JEE Main 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. How is a complex number defined as an ordered pair of real numbers, and how are equality, addition, and multiplication defined?

   A complex number is an ordered pair $(a,b)$ of reals. Equality: $(a,b)=(c,d)\iff a=c,\ b=d$. Addition: $(a,b)+(c,d)=(a+c,\ b+d)$. Multiplication: $(a,b)\cdot(c,d)=(ac-bd,\ ad+bc)$.

2. Write a complex number in the form $a+ib$, and identify the real and imaginary parts. What is $i$?

   $z=a+ib$ where $a,b\in\mathbb{R}$, $i=\sqrt{-1}$ with $i^{2}=-1$. Here $\operatorname{Re}(z)=a$ is the real part and $\operatorname{Im}(z)=b$ is the imaginary part.

3. What are the values of $i^{2}$, $i^{3}$, $i^{4}$, and the general value of $i^{n}$?

   $i^{2}=-1$, $i^{3}=-i$, $i^{4}=1$. The powers cycle with period 4, so $i^{n}=i^{n\bmod 4}$ (e.g. $i^{4k}=1,\ i^{4k+1}=i,\ i^{4k+2}=-1,\ i^{4k+3}=-i$).

4. What is the Argand diagram, and how is a complex number $z=a+ib$ represented in it?

   The Argand diagram (complex plane) represents $z=a+ib$ as the point $(a,b)$, with the real part on the horizontal (real) axis and the imaginary part on the vertical (imaginary) axis.

5. Define the conjugate $\bar{z}$ of $z=a+ib$ and state $z\bar{z}$.

   The conjugate is $\bar{z}=a-ib$ (reflection of $z$ across the real axis). Then $z\bar{z}=a^{2}+b^{2}=|z|^{2}$, a non-negative real number.

6. State the properties of conjugates: $\overline{z_1+z_2}$, $\overline{z_1 z_2}$, and $\overline{\left(\frac{z_1}{z_2}\right)}$.

   $\overline{z_1+z_2}=\bar{z_1}+\bar{z_2}$, $\overline{z_1 z_2}=\bar{z_1}\,\bar{z_2}$, and $\overline{\left(\dfrac{z_1}{z_2}\right)}=\dfrac{\bar{z_1}}{\bar{z_2}}$. Also $\overline{\bar{z}}=z$ and $z+\bar{z}=2\operatorname{Re}(z)$.

7. Define the modulus of a complex number $z=a+ib$.

   The modulus is $|z|=\sqrt{a^{2}+b^{2}}$, the distance of the point $z$ from the origin in the Argand plane. Note $|z|=|\bar{z}|$ and $|z|^{2}=z\bar{z}$.

8. State the modulus properties: $|z_1 z_2|$, $\left|\frac{z_1}{z_2}\right|$, and the triangle inequality.

   $|z_1 z_2|=|z_1||z_2|$ and $\left|\dfrac{z_1}{z_2}\right|=\dfrac{|z_1|}{|z_2|}$. Triangle inequality: $|z_1+z_2|\leq|z_1|+|z_2|$ and $|z_1-z_2|\geq \big||z_1|-|z_2|\big|$.

9. Define the argument (amplitude) of a complex number $z=a+ib$.

   The argument $\theta=\arg(z)$ is the angle the line joining $z$ to the origin makes with the positive real axis, given by $\tan\theta=\dfrac{b}{a}$. The principal value lies in $(-\pi,\pi]$.

10. Write the polar (trigonometric) form of a complex number and its Euler form.

    $z=r(\cos\theta+i\sin\theta)$, where $r=|z|$ and $\theta=\arg(z)$. In Euler form, $z=re^{i\theta}$.

11. How do you compute the multiplicative inverse $z^{-1}$ of a non-zero complex number $z=a+ib$?

    $z^{-1}=\dfrac{1}{z}=\dfrac{\bar{z}}{|z|^{2}}=\dfrac{a-ib}{a^{2}+b^{2}}=\dfrac{a}{a^{2}+b^{2}}-i\dfrac{b}{a^{2}+b^{2}}$.

12. How is division of complex numbers $\dfrac{z_1}{z_2}$ carried out?

    Multiply numerator and denominator by the conjugate of the denominator: $\dfrac{z_1}{z_2}=\dfrac{z_1\,\bar{z_2}}{z_2\,\bar{z_2}}=\dfrac{z_1\,\bar{z_2}}{|z_2|^{2}}$, which removes $i$ from the denominator.

13. State the general form of a quadratic equation and its standard solution (quadratic formula).

    $ax^{2}+bx+c=0$ with $a\neq 0$. Its roots are $x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$.

14. What is the discriminant of a quadratic equation, and what does it determine?

    The discriminant is $D=b^{2}-4ac$. Its sign determines the nature (real/complex, equal/distinct) of the roots of $ax^{2}+bx+c=0$.

15. For a quadratic with real coefficients, what is the nature of roots when $D>0$, $D=0$, and $D<0$?

    If $D>0$: roots are real and distinct. If $D=0$: roots are real and equal ($x=-\dfrac{b}{2a}$). If $D<0$: roots are complex (non-real) and occur as conjugate pairs.

16. When are the roots of $ax^{2}+bx+c=0$ (with rational coefficients) rational, and when irrational?

    If $D=b^{2}-4ac$ is a perfect square (and $\geq 0$), the roots are rational; if $D>0$ but not a perfect square, the roots are irrational and occur as conjugate surds $p\pm\sqrt{q}$.

17. State the relations between the roots $\alpha,\beta$ and the coefficients of $ax^{2}+bx+c=0$.

    Sum of roots: $\alpha+\beta=-\dfrac{b}{a}$. Product of roots: $\alpha\beta=\dfrac{c}{a}$.

18. How do you form a quadratic equation given its roots $\alpha$ and $\beta$?

    Use $x^{2}-(\text{sum})x+(\text{product})=0$, i.e. $x^{2}-(\alpha+\beta)x+\alpha\beta=0$.

19. Express $\alpha^{2}+\beta^{2}$ and $\alpha-\beta$ in terms of the sum and product of roots.

    $\alpha^{2}+\beta^{2}=(\alpha+\beta)^{2}-2\alpha\beta$, and $(\alpha-\beta)^{2}=(\alpha+\beta)^{2}-4\alpha\beta$, so $|\alpha-\beta|=\dfrac{\sqrt{D}}{|a|}$.

20. For a complex (non-real) root of a quadratic with real coefficients, what can be said about the other root?

    Complex roots occur in conjugate pairs: if $p+iq$ is a root, then $p-iq$ is also a root (provided the coefficients are real).

21. What is the condition for the roots of $ax^{2}+bx+c=0$ to be equal in magnitude but opposite in sign, and reciprocal of each other?

    Roots are equal and opposite (sum $=0$) when $b=0$. Roots are reciprocals of each other (product $=1$) when $c=a$, i.e. $a=c$.

22. Solve $x^{2}+1=0$ over the complex number system.

    $x^{2}=-1 \implies x=\pm\sqrt{-1}=\pm i$. The roots are $i$ and $-i$.

23. How do you compute the square root of a complex number, e.g. $\sqrt{a+ib}$ method?

    Set $\sqrt{a+ib}=x+iy$, then square to get $x^{2}-y^{2}=a$ and $2xy=b$. Solving these (together with $x^{2}+y^{2}=\sqrt{a^{2}+b^{2}}$) gives the two square roots $\pm(x+iy)$.

24. What is the geometric meaning of $|z-z_0|=r$ and $|z-z_1|=|z-z_2|$ in the Argand plane?

    $|z-z_0|=r$ represents a circle of radius $r$ centered at $z_0$. $|z-z_1|=|z-z_2|$ represents the perpendicular bisector of the segment joining points $z_1$ and $z_2$.

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