Question
✓
Answer
(a) Yes
Reason: Every element of set A has been assigned a unique element in set B.
Reason: Every element of set A has been assigned a unique element in set B.
(b) No
Reason: An element of set A has been assigned more than one element from set B.
(c) No
Reason: Not every element of set A has been assigned an image from set B.
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If $A=\{1,2,3\}, B=\{4,5,6\}$, which of the following are relations from $A$ to $B$.
(i) $R_1=\{(1,4),(1,5),(1,6)\}$
(ii) $R _2=\{(1,5),(2,4),(3,6)\}$
(iii) $R_3=\{(1,4),(1,5),(3,6),(2,6),(3,4)\}$
(iv) $R_4=\{(4,2),(2,6),(5,1),(2,4)\}$
→(i) $R_1=\{(1,4),(1,5),(1,6)\}$
(ii) $R _2=\{(1,5),(2,4),(3,6)\}$
(iii) $R_3=\{(1,4),(1,5),(3,6),(2,6),(3,4)\}$
(iv) $R_4=\{(4,2),(2,6),(5,1),(2,4)\}$
If $\mathrm{f}(\mathrm{x})=\left\{\begin{array}{cl}4 x-2, & x \leq-3 \\ 5, & -3<x<3, \text { then fmd } \\ x^2, & x \geq 3\end{array}\right.$
(i) $\mathrm{f}(-4)$
(ii) $f(-3)$
(iii) $f(1)$
(iv) $f(5)$
→(i) $\mathrm{f}(-4)$
(ii) $f(-3)$
(iii) $f(1)$
(iv) $f(5)$
Find the value(s) of $x$, if
i) $\left|\begin{array}{ll}2 & 3 \\ 4 & 5\end{array}\right|=\left|\begin{array}{cc}x & 3 \\ 2 x & 5\end{array}\right|$
ii) $\left|\begin{array}{ccc}2 & 1 & x+1 \\ -1 & 3 & -4 \\ 0 & -5 & 3\end{array}\right|=0$
iii) $\left|\begin{array}{ccc}x-1 & x & x-2 \\ 0 & x-2 & x-3 \\ 0 & 0 & x-3\end{array}\right|=0$
→i) $\left|\begin{array}{ll}2 & 3 \\ 4 & 5\end{array}\right|=\left|\begin{array}{cc}x & 3 \\ 2 x & 5\end{array}\right|$
ii) $\left|\begin{array}{ccc}2 & 1 & x+1 \\ -1 & 3 & -4 \\ 0 & -5 & 3\end{array}\right|=0$
iii) $\left|\begin{array}{ccc}x-1 & x & x-2 \\ 0 & x-2 & x-3 \\ 0 & 0 & x-3\end{array}\right|=0$
D(-1, 8), E(4, -2), F(-5, -3) are midpoints of sides BC, CA and AB of ∆ABC. Find
(i) equations of sides of ∆ABC.
(ii) co-ordinates of the circumcentre of ∆ABC.
(i) equations of sides of ∆ABC.
(ii) co-ordinates of the circumcentre of ∆ABC.
If $A=\left\{x / 6 x^2+x-15=0\right\}, B=\left\{x / 2 x^2-5 x-3=0\right\}, C=\left\{x / 2 x^2-x-3=0\right\}$, then find (i) $(A \cup B \cup C)($ ii $)(A \cap B \cap C)$
→Write the domain and range of the following relations.
(i) {(a, b) / a ∈ N, a < 6 and b = 4}
(ii) {(a, b) / a, b ∈ N, a + b = 12}
(iii) {(2, 4), (2, 5), (2, 6), (2, 7)}
(i) {(a, b) / a ∈ N, a < 6 and b = 4}
(ii) {(a, b) / a, b ∈ N, a + b = 12}
(iii) {(2, 4), (2, 5), (2, 6), (2, 7)}
Without expanding the determinants, show that:
i) $\left|\begin{array}{lll}b+c & b c & b^2 c^2 \\ c+a & c a & c^2 a^2 \\ a+b & a b & a^2 b^2\end{array}\right|=0$
ii)$\left|\begin{array}{ccc}x a & y b & z c \\ a^2 & b^2 & c^2 \\ 1 & 1 & 1\end{array}\right|$ =$\left|\begin{array}{ccc}x & y & z \\ a & b & c \\ b c & c a & a b\end{array}\right|$
iii) $\left|\begin{array}{lll}l & m & n \\ e & d & f \\ u & v & w\end{array}\right|=\left|\begin{array}{lll}n & f & w \\ l & e & u \\ m & d & v\end{array}\right|$
iv) $\left|\begin{array}{ccc}0 & a & b \\ -a & 0 & c \\ -b & -c & 0\end{array}\right|=0$
→i) $\left|\begin{array}{lll}b+c & b c & b^2 c^2 \\ c+a & c a & c^2 a^2 \\ a+b & a b & a^2 b^2\end{array}\right|=0$
ii)$\left|\begin{array}{ccc}x a & y b & z c \\ a^2 & b^2 & c^2 \\ 1 & 1 & 1\end{array}\right|$ =$\left|\begin{array}{ccc}x & y & z \\ a & b & c \\ b c & c a & a b\end{array}\right|$
iii) $\left|\begin{array}{lll}l & m & n \\ e & d & f \\ u & v & w\end{array}\right|=\left|\begin{array}{lll}n & f & w \\ l & e & u \\ m & d & v\end{array}\right|$
iv) $\left|\begin{array}{ccc}0 & a & b \\ -a & 0 & c \\ -b & -c & 0\end{array}\right|=0$
If U = {x / x ∈ N, 1 ≤ x ≤ 12}, A = {1, 4, 7, 10}, B = {2, 4, 6, 7, 11}, C = {3, 5, 8, 9, 12}.
Write the sets
(i) A ∪ B
(ii) B ∩ C
(iii) A – B
(iv) B – C
(v) A ∪ B ∪ C
(vi) A ∩ (B ∪ C)
Write the sets
(i) A ∪ B
(ii) B ∩ C
(iii) A – B
(iv) B – C
(v) A ∪ B ∪ C
(vi) A ∩ (B ∪ C)
Which sets of ordered pairs represent functions from A = {1, 2, 3, 4} to B = {-1, 0, 1, 2, 3}? Justify.
(i) {(1, 0), (3, 3), (2, -1), (4, 1), (2, 2)}
(ii) {(1, 2), (2, -1), (3, 1), (4, 3)}
(iii) {(1, 3), (4, 1), (2, 2)}
(iv) {(1, 1), (2, 1), (3, 1), (4, 1)}
(i) {(1, 0), (3, 3), (2, -1), (4, 1), (2, 2)}
(ii) {(1, 2), (2, -1), (3, 1), (4, 3)}
(iii) {(1, 3), (4, 1), (2, 2)}
(iv) {(1, 1), (2, 1), (3, 1), (4, 1)}
Find the slope, x-intercept, y-intercept of each of the following lines.
(a) 2x + 3y – 6 = 0
(b) x + 2y = 0
(a) 2x + 3y – 6 = 0
(b) x + 2y = 0