Quiz: Real Analysis and Elementary Functions — 67 questions

1. Which expression is a mathematical assertion?

A. The integer 7 is prime

B. Find an integer satisfying the equation

C. Is the variable x positive?

D. For every integer n

Answer: (A)

Explanation

A mathematical assertion is unambiguously either true or false, and “the integer 7 is prime” has that property. The other expressions are incomplete requests or questions rather than truth-valued statements.

2. When is the inclusive disjunction A or B true?

A. When both A and B are false

B. When A and B have the same truth value

C. When at least one of A and B is true

D. When A is true and B is false

Answer: (C)

Explanation

An inclusive disjunction is true when A is true, B is true, or both are true. Requiring both assertions to be true describes conjunction instead.

3. In the implication A ⇒ B, what is the logical relationship between A and B?

A. A and B are equivalent assertions

B. A and B must have identical truth values

C. A is sufficient for B, and B is necessary for A

D. A is necessary for B, and B is sufficient for A

Answer: (C)

Explanation

The implication requires B whenever A holds, so A guarantees B and is sufficient for it; consequently, B is necessary for A. The implication does not require the two assertions to be equivalent.

4. Which statement is equivalent to the negation of ∀x,A(x)?

A. ¬∃x,A(x)

B. ∀x,¬A(x)

C. ∃x,¬A(x)

D. ∃x,A(x)

Answer: (C)

Explanation

Negating a universal quantifier produces an existential quantifier and negates the assertion, giving ∃x,¬A(x). The statement ∀x,¬A(x) is instead equivalent to negating an existential assertion.

5. If every element of A belongs to B, which relation is established?

A. B is included in A

B. A is included in B

C. A and B are necessarily equal

D. A and B have no common elements

Answer: (B)

Explanation

The inclusion A ⊂ B means that every element of A is also an element of B. Equality would additionally require every element of B to belong to A.

6. Which elements belong to A ∩ B?

A. Elements belonging to A but not B

B. Elements belonging to A or B

C. Elements belonging to both A and B

D. Elements belonging to neither A nor B

Answer: (C)

Explanation

The intersection contains elements that are members of both sets simultaneously. Membership in at least one set describes the union instead.

7. If f:E→F and g:F→G, how is the composition g∘f evaluated at x?

A. $g(f(x))$

B. $f(x)+g(x)$

C. $f(g(x))$

D. $g(x)-f(x)$

Answer: (A)

Explanation

Composition applies f first and then g, so $(g\circ f)(x)=g(f(x))$. Reversing the order would describe a different composition and may not even be defined.

8. What property does an application have if distinct inputs always produce distinct images and every target element has an antecedent?

A. It is bijective

B. It is injective but not surjective

C. It is surjective but not injective

D. It is neither injective nor surjective

Answer: (A)

Explanation

Distinct inputs having distinct images gives injectivity, while every target element having an antecedent gives surjectivity. An application with both properties is bijective.

9. Why is the remainder constraint r<b essential in Euclidean division?

A. It guarantees that the quotient and remainder are unique

B. It makes the remainder equal to zero

C. It guarantees that the quotient is larger than the divisor

D. It permits the divisor to be zero

Answer: (A)

Explanation

For a nonzero divisor, the condition r<b selects a unique quotient-remainder pair satisfying a=bq+r. Without that bound, multiple decompositions can represent the same number.

10. Which statement characterizes the prime factorization of a natural number n≥2?

A. It is unique when prime factors are ordered increasingly

B. It contains every prime number below n with a positive exponent

C. It is an arbitrary product of prime and composite factors

D. It depends on the order in which factors are discovered

Answer: (A)

Explanation

Every natural number at least 2 has a unique factorization into prime powers, apart from the ordering of the factors. An arbitrary product decomposition can be nonunique because composite factors may be grouped differently.

11. Which two steps are required for an induction proof beginning at n=0?

A. Prove P(n) for one large value and factor the result

B. Prove P(0) and establish P(n)⇒P(n+1) for every n

C. Prove P(1) and assume P(n+1) to prove P(n)

D. Assume P(0) and verify P(n)⇒P(n−1) for every n

Answer: (B)

Explanation

Induction requires an initialization proving P(0) and a hereditary step showing that P(n) implies P(n+1) for every natural number n. Proving one isolated case does not establish the statement for all natural numbers.

12. How many ways are there to choose p elements from a set of n elements?

A. $\frac{n!}{p!(n+p)!}$

B. $\frac{(n-p)!}{n!p!}$

C. $\frac{p!}{n!(p-n)!}$

D. $\frac{n!}{p!(n-p)!}$

Answer: (D)

Explanation

The number of combinations is $\binom{n}{p}=\frac{n!}{p!(n-p)!}$ for $0\le p\le n$. The other expressions do not represent the standard count of unordered selections.

13. Why is the relation $x\le y\Longleftrightarrow y-x\in[0,+\infty[$ a total order on $\mathbb R$?

A. It is reflexive, symmetric, transitive, and applies only to positive real numbers.

B. It is reflexive, transitive, antisymmetric, and compares every pair of real numbers.

C. It is antisymmetric and transitive, but it does not compare every pair of real numbers.

D. It is symmetric, transitive, antisymmetric, and compares every pair of real numbers.

Answer: (B)

Explanation

The relation $\le$ has reflexivity, transitivity, antisymmetry, and comparability for every pair of real numbers, which are the defining properties of a total order. The relation $<$ is not reflexive, so it cannot satisfy the requirements of an order relation.

14. If $-3<2$, what inequality results after multiplying both sides by $-4$?

A. $12< -8$

B. $-12<8$

C. $-12>8$

D. $12>-8$

Answer: (D)

Explanation

Multiplication by a negative number reverses the direction of an inequality, so multiplying $-3<2$ by $-4$ gives $12>-8$. Multiplication by a positive number would preserve the direction instead.

15. For a nonnegative real number $a$, which condition is equivalent to $|x|>a$?

A. $-a<x<a$

B. $x<-a\text{ or }x>a$

C. $-a\le x\le a$

D. $x\le-a\text{ or }x\ge a$

Answer: (B)

Explanation

The inequality $|x|>a$ means that $x$ lies farther from zero than $a$, giving $x<-a$ or $x>a$. The interval $-a\le x\le a$ instead describes the condition $|x|\le a$.

16. What distinguishes a maximum from a supremum of a set?

A. A maximum belongs to the set, while a supremum may be approached without belonging to it.

B. A maximum is approached by the set, while a supremum must be an element of the set.

C. A maximum may lie outside the set, while a supremum must lie inside the set.

D. A maximum and a supremum are both required to be elements of the set.

Answer: (A)

Explanation

A maximum is an element of the set that is at least as large as every other element, whereas a supremum is the least upper bound and need not belong to the set. Thus a set can approach its supremum without attaining it.

17. How can a real polynomial be represented algebraically?

A. As a sequence whose coefficients are multiplied term by term to define polynomial products.

B. As an infinite real coefficient sequence that becomes zero from some index onward.

C. As a function whose coefficients are defined only after substituting a real variable.

D. As a finite sequence of nonzero coefficients with no zero coefficients after its degree.

Answer: (B)

Explanation

A real polynomial can be encoded by an infinite coefficient sequence with all sufficiently late coefficients equal to zero; addition is coefficientwise and multiplication uses convolution. The indeterminate $X$ is formal, whereas a real variable is used when evaluating the polynomial.

18. If $A$ and $B$ are nonzero polynomials with degrees $4$ and $7$, respectively, what can be concluded about their sum and product?

A. $\deg(A+B)\le11$ and $\deg(AB)=7$

B. $\deg(A+B)\le7$ and $\deg(AB)=11$

C. $\deg(A+B)=7$ and $\deg(AB)=28$

D. $\deg(A+B)=11$ and $\deg(AB)\le7$

Answer: (B)

Explanation

The degree of a sum is at most the larger degree, so $\deg(A+B)\le\max(4,7)=7$, while the product degree is additive, giving $\deg(AB)=4+7=11$. Cancellation can reduce the degree of a sum but cannot reduce the degree of a product of nonzero real polynomials.

19. Which statement describes the Euclidean division of a polynomial $A$ by a nonzero polynomial $B$?

A. There is a unique pair $Q,R$ such that $A=BQ+R$ and $\deg(R)\le\deg(B)$.

B. There is a unique pair $Q,R$ such that $A=BQ+R$ and $\deg(B)<\deg(R)$.

C. There is a pair $Q,R$ such that $A=BQ+R$ and $\deg(Q)<\deg(B)$.

D. There is a unique pair $Q,R$ such that $A=BQ+R$ and $\deg(R)<\deg(B)$.

Answer: (D)

Explanation

Polynomial Euclidean division produces a unique quotient and remainder satisfying $A=BQ+R$ with the remainder degree strictly smaller than the divisor degree. A non-strict degree bound would not ensure the standard division form.

20. Which condition is equivalent to saying that a real number $a$ is a root of a polynomial $P$?

A. The remainder of dividing $P$ by $X+a$ is zero.

B. The polynomial $X-a$ divides $P$.

C. The polynomial $X+a$ divides $P$.

D. The value $P(a)$ equals the degree of $P$.

Answer: (B)

Explanation

The factor theorem states that $a$ is a root of $P$ exactly when $X-a$ divides $P$. Using $X+a$ would correspond to testing whether $-a$ is a root.

21. On which domain is the natural logarithm defined by $\ln(x)=\int_1^x\frac{1}{t}\,dt$?

A. The positive real numbers $]0,+\infty[$

B. The nonnegative real numbers $[0,+\infty[$

C. All real numbers including zero

D. The real numbers less than or equal to zero

Answer: (A)

Explanation

The natural logarithm is defined on positive arguments, namely $]0,+\infty[$, because the integrand $1/t$ is not defined at zero. The exponential function, in contrast, is defined for every real input.

22. For positive numbers $a$ and $b$, how does the logarithm simplify $\ln\left(\frac{a}{b}\right)$?

A. $\ln a-\ln b$

B. $\frac{\ln a}{\ln b}$

C. $\ln a+\ln b$

D. $\ln(ab)$

Answer: (A)

Explanation

Logarithms transform division into subtraction, so $\ln(a/b)=\ln a-\ln b$ for positive $a$ and $b$. Addition of logarithms corresponds instead to the logarithm of a product.

23. Which property identifies the exponential function as the inverse of the natural logarithm?

A. It satisfies $\exp(1)=1$ and decreases as its input increases.

B. It satisfies $\exp(0)=1$ and $\exp'(x)=\exp(x)>0$.

C. It is defined only for positive real inputs and has derivative $1/x$.

D. It satisfies $\exp(0)=0$ and $\exp'(x)=x$.

Answer: (B)

Explanation

The exponential is the inverse of the natural logarithm, with $\exp(0)=1$ and derivative $\exp'(x)=\exp(x)>0$, so it is strictly increasing. The derivative $1/x$ belongs to the natural logarithm, not the exponential.

24. For positive $x$ and real $\alpha$, what is the derivative of $x^\alpha$?

A. $x^{\alpha-1}/\alpha$

B. $\alpha x^{\alpha-1}$

C. $x^{\alpha+1}$

D. $\alpha x^\alpha$

Answer: (B)

Explanation

The power rule gives $\frac{d}{dx}x^\alpha=\alpha x^{\alpha-1}$ for positive $x$ and real $\alpha$. Multiplying by $x^\alpha$ would omit the required decrease of the exponent by one.

25. On the unit circle, what do $\cos x$ and $\sin x$ represent for the point associated with the oriented angle $x$?

A. The horizontal and vertical coordinates, respectively

B. The vertical and horizontal coordinates, respectively

C. The distance from the origin and the horizontal coordinate

D. The distance from the point and the vertical coordinate

Answer: (A)

Explanation

On the unit circle, $\cos x$ is the abscissa, or horizontal coordinate, and $\sin x$ is the ordinate, or vertical coordinate. Reversing these coordinates confuses the roles of sine and cosine.

26. Which derivative pair correctly describes the sine and cosine functions?

A. $\sin'(x)=\cos x$ and $\cos'(x)=-\sin x$

B. $\sin'(x)=\cos x$ and $\cos'(x)=\sin x$

C. $\sin'(x)=-\cos x$ and $\cos'(x)=\sin x$

D. $\sin'(x)=\sin x$ and $\cos'(x)=\cos x$

Answer: (A)

Explanation

Differentiating sine gives cosine, while differentiating cosine gives negative sine. The pair with a positive sine derivative for cosine omits the required negative sign.

27. For which real values of $x$ is the tangent function defined by $\tan x=\frac{\sin x}{\cos x}$?

A. For $x\notin\left\{\frac\pi4+k\pi:k\in\mathbb Z\right\}$

B. For $x\notin\left\{k\pi:k\in\mathbb Z\right\}$

C. For every real number except multiples of $2\pi$

D. For $x\notin\left\{\frac\pi2+k\pi:k\in\mathbb Z\right\}$

Answer: (D)

Explanation

The quotient is defined when $\cos x\ne0$, which excludes $x=\frac\pi2+k\pi$ for integer $k$. Multiples of $\pi$ have cosine equal to $\pm1$, so tangent is defined there.

28. Which identity gives the correct expansion of $\sin(a+b)$?

A. $\sin a\cos b+\cos a\sin b$

B. $\cos a\cos b+\sin a\sin b$

C. $\sin a\sin b+\cos a\cos b$

D. $\sin a\cos b-\cos a\sin b$

Answer: (A)

Explanation

The sine addition formula is $\sin(a+b)=\sin a\cos b+\cos a\sin b$. The subtraction of the cross terms belongs to the cosine addition formula, not the sine formula.

29. A complex number is written as $z=x+iy$ with real $x$ and $y$. When is $z$ a real number?

A. When $x=y$

B. When $x=0$

C. When $y=0$

D. When $y=1$

Answer: (C)

Explanation

The imaginary part of $z=x+iy$ is $y$, so the number is real when that part equals zero. If $x=0$ instead, the number is generally purely imaginary.

30. If $z=x+iy$ and $z'=x'+iy'$, what is the product $zz'$ in algebraic form?

A. $(x+x')+i(y+y')$

B. $(xx'+yy')+i(xy'-x'y)$

C. $(xx'-yy')+i(xy'+x'y)$

D. $(xx'-xy')+i(yx'+yy')$

Answer: (C)

Explanation

Expanding the product and using $i^2=-1$ gives the real part $xx'-yy'$ and imaginary part $xy'+x'y$. The expression with sums in both real terms incorrectly treats $i^2$ as positive.

31. What is the conjugate of the complex number $z=x+iy$?

A. $\bar z=x-iy$

B. $\bar z=-x-iy$

C. $\bar z=x+iy$

D. $\bar z=-x+iy$

Answer: (A)

Explanation

Conjugation preserves the real part and changes the sign of the imaginary part, producing $x-iy$. Negating the real part instead describes reflection across the imaginary axis rather than complex conjugation.

32. What is the modulus of $z=x+iy$?

A. $|z|=\sqrt{x^2+y^2}$

B. $|z|=x+y$

C. $|z|=x^2+y^2$

D. $|z|=\sqrt{x^2-y^2}$

Answer: (A)

Explanation

The modulus is the distance from the origin to the point $(x,y)$ in the complex plane, so it equals $\sqrt{x^2+y^2}$ and is nonnegative. The expression $x^2+y^2$ is the squared modulus, not the modulus itself.

33. Which statement correctly describes the arguments of a nonzero complex number $z$?

A. They consist of one fixed real number determined by the modulus

B. They are all values of the form $\theta+k\pi$, where $k\in\mathbb{Z}$

C. They are defined only when the complex number lies on the real axis

D. They are all values of the form $\theta+2k\pi$, where $k\in\mathbb{Z}$

Answer: (D)

Explanation

For a nonzero complex number, every argument differs from any chosen argument by an integer multiple of $2\pi$. The modulus determines distance from the origin, not the set of angular directions.

34. A nonzero complex number has modulus $5$ and argument $\frac{\pi}{3}$. Which trigonometric form represents it?

A. $5\left(\cos\frac{\pi}{6}+i\sin\frac{\pi}{6}\right)$

B. $\frac{\pi}{3}(\cos 5+i\sin 5)$

C. $5\left(\cos\frac{\pi}{3}+i\sin\frac{\pi}{3}\right)$

D. $\frac{1}{5}\left(\cos\frac{\pi}{3}+i\sin\frac{\pi}{3}\right)$

Answer: (C)

Explanation

The trigonometric form is $z=r(\cos\theta+i\sin\theta)$, with modulus $r=5$ and argument $\theta=\frac{\pi}{3}$. The alternative using $\frac{\pi}{3}$ as the radius confuses the roles of modulus and argument.

35. If $\arg(z)=\frac{\pi}{4}$ and $\arg(z')=\frac{\pi}{6}$, what is an argument of $\frac{z}{z'}$?

A. $\frac{\pi}{12}$

B. $\frac{2\pi}{3}$

C. $\frac{\pi}{24}$

D. $\frac{5\pi}{12}$

Answer: (A)

Explanation

Arguments of quotients are found by subtraction, so an argument is $\frac{\pi}{4}-\frac{\pi}{6}=\frac{\pi}{12}$ modulo $2\pi$. Adding the two arguments would apply the product rule instead.

36. How many distinct fourth roots does every nonzero complex number have?

A. Eight

B. Four

C. Two

D. One

Answer: (B)

Explanation

The root formula gives $n$ distinct $n$-th roots, so a nonzero complex number has four distinct fourth roots. The roots are separated by angular increments of $\frac{2\pi}{4}$.

37. What does it mean for a function $f$ to have finite limit $\ell$ at $a$?

A. The value $f(a)$ determines nearby values of the function exactly

B. Values of $f(x)$ can be made arbitrarily close to $\ell$ by taking $x$ sufficiently close to $a$

C. The function must equal $\ell$ throughout an interval containing $a$

D. The function must be defined at $a$ and satisfy $f(a)=\ell$

Answer: (B)

Explanation

A finite limit describes how function values behave when inputs approach $a$, allowing their distance from $\ell$ to become arbitrarily small. Defining the function at $a$ with matching value is an additional condition for continuity.

38. A function is not defined at $a$, but its left-hand and right-hand limits at $a$ both equal $7$. What can be concluded?

A. The function must have value $0$ at $a$

B. The two-sided limit at $a$ exists and equals $7$

C. The function is continuous at $a$ with value $7$

D. The two-sided limit fails because the function is undefined at $a$

Answer: (B)

Explanation

Equal left-hand and right-hand limits establish the two-sided limit, even when the function is not defined at the point. Continuity cannot be concluded because continuity also requires the function to be defined there.

39. Which condition is required for a function to be continuous at $a$?

A. Its limit at $a$ exists even if $f(a)$ is undefined

B. Its values remain constant on a neighborhood of $a$

C. It is defined at $a$ and $\lim_{x\to a}f(x)=f(a)$

D. Its left-hand and right-hand limits are unequal at $a$

Answer: (C)

Explanation

Continuity at $a$ requires both that $f(a)$ exists and that the limit as $x\to a$ equals this value. Existence of a limit without a defined matching function value describes a possible removable discontinuity instead.

40. A continuous function satisfies $f(1)=-3$ and $f(4)=2$. Which conclusion follows?

A. There is some $c\in(1,4)$ such that $f(c)=5$

B. There is some $c\in(1,4)$ such that $f(c)=0$

C. The function has no zero between $1$ and $4$

D. The function must be linear between $1$ and $4$

Answer: (B)

Explanation

Because the continuous function takes values of opposite signs at the endpoints, the intermediate value theorem guarantees a zero between them. Continuity does not imply that the function is linear or that it takes every value outside the endpoint range.

41. Suppose that near a point, $h(x)\le f(x)\le g(x)$ and both $h(x)$ and $g(x)$ tend to $4$. What follows from the squeeze theorem?

A. The limit of $f(x)$ cannot be determined from the bounds

B. $f(x)$ tends to $0$ because it is bounded

C. $f(x)$ tends to the larger bounding limit

D. $f(x)$ tends to $4$

Answer: (D)

Explanation

When the lower and upper functions have the same limit, every value of $f(x)$ is squeezed toward that common limit, so $f(x)\to4$. If the bounding limits were different, the inequalities would not determine a unique limit.

42. Under what additional condition is the reciprocal of a function continuous near a point where the original function is continuous?

A. The function in the denominator changes sign near that point

B. The reciprocal has the same value as the original function

C. The function in the denominator does not vanish near that point

D. The denominator has a higher degree than the numerator

Answer: (C)

Explanation

The reciprocal operation is continuous where its denominator is nonzero, so the denominator must not vanish near the point under consideration. Sums and products do not require this nonvanishing condition, which is why reciprocals need special care.

43. If $f$ is continuous at $a$ and $g$ is continuous at $f(a)$, what can be concluded about $g\circ f$ at $a$?

A. The composition is discontinuous unless $g(a)=f(a)$

B. The composition is defined only when both functions are linear

C. The composition $g\circ f$ is continuous at $a$

D. The composition is continuous only if $f(a)=a$

Answer: (C)

Explanation

Continuity of $f$ at $a$ sends inputs near $a$ to values near $f(a)$, where continuity of $g$ applies; therefore $g\circ f$ is continuous at $a$. The inner and outer functions need not be linear or share the same input value.

44. For a reduced rational function $\frac{P(x)}{Q(x)}$ with numerator degree smaller than denominator degree, what is its limit as $x\to\pm\infty$?

A. The ratio of the leading coefficients

B. An infinite value determined by the leading terms

C. The constant term of the numerator

D. $0$

Answer: (D)

Explanation

When the numerator degree is less than the denominator degree, the denominator grows faster in magnitude, so the quotient tends to $0$ at both ends. The ratio of leading coefficients applies when the two degrees are equal.

45. Which condition defines the line $y=ax+b$ as an asymptote to the graph of $f$ at $+\infty$?

A. The function $f(x)$ approaches the constant value $b$ as $x\to+\infty$

B. The difference $f(x)-(ax+b)$ tends to zero as $x\to+\infty$

C. The derivative $f'(x)$ tends to the constant $a$ as $x\to+\infty$

D. The quotient $f(x)/(ax+b)$ tends to zero as $x\to+\infty$

Answer: (B)

Explanation

An asymptote at $+\infty$ means that the vertical difference between the function and the line tends to zero. A horizontal asymptote is the special case with slope zero, so approaching $b$ alone does not describe a general oblique asymptote.

46. If an oblique asymptote of $f$ at $+\infty$ is written as $y=ax+b$, how are its coefficients determined?

A. Compute $a=\lim_{x\to+\infty}f(x)/x$ and then $b=\lim_{x\to+\infty}(f(x)-ax)$

B. Compute $a=\lim_{x\to+\infty}(f(x)-x)$ and then $b=\lim_{x\to+\infty}f(x)/x$

C. Compute $a=\lim_{x\to+\infty}f(x)$ and then $b=\lim_{x\to+\infty}f(x)/x$

D. Compute $a=\lim_{x\to+\infty}f'(x)$ and then $b=\lim_{x\to+\infty}f(x)$

Answer: (A)

Explanation

The slope is obtained from the leading growth rate $f(x)/x$, and the intercept is found after removing the linear term through $f(x)-ax$. Reversing these limits or replacing the corrected difference with $f(x)$ does not identify the asymptote coefficients.

47. What are the vertical and oblique asymptotes of $f(x)=\frac{(x+2)^2}{3(x+1)}$?

A. The vertical asymptote is $x=-1$, and the oblique asymptote is $y=3x-1$

B. The vertical asymptote is $x=-2$, and the oblique asymptote is $y=3x+1$

C. The vertical asymptote is $x=1$, and the oblique asymptote is $y=\frac{x}{3}-1$

D. The vertical asymptote is $x=-1$, and the oblique asymptote is $y=\frac{x}{3}+1$

Answer: (D)

Explanation

The denominator vanishes at $x=-1$, giving the vertical asymptote, while division produces the linear behavior $y=\frac{x}{3}+1$ at both ends. The other choices use incorrect denominator roots or incorrect coefficients in the oblique line.

48. Which condition is required for a function to be differentiable at $a$?

A. The finite two-sided limit $\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$ must exist

B. The function must have equal one-sided values near $a$ without a derivative limit

C. The function must approach a finite slope from the right while the left slope may differ

D. The function must be continuous on an interval without requiring a finite difference quotient

Answer: (A)

Explanation

Differentiability at $a$ requires the difference quotient to have a finite two-sided limit. Matching one-sided behavior or continuity on an interval does not by itself establish the existence of that derivative.

49. What is the equation of the tangent to the graph of $f$ at the point $(a,f(a))$?

A. $y=f(a)+f'(a)(x-a)$

B. $y=f(a)+a f'(x)$

C. $y=f(x)+f'(a)(x-a)$

D. $y=f'(a)+f(a)(x-a)$

Answer: (A)

Explanation

The tangent passes through $(a,f(a))$ and has slope $f'(a)$, giving the stated point-slope equation. An asymptote describes behavior far from a point, whereas this formula describes local behavior at the specified point.

50. Which statement correctly describes the relationship between continuity and differentiability at a point?

A. Continuity implies differentiability, but differentiability does not imply continuity

B. The two properties are equivalent whenever the function has a finite value

C. Differentiability implies continuity, but continuity does not imply differentiability

D. Neither property places a condition on the function near the point

Answer: (C)

Explanation

A finite derivative forces the function to be continuous at that point. The function $|x|$ is continuous at $0$ but has a corner there, so it is not differentiable at $0$.

51. If a function has a differentiable local extremum at an interior point $c$, what must be true?

A. $f(c)=0$

B. $f'(c)=1$

C. $f''(c)>0$

D. $f'(c)=0$

Answer: (D)

Explanation

At a differentiable local maximum or minimum located inside the domain, the tangent is horizontal, so $f'(c)=0$. The function value need not vanish, and the second derivative condition is not required by this rule.

52. What does the norm of a vector represent?

A. Its nonnegative length

B. Its angle measured from the positive horizontal axis

C. Its signed displacement along a direction

D. Its orientation relative to an axis

Answer: (A)

Explanation

The norm is the vector's nonnegative length, and a vector with norm $1$ is called unitary. Direction and sense describe orientation rather than the size measured by the norm.

53. How can two nonzero vectors be recognized as collinear?

A. One vector is a real scalar multiple of the other

B. Their scalar product is zero

C. They have equal norms and opposite senses

D. Their endpoints form a right angle with the origin

Answer: (A)

Explanation

Collinear vectors have the same direction and can therefore be expressed as real scalar multiples of one another. A zero scalar product instead indicates orthogonality, which concerns a right angle rather than common direction.

54. Why does a Cartesian frame with noncollinear basis vectors give unique coordinates to every vector?

A. The basis vectors have equal lengths and point in opposite directions

B. The origin fixes the direction of every vector in the plane

C. The basis vectors are independent and span the plane

D. Every vector in the frame must have a norm equal to one

Answer: (C)

Explanation

Noncollinear basis vectors are linearly independent and span the plane, so each vector has one and only one pair of coefficients. Equal lengths or unit norms are not what guarantees coordinate uniqueness.

55. How does a geometric angle differ from an oriented angle between two nonzero vectors?

A. A geometric angle depends on a chosen orientation, whereas an oriented angle is independent of orientation

B. A geometric angle is the shortest measure in $[0,\pi]$, whereas an oriented angle is defined modulo $2\pi$

C. A geometric angle is defined modulo $2\pi$, whereas an oriented angle always lies in $[0,\pi]$

D. A geometric angle applies to collinear vectors, whereas an oriented angle applies to vectors with a right angle

Answer: (B)

Explanation

The geometric angle records the shortest angular separation and lies in $[0,\pi]$. An oriented angle incorporates a chosen orientation of the plane and is determined modulo $2\pi$, so it is not restricted to the shortest separation.

56. What distinguishes the principal measure of an oriented angle from all of its measures?

A. It is defined only when the vectors are orthogonal

B. It is a single signed representative of the angle

C. It is always measured in the direct direction

D. It includes every value obtained by adding complete turns

Answer: (B)

Explanation

The principal measure is one signed angular gap, positive in the direct direction and negative otherwise. The family of all measures is obtained by adding integer multiples of $2\pi$, so it contains more than one representative.

57. If an oriented angle has principal measure $\alpha$, which expression gives all of its measures?

A. $\alpha+2k\pi$ with $k\in\mathbb{Z}$

B. $\alpha+k\pi$ with $k\in\mathbb{R}$

C. $2\alpha+k\pi$ with $k\in\mathbb{Z}$

D. $-\alpha+2k\pi$ with $k\in\mathbb{Z}$

Answer: (A)

Explanation

All measures of the oriented angle differ from its principal measure by complete turns, giving $\alpha+2k\pi$ for integer $k$. Reversing orientation instead changes the angle to $-\alpha$ modulo $2\pi$.

58. For three nonzero vectors, which relation expresses the Chasles relation for oriented angles?

A. $(\vec u,\vec v)=(\vec u,\vec w)-(\vec w,\vec v)\ [2\pi]$

B. $(\vec u,\vec v)=(\vec u,\vec w)(\vec w,\vec v)\ [2\pi]$

C. $(\vec u,\vec v)=(\vec u,\vec w)+(\vec w,\vec v)\ [2\pi]$

D. $(\vec u,\vec v)=(\vec w,\vec u)+(\vec v,\vec w)\ [\pi]$

Answer: (C)

Explanation

The Chasles relation states that the oriented angle from $\vec u$ to $\vec v$ is the sum of the angles from $\vec u$ to $\vec w$ and from $\vec w$ to $\vec v$, modulo $2\pi$. Subtraction, reversal, or multiplication does not express this angle-composition rule.

59. How can orthogonality and collinearity of two nonzero vectors be characterized using their oriented angle?

A. Both properties correspond to $\frac{\pi}{2}\ [2\pi]$

B. Both properties correspond to $0\ [2\pi]$

C. Orthogonality corresponds to $\frac{\pi}{2}\ [\pi]$, while collinearity corresponds to $0\ [\pi]$

D. Orthogonality corresponds to $0\ [\pi]$, while collinearity corresponds to $\frac{\pi}{2}\ [\pi]$

Answer: (C)

Explanation

Two nonzero vectors are orthogonal exactly when their oriented angle is $\frac{\pi}{2}$ modulo $\pi$, and collinear exactly when it is $0$ modulo $\pi$. The two conditions therefore correspond to different angle classes.

60. Which expression gives the Euclidean scalar product of two nonzero vectors?

A. $\|\vec u\|\,\|\vec v\|\cos(\vec u,\vec v)$

B. $\|\vec u\|\,\|\vec v\|\sin(\vec u,\vec v)$

C. $\|\vec u\|\,\|\vec v\|\tan(\vec u,\vec v)$

D. $\|\vec u\|+\|\vec v\|+\cos(\vec u,\vec v)$

Answer: (A)

Explanation

The scalar product uses the product of the vector norms multiplied by the cosine of their angle. The sine expression defines the determinant instead, while the other expressions are not the Euclidean scalar product.

61. What happens to the determinant when the order of two vectors is exchanged?

A. It becomes zero because the vectors are listed in reverse order

B. Its magnitude changes according to the sum of the vector norms

C. Its sign changes because the determinant depends on orientation

D. Its value remains unchanged because it is symmetric

Answer: (C)

Explanation

The determinant is based on the sine of the oriented angle, so exchanging the vectors reverses orientation and changes its sign. The scalar product, unlike the determinant, is symmetric under exchanging the vectors.

62. A pair of vectors has scalar product zero but a nonzero determinant; what relationship do the vectors have?

A. They have equal norms and opposite orientations

B. They are orthogonal but not collinear

C. They are identical in direction and length

D. They are collinear but not orthogonal

Answer: (B)

Explanation

A zero scalar product characterizes orthogonality, while a nonzero determinant rules out collinearity. Thus the vectors are orthogonal without being collinear.

63. In a direct orthonormal basis, what are the coordinate formulas for the scalar product and determinant of $\vec u=(x,y)$ and $\vec v=(x',y')$?

A. $\vec u\cdot\vec v=xx'-yy'$ and $\det(\vec u,\vec v)=xy'+yx'$

B. $\vec u\cdot\vec v=x+y+x'+y'$ and $\det(\vec u,\vec v)=xx'yy'$

C. $\vec u\cdot\vec v=xx'+yy'$ and $\det(\vec u,\vec v)=xy'-yx'$

D. $\vec u\cdot\vec v=xy'-yx'$ and $\det(\vec u,\vec v)=xx'+yy'$

Answer: (C)

Explanation

In an orthonormal basis, the scalar product is the sum of coordinatewise products, while the determinant is the cross-difference $xy'-yx'$ in a direct basis. The second listed pair incorrectly swaps these two operations.

64. Which parameterization represents the complete line through A with direction vector $\vec u=(u_x,u_y)$?

A. $x=x_A+u_x,\ y=y_A+u_y$ with $\alpha\in\mathbb{R}$

B. $x=\alpha x_A+u_x,\ y=\alpha y_A+u_y$ with $\alpha\in\mathbb{R}$

C. $x=x_A+\alpha u_x,\ y=y_A+\alpha u_y$ with $\alpha\in\mathbb{R}$

D. $x=x_A+\alpha u_x,\ y=y_A+\alpha u_y$ with $0\leq\alpha\leq1$

Answer: (C)

Explanation

A complete line is obtained by allowing the real parameter $\alpha$ to take every real value in the displacement $\alpha\vec u$ from A. Restricting $\alpha$ to an interval produces only part of the line, such as a segment.

65. For the line $ax+by+c=0$ with $(a,b)\ne(0,0)$, which pair gives a normal vector and a direction vector, respectively?

A. $(-b,a)$ and $(a,b)$

B. $(a,c)$ and $(b,c)$

C. $(b,-a)$ and $(-a,-b)$

D. $(a,b)$ and $(-b,a)$

Answer: (D)

Explanation

The coefficient vector $(a,b)$ is normal to the line, and rotating it by a right angle gives the direction vector $(-b,a)$. Reversing their roles confuses perpendicular and parallel directions.

66. What is the distance from $A(x_A,y_A)$ to the line $ax+by+c=0$?

A. $\frac{|ax_A+by_A+c|}{\sqrt{a^2+b^2}}$

B. $|ax_A+by_A+c|\sqrt{a^2+b^2}$

C. $\frac{ax_A+by_A+c}{\sqrt{|a|+|b|}}$

D. $\frac{|ax_A+by_A+c|}{a^2+b^2}$

Answer: (A)

Explanation

The point-to-line distance is the absolute value of the line expression evaluated at the point, divided by the norm of the normal vector. The denominator must therefore be $\sqrt{a^2+b^2}$, not its square or a different coordinate expression.

67. What equation represents the circle with centre $A(x_A,y_A)$ and radius $R$ in an orthonormal coordinate system?

A. $(x-x_A)^2+(y-y_A)^2=R^2$

B. $(x+x_A)^2+(y+y_A)^2=R^2$

C. $x^2+y^2-x_A-y_A=R$

D. $(x-x_A)+(y-y_A)=R^2$

Answer: (A)

Explanation

A point lies on the circle when its squared distance from the centre equals the squared radius, giving $(x-x_A)^2+(y-y_A)^2=R^2$. The other expressions do not represent the Euclidean distance condition for a circle with that centre.