The marks scored by 750 students in an examination
Question: The marks scored by 750 students in an examination are given in the form of a frequency distribution table: Prepare a cumulative frequency table by less than method and draw an ogive. Solution: Now, we draw the less than ogive with suitable points....
Read More →Draw an ogive by less than method for the following data:
Question: Draw an ogive by less than method for the following data:/spanbr data-mce-bogus="1"/ppimg src="https://www.esaral.com/qdb/uploads/2022/01/01/image96263.png" alt="" Solution: Firstly, we prepare the cumulative frequency table by less than method. Now, plot the less than ogive using the points (1, 4), (2, 13), (3, 35), (4, 63), (5, 87), (6, 99), (7, 107), (8, 113), (9, 118) and (10, 120)....
Read More →If the mean of the following data is 18.75. Find the value of p.
Question: If the mean of the following data is 18.75. Find the value ofp. Solution: Given: Mean $=18.75$ First of all prepare the frequency table in such a way that its first column consist of the values of the variate $\left(x_{i}\right)$ and the second column the corresponding frequencies $\left(f_{i}\right)$. Thereafter multiply the frequency of each row with corresponding values of variable to obtain third column containing $\left(f_{i} x_{i}\right)$. Then, sum of all entries in the column s...
Read More →The arithmetic mean of the following data is 25,
Question: The arithmetic mean of the following data is 25, find the value ofk. Solution: Given: Mean $=25$ First of all prepare the frequency table in such a way that its first column consist of the values of the variate $\left(x_{i}\right)$ and the second column the corresponding frequencies $\left(f_{i}\right)$. Thereafter multiply the frequency of each row with corresponding values of variable to obtain third column containing $\left(f_{i} x_{i}\right)$. Then, sum of all entries in the column...
Read More →Using factor theorem, show that g(x) is a factor of p(x), when
Question: Using factor theorem, show that g(x) is a factor of p(x), when $p(x)=2 \sqrt{2} x^{2}+5 x+\sqrt{2}, g(x)=x+\sqrt{2}$ Solution: Let: $p(x)=2 \sqrt{2} x^{2}+5 x+\sqrt{2}$ Here, $x+\sqrt{2}=0 \Rightarrow x=-\sqrt{2}$ By the factor theorem, $(x+\sqrt{2})$ will be a factor of the given polynomial if $p(-\sqrt{2})=0$. Thus, we have: $p(-\sqrt{2})=\left[2 \sqrt{2} \times(-\sqrt{2})^{2}+5 \times(-\sqrt{2})+\sqrt{2}\right]$ $=(4 \sqrt{2}-5 \sqrt{2}+\sqrt{2})$ $=0$ Hence, $(x+\sqrt{2})$ is a fac...
Read More →The arithmetic mean of the following data is 14. Find the value of k.
Question: The arithmetic mean of the following data is 14. Find the value ofk. Solution: Given: Mean $=14$ First of all prepare the frequency table in such a way that its first column consist of the values of the variate $\left(x_{i}\right)$ and the second column the corresponding frequencies $\left(f_{i}\right)$. Thereafter multiply the frequency of each row with corresponding values of variable to obtain third column containing $\left(f_{i} x_{i}\right)$. Then, sum of all entries in the column...
Read More →Write the smallest equivalence relation on the set A = {1, 2, 3}.
Question: Write the smallest equivalence relation on the setA= {1, 2, 3}. Solution: The smallest equivalence relation on the setA= {1, 2, 3} isR= {(1, 1), (2, 2), (3, 3)}...
Read More →Let the relation R be defined on N by aRb iff 2a + 3b = 30.
Question: Let the relation $R$ be defined on $\mathbf{N}$ by $a R b$ iff $2 a+3 b=30$. Then write $R$ as a set of ordered pairs. Solution: As, $R=\{(a, b): 2 a+3 b=30 ; a, b \in \mathbf{N}\}$ So, $R=\{(3,8),(6,6),(9,4),(12,2)\}$...
Read More →Using factor theorem, show that g(x) is a factor of p(x), when
Question: Using factor theorem, show that g(x) is a factor of p(x), when $p(x)=7 x^{2}-4 \sqrt{2} x-6, g(x)=x-\sqrt{2}$ Solution: Let: $p(x)=7 x^{2}-4 \sqrt{2} x-6$ Here, $x-\sqrt{2}=0 \Rightarrow x=\sqrt{2}$ By the factor theorem, $(x-\sqrt{2})$ is a factor of the given polynomial if $p(\sqrt{2})=0$ Thus, we have: $p(\sqrt{2})=\left[7 \times(\sqrt{2})^{2}-4 \sqrt{2} \times \sqrt{2}-6\right]$ $=(14-8-6)$ $=0$ Hence, $(x-\sqrt{2})$ is a factor of the given polynomial....
Read More →Find the missing frequencies in the following frequency
Question: Find the missing frequencies in the following frequency distribution if it is known that the mean of the distribution is 50. Solution: Given: Mean $=50$ First of all prepare the frequency table in such a way that its first column consists of the values of the variate $\left(x_{i}\right)$ and the second column the corresponding frequencies $\left(f_{i}\right)$. Thereafter multiply the frequency of each row with corresponding values of variable to obtain third column containing $\left(f_...
Read More →Using factor theorem, show that g(x) is a factor of p(x), when
Question: Using factor theorem, show that g(x) is a factor of p(x), when $p(x)=3 x^{3}+x^{2}-20 x+12, g(x)=3 x-2$ Solution: $p(x)=3 x^{3}+x^{2}-20 x+12$ $g(x)=3 x-2=3\left(x-\frac{2}{3}\right)$ Putting $x=\frac{2}{3}$ in $p(x)$, we get $p\left(\frac{2}{3}\right)=3 \times\left(\frac{2}{3}\right)^{3}+\left(\frac{2}{3}\right)^{2}-20 \times \frac{2}{3}+12=\frac{8}{9}+\frac{4}{9}-\frac{40}{3}+12=\frac{8+4-120+108}{9}=\frac{120-120}{9}=0$ Therefore, by factor theorem, (3x 2) is a factor ofp(x).Hence,g...
Read More →Let the relation $R$ be defined on the set
Question: Let the relation $R$ be defined on the set $A=\{1,2,3,4,5\}$ by $R=\left\{(a, b):\left|a^{2}-b^{2}\right|8\right\}$.WriteRas a set of ordered pairs. Solution: As, $R=\left\{(a, b):\left|a^{2}-b^{2}\right|8\right\}$ So, $R=\{(1,1),(1,2),(2,1),(2,2),(2,3),(3,2),(3,3),(3,4),(4,3),(4,4),(5,5)\}$...
Read More →Let A = {0, 1, 2, 3} and R be a relation on A defined as
Question: LetA= {0, 1, 2, 3} andRbe a relation onAdefined asR= {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (2, 2), (3, 0), (3, 3)}IsRreflexive? symmetric? transitive? Solution: We have, $R=\{(0,0),(0,1),(0,3),(1,0),(1,1),(2,2),(3,0),(3,3)\}$ As, $(a, a) \in R \forall a \in A$ So, $R$ is a reflexive relation Also, $(a, b) \in R$ and $(b, a) \in R$ So, $R$ is a symmetric relation as well And, $(0,1) \in R$ but $(1,2) \notin R$ and $(2,3) \notin R$ So, $R$ is not a transitive relation...
Read More →Using factor theorem, show that g(x) is a factor of p(x), when
Question: Using factor theorem, show that g(x) is a factor of p(x), when $p(x)=2 x^{4}+x^{3}-8 x^{2}-x+6, g(x)=2 x-3$ Solution: Let: $p(x)=2 x^{4}+x^{3}-8 x^{2}-x+6$ Here, $2 x-3=0 \Rightarrow x=\frac{3}{2}$ By the factor theorem, $(2 x-3)$ is a factor of the given polynomial if $p\left(\frac{3}{2}\right)=0$. Thus, we have: $p\left(\frac{3}{2}\right)=\left[2 \times\left(\frac{3}{2}\right)^{4}+\left(\frac{3}{2}\right)^{3}-8 \times\left(\frac{3}{2}\right)^{2}-\left(\frac{3}{2}\right)+6\right]$ $=\...
Read More →Solve the following
Question: Write $\left(i^{25}\right)^{3}$ in polar form. Solution: $\left(i^{25}\right)^{3}=i^{75}$ $=i^{4 \times 18+3}$ $=\left(i^{4}\right)^{18} \cdot i^{3}$ $=i^{3} \quad\left[\because i^{4}=1\right]$ $=-i \quad\left[\because i^{3}=-i\right]$ Let $z=0-i$ Then, $|z|=\sqrt{0^{2}+(-1)^{2}}=1$ Let $\theta$ be the argument of $z$ and $\alpha$ be the acute angle given by $\tan \alpha=\frac{|\operatorname{Im}(z)|}{|\operatorname{Re}(z)|}$. Then, $\tan \alpha=\frac{1}{0}=\infty$ $\Rightarrow \alpha=\...
Read More →Write $left(i^{25} ight)^{3}$ in polar form.
Question: Write $\left(i^{25}\right)^{3}$ in polar form. Solution: $\left(i^{25}\right)^{3}=i^{75}$ $=i^{4 \times 18+3}$ $=\left(i^{4}\right)^{18} \cdot i^{3}$ $=i^{3} \quad\left[\because i^{4}=1\right]$ $=-i \quad\left[\because i^{3}=-i\right]$ Let $z=0-i$ Then, $|z|=\sqrt{0^{2}+(-1)^{2}}=1$ Let $\theta$ be the argument of $z$ and $\alpha$ be the acute angle given by $\tan \alpha=\frac{|\operatorname{Im}(z)|}{|\operatorname{Re}(z)|}$. Then, $\tan \alpha=\frac{1}{0}=\infty$ $\Rightarrow \alpha=\...
Read More →For the set $A={1,2,3}$, define a relation $R$ on the set $A$ as follows:
Question: For the setA= {1, 2, 3}, define a relationRon the setAas follows:R= {(1, 1), (2, 2), (3, 3), (1, 3)}Write the ordered pairs to be added toRto make the smallest equivalence relation. Solution: We have, $R=\{(1,1),(2,2),(3,3),(1,3)\}$ As, $(a, a) \in R$, for all values of $a \in A$ So,Ris a reflexive relation Rcan be a symmetric and transitive relation only when element (3, 1) is addedHence,the ordered pairs to be added toR to make the smallest equivalence relation is (3, 1)....
Read More →Write $(25)^{3}$ in polar form.
Question: Write $(25)^{3}$ in polar form. Solution: $\left(i^{25}\right)^{3}=i^{75}$ $=i^{4 \times 18+3}$ $=\left(i^{4}\right)^{18} \cdot i^{3}$ $=i^{3} \quad\left[\because i^{4}=1\right]$ $=-i \quad\left[\because i^{3}=-i\right]$ Let $z=0-i$ Then, $|z|=\sqrt{0^{2}+(-1)^{2}}=1$ Let $\theta$ be the argument of $z$ and $\alpha$ be the acute angle given by $\tan \alpha=\frac{|\operatorname{Im}(z)|}{|\operatorname{Re}(z)|}$. Then, $\tan \alpha=\frac{1}{0}=\infty$ $\Rightarrow \alpha=\frac{\pi}{2}$ C...
Read More →Using factor theorem, show that g(x) is a factor of p(x), when
Question: Using factor theorem, show that g(x) is a factor of p(x), when $p(x)=2 x^{3}+9 x^{2}-11 x-30, g(x)=x+5$ Solution: Let: $p(x)=2 x^{3}+9 x^{2}-11 x-30$ Here, $x+5=0 \Rightarrow x=-5$ By the factor theorem, (x+ 5) is a factor ofthe given polynomialifp(-5) = 0.Thus, we have: $p(-5)=\left[2 \times(-5)^{3}+9 \times(-5)^{2}-11 \times(-5)-30\right]$ $=(-250+225+55-30)$ $=0$ Hence,(x+ 5) is a factor of the given polynomial....
Read More →Let R be the equivalence relation on the set Z of the integers given by R
Question: LetRbe the equivalence relation on the setZof the integers given byR= {(a,b) : 2 dividesa-b}. Write the equivalence class [0]. [NCERT EXEMPLAR] Solution: We have,An equivalence relation,R= {(a,b) : 2 dividesa-b} If $b=0$, then $a-b=a-0=a$ As, 2 divides $a-b$ And, the set of integers which are divided by 2 is $\{0, \pm 2, \pm 4, \pm 6, \ldots\}$ So, the equivalence class $[0]=\{0, \pm 2, \pm 4, \pm 6, \ldots\}$...
Read More →Using factor theorem, show that g(x) is a factor of p(x), when
Question: Using factor theorem, show that g(x) is a factor of p(x), when $p(x)=69+11 x-x^{2}+x^{3}, g(x)=x+3$ Solution: $p(x)=69+11 x-x^{2}+x^{3}$ $g(x)=x+3$ Putting $x=-3$ in $p(x)$, we get $p(-3)=69+11 \times(-3)-(-3)^{2}+(-3)^{3}=69-33-9-27=0$ Therefore, by factor theorem, (x+ 3) is a factor ofp(x).Hence,g(x) is a factor ofp(x)....
Read More →Using factor theorem, show that g(x) is a factor of p(x), when
Question: Using factor theorem, show that g(x) is a factor of p(x), when $p(x)=x^{4}-x^{2}-12, g(x)=x+2$ Solution: Let: $p(x)=x^{4}-x^{2}-12$ Here, $x+2=0 \Rightarrow x=-2$ By the factor theorem, (x+ 2) is a factor ofthe given polynomial ifp(-2) = 0.Thus, we have: $p(-2)=\left[(-2)^{4}-(-2)^{2}-12\right]$ $=(16-4-12)$ $=0$ Hence,(x+ 2) is a factor of the given polynomial....
Read More →Five coins were simultaneously tossed 1000 times
Question: Five coins were simultaneously tossed 1000 times and at each toss the number of heads were observed. The number of tosses during which 0, 1, 2, 3, 4 and 5 heads were obtained are shown in the table below. Find the mean number of heads per toss. Solution: Given: First of all prepare the frequency table in such a way that its first column consist of the numnber of heads per tosses $\left(x_{i}\right)$ and the second column the corresponding number of tosses $\left(f_{i}\right)$. Thereaft...
Read More →Candidate of four schools appear in a mathematics test.
Question: Candidate of four schools appear in a mathematics test. The data were as follows: If the average score of the candidates of all the four schools is 66, find the number of candidates that appeared from school III. Solution: Given: Mean score of the candidates $=66$ Let the number of candidates that appeared from school $I / I$ be $x$. First of all prepare the frequency table in such a way that its first column consists of the values of the variate $\left(x_{i}\right)$ and the second col...
Read More →State the reason for the relation R on the set
Question: State the reason for the relationRon the set {1, 2, 3} given byR= {(1, 2), (2, 1)} to be transitive. Solution: Since $(1,2) \in R,(2,1) \in R$ but $(1,1) \notin R, R$ is not transitive on the set $\{1,2,3\}$. For $R$ to be in a transitive relation, we must have $(1,1) \in R$....
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