SRMJEEE 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. State the formula for the inverse of a square matrix $A$ in terms of its adjoint.

   $A^{-1} = \frac{1}{\det(A)} \,\text{adj}(A)$, valid when $\det(A) \neq 0$ (i.e. $A$ is non-singular).

2. For a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, give its inverse.

   $A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$, provided $ad - bc \neq 0$.

3. What is the determinant of a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$?

   $\det = ad - bc$.

4. State the multiplicative property of determinants for square matrices $A$ and $B$.

   $\det(AB) = \det(A)\,\det(B)$. Also $\det(A^T) = \det(A)$ and $\det(kA) = k^n \det(A)$ for an $n \times n$ matrix.

5. What happens to a determinant when two rows (or columns) are interchanged?

   The determinant changes sign (is multiplied by $-1$). If two rows or columns are identical, the determinant is $0$.

6. State Cramer's rule for solving the system $a_1x + b_1y = c_1$, $a_2x + b_2y = c_2$.

   $x = \frac{D_x}{D}$, $y = \frac{D_y}{D}$, where $D = \begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}$, and $D_x, D_y$ replace the respective coefficient column with the constants. Requires $D \neq 0$.

7. Using the matrix method, how is the system $AX = B$ solved when $A$ is invertible?

   $X = A^{-1}B$, valid when $\det(A) \neq 0$. The solution is then unique.

8. For a system $AX = B$, classify solutions by $\det(A)$ (Cramer's rule perspective).

   If $\det(A) \neq 0$: unique solution (consistent). If $\det(A) = 0$ and $(\text{adj}\,A)B = 0$: infinitely many or none. If $\det(A)=0$ and $(\text{adj}\,A)B \neq 0$: no solution (inconsistent).

9. State the formula for the $n$-th term of an arithmetic progression (AP).

   $a_n = a + (n-1)d$, where $a$ is the first term and $d$ is the common difference.

10. State the formula for the sum of the first $n$ terms of an AP.

    $S_n = \frac{n}{2}\big[2a + (n-1)d\big] = \frac{n}{2}(a + l)$, where $l$ is the last term.

11. State the formula for the $n$-th term and sum of a geometric progression (GP).

    $n$-th term: $a_n = ar^{n-1}$. Sum: $S_n = \frac{a(r^n - 1)}{r - 1}$ for $r \neq 1$.

12. What is the sum to infinity of a GP, and when does it exist?

    $S_\infty = \frac{a}{1 - r}$, valid only when $|r| < 1$.

13. Define a harmonic progression (HP) and how its $n$-th term is found.

    An HP is a sequence whose reciprocals form an AP. Its $n$-th term is $\frac{1}{a + (n-1)d}$, where $a, d$ are the first term and common difference of the corresponding AP of reciprocals.

14. State the relationship (inequality) between the arithmetic mean (AM), geometric mean (GM) and harmonic mean (HM) of positive numbers.

    $\text{AM} \geq \text{GM} \geq \text{HM}$, with equality when all numbers are equal. Also $\text{GM}^2 = \text{AM} \times \text{HM}$ for two numbers.

15. State the general binomial theorem for $(a+b)^n$ where $n$ is a positive integer.

    $(a+b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^{r}$, where $\binom{n}{r} = \frac{n!}{r!(n-r)!}$.

16. Write the general (general $(r+1)$-th) term in the expansion of $(a+b)^n$.

    $T_{r+1} = \binom{n}{r} a^{n-r} b^{r}$.

17. How many terms are in the expansion of $(a+b)^n$ and what is the sum of all binomial coefficients?

    There are $n+1$ terms. The sum of all binomial coefficients is $\sum_{r=0}^{n}\binom{n}{r} = 2^n$.

18. How do you find the middle term(s) in the expansion of $(a+b)^n$?

    If $n$ is even, there is one middle term: $T_{\frac{n}{2}+1}$. If $n$ is odd, there are two middle terms: $T_{\frac{n+1}{2}}$ and $T_{\frac{n+3}{2}}$.

19. State the formula for the number of permutations of $n$ distinct objects taken $r$ at a time.

    $^nP_r = \frac{n!}{(n-r)!}$.

20. State the formula for the number of combinations of $n$ distinct objects taken $r$ at a time.

    $^nC_r = \frac{n!}{r!\,(n-r)!}$, and $^nP_r = r! \cdot {}^nC_r$.

21. State the symmetry and Pascal's rule properties of combinations.

    Symmetry: $^nC_r = {}^nC_{n-r}$. Pascal's rule: $^nC_r + {}^nC_{r-1} = {}^{n+1}C_r$.

22. State the classical (theoretical) definition of the probability of an event $E$.

    $P(E) = \frac{\text{number of favourable outcomes}}{\text{total number of equally likely outcomes}} = \frac{n(E)}{n(S)}$, with $0 \leq P(E) \leq 1$.

23. State the addition theorem of probability for two events $A$ and $B$.

    $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. For mutually exclusive events, $P(A \cup B) = P(A) + P(B)$.

24. State the conditional probability and multiplication theorem for two events.

    $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$ (for $P(B) > 0$). Multiplication: $P(A \cap B) = P(B)\,P(A \mid B)$. For independent events, $P(A \cap B) = P(A)P(B)$.

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