Using the property of determinants and without expanding, prove that:
Question: Using the property of determinants and without expanding, prove that: $\left|\begin{array}{lll}b+c q+r y+z \\ c+a r+p z+x \\ a+b p+q x+y\end{array}\right|=2\left|\begin{array}{lll}a p x \\ b q y \\ c r z\end{array}\right|$ Solution: $\Delta=\left|\begin{array}{ccc}b+c q+r y+z \\ c+a r+p z+x \\ a+b p+q x+y\end{array}\right|$ $=\left|\begin{array}{lll}b+c q+r y+z \\ c+a r+p z+x \\ a p x\end{array}\right|+\left|\begin{array}{lll}b+c q+r y+z \\ c+a r+p z+x \\ b q y\end{array}\right|$ $=\De...
Read More →At what angle should a ray of light be incident on the face
Question: At what angle should a ray of light be incident on the face of a prism of refracting angle 60 so that it just suffers total internal reflection at the other face? The refractive index of the material of the prism is 1.524. Solution: The incident, refracted, and emergent rays associated with a glass prism ABC are shown in the given figure. Angle of prism,A= 60 Refractive index of the prism,= 1.524 $i_{1}=$ Incident angle $r_{1}=$ Refracted angle $r_{2}=$ Angle of incidence at the face $...
Read More →How many terms of the A.P. are needed to give the sum –25?
Question: How many terms of the A.P. $-6,-\frac{11}{2},-5, \ldots$ are needed to give the sum $-25 ?$ Solution: Let the sum ofnterms of the given A.P. be 25. It is known that, $S_{n}=\frac{n}{2}[2 a+(n-1) d]$, where $n=$ number of terms, $a=$ first term, and $d=$ common difference Here, $a=-6$ $d=-\frac{11}{2}+6=\frac{-11+12}{2}=\frac{1}{2}$ Therefore, we obtain $-25=\frac{n}{2}\left[2 \times(-6)+(n-1)\left(\frac{1}{2}\right)\right]$ $\Rightarrow-50=n\left[-12+\frac{n}{2}-\frac{1}{2}\right]$ $\R...
Read More →In an A.P, the first term is 2 and the sum of the first five terms is one-fourth of the next five terms. Show that 20th term is –112.
Question: In an A.P, the first term is 2 and the sum of the first five terms is one-fourth of the next five terms. Show that$20^{\text {th }}$ term is $-112$. Solution: First term = 2 Letdbe the common difference of the A.P. Therefore, the A.P. is 2, 2 +d, 2 + 2d, 2 + 3d, Sum of first five terms = 10 + 10d Sum of next five terms = 10 + 35d According to the given condition, $10+10 d=\frac{1}{4}(10+35 d)$ $\Rightarrow 40+40 d=10+35 d$ $\Rightarrow 30=-5 d$ $\Rightarrow d=-6$ $\therefore a_{20}=a+(...
Read More →A certain salt X, gives the following results.
Question: A certain saltX, gives the following results. (i) Its aqueous solution is alkaline to litmus. (ii) It swells up to a glassy materialYon strong heating. (iii) When conc. H2SO4is added to a hot solution ofX, white crystal of an acidZseparates out. Write equations for all the above reactions and identifyX,YandZ. Solution: The givensalt is alkaline to litmus. Therefore,Xis a salt of a strong base and a weak acid. Also, whenXis strongly heated, it swells to form substanceY. Therefore,Xmust ...
Read More →(a) Determine the ‘effective focal length’
Question: (a)Determine the effective focal length of the combination of the two lenses in Exercise 9.10, if they are placed 8.0 cm apart with their principal axes coincident. Does the answer depend on which side of the combination a beam of parallel light is incident? Is the notion of effective focal length of this system useful at all? (b)An object 1.5 cm in size is placed on the side of the convex lens in the arrangement (a) above. The distance between the object and the convex lens is 40 cm. ...
Read More →Using the property of determinants and without expanding, prove that:
Question: Using the property of determinants and without expanding, prove that: $\left|\begin{array}{lll}1 b c a(b+c) \\ 1 c a b(c+a) \\ 1 a b c(a+b)\end{array}\right|=0$ Solution: $\Delta=\left|\begin{array}{lll}1 b c a(b+c) \\ 1 c a b(c+a) \\ 1 a b c(a+b)\end{array}\right|$ By applying $\mathrm{C}_{3} \rightarrow \mathrm{C}_{3}+\mathrm{C}_{2}$, we have: $\Delta=\left|\begin{array}{lll}1 b c a b+b c+c a \\ 1 c a a b+b c+c a \\ 1 a b a b+b c+c a\end{array}\right|$ Here, two columns C1and C3are p...
Read More →What do you understand by
Question: What do you understand by (a) inert pair effect (b) allotropy and (c) catenation? Solution: (a)Inert pair effect As one moves down the group, the tendency of $s$-block electrons to participate in chemical bonding decreases. This effect is known as inert pair effect. In case of group 13 elements, the electronic configuration is $n s^{2} n p^{1}$ and their group valency is $+3$. However, on moving down the group, the $+1$ oxidation state becomes more stable. This happens because of the p...
Read More →Using the property of determinants and without expanding, prove that:
Question: Using the property of determinants and without expanding, prove that: $\left|\begin{array}{lll}2 7 65 \\ 3 8 75 \\ 5 9 86\end{array}\right|=0$ Solution: $\left|\begin{array}{lll}2 7 65 \\ 3 8 75 \\ 5 9 86\end{array}\right|=\left|\begin{array}{lll}2 1 65+2 \\ 3 8 72+3 \\ 5 9 81+5\end{array}\right|$ $=\left|\begin{array}{lll}2 7 63 \\ 3 8 72 \\ 5 9 81\end{array}\right|+\left|\begin{array}{lll}2 7 2 \\ 3 8 3 \\ 5 9 5\end{array}\right|$ $=\left|\begin{array}{lll}2 7 9(7) \\ 3 8 9(8) \\ 5 9...
Read More →Find the sum of all natural numbers lying between 100 and 1000, which are multiples of 5.
Question: Find the sum of all natural numbers lying between 100 and 1000, which are multiples of 5. Solution: The natural numbers lying between 100 and 1000, which are multiples of 5, are 105, 110, 995. Here, $a=105$ and $d=5$ $a+(n-1) d=995$ $\Rightarrow 105+(n-1) 5=995$ $\Rightarrow(n-1) 5=995-105=890$ $\Rightarrow 105+(n-1) 5=995$ $\Rightarrow n-1=178$ $\Rightarrow n=179$ $\therefore S_{n}=\frac{179}{2}[2(105)+(179-1)(5)]$ $=\frac{179}{2}[2(105)+(178)(5)]$ $=179[105+(89) 5]$ $=(179)(550)$ $=(...
Read More →When metal X is treated with sodium hydroxide, a white precipitate (A) is obtained, which is soluble in excess of NaOH to give soluble complex (B).
Question: When metal X is treated with sodium hydroxide, a white precipitate (A) is obtained, which is soluble in excess of NaOH to give soluble complex (B). Compound (A) is soluble in dilute HCl to form compound (C). The compound (A) when heated strongly gives (D), which is used to extract metal. Identify (X), (A), (B), (C) and (D). Write suitable equations to support their identities. Solution: The given metal X gives a white precipitate with sodium hydroxide and the precipitate dissolves in e...
Read More →In some of the reactions thallium resembles aluminium, whereas in others it resembles with group I metals.
Question: In some of the reactions thallium resembles aluminium, whereas in others it resembles with group I metals. Support this statement by giving some evidences. Solution: Thallium belongs to group 13 of the periodic table. The most common oxidation state for this group is +3. However, heavier members of this group also displaythe +1 oxidation state. This happens because of the inert pair effect. Aluminium displays the +3 oxidation state and alkali metals display the +1 oxidation state. Thal...
Read More →In some of the reactions thallium resembles aluminium, whereas in others it resembles with group I metals.
Question: In some of the reactions thallium resembles aluminium, whereas in others it resembles with group I metals. Support this statement by giving some evidences. Solution: Thallium belongs to group 13 of the periodic table. The most common oxidation state for this group is +3. However, heavier members of this group also displaythe +1 oxidation state. This happens because of the inert pair effect. Aluminium displays the +3 oxidation state and alkali metals display the +1 oxidation state. Thal...
Read More →Using the property of determinants and without expanding, prove that:
Question: Using the property of determinants and without expanding, prove that: $\left|\begin{array}{lll}a-b b-c c-a \\ b-c c-a a-b \\ c-a a-b b-c\end{array}\right|=0$ Solution: $\Delta=\left|\begin{array}{lll}a-b b-c c-a \\ b-c c-a a-b \\ c-a a-b b-c\end{array}\right|$ Applying $R_{1} \rightarrow R_{1}+R_{2}$, we have: $\Delta=\left|\begin{array}{lll}a-c b-a c-b \\ b-c c-a a-b \\ -(a-c) -(b-a) -(c-b)\end{array}\right|$ $=-\left|\begin{array}{ccc}a-c b-a c-b \\ b-c c-a a-b \\ a-c b-a c-b\end{arr...
Read More →Find the sum of odd integers from 1 to 2001.
Question: Find the sum of odd integers from 1 to 2001. Solution: The odd integers from 1 to 2001 are 1, 3, 5, 1999, 2001. This sequence forms an A.P. Here, first term,a= 1 Common difference,d= 2 Here, $a+(n-1) d=2001$ $\Rightarrow 1+(n-1)(2)=2001$ $\Rightarrow 2 n-2=2000$ $\Rightarrow n=1001$ $S_{n}=\frac{n}{2}[2 a+(n-1) d]$ $\therefore S_{n}=\frac{1001}{2}[2 \times 1+(1001-1) \times 2]$ $=\frac{1001}{2}[2+1000 \times 2]$ $=\frac{1001}{2} \times 2002$ $=1001 \times 1001$ $=1002001$ Thus, the sum...
Read More →A screen is placed 90 cm from an object.
Question: A screen is placed 90 cm from an object. The image of the object on the screen is formed by a convex lens at two different locations separated by 20 cm. Determine the focal length of the lens. Solution: Distance between the image (screen) and the object,D= 90 cm Distance between two locations of the convex lens,d= 20 cm Focal length of the lens =f Focal length is related todandDas: $f=\frac{D^{2}-d^{2}}{4 D}$ $=\frac{(90)^{2}-(20)^{2}}{4 \times 90}=\frac{770}{36}=21.39 \mathrm{~cm}$ Th...
Read More →The Fibonacci sequence is defined by
Question: The Fibonacci sequence is defined by $\mathrm{l}=\mathrm{a}_{1}=\mathrm{a}_{2}$ and $\mathrm{a}_{\mathrm{n}}=\mathrm{a}_{\mathrm{n}-1}+\mathrm{a}_{\mathrm{n}-2}, \mathrm{n}2$ Find $\frac{\mathrm{a}_{\mathrm{n}+1}}{\mathrm{a}_{\mathrm{n}}}$, for $\mathrm{n}=1,2,3,4,5$ Solution: $\mathrm{l}=\mathrm{a}_{1}=\mathrm{a}_{2}$ $\mathrm{a}_{\mathrm{n}}=\mathrm{a}_{\mathrm{n}-1}+\mathrm{a}_{\mathrm{n}-2}, \mathrm{n}2$ $\therefore \mathrm{a}_{3}=\mathrm{a}_{2}+\mathrm{a}_{1}=1+\mathrm{l}=2$ $\mat...
Read More →Classify following oxides as neutral, acidic, basic or amphoteric:
Question: (a)Classify following oxides as neutral, acidic, basic or amphoteric: $\mathrm{CO}, \mathrm{B}_{2} \mathrm{O}_{3}, \mathrm{SiO}_{2}, \mathrm{CO}_{2}, \mathrm{Al}_{2} \mathrm{O}_{3}, \mathrm{PbO}_{2}, \mathrm{Tl}_{2} \mathrm{O}_{3}$ (b)Write suitable chemical equations to show their nature. Solution: (1)CO = Neutral (2) $\mathrm{B}_{2} \mathrm{O}_{3}=$ Acidic Being acidic, it reacts with bases to form salts. It reacts with NaOH to form sodium metaborate. $\mathrm{B}_{2} \mathrm{O}_{3}+2...
Read More →The image of a small electric bulb fixed on the wall
Question: The image of a small electric bulb fixed on the wall of a room is to be obtained on the opposite wall 3 m away by means of a large convex lens. What is the maximum possible focal length of the lens required for the purpose? Solution: Distance between the object and the image,d= 3 m Maximum focal length of the convex lens $=f_{\max }$ For real images, the maximum focal length is given as: $f_{\max }=\frac{d}{4}$ $=\frac{3}{4}=0.75 \mathrm{~m}$ Hence, for the required purpose, the maximu...
Read More →Using the property of determinants and without expanding, prove that:
Question: Using the property of determinants and without expanding, prove that: $\left|\begin{array}{lll}x a x+a \\ y b y+b \\ z c z+c\end{array}\right|=0$ Solution: $\left|\begin{array}{lll}x a x+a \\ y b y+b \\ z c z+c\end{array}\right|=0$ $\left|\begin{array}{lll}x a x+a \\ y b y+b \\ z c z+c\end{array}\right|=\left|\begin{array}{lll}x a x \\ y b y \\ z c z\end{array}\right|+\left|\begin{array}{lll}x a a \\ y b b \\ z c c\end{array}\right|=0+0=0$ [Here, the two columns of the determinants are...
Read More →Write the first five terms of the following sequence and obtain the corresponding series:
Question: Write the first five terms of the following sequence and obtain the corresponding series: $a_{1}=a_{2}=2, a_{n}=a_{a-1}-1, n2$ Solution: $a_{1}=a_{2}=2, a_{n}=a_{n-1}-1, n2$ $\Rightarrow a_{3}=a_{2}-1=2-1=1$ $a_{4}=a_{3}-1=1-1=0$ $a_{5}=a_{4}-1=0-1=-1$ Hence, the first five terms of the sequence are 2, 2, 1, 0, and 1. The corresponding series is 2 + 2 + 1 + 0 + (1) +...
Read More →Answer the following questions:
Question: Answer the following questions: (a)You have learnt that plane and convex mirrors produce virtual images of objects. Can they produce real images under some circumstances? Explain. (b)A virtual image, we always say, cannot be caught on a screen. Yet when we see a virtual image, we are obviously bringing it on to the screen (i.e., the retina) of our eye. Is there a contradiction? (c)A diver under water, looks obliquely at a fisherman standing on the bank of a lake. Would the fisherman lo...
Read More →Write the first five terms of the following sequence and obtain the corresponding series:
Question: Write the first five terms of the following sequence and obtain the corresponding series: $a_{1}=-1, a_{n}=\frac{a_{n-1}}{n}, n \geq 2$ Solution: $a_{1}=-1, a_{n}=\frac{a_{n-1}}{n}, n \geq 2$ $\Rightarrow a_{2}=\frac{a_{1}}{2}=\frac{-1}{2}$ $a_{3}=\frac{a_{2}}{3}=\frac{-1}{6}$ $a_{4}=\frac{a_{3}}{4}=\frac{-1}{24}$ $a_{5}=\frac{a_{4}}{4}=\frac{-1}{120}$ Hence, the first five terms of the sequence are $-1, \frac{-1}{2}, \frac{-1}{6}, \frac{-1}{24}$, and $\frac{-1}{120}$. The correspondin...
Read More →If, then x is equal to
Question: If $\left|\begin{array}{cc}x 2 \\ 18 x\end{array}\right|=\left|\begin{array}{cc}6 2 \\ 18 6\end{array}\right|$, then $x$ is equal to (A) 6 (B) $\pm 6$ (C) $-6$ (D) 0 Solution: Answer: B $\left|\begin{array}{cc}x 2 \\ 18 x\end{array}\right|=\left|\begin{array}{cc}6 2 \\ 18 6\end{array}\right|$ $\Rightarrow x^{2}-36=36-36$ $\Rightarrow x^{2}-36=0$ $\Rightarrow x^{2}=36$ $\Rightarrow x=\pm 6$ Hence, the correct answer is B....
Read More →Write the first five terms of the following sequence and obtain the corresponding series:
Question: Write the first five terms of the following sequence and obtain the corresponding series: $a_{1}=3, a_{n}=3 a_{n-1}+2$ for all $n1$ Solution: $a_{1}=3, a_{n}=3 a_{n-1}+2$ for all $n1$ $\Rightarrow a_{2}=3 a_{1}+2=3(3)+2=11$ $a_{3}=3 a_{2}+2=3(11)+2=35$ $a_{4}=3 a_{3}+2=3(35)+2=107$ $a_{5}=3 a_{4}+2=3(107)+2=323$ Hence, the first five terms of the sequence are 3, 11, 35, 107, and 323. The corresponding series is 3 + 11 + 35 + 107 + 323 +...
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