The variance of 15 observations is 6.
Question: The variance of 15 observations is 6. If each observation is increased by 8, find the variance of the resulting observations. Solution: Let the observations are $\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3}, \mathrm{x}_{4}, \ldots, \mathrm{x}_{15}$ and Let mean $=\overline{\mathrm{X}}$ Given: Variance $=6$ and $n=15$ We know that, Variance, $\sigma^{2}=\frac{1}{\mathrm{n}} \sum\left(\mathrm{x}_{\mathrm{i}}-\overline{\mathrm{x}}\right)^{2}$ Putting the given values, we get $6=\frac{1}...
Read More →Consider an extended object immersed in water contained
Question: Consider an extended object immersed in water contained in a plane trough. When seen from close to the edge of the trough the object looks distorted because (a) the apparent depth of the points close to the edge is nearer the surface of the water compared to the points away from the edge. (b) the angle subtended by the image of the object at the eye is smaller than the actual angle subtended by the object in the air. (c) some of the points of the object far away from the edge may not b...
Read More →There are certain material developed in laboratories
Question: There are certain material developed in laboratories which have a negative refractive index (Fig. 9.3). A ray incident from the air (medium 1) into such a medium (medium 2) shall follow a path given by Solution: (a) The speed of the car in the rear is 65 km h1....
Read More →If the standard deviation of the numbers 2, 3, 2x, 11 is 3.5,
Question: If the standard deviation of the numbers 2, 3, 2x, 11 is 3.5, calculate the possible values of x. Solution: Given: Standard Deviation, $\sigma=3.5$ and Numbers are 2, 3, 2x, 11 We know that, Mean $(\overline{\mathrm{x}})=\frac{\text { Sum of observations }}{\text { Total number of observations }}$ $=\frac{2+3+2 x+11}{4}$ $=\frac{16+2 x}{4}$ $\overline{\mathrm{x}}=\frac{8+\mathrm{x}}{2}$ Variance, $\sigma^{2}=\frac{1}{\mathrm{n}} \sum\left(\mathrm{x}_{\mathrm{i}}-\overline{\mathrm{x}}\r...
Read More →Using short cut method, find the mean, variation and standard deviation for the data :
Question: Using short cut method, find the mean, variation and standard deviation for the data : Solution: Here, we apply the step deviation method with A = 50 and h = 10 To find: MEAN Now, $\operatorname{Mean}(\overline{\mathrm{x}})=\mathrm{a}+\mathrm{h}\left(\frac{\sum \mathrm{f}_{\mathrm{i}} \mathrm{y}_{\mathrm{i}}}{\sum \mathrm{f}_{\mathrm{i}}}\right)$ $\Rightarrow \overline{\mathrm{x}}=50+10\left(\frac{18}{45}\right)$ $\Rightarrow \overline{\mathrm{x}}=50+\frac{2 \times 18}{9}$ $\Rightarrow...
Read More →Using short cut method, find the mean, variation and standard deviation for the data :
Question: Using short cut method, find the mean, variation and standard deviation for the data : Solution: Here, we apply the step deviation method with A = 65 and h = 10 To find: MEAN Now, $\operatorname{Mean}(\overline{\mathrm{x}})=\mathrm{a}+\mathrm{h}\left(\frac{\sum \mathrm{f}_{\mathrm{i}} \mathrm{y}_{\mathrm{i}}}{\sum \mathrm{f}_{\mathrm{i}}}\right)$ $\Rightarrow \overline{\mathrm{x}}=65+10\left(\frac{-15}{50}\right)$ $\Rightarrow \overline{\mathrm{x}}=65-\frac{150}{50}$ $\Rightarrow \over...
Read More →Using short cut method, find the mean, variation and standard deviation for the data :
Question: Using short cut method, find the mean, variation and standard deviation for the data : Solution: Here, we apply the step deviation method with A = 25 and h = 10 To find: MEAN Now, $\quad$ Mean $(\overline{\mathrm{x}})=\mathrm{a}+\mathrm{h}\left(\frac{\sum \mathrm{f}_{\mathrm{i}} \mathrm{y}_{\mathrm{i}}}{\sum \mathrm{f}_{\mathrm{i}}}\right)$ $\Rightarrow \overline{\mathrm{x}}=25+10\left(\frac{10}{50}\right)$ $\Rightarrow \overline{\mathrm{X}}=25+\frac{100}{50}$ $\Rightarrow \overline{\m...
Read More →Evaluate the integral:
Question: Evaluate the integral: $\int \frac{2 x+5}{x^{2}-x-2} d x$ Solution: $I=\int \frac{2 x+5}{x^{2}-x-2} d x$ As we can see that there is a term of $x$ in numerator and derivative of $x^{2}$ is also $2 x$. So there is a chance that we can make substitution for $x^{2}-x-2$ and I can be reduced to a fundamental integration. As, $\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}^{2}-\mathrm{x}-2\right)=2 \mathrm{x}-1$ $\therefore$ Let, $2 x+5=A(2 x-1)+B$ $\Rightarrow 2 x+5=2 A x-A+B$ On comparing...
Read More →Using short cut method, find the mean, variation and standard deviation for the data :
Question: Using short cut method, find the mean, variation and standard deviation for the data : Solution: To find: MEAN Now, $\operatorname{Mean}(\overline{\mathrm{x}})=\frac{\sum \mathrm{f}_{\mathrm{i}} \mathrm{x}_{\mathrm{i}}}{\sum \mathrm{f}_{\mathrm{i}}}$ $=\frac{2200}{22}$ $=100$ To find: VARIANCE Variance, $\sigma^{2}=\frac{\sum \mathrm{f}_{\mathrm{i}}\left(\mathrm{x}_{\mathrm{i}}-\overline{\mathrm{x}}\right)^{2}}{\mathrm{~N}}$ $=\frac{640}{22}$ $=29.09$ To find: STANDARD DEVIATION Standa...
Read More →Using short cut method, find the mean, variation and standard deviation for the data :
Question: Using short cut method, find the mean, variation and standard deviation for the data : Solution: To find: MEAN Now, $\operatorname{Mean}(\overline{\mathrm{x}})=\frac{\sum \mathrm{f}_{\mathrm{i}} \mathrm{x}_{\mathrm{i}}}{\sum \mathrm{f}_{\mathrm{i}}}$ $=\frac{390}{20}$ $=19.5$ To find: VARIANCE Variance, $\sigma^{2}=\frac{\sum \mathrm{f}_{\mathrm{i}}\left(\mathrm{x}_{\mathrm{i}}-\overline{\mathrm{x}}\right)^{2}}{\mathrm{~N}}$ $=\frac{385}{20}$ $=19.25$ To find: STANDARD DEVIATION Standa...
Read More →Evaluate the integral:
Question: Evaluate the integral: $\int \frac{1-3 x}{3 x^{2}+4 x+2} d x$ Solution: $I=\int \frac{1-3 x}{3 x^{2}+4 x+2} d x$ As we can see that there is a term of $x$ in numerator and derivative of $x^{2}$ is also $2 x$. So there is a chance that we can make substitution for $3 x^{2}+4 x+2$ and I can be reduced to a fundamental integration. As, $\frac{\mathrm{d}}{\mathrm{dx}}\left(3 \mathrm{x}^{2}+4 \mathrm{x}+2\right)=6 \mathrm{x}+4$ $\therefore$ Let, $1-3 x=A(6 x+4)+B$ $\Rightarrow 1-3 x=6 A x+4...
Read More →A car is moving with at a constant speed of 60 km h–1
Question: A car is moving with at a constant speed of 60 km h1 on a straight road. Looking at the rearview mirror, the driver finds that the car following him is at a distance of 100 m and is approaching with a speed of 5 km h 1. In order to keep track of the car in the rear, the driver begins to glance alternatively at the rear and side mirror of his car after every 2 still the other car overtakes. If the two cars were maintaining their speeds, which of the following statement (s) is/are correc...
Read More →The optical density of turpentine is higher
Question: The optical density of turpentine is higher than that of water while its mass density is lower. The figure shows a layer of turpentine floating over water in a container. For which one of the four rays incident on turpentine in the figure, the path shown is correct? (a) 1 (b) 2 (c) 3 (d) 4 Solution: (b) 2...
Read More →The direction of a ray of light incident
Question: The direction of a ray of light incident on a concave mirror as shown by PQ while directions in which the ray would travel after reflection is shown by four rays marked 1, 2, 3, and 4. Which of the four rays correctly shows the direction of reflected ray? (a) 1 (b) 2 (c) 3 (d) 4 Solution: (b) 2...
Read More →Using short cut method, find the mean, variation and standard deviation for the data :
Question: Using short cut method, find the mean, variation and standard deviation for the data : Solution: To find: MEAN Now, $\operatorname{Mean}(\overline{\mathrm{x}})=\frac{\sum \mathrm{f}_{\mathrm{i}} \mathrm{x}_{\mathrm{i}}}{\sum \mathrm{f}_{\mathrm{i}}}$ $=\frac{760}{40}$ $=19$ To find: VARIANCE Variance, $\sigma^{2}=\frac{\sum \mathrm{f}_{\mathrm{i}}\left(\mathrm{x}_{\mathrm{i}}-\overline{\mathrm{x}}\right)^{2}}{\mathrm{~N}}$ $=\frac{1736}{40}$ $=43.4$ To find: STANDARD DEVIATION Standard...
Read More →The phenomena involved in the reflection
Question: The phenomena involved in the reflection of radiowaves by ionosphere is similar to (a) reflection of light by a plane mirror (b) total internal reflection of light in the air during a mirage (c) dispersion of light by water molecules during the formation of a rainbow (d) scattering of light by the particles of air Solution: (b) total internal reflection of light in the air during a mirage...
Read More →The radius of curvature of the curved surface
Question: The radius of curvature of the curved surface of a plano-convex lens is 20 cm. If the refractive index of the material of the lens be 1.5, it will (a) act as a convex lens only for the objects that lie on its curved side (b) act as a concave lens only for the objects that lie on its curved side (c) act as a convex lens irrespective of the side on which the object lies (d) act as a concave lens irrespective of the side on which the object lies Solution: (c) act as a convex lens irrespec...
Read More →You are given four sources of light
Question: You are given four sources of light each one providing a light of a single colour- red, blue, green, and yellow. Suppose the angle of refraction for a beam of yellow light corresponding to a particular angle of incidence at the interface of two media is 90o. Which of the following statements is correct if the source of yellow light is replaced with that of other lights without changing the angle of incidence? (a) the beam of red light would undergo total internal reflection (b) the bea...
Read More →A passenger in an aeroplane shall
Question: A passenger in an aeroplane shall (a) never see a rainbow (b) may see a primary and a secondary rainbow as concentric circles (c) may see a primary and a secondary rainbow as concentric arcs (d) shall never see a secondary rainbow Solution: (b) may see a primary and a secondary rainbow as concentric circles...
Read More →An object approaches a convergent lens from
Question: An object approaches a convergent lens from the left of the lens with a uniform speed 5 m/s and stops at the focus. The image (a) moves away from the lens with a uniform speed 5 m/s (b) moves away from the lens with a uniform acceleration (c) moves away from the lens with a non-uniform acceleration (d) moves towards the lens with a non-uniform acceleration Solution: (c) moves away from the lens with a non-uniform acceleration...
Read More →A short pulse of white light is incident
Question: A short pulse of white light is incident from air to a glass slab at normal incidence. After travelling through the slab, the first colour to emerge is (a) blue (b) green (c) violet (d) red Solution: (d) red...
Read More →A ray of light incident at an angle θ on a refracting
Question: A ray of light incident at an angle on a refracting face of a prism emerges from the other face normally. If the angle of the prism is 5oand the prism is made of a material of refractive index 1.5, the angle of incidence is (a) 7.5o (b) 5o (c) 15o (d) 2.5o Solution: (a) 7.5o...
Read More →Using short cut method, find the mean, variation and standard deviation for the data
Question: Using short cut method, find the mean, variation and standard deviation for the data Solution: To find: MEAN Now, $\operatorname{Mean}(\overline{\mathrm{x}})=\frac{\sum \mathrm{f}_{\mathrm{i}} \mathrm{x}_{\mathrm{i}}}{\sum \mathrm{f}_{\mathrm{i}}}$ $=\frac{420}{30}$ $=14$ To find: VARIANCE Variance, $\sigma^{2}=\frac{\sum \mathrm{f}_{\mathrm{i}}\left(\mathrm{x}_{\mathrm{i}}-\overline{\mathrm{x}}\right)^{2}}{\mathrm{~N}}$ $=\frac{1374}{30}$ $=45.8$ To find: STANDARD DEVIATION Standard D...
Read More →Find the mean, variance and standard deviation for the first six odd natural numbers
Question: Find the mean, variance and standard deviation for the first six odd natural numbers Solution: Odd natural numbers = 1, 3, 5, 7, 9, First Six Odd Natural Numbers = 1, 3, 5, 7, 9, 11 To find: MEAN We know that $\operatorname{Mean}(\overline{\mathrm{x}})=\frac{\text { Sum of observations }}{\text { Total number of observations }}$ $=\frac{1+3+5+7+9+11}{6}$ $=\frac{36}{6}$ $\overline{\mathrm{x}}=6$ To find: VARIANCE Variance, $\sigma^{2}=\frac{\sum\left(\mathrm{x}_{\mathrm{i}}-\overline{\...
Read More →Evaluate the integral:
Question: Evaluate the integral: $\int \frac{2 x}{2+x-x^{2}} d x$ Solution: $1=\int \frac{2 x}{2+x-x^{2}} d x$ As we can see that there is a term of $x$ in numerator and derivative of $x^{2}$ is also $2 x$. So there is a chance that we can make substitution for $-x^{2}+x+2$ and I can be reduced to a fundamental integration. As, $\frac{\mathrm{d}}{\mathrm{dx}}\left(-\mathrm{x}^{2}+\mathrm{x}+2\right)=-2 \mathrm{x}+1$ $\therefore$ Let, $2 x=A(-2 x+1)+B$ $\Rightarrow 2 x=-2 A x+A+B$ On comparing bo...
Read More →