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AP Calculus AB Unit 1B1: Limit Concepts and Graphical Limits — practice questions — page 3

Explore 73 free Unit 1B1: Limit Concepts and Graphical Limits questions in AP Calculus AB Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Unit 1B1: Limit Concepts and Graphical Limits questions and exercises with answers

Showing questions 61–73 of 73.

If $p$ is a prime number and $p$ divides $a^2$, then which conclusion must be true?Read the answer and worked solution →A square matrix $A$ satisfies $A^T A = I$. Such a matrix is called:Read the answer and worked solution →Which of the following is not a polynomial in the variable $x$?Read the answer and worked solution →If $A$ is a matrix of order $3 \times 4$ and $B$ is of order $4 \times 5$, then the order of the product $AB$ is:Read the answer and worked solution →For any square matrix $A$, the matrix $A + A'$ is:Read the answer and worked solution →The diagonal entries of any skew-symmetric matrix are:Read the answer and worked solution →If $A$ and $B$ are matrices for which $AB$ is defined, then $(AB)'$ equals:Read the answer and worked solution →A square matrix $A$ is **invertible** if and only if:Read the answer and worked solution →For a non-singular $n \times n$ matrix $A$, the relation between $\det(\operatorname{adj} A)$ and $\det A$ is:Read the answer and worked solution →If $A$ is a square matrix with $|A| = 3$ and order $n = 3$, then $|\operatorname{adj}(A)|$ equals:Read the answer and worked solution →If $(d,m,n)$ are direction cosines of a line, which relation must hold?Read the answer and worked solution →Which statement is true for a polynomial of degree $n$?Read the answer and worked solution →Which of the following is a rational number?Read the answer and worked solution →

Worked practice questions

If $p$ is a prime number and $p$ divides $a^2$, then which conclusion must be true?

  1. $p$ divides $a$
  2. $a$ must be prime
  3. $a$ must equal $p$
  4. $p$ divides only even numbers
Show answer and solution

Correct answer: A. $p$ divides $a$

This is a standard theorem based on prime factorisation: if a prime divides $a^2$ then that prime must already occur in the factorisation of $a$. **Hence $p$ divides $a$.**

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A square matrix $A$ satisfies $A^T A = I$. Such a matrix is called:

  1. Symmetric
  2. Skew-symmetric
  3. Orthogonal
  4. Nilpotent
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Correct answer: C. Orthogonal

An orthogonal matrix satisfies $A^T A = A A^T = I$, equivalently $A^{-1} = A^T$. Its rows (and columns) form an orthonormal set. Rotation and reflection matrices in $\mathbb{R}^n$ are orthogonal. $|A|$ for an orthogonal matrix is always $\pm 1$.

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Which of the following is not a polynomial in the variable $x$?

  1. $5x^3-4x^2+x-\sqrt{2}$
  2. $4x+2$
  3. $\dfrac{1}{x-1}$
  4. $x^2-3x-4$
Show answer and solution

Correct answer: C. $\dfrac{1}{x-1}$

A polynomial in $x$ can have only non-negative integer powers of $x$. In $\dfrac{1}{x-1}$, the variable appears in the denominator, so it is not a polynomial.

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More Mathematics topics

Unit 3B: Implicit Differentiation and Inverse FunctionsUnit 3A: Chain Rule and Composite FunctionsUnit 1A3: Function Composition and InversesUnit 1A1: Relations, Functions and GraphsUnit 1A2: Trigonometric Functions & Inverse TrigUnit 2: Differentiation and Fundamental Derivative RulesUnit 6A: Indefinite Integrals & Fundamental TheoremUnit 1B3: Asymptotic Behavior and Infinite Limits