Find the value of a for which (x + 2a) is a factor of
Question: Find the value of $a$ for which $(x+2 a)$ is a factor of $\left(x^{5}-4 a^{2} x^{3}+2 x+2 a+3\right)$. Solution: Let $f(x)=x^{5}-4 a^{2} x^{3}+2 x+2 a+3$ It is given that $(x+2 a)$ is a factor of $f(x)$ Using factor theorem, we have $f(-2 a)=0$ $\Rightarrow(-2 a)^{5}-4 a^{2} \times(-2 a)^{3}+2 \times(-2 a)+2 a+3=0$ $\Rightarrow-32 a^{5}-4 a^{2} \times\left(-8 a^{3}\right)+2 \times(-2 a)+2 a+3=0$ $\Rightarrow-32 a^{5}+32 a^{5}-4 a+2 a+3=0$ $\Rightarrow-2 a+3=0$ $\Rightarrow 2 a=3$ $\Rig...
Read More →Write the modal class for the following frequency distribution:
Question: Write the modal class for the following frequency distribution: Solution: Here, the maximum frequency is 75 and the corresponding class-interval is 2025. Therefore, 2025 is the modal class....
Read More →Which measure of central tendency can be determine graphically?
Question: Which measure of central tendency can be determine graphically? Solution: Median can be determined graphically....
Read More →The value of
Question: The value of $(1+i)\left(1+i^{2}\right)\left(1+i^{3}\right)\left(1+i^{4}\right)$ is (a) 2 (b) 0 (c) 1 (d)i Solution: (b) 0 $(1+i)\left(1+i^{2}\right)\left(1+i^{3}\right)\left(1+i^{4}\right)$ $=(1+i)(1-1)(1-i)(1+1) \quad\left(\because i^{2}=-1, i^{3}=-i\right.$ and $\left.i^{4}=1\right)$ = (1 +i) (0) (1-i) (2) = 0...
Read More →Write the empirical relation between mean, mode and median.
Question: Write the empirical relation between mean, mode and median. Solution: The empirical relation between mean, median and mode is Mode = 3 Median 2 Mean...
Read More →What is the value of the median of the data using the
Question: What is the value of the median of the data using the graph in the following figure of less than ogive and more than ogive? Solution: We know that the abscissa of the point of intersection of two ogives gives the median. From the given figure, it can be seen that both the ogives intersect at the point (4, 15). $\therefore$ Median of the data $=4$...
Read More →Solve the following
Question: Express $\sin \frac{\pi}{5}+i\left(1-\cos \frac{\pi}{5}\right)$ in polar form. Solution: Let $z=\sin \frac{\pi}{5}+i\left(1-\cos \frac{\pi}{5}\right)$ $\Rightarrow|z|=\sqrt{\left(\sin \frac{\pi}{5}\right)^{2}+\left(1-\cos \frac{\pi}{5}\right)^{2}}$ $=\sqrt{\sin ^{2} \frac{\pi}{5}+1+\cos ^{2} \frac{\pi}{5}-2 \cos \frac{\pi}{5}}$ $=\sqrt{2-2 \cos \frac{\pi}{5}}$ $=\sqrt{2}\left(\sqrt{1-\cos \frac{\pi}{5}}\right)$ $=\sqrt{2}\left(\sqrt{2 \sin ^{2} \frac{\pi}{10}}\right)$ $=2 \sin \frac{\p...
Read More →Which measure of central tendency is given by the x-coordinate
Question: Which measure of central tendency is given by thex-coordinate of the point of intersection of the 'more than' ogive and 'less than' ogive? Solution: Median...
Read More →Find the value of a for which (x + 1) is a factor of
Question: Find the value of $a$ for which $(x+1)$ is a factor of $\left(a x^{3}+x^{2}-2 x+4 a-9\right)$. Solution: Let $f(x)=a x^{3}+x^{2}-2 x+4 a-9$ It is given that(x+ 1) is a factor off(x).Using factor theorem, we have $f(-1)=0$ $\Rightarrow a \times(-1)^{3}+(-1)^{2}-2 \times(-1)+4 a-9=0$ $\Rightarrow-a+1+2+4 a-9=0$ $\Rightarrow 3 a-6=0$ $\Rightarrow 3 a=6$ $\Rightarrow a=2$ Thus, the value ofais 2....
Read More →What is the algebraic sum of deviation of a frequency distribution about its mean?
Question: What is the algebraic sum of deviation of a frequency distribution about its mean? Solution: The algebraic sum of deviation of a frequency distribution about its mean is zero....
Read More →Give an example of a function
Question: Give an example of a function(i) which is one-one but not onto(ii) which is not one-one but onto(iii) which is neither one-one nor onto Solution: (i) which is one-one but not onto. $f: Z \rightarrow Z$ given by $f(x)=3 x+2$ Injectivity: Let $x$ and $y$ be any two elements in the domain $(Z)$, such that $f(x)=f(y)$. $f(x)=f(y)$ $\Rightarrow 3 x+2=3 y+2$ $\Rightarrow 3 x=3 y$ $\Rightarrow x=y$ $\Rightarrow f(x)=f(y) \Rightarrow x=y$ So,fis one-one. Surjectivity: Let $y$ be any element in...
Read More →Define mean.
Question: Define mean. Solution: Mean The mean of a set of observation is equal to sum of observations divided by the number of observations. $\bar{x}=\frac{\sum x_{i}}{n}$ (Where $\sum x_{i}=$ sum of the observation and $n=$ number of observations)...
Read More →Solve the following
Question: If $z_{1}, z_{2}$ and $z_{3}, z_{4}$ are two pairs of conjugate complex numbers, prove that $\arg \left(\frac{z_{1}}{z_{4}}\right)+\arg \left(\frac{z_{2}}{z_{3}}\right)=0$. Solution: Given that $z_{1}, z_{2}$ and $z_{3}, z_{4}$ are two pairs of conjugate complex numbers. $\therefore z_{1}=r_{1} e^{i \theta_{1}}, z_{2}=r_{1} e^{-i \theta_{1}}, z_{3}=r_{2} e^{i \theta_{2}}$ and $z_{4}=r_{2} e^{-i \theta_{2}}$ Then, $\frac{z_{1}}{z_{4}}=\frac{r_{1} e^{i \theta_{1}}}{r_{2} e^{-i \theta_{2}...
Read More →During the medical check up of 35 students of a class,
Question: During the medical check up of 35 students of a class, their weights were recorded as follows: Draw a less than type ogive for the given data. Hence, obtain the median weight from the graph and verify the result by using the formula. Solution: Prepare the table for cumulative frequency for less than type. Now, draw the less than ogive using suitable points. Here, $\frac{N}{2}=\frac{\sum f}{2}==\frac{35}{2}=17.5$ From (0, 17.5) draw a line parallel to horizontal axis, which intersects t...
Read More →Find the value of a for which (x − 4) is a factor of
Question: Find the value of $a$ for which $(x-4)$ is a factor of $\left(2 x^{3}-3 x^{2}-18 x+a\right)$. Solution: Let: $f(x)=2 x^{3}-3 x^{2}-18 x+a$ $(x-4)$ is a factor of $f(x)=2 x^{3}-3 x^{2}-18 x+a$. $\Rightarrow f(4)=0$ $\Rightarrow 2 \times 4^{3}-3 \times 4^{2}-18 \times 4+a=0$ $\Rightarrow 8+a=0$ $\Rightarrow a=-8$ Hence, the required value ofais-8....
Read More →Solve the following
Question: If $z_{1}$ and $z_{2}$ are two complex numbers such that $\left|z_{1}\right|=\left|z_{2}\right|$ and $\arg \left(z_{1}\right)+\arg \left(z_{2}\right)=\pi$, then show that $z_{1}=-\overline{z_{2}}$. Solution: Let $\theta_{1}$ be the $\arg \left(z_{1}\right)$ and $\theta_{2}$ be the $\arg \left(z_{2}\right)$. It is given that $\left|z_{1}\right|=\left|z_{2}\right|$ and $\arg \left(z_{1}\right)+\arg \left(z_{2}\right)=\pi$. Since, $z_{1}$ is a complex number. $z_{1}=\left|z_{1}\right|\lef...
Read More →The following table gives production yield per hectare of wheat of 100 farms of a village:
Question: The following table gives production yield per hectare of wheat of 100 farms of a village: Solution: Prepare a table for less than type. Now, plot the less than ogive using suitable points....
Read More →The following distribution gives the daily income of 50 workers of a factory:
Question: The following distribution gives the daily income of 50 workers of a factory: Convert the above distribution to a less than type cumulative frequency distribution and draw its ogive. Solution: Now, plot the less than ogive with suitable points....
Read More →Express the following complex in the form r(cos θ + i sin θ):
Question: Express the following complex in the formr(cos + i sin ): (i) $1+i \tan a$ (ii) $\tan \alpha-i$ (iii) $1-\sin \alpha+i \cos \alpha$ (iv) $\frac{1-i}{\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}}$ Solution: (i) Let $\mathrm{z}=1+i \tan \alpha$ $\because \tan \alpha$ is periodic with period $\pi .$ So, let us take $\alpha \in[0, \pi / 2) \cup(\pi / 2, \pi]$ Case I : When $\alpha \in[0, \pi / 2)$ $z=1+i \tan \alpha$ $\Rightarrow|\mathrm{z}|=\sqrt{1+\tan ^{2} \alpha}$ $=|\sec \alpha|$ $\left[\b...
Read More →Find the value of k for which (x − 1) is a factor of
Question: Find the value of $k$ for which $(x-1)$ is a factor of $\left(2 x^{3}+9 x^{2}+x+k\right)$. Solution: Let: $f(x)=2 x^{3}+9 x^{2}+x+k$ $(x-1)$ is a factor of $f(x)=2 x^{3}+9 x^{2}+x+k$. $\Rightarrow f(1)=0$ $\Rightarrow 2 \times 1^{3}+9 \times 1^{2}+1+k=0$ $\Rightarrow 12+k=0$ $\Rightarrow k=-12$ Hence, the required value ofkis-12....
Read More →Express the following complex in the form r(cos θ + i sin θ):
Question: Express the following complex in the formr(cos + i sin ): (i) $1+i \tan a$ (ii) $\tan \alpha-i$ (iii) $1-\sin \alpha+i \cos \alpha$ (iv) $\frac{1-i}{\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}}$ Solution: (i) Let $\mathrm{z}=1+i \tan \alpha$ $\because \tan \alpha$ is periodic with period $\pi .$ So, let us take $\alpha \in[0, \pi / 2) \cup(\pi / 2, \pi]$ Case I : When $\alpha \in[0, \pi / 2)$ $z=1+i \tan \alpha$ $\Rightarrow|\mathrm{z}|=\sqrt{1+\tan ^{2} \alpha}$ $=|\sec \alpha|$ $\left[\b...
Read More →The annual profits earned by 30 shops of a shopping complex in a locality give rise to the following distribution:
Question: The annual profits earned by 30 shops of a shopping complex in a locality give rise to the following distribution: Draw both ogives for the above data and hence obtain the median. Solution: Firstly, we prepare the cumulative frequency table for less than type. Again, prepare the cumulative frequency table for more than type Now, more than ogive and less than ogive can be drawn as follows: Thex-coordinate of the point of intersection of the more-than ogive and less-than ogive gives the ...
Read More →The following table gives the height of trees:
Question: The following table gives the height of trees: Draw 'less than' ogive and 'more than' ogive. Solution: Consider the following table. Now, draw the less than ogive using suitable points. Now, prepare the cumulative frequency table for more than series. Now, draw the more than ogive using suitable points....
Read More →The monthly profits (in Rs.) of 100 shops are distributed as follows:
Question: The monthly profits (in Rs.) of 100 shops are distributed as follows: Draw the frequency polygon for it. Solution: Firstly, we make a cumulative frequency table. Now, plot the frequency polygon (or more than ogive) using suitable points....
Read More →Draw an ogive to represent the following frequency distribution:
Question: Draw an ogive to represent the following frequency distribution: Solution: Firstly, prepare the cumulative frequency table. Now, plot the less than ogive using the suitable points....
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