
y = 2x + 1
y = -3x + 7
Answer: x = 2, y = 5

Explanation:
To solve a system of equations, we can substitute one equation into the other and solve for a
variable
...
Solving for x, we find that x = 2
...
Differentiate y = sin^2(x) with respect to x
...
So we have d/dx(sin^2(x)) = d/dx(sin(x) * sin(x)) = 2*sin(x)*cos(x)

Q:
5
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Answer: y = -3x + 19

Explanation:
To find the equation of a line given a point and a slope, we can use the point-slope form of a
line, which is y - y1 = m(x - x1), where (x1, y1) is a point on the line, and m is the slope
...
So we have y - 7 = -3(x - 4)
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Q:
6
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Find the area of the circle with radius 5
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In this case, the radius is 5, so the area is
π * 5^2 = 25π
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What is the dot product of the vectors u = [1, 2, 3] and v = [4, 5, 6]?
Answer: 32

Explanation:
The dot product of two vectors u and v is given by u · v = u1v1 + u2v2 + u3v3
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What is the length of the diagonal of a rectangle with sides of length 5 and 8?
Answer: √(5^2 + 8^2)

Explanation:
The length of the diagonal of a rectangle can be found using the Pythagorean theorem
...
Evaluate the following definite integral:
∫(x^2 + 1) dx from 0 to 2
Answer: 9

Explanation:
To evaluate the definite integral ∫(x^2 + 1)dx from 0 to 2, we can use the fundamental
theorem of calculus, which states that the definite integral of a function f(x) from a to b is
equal to F(b) - F(a), where F(x) is any antiderivative of f(x)
...
So, the definite integral from 0 to 2 is:
∫(x^2 + 1)dx from 0 to 2 = (x^3/3 + x)|2 - (x^3/3 + x)|0 = (8/3 + 2) - (0 + 0) = 8/3 + 2 = 2 +
(8/3) = 2 + 2
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666 ≈ 9
so the definite integral of the function equals 9

Q:
11
...

Answer: (x-4)^2 + (y-5)^2 = 9

Explanation:
The equation of a circle with center (h, k) and radius r is given by (x-h)^2 + (y-k)^2 = r^2
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What is the derivative of f(x) = x^3 + 2x^2 -5x + 6?
Answer: 3x^2 + 4x -5

Explanation:
The derivative of a polynomial can be obtained by applying the power rule repeatedly
...


Q:
13
...
In this case, dy/dx = x^2 - y^2, is a
separable differential equation, this mean that we can separate the variable y from x
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Evaluate the definite integral ∫(x^3 + 4x^2) dx from -2 to 3
Answer: 40

Explanation:
To evaluate a definite integral, you need to find the antiderivative of the function and
calculate the value of this antiderivative for the given limits
...
Find the equation of the tangent line to the curve y = x^3 at the point (1,1)
Answer: y = 4x-3

Explanation:
To find the equation of the tangent line to a curve y = f(x) at point (a,b), we need to find the
slope of the tangent line at that point
...
In this case, y = x^3, so f'(x) = 3x^2, then for x =1, the slope is 3(1)^2
=3
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What is the dot product of the vectors u = [3,4,5] and v = [-1,2,-1]?
Answer: 0

Explanation:
The dot product of two vectors u and v is given by u · v = u1v1 + u2v2 + u3v3, In this case, u
= [3,4,5] and v = [-1,2,-1] so the dot product is (3)(-1) + (4)(2) + (5)(-1) = -3 + 8 - 5 = 0

Q:
17
...
In this
case, the line y = 3x + 2 has a slope of 3, so we can use the point-slope form of a line, which
is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope
...

This is the equation of the line that is parallel to y = 3x + 2 and passes through the point
(4,7)
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Additionally, as the new line goes through the point (4,7) it guarantees
that this line also pass through that point
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What is the length of the hypotenuse of a right triangle with legs of length 8 and 15?
Answer: √(8^2 + 15^2)

Explanation:
The hypotenuse of a right triangle is the side opposite the right angle, this side can be
calculated using the Pythagorean theorem, which states that the square of the length of the
hypotenuse equals the sum of the squares of the other two sides
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Solve the following system of equations:
2x + 3y = 12
4x - y = 2
Answer: x = 2, y = 2

Explanation:
To solve a system of equations, we can use different techniques, one of them is substitution
...
Then we can substitute this value of x back into either
of the original equations to find the value of y, in this case y=2

Q:
20
...

Answer: 1/(1 + x^2)

Explanation:

To differentiate y = tan^-1(x) with respect to x, we can use the chain rule of differentiation
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Solve the following system of equations:
2x + y = 4
-x + y = 2
Answer: x = 1, y = 3

Explanation:
To solve a system of equations, one possible method is to use substitution
...
Then we can substitute this value of x back into either of the original
equations to find the value of y, in this case y=3

Q:
22
...
In this case, we have g(x) = x^2+1 and f(x) = ln(x),
and we know that the derivative of f(x) is 1/x, and the derivative of g(x) is 2x, so we can
apply the chain rule: dy/dx = f'(g(x))*g'(x) = (1/(x^2+1)) * 2x = 2x / (x^2 + 1)

Q:
23
...
5

Explanation:
To find the area of a triangle with vertices at given coordinates, we can use the Shoelace
Theorem, which states that the area of the triangle is equal to half the absolute value of the
determinant of the matrix formed by the coordinates
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The determinant of the matrix is (-2 + 1) - (2 + 4) = -5, so the area of the
triangle is |-5|/2 = 2
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Evaluate the definite integral ∫(e^x) dx from 0 to 2
Answer: e^2 - e^0

Explanation:
To evaluate a definite integral, you need to find the antiderivative of the function and
calculate the value of this antiderivative for the given limits
...
Find the equation of the line that is perpendicular to y = 2x +1 and passes through
the point (4,7)
Answer: y = -0
...
5

Explanation:
To find the equation of a line that is perpendicular to another line, we can use the negative
reciprocal of the slope of the given line
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In this case, we have m = -1/2 and (x1, y1) = (4,7)
so we have y - 7 = -1/2(x - 4) which can be simplified as y = -0
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5

Q:
26
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What is the length of the side of a square with area 25?
Answer: 5

Explanation:
To find the length of a side of a square with a given area, we can take the square root of the
area
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Solve the following differential equation: dy/dx = x - y
Answer: y = x + Ce^-x

Explanation:
To solve a differential equation, we can try to find a solution by guessing
...
Differentiate y = csc^2(x) with respect to x
Answer: -2csc(x)cot(x)

Explanation:
To differentiate y = csc^2(x) with respect to x, we use the product rule
...
In this case, y = csc^2(x) =
1/sin^2(x) and we know that the derivative of csc(x) is -csc(x)cot(x) and the derivative of
sin^-2(x) = -2cos(x)sin(x), so we can apply the product rule: dy/dx = -csc(x)cot(x) *
(-2cos(x)sin(x)) = -2csc(x)cot(x)

Q:
30
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The vertex form of a parabola is of the form y = a(x-h)^2 +
k , where (h,k) is the vertex of the parabola
...
it gives us y =
-(x+1)^2 -1

Q:
31
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In this case, we
can substitute the first equation into the second, we obtain x + y = 4 -> x + (6 - 2x) = 4 and
solving for x = 1
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Using the chain rule, differentiate y = ln(x^3 + 1) with respect to x
Answer: 3x^2 / (x^3 + 1)

Explanation:
To differentiate y = ln(x^3 + 1) with respect to x, we can use the chain rule which states that
the derivative of f(g(x)) is f'(g(x))*g'(x)
...
Find the volume of a sphere with radius 5
Answer: 523
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In this case, the
radius is 5 so the volume is (4/3)π(5^3) = (4/3)π(125) = (500/3)π = 523
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Evaluate the definite integral ∫(sin^3(x)) dx from pi/2 to pi
Answer: -1/8

Explanation:
To evaluate a definite integral, you need to find the antiderivative of the function and
calculate the value of this antiderivative for the given limits
...
Find the equation of the line that is perpendicular to y = -x + 1 and passes through
the point (3, 4)
Answer: y = x + 1

Explanation:
To find the equation of a line that is perpendicular to another line, we can use the negative
reciprocal of the slope of the given line
...
In this case, we have m = 1 and (x1, y1) = (3,4) so
we have y - 4 = 1(x - 3) which can be simplified as y = x + 1

Q:
36
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In this case, u = [2, 4, 1] and v = [1, 3, -1] so the cross product is [4*-1 - 11,
-2-1+14, 23-4*1] = [-7, -11, 5]

Q:
37
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In this case, the two
points are (4, -3) and (-2, 1), so the slope is (1 - (-3))/(-2 - 4) = -2

Q:
38
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In this case, dy/dx
= y^2 - x^2, this is a non-linear differential equation and can be solved by integrating factors
or by separation of variables
...
Differentiate y = sec^3(x) with respect to x
Answer: 3sec^2(x)tan(x)sec(x)

Explanation:
To differentiate y = sec^3(x) with respect to x, we use the product rule
...
In this case, y = sec^3(x) =
1/cos^3(x) and we know that the derivative of sec(x) is sec(x)tan(x) and the derivative of
cos^-3(x) = -3sin(x)cos^-2(x), so we can apply the product rule: dy/dx = sec(x)tan(x) *
(-3sin(x)cos^-2(x)) = 3sec^2(x)tan(x)sec(x)

Q:
40
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Answer: (x^2/4) + (y^2/2) = 1

Explanation:
The equation of an ellipse centered at the origin is of the form (x^2/a^2) + (y^2/b^2) = 1,
where a and b are the lengths of the major and minor axes, respectively

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