JEE Main Paper 2 (B.Arch) 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. What is a symmetric matrix and a skew-symmetric matrix?

   Symmetric: $A^{T} = A$ (i.e. $a_{ij} = a_{ji}$). Skew-symmetric: $A^{T} = -A$ (i.e. $a_{ij} = -a_{ji}$), which forces all diagonal entries to be $0$.

2. When is the product $AB$ of two matrices defined, and what is the order of the result?

   $AB$ is defined only when the number of columns of $A$ equals the number of rows of $B$. If $A$ is $m \times n$ and $B$ is $n \times p$, then $AB$ has order $m \times p$.

3. Give the formula for the determinant of a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$.

   $\det = ad - bc$.

4. State three key properties of determinants.

   (1) Interchanging two rows (or columns) changes the sign of the determinant. (2) If two rows/columns are identical, the determinant is $0$. (3) Multiplying a row/column by $k$ multiplies the determinant by $k$.

5. What is the formula for the area of a triangle with vertices $(x_1,y_1), (x_2,y_2), (x_3,y_3)$ using a determinant?

   $\text{Area} = \frac{1}{2} \left| \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \right|$. The points are collinear when this determinant equals $0$.

6. Define the adjoint (adjugate) of a square matrix $A$.

   The adjoint $\text{adj}(A)$ is the transpose of the cofactor matrix of $A$. It satisfies $A \cdot \text{adj}(A) = \text{adj}(A) \cdot A = |A| I$.

7. State the formula for the inverse of a square matrix $A$.

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

8. What is the condition for a square matrix to be invertible (non-singular)?

   A square matrix $A$ is invertible if and only if its determinant is non-zero, $|A| \neq 0$. If $|A| = 0$, the matrix is singular and has no inverse.

9. Using matrices, how do you solve the linear system $AX = B$ when $A$ is invertible?

   Multiply both sides by $A^{-1}$ to get the unique solution $X = A^{-1}B$, valid when $|A| \neq 0$.

10. In Cramer's rule, what determines whether a system has a unique solution, no solution, or infinitely many?

    Let $D = |A|$. If $D \neq 0$: unique solution. If $D = 0$ and all $D_i = 0$: infinitely many solutions (consistent). If $D = 0$ and at least one $D_i \neq 0$: no solution (inconsistent).

11. State the fundamental principle of counting (multiplication rule).

    If one event can occur in $m$ ways and a second independent event in $n$ ways, then both events together can occur in $m \times n$ ways. The addition rule applies to mutually exclusive choices: $m + n$ ways.

12. What is the formula for permutations ${}^{n}P_{r}$ and what does it count?

    ${}^{n}P_{r} = \frac{n!}{(n-r)!}$. It counts the number of arrangements (where order matters) of $r$ objects chosen from $n$ distinct objects.

13. What is the formula for combinations ${}^{n}C_{r}$ and what does it count?

    ${}^{n}C_{r} = \frac{n!}{r!(n-r)!}$. It counts the number of selections (where order does not matter) of $r$ objects from $n$ distinct objects.

14. State the relationship between ${}^{n}P_{r}$ and ${}^{n}C_{r}$.

    ${}^{n}P_{r} = {}^{n}C_{r} \times r!$, because each combination of $r$ objects can be arranged in $r!$ ways.

15. State two important identities for combinations.

    (1) ${}^{n}C_{r} = {}^{n}C_{n-r}$ (symmetry). (2) ${}^{n}C_{r} + {}^{n}C_{r-1} = {}^{n+1}C_{r}$ (Pascal's rule).

16. State the binomial theorem for a positive integral index $n$.

    $(a+b)^{n} = \sum_{r=0}^{n} {}^{n}C_{r}\, a^{n-r} b^{r} = {}^{n}C_0 a^n + {}^{n}C_1 a^{n-1}b + \cdots + {}^{n}C_n b^n$. The expansion has $n+1$ terms.

17. What is the general term in the binomial expansion of $(a+b)^{n}$?

    $T_{r+1} = {}^{n}C_{r}\, a^{n-r} b^{r}$, the $(r+1)$th term, with $r = 0, 1, 2, \ldots, n$.

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

    If $n$ is even: one middle term, the $\left(\frac{n}{2}+1\right)$th term. If $n$ is odd: two middle terms, the $\left(\frac{n+1}{2}\right)$th and $\left(\frac{n+3}{2}\right)$th terms.

19. What is the sum of the binomial coefficients in the expansion of $(1+x)^{n}$?

    Setting $x = 1$ gives $\sum_{r=0}^{n} {}^{n}C_{r} = 2^{n}$. Setting $x = -1$ gives $\sum_{r=0}^{n} (-1)^r {}^{n}C_{r} = 0$.

20. Give 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.

21. Give the formula for the sum of the first $n$ terms of an AP.

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

22. Give the formula for the $n$th term and the sum of $n$ terms of a geometric progression (GP).

    $n$th term: $a_n = a r^{n-1}$. Sum: $S_n = a\dfrac{r^{n}-1}{r-1}$ for $r \neq 1$, where $a$ is the first term and $r$ the common ratio.

23. What is the sum of an infinite geometric progression and its condition for convergence?

    $S_\infty = \frac{a}{1-r}$, valid only when $|r| < 1$. If $|r| \geq 1$, the infinite GP does not converge to a finite sum.

24. What are the arithmetic mean (AM) and geometric mean (GM) of two positive numbers $a$ and $b$, and how are they related?

    $\text{AM} = \frac{a+b}{2}$, $\text{GM} = \sqrt{ab}$. They satisfy the inequality $\text{AM} \geq \text{GM}$, with equality if and only if $a = b$.

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