Using Cofactors of elements of second row, evaluate.
Question: Using Cofactors of elements of second row, evaluate $\Delta=\left|\begin{array}{lll}5 3 8 \\ 2 0 1 \\ 1 2 3\end{array}\right|$. Solution: The given determinant is $\left|\begin{array}{lll}5 3 8 \\ 2 0 1 \\ 1 2 3\end{array}\right|$. We have: $\mathrm{M}_{21}=\left|\begin{array}{ll}3 8 \\ 2 3\end{array}\right|=9-16=-7$ $\therefore \mathrm{A}_{21}=$ cofactor of $\mathrm{a}_{21}=(-1)^{2+1} \mathrm{M}_{21}=7$ $\mathrm{M}_{22}=\left|\begin{array}{ll}5 8 \\ 1 3\end{array}\right|=15-8=7$ $\the...
Read More →The 5th, 8th and 11th terms of a G.P. are p, q and s, respectively. Show that q2 = ps.
Question: The $5^{\text {th }}, 8^{\text {th }}$ and $11^{\text {th }}$ terms of a G.P. are $p, q$ and $s$, respectively. Show that $q^{2}=p s$. Solution: Letabe the first term andrbe the common ratio of the G.P. According to the given condition, $a_{5}=a r^{5-1}=a r^{4}=p \ldots(1)$ $a_{8}=a r^{8-1}=a r^{7}=q \ldots(2)$ $a_{11}=a r^{11-1}=a r^{10}=s \ldots(3)$ Dividing equation (2) by (1), we obtain $\frac{a r^{7}}{a r^{4}}=\frac{q}{p}$ $r^{3}=\frac{q}{p}$ $\ldots(4)$ Dividing equation (3) by (...
Read More →In a Young’s double-slit experiment, the slits are separated by 0.28 mm and the screen is placed 1.4 m away.
Question: In a Youngs double-slit experiment, the slits are separated by 0.28 mm and the screen is placed 1.4 m away. The distance between the central bright fringe and the fourth bright fringe is measured to be 1.2 cm. Determine the wavelength of light used in the experiment. Solution: Distance between the slits, $d=0.28 \mathrm{~mm}=0.28 \times 10^{-3} \mathrm{~m}$ Distance between the slits and the screen,D= 1.4 m Distance between the central fringe and the fourth (n= 4) fringe, u= 1.2 cm = 1...
Read More →Identify the functional groups in the following compounds
Question: Identify the functional groups in the following compounds (a) (b) (c) Solution: The functional groups present in the given compounds are: (a)Aldehyde (CHO), Hydroxyl(OH), Methoxy (OMe), $\mathrm{C}=\mathrm{C}$ double bond $(-\stackrel{1}{\mathrm{C}}=\stackrel{1}{\mathrm{C}}-)$ (b)Amino (NH2); primary amine, Ester (-O-CO-), Triethylamine (N(C2H5)2); tertiary amine (c)Nitro (NO2), $\mathrm{C}=\mathrm{C}$ double bond $(-\mathrm{C}=\mathrm{C}-)$...
Read More →(a) The refractive index of glass is 1.5. What is the speed of light in glass?
Question: (a)The refractive index of glass is 1.5. What is the speed of light in glass? Speed of light in vacuum is 3.0 108m s1) (b)Is the speed of light in glass independent of the colour of light? If not, which of the two colours red and violet travels slower in a glass prism? Solution: (a)Refractive index of glass,= 1.5 Speed of light, c = 3 108m/s Speed of light in glass is given by the relation, $v=\frac{c}{\mu}$ $=\frac{3 \times 10^{8}}{1.5}=2 \times 10^{8} \mathrm{~m} / \mathrm{s}$ Hence,...
Read More →Find the 12th term of a G.P. whose 8th term is 192 and the common ratio is 2.
Question: Find the $12^{\text {th }}$ term of a G.P. whose $8^{\text {th }}$ term is 192 and the common ratio is $2 .$ Solution: Common ratio, $r=2$ Letabe the first term of the G.P. $\therefore a_{8}=a r^{8-1}=a r^{7}$ $\Rightarrow a r^{7}=192$ $a(2)^{7}=192$ $a(2)^{7}=(2)^{6}(3)$ $\Rightarrow a=\frac{(2)^{6} \times 3}{(2)^{7}}=\frac{3}{2}$ $\therefore a_{12}=a r^{12-1}=\left(\frac{3}{2}\right)(2)^{11}=(3)(2)^{10}=3072$...
Read More →(i) find solution
Question: (i) $\left|\begin{array}{lll}1 0 0 \\ 0 1 0 \\ 0 0 1\end{array}\right|$ (ii) $\left|\begin{array}{rrr}1 0 4 \\ 3 5 -1 \\ 0 1 2\end{array}\right|$ Solution: (i) The given determinant is $\left|\begin{array}{lll}1 0 0 \\ 0 1 0 \\ 0 0 1\end{array}\right|$. By the definition of minors and cofactors, we have: $M_{11}=$ minor of $a_{11}=\left|\begin{array}{ll}1 0 \\ 0 1\end{array}\right|=1$ $M_{12}=$ minor of $a_{12}=\left|\begin{array}{ll}0 0 \\ 0 1\end{array}\right|=0$ $M_{13}=$ minor of $...
Read More →What is the shape of the wavefront in each of the following cases:
Question: What is the shape of the wavefront in each of the following cases: (a)Light diverging from a point source. (b)Light emerging out of a convex lens when a point source is placed at its focus. (c)The portion of the wavefront of light from a distant star intercepted by the Earth. Solution: (a)The shape of the wavefront in case of a light diverging from a point source is spherical. The wavefront emanating from a point source is shown in the given figure. (b)The shape of the wavefront in cas...
Read More →Find the 20th and nthterms of the G.P.
Question: Find the $20^{\text {th }}$ and $n^{\text {thterms }}$ of the G.P. $=\frac{5}{2}, \frac{5}{4}, \frac{5}{8} \ldots$ Solution: The given G.P. is $\frac{5}{2}, \frac{5}{4}, \frac{5}{8}, \ldots$ Here, $a=$ First term $=\frac{5}{2}$ $r=$ Common ratio $=\frac{\frac{5}{4}}{\frac{5}{2}}=\frac{1}{2}$ $a_{20}=a r^{20-1}=\frac{5}{2}\left(\frac{1}{2}\right)^{19}=\frac{5}{(2)(2)^{19}}=\frac{5}{(2)^{20}}$ $a_{n}=a r^{n-1}=\frac{5}{2}\left(\frac{1}{2}\right)^{n-1}=\frac{5}{(2)(2)^{n-1}}=\frac{5}{(2)^...
Read More →Monochromatic light of wavelength 589 nm is incident from air on a water surface.
Question: Monochromatic light of wavelength 589 nm is incident from air on a water surface. What are the wavelength, frequency and speed of (a) reflected, and (b) refracted light? Refractive index of water is 1.33. Solution: Wavelength of incident monochromatic light, = 589 nm = 589 109m Speed of light in air,c= 3 108m/s Refractive index of water,= 1.33 (a)The ray will reflect back in the same medium as that of incident ray. Hence, the wavelength, speed, and frequency of the reflected ray will b...
Read More →Give condensed and bond line structural formulas and identify the functional group
Question: Give condensed and bond line structural formulas and identify the functional group(s) present, if any, for : (a) 2,2,4-Trimethylpentane (b) 2-Hydroxy-1,2,3-propanetricarboxylic acid (c) Hexanedial Solution: (a)2, 2, 4trimethylpentane Condensed formula: $\left(\mathrm{CH}_{3}\right)_{2} \mathrm{CHCH}_{2} \mathrm{C}\left(\mathrm{CH}_{3}\right)_{3}$ Bond line formula: (b)2hydroxy1, 2, 3propanetricarboxylic acid Condensed Formula: $(\mathrm{COOH}) \mathrm{CH}_{2} \mathrm{C}(\mathrm{OH})(\m...
Read More →Write Minors and Cofactors of the elements of following determinants:
Question: Write Minors and Cofactors of the elements of following determinants: (i) $\left|\begin{array}{rr}2 -4 \\ 0 3\end{array}\right|$ (ii) $\left|\begin{array}{ll}a c \\ b d\end{array}\right|$ Solution: (i) The given determinant is $\left|\begin{array}{rr}2 -4 \\ 0 3\end{array}\right|$. Minor of element $a_{i j}$ is $M_{i j}$. $\therefore \mathrm{M}_{11}=$ minor of element $a_{11}=3$ $M_{12}=$ minor of element $a_{12}=0$ $M_{21}=$ minor of element $a_{21}=-4$ $M_{22}=$ minor of element $a_{...
Read More →The difference between any two consecutive interior angles of a polygon is 5°
Question: The difference between any two consecutive interior angles of a polygon is $5^{\circ}$. If the smallest angle is $120^{\circ}$, find the number of the sides of the polygon. Solution: The angles of the polygon will form an A.P. with common difference $d$ as $5^{\circ}$ and first term $a$ as $120^{\circ}$. It is known that the sum of all angles of a polygon with $n$ sides is $180^{\circ}(n-2)$. $\therefore S_{n}=180^{\circ}(n-2)$ $\Rightarrow \frac{n}{2}[2 a+(n-1) d]=180^{\circ}(n-2)$ $\...
Read More →A man starts repaying a loan as first installment of Rs. 100.
Question: A man starts repaying a loan as first installment of Rs. 100 . If he increases the installment by Rs 5 every month, what amount he will pay in the $30^{\text {th }}$ installment? Solution: The first installment of the loan is Rs 100. The second installment of the loan is Rs 105 and so on. The amount that the man repays every month forms an A.P. The A.P. is 100, 105, 110, First term,a= 100 Common difference,d= 5 $\mathrm{A}_{30}=a+(30-1) d$ $=100+(29)(5)$ $=100+145$ $=245$ Thus, the amo...
Read More →Figure 9.37 shows an equiconvex lens (of refractive index 1.50) in contact with a liquid layer on top of a plane mirror.
Question: Figure 9.37 shows an equiconvex lens (of refractive index 1.50) in contact with a liquid layer on top of a plane mirror. A small needle with its tip on the principal axis is moved along the axis until its inverted image is found at the position of the needle. The distance of the needle from the lens is measured to be 45.0 cm. The liquid is removed and the experiment is repeated. The new distance is measured to be 30.0 cm. What is the refractive index of the liquid? Solution: Focal leng...
Read More →If area of triangle is 35 square units with vertices (2, −6), (5, 4), and (k, 4). Then k is
Question: If area of triangle is 35 square units with vertices (2, 6), (5, 4), and (k, 4). Thenkis A.12 B.2 C.12, 2 D.12, 2 Solution: Answer: D The area of the triangle with vertices (2, 6), (5, 4), and (k, 4) is given by the relation, $\Delta=\frac{1}{2}\left|\begin{array}{ccc}2 -6 1 \\ 5 4 1 \\ k 4 1\end{array}\right|$ $=\frac{1}{2}[2(4-4)+6(5-k)+1(20-4 k)]$ $=\frac{1}{2}[30-6 k+20-4 k]$ $=\frac{1}{2}[50-10 k]$ $=25-5 k$ It is given that the area of the triangle is 35. Therefore, we have: $\Ri...
Read More →Between 1 and 31, m numbers have been inserted in such a way that the resulting sequence is an A.P.
Question: Between 1 and 31,mnumbers have been inserted in such a way that the resulting sequence is an A.P.and the ratio of $7^{\text {th }}$ and $(m-1)^{\text {th }}$ numbers is $5: 9$. Find the value of $m$. Solution: Let $\mathrm{A}_{1}, \mathrm{~A}_{2}, \ldots \mathrm{A}_{m}$ be $m$ numbers such that $1, \mathrm{~A}_{1}, \mathrm{~A}_{2}, \ldots \mathrm{A}_{m}, 31$ is an $\mathrm{A} . \mathrm{P}$. Here, $a=1, b=31, n=m+2$ $\therefore 31=1+(m+2-1)(d)$ $\Rightarrow 30=(m+1) d$ $\Rightarrow d=\f...
Read More →Draw formulas for the first five members of each homologous series beginning with the following compounds.
Question: Draw formulas for the first five members of each homologous series beginning with the following compounds. (a) HCOOH (b)CH3COCH3 (c)HCH=CH2 Solution: The first five members of each homologous series beginning with the given compounds areshown as follows: (a) H-COOH : Methanoic acid $\mathrm{CH}_{3}-\mathrm{COOH}$ : Ethanoic acid $\mathrm{CH}_{3}-\mathrm{CH}_{2}-\mathrm{COOH}$ : Propanoic acid $\mathrm{CH}_{3}-\mathrm{CH}_{2}-\mathrm{CH}_{2}-\mathrm{COOH}:$ Butanoic acid $\mathrm{CH}_{3...
Read More →Light incident normally on a plane mirror attached to a galvanometer coil retraces backwards as shown in Fig. 9.36.
Question: Light incident normally on a plane mirror attached to a galvanometer coil retraces backwards as shown in Fig. 9.36. A current in the coil produces a deflection of 3.5of the mirror. What is the displacement of the reflected spot of light on a screen placed 1.5 m away? Solution: Angle of deflection,= 3.5 Distance of the screen from the mirror,D= 1.5 m The reflected rays get deflected by an amount twice the angle of deflection i.e., 2= 7.0 The displacement (d) of the reflected spot of lig...
Read More →Light incident normally on a plane mirror attached to a galvanometer coil retraces backwards as shown in Fig. 9.36.
Question: Light incident normally on a plane mirror attached to a galvanometer coil retraces backwards as shown in Fig. 9.36. A current in the coil produces a deflection of 3.5of the mirror. What is the displacement of the reflected spot of light on a screen placed 1.5 m away? Solution: Angle of deflection,= 3.5 Distance of the screen from the mirror,D= 1.5 m The reflected rays get deflected by an amount twice the angle of deflection i.e., 2= 7.0 The displacement (d) of the reflected spot of lig...
Read More →(i) Find equation of line joining (1, 2) and (3, 6) using determinants
Question: (i) Find equation of line joining (1, 2) and (3, 6) using determinants (ii) Find equation of line joining (3, 1) and (9, 3) using determinants Solution: (i) Let P (x,y) be any point on the line joining points A (1, 2) and B (3, 6). Then, the points A, B, and P are collinear. Therefore, the area of triangle ABP will be zero. $\therefore \frac{1}{2}\left|\begin{array}{lll}1 2 1 \\ 3 6 1 \\ x y 1\end{array}\right|=0$ $\Rightarrow \frac{1}{2}[1(6-y)-2(3-x)+1(3 y-6 x)]=0$ $\Rightarrow 6-y-6...
Read More →A Cassegrain telescope uses two mirrors as shown in Fig. 9.33. Such a telescope is built with the mirrors 20 mm apart.
Question: A Cassegrain telescope uses two mirrors as shown in Fig. 9.33. Such a telescope is built with the mirrors 20 mm apart. If the radius of curvature of the large mirror is 220 mm and the small mirror is 140 mm, where will the final image of an object at infinity be? Solution: The following figure shows a Cassegrain telescope consisting of a concave mirror and a convex mirror. Distance between the objective mirror and the secondary mirror,d= 20 mm Radius of curvature of the objective mirro...
Read More →Which of the following represents the correct IUPAC name for the compounds concerned?
Question: Which of the following represents the correct IUPAC name for the compounds concerned? (a) 2,2-Dimethylpentane or 2-Dimethylpentane (b) 2,4,7-Trimethyloctane or 2,5,7-Trimethyloctane (c) 2-Chloro-4-methylpentane or 4-Chloro-2-methylpentane (d) But-3-yn-1-ol or But-4-ol-1-yne Solution: (a)The prefixdiin the IUPAC name indicates that two identical substituent groups are present in the parent chain. Since two methyl groups are present in the C2 of the parent chain of the given compound, th...
Read More →If is the A.M. between a and b, then find the value of n.
Question: If $\frac{a^{n}+b^{n}}{a^{n-1}+b^{n-1}}$ isthe A.M. betweenaandb, then find the value ofn. Solution: A.M. of $a$ and $b=\frac{a+b}{2}$ According to the given condition, $\frac{a+b}{2}=\frac{a^{n}+b^{n}}{a^{n-1}+b^{n-1}}$ $\Rightarrow(a+b)\left(a^{n-1}+b^{n-1}\right)=2\left(a^{n}+b^{n}\right)$ $\Rightarrow a^{n}+a b^{n-1}+b a^{n-1}+b^{n}=2 a^{n}+2 b^{n}$ $\Rightarrow a b^{n-1}+a^{n-1} b=a^{n}+b^{n}$ $\Rightarrow a b^{n-1}-b^{n}=a^{n}-a^{n-1} b$ $\Rightarrow b^{n-1}(a-b)=a^{n-1}(a-b)$ $\...
Read More →Find values of k if area of triangle is 4 square units and vertices are
Question: Find values ofkif area of triangle is 4 square units and vertices are (i) $(k, 0),(4,0),(0,2)$ (ii) $(-2,0),(0,4),(0, k)$ Solution: We know that the area of a trianglewhose vertices are (x1,y1), (x2,y2), and (x3,y3)is the absolute value of the determinant (Δ), where $\Delta=\frac{1}{2}\left|\begin{array}{lll}x_{1} y_{1} 1 \\ x_{2} y_{2} 1 \\ x_{3} y_{3} 1\end{array}\right|$ It is given that the area of triangle is 4 square units. $\therefore \Delta=\pm 4$ (i) The area of the triangle w...
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