I85PREP
⌘ K
Kaoyan Mathematics III study materials
Kaoyan Mathematics III study resources

Kaoyan Mathematics III practice questions

One question at a time. Build your Kaoyan Mathematics III understanding.

First 50 of 190 questions. Each one lists the choices, the correct answer, and an explanation. The checker at the top of this page saves progress in this browser when it is available.

Continue in the practice tool above to load more questions, 50 at a time.

Question 1Calculus

Evaluate the limit lim_{x -> 0} (e^x - 1) / x.

  1. 1 Correct answer.
  2. 0
  3. e
  4. Does not exist

Explanation

The standard exponential limit is lim_{x -> 0} (e^x - 1) / x = 1. Equivalently, it is the derivative of e^x at x = 0.

Question 2Calculus

Compute the integral int_0^2 3x^2 dx.

  1. 6
  2. 8 Correct answer.
  3. 12
  4. 24

Explanation

An antiderivative of 3x^2 is x^3. Evaluating from 0 to 2 gives 2^3 - 0 = 8.

Question 3Calculus

For f(x) = x^2 + 4x + 5, what is the minimum value?

  1. 0
  2. 1 Correct answer.
  3. 4
  4. 5

Explanation

Complete the square: x^2 + 4x + 5 = (x + 2)^2 + 1. The minimum value is 1 at x = -2.

Question 4Calculus

If f'(a) = 0 and f''(a) > 0, what does the second derivative test indicate?

  1. A local maximum at a
  2. A local minimum at a Correct answer.
  3. No possible extremum at a
  4. A vertical asymptote at a

Explanation

A positive second derivative means the function is locally convex at the stationary point, so f has a local minimum at a.

Question 5Calculus

Evaluate lim_{x -> 0} (ln(1+x) - x) / x^2.

  1. -1/2 Correct answer.
  2. 0
  3. 1/2
  4. 1

Explanation

Using ln(1+x) = x - x^2/2 + o(x^2), the numerator is -x^2/2 + o(x^2). Dividing by x^2 gives -1/2.

Question 6Calculus

For f(x) = x^x with x > 0, what is f'(e)?

  1. e^e
  2. 2e^e Correct answer.
  3. e^{e-1}
  4. 0

Explanation

Log-differentiate: ln f = x ln x, so f' = x^x(ln x + 1). At x = e, f'(e) = e^e(1+1) = 2e^e.

Question 7Calculus

Compute int_0^infinity x e^{-x^2} dx.

  1. 1
  2. 1/2 Correct answer.
  3. sqrt(pi)/2
  4. 0

Explanation

Let u = x^2, so du = 2x dx. The integral becomes (1/2) int_0^infinity e^{-u} du = 1/2.

Question 8Calculus

What is the area between y = x and y = x^2 on [0,1]?

  1. 1/2
  2. 1/3
  3. 1/6 Correct answer.
  4. 2/3

Explanation

On [0,1], x is above x^2. The area is int_0^1 (x - x^2) dx = 1/2 - 1/3 = 1/6.

Question 9Calculus

Maximize xy subject to x + y = 10 with x,y > 0. What is the maximum value?

  1. 10
  2. 20
  3. 25 Correct answer.
  4. 100

Explanation

For a fixed sum, the product is maximized when x = y = 5. The maximum is 5*5 = 25.

Question 10Calculus

A twice differentiable function has Hessian [[2,0],[0,2]] at a critical point. What does this indicate?

  1. Strict local minimum Correct answer.
  2. Strict local maximum
  3. Saddle point
  4. No conclusion because determinant is negative

Explanation

The Hessian is positive definite because both eigenvalues are 2. At a critical point, that indicates a strict local minimum.

Question 11Calculus

Compute sum_{n=1}^infinity (1/2)^n.

  1. 1/2
  2. 1 Correct answer.
  3. 2
  4. Diverges

Explanation

This is a geometric series with first term 1/2 and ratio 1/2. The sum is (1/2)/(1-1/2) = 1.

Question 12Calculus

The second-order Taylor polynomial of e^x at x = 0 is:

  1. 1 + x
  2. 1 + x + x^2/2 Correct answer.
  3. x + x^2
  4. 1 + x + x^2

Explanation

The Maclaurin expansion of e^x begins 1 + x + x^2/2 + ... . Keeping through degree 2 gives 1 + x + x^2/2.

Question 13Calculus

Compute int_0^1 int_0^x y dy dx.

  1. 1/2
  2. 1/3
  3. 1/6 Correct answer.
  4. 1/8

Explanation

The inner integral is int_0^x y dy = x^2/2. Then int_0^1 x^2/2 dx = 1/6.

Question 14Calculus

The equation x^2 + y^2 + xy = 7 defines y as a function of x near (1,2). What is dy/dx at (1,2)?

  1. -4/5 Correct answer.
  2. -5/4
  3. 4/5
  4. -1

Explanation

Implicit differentiation gives 2x + 2y y' + y + x y' = 0, so y'=-(2x+y)/(2y+x). At (1,2), y'=-4/5.

Question 15Calculus

At (0,0), the function f(x,y)=x^2-y^2 has what type of critical point?

  1. Strict local minimum
  2. Strict local maximum
  3. Saddle point Correct answer.
  4. Not a critical point

Explanation

The gradient is (2x,-2y), so (0,0) is critical. Along the x-axis f is positive, while along the y-axis f is negative, so it is a saddle point.

Question 16Calculus

Compute the double integral of x + y over the triangle D = {(x,y): x >= 0, y >= 0, x + y <= 1}.

  1. 1/6
  2. 1/3 Correct answer.
  3. 1/2
  4. 2/3

Explanation

By symmetry, integral_D x dA = integral_D y dA. Over the unit right triangle, each equals 1/6, so the total is 1/3.

Question 17Calculus

What is the radius of convergence of sum_{n=1}^infinity n x^n / 3^n?

  1. 1/3
  2. 1
  3. 3 Correct answer.
  4. Infinity

Explanation

The series is sum n (x/3)^n. Since sum n r^n converges for |r| < 1, we need |x/3| < 1, so the radius is 3.

Question 18Calculus

Solve y' + y = e^x with y(0)=0.

  1. y = e^x - e^{-x}
  2. y = (e^x - e^{-x})/2 Correct answer.
  3. y = e^x + e^{-x}
  4. y = x e^{-x}

Explanation

Using integrating factor e^x gives (e^x y)' = e^{2x}. Thus y = (1/2)e^x + C e^{-x}. The condition y(0)=0 gives C=-1/2.

Question 19Linear algebra

What is det([[2,1],[3,4]])?

  1. 5 Correct answer.
  2. 8
  3. 11
  4. -5

Explanation

For a 2 by 2 matrix [[a,b],[c,d]], the determinant is ad - bc. Here 2*4 - 1*3 = 5.

Question 20Linear algebra

What is the rank of [[1,2],[2,4]]?

  1. 0
  2. 1 Correct answer.
  3. 2
  4. Cannot be determined

Explanation

The second row is twice the first row, and the matrix is not zero. Therefore it has exactly one linearly independent row, so rank is 1.

Question 21Linear algebra

Solve the system x + y = 5 and x - y = 1.

  1. x = 2, y = 3
  2. x = 3, y = 2 Correct answer.
  3. x = 4, y = 1
  4. x = 1, y = 4

Explanation

Adding the equations gives 2x = 6, so x = 3. Then y = 2 from x + y = 5.

Question 22Linear algebra

What are the eigenvalues of the diagonal matrix diag(2,5)?

  1. 2 and 5 Correct answer.
  2. 7 and 10
  3. 1 and 1
  4. Only 5

Explanation

For a diagonal matrix, the eigenvalues are the diagonal entries. Thus they are 2 and 5.

Question 23Linear algebra

A square matrix with determinant 0 is:

  1. Always invertible
  2. Singular Correct answer.
  3. Orthogonal
  4. The identity matrix

Explanation

A square matrix is invertible if and only if its determinant is nonzero. Determinant 0 means it is singular.

Question 24Linear algebra

If A and B are square matrices of the same size, which trace identity is valid?

  1. tr(AB) = tr(BA) Correct answer.
  2. tr(AB) = tr(A)tr(B) always
  3. tr(A+B) = tr(AB)
  4. tr(A^{-1}) = 1/tr(A) always

Explanation

For compatible square matrices, tr(AB) = tr(BA). This cyclic property is frequently useful in matrix calculus and statistics.

Question 25Linear algebra

What is the determinant of an upper triangular matrix with diagonal entries 2, -1, and 4?

  1. -8 Correct answer.
  2. 8
  3. 5
  4. 0

Explanation

The determinant of a triangular matrix is the product of its diagonal entries: 2*(-1)*4 = -8.

Question 26Linear algebra

A 2 by 2 linear system Ax = b has a unique solution for every b exactly when:

  1. det(A) = 0
  2. det(A) != 0 Correct answer.
  3. A has a zero row
  4. b = 0

Explanation

A square system has a unique solution for every right-hand side precisely when A is invertible, which for 2 by 2 matrices is equivalent to det(A) != 0.

Question 27Linear algebra

Solve 2x + y = 7 and x + y = 5.

  1. x=1, y=4
  2. x=2, y=3 Correct answer.
  3. x=3, y=2
  4. x=4, y=1

Explanation

Subtract the second equation from the first to get x = 2. Then x + y = 5 gives y = 3.

Question 28Linear algebra

What are the eigenvalues of [[2,1],[1,2]]?

  1. 2 and 2
  2. 3 and 1 Correct answer.
  3. 4 and 0
  4. 1 and -1

Explanation

The characteristic polynomial is (2-lambda)^2 - 1 = 0, so 2-lambda = +/-1. The eigenvalues are 3 and 1.

Question 29Linear algebra

A 2 by 2 matrix with two distinct real eigenvalues is:

  1. Never diagonalizable
  2. Always diagonalizable over R Correct answer.
  3. Diagonalizable only if symmetric
  4. Always orthogonal

Explanation

Distinct eigenvalues have linearly independent eigenvectors. A 2 by 2 matrix with two distinct real eigenvalues is diagonalizable over the real numbers.

Question 30Linear algebra

The quadratic form 2x^2 + 4xy + 5y^2 has matrix [[2,2],[2,5]]. Which statement is true?

  1. It is positive definite Correct answer.
  2. It is negative definite
  3. It is indefinite
  4. It is identically zero

Explanation

For [[2,2],[2,5]], the leading principal minor is 2 > 0 and the determinant is 10 - 4 = 6 > 0. Hence it is positive definite.

Question 31Linear algebra

What is the projection of vector (3,4) onto the x-axis?

  1. (0,4)
  2. (3,0) Correct answer.
  3. (3,4)
  4. (0,0)

Explanation

The x-axis keeps the x-component and sets the y-component to zero. The projection is (3,0).

Question 32Linear algebra

If det(A) = 4 for an invertible square matrix A, what is det(A^{-1})?

  1. 4
  2. -4
  3. 1/4 Correct answer.
  4. 0

Explanation

The determinant of the inverse is the reciprocal of the determinant: det(A^{-1}) = 1/det(A) = 1/4.

Question 33Linear algebra

A linear map from R^5 to R^3 has rank 3. What is the nullity?

  1. 0
  2. 2 Correct answer.
  3. 3
  4. 5

Explanation

By rank-nullity, rank + nullity = domain dimension = 5. Thus nullity = 5 - 3 = 2.

Question 34Linear algebra

If A is a nonzero 3 by 3 matrix with rank 1, what is the adjugate matrix adj(A)?

  1. The zero matrix Correct answer.
  2. A^{-1}
  3. A itself
  4. A diagonal matrix with rank 3

Explanation

For an n by n matrix with rank at most n-2, the adjugate is the zero matrix. Here n=3 and rank 1 <= 1, so adj(A)=0.

Question 35Linear algebra

The matrix equation AX = B is solvable exactly when:

  1. rank(A) = rank([A B]) Correct answer.
  2. rank(A) = 0
  3. det(B) != 0
  4. A has more columns than rows

Explanation

The consistency condition is that augmenting A with B does not increase rank: rank(A)=rank([A B]).

Question 36Linear algebra

For the symmetric matrix [[a,1],[1,a]], for which values of a is it positive definite?

  1. a > 1 Correct answer.
  2. a > 0
  3. a < -1
  4. -1 < a < 1

Explanation

Positive definiteness requires the first leading minor a > 0 and determinant a^2 - 1 > 0. Together these give a > 1.

Question 37Probability and statistics

If X is Bernoulli with success probability p = 0.3, what is Var(X)?

  1. 0.09
  2. 0.21 Correct answer.
  3. 0.3
  4. 0.7

Explanation

For a Bernoulli variable, Var(X) = p(1-p). With p = 0.3, variance is 0.3*0.7 = 0.21.

Question 38Probability and statistics

For a standard normal random variable Z, Phi(1.96) is approximately:

  1. 0.025
  2. 0.500
  3. 0.950
  4. 0.975 Correct answer.

Explanation

The standard normal value 1.96 cuts off about 2.5% in the upper tail, so Phi(1.96) is about 0.975.

Question 39Probability and statistics

If X_1,...,X_n are sampled from a population with mean mu, what is E[Xbar]?

  1. 0
  2. mu Correct answer.
  3. n mu
  4. mu/n

Explanation

The sample mean is an unbiased estimator of the population mean, so E[Xbar] = mu.

Question 40Probability and statistics

Which estimator is unbiased for a population mean mu under random sampling?

  1. The sample mean Xbar Correct answer.
  2. Twice the sample mean
  3. The sample maximum
  4. The sample size n

Explanation

Under random sampling, E[Xbar] = mu, so the sample mean is unbiased for the population mean.

Question 41Probability and statistics

If X has an exponential distribution with mean theta = 5, what is Var(X)?

  1. 5
  2. 10
  3. 25 Correct answer.
  4. 1/25

Explanation

For an exponential distribution parameterized by its mean theta, the variance is theta^2. With theta = 5, Var(X)=25.

Question 42Probability and statistics

If X is uniformly distributed on (0,4), what is P(X > 1)?

  1. 1/4
  2. 1/2
  3. 3/4 Correct answer.
  4. 1

Explanation

The interval length above 1 is 3, and the total interval length is 4. Therefore P(X>1)=3/4.

Question 43Probability and statistics

If X ~ Poisson(2), Y ~ Poisson(3), and X and Y are independent, what is the distribution of X + Y?

  1. Poisson(5) Correct answer.
  2. Poisson(6)
  3. Binomial(5,0.4)
  4. Normal(5,1)

Explanation

The sum of independent Poisson variables is Poisson with parameter equal to the sum of the parameters, so X+Y ~ Poisson(5).

Question 44Probability and statistics

Why is sample variance commonly computed with denominator n - 1 rather than n?

  1. To make it unbiased for population variance Correct answer.
  2. To make it equal to the sample mean
  3. To force all residuals to zero
  4. To remove random sampling

Explanation

Using n - 1 corrects for estimating the mean from the same data and makes the usual sample variance unbiased for the population variance.

Question 45Probability and statistics

For an exponential distribution with rate lambda and observations x_1,...,x_n, what is the MLE of lambda?

  1. sum x_i / n
  2. n / sum x_i Correct answer.
  3. 1 / sum x_i
  4. max x_i

Explanation

The likelihood is lambda^n exp(-lambda sum x_i). The log-likelihood derivative gives n/lambda - sum x_i = 0, so lambda_hat = n/sum x_i.

Question 46Probability and statistics

For X ~ Uniform(0, theta), what method-of-moments estimator follows from E[X] = theta/2?

  1. theta_hat = Xbar/2
  2. theta_hat = 2Xbar Correct answer.
  3. theta_hat = Xbar^2
  4. theta_hat = max X_i / 2

Explanation

Set the sample mean Xbar equal to theta/2 and solve for theta. This gives theta_hat = 2Xbar.

Question 47Probability and statistics

By the central limit theorem, for iid variables with mean mu and variance sigma^2, the sample mean is approximately:

  1. N(mu, sigma^2/n) for large n Correct answer.
  2. N(0,1) for every n
  3. Poisson(mu) for large n
  4. Uniform(mu-sigma, mu+sigma)

Explanation

The CLT gives Xbar approximately normal with mean mu and variance sigma^2/n for large n under standard conditions.

Question 48Probability and statistics

If Y = 3X + 2 and Z = 5W - 1, with positive scale factors, what happens to the correlation between X and W?

  1. It remains the same for Y and Z Correct answer.
  2. It becomes zero
  3. It is multiplied by 15
  4. It changes sign

Explanation

Correlation is unchanged by adding constants and multiplying both variables by positive constants. Thus corr(Y,Z)=corr(X,W).

Question 49Probability and statistics

Two discrete random variables X and Y are independent if and only if:

  1. P(X=x,Y=y)=P(X=x)P(Y=y) for all x,y Correct answer.
  2. E[X]=E[Y]
  3. Var(X)=Var(Y)
  4. X always equals Y

Explanation

Independence for discrete variables requires the joint probability to factor into the product of marginals for every pair of values.

Question 50Probability and statistics

In hypothesis testing, a Type I error means:

  1. Failing to reject a false null hypothesis
  2. Rejecting a true null hypothesis Correct answer.
  3. Rejecting a false null hypothesis
  4. Choosing a larger sample size

Explanation

A Type I error occurs when the null hypothesis is true but the test rejects it.