JEE Advanced Mathematics Flashcards

24 sample cards from the Mathematics deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. Define a bijective function and state the condition for invertibility.

   A function is bijective if it is both one-to-one and onto. A function $f$ is invertible if and only if it is bijective; then a unique inverse $f^{-1}: B \to A$ exists with $f^{-1}(f(x)) = x$.

2. For finite sets with $n(A) = m$ and $n(B) = n$, give the count of total functions and of one-to-one functions from $A$ to $B$.

   Total functions: $n^{m}$. One-to-one functions (require $n \geq m$): $n(n-1)(n-2)\cdots(n-m+1) = \dfrac{n!}{(n-m)!}$.

3. Define even and odd functions and give one example of each.

   Even: $f(-x) = f(x)$ for all $x$ (graph symmetric about the $y$-axis), e.g. $f(x) = x^{2}$ or $\cos x$. Odd: $f(-x) = -f(x)$ (symmetric about origin), e.g. $f(x) = x^{3}$ or $\sin x$.

4. How do you express any function as the sum of an even and an odd function?

   $f(x) = \underbrace{\frac{f(x)+f(-x)}{2}}_{\text{even part}} + \underbrace{\frac{f(x)-f(-x)}{2}}_{\text{odd part}}$.

5. Define the greatest integer (floor) function and state its range.

   $f(x) = \lfloor x \rfloor$ is the greatest integer not exceeding $x$, e.g. $\lfloor 2.7 \rfloor = 2$, $\lfloor -1.3 \rfloor = -2$. Domain $\mathbb{R}$, range $\mathbb{Z}$ (it is a step function).

6. Define the absolute value (modulus) function and give its domain and range.

   $f(x) = |x| = \begin{cases} x, & x \geq 0 \\ -x, & x < 0 \end{cases}$. Domain $\mathbb{R}$, range $[0, \infty)$. It is an even function.

7. State the domain and range of the exponential function $f(x) = a^{x}$ ($a>0,\ a \neq 1$) and the logarithmic function $g(x) = \log_a x$.

   Exponential $a^{x}$: domain $\mathbb{R}$, range $(0, \infty)$. Logarithm $\log_a x$: domain $(0, \infty)$, range $\mathbb{R}$. They are inverses of each other.

8. What is a polynomial function, and what is the domain of a power function $f(x)=x^{n}$?

   A polynomial function is $f(x) = a_n x^{n} + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0$ with constant coefficients and non-negative integer powers. A power function $x^{n}$ with $n \in \mathbb{N}$ has domain $\mathbb{R}$.

9. State the domains and ranges of the basic trigonometric functions $\sin x$ and $\tan x$.

   $\sin x$: domain $\mathbb{R}$, range $[-1,1]$. $\tan x$: domain $\mathbb{R} \setminus \{(2n+1)\frac{\pi}{2}\}$, range $\mathbb{R}$.

10. Define the sum, difference and product of two functions $f$ and $g$, and state their domain.

    $(f+g)(x)=f(x)+g(x)$, $(f-g)(x)=f(x)-g(x)$, $(fg)(x)=f(x)\,g(x)$. Each has domain $\text{dom}(f) \cap \text{dom}(g)$.

11. Define the quotient $\left(\frac{f}{g}\right)(x)$ of two functions and its domain.

    $\left(\dfrac{f}{g}\right)(x) = \dfrac{f(x)}{g(x)}$, with domain $\{x \in \text{dom}(f) \cap \text{dom}(g) : g(x) \neq 0\}$.

12. Define the composition of functions $(f \circ g)(x)$ and state whether composition is commutative.

    $(f \circ g)(x) = f(g(x))$, applying $g$ first then $f$. Composition is generally not commutative: $f \circ g \neq g \circ f$ in general, though it is associative.

13. Define a complex number and its real and imaginary parts, with the value of $i$.

    A complex number is $z = a + bi$ where $a, b \in \mathbb{R}$ and $i = \sqrt{-1}$ (so $i^{2} = -1$). $\operatorname{Re}(z) = a$ and $\operatorname{Im}(z) = b$.

14. State the condition for equality of two complex numbers $a+bi$ and $c+di$.

    $a + bi = c + di$ if and only if $a = c$ and $b = d$ (real parts equal and imaginary parts equal).

15. How do you add and subtract two complex numbers $z_1 = a+bi$ and $z_2 = c+di$?

    $z_1 + z_2 = (a+c) + (b+d)i$ and $z_1 - z_2 = (a-c) + (b-d)i$ — add/subtract real and imaginary parts separately.

16. How do you multiply two complex numbers $(a+bi)(c+di)$?

    Expand using $i^{2} = -1$: $(a+bi)(c+di) = (ac - bd) + (ad + bc)i$.

17. List the algebraic properties (closure, commutativity, associativity, identities) of complex number addition.

    Addition of complex numbers is closed, commutative ($z_1+z_2=z_2+z_1$), and associative. The additive identity is $0 = 0+0i$, and the additive inverse of $a+bi$ is $-a-bi$.

18. What are the powers of $i$ in cyclic form?

    $i^{1}=i$, $i^{2}=-1$, $i^{3}=-i$, $i^{4}=1$, and the pattern repeats with period 4: $i^{4k}=1$, $i^{4k+1}=i$, $i^{4k+2}=-1$, $i^{4k+3}=-i$.

19. Define the conjugate $\bar{z}$ of a complex number and state $z\bar{z}$.

    For $z = a+bi$, the conjugate is $\bar{z} = a - bi$. Their product $z\bar{z} = a^{2} + b^{2} = |z|^{2}$, a non-negative real number.

20. How do you divide complex numbers $\dfrac{a+bi}{c+di}$?

    Multiply numerator and denominator by the conjugate of the denominator: $\dfrac{a+bi}{c+di} = \dfrac{(a+bi)(c-di)}{c^{2}+d^{2}} = \dfrac{(ac+bd)+(bc-ad)i}{c^{2}+d^{2}}$.

21. Compare an into function and an onto function.

    Onto (surjective): range $= $ codomain — every codomain element is hit. Into: range $\subset$ codomain — at least one codomain element has no preimage. A function is either onto or into, never both.

22. Why is the constant function $f(x) = c$ both an example function and not one-to-one (for a domain with $>1$ element)?

    It maps every input to the same output $c$, so it is a valid function (each input has one image), but $f(x_1)=f(x_2)=c$ for distinct inputs, violating injectivity; its range is the singleton $\{c\}$.

23. State the relationship between a relation being a function and the 'vertical line test' / pre-image condition.

    A relation is a function iff no input has more than one output. Graphically (vertical line test), every vertical line meets the graph in at most one point; set-theoretically, no two ordered pairs share the same first component.

24. Distinguish the domain of a relation from the domain of the function it may define.

    For a relation $R \subseteq A \times B$, the domain is only the set of first elements appearing in $R$ (a subset of $A$). For a function $f: A \to B$, the domain is the entire set $A$, since every element of $A$ must have an image.

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