Construct a 2 × 2 matrix, A = [aij], whose elements are given by:
Answer :-
language - Hindi
Saturday, April 27, 2013
Friday, April 26, 2013
matrix , chapter 3 mathematics class 12
| S.No. | Matrix chapter 3 |
| EXERCISE 3.1 | |
| 1 | In the matrix A, write (i) The order of the matrix, (ii) The number of elements, (iii) Write the elements a13, a21, a33, a24, a23. |
| 2 | If a matrix has 24 elements, what are the possible orders it can have? What, if ithas 13 elements? |
| 3 | If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements? |
| 4 | Construct a 2 × 2 matrix, A = [aij], whose elements are given by: |
| 5 | Construct a 3 × 4 matrix, whose elements are given by: |
| 6 | Find the values of x, y and z from the following equations: |
| 7 | Find the value of a, b, c and d from the equation: |
| 8 | A = [aij]m × n\ is a square matrix, if (A) m < n (B) m > n (C) m = n (D) None of these |
| 9 | Which of the given values of x and y make the following pair of matrices equal |
| 10 | The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is: (A) 27 (B) 18 (C) 81 (D) 512 |
| EXERCISE 3.2 | |
| 1 | Let A= , B= , and C= Find each of the following: (i) A + B (ii) A – B (iii) 3A – C (iv) AB (v) BA |
| 2 | Compute the following: |
| 3 | Compute the indicated products. |
| 4 | If A= , B= , and C= then compute (A+B) and (B – C). Also, verify that A + (B – C) = (A + B) – C. |
| 5 | If A = , B = , then compute the 3A- 5B |
| 6 | Simplify |
| 7 | Find X and Y, if |
| 8 | Find X, if Y = and and 2X + Y = |
| 9 | Find x and y, if |
| 10 | Solve the equation for x, y, z and t, if |
| 11 | if find the values of x and y. |
| 12 | Given , find the values of x, y, z and w. |
| 13 | If F(x) = show that F(x) F(y) = F(x + y). |
| 14 | Show tha |
| 15 | Find A2 – 5A + 6I, if A= |
| 16 | If A = prove that A3 – 6A2 + 7A + 2I = 0 |
| 17 | If A = , k = find k so that A2 = kA – 2I |
| 18 | If A = and I is the identity matrix of order 2, show that |
| 19 | A trust fund has Rs 30,000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide Rs 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of: (a) Rs 1800 (b) Rs 2000 |
| 20 | The bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books. Their selling prices are Rs 80, Rs 60 and Rs 40 each respectively. Find the total amount the bookshop will receive from selling all the books using matrix algebra. |
| 21 | Assume X, Y, Z, W and P are matrices of order 2 × n, 3 × k, 2 × p, n × 3 and p × k, respectively. Choose the correct answer in Exercises 21 and 22. |
| 22 | 21. The restriction on n, k and p so that PY + WY will be defined are: (A) k = 3, p = n (B) k is arbitrary, p = 2 (C) p is arbitrary, k = 3 (D) k = 2, p = 3 |
| 23 | If n = p, then the order of the matrix 7X – 5Z is: (A) p × 2 (B) 2 × n (C) n × 3 (D) p × n |
| EXERCISE 3.3 | |
| 1 | Find the transpose of each of the following matrices: |
| 2 | If A = and B = , then verify that (i) (A + B)′ = A′ + B′, (ii) (A – B)′ = A′ – B′ |
| 3 | If A′ = and B = ,then verify that (i) (A + B)′ = A′ + B′ (ii) (A – B)′ = A′ – B′ |
| 4 | If A′ = and B = ,then find (A + 2B)′ |
| 5 | For the matrices A and B, verify that (AB)′ = B′A′, where |
| 6 | If A = then verify that A′ A = I |
| 7 | (i) Show that the matrix A= is a symmetric matrix. (ii) Show that the matrix A = is a skew symmetric matrix |
| 8 | For the matrix A= , verify that (i) (A + A′) is a symmetric matrix (ii) (A – A′) is a skew symmetric matrix |
| 9 | Find when A = |
| 10 | Express the following matrices as the sum of a symmetric and a skew symmetric matrix: |
| EXERCISE 3.4 | |
| 1 | Using elementary transformations, find the inverse of each of the matrices, if it exists in Exercises 1 to 17. |
| 1 | Question 1 |
| 2 | Question 2 |
| 3 | Question 3 |
| 4 | Question 4 |
| 5 | Question 5 |
| 6 | Question 6 |
| 7 | Question 7 |
| 8 | Question 8 |
| 9 | Question 9 |
| 10 | Question 10 |
| 11 | Question 11 |
| 12 | Question 12 |
| 13 | Question 13 |
| 14 | Question 14 |
| 15 | Question 15 |
| 16 | Question 16 |
| 17 | Question 17 |
| 18 | Matrices A and B will be inverse of each other only if (A) AB = BA (B) AB = BA = 0 (C) AB = 0, BA = I (D) AB = BA = I |
| Miscellaneous Exercise on Chapter 3 | |
| 1 | Let A=, show that (aI + bA)n = an I + nan – 1 bA, where I is the identity matrix of order 2 and n ∈ N. |
| 2 | If A = , prove that n ∈N |
| 3 | If A = , then prove that An = , where n is any positive integer. |
| 4 | If A and B are symmetric matrices, prove that AB – BA is a skew symmetric matrix. |
| 5 | Show that the matrix B′AB is symmetric or skew symmetric according as A is symmetric or skew symmetric. |
| 6 | Find the values of x, y, z if the matrix A= satisfy the equation A′A = I. |
| 7 | For what values of x : |
| 8 | If A = , show that A2 – 5A + 7I = 0. |
| 9 | Find x, if |
| 10 | A manufacturer produces three products x, y, z which he sells in two markets. Annual sales are indicated below: Market Products I 10,000 2,000 18,000 II 6,000 20,000 8,000 (a) If unit sale prices of x, y and z are Rs 2.50, Rs 1.50 and Rs 1.00, respectively, find the total revenue in each market with the help of matrix algebra. (b) If the unit costs of the above three commodities are Rs 2.00, Rs 1.00 and 50 paise respectively. Find the gross profit. |
| 11 | Find the matrix X so that |
| 12 | If A and B are square matrices of the same order such that AB = BA, then prove by induction that ABn = BnA. Further, prove that (AB)n = AnBn for all n ∈ N. |
| Choose the correct answer in the following questions: | |
| 13 | If A =is such that A² = I, then(A) 1 + α² + βγ = 0 (B) 1 – α² + βγ = 0 (C) 1 – α² – βγ = 0 (D) 1 + α² – βγ = 0 |
| 14 | If the matrix A is both symmetric and skew symmetric, then (A) A is a diagonal matrix (B) A is a zero matrix (C) A is a square matrix (D) None of these |
| 15 | If A is square matrix such that A2 = A, then (I + A)³ – 7 A is equal to (A) A (B) I – A (C) I (D) 3A |
Thursday, April 25, 2013
With the help of a diagram show that how breakdown of glucose done through various pathways
With the help of a diagram show that how breakdown of glucose done through various pathways.
Breakdown of glucose done through various pathways.
Breakdown of glucose done through various pathways.
 |
| diagram of breakdown of glucose by various pathways |
In the very first stage the Glucose which is a 6-carbon molecule is breaks into pyruvate which is a 3-carbon molecule in the cytoplasm. After that they are broken down by three different pathways to release energy.
(i) In the absence of oxygen in Yeast.
(ii) In the lack of Oxygen in Human Muscles.
(iii) In the presence of oxygen in mitochondria.
How do animal muscles move in order to perform an action or movement?
How do animal muscles move in order to perform an action or movement?
(Ans) When a nerve impulse reaches the muscle. At cellular level muscle cell changes its shape and shorten.
The muscle cells have special proteins, which in response to nervous electrical impulses change their shape and their arrangement in the muscle cell. The new arrangement of proteins makes the muscle cells shorter resulting in the contraction of muscles.
What are the two ways of control and coordination in animals
What are the two ways of control and coordination in animals?
(Ans) The two ways for animals control and coordination are; the functions of the nervous system and hormones. How are the time and amount of hormone released, controlled? Explain with example.
How are the time and amount of hormone released, controlled? Explain with example.
(Ans) The timing and amount of hormone released are regulated by feed-backmechanisms, e.g. if sugar level in blood increases, the pancreatic cells secrete more insulin. If sugar level falls, insulin secretion is reduced.
What are plant hormones? What is the relationship between their site of production and site of action?
What are plant hormones? What is the relationship between their site of production and site of action?
(Ans) Plant hormones are the chemicals released by stimulated cells. These chemical compounds help in control and coordination of growth, development and response to environment.
They are synthesized at places away from where they act. They simply diffuse to the area of action.
They are synthesized at places away from where they act. They simply diffuse to the area of action.
Why do stars twinkle
Why do stars twinkle?
(Ans) The twinkling of stars is due to atmospheric refraction of star-light. The atmosphere is made of several layers and the refractive indices which keep on changing continuously due to this path of light rays from the star keeps on changing their path continuously. As a consequence the number of rays entering, the pupil of eye goes on changing with time and the stars appear twinkling.
How will you say that white light of the sun is made of seven colours
How will you say that white light of the sun is made of seven colours?
(Ans) When a ray of white light of the sun is passed through a glass prism, it is dispersed into its seven colour components i.e. violet, indigo, blue, green, yellow, orange and red (VIBGYOR) Different colours of light bend through different angles with respect to the incident ray, as they pass through the prism. Hence, white light is composed of seven colours. The violet colour bends most while the red colour bends the least.
Define dispersion and spectrum
Define dispersion and spectrum?
(Ans) Dispersion—The splitting of white light into its component colours is known as dispersion.
Spectrum-The band of the coloured components of a light beam is called its spectrum.
Why does the sky appear dark instead of blue to an astronaut
Why does the sky appear dark instead of blue to an astronaut?
(Ans) The sky appears dark instead of blue to an astronaut, as scattering of light does not take place outside earth’s atmosphere.
Show that the relation R in the set A of points in a plane given by R = {(P, Q): distance of the point P from the origin is same as the distance of the point Q from the origin
Question 11: Show that the relation R in the set A of points in a plane given by R = {(P, Q): distance of the point P from the origin is same as the distance of the point Q from the origin}, is an equivalence relation. Further, show that the set of all point related to a point P ≠ (0, 0) is the circle passing through P with origin as centre.
Answer: R = {(P, Q): distance of point P from the origin is the same as the distance of point Q from the origin}
Clearly, (P, P) ∈R since the distance of point P from the origin is always the same as the distance of the same point P from the origin.
∴R is reflexive.
Now,
Let (P, Q) ∈R.
⇒The distance of point P from the origin is the same as the distance of point Q from the origin.
⇒The distance of point Q from the origin is the same as the distance of point P from the origin.
⇒(Q, P) ∈R
∴R is symmetric.
Now,
Let (P, Q), (Q, S) ∈ R.
⇒The distance of points P and Q from the origin is the same and also, the distance of points Q and S from the origin is the same.
⇒The distance of points P and S from the origin is the same.
⇒(P, S) ∈R
∴R is transitive.
Therefore, R is an equivalence relation.
The set of all points related to P ≠ (0, 0) will be those points whose distance from the origin is the same as the distance of point P from the origin.
In other words, if O (0, 0) is the origin and OP = k, then the set of all points related to P is at a distance of k from the origin.
Therefore, this set of points forms a circle with the centre as the origin and this circle passes through point P.
Show that the relation R in the set A of all the books in a library of a college, given by R = {(x, y): x and y have same number of pages} is an equivalence relation.
Question 7: Show that the relation R in the set A of all the books in a library of a college, given by R = {(x, y): x and y have same number of pages} is an equivalence relation.
Answer: Set A is the set of all books in the library of a college.
R = {x, y): x and y have the same number of pages}
Now, R is reflexive since (x, x) ∈ R as x and x has the same number of pages.
Let (x, y) ∈R ⇒ x and y have the same number of pages.
⇒ y and x have the same number of pages.
⇒(y, x) ∈ R
∴R is symmetric.
Now, let (x, y) ∈R and (y, z) ∈ R.
⇒ x and y and have the same number of pages and y and z have the same number of pages.
⇒ x and z have the same number of pages.
⇒(x, z) ∈ R
∴R is transitive.
Therefore, R is an equivalence relation.
Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive
Question 6: Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.
Answer:
Let A = {1, 2, 3}.
A relation R on A is defined as R = {(1, 2), (2, 1)}.
Here (1, 1), (2, 2), (3, 3) ∉R.
∴R is not reflexive.
Now, as (1, 2) ∈R and (2, 1) ∈R, then R is symmetric.
Now, (1, 2) and (2, 1) ∈ R
However,
(1, 1) ∉ R
∴R is not transitive.
Therefore, R is symmetric but neither reflexive nor transitive.
Draw a well labeled sectional view of human heart showing the passage of flow of blood
Draw a well labeled sectional view of human heart showing the passage of flow of blood.
Describe the structure and functioning of nephron in the human body
(Q.78) Describe the structure and functioning of nephron in the human body.
(Ans) Nephrons are the filtration units of kidney. Each kidney has a large no. of nephrons. The process starts with filteration of urine.
Cup-shaped structure associated with capillaries is called Bowmans capsule. This collects filtered urine. As this urine flows through the tubular part of nephron, reabsorption of glucose, amino acids, salts and water take place. The amount of water and other substances reabsorbed depends upon how in excess they are present in the body.The urine formed enters into a long tube called ureter through the collecting ducts of kidney. Then urine is collected in urinary bladder.
How is the small intestine designed to absorb digested food
How is the small intestine designed to absorb digested food?
(Ans) To digest the carbohydrates, fats and proteins, the small intestine receives the secretions of the liver and pancrease. The acidic food coming from the stomach is made alkaline for the action of pancreatic enzymes. The liver secretions have bile juice. The bile salts break the larger fat globules into the smaller one for the effective action of enzymes (emulsification of fats).
The walls of small intestine contain glands which secrete intestinal juice. This contains enzymes to convert:
Proteins àamino acids
Complex Carbohydrates àglucose
Fats àfatly acids & glycerol
The digested food components are absorbed into the walls of the intestine.
(Ans) To digest the carbohydrates, fats and proteins, the small intestine receives the secretions of the liver and pancrease. The acidic food coming from the stomach is made alkaline for the action of pancreatic enzymes. The liver secretions have bile juice. The bile salts break the larger fat globules into the smaller one for the effective action of enzymes (emulsification of fats).
The walls of small intestine contain glands which secrete intestinal juice. This contains enzymes to convert:
Proteins àamino acids
Complex Carbohydrates àglucose
Fats àfatly acids & glycerol
The digested food components are absorbed into the walls of the intestine.
Draw the figures of an open and a closed stomata
| (Q.76) Draw the figures of an open and a closed stomata.  | | |
How does the process of nutrition take place in amoeba
(Q.75) How does the process of nutrition take place in amoeba? Explain with the help of labeled figures.
(Ans).
(Ans).
Amoeba takes food using finger like extension of cell surface called pseudopodia. These pseudopodia fuse to form food vacuole. Food is digested in food vacuole and then diffuses into the cytoplasm. The undigested food moves on surface and thrown out.
Why is the process of simple diffusion sufficient for taking in food, exchange of gases and removal of waste in unicellular organism but not sufficient in multicellular organism
| (Q.74) Why is the process of simple diffusion sufficient for taking in food, exchange of gases and removal of waste in unicellular organism but not sufficient in multicellular organism? | | |
(Ans) The process of simple diffusion is sufficient to carry different activities of life in unicellular organisms as in them, the entire surface of the organism is in contact of the environment.
In case of multicellular organisms all the cells are not in direct contact of the environment so simple diffusion will not work.
In case of multicellular organisms all the cells are not in direct contact of the environment so simple diffusion will not work.
Describe what happens to the eaten food in stomach?
| (Q.72) Describe what happens to the eaten food in stomach? | | |
(Ans) The gastric glands present in the wall of stomach release HCl, pepsin, and mucus.
1. HCl creates acidic medium in the stomach to facilitate the action of enzyme pepsin.
2. Pepsin is protein-digesting enzyme.
3. Mucus protects the lining of stomach from the acidic action of HCl.
How do the alveoli of lungs in human body help in the exchange of gases
| (Q.71) How do the alveoli of lungs in human body help in the exchange of gases? | | |
(Ans) Alveoli provide a surface for the exchange of gases. An extensive network of blood vessels is present in the wall of the alveoli. By lifting our ribs and flatten the diaphragm, the chest cavity becomes spacious. Air
is sucked into the lungs and alveoli. The oxygen from the breath, diffuses into the blood and CO2from the blood brought from the body, diffuses out into the air.
How does the transportation of food take place in plants?
| (Q.70) How does the transportation of food take place in plants? | | |
How is lymph formed? Write its functions.
| (Q.69) How is lymph formed? Write its functions. | | |
(Ans) Through the pores present in the walls of capillaries, some amount of plasma, proteins and blood cells escape into intercellular spaces in the tissues to form the tissue fluid or lymph. From intercellular spaces, it enters into lymphatic capillaries, which join to form lymph vessels, which open into larger veins. Lymph carries digested and absorbed fat from intestine and drains excess fluid from extra cellular space back into blood.
What are the different ways in which glucose is oxidised to provideenergy in various organisms
(Q.68) Sketch a flow diagram for the various pathways for breakdown of glucose. (Ans)  |
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