The function f (x) = |cos x| is
Question: The function $f(x)=|\cos x|$ is (a) everywhere continuous and differentiable (b) everywhere continuous but not differentiable at $(2 n+1) \pi / 2, n \in Z$ (c) neither continuous nor differentiable at $(2 n+1) \pi / 2, n \in Z$ (d) none of these Solution: (b) everywhere continuous but not differentiable at $(2 n+1) \pi / 2, n \in Z$ We have, $f(x)=|\cos x|$ $\Rightarrow f(x)=\left\{\begin{array}{rc}\cos x, 2 n \pi \leq x(4 n+1) \frac{\pi}{2} \\ 0, x=(4 n+1) \frac{\pi}{2} \\ -\cos x, (4...
Read More →Let A = {0, 1, 2, 3, 4, 5, 6, 7, 8}
Question: Let $A=\{0,1,2,3,4,5,6,7,8\}$ and let $R=\{(a, b): a, b \in A$ and $2 a+3 b=12\}$. Express R as a set of ordered pairs. Show that R is a binary relation on A. Find its domain and range. Solution: $A=\{0,1,2,3,4,5,6,7,8\}$ 2a + 3b = 12 $b=\frac{12-2 a}{3}$ $a=0$ $b=4$ $a=3$ $b=2$ $a=6$ $b=0$ $R=\{(0,4),(3,2),(6,0)\}$ Since, R is a subset of A A, it a relation to A The domain of R is the set of first co-ordinates of R Dom(R) = {0, 3, 6} The range of R is the set of second co-ordinates of...
Read More →Evaluate:
Question: Evaluate: (i) $4^{-3}$ (ii) $\left(\frac{1}{2}\right)^{-5}$ (iii) $\left(\frac{4}{3}\right)^{-3}$ (iv) $(-3)^{-4}$ (v) $\left(\frac{-2}{3}\right)^{-5}$ Solution: (i) $4^{-3}=\frac{1}{4^{3}}=\frac{1}{64}$ (ii) $\left(\frac{1}{2}\right)^{-5}=2^{5}=32$ (iii) $\left(\frac{4}{3}\right)^{-3}=\left(\frac{3}{4}\right)^{3}=\frac{3^{3}}{4^{3}}=\frac{27}{64}$ (iv) $(-3)^{-4}=\left(\frac{-1}{3}\right)^{4}=\frac{(-1)^{4}}{3^{4}}=\frac{1}{81}$ (v) $\left(\frac{-2}{3}\right)^{-5}=\left(\frac{-3}{2}\r...
Read More →Let A = {2, 3, 5} and R = {(2, 3), (2, 5), (3, 3), (3, 5)}.
Question: Let A = {2, 3, 5} and R = {(2, 3), (2, 5), (3, 3), (3, 5)}. Show that R is a binary relation on A. Find its domain and range. Solution: First, calculate AA. AA = {(2, 2), (2, 3), (2, 5), (3, 2), (3, 3), (3, 5), (5, 2), (5, 3), (5, 5)} Since, $R$ is a subset of $A \times A$, it's a binary relation on $A$. The domain of $R$ is the set of first co-ordinates of $R$ Dom(R) = {2, 3} The range of R is the set of second co-ordinates of R Range(R) = {3, 5}...
Read More →What do you mean by a binary relation on a set A?
Question: What do you mean by a binary relation on a set A? Define the domain and range of relation on A. Solution: Any subset of (A A) is called a binary relation to A. Here, (A A) is the cartesian product of A with A. Let $A=\{4,5,6)$ and $R=\{(4,5),(6,4),(5,6)\}$ Here, R is a binary relation to A. The domain of R is the set of first co-ordinates of R Dom(R) = {4, 6, 5} The range of R is the set of second co-ordinates of R Range(R) = {5, 4, 6}...
Read More →From the top of a tower h m high,
Question: From the top of a tower h m high, angles of depression of two objects, which are in line with the foot of the tower are a and ( a). Find the distance between the two objects. Solution: Let the distance between two objects is x m, and CD = y m. Given that, BAX = = ABD, [alternate angle] CAY = p = ACD [alternate angle] Now, in $\triangle A C D$, $\tan \beta=\frac{A D}{C D}=\frac{h}{y}$ $\Rightarrow$$y=\frac{h}{\tan \beta}$ .....(i) and in $\triangle A B D$, $\tan \alpha=\frac{A D}{B D} \...
Read More →The function
Question: The function $f(x)=e^{-|x|}$ is (a) continuous everywhere but not differentiable atx= 0(b) continuous and differentiable everywhere(c) not continuous atx= 0(d) none of these Solution: (a) continuous everywhere but not differentiable atx= 0 Given: $f(x)=e^{-|x|}= \begin{cases}e^{x}, x \geq 0 \\ e^{-x}, x0\end{cases}$ Continuity : $\lim _{x \rightarrow 0^{-}} f(x)$ $=\lim _{h \rightarrow 0} f(0-h)$ $=\lim _{h \rightarrow 0} e^{-(0-h)}$ $=\lim _{h \rightarrow 0} e^{h}$ $=1$ RHL atx= 0 $\l...
Read More →Let A = {3, 4} and B = {7, 9}.
Question: Let $A=\{3,4\}$ and $B=\{7,9\} .$ Let $R=\{(a, b): a \in A, b \in B$ and $(a-b)$ is odd $\} .$ Show that $R$ is an empty relation from $A$ to $B$. Solution: Given: A = {3, 4} and B = {7, 9} $\mathrm{R}=\{(\mathrm{a}, \mathrm{b}): \mathrm{a} \in \mathrm{A}, \mathrm{b} \in \mathrm{B}$ and $(\mathrm{a}-\mathrm{b})$ is odd $\}$ So, $R=\{(4,7),(4,9)\}$ An empty relation means there is no elements in the relation set. Here we get two relations which satisfy the given conditions. Therefore, t...
Read More →Write 'T' for true and 'F' for false for each of the following:
Question: Write 'T' for true and 'F' for false for each of the following: (i) Rational numbers are always closed under subtraction. (ii) Rational numbers are aways closed under division. (iii) 1 0 = 0. (iv) Subtraction is commutative on rational numbers. (v) $-\left(\frac{-7}{8}\right)=\frac{7}{8}$ Solution: (i) T If $\frac{a}{b}$ and $\frac{c}{d}$ are rational numbers, then $\frac{a}{b}-\frac{c}{d}=\frac{a d-b c}{b d}$ is also a rational number because $a d, b c$ and $b d$ are all rational numb...
Read More →The function
Question: The function $f(x)=e^{-|x|}$ is (a) continuous everywhere but not differentiable atx= 0(b) continuous and differentiable everywhere(c) not continuous atx= 0(d) none of these Solution: (a) continuous everywhere but not differentiable atx= 0 Given: $f(x)=e^{-|x|}= \begin{cases}e^{x}, x \geq 0 \\ e^{-x}, x0\end{cases}$ Continuity : $\lim _{x \rightarrow 0^{-}} f(x)$ $=\lim _{h \rightarrow 0} f(0-h)$ $=\lim _{h \rightarrow 0} e^{-(0-h)}$ $=\lim _{h \rightarrow 0} e^{h}$ $=1$ RHL atx= 0 $\l...
Read More →Fill in the blanks.
Question: Fill in the blanks. (i) $\frac{25}{8} \div(\ldots \ldots)=-10$. (ii) $\frac{-8}{9} \times(\ldots \ldots)=\frac{-2}{3}$. (iii) $(-1)+(\ldots \ldots)=\frac{-2}{9}$. (iv) $\frac{2}{3}-(\ldots \ldots)=\frac{1}{15}$. Solution: (i) Let the number be $x$. Now, we have : $\frac{25}{8} \div x=-10$ $\Rightarrow \frac{25}{8} \times \frac{1}{x}=-10$ $\Rightarrow \frac{1}{x}=-10 \div \frac{25}{8}$ $\Rightarrow \frac{1}{x}=-10 \times \frac{8}{25}$ $\Rightarrow \frac{1}{x}=\frac{-80}{25}$ $\Rightarro...
Read More →Solve this
Question: Let $A=\{2,3\}$ and $B=\{3,5\}$ (i) Find $(A \times B)$ and $n(A \times B)$. (ii) How many relations can be defined from $A$ to $B$ ? Solution: Given: A = {2, 3} and B= {3, 5} (i) $(A \times B)=\{(2,3),(2,5),(3,3),(3,5)\}$ Therefore, n(A B) = 4 (ii) No. of relation from $A$ to $B$ is a subset of Cartesian product of $(A \times B)$. Here no. of elements in A = 2 and no. of elements in B = 2. So, (A B) = 2 2 = 4 So, the total number of relations can be defined from A to B is $=2^{4}=16$...
Read More →The angle of elevation of the top of a tower 30 m
Question: The angle of elevation of the top of a tower 30 m high from the foot of another tower in the same plane is 60 and the angle of elevation of the top of the second tower from the foot of the first tower is 30. Find the distance between the two towers and also the height of the tower. Solution: Let distance between the two towers = AB = x m and height of the other tower = PA = h m Given that, height of the tower = QB = 30 m and QAB = 60, PBA = 30 Now, in $\triangle Q A B, \quad \tan 60^{\...
Read More →Solve this
Question: Let $f(x)=(x+|x|)|x|$. Then, for all $x$ (a)fis continuous(b)fis differentiable for somex(c)f' is continuous(d)f'' is continuous Solution: (a)fis continuous(c)f' is continuous We have, $f(x)=(x+|x|)|x|$ $=x|x|+(|x|)^{2}$ $=x|x|+x^{2}$ $f(x)= \begin{cases}2 x^{2} x \geq 0 \\ 0 x0\end{cases}$ To check continuity of $f(x)$ at $x=0$ $(\mathrm{LHL}$ at $x=0)=\lim _{x \rightarrow 0^{-}} f(x)$ $=\lim _{x \rightarrow 0^{+}} 2 x^{2}$ $=0$ And $f(0)=0$ Here, $\mathrm{LHL}=\mathrm{RHL}=f(0)$ Ther...
Read More →Solve this
Question: Let $R=\left\{(x, y): x, y \in Z\right.$ and $\left.x^{2}+y^{2} \leq 4\right\}$. (i) Write $\mathbf{R}$ in roster form. (ii) Find dom (R) and range (R). Solution: Given: $R=\left\{(x, y): x, y \in Z\right.$ and $\left.x^{2}+y^{2} \leq 4\right\}$ (i) R is Foster Form is $R=\{(-2,0),(-1,-1),(-1,0),(-1,1),(0,-2),(0,-1),(0,0),(0,1),(0,2),(1,-1),(1,0),(1,$, 1), $(2,0)\}$ (ii) $\operatorname{Dom}(R)=\{-2,-1,0,1,2\}$ Range $(R)=\{-2,-1,0,1,2\}$...
Read More →Solve this
Question: Let $f(x)=(x+|x|)|x|$. Then, for all $x$ (a)fis continuous(b)fis differentiable for somex(c)f' is continuous(d)f'' is continuous Solution: (a)fis continuous(c)f' is continuous We have, $f(x)=(x+|x|)|x|$ $=x|x|+(|x|)^{2}$ $=x|x|+x^{2}$ $f(x)= \begin{cases}2 x^{2} x \geq 0 \\ 0 x0\end{cases}$ To check continuity of $f(x)$ at $x=0$ $(\mathrm{LHL}$ at $x=0)=\lim _{x \rightarrow 0^{-}} f(x)$ $=\lim _{x \rightarrow 0^{+}} 2 x^{2}$ $=0$ And $f(0)=0$ Here, $\mathrm{LHL}=\mathrm{RHL}=f(0)$ Ther...
Read More →A rational number between
Question: A rational number between $\frac{-2}{3}$ and $\frac{1}{2}$ is (a) $\frac{-1}{6}$ (b) $\frac{-1}{12}$ (C) $\frac{-5}{6}$ (d) $\frac{5}{6}$ Solution: (b) $\frac{-1}{12}$ Number between $\frac{-2}{3}$ and $\frac{1}{2}=\frac{1}{2} \times\left(\frac{-2}{3}+\frac{1}{2}\right)$ $=\frac{1}{2} \times\left(\frac{-2 \times 2}{3 \times 2}+\frac{1 \times 3}{2 \times 3}\right)$ $=\frac{1}{2} \times\left(\frac{-4}{6}+\frac{3}{6}\right)$ $=\frac{1}{2} \times\left(\frac{-4+3}{6}\right)$ $=\frac{-1}{12}...
Read More →Prove the following
Question: Prove that $\frac{1+\sec \theta-\tan \theta}{1+\sec \theta+\tan \theta}=\frac{1-\sin \theta}{\cos \theta}$ Solution: LHS $=\frac{1+\sec \theta-\tan \theta}{1+\sec \theta+\tan \theta}$ $=\frac{1+1 / \cos \theta-\sin \theta / \cos \theta}{1+1 / \cos \theta+\sin \theta / \cos \theta}$ $\left[\because \sec \theta=\frac{1}{\cos \theta}\right.$ and $\left.\tan \theta=\frac{\sin \theta}{\cos \theta}\right]$ $=\frac{\cos \theta+1-\sin \theta}{\cos \theta+1+\sin \theta}=\frac{(\cos \theta+1)-\s...
Read More →Define a relation R from Z to Z, given by
Question: Define a relation R from Z to Z, given by $\mathbf{R}=\{(\mathbf{a}, \mathbf{b}): \mathbf{a}, \mathbf{b} \in \mathbf{Z}$ and $(\mathbf{a}-\mathbf{b})$ is an integer. Find dom (R) and range (R). Solution: Given: $R=\{(a, b): a, b \in Z$ and $(a-b)$ is an integer The condition satisfies for all the values of a and b to be any integer. So, $R=\{(a, b):$ for all $a, b \in(-\infty, \infty)\}$ $\operatorname{Dom}(R)=\{-\infty, \infty\}$ Range $(R)=\{-\infty, \infty\}$...
Read More →Mark (✓) against the correct answer
Question: Mark (✓) against the correct answer Reciprocal of $\frac{-7}{9}$ is (a) $\frac{9}{7}$ (b) $\frac{-9}{7}$ (c) $\frac{7}{9}$ (d) none of these Solution: (b) $\frac{-9}{7}$ Reciprocal of $\frac{-7}{9}=\left(\frac{-7}{9}\right)^{-1}$ Now, we have : $\frac{1}{\left(\frac{-7}{9}\right)}$ $=\frac{9}{-7}$ $=\frac{9 \times-1}{-7 \times-1}$ $=\frac{-9}{7}$...
Read More →Prove the following
Question: If $a \sin \theta+b \cos \theta=c$, then prove that $a \cos \theta-b \sin \theta=\sqrt{a^{2}+b^{2}-c^{2}}$. Solution: Given that, $a \sin \theta+b \cos \theta=c$ On squaring both sides, $(a \cdot \sin \theta+\cos \theta \cdot b)^{2}=c^{2}$ $\Rightarrow \quad a^{2} \sin ^{2} \theta+b^{2} \cos ^{2} \theta+2 a b \sin \theta \cdot \cos \theta=c^{2} \quad\left[\because(x+y)^{2}=x^{2}+2 x y+y^{2}\right]$ $\Rightarrow \quad a^{2}\left(1-\cos ^{2} \theta\right)+b^{2}\left(1-\sin ^{2} \theta\ri...
Read More →Let A = {1, 2, 3, 4, 6} and
Question: Let $A=\{1,2,3,4,6\}$ and $R=\{(a, b): a, b \in A$, and a divides b $\} .$ (i) Write $\mathbf{R}$ in roster form. (ii) Find dom (R) and range (R). Solution: Given: A = {1, 2, 3, 4, 6} (i) $R=\{(a, b): a, b \in A$, and a divides $b\}$ R is Foster Form is, $R=\{(1,2),(1,3),(1,4),(1,6),(2,2),(2,4),(2,6),(3,3),(3,6),(4,4),(6,6)\}$ (ii) Dom(R) $=\{1,2,3,4,6\}$ Range(R) = {2, 3, 4, 6}...
Read More →Mark (✓) against the correct answer
Question: Mark (✓) against the correct answer $\frac{4}{3} \div ?=\frac{-5}{2}$ (a) $\frac{-8}{5}$ (b) $\frac{8}{5}$ (C) $\frac{-8}{15}$ (d) $\frac{8}{15}$ Solution: (c) $\frac{-8}{15}$ We have: $\frac{4}{3} \div x=\frac{-5}{2}$ $\Rightarrow \frac{4}{3} \times \frac{1}{x}=\frac{-5}{2}$ $\Rightarrow \frac{1}{x}=\frac{\left(\frac{-5}{2}\right)}{\left(\frac{4}{3}\right)}$ $\Rightarrow \frac{1}{x}=\left(\frac{-5}{2}\right) \times\left(\frac{3}{4}\right)$ $\Rightarrow \frac{1}{x}=\frac{-15}{8}$ $\Rig...
Read More →If sinθ + cosθ = p and sec θ + cosec θ = q,
Question: If sin + cos = p and sec + cosec = q, then prove that q(p2-1) = 2p. Solution: Given that, $\quad \sin \theta+\cos \theta=p$ $\ldots(i)$ and $\quad \sec \theta+\operatorname{cosec} \theta=q$ $\Rightarrow$ $\frac{1}{\cos \theta}+\frac{1}{\sin \theta}=q$ $\left[\because \sec \theta=\frac{1}{\cos \theta}\right.$ and $\left.\operatorname{cosec} \theta=\frac{1}{\sin \theta}\right]$ $\Rightarrow$ $\frac{\sin \theta+\cos \theta}{\sin \theta \cdot \cos \theta}=q$ $\Rightarrow$ $\frac{p}{\sin \t...
Read More →Mark (✓) against the correct answer
Question: Mark (✓) against the correct answer $\left(\frac{-5}{6} \div \frac{-2}{3}\right)=?$ (a) $\frac{-5}{4}$ (b) $\frac{5}{4}$ (c) $\frac{-4}{5}$ (d) $\frac{4}{5}$ Solution: (b) $\frac{5}{4}$ We have: $\left(\frac{-5}{6} \div \frac{-2}{3}\right)$ $=\frac{-5}{6} \times \frac{3}{-2}$ $=\frac{15}{12}$ $=\frac{5}{4}$...
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