A shopkeeper offers 10% off-season discount to the customers and still makes a profit of 26%.
Question: A shopkeeper offers 10% off-season discount to the customers and still makes a profit of 26%. What is the cost price for the shopkeeper on a pair of shoes marked at Rs 1120? Solution: Given, MP of the pair of shoes $=$ Rs. 1120 Discount $=10 \%$ So, SP $=$ MP $\left(\frac{100-\text { Discount } \%}{100}\right)$ $=1120 \times \frac{90}{100}$ $=$ Rs. 1008 Now, Profit $=26 \%$ $\mathrm{SP}=$ Rs. 1008 Therefore, $\mathrm{CP}=\left(\frac{100 \times \mathrm{SP}}{100+\text { Profit\% }}\right...
Read More →If A is an invertible matrix, then which of the following is not true
Question: IfAis an invertible matrix, then which of the following is not true (a) $\left(A^{2}\right)^{-1}=\left(A^{-1}\right)^{2}$ (b) $\left|A^{-1}\right|=|A|^{-1}$ (c) $\left(A^{T}\right)^{-1}=\left(A^{-1}\right)^{T}$ (d) $|A| \neq 0$ Solution: (a) $\left(A^{2}\right)^{-1}=\left(A^{-1}\right)^{2}$ We have, $\left|A^{-1}\right|=|A|^{-1},\left(A^{T}\right)^{-1}=\left(A^{-1}\right)^{T}$ and $|A| \neq 0$ all are the properties of the inverse of a matrix $A$...
Read More →If A is an invertible matrix, then which of the following is not true
Question: IfAis an invertible matrix, then which of the following is not true (a) $\left(A^{2}\right)^{-1}=\left(A^{-1}\right)^{2}$ (b) $\left|A^{-1}\right|=|A|^{-1}$ (c) $\left(A^{T}\right)^{-1}=\left(A^{-1}\right)^{T}$ (d) $|A| \neq 0$ Solution: (a) $\left(A^{2}\right)^{-1}=\left(A^{-1}\right)^{2}$ We have, $\left|A^{-1}\right|=|A|^{-1},\left(A^{T}\right)^{-1}=\left(A^{-1}\right)^{T}$ and $|A| \neq 0$ all are the properties of the inverse of a matrix $A$...
Read More →As the number of tosses of a coin increases,
Question: As the number of tosses of a coin increases, the ratio of the number of heads to the total number of tosses will be . Is it correct? If not, write the correct one. Solution: No, since the number of coin increases, the ratio of the number of heads to the total number of tosses will be nearer to but not exactly ....
Read More →A solid metallic right circular cone 20 cm high and whose vertical angle is 60°,
Question: A solid metallic right circular cone 20 cm high and whose vertical angle is 60, is cut into two parts at the middle of its height by a plane parallel to its base. If the frustum so obtained be drawn into a wire of diameter $\frac{1}{12} \mathrm{~cm}$, then find the length of the wire. Solution: We have, Height of the solid metallic cone, $H=20 \mathrm{~cm}$, Height of the frustum, $h=\frac{20}{2}=10 \mathrm{~cm}$ and Radius of the wire $=\frac{1}{24} \mathrm{~cm}$ Let the length of the...
Read More →Jasmine allows 4% discount on the marked price of her goods and still earns a profit of 20%
Question: Jasmine allows 4% discount on the marked price of her goods and still earns a profit of 20%. What is the cost price of a shirt for her marked at Rs 850? Solution: Given, MP of the shirt $=$ Rs. 850 Discount $=4 \%$ Discount allowed $=$ Rs. $\left(\frac{4}{100} \times 850\right)$ $=$ Rs. 34 Thus, SP of the shirt $=$ Rs. $(850-34)=$ Rs. 816 Now, Profi $t$ earned by Jasmine $=20 \%$ Thus, CP $=\frac{100 \times \text { SP }}{(100+\text { Profit } \%)}$ $=$ Rs. $\left(\frac{100 \times 816}{...
Read More →Find the inverse of each of the following matrices by using elementary row transformations:
Question: Find the inverse of each of the following matrices by using elementary row transformations: $\left[\begin{array}{ccc}2 -3 5 \\ 3 2 -4 \\ 1 1 -2\end{array}\right]$ Solution: Let $A=\left[\begin{array}{ccc}2 -3 5 \\ 3 2 -4 \\ 1 1 -2\end{array}\right]$ $A=I A$ $\Rightarrow\left[\begin{array}{ccc}2 -3 5 \\ 3 2 -4 \\ 1 1 -2\end{array}\right]=\left[\begin{array}{lll}1 0 0 \\ 0 1 0 \\ 0 0 1\end{array}\right] A$ Applying $R_{1} \leftrightarrow R_{3}$ $\Rightarrow\left[\begin{array}{ccc}1 1 -2 ...
Read More →Can the experimental probability
Question: Can the experimental probability of an event be greater than 1? Justify your answer. Solution: No, since the number of trials in which the event can happen cannot be greater than the total number of trials....
Read More →Can the experimental probability
Question: Can the experimental probability of an event be a negative number? If not, why? Solution: No, since the number of trials in which the event can happen cannot be negative and the total number of trials is always positive....
Read More →The class marks of a continuous distribution are 1.04,
Question: The class marks of a continuous distribution are 1.04, 1.14, 1.24, 1.34, 1.44,1.54 and 1.64. Is it correct to say the last interval will be 1.55-1.73? Justify your answer. Solution: It is not correct. Because the difference between two consecutive class marks should be equal to the class size. Here, difference between two consecutive marks is 0.1 and class size of 1.55-1.73 is 0.18, which are not equal....
Read More →What price should Aslam mark on a pair of shoes which costs him Rs 1200
Question: What price should Aslam mark on a pair of shoes which costs him Rs 1200 so as to gain 12% after allowing a discount of 16%? Solution: Given, CP of the pair of shoes $=$ Rs. 1470 Gain $=12 \%$ Discount $=16 \%$ So, $S P=$ Rs. $\left(\frac{100+\text { Gain }}{100} \times C P\right)$ $=$ Rs. $\left(\frac{100+12}{100} \times 1200\right)$ $=$ Rs. 1344 Now, SP of the pair of shoes $=$ Rs. 1344 Discount $=16 \%$ So, $\mathrm{MP}=$ Rs. $\left(\frac{100 \times \mathrm{SP}}{100-\text { Discount ...
Read More →Is it correct to say that in a histogram,
Question: Is it correct to say that in a histogram, the area of each rectangle is proportional to the class size of the corresponding class interval? If not, correct the statement. Solution: It is not correct. Because in a histogram, the area of each rectangle is proportional to the corresponding frequency of its class....
Read More →A child says that the median of 3, 14, 18, 20
Question: A child says that the median of 3, 14, 18, 20 and 5 is 18. What does not the child understand about finding the median? Solution: The child does not understand, that data has to be arranged in ascending or descending order before finding the median...
Read More →The height of a right circular cone is 20 cm.
Question: The height of a right circular cone is $20 \mathrm{~cm}$. A small cone is cut off at the top by a plane parallel to the base. If its volume be $\frac{1}{8}$ of the volume of the givencone, then at what height above the base is the section made? Solution: We have, Height of the given cone, $H=20 \mathrm{~cm}$ Let the radius of the given cone be $R$, the height of the smaller cone be $h$ and the radius of the smaller cone be $r$. Now, in $\Delta \mathrm{AQD}$ and $\Delta \mathrm{APC}$, $...
Read More →In a diagnostic test in mathematics given to students,
Question: In a diagnostic test in mathematics given to students, the following marks (out of 100) are recorded 46, 52, 48, 11, 41, 62, 54, 53, 96, 40, 98 and 44. Which average will be a good representative of the above data and why? Solution: Median will be a good representative of the data, because each value occurs once. the data is influenced by extreme values....
Read More →Find the inverse of each of the following matrices by using elementary row transformations:
Question: Find the inverse of each of the following matrices by using elementary row transformations: $\left[\begin{array}{ccc}1 2 3 \\ 2 5 7 \\ -2 -4 -5\end{array}\right]$ Solution: $A=\mid A$ $A=\left[\begin{array}{ccc}1 2 3 \\ 2 5 7 \\ -2 -4 -5\end{array}\right]$ $\left[\begin{array}{ccc}1 2 3 \\ 2 5 7 \\ -2 -4 -5\end{array}\right]=\left[\begin{array}{lll}1 0 0 \\ 0 1 0 \\ 0 0 1\end{array}\right] A$ $R_{2} \rightarrow R_{2}-2 R_{1}$ $\left[\begin{array}{ccc}1 2 3 \\ 0 1 1 \\ -2 -4 -5\end{arra...
Read More →Refer to Q. 29. The probability that
Question: Refer to Q. 29. The probability that bulbs selected randomly from the lot has life less than 900 h, is (a)11/40 (b)5/16 (c)7/16 (d)9/16 Solution: (d)Total number of bulbs in a lot, n(S) = 80 Number of bulbs whose life time is less than 900 h, n(E) = 10 + 12 + 23 = 45 Probability that bulbs has life time less than 900 h =n(E)/n(S) = 45/80 = 9/16 Hence, the probability that bulb has life time less than 900 is 9/16....
Read More →Jyoti and Meena run a ready-made garment shop.
Question: Jyoti and Meena run a ready-made garment shop. They mark the garments at such a price that even after allowing a discount of 12.5%, they make a profit of 10%. Find the marked price of a suit which costs them Rs 1470 Solution: Given, $C \mathrm{P}$ of the suit $=$ Rs. 1470 Gain $=10 \%$ So, $S P=$ Rs. $\left(\frac{100+\text { Gain }}{100} \times C P\right)$ $=$ Rs. $\left(\frac{100+10}{100} \times 1470\right)$ $=$ Rs. 1617 Now, $\mathrm{SP}=$ Rs. 1617 Discount $=12.5 \%$ So, $\mathrm{MP...
Read More →In a survey of 364 children aged 19-36 months,
Question: In a survey of 364 children aged 19-36 months, it was found that 91 liked to eat potato chips. If a child is selected at random, the probability that he/she does not like to eat potato chips, is (a)0.25 (b)0.50 (c)0.75 (d)0.80 Solution: (c)Total number of survey childrens age from 19-36 months, n(S) = 364 In those of them 91 out of them liked to eat potato chips. Number of children who do not like to eat potato chips, n(E) = 364 91 = 273 Probability that he/she does not like to eat pot...
Read More →In a sample study of 642 people,
Question: In a sample study of 642 people, it was found that 514 people have a high school certificate. If a person is selected at random, the probability that the person has a high school certificate, is (a)0.5 (b)0.6 (c)0.7 (d)0.8 Solution: (d)The total number of people in sample study, n(S) = 642. The number of people who have high school certificate, n(E) = 514. So, the probability that the person selected has a high school certificate = n(E)/n(S)=514/642 = 0.8 Hence, the probability that th...
Read More →A cycle merchant allows 20% discount on the marked price of the cycles and still makes a profit of 20%.
Question: A cycle merchant allows 20% discount on the marked price of the cycles and still makes a profit of 20%. If he gains Rs 360 over the sale of one cycle, find the marked price of the cycle. Solution: Given, Gain on one cycle $=$ Rs. 360 Gain $=20 \%$ Gain $\%=\frac{\text { Gain }}{\mathrm{CP}} \times 100$ $20=\frac{360}{\mathrm{CP}} \times 100$ $\mathrm{CP}=$ Rs. 1800 $\mathrm{SP}=\frac{100+\text { Gain } \%}{100} \times \mathrm{CP}$ $\mathrm{SP}=\frac{120}{100} \times 1800=$ Rs. 2160 $\m...
Read More →The mode of given data 15,
Question: The mode of given data 15, 14, 19, 20, 14, 15, 16, 14, 15, 18, 14, 19, 15,17 and 15 is (a)14 (b)15 (c)16 (d)17 Thinking Process Find the maximum number of repeated observation to get the value of mode. Solution: (b)We first arrange the given data in ascending order as follows 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 17, 18, 19, 19, 20 From above, we see that 15 occurs most frequently i.e., 5 times. Hence, the mode of the given data is 15....
Read More →Find the inverse of each of the following matrices by using elementary row transformations:
Question: Find the inverse of each of the following matrices by using elementary row transformations: $\left[\begin{array}{ccc}-1 1 2 \\ 1 2 3 \\ 3 1 1\end{array}\right]$ Solution: Let $A=\left[\begin{array}{ccc}-1 1 2 \\ 1 2 3 \\ 3 1 1\end{array}\right]$ To find inverse, first write $A=I A$. i.e., $\left[\begin{array}{ccc}-1 1 2 \\ 1 2 3 \\ 3 1 1\end{array}\right]=\left[\begin{array}{lll}1 0 0 \\ 0 1 0 \\ 0 0 1\end{array}\right] A$ $\Rightarrow\left[\begin{array}{ccc}1 -1 -2 \\ 1 2 3 \\ 3 1 1\e...
Read More →A solid cone of base radius 10 cm is cut into two parts through the midpoint of its height, by a plane parallel to its base.
Question: A solid cone of base radius 10 cm is cut into two parts through the midpoint of its height, by a plane parallel to its base. Find the ratio of the volumes of the two parts of the cone. Solution: We have, Radius of solid cone, $R=\mathrm{CP}=10 \mathrm{~cm}$, Let the height of the solid cone be, $\mathrm{AP}=H$, the radius of the smaller cone, $\mathrm{QD}=r$ and the height of the smaller cone be, $\mathrm{AQ}=h$. Also, $\mathrm{AQ}=\frac{\mathrm{AP}}{2}$ i. e. $h=\frac{H}{2}$ or $H=2 h...
Read More →For drawing a frequency polygon of a continuous
Question: For drawing a frequency polygon of a continuous frequency distribution, we plot the points whose ordinates are the frequencies of the respective classes and abscissae are, respectively (a)upper limits of the classes (b)lower limits of the classes (c)class marks of the classes (d)upper limits of proceeding classes Solution: (c)Class marks i.e., the mid-point of the classes are abscissa of the points, which we plot for frequency polygon....
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