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Challenging College Mathematics Exam: Putting
Your Knowledge to the Test
1
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What is the difference quotient for the function f(x) = x^2?
a) (x+h)^2 - x^2/h
b) (x+h)^2 - x^2
c) (x+h)^2 - (x-h)^2
d) (x+h)^2 - 2xh + h^2
Answer: d) (x+h)^2 - 2xh + h^2
3
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Solve the equation log5(x+3) = 2
a) x = 8
b) x = -1
c) x = 11
d) x = 25
Answer: a) x = 8
5
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Simplify (3x+2y)^2
a) 9x^2 + 12xy + 4y^2
b) 6x^2 + 4xy + 4y^2
c) 9x^2 + 12xy + 4y^2

d) 6x^2 + 12xy + 4y^2
Answer: a) 9x^2 + 12xy + 4y^2
7
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What is the equation of the line that passes through the point (5, -2) and has a
slope of -3/4?
a) y = -3/4x + 7/4
b) y = -3/4x + 5/2
c) y = -3x + 5/4
d) y = -4x + 7/3
Answer: b) y = -3/4x + 5/2
9
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What is the area of the triangle with base b = 8 and height h = 4?
a) 12
b) 16
c) 24
d) 32
Answer: b) 16

In any case, here is a brief explanation of why the answers above
are correct:
1
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2
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a) The equation 5x - 2y = 10 is in the form of 5x - 2y = constant, it is always
satisfied for infinitely many values of x and y,

4
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c) x^3/3 + 1 is the antiderivative of x^2+1 dx
6
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a) Using the chain rule, d/dx (sin(x^2)) = d/du (sin(u)) * du/dx = cos(x^2)* 2x
8
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a) 3x + 7 = 2x - 5 add x to both sides, 5x = -12, x = -12/5 = -3
10
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What is the domain of the function f(x) = 1/(x-3)?
a) x < 3
b) x > 3
c) x ≠ 3
d) all real numbers
Answer: c) x ≠ 3
2
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How many solutions does the equation x^2 + 4x + 4 = 0 have?
a) infinitely many
b) no solutions
c) one solution
d) two solutions
Answer: d) two solutions
4
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Find the definite integral of x^3 from -1 to 1
a) 0
b) 2
c) 4
d) -2
Answer: a) 0
6
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What is the second derivative of y = x^3
a) 6x
b) 6
c) 3x^2
d) 3x
Answer: c) 3x^2
8
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Solve for x in the equation 5/x + 2x = 8
a) x = -5
b) x = -2
c) x = 2
d) x = 5
Answer: c) x = 2
10
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c) x ≠ 3 as the function is not defined for x = 3
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a) e^x is the derivative of e^x, this can be derived using the limit definition of
derivative and the properties of exponential functions
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d) two solutions the equation x^2 + 4x + 4 = 0 is a quadratic equation in
standard form, (ax^2 + bx + c = 0) and since b^2 - 4ac > 0, the solutions are
given by (-b ± √(b^2 - 4ac))/ 2a
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a) x = 5, 2^5 = 32
5
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6
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c) 3x^2, The second derivative of a function y = x^n (n is a constant) is given
by n(n-1)x^(n-2)
8
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c) x = 2, it can be derived by multiplying both sides by x, and simplifying the
equation
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c) 27π, The volume of a sphere is given by V = (4/3)πr^3, with r = 3, the
volume is 27π
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What is the sum of the first 100 positive integers?
a) 5050
b) 100
c) 4950
d) 5000
Answer: a) 5050

2
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What is the equation of the line that passes through the point (2, -3) and has a
slope of -1/2?
a) y = -1/2x + 2
b) y = -1/2x -3
c) y = 1/2x -3
d) y = 1/2x +2
Answer: b) y = -1/2x -3
4
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What is the product of the roots of the equation x^2+4x+3 =0 ?
a) 3
b) -3
c) -1
d) 1
Answer: b) -3
6
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What is the equation of the tangent line to the curve y = x^2 at the point (2,4)?
a) y = 2x + 2
b) y = 4x - 2
c) y = 4x + 2
d) y = 2x - 2
Answer: a) y = 2x + 2

8
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What is the equation of the circle with center (3,4) and radius 2?
a) (x-3)^2 + (y-4)^2 = 4
b) (x-3)^2 + (y-4)^2 = 2
c) (x-4)^2 + (y-3)^2 = 2
d) (x-4)^2 + (y-3)^2 = 4
Answer: b) (x-3)^2 + (y-4)^2 = 2
10
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Simplify the expression: 3(x+2) + 4(x-1)
a) 7x + 1
b) 7x + 2
c) 7x + 4
d) 7x + 8
Answer: c) 7x + 4
12
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Find the equation of the line that is perpendicular to y = 2x + 1 and passes
through the point (3,1)
a) y = -0
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5x + 4
d) y = -2x + 1
Answer: a) y = -0
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What is the value of lim x->0 (sin(x))/x?
a) 1
b) 0
c) undefined
d) -1
Answer: b) 0
15
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a) 5050, This can be easily computed by adding all positive integers from 1 to
100
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c) 12, (2x^2+3x) dx is a polynomial function, the integral of that is defined as
the anti-derivative of the function, which is (2/3)x^3+ (3/2)x^2+C, so the
definite integral between 0 and 2 is (2/3)*8+(3/2)*4=12
3
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These two pieces of information are enough to find the equation of the line
using the point-slope form of the equation of a line, which is y - y1 = m(x - x1)
where (x1, y1) is a point on the line and m is the slope of the line
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a) (2,3) this is the solution of the system of linear equations that can be solved
using substitution, elimination or matrix method
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b) -3, The product of the roots of the equation x^2+4x+3 =0 is given by -c/a,
where a,b and c are coefficients of the equation and x1 and x2 are the roots of
the equation
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c) 16, The area of a rectangle is given by the product of the width and the
length
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a) y = 2x + 2, this can be derived by applying the limit definition of the
derivative, and noting that at the point (2,4), the function y = x^2 has a slope
of 4, and we use the point-slope formula to write the equation of the tangent
line, which is y - y1 = m(x - x1) where m is the slope of the tangent line, x1
and y1 are the coordinates of the point
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d) two solutions, the equation 2x^2 - 3x + 2 = 0 is a quadratic equation in
standard form, (ax^2 + bx + c = 0) and since b^2 - 4ac > 0, the solutions are
given by (-b ± √(b^2 - 4ac))/ 2a
...
b) (x-3)^2 + (y-4)^2 = 2 is the equation of the circle, with center (3,4) and
radius 2
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10
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11
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12
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a) y = -0
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15
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And also, it's always good practice to explain the
reasoning behind your answers in an exam

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