The function f(x) = [x] is continuous at
Question: The functionf(x) = [x] is continuous at (a) 4 (b) $-2$ (c) 1 (d) $1.5$ Solution: The graph off(x) = [x] is shown below. It can be seen that, the functionf(x) = [x] is discontinuous at all integral values ofx. It is continuous at all points except the integer points.Thus, the functionf(x) = [x] is continuous atx= 1.5 and discontinuous atx= 4,x= 2 andx= 1.Hence, the correct answer is option (d)....
Read More →Let A = {x : x = 6n N) and B = {x : x = 9n, n ϵ N}, find A ∩ B.
Question: Let $A=\{x: x=6 n \in N$ ) and $B=\{x: x=9 n, n \in N\}$, find $A \cap B$. Solution: $A=\{x: x=6 n \forall n \in N)$ As $x=6 n$ hence for $n=1,2,3,4,5,6 \ldots x=6,12,18,24,30,36 \ldots$ Hence $A=\{6,12,18,24,30,36 \ldots\}$ $B=\{x: x=9 n \forall n \in N)$ As $x=9 n$ hence for $n=1,2,3,4 \ldots x=9,18,27,36 \ldots$ Hence $B=\{9,18,27,36 \ldots\}$ $A \cap B$ means common elements to both sets The common elements are $18,36,54, \ldots$ Hence $A \cap B=\{18,36,54, \ldots\}$ All the elemen...
Read More →What number should be added to
Question: What number should be added to $\frac{-5}{8}$ so as to get $\frac{-3}{2} ?$ Solution: Let the required number bex. Now, $\frac{-5}{8}+x=\frac{-3}{2}$ $\Rightarrow \frac{-5}{8}+x+\frac{5}{8}=\frac{-3}{2}+\frac{5}{8} \quad$ (Adding $\frac{5}{8}$ to both the sides) $\Rightarrow x=\left(\frac{-3}{2}+\frac{5}{8}\right)$ $\Rightarrow x=\left(\frac{-12}{8}+\frac{5}{8}\right)$ $\Rightarrow x=\left(\frac{-12+5}{8}\right)$ $\Rightarrow x=\frac{-7}{8}$ Hence, the required number is $\frac{-7}{8}$...
Read More →The number of points at which the function
Question: The number of points at which the function $f(x)=\frac{1}{x-[x]}$ is not continuous is (a) 1(b) 2(c) 3(d) none of these Solution: The function $f(x)=\frac{1}{x-[x]}$ is discontinuous when $x-[x]=0$. $x-[x]=0$ $\Rightarrow x=[x]$ $\Rightarrow x$ is an integer So, $f(x)$ is not continuous for all $x \in Z$. Thus, the function $f(x)=\frac{1}{x-[x]}$ is not continuous at infinitely many points. Hence, the correct answer is option (d)....
Read More →If (a, b) is the mid-point of the line segment joining
Question: If (a, b) is the mid-point of the line segment joining the points A(10, 6), B(k, 4) and a 2b= 18, then find the value of k and the distance AB. Solution: Since, (a, b) is the mid-point of line segment AB. $\therefore$$(a, b)=\left(\frac{10+k}{2}, \frac{-6+4}{2}\right)$ $\left[\right.$ since, mid-point of a line segment having points $\left(x_{1}, y_{1}\right)$ and $\left.\left(x_{2}, y_{2}\right)=\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)\right]$ $\Rightarrow$ $(a, b)=\l...
Read More →If A ⊂ B, show that (B’ – A’) = ϕ.
Question: If $A \subset B$, show that $\left(B^{\prime}-A^{\prime}\right)=\phi$ Solution: As $A \subset B$ the set $A$ is inside set $B$ Hence $A \cup B=B$ Taking compliment $\Rightarrow(\mathrm{A} \cup \mathrm{B})^{\prime}=\mathrm{B}^{\prime}$ Using De-Morgan's law $(A \cup B)^{\prime}=A^{\prime} \cap B^{\prime}$ $\Rightarrow A^{\prime} \cap B^{\prime}=B^{\prime} \ldots$ (i) Now we know that $B^{\prime}=\left(B^{\prime}-A^{\prime}\right)+\left(A^{\prime} \cap B^{\prime}\right)$ Using (i) $\Righ...
Read More →The sum of two rational numbers is
Question: The sum of two rational numbers is $\frac{-1}{2}$. If one of the numbers is $\frac{5}{6}$, find the other. Solution: Let the other number be $x$. Now, $x+\frac{5}{6}=\frac{-1}{2}$ $\Rightarrow x=-\frac{1}{2}-\frac{5}{6}$ $\Rightarrow x=\frac{-3-5}{6}$ $\Rightarrow x=\frac{-8}{6}$ $\Rightarrow x=\frac{-4}{3}$...
Read More →Solve this
Question: If $f(x)=x^{2} \sin \frac{1}{x}$, where $x \neq 0$, then the value of the function $f$ at $x=0$, so that the function is continuous at $x=0$, is (a) 0 (b) $-1$ (c) 1 (d) none Solution: The given function is $f(x)=x^{2} \sin \frac{1}{x}$, where $x \neq 0$. Now, $f(x)$ is continuous at $x=0$. $\therefore f(0)=\lim _{x \rightarrow 0} f(x)$ $\Rightarrow f(0)=\lim _{x \rightarrow 0} x^{2} \sin \frac{1}{x}$ $\Rightarrow f(0)=\lim _{x \rightarrow 0} x^{2} \times \lim _{x \rightarrow 0} \sin \...
Read More →The sum of two rational numbers is 2.
Question: The sum of two rational numbers is $-2$. If one of the numbers is $\frac{-14}{5}$, find the other. Solution: Let the other number be $x$. Now, $\Rightarrow x+\frac{-14}{5}=-2$ $\Rightarrow x-\frac{14}{5}=-2$ $\Rightarrow x=-2+\frac{14}{5}$ $\Rightarrow x=\frac{(-2) \times 5+14}{5}$ $\Rightarrow x=\frac{-10+14}{5}$ $\Rightarrow x=\frac{4}{5}$...
Read More →Using the rearrangement property find the sum:
Question: Using the rearrangement property find the sum: (i) $\frac{4}{3}+\frac{3}{5}+\frac{-2}{3}+\frac{-11}{5}$ (ii) $\frac{-8}{3}+\frac{-1}{4}+\frac{-11}{6}+\frac{3}{8}$ (iii) $\frac{-13}{20}+\frac{11}{14}+\frac{-5}{7}+\frac{7}{10}$ (iv) $\frac{-6}{7}+\frac{-5}{6}+\frac{-4}{9}+\frac{-15}{7}$ Solution: (i) $\left(\frac{4}{3}+\frac{-2}{3}\right)+\left(\frac{3}{5}+\frac{-11}{5}\right)$ $=\left(\frac{4-2}{3}\right)+\left(\frac{3-11}{5}\right)$ $=\left(\frac{2}{3}+\frac{-8}{5}\right)$ $=\left(\fra...
Read More →The function f(x) = cot x is discontinuous on the set
Question: The functionf(x) = cotxis discontinuous on the set' (a) $\{x: x=n \pi, n \in Z\}$ (b) $\{x: x=2 m \pi, n \in Z\}$ (C) $\left\{x: x=(2 n+1) \frac{\pi}{2}, n \in Z\right\}$ (d) $\left\{x: x=\frac{n \pi}{2}, n \in Z\right\}$ Solution: $f(x)=\cot x=\frac{\cos x}{\sin x}$ Now, $f(x)$ is discontinuous when $\sin x=0$. $\sin x=0$ $\Rightarrow x=n \pi, n \in Z$ So, $f(x)=\cot x$ is discontinuous on the set $\{x: x=n \pi, n \in Z\}$ Hence, the correct answer is option (a)....
Read More →The function f(x) = cot x is discontinuous on the set
Question: The functionf(x) = cotxis discontinuous on the set' (a) $\{x: x=n \pi, n \in Z\}$ (b) $\{x: x=2 m \pi, n \in Z\}$ (C) $\left\{x: x=(2 n+1) \frac{\pi}{2}, n \in Z\right\}$ (d) $\left\{x: x=\frac{n \pi}{2}, n \in Z\right\}$ Solution: $f(x)=\cot x=\frac{\cos x}{\sin x}$ Now, $f(x)$ is discontinuous when $\sin x=0$. $\sin x=0$ $\Rightarrow x=n \pi, n \in Z$ So, $f(x)=\cot x$ is discontinuous on the set $\{x: x=n \pi, n \in Z\}$ Hence, the correct answer is option (a)....
Read More →If P(9a -2, – b) divides line segment joining
Question: If P(9a -2, b) divides line segment joining A(3a + 1,-3) and B(8a, 5) in the ratio 3 : 1, then find the values of a and b. Solution: Let P(9a 2, b) divides AS internally in the ratio 3:1. By section formula, $9 a-2=\frac{3(8 a)+1(3 a+1)}{3+1}$ $[\because$ internal section formula, the coordinates of point $P$ divides the line segment joiningpoint $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ in the ratio $m_{1}: m_{2}$ internally is $\left.\left(\frac{m_{2} x_{1}+m_{1} x_...
Read More →. If A ⊂ B, prove that B’ ⊂ A’.
Question: If $A \subset B$, prove that $B^{\prime} \subset A^{\prime} .$ Solution: As $A \subset B$ the set $A$ is inside set $B$ Hence $A \cup B=B$ Taking compliment $\Rightarrow(A \cup B)^{\prime}=B .^{\prime}$ Using de-morgans law $(A \cup B)^{\prime}=A^{\prime} \cap B .^{\prime}$ $\Rightarrow A^{\prime} \cap B^{\prime}=B .^{\prime}$ $A^{\prime} \cap B^{\prime}=B^{\prime}$ means that the set $B^{\prime}$ is inside the set $A . .^{\prime}$ Representing in Venn diagram, As seen from Venn diagra...
Read More →Solve this
Question: If $f(x)=2 x$ and $g(x)=\frac{x^{2}}{2}+1$, then which of the following can be a discontinuous function (a) $f(x)+g(x)$ (b) $f(x)-g(x)$ (c) $f(x) g(x)$ (d) $\frac{g(x)}{f(x)}$ Solution: $f(x)=2 x$ and $g(x)=\frac{x^{2}}{2}+1$ are polynomial functions. We know polynomial functions are continuous for all values of $x$. So, $f(x)=2 x$ and $g(x)=\frac{x^{2}}{2}+1$ are continuous functions. Also, sum, difference and product of continuous functions is continuous functions. $\therefore f(x)+g...
Read More →Solve the following
Question: Subtract: (i) $\frac{3}{4}$ from $\frac{1}{3}$ (ii) $\frac{-5}{6}$ from $\frac{1}{3}$ (iii) $\frac{-8}{9}$ from $\frac{-3}{5}$ (iv) $\frac{-9}{7}$ from $-1$ (v) $\frac{-18}{11}$ from 1 (vi) $\frac{-13}{9}$ from 0 (vii) $\frac{-32}{13}$ from $\frac{-6}{5}$ (viii) $-7$ from $\frac{-4}{7}$ Solution: (i) $\left(\frac{1}{3}-\frac{3}{4}\right)=\frac{1}{3}+\left(\right.$ Additive inverse of $\left.\frac{3}{4}\right)$ $=\left(\frac{1}{3}+\frac{-3}{4}\right)=\left(\frac{4}{12}+\frac{-9}{12}\rig...
Read More →Solve this
Question: If $f(x)=\left\{\begin{array}{cc}\frac{\sin (\cos x)-\cos x}{(\pi-2 x)^{2}}, x \neq \frac{\pi}{2} \\ k , x=\frac{\pi}{2}\end{array}\right.$ is continuous at $x=\pi / 2$, then $k$ is equal to (a) 0 (b) $\frac{1}{2}$ (C) 1 (d) $-1$ Solution: (a) 0 Given: $f(x)=\left\{\begin{array}{l}\frac{\sin (\cos x)-\cos x}{(\pi-2 \mathrm{x})^{2}}, x \neq \frac{\pi}{2} \\ k, x=\frac{\pi}{2}\end{array}\right.$ If $f(x)$ is continuous at $x=\frac{\pi}{2}$, then $\lim _{x \rightarrow \frac{\pi}{2}} f(x)=...
Read More →Solve this
Question: If $f(x)=\left\{\begin{array}{cc}\frac{\sin (\cos x)-\cos x}{(\pi-2 x)^{2}}, x \neq \frac{\pi}{2} \\ k , x=\frac{\pi}{2}\end{array}\right.$ is continuous at $x=\pi / 2$, then $k$ is equal to (a) 0 (b) $\frac{1}{2}$ (C) 1 (d) $-1$ Solution: (a) 0 Given: $f(x)=\left\{\begin{array}{l}\frac{\sin (\cos x)-\cos x}{(\pi-2 \mathrm{x})^{2}}, x \neq \frac{\pi}{2} \\ k, x=\frac{\pi}{2}\end{array}\right.$ If $f(x)$ is continuous at $x=\frac{\pi}{2}$, then $\lim _{x \rightarrow \frac{\pi}{2}} f(x)=...
Read More →If A and B are two sets such than
Question: If A and B are two sets such than n(A) = 54, n(B) = 39 and n(B A) = 13 then find n(A B). Hint $n(B)=n(B-A)+n(A \cap B) \Rightarrow n(A \cap B)=(39-13)=26$ Solution: Given: n(A) = 54, n(B) = 39, n(B A) = 13 Using the hint $n(B)=n(B-A)+n(A \cap B)$ $\Rightarrow 39=13+n(A \cap B)$ $\Rightarrow n(A \cap B)=39-13$ $\Rightarrow n(A \cap B)=26$ Visualizing the hint given We know that $n(A \cup B)=n(A)+n(B)-n(A \cap B)$ $\Rightarrow \mathrm{n}(\mathrm{A} \cup \mathrm{B})=54+39-26$ $\Rightarrow...
Read More →Find the additive inverse of each of the following:
Question: Find the additive inverse of each of the following: (i) $\frac{1}{3}$ (ii) $\frac{23}{9}$ (iii) $-18$ (iv) $\frac{-17}{8}$ (v) $\frac{15}{-4}$ (vi) $\frac{-16}{-5}$ (vii) $\frac{-3}{11}$ (viii) 0 (ix) $\frac{19}{-6}$ (x) $\frac{-8}{-7}$ Solution: The additive inverse of $\frac{a}{b}$ is $\frac{-a}{b}$. Therefore, $\frac{a}{b}+\left(\frac{-a}{b}\right)=0$ (i) Additive inverse of $\frac{1}{3}$ is $\frac{-1}{3}$. (ii) Additive inverse of $\frac{23}{9}$ is $\frac{-23}{9}$. (iii) Additive i...
Read More →Find the ratio in which the point
Question: Find the ratio in which the point $P\left(\frac{3}{4}, \frac{5}{12}\right)$ divides the line segment joinnig the points $A\left(\frac{1}{2}, \frac{3}{2}\right)$ and $B(2,5)$. Solution: Let $\mathrm{P}\left(\frac{3}{4}, \frac{5}{12}\right)$ divide $\mathrm{AB}$ internally in the ratio $\mathrm{m}$ :n usina the section formula we get. $\left(\frac{3}{4}, \frac{5}{12}\right)=\left(\frac{2 m-\frac{n}{2}}{m+n}, \frac{-5 m+\frac{3}{2} n}{m+n}\right)$ $[\because$ internal section formula, the...
Read More →Fill in the blanks.
Question: Fill in the blanks. (i) $\left(\frac{-3}{17}\right)+\left(\frac{-12}{5}\right)=\left(\frac{-12}{5}\right)+(\ldots \ldots)$ (ii) $-9+\frac{-21}{8}=(\ldots \ldots)+(-9)$ (iii) $\left(\frac{-8}{13}+\frac{3}{7}\right)+\left(\frac{-13}{4}\right)=(\ldots \ldots)+\left[\frac{3}{7}+\left(\frac{-13}{4}\right)\right]$ (iv) $-12+\left(\frac{7}{12}+\frac{-9}{11}\right)=\left(-12+\frac{7}{12}\right)+(\ldots \ldots)$ (v) $\frac{19}{-5}+\left(\frac{-3}{11}+\frac{-7}{8}\right)=\left\{\frac{19}{-5}+(\l...
Read More →Fill in the blanks.
Question: Fill in the blanks. (i) $\left(\frac{-3}{17}\right)+\left(\frac{-12}{5}\right)=\left(\frac{-12}{5}\right)+(\ldots \ldots)$ (ii) $-9+\frac{-21}{8}=(\ldots \ldots)+(-9)$ (iii) $\left(\frac{-8}{13}+\frac{3}{7}\right)+\left(\frac{-13}{4}\right)=(\ldots \ldots)+\left[\frac{3}{7}+\left(\frac{-13}{4}\right)\right]$ (iv) $-12+\left(\frac{7}{12}+\frac{-9}{11}\right)=\left(-12+\frac{7}{12}\right)+(\ldots \ldots)$ (v) $\frac{19}{-5}+\left(\frac{-3}{11}+\frac{-7}{8}\right)=\left\{\frac{19}{-5}+(\l...
Read More →In what ratio does the X-axis divide the line segment
Question: In what ratio does the X-axis divide the line segment joining the points (- 4, 6) and (- 1, 7)? Find the coordinates of the points of division. Solution: Let the required ratio be : 1. So, the coordinates of the point M of division A (- 4, 6) and B(-1,7) are $\left\{\frac{\lambda x_{2}+1 \cdot x_{1}}{\lambda+1}, \frac{\lambda y_{2}+1 \cdot y_{1}}{\lambda+1}\right\}$ Here, $x_{1}=-4, x_{2}=-1$ and $y_{1}=-6, y_{2}=7$ i.e., $\quad\left(\frac{\lambda(-1)+1(-4)}{\lambda+1}, \frac{\lambda(7...
Read More →Verify the following:
Question: Verify the following: (i) $\left(\frac{3}{4}+\frac{-2}{5}\right)+\frac{-7}{10}=\frac{3}{4}+\left(\frac{-2}{5}+\frac{-7}{10}\right)$ (ii) $\left(\frac{-7}{11}+\frac{2}{-5}\right)+\frac{-13}{22}=\frac{-7}{11}+\left(\frac{2}{-5}+\frac{-13}{22}\right)$ (iii) $-1+\left(\frac{-2}{3}+\frac{-3}{4}\right)=\left(-1+\frac{-2}{3}\right)+\frac{-3}{4}$ Solution: 1. $\mathrm{LHS}=\left\{\left(\frac{3}{4}+\frac{-2}{5}\right)+\frac{-7}{10}\right\}$ $\left\{\left(\frac{15-8}{20}\right)+\frac{-7}{10}\rig...
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