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Kaoyan Mathematics II - independent mathematical foundations study materials
Kaoyan Mathematics II - independent mathematical foundations study resources

Kaoyan Mathematics II - independent mathematical foundations practice questions

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Question 1Linear algebra

Find 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 it is 2*4 - 1*3 = 5.

Question 2Linear algebra

For A = [[a,1],[2,3]], for which condition is A invertible?

  1. a = 2/3
  2. a != 2/3 Correct answer.
  3. a = 0
  4. a != 3/2

Explanation

A 2 by 2 matrix is invertible exactly when its determinant is nonzero. det A = 3a - 2, so A is invertible when a != 2/3.

Question 3Linear algebra

Compute [[1,2],[0,1]] times the column vector (3,4).

  1. (11,4) Correct answer.
  2. (3,7)
  3. (7,4)
  4. (11,7)

Explanation

The first component is 1*3 + 2*4 = 11, and the second is 0*3 + 1*4 = 4.

Question 4Linear algebra

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

  1. 2 and 5 Correct answer.
  2. 0 and 7
  3. 10 and 1
  4. Only 2

Explanation

For a diagonal matrix, the eigenvalues are its diagonal entries. Therefore the eigenvalues are 2 and 5.

Question 5Linear algebra

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

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

Explanation

The second row is 2 times the first row, so there is only one independent row. The rank is 1.

Question 6Linear algebra

Which pair is linearly independent in R^2?

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

Explanation

The vectors (1,0) and (0,1) are the standard basis vectors and neither is a scalar multiple of the other. They are linearly independent.

Question 7Linear algebra

Solve the system x + y = 3 and x - y = 1.

  1. x = 1, y = 2
  2. x = 2, y = 1 Correct answer.
  3. x = 3, y = 1
  4. x = 0, y = 3

Explanation

Adding the equations gives 2x = 4, so x = 2. Then y = 1 from x + y = 3.

Question 8Linear algebra

Find the determinant of an upper triangular 3 by 3 matrix with diagonal entries 2, 3, and 4.

  1. 9
  2. 24 Correct answer.
  3. 12
  4. 0

Explanation

The determinant of a triangular matrix is the product of its diagonal entries. Thus det = 2*3*4 = 24.

Question 9Linear algebra

A 3 by 2 matrix has rank 2. What can be concluded about its two columns?

  1. They are linearly independent Correct answer.
  2. They are both zero
  3. They are equal
  4. They span R^3

Explanation

Rank 2 means there are two independent columns. Since the matrix has exactly two columns, both columns are linearly independent.

Question 10Linear algebra

For which k are the rows of [[1,k],[2,6]] linearly dependent?

  1. k = 2
  2. k = 3 Correct answer.
  3. k = 6
  4. No value of k

Explanation

Rows are dependent when the determinant is zero: 1*6 - 2k = 0. Thus k = 3.

Question 11Linear algebra

Find the determinant of a triangular 3 by 3 matrix with diagonal entries 1, -2, and 5.

  1. -10 Correct answer.
  2. 10
  3. -6
  4. 0

Explanation

For a triangular matrix, the determinant is the product of diagonal entries: 1*(-2)*5 = -10.

Question 12Linear algebra

Rows r1=(1,0,1), r2=(0,1,1), r3=(1,1,2) form a 3 by 3 matrix. What is its rank?

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

Explanation

The third row equals r1 + r2, while r1 and r2 are independent. Therefore the rank is 2.

Question 13Linear algebra

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

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

Explanation

A square matrix gives a unique solution for every right-hand side exactly when A is invertible. This is equivalent to det A != 0.

Question 14Linear algebra

Which statement is true for A = [[1,1],[0,1]]?

  1. A has two distinct eigenvalues
  2. A has eigenvalue 1 with algebraic multiplicity 2 Correct answer.
  3. A is diagonal
  4. A has no real eigenvalues

Explanation

The characteristic polynomial is (lambda - 1)^2. Thus the only eigenvalue is 1, with algebraic multiplicity 2.

Question 15Linear algebra

Find the characteristic polynomial det(lambda I - A) for A = [[0,1],[-2,3]].

  1. lambda^2 + 3lambda + 2
  2. lambda^2 - 3lambda + 2 Correct answer.
  3. lambda^2 - 2lambda + 3
  4. lambda^2 + 2lambda - 3

Explanation

lambda I - A = [[lambda,-1],[2,lambda-3]]. Its determinant is lambda(lambda-3) + 2 = lambda^2 - 3lambda + 2.

Question 16Linear algebra

A 3 by 4 matrix has rank 2. What is the nullity of the associated linear map from R^4 to R^3?

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

Explanation

By rank-nullity, nullity = number of columns - rank = 4 - 2 = 2.

Question 17Linear algebra

Are v1=(1,1,0), v2=(1,0,1), and v3=(0,1,1) linearly independent in R^3?

  1. Yes, because the determinant with these as columns is nonzero Correct answer.
  2. No, because there are three vectors
  3. No, because their entries sum to 2
  4. Yes, because each has two nonzero entries

Explanation

Using the vectors as columns gives determinant -2, which is nonzero. Therefore the three vectors are linearly independent.

Question 18Linear algebra

If A = diag(2, -1), what is A^2?

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

Explanation

Squaring a diagonal matrix squares each diagonal entry. Thus A^2 = diag(2^2, (-1)^2) = diag(4,1).

Question 19Linear algebra

If B = P^(-1)AP, which quantity is always the same for A and B?

  1. Trace Correct answer.
  2. Every entry
  3. Every row sum
  4. Every column

Explanation

Similar matrices have the same trace, determinant, characteristic polynomial, and eigenvalues. Their individual entries, rows, and columns can change.

Question 20Linear algebra

A linear system Ax=b is consistent exactly when:

  1. rank(A) = rank([A|b]) Correct answer.
  2. rank(A) < rank([A|b])
  3. det A = 0 always
  4. b = 0 only

Explanation

The consistency criterion says Ax=b is consistent if and only if the coefficient matrix and augmented matrix have the same rank.

Question 21Linear algebra

The quadratic form q(x,y)=x^2+2xy+y^2 is:

  1. Positive definite
  2. Negative definite
  3. Positive semidefinite but not positive definite Correct answer.
  4. Indefinite

Explanation

q(x,y) = (x+y)^2, so it is always nonnegative. It equals 0 for nonzero vectors such as (1,-1), so it is not positive definite.

Question 22Linear algebra

Which describes the symmetric matrix [[2,1],[1,2]]?

  1. Positive definite Correct answer.
  2. Negative definite
  3. Singular
  4. Indefinite

Explanation

The leading principal minors are 2 and det = 3, both positive. Therefore the matrix is positive definite.

Question 23Linear algebra

A 2 by 2 matrix with two distinct eigenvalues is:

  1. Always diagonalizable over the field containing those eigenvalues Correct answer.
  2. Never diagonalizable
  3. Diagonalizable only if it is symmetric
  4. Always the identity matrix

Explanation

Distinct eigenvalues have linearly independent eigenvectors. A 2 by 2 matrix with two distinct eigenvalues therefore has an eigenbasis and is diagonalizable over that field.

Question 24Linear algebra

For an overdetermined least-squares problem Ax approximately equals b, the normal equations are:

  1. A^T A x = A^T b Correct answer.
  2. A A^T x = b
  3. A x = A^T b
  4. A^T x = b

Explanation

The least-squares residual is orthogonal to the column space of A, which gives A^T(b - Ax)=0. Rearranging yields A^T A x = A^T b.

Question 25Linear algebra

If A is 3 by 3 and det A = 4, what is det(2A)?

  1. 8
  2. 16
  3. 32 Correct answer.
  4. 4

Explanation

Multiplying a 3 by 3 matrix by scalar 2 multiplies the determinant by 2^3. Therefore det(2A)=8 det A = 32.

Question 26Linear algebra

For A = [[1,1],[0,1]], what is A^n for positive integer n?

  1. [[1,n],[0,1]] Correct answer.
  2. [[1,1],[0,n]]
  3. [[n,1],[0,n]]
  4. [[1,n^2],[0,1]]

Explanation

Write A = I + N where N = [[0,1],[0,0]] and N^2 = 0. Then (I+N)^n = I + nN = [[1,n],[0,1]].

Question 27Linear algebra

Why is A = [[2,1],[0,2]] not diagonalizable?

  1. It has no eigenvalues
  2. Its only eigenspace is one-dimensional Correct answer.
  3. Its determinant is zero
  4. It is symmetric

Explanation

The only eigenvalue is 2. Solving (A-2I)v=0 gives v2=0, so the eigenspace has dimension 1, not enough for a 2 by 2 eigenbasis.

Question 28Linear algebra

For M = [[1,a,0],[0,1,a],[a,0,1]], when is M singular over the real numbers?

  1. a = 0
  2. a = 1
  3. a = -1 Correct answer.
  4. All real a

Explanation

Expanding the determinant gives det M = 1 + a^3. Over the real numbers, this is zero exactly when a = -1.

Question 29Linear algebra

If A has full column rank n, what is true about A^T A?

  1. It is singular
  2. It is invertible Correct answer.
  3. It is the zero matrix
  4. It has rank 0

Explanation

For any nonzero x, x^T A^T A x = ||Ax||^2 > 0 because full column rank makes Ax nonzero. Thus A^T A is positive definite and invertible.

Question 30Linear algebra

The quadratic form 2x^2 + 4xy + 5y^2 has matrix [[2,2],[2,5]]. What is its definiteness?

  1. Positive definite Correct answer.
  2. Positive semidefinite only
  3. Indefinite
  4. Negative definite

Explanation

For the symmetric matrix, the leading principal minors are 2 and 2*5 - 2*2 = 6, both positive. Hence the form is positive definite.

Question 31Linear algebra

For the symmetric matrix [[2,1],[1,2]], which eigenvector corresponds to eigenvalue 3?

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

Explanation

Multiplying gives [[2,1],[1,2]](1,1) = (3,3) = 3(1,1). Thus (1,1) is an eigenvector for eigenvalue 3.

Question 32Linear algebra

For block upper triangular matrix [[A,B],[0,D]], with square A and D, what is its determinant?

  1. det(A) + det(D)
  2. det(A)det(D) Correct answer.
  3. det(B)
  4. 0 always

Explanation

A block triangular matrix has determinant equal to the product of the determinants of its diagonal blocks. The off-diagonal block B does not affect the determinant.

Question 33Multivariable calculus

For f(x,y) = x^2 + y^2, what is grad f at (1,-2)?

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

Explanation

The gradient is (partial f/partial x, partial f/partial y) = (2x, 2y). At (1,-2), this is (2,-4).

Question 34Multivariable calculus

For f(x,y) = xy^2 + 3x, what is partial f/partial y at (2,1)?

  1. 2
  2. 4 Correct answer.
  3. 7
  4. 10

Explanation

Treat x as constant when differentiating with respect to y. partial f/partial y = 2xy, so at (2,1) the value is 4.

Question 35Multivariable calculus

If grad f = (1,2) and u = (3/5,4/5), what is the directional derivative in direction u?

  1. 7/5
  2. 11/5 Correct answer.
  3. 5
  4. 1

Explanation

The directional derivative is grad f dot u. Compute 1*(3/5) + 2*(4/5) = 3/5 + 8/5 = 11/5.

Question 36Multivariable calculus

Compute the double integral over 0 <= x <= 1, 0 <= y <= 2 of (x + y) dA.

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

Explanation

Integrate with respect to y first: integral_0^2 (x+y) dy = 2x + 2. Then integral_0^1 (2x+2) dx = 1 + 2 = 3.

Question 37Multivariable calculus

Change the order of integration for integral_0^1 integral_0^x f(x,y) dy dx.

  1. integral_0^1 integral_y^1 f(x,y) dx dy Correct answer.
  2. integral_0^1 integral_0^y f(x,y) dx dy
  3. integral_0^x integral_0^1 f(x,y) dx dy
  4. integral_0^1 integral_0^1 f(x,y) dx dy

Explanation

The region is 0 <= y <= x <= 1. For a fixed y between 0 and 1, x runs from y to 1, giving integral_0^1 integral_y^1 f(x,y) dx dy.

Question 38Multivariable calculus

For z = x^2 y + sin y, what is partial z/partial x at (2,0)?

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

Explanation

Treat y as constant. partial z/partial x = 2xy, and at (2,0) this equals 0.

Question 39Multivariable calculus

Use differentials to approximate the change in z = x^2 y at (1,2) when dx = 0.1 and dy = 0.2.

  1. 0.2
  2. 0.4
  3. 0.6 Correct answer.
  4. 0.8

Explanation

dz = z_x dx + z_y dy = (2xy)dx + (x^2)dy. At (1,2), dz = 4(0.1) + 1(0.2) = 0.6.

Question 40Multivariable calculus

For f(x,y)=x^2+xy+y^2, what is grad f at (1,1)?

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

Explanation

f_x = 2x + y and f_y = x + 2y. At (1,1), both components equal 3, so grad f = (3,3).

Question 41Multivariable calculus

What type of critical point does f(x,y)=x^2+y^2 have at (0,0)?

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

Explanation

The function is nonnegative everywhere and equals 0 only at (0,0). Therefore the origin is a strict local minimum.

Question 42Multivariable calculus

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

  1. 10
  2. 20
  3. 25 Correct answer.
  4. 50

Explanation

With fixed sum 10, the product xy is maximized at x = y = 5. The maximum is 25.

Question 43Multivariable calculus

Compute integral_0^1 integral_0^x x y dy dx.

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

Explanation

For fixed x, integral_0^x x y dy = x*(x^2/2)=x^3/2. Then integral_0^1 x^3/2 dx = 1/8.

Question 44Multivariable calculus

Find the minimum of x^2 + y^2 subject to x + 2y = 5.

  1. 1
  2. 5 Correct answer.
  3. 25
  4. sqrt(5)

Explanation

The closest point on the line to the origin is parallel to the normal vector (1,2). Setting (x,y)=t(1,2) and x+2y=5 gives t=1, so (x,y)=(1,2) and the minimum value is 5.

Question 45Multivariable calculus

What type of critical point does f(x,y)=x^2-y^2 have at (0,0)?

  1. Strict local minimum
  2. Strict local maximum
  3. Saddle point Correct answer.
  4. No critical point

Explanation

The gradient is zero at the origin. Along the x-axis f=x^2 is positive, while along the y-axis f=-y^2 is negative, so the origin is a saddle point.

Question 46Limits and differential calculus

For f(x) = x^3 - 2x + 1, what is f'(2)?

  1. 8
  2. 10 Correct answer.
  3. 11
  4. 5

Explanation

Differentiate first: f'(x) = 3x^2 - 2. Substituting x = 2 gives 12 - 2 = 10.

Question 47Limits and differential calculus

What is the second derivative of e^(2x) at x = 0?

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

Explanation

The first derivative is 2e^(2x), and the second derivative is 4e^(2x). At x = 0 this equals 4.

Question 48Limits and differential calculus

What is d/dx ln x at x = 1?

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

Explanation

The derivative of ln x is 1/x. At x = 1, the derivative equals 1.

Question 49Limits and differential calculus

Where does f(x) = x^2 + 4x have its critical point?

  1. x = -4
  2. x = -2 Correct answer.
  3. x = 0
  4. x = 2

Explanation

The derivative is f'(x) = 2x + 4. Setting it equal to zero gives x = -2.

Question 50Limits and differential calculus

What is the slope of the tangent line to y = x^2 at x = 3?

  1. 3
  2. 6 Correct answer.
  3. 9
  4. 12

Explanation

The slope is the derivative y' = 2x. At x = 3, the slope is 6.