The mean of n observation is X
Question: The mean of $n$ observation is $\bar{X}$. If the first item is increased by 1 , second by 2 and so on, then the new mean is (a) $\overline{\mathrm{X}}+n$ (b) $\overline{\mathrm{X}}+\frac{n}{2}$ (c) $\overline{\mathrm{X}}+\frac{n+1}{2}$ (d) None of these Solution: Let $x_{1}, x_{2}, x_{3}, \ldots, x_{n}$ be the $n$ observations. Mean $=\bar{X}=\frac{x_{1}+x_{2}+\ldots+x_{n}}{n}$ $\Rightarrow x_{1}+x_{2}+x_{3}+\ldots+x_{n}=n \bar{X}$ If the first item is increased by 1, second by 2 and s...
Read More →The polar form of
Question: The polar form of $\left(i^{25}\right)^{3}$ is (a) $\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}$ (b) $\cos \pi+i \sin \pi$ (c) $\cos \pi-i \sin \pi$ (d) $\cos \frac{\pi}{2}-i \sin \frac{\pi}{2}$ Solution: (d) $\cos \frac{\pi}{2}-i \sin \frac{\pi}{2}$ $\left(i^{25}\right)^{3}=(i)^{75}$ $=(i)^{4 \times 18+3}$ $=(i)^{3}$ $=-i \quad\left(\because i^{4}=1\right)$ Let $z=0-i$ Since, the point $(0,-1)$ lies on the negative direction of imaginary axis. Therefore, $\arg (z)=\frac{-\pi}{2}$ Modulus,...
Read More →Mode is
Question: Mode is(a) least frequency value(b) middle most value(c) most frequent value(d) None of these Solution: Mode is Most frequent value. Hence, the correct option is (c)....
Read More →Without actual division, show that
Question: Without actual division, show that $\left(x^{3}-3 x^{2}-13 x+15\right)$ is exactly divisible by $\left(x^{2}+2 x-3\right)$ Solution: Let: $f(x)=x^{3}-3 x^{2}-13 x+15$ And, $g(x)=x^{2}+2 x-3$ $=x^{2}+x-3 x-3$ $=x(x-1)+3(x-1)$ $=(x-1)(x+3)$ Now, $f(x)$ will be exactly divisible by $g(x)$ if it is exactly divisible by $(x-1)$ as well as $(x+3)$. For this, we must have: $f(1)=0$ and $f(-3)=0$ Thus, we have $f(1)=\left(1^{3}-3 \times 1^{2}-13 \times 1+15\right)$ $=(1-3-13+15)$ $=0$ And, $f(...
Read More →The mode of a frequency distribution can be determined graphically from
Question: The mode of a frequency distribution can be determined graphically from (a) Histogram(b) Frequency polygon(c) Ogive(d) Frequency curve Solution: The mode of frequency distribution can be determined graphically from Histogram. Hence, the correct option is (a)....
Read More →The median of a given frequency distribution is found graphically with the help of
Question: The median of a given frequency distribution is found graphically with the help of (a) Histogram(b) Frequency curve(c) Frequency polygon(d) Ogive Solution: The median of a given frequency distribution is found graphically with the help of Ogive. Hence, the correct option is (d)....
Read More →Which of the following cannot be determined graphically?
Question: Which of the following cannot be determined graphically?(a) Mean(b) Median(c) Mode(d) None of these Solution: Mean cannot be determined by graphically. Hence, the correct option is (a)....
Read More →For a frequency distribution, mean,
Question: For a frequency distribution, mean, median and mode are connected by the relation(a) Mode = 3 Mean 2 Median(b) Mode = 2 Median 3 Mean(c) Mode = 3 Median 2 Mean(d) Mode = 3 Median + 2 Mean Solution: The relation between mean, median and mode is Mode = 3 Median 2 Mean Hence, the correct option is (c)....
Read More →Solve the following
Question: If $z=\cos \frac{\pi}{4}+i \sin \frac{\pi}{6}$, then (a) $|z|=1, \arg (z)=\frac{\pi}{4}$ (b) $|z|=1, \arg (z)=\frac{\pi}{6}$ (c) $|z|=\frac{\sqrt{3}}{2}, \arg (z)=\frac{5 \pi}{24}$ (d) $|z|=\frac{\sqrt{3}}{2}, \arg (z)=\tan ^{-1} \frac{1}{\sqrt{2}}$ Solution: (d) $|z|=\frac{\sqrt{3}}{2}, \arg (z)=\tan ^{-1} \frac{1}{\sqrt{2}}$ $z=\cos \frac{\pi}{4}+i \sin \frac{\pi}{6}$ $\Rightarrow z=\frac{1}{\sqrt{2}}+\frac{1}{2} i$ $\Rightarrow|z|=\sqrt{\left(\frac{1}{\sqrt{2}}\right)^{2}+\frac{1}{4...
Read More →Prove that the function $f: N ightarrow N$, defined by
Question: Prove that the function $f: N \rightarrow N$, defined by Solution: $f: N \rightarrow N$, defined by $f(x)=x^{2}+x+1$ Injectivity: Letxandybe any two elements in the domain (N), such thatf(x) = f(y). $\Rightarrow x^{2}+x+1=y^{2}+y+1$ $\Rightarrow\left(x^{2}-y^{2}\right)+(x-y)=0$ $\Rightarrow(x+y)(x-y)+(x-y)=0$ $\Rightarrow(x-y)(x+y+1)=0$ $\Rightarrow x-y=0 \quad[(\mathrm{x}+\mathrm{y}+1)$ cannot be zero because $x$ and $y$ are natural numbers $]$ $\Rightarrow x=y$ So,fis one-one. Surjec...
Read More →The arithmetic mean of 1, 2, 3, ... , n is
Question: The arithmetic mean of 1, 2, 3, ... ,nis (a) $\frac{n+1}{2}$ (b) $\frac{n-1}{2}$ (c) $\frac{n}{2}$ (d) $\frac{n}{2}+1$ Solution: Arithmetic mean of 1, 2, 3, ... ,n $=\frac{1+2+3+\ldots+n}{n}$ $=\frac{\frac{n(n+1)}{2}}{n}$ $=\frac{n+1}{2}$ Hence, the correct option is (a)....
Read More →The algebraic sum of the deviations of a frequency distribution from its mean is
Question: The algebraic sum of the deviations of a frequency distribution from its mean is (a) always positive(b) always negative(c) 0(d) a non-zero number Solution: The algebraic sum of the deviations of a frequency distribution from its mean is zero. Hence, the correct option is (c)....
Read More →Solve the following
Question: If $\sqrt{a+i b}=x+i y$, then possible value of $\sqrt{a-i b}$ is (a) $x^{2}+y^{2}$ (b) $\sqrt{x^{2}+y^{2}}$ (c) $x+i y$ (d) $x-i y$ (e) $\sqrt{x^{2}-y^{2}}$ Solution: (d) $x-i y$ $\sqrt{a+i b}=x+i y$ Squaring on both the sides, we get, $a+i b=x^{2}+(i y)^{2}+2 i x y$ $\Rightarrow a+i b=\left(x^{2}-y^{2}\right)+2 i x y$ $\therefore a=\left(x^{2}-y^{2}\right)$ and $b=2 x y$ $\therefore a-i b=\left(x^{2}-y^{2}\right)-2 i x y$ $\Rightarrow a-i b=x^{2}+i^{2} y^{2}-2 i x y \quad\left[\becau...
Read More →Which of the following is not a measure of central tendency?
Question: Which of the following is not a measure of central tendency? (a) Mean(b) Median(c) Mode(d) Standard deviation Solution: Standard deviation is not a measure of central tendency. Hence, correct option is (d)....
Read More →Find the value of a for which the polynomial
Question: Find the value of $a$ for which the polynomial $\left(x^{4}-x^{3}-11 x^{2}-x+a\right)$ is divisible by $(x+3)$. Solution: Let: $f(x)=x^{4}-x^{3}-11 x^{2}-x+a$ Now, $x+3=0 \Rightarrow x=-3$ By the factor theorem, $f(x)$ is exactly divisible by $(x+3)$ if $f(-3)=0$. Thus, we have: $f(-3)=\left[(-3)^{4}-(-3)^{3}-11 \times(-3)^{2}-(-3)+a\right]$ $=(81+27-99+3+a)$ $=12+a$ Also $f(-3)=0$ $\Rightarrow 12+a=0$ $\Rightarrow a=-12$ Hence, $f(x)$ is exactly divisible by $(x+3)$ when $a$ is $-12$....
Read More →Write the median class of the following distribution:
Question: Write the median class of the following distribution: Solution: Consider the following distribution table. Here, $N=50$ $\frac{N}{2}=25$ The cumulative frequency just greater than 25 is 26. So, the median class is 3040....
Read More →Find the class marks of classes 10−25 and 35−55.
Question: Find the class marks of classes 1025 and 3555. Solution: Class mark of 1025 $=\frac{10+25}{2}$ $=\frac{35}{2}$ $=17.5$ Class mark of 3555 $=\frac{35+55}{2}$ $=\frac{90}{2}$ $=45$...
Read More →Which of the following functions from A to B are one-one and onto?
Question: Which of the following functions fromAtoBare one-one and onto? (i)f1= {(1, 3), (2, 5), (3, 7)} ;A= {1, 2, 3},B= {3, 5, 7}(ii)f2= {(2,a), (3,b), (4,c)} ;A= {2, 3, 4},B= {a,b,c} (iii)f3= {(a,x), (b,x), (c,z), (d,z)} ;A= {a,b,c,d,},B= {x,y,z} Solution: (i) $f_{1}=\{(1,3),(2,5),(3,7)\} ; A=\{1,2,3\}, B=\{3,5,7\}$ Injectivity: $f_{1}(1)=3$ $f_{1}(2)=5$ $f_{1}(3)=7$ $\Rightarrow$ Every element of $A$ has different images in $B$. So, $f_{1}$ is one-one. Surjectivity: Co-domain of $f_{1}=\{3,5...
Read More →In the graphical representation of a frequency distribution,
Question: In the graphical representation of a frequency distribution, if the distance between mode and mean isktimes the distance between median and mean, then write the value ofk. Solution: We have,Mode = 3 Median 2 Mean⇒Mode Mean = 3 Median 3 Mean⇒Mode Mean = 3(Median Mean) .....(1) It is given that,Mode Mean =k(Median Mean) .....(2)From (1) and (2), we getk= 3...
Read More →Solve the following
Question: If $(1+i)(1+2 i)(1+3 i) \ldots(1+n i)=a+i b$, then $2 \times 5 \times 10 \times \ldots \times\left(1+n^{2}\right)$ is equal to (a) $\sqrt{a^{2}+b^{2}}$ (b) $\sqrt{a^{2}-b^{2}}$ (c) $a^{2}+b^{2}$ (d) $a^{2}-b^{2}$ (e) $a+b$ Solution: (c) $a^{2}+b^{2}$ $(1+i)(1+2 i)(1+3 i) \ldots \ldots(1+n i)=a+i b$ Taking modulus on both the sides, we get: $|(1+i)(1+2 i)(1+3 i) \ldots \ldots(1+n i)|=|a+i b|$ $|(1+i)(1+2 i)(1+3 i) \ldots \ldots(1+n i)|$ can be written as $|(1+i)||(1+2 i)||(1+3 i)| \ldot...
Read More →Find the value of m for which (2x – 1) is a factor of
Question: Find the value of $m$ for which $(2 x-1)$ is a factor of $\left(8 x^{4}+4 x^{3}-16 x^{2}+10 x+m\right)$. Solution: Let $f(x)=8 x^{4}+4 x^{3}-16 x^{2}+10 x+m$ It is given that $(2 x-1)=2\left(x-\frac{1}{2}\right)$ is a factor of $f(x)$. Using factor theorem, we have $f\left(\frac{1}{2}\right)=0$ $\Rightarrow 8 \times\left(\frac{1}{2}\right)^{4}+4 \times\left(\frac{1}{2}\right)^{3}-16 \times\left(\frac{1}{2}\right)^{2}+10 \times \frac{1}{2}+m=0$ $\Rightarrow \frac{1}{2}+\frac{1}{2}-4+5+m...
Read More →Write the median class for the following frequency distribution:
Question: Write the median class for the following frequency distribution: Solution: We are given the following table. Here,N= 100 $\therefore \frac{N}{2}=50$ The cumulative frequency just greater than 50 is 60.So, the median class is 4050....
Read More →Solve the following
Question: If $\frac{3+2 i \sin \theta}{1-2 i \sin \theta}$ is a real number and $0\theta2 \pi$, then $\theta=$ (a) $\pi$ (b) $\frac{\pi}{2}$ (c) $\frac{\pi}{3}$ (d) $\frac{\pi}{6}$ Solution: (a) $\pi$ Given: $\frac{3+2 i \sin \theta}{1-2 i \sin \theta}$ is a real number On rationalising, we get, $\frac{3+2 i \sin \theta}{1-2 i \sin \theta} \times \frac{1+2 i \sin \theta}{1+2 i \sin \theta}$ $=\frac{(3+2 i \sin \theta)(1+2 i \sin \theta)}{(1)^{2}-(2 i \sin \theta)^{2}}$ $=\frac{3+2 i \sin \theta+...
Read More →Solve the following
Question: If $\frac{3+2 i \sin \theta}{1-2 i \sin \theta}$ is a real number and $0\theta2 \pi$, then $\theta=$ (a) $\pi$ (b) $\frac{\pi}{2}$ (c) $\frac{\pi}{3}$ (d) $\frac{\pi}{6}$ Solution: (a) $\pi$ Given: $\frac{3+2 i \sin \theta}{1-2 i \sin \theta}$ is a real number On rationalising, we get, $\frac{3+2 i \sin \theta}{1-2 i \sin \theta} \times \frac{1+2 i \sin \theta}{1+2 i \sin \theta}$ $=\frac{(3+2 i \sin \theta)(1+2 i \sin \theta)}{(1)^{2}-(2 i \sin \theta)^{2}}$ $=\frac{3+2 i \sin \theta+...
Read More →A student draws a cumulative frequency curve for the marks obtained
Question: A student draws a cumulative frequency curve for the marks obtained by 40 students of a class as shown below. Find the median marks obtained by the students of the class. Solution: Here,N= 40 So, $\left(\frac{N}{2}\right)=20$ Draw a line parallel tox-axis from the point (0, 20), intersecting the graph at point P. Now, draw PM from P on thex-axis. Thex-coordinate of M gives us the median. $\therefore$ Median $=50$...
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