Solve the given inequality graphically in two-dimensional plane: 2x – 3y > 6
Question: Solve the given inequality graphically in two-dimensional plane: 2x 3y 6 Solution: The graphical representation of 2x 3y= 6 is given as dotted line in the figure below. This line divides thexy-plane in two half planes. Select a point (not on theline), which lies in one of the half planes, to determine whether the point satisfies the given inequality or not. We select the point as (0, 0). It is observed that, 2(0) 3(0) 6 or 0 6, which is false Therefore, the upper half plane is not the ...
Read More →Find each of the following
Question: Let $A=\left[\begin{array}{ll}2 4 \\ 3 2\end{array}\right], B=\left[\begin{array}{rr}1 3 \\ -2 5\end{array}\right], C=\left[\begin{array}{rr}-2 5 \\ 3 4\end{array}\right]$ Find each of the following (i) $A+B$ (ii) $A-B$ (iii) $3 A-C$ (iv) $A B$ (v) $B A$ Solution: (i) $A+B=\left[\begin{array}{ll}2 4 \\ 3 2\end{array}\right]+\left[\begin{array}{cc}1 3 \\ -2 5\end{array}\right]=\left[\begin{array}{cc}2+1 4+3 \\ 3-2 2+5\end{array}\right]=\left[\begin{array}{ll}3 7 \\ 1 7\end{array}\right]...
Read More →Find each of the following
Question: Let $A=\left[\begin{array}{ll}2 4 \\ 3 2\end{array}\right], B=\left[\begin{array}{rr}1 3 \\ -2 5\end{array}\right], C=\left[\begin{array}{rr}-2 5 \\ 3 4\end{array}\right]$ Find each of the following (i) $A+B$ (ii) $A-B$ (iii) $3 A-C$ (iv) $A B$ (v) $B A$ Solution: (i) $A+B=\left[\begin{array}{ll}2 4 \\ 3 2\end{array}\right]+\left[\begin{array}{cc}1 3 \\ -2 5\end{array}\right]=\left[\begin{array}{cc}2+1 4+3 \\ 3-2 2+5\end{array}\right]=\left[\begin{array}{ll}3 7 \\ 1 7\end{array}\right]...
Read More →A magnetic field set up using Helmholtz coils (described in Exercise 4.16) is uniform in a small region and has a magnitude of 0.75 T.
Question: A magnetic field set up using Helmholtz coils (described in Exercise 4.16) is uniform in a small region and has a magnitude of 0.75 T. In the same region, a uniform electrostatic field is maintained in a direction normal to the common axis of the coils. A narrow beam of (single species) charged particles all accelerated through $15 \mathrm{kV}$ enters this region in a direction perpendicular to both the axis of the coils and the electrostatic field. If the beam remains undeflected when...
Read More →Define HCF of two positive integers and find the HCF of the following pairs of numbers:
Question: Define HCF of two positive integers and find the HCF of the following pairs of numbers: (i) 32 and 54 (ii) 18 and 24 (iii) 70 and 30 (iv) 56 and 88 (v) 475 and 495 (vi) 75 and 243. (vii) 240 and 6552 (viii) 155 and 1385 (ix) 100 and 190 (x) 105 and 120 Solution: (i) We need to find H.C.F. of 32 and 54. By applying division lemma $54=32 \times 1+22$ Since remainder $\neq 0$, apply division lemma on 32 and remainder 22 $32=22 \times 1+10$ Since remainder $\neq 0$, apply division lemma on...
Read More →Solve the given inequality graphically in two-dimensional plane: x – y ≤ 2
Question: Solve the given inequality graphically in two-dimensional plane: $x-y \leq 2$ Solution: The graphical representation ofxy= 2 is given in the figure below. This line divides thexy-plane in two half planes. Select a point (not on theline), which lies in one of the half planes, to determine whether the point satisfies the given inequality or not. We select the point as (0, 0). It is observed that, $0-0 \leq 2$ or $0 \leq 2$, which is true Therefore, the lower half plane is not the solutio...
Read More →The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is:
Question: The number of all possible matrices of order 33 with each entry 0 or 1 is: (A)27 (B)18 (C)81 (D)512 Solution: The correct answer is D. The givenmatrix of the order 33 has 9 elements and each of these elements can be either 0 or 1. Now, each of the 9 elements can be filled in two possible ways. Therefore, by the multiplication principle, the required number of possible matrices is $2^{9}=512$...
Read More →Solve the given inequality graphically in two-dimensional plane: y + 8 ≥ 2x
Question: Solve the given inequality graphically in two-dimensional plane: $y+8 \geq 2 x$ Solution: The graphical representation ofy+ 8 = 2xis given in the figure below. This line divides thexy-plane in two half planes. Select a point (not on theline), which lies in one of the half planes, to determine whether the point satisfies the given inequality or not. We select the point as (0, 0). It is observed that, $0+8 \geq 2(0)$ or $8 \geq 0$, which is true Therefore, lower half plane is not the sol...
Read More →In Ostwald’s process for the manufacture of nitric acid,
Question: In Ostwalds process for the manufacture of nitric acid, the first step involves the oxidation of ammonia gas by oxygen gas to give nitric oxide gas and steam. What is the maximum weight of nitric oxide that can be obtained starting only with 10.00 g. of ammonia and 20.00 g of oxygen? Solution: The balanced chemical equation for the given reaction is given as: Thus, $68 \mathrm{~g}$ of $\mathrm{NH}_{3}$ reacts with $160 \mathrm{~g}$ of $\mathrm{O}_{2}$. Therefore, $10 \mathrm{~g}$ of $\...
Read More →Solve the given inequality graphically in two-dimensional plane: 3x + 4y ≤ 12
Question: Solve the given inequality graphically in two-dimensional plane: $3 x+4 y \leq 12$ Solution: $3 x+4 y \leq 12$ The graphical representation of $3 x+4 y=12$ is given in the figure below. This line divides thexy-plane in two half planes,IandII. Select a point (not on theline), which lies in one of the half planes, to determine whether the point satisfies the given inequality or not. We select the point as (0, 0). It is observed that, $3(0)+4(0) \leq 12$ or $0 \leq 12$, which is true Ther...
Read More →An electron emitted by a heated cathode and accelerated through a potential difference of 2.0 kV,
Question: An electron emitted by a heated cathode and accelerated through a potential difference of 2.0 kV, enters a region with uniform magnetic field of 0.15 T. Determine the trajectory of the electron if the field (a) is transverse to its initial velocity, (b) makes an angle of 30 with the initial velocity. Solution: Magnetic field strength,B= 0.15 T Charge on the electron,e= 1.6 1019C Mass of the electron,m= 9.1 1031kg Potential difference,V= 2.0 kV = 2 103V Thus, kinetic energy of the elect...
Read More →Which of the given values of x and y make the following pair of matrices equal
Question: Which of the given values ofxandymake the following pair of matrices equal $\left[\begin{array}{ll}3 x+7 5 \\ y+1 2-3 x\end{array}\right]=\left[\begin{array}{ll}0 y-2 \\ 8 4\end{array}\right]$ (A) $x=\frac{-1}{3}, y=7$ (B) Not possible to find (C) $y=7, x=\frac{-2}{3}$ (D) $x=\frac{-1}{3}, y=\frac{-2}{3}$ Solution: The correct answer is B. It is given that $\left[\begin{array}{ll}3 x+7 5 \\ y+1 2-3 x\end{array}\right]=\left[\begin{array}{ll}0 y-2 \\ 8 4\end{array}\right]$ Equatingthe c...
Read More →An electron emitted by a heated cathode and accelerated through a potential difference of 2.0 kV,
Question: An electron emitted by a heated cathode and accelerated through a potential difference of 2.0 kV, enters a region with uniform magnetic field of 0.15 T. Determine the trajectory of the electron if the field (a) is transverse to its initial velocity, (b) makes an angle of 30 with the initial velocity. Solution: Magnetic field strength,B= 0.15 T Charge on the electron,e= 1.6 1019C Mass of the electron,m= 9.1 1031kg Potential difference,V= 2.0 kV = 2 103V Thus, kinetic energy of the elect...
Read More →Refer to the periodic table given in your book and now answer the following questions:
Question: Refer to the periodic table given in your book and now answer the following questions: (a) Select the possible non metals that can show disproportionation reaction. (b) Select three metals that can show disproportionation reaction. Solution: In disproportionation reactions, one of the reacting substances always contains an element that can exist in at least three oxidation states. (a)P, Cl, and S can show disproportionation reactions as these elements can exist in three or more oxidati...
Read More →$A=left[a_{i j} ight]_{m imes n}$ is a square matrix, if
Question: $A=\left[a_{i j}\right]_{m \times n}$ is a square matrix, if (A) $mn$ (B) $mn$ (C) $m=n$ (D) None of these Solution: The correct answer is C. It is known that a given matrixis said to be a square matrix if the number of rows is equal to the number of columns. Therefore, $A=\left[a_{i j}\right]_{m \times n}$ is a square matrix, if $m=n$....
Read More →Find the value of $a, b, c$, and $d$ from the equation:
Question: Find the value of $a, b, c$, and $d$ from the equation: $\left[\begin{array}{ll}a-b 2 a+c \\ 2 a-b 3 c+d\end{array}\right]=\left[\begin{array}{ll}-1 5 \\ 0 13\end{array}\right]$ Solution: $\left[\begin{array}{ll}a-b 2 a+c \\ 2 a-b 3 c+d\end{array}\right]=\left[\begin{array}{ll}-1 5 \\ 0 13\end{array}\right]$ As the two matrices are equal, their corresponding elementsare also equal. Comparing the corresponding elements, we get: ab= 1 (1) 2ab= 0 (2) 2a+c= 5 (3) 3c+d= 13 (4) From (2), we ...
Read More →Chlorine is used to purify drinking water. Excess of chlorine is harmful.
Question: Chlorine is used to purify drinking water. Excess of chlorine is harmful. The excess of chlorine is removed by treating with sulphur dioxide. Present a balanced equation for this redox change taking place in water. Solution: The given redox reaction can be represented as: The oxidation half reaction is: The oxidation number is balanced by adding two electrons as: The charge is balanced by adding 4H+ions as: The O atoms and H+ions are balanced by adding 2H2O molecules as: The reduction ...
Read More →Solve the given inequality graphically in two-dimensional plane: 2x + y ≥ 6
Question: Solve the given inequality graphically in two-dimensional plane: $2 x+y \geq 6$ Solution: The graphical representation of 2x+y= 6 is given in the figure below. This line divides thexy-plane in two half planes,IandII. Select a point (not on theline), which lies in one of the half planes, to determine whether the point satisfies the given inequality or not. We select the point as (0, 0). It is observed that, $2(0)+0 \geq 6$ or $0 \geq 6$, which is false Therefore, half planeIis not the s...
Read More →Answer the following questions:
Question: Answer the following questions: (a)A magnetic field that varies in magnitude from point to point but has a constant direction (east to west) is set up in a chamber. A charged particle enters the chamber and travels undeflected along a straight path with constant speed. What can you say about the initial velocity of the particle? (b)A charged particle enters an environment of a strong and non-uniform magnetic field varying from point to point both in magnitude and direction, and comes o...
Read More →Solve the given inequality graphically in two-dimensional plane: x + y < 5
Question: Solve the given inequality graphically in two-dimensional plane: $x+y5$ Solution: The graphical representation ofx+y= 5 is given as dotted line in the figure below. This line divides thexy-plane in two half planes,IandII. Select a point (not on theline), which lies in one of the half planes, to determine whether the point satisfies the given inequality or not. We select the point as (0, 0). It is observed that, $0+05$ or, $05$, which is true Therefore, half plane II is not the solution...
Read More →A toroid has a core (non-ferromagnetic) of inner radius 25 cm and outer radius 26 cm,
Question: A toroid has a core (non-ferromagnetic) of inner radius 25 cm and outer radius 26 cm, around which 3500 turns of a wire are wound. If the current in the wire is 11 A, what is the magnetic field (a) outside the toroid, (b) inside the core of the toroid, and (c) in the empty space surrounded by the toroid. Solution: Inner radius of the toroid,r1= 25 cm = 0.25 m Outer radius of the toroid,r2= 26 cm = 0.26 m Number of turns on the coil,N= 3500 Current in the coil,I= 11 A (a)Magnetic field ...
Read More →A toroid has a core (non-ferromagnetic) of inner radius 25 cm and outer radius 26 cm,
Question: A toroid has a core (non-ferromagnetic) of inner radius 25 cm and outer radius 26 cm, around which 3500 turns of a wire are wound. If the current in the wire is 11 A, what is the magnetic field (a) outside the toroid, (b) inside the core of the toroid, and (c) in the empty space surrounded by the toroid. Solution: Inner radius of the toroid,r1= 25 cm = 0.25 m Outer radius of the toroid,r2= 26 cm = 0.26 m Number of turns on the coil,N= 3500 Current in the coil,I= 11 A $B=\frac{\mu_{0} N...
Read More →Consider the elements:
Question: Consider the elements: Cs, Ne, I and F (a) Identify the element that exhibits only negative oxidation state. (b) Identify the element that exhibits only postive oxidation state. (c) Identify the element that exhibits both positive and negative oxidation states. (d) Identify the element which exhibits neither the negative nor does the positive oxidation state. Solution: (a)F exhibits only negative oxidation state of 1. (b)Cs exhibits positive oxidation state of +1. (c)I exhibits both po...
Read More →A man wants to cut three lengths from a single piece of board of length 91 cm.
Question: A man wants to cut three lengths from a single piece of board of length 91 cm. The second length is to be 3 cm longer than the shortest and the third length is to be twice as long as the shortest. What are the possible lengths of the shortest board if the third piece is to be at least 5 cm longer than the second? [Hint: If $x$ is the length of the shortest board, then $x,(x+3)$ and $2 x$ are the lengths of the second and third piece, respectively. Thus, $x=(x+3)+2 x \leq 91$ and $2 x \...
Read More →Find the value of $x, y$, and $z$ from the following equation:
Question: Find the value of $x, y$, and $z$ from the following equation: (i) $\left[\begin{array}{ll}4 3 \\ x 5\end{array}\right]=\left[\begin{array}{ll}y z \\ 1 5\end{array}\right]$ (ii) $\left[\begin{array}{ll}x+y 2 \\ 5+z x y\end{array}\right]=\left[\begin{array}{ll}6 2 \\ 5 8\end{array}\right]$ (iii) $\left[\begin{array}{c}x+y+z \\ x+z \\ y+z\end{array}\right]=\left[\begin{array}{l}9 \\ 5 \\ 7\end{array}\right]$ Solution: (i) $\left[\begin{array}{ll}4 3 \\ x 5\end{array}\right]=\left[\begin{...
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