If A = {5, 7), find (i) A × A × A.
Question: If A = {5, 7), find (i) A A A. Solution: We have, A = {5, 7} So, By the definition of the Cartesian product, Given two non empty sets P and Q. The Cartesian product P Q is the set of all ordered pairs of elements from P and Q, .i.e. $P \times Q=\{(p, q): p \in P, q \in Q\}$ Here, $A=\{5,7\}$ and $A=\{5,7\} .$ So, $A \times A=\{(5,5),(5,7),(7,5),(7,7)\}$ Now again, we apply the definition of Cartesian product to find $A \times A \times A$ Here, $A=\{5,7\}$ and $A \times A=\{(5,5),(5,7),...
Read More →Let A = {–2, 2} and B = (0, 3, 5). Find:
Question: Let $A=\{-2,2\}$ and $B=(0,3,5)$. Find: (i) $A \times B$ (ii) $\mathbf{B} \times \mathbf{A}$ (iii) $\mathbf{A} \times \mathbf{A}$ (iv) $\mathbf{B} \times \mathbf{B}$ Solution: (i) Given: $A=\{-2,2\}$ and $B=\{0,3,5\}$ To find: A B By the definition of the Cartesian product, Given two non empty sets P and Q. The Cartesian product P Q is the set of all ordered pairs of elements from P and Q, .i.e. $P \times Q=\{(p, q): p \in P, q \in Q\}$ Here, $A=\{-2,2\}$ and $B=\{0,3,5\}$. So, $A \tim...
Read More →Let A and B be two sets such that n(A) = 3 and n(B) = 2.
Question: Let A and B be two sets such that n(A) = 3 and n(B) = 2. If $a \neq b \neq c$ and $(a, 0),(b, 1),(c, 0)$ is in $A \times B$, find $A$ and $B$. Solution: Since, (a, 0), (b, 1), (c, 0) are the elements of A B. $\therefore a, b, c \in A$ and $0,1 \in B$ It is given that $n(A)=3$ and $n(B)=2$ $\therefore a, b, c \in A$ and $n(A)=3$ $\Rightarrow A=\{a, b, c\}$ and 0, 1 Є B and n(B) = 2 ⇒ B = {0, 1}...
Read More →Find the value of k for which the function
Question: Find the value of $k$ for which the function $f(x)=\left\{\begin{array}{cl}\frac{x^{2}+3 x-10}{x-2}, x \neq 2 \\ k, x=2\end{array}\right.$ is continuous at $x=2$ Solution: Given, $f(x)=\left\{\begin{array}{cc}\frac{x^{2}+3 x-10}{x-2}, x \neq 2 \\ k, x=2\end{array}\right.$ $\lim _{x \rightarrow 2^{-}}\left(\frac{x^{2}+3 x-10}{x-2}\right)=\lim _{x \rightarrow 2^{-}}(x+5)=7$ $f(2)=k$ $\lim _{x \rightarrow 2^{+}}\left(\frac{x^{2}+3 x-10}{x-2}\right)=\lim _{x \rightarrow 2^{+}}(x+5)=7$ If $...
Read More →Let A × B = {(a, b): b = 3a – 2}.
Question: Let $A \times B=\{(a, b): b=3 a-2\}$. if $(x,-5)$ and $(2, y)$ belong to $A \times B$, find the values of x and y. Solution: Given: $A \times B=\{(a, b): b=3 a-2\}$ and $\{(x,-5),(2, y)\} \in A \times B$ For $(x,-5) \in A \times B$ $b=3 a-2$ $\Rightarrow-5=3(x)-2$ $\Rightarrow-5+2=3 x$ $\Rightarrow-3=3 x$ $\Rightarrow x=-1$ For $(2, y) \in A \times B$ $b=3 a-2$ $\Rightarrow y=3(2)-2$ $\Rightarrow y=6-2$ $\Rightarrow y=4$ Hence, the value of x = -1 and y = 4...
Read More →Determine the value of the constant ' k ' so that function
Question: Determine the value of the constant ' $k$ ' so that function $f(x)=\left\{\begin{array}{cl}\frac{k x}{|x|}, \text { if } x0 \\ 3, \text { if } x \geq 0\end{array}\right.$ is continuous at $x=0 .$ Solution: Given, $f(x)= \begin{cases}\frac{k x}{|x|} , \text { if } x0 \\ 3 , \text { if } x \geqslant 0\end{cases}$ Since the function is continuous at $x=0$, therefore, $\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{+}} f(x)=f(0)$ $\Rightarrow \lim _{x \rightarrow 0} \frac{-k x}{x...
Read More →The value of
Question: The value of $\left(\tan 1^{\circ} \tan 2^{\circ} \tan 3^{\circ} \ldots \tan 89^{\circ}\right)$ is (a) 0 (b) 1 (c) 2 (d) $\frac{1}{2}$ Solution: (b)tan1-tan2-tan3 tan 89 = tan1-tan2-tan3 tan44 . tan 45 . tan 46 tan 87-tan 88tan 89 = tan 1- tan2 - tan 3 tan 44 . (1)- tan (90 44) tan (90 3) tan (90 -2)- tan (90 -1) ( tan 45 = 1) = tan1-tan2-tan3. tan44 (1) . cot 44. cot3-cot2-cot1 $\left[\because \tan \left(90^{\circ}-\theta\right)=\cot \theta\right]$ $=\tan 1^{\circ} \cdot \tan 2^{\circ...
Read More →Write the value of b for which
Question: Write the value of $b$ for which $f(x)=\left\{\begin{array}{rl}5 x-4 0x \leq 1 \\ 4 x^{2}+3 b x 1x2\end{array}\right.$ is continuous at $x=1$ Solution: Given: $f(x)=\left\{\begin{array}{l}5 x-4,0x \leq 1 \\ 4 x^{2}+3 b x, 1x2\end{array}\right.$ If $f(x)$ is continuous at $x=1$, then $\lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1^{+}} f(x)=f(1)$ ....(1) Now, $\lim _{x \rightarrow 1^{-}} f(x)=\lim _{h \rightarrow 0} f(1-h)=\lim _{h \rightarrow 0} 5(1-h)-4=5-4=1$ $\lim _{x \righ...
Read More →Write the value of b for which
Question: Write the value of $b$ for which $f(x)=\left\{\begin{array}{rl}5 x-4 0x \leq 1 \\ 4 x^{2}+3 b x 1x2\end{array}\right.$ is continuous at $x=1$ Solution: Given: $f(x)=\left\{\begin{array}{l}5 x-4,0x \leq 1 \\ 4 x^{2}+3 b x, 1x2\end{array}\right.$ If $f(x)$ is continuous at $x=1$, then $\lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1^{+}} f(x)=f(1)$ ....(1) Now, $\lim _{x \rightarrow 1^{-}} f(x)=\lim _{h \rightarrow 0} f(1-h)=\lim _{h \rightarrow 0} 5(1-h)-4=5-4=1$ $\lim _{x \righ...
Read More →Prove the following
Question: If $\cos (\alpha+\beta)=0$, then $\sin (\alpha-\beta)$ can be reduced to (a) $\cos \beta$ (b) $\cos 2 \beta$ (c) $\sin \alpha$ (d) $\sin 2 a$ Solution: (b) Given, $\cos (\alpha+\beta)=0=\cos 90^{\circ}$ $\left[\because \cos 90^{\circ}=0\right]$ $\Rightarrow \quad \alpha+\beta=90^{\circ}$ $\Rightarrow \quad \alpha=90^{\circ}-\beta$ ...(i) Now, $\quad \sin (\alpha-\beta)=\sin \left(90^{\circ}-\beta-\beta\right) \quad$ [put the value from Eq. (i)] $=\sin \left(90^{\circ}-2 \beta\right)$ $...
Read More →Let A = {2, 3} and B = {4, 5}.
Question: Let $A=\{2,3\}$ and $B=\{4,5\} .$ Find $(A \times B) .$ How many subsets will $(A \times B)$ have? Solution: Given: A = {2, 3} and B = {4, 5} To find: A B By the definition of the Cartesian product, Given two non - empty sets $P$ and $Q$. The Cartesian product $P \times Q$ is the set of all ordered pairs of elements from $P$ and $Q$, . i.e. $P \times Q=\{(p, q): p \in P, q \in Q\}$ Here, $A=\{2,3\}$ and $B=\{4,5\}$. So, $A \times B=(2,3) \times(4,5)$ $=\{(2,4),(2,5),(3,4),(3,5)\}$ $\th...
Read More →Are rational numbers always closed under division?
Question: (i) Are rational numbers always closed under division? (ii) Are rational numbers always commutative under division? (iii) Are rational numbers always associative under division? (iv) Can we divide 1 by 0? Solution: (i) No, rational numbers are not closed under division in general. $\frac{a}{0}=\infty$; it is not a rational number. (ii) No $\frac{a}{b} \div \frac{c}{d}=\frac{a}{b} \times \frac{d}{c}=\frac{a d}{b c}$ Also, $\frac{c}{d} \div \frac{a}{b}=\frac{c}{d} \times \frac{b}{a}=\fr...
Read More →Solve this
Question: If $f(x)=\left\{\begin{array}{cl}\frac{\sin ^{-1} x}{x}, x \neq 0 \\ k , x=0\end{array}\right.$ is continuous at $x=0$, write the value of $k$. Solution: Given, $f(x)=\left\{\begin{array}{l}\frac{\sin ^{-1} x}{x}, x \neq 0 \\ k, x=0\end{array}\right.$ If $f(x)$ is continuous at $x=0$, then $\lim _{x \rightarrow 0} f(x)=f(0)$ $\Rightarrow \lim _{x \rightarrow 0}\left(\frac{\sin ^{-1} x}{x}\right)=f(0)$ $\Rightarrow \lim _{x \rightarrow 0}\left(\frac{\sin ^{-1} x}{x}\right)=k$ $\Rightarr...
Read More →Prove the following
Question: If $\sin \theta=\frac{3}{5}$, then $\cos \theta$ is equal to (a) $\frac{b}{\sqrt{b^{2}-a^{2}}}$ (b) $\frac{b}{a}$ (c) $\frac{\sqrt{b^{2}-a^{2}}}{b}$ (d) $\frac{a}{\sqrt{b^{2}-a^{2}}}$ Solution: (c) Given, $\sin \theta=\frac{a}{b}$ $\left[\because \sin ^{2} \theta+\cos ^{2} \theta=1 \Rightarrow \cos \theta=\sqrt{1-\sin ^{2} \theta}\right]$ $\therefore \quad \cos \theta=\sqrt{1-\sin ^{2} \theta}$ $=\sqrt{1-\left(\frac{a}{b}\right)^{2}}=\sqrt{1-\frac{a^{2}}{b^{2}}}=\frac{\sqrt{b^{2}-a^{2}...
Read More →Solve this
Question: If $f(x)=\left\{\begin{aligned} \frac{1-\cos x}{x^{2}}, x \neq 0 \\ k , x=0 \end{aligned}\right.$ is continuous at $x=0$, find $k$. Solution: Given: $f(x)=\left\{\begin{array}{l}\frac{1-\cos x}{x^{2}}, x \neq 0 \\ k, x=0\end{array}\right.$ If $f(x)$ is continuous at $x=0$, then $\lim _{x \rightarrow 0} f(x)=f(0)$ $\Rightarrow \lim _{x \rightarrow 0}\left(\frac{1-\cos x}{x^{2}}\right)=k$ $\Rightarrow \lim _{x \rightarrow 0}\left(\frac{2\left[\sin \left(\frac{x}{2}\right)\right]^{2}}{4\l...
Read More →The value of the expression
Question: The value of the expression $\operatorname{cosec}\left(75^{\circ}+0\right)-\sec \left(15^{\circ}-0\right)-\tan \left(55^{\circ}+0\right)+\cot \left(35^{\circ}-0\right)$ is (a) $-1$ (b) 0 (c) 1 (d) $\frac{3}{2}$ Solution: (b)Given, expression = cosec (75 + 0) sec (15 0) tan (55 + 0) + cot (35 0) =cosec [90 (15 0)] sec (15 0)- tan (55 + 0) + cot (90 (55 + 0)} = sec (15 0) sec (15 0) tan (55 + 0) + tan (55 + 0) [ cosec (90 0) = sec 0 and cot (90 0) = tan 0] = 0Hence, the required value of...
Read More →Fill in the blanks:
Question: Fill in the blanks: (i) $\frac{9}{8}+(\ldots)=\frac{-3}{2}$ (ii) $(\ldots) \div\left(\frac{-7}{5}\right)=\frac{10}{19}$ (iii) $(\ldots) \div(-3)=\frac{-4}{15}$ (iv) $(-12) \div(\ldots)=\frac{-6}{5}$ Solution: (i) Let $\frac{9}{8} \div \mathrm{x}=\frac{-3}{2}$ $\Rightarrow \frac{9}{8} \times \frac{1}{\mathrm{x}}=\frac{-3}{2}$ $\Rightarrow \frac{1}{\mathrm{x}}=\frac{-3}{2} \div \frac{9}{8}$ $\Rightarrow \frac{1}{\mathrm{x}}=\frac{-3}{2} \times \frac{8}{9}$ $\Rightarrow \frac{1}{\mathrm{x...
Read More →Determine whether
Question: Determine whether $f(x)=\left\{\begin{array}{cl}\frac{\sin x^{2}}{x}, x \neq 0 \\ 0 , x=0\end{array}\right.$ is continuous at $x=0$ or not. Solution: Given: $f(x)=\left\{\begin{array}{l}\frac{\sin x^{2}}{x}, x \neq 0 \\ 0, x=0\end{array}\right.$ We have $\lim _{x \rightarrow 0} f(x)=\lim _{x \rightarrow 0} \frac{\sin x^{2}}{x}$ $=\lim _{x \rightarrow 0} \frac{x \sin x^{2}}{x^{2}}$ $=\lim _{x \rightarrow 0} \frac{\sin x^{2}}{x^{2}} \lim _{x \rightarrow 0} x$ $=1 \times 0$ $=0$ $=f(0)$ $...
Read More →If A × B = {(–2, 3), (–2, 4), (0, 4), (3, 3), (3, 4), find A and B.
Question: If $A \times B=\{(-2,3),(-2,4),(0,4),(3,3),(3,4)$, find $A$ and $B$. Solution: Here, A B = {(2, 3), (2, 4), (0, 4), (3, 3), (3, 4)} To find: A and B Clearly, A is the set of all first entries in ordered pairs in A B and B is the set of all second entries in ordered pairs in A B...
Read More →Prove the following
Question: If $\sin A=\frac{1}{2}$ then the value of $\cot A$ is (a) $\sqrt{3}$ (b) $\frac{1}{\sqrt{3}}$ (c) $\frac{\sqrt{3}}{1}$ (d) 1 Solution: (a) Given, $\sin A=\frac{1}{2}$ $\therefore \quad \cos A=\sqrt{1-\sin ^{2} A}=\sqrt{1-\left(\frac{1}{2}\right)^{2}}$ $=\sqrt{1-\frac{1}{4}}=\sqrt{\frac{3}{4}}=\frac{\sqrt{3}}{2}$ $\left[\because \sin ^{2} A+\cos ^{2}=1 \Rightarrow \cos A=\sqrt{1-\sin ^{2} A}\right]$ Now, $\cot A=\frac{\cos A}{\sin A}=\frac{\frac{\sqrt{3}}{2}}{1}=\sqrt{3}$...
Read More →Solve this
Question: If $f(x)=\left\{\begin{array}{cc}\frac{x^{2}-16}{x-4}, \text { if } x \neq 4 \\ k, \text { if } x=4\end{array}\right.$ is continuous at $x=4$, find $k$. Solution: Given: $f(x)=\left\{\begin{array}{l}\frac{x^{2}-16}{x-4}, \text { if } x \neq 4 \\ k, \text { if } x=4\end{array}\right.$ If $f(x)$ is continuous at $x=4$, then $\lim _{x \rightarrow 4} f(x)=f(4)$ $\Rightarrow \lim _{x \rightarrow 4}\left(\frac{x^{2}-16}{x-4}\right)=k$ $\Rightarrow \lim _{x \rightarrow 4} \frac{(x+4)(x-4)}{(x...
Read More →Divide the sum of
Question: Divide the sum of $\frac{65}{12}$ and $\frac{8}{3}$ by their difference. Solution: $\left(\frac{65}{12}+\frac{8}{3}\right) \div\left(\frac{65}{12}-\frac{8}{3}\right)$ $=\left(\frac{65}{12}+\frac{32}{12}\right) \div\left(\frac{65}{12}-\frac{32}{12}\right)$ $=\left(\frac{97}{12}\right) \div\left(\frac{33}{12}\right)$ $=\frac{97}{12} \times \frac{12}{33}$ $=\frac{97}{33}$...
Read More →Let A = {x ϵ W : x < 2}, B = {x ϵ N : 1 < x ≤ 4} and C = {3, 5}
Question: Let $A=\{x \in W: x2\}, B=\{x \in N: 1x \leq 4\}$ and $C=\{3,5\} .$ Verify that: (i) $A \times(B \cup C)=(A \times B) \cup(A \times C)$ (ii) $A \times(B \cap C)=(A \times B) \cap(A \times C)$ Solution: (i) Given: $A=\{x \in W: x2\}$ Here, W denotes the set of whole numbers (non negative integers). $\therefore A=\{0,1\}$ $[\because$ It is given that $x2$ and the whole numbers which are less than 2 are $0 \ 1]$ $B=\{x \in N: 1x \leq 4\}$ Here, N denotes the set of natural numbers. $\ther...
Read More →Prove the following
Question: If $\cos A=\frac{4}{5}$, then the value of $\tan A$ is (a) $\frac{3}{5}$ (b) $\frac{3}{4}$ (C) $\frac{4}{3}$ (d) $\frac{5}{3}$ Solution: (b) Given, $\cos A=\frac{4}{5}$ $\therefore \quad \sin A=\sqrt{1-\cos ^{2} A}$$\left[\begin{array}{l}\because \sin ^{2} A+\cos ^{2} A=1 \\ \therefore \sin A=\sqrt{1-\cos ^{2} A}\end{array}\right]$ $=\sqrt{1-\left(\frac{4}{5}\right)^{2}}=\sqrt{1-\frac{16}{25}}=\sqrt{\frac{9}{25}}=\frac{3}{5}$ Now, $\tan A=\frac{\sin A}{\cos A}=\frac{\frac{3}{5}}{\frac{...
Read More →Divide the sum of
Question: Divide the sum of $\frac{13}{5}$ and $\frac{-12}{7}$ by the product of $\frac{-31}{7}$ and $\frac{1}{-2}$. Solution: $\left(\frac{13}{5}+\frac{-12}{7}\right) \div\left(\frac{-31}{7} \times \frac{1}{-2}\right)$ $=\left(\frac{91-60}{35}\right) \div\left(\frac{-31}{-14}\right)$ $=\left(\frac{31}{35}\right) \div\left(\frac{31}{14}\right)$ $=\left(\frac{31}{35}\right) \times\left(\frac{14}{31}\right)$ $=\frac{14}{35}$ $=\frac{14 \div 7}{35 \div 7}$ $=\frac{2}{5}$...
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