holidays homework(2015-16) class xii english · holidays homework(2015-16) class xii ... project...
TRANSCRIPT
1
HOLIDAYS HOMEWORK(2015-16)
CLASS XII
ENGLISH
Q1. Read the novel “The Invisible Man” at home and write its summary in your English notebook.
Q2. Draw the character sketch of
Hr. Griffin
Dr. Kamp
Mrs. Hall
Mr. Thomas Marvel
Q3. How did the invisible man, Mr. Griffin, meet his end?
Q4. Why was the invisible man, Mr. Griffin fearful of dogs?
Q5. Do you think that the end of the Novel was ‘first and fair’?
Q6. Try to give another twist to the end in your own words. (Write the epilogue (end) in your own words)
Q7. Read the newspaper daily and paste at least four reports and four articles in your notebook. Frame four
questions answers from them and four vocabularies from each of them.
Q8. Paste any two public awareness posters in a notebook.
2
ECONOMICS
1. What is fundamental psychological law.
2. Give the reason for the operation of law of diminishing marginal utility.
3. Is the consumption a continuous process? Give reasons.
4. Give the various properties of indifference curve.
5. What is the law of demand. Explain with schedule and curve.
6. Differentiate between:
a) Normal goods and inferior goods
b) Intermediate goods and final goods
7. How can one good be intermediate and final both? Explain with example.
8. Classify the term movement along with demand curve and shift in demand curve.
9. Explain the various methods of measuring elasticity of demand.
10. Solve 10 numerical problems based on calculation of Ed.
3
ACCOUNTANCY
1. Alka , Barkha and charu are partners in a firm having no partnership deed. Alka , barkha and charu
contributed 2,00,000
3,00,000 and 1,00,000 respectively. Alka and barkha desire that the profit should be divided in the
ratio of capital contribution. Charu does not agree to this. How will you settle the dispute.
2. A and B started a partnership business on 1st april’13. They contributed 6,00,000 and 4,00,000
respectively as their capitals. The terms of the partnership agreement are as under:
a. Interest on capital and drawing @ 6% p.a
b. B is to get a monthly salary of 2500
c. Sharing of profit or less will be in the ratio of their capital contribution.
The profit for the year ended 31st mar’14 , before making above appropriate was 2,07,400
The drawing of A and B were 48,000 and 40,000 respectively. Interest on drawing amounted to
1,500 for A and 1,100 for B. prepare profit and loss appropriation account and partner’s capital
accounts assuming that their capital are fluctuating.
3. P, Q and R are in partnership. P and Q sharing profits in the ratio of 4:3 and R is receiving a salary of
20,000 p.a plus 10% of profits after charging his salary and commission or 1/6 th of the profits of the
firm whichever is more. Any excess of the later over4 the former received by R is under the partnership
deed , to be borne by P and Q in the ratio of 3:2. The profit for the year ending 31st march’12 came to
3,85,000 after charging R’s salary . divide the profit among partners.
4. Name any two factors affecting goodwill of a partnership firm.
5. P,Q and R are in partnership sharing profits and losses in the ratio of 5:4:3 on 31st march’13 their
balance sheet was as follow:
liabilities amount assest amount
Sundry creditors 50,000 Cash at bank 40,000
o/s expenses 5,000 Sundry debtors 2,10,000
General reserve 75,000 Stock 3,00,000
Capital amount P 4,00,000 Q 3,00,000 R 2,00,000
9,00,00
Furniture 60,000
Plant and m/c 4,20,000
10,30,000 10,30,000
It was decided that w.e.f 1st april’13 the profit sharing will be 4:3:2 . for their purpose the following revaluation
were under:
a. Furniture be taken at 80% of its value
b. Stock be appreciated by 20%
c. Plant and machinery be valued at 4,00,000
d. Create provision for doubtful debts for 10,000 on debtors
4
e. o/s expanses be increased by 3,000. Partners agreed that altered values are not be recorded in the
books and they also do not want to distribute the general reserve.
f. You are required to port a single journal entry to give effect to the above. Also prepare the required
balance sheet.
6. A,B and C are partners in a firm sharing profit and losses in the ratio of 3:2:1 . They decided to take D
into partnership for ¼ th share on 1st april’11. For this purpose, goodwill is to be valued at 3 times the
average annual profit of the previous four or five years whichever is higher. The agreed profits gor
goodwill purpose of the past five years are as follows:
Year ending on 31st mar’07 1,30,000
On 31st mar’08 1,20,000
On 31st mar’09 1,50,000
On 31st mar’ 10 1,10,000
On 31st mar’11 2,00,000 Calculate the value of goodwill.
7. Calculate the interest on drawing of sh. Ganesh @ 9% p.a for the year ended 31st mar’07 , in each of
the following alternative cases:
Case 1:
a. if he withdraw 4,000 p.m in the beginning of every month
b. if he withdraw 5,000 p.m at the end of every month
c. if he withdraw 6,000 p.m
d. if he withdraw 72,000 during the year
e. if he withdraw as follows
30th april’2006 10,000
1st july’2006 15,000
1st oct’2006 18,000
30th nov 2006 12,000
31st mar’2007 20,000
8. The partners of a firm distributed the profits for the year ended 31st mar’03, 1,50,000 in the ratio of
2:2:1 without providing for the following adjustment:
a. A and B were entitled to a salary of 1,500 per quarter
b. C was entitled to commission of 18,000
c. A and C had guaranteed a minimum profit of 50,000 p.a to B
d. Profit were to be shared in the ration of 3:3:2.
e. Pass necessary journal entry for the above adjustment in the books.
9. The following information relates toa firm of yuvraj, maharaj and raghuraj:
a. Profits for the last 4 years:
2000 2,50,000 (profit)
2001 2,70,000( profit)
2002 1,80,000( loss)
5
2003 5,24,000( profit)
b. Remuneration to each partner 1000 p.m
c. Average capital employed in the business 8,00,000
d. normal rate of return 15%
e. assest 8,75,000, liabilities 32,000
you are required to calculate the value of goodwill
i. At 2 years purchase of average profits
ii. At 3 years’ purchase of super profit
iii. On the basis of capitalization of super profits
iv. On the basis of capitalization of average profits
10. How would you calculate interes on drawings of equal amount drawings of equal amount drawn on the
first day of every month.
Project : prepare comprehensive project.
6
BUSINESS STUDIES
Q1. To meet the objectives of firm, the management of JMD ltd. Offers employment to the physically
challenged persons. Identify the managerial objectives it is trying to achieve.
Q2. Aman is working as File manager executive in ABC ltd. At what level of management is he working?
Q3. Name the technique of Taylor which is the strongest motivator for a worker to reach standard
performance.
Q4. “ Management principles have a scope of modification depending upon the demands of situation.”
Identify the characteristics of management principles.
Q5. What is meant by Gang Plank?
Q6. In fashion industries, it is difficult to predict what is going to happen in future. Identify the characteristics
of business environment.
Q7. Mrs. Renu and Mr. Mohit are data entry operators in a company having same educational qualifications.
Renu is getting Rs. 10,000 per month and Mohit gets Rs. 15000 per month as salary for same working
hours.
a) Which principle of management is violated in this case? Name and explain which principle of
management is violated in this case? Name and explain which principle of management does
functional foremanship violate?
Q8. Karan enterprise limited is facing a lot of problems. It manufactures pens. It is suffering losses due to
surplus of production of pens. The production department produces more of pens than required and
sales department is able to sell those many pens. What quality of management, Do you think the
company is lacking?
Q9. ABC ltd is running very smoothly. It is making huge profits. The reason behind this is the relation
between workers and management. Workers and managers carry on with their respective works in
cooperation with each other. There is existence of mutual confidence and understanding for each other.
The management even takes worker into confidence before setting up standard for their task.
a) Which principle of Taylor is applied by ABC ltd?
b) Explain the principle and two consequences of violating it.
Q10. The court passed on order to ban polythene bags as
a) These bags are creating many environmental problems which affects the life of people in general.
b) Society in general is more concerned about quality of life.
c) The government decided to give subsidy to jute industry to promote this business. Innovative
techniques are being developed to manufacture jute bags at low rates. Incomes are rising and
7
people can afford to buy these bags. Identify the different dimensions of business environment by
quoting lines from the above particulars.
Project work on principles of management or business environment as per CBSE guidelines.
8
MATHS
MATRICES
Ex 1
Q1 If A is a matrix of type p × q and R is a row of A,
then what is the type of R as a matrix ?
Q2 If A is a column matrix with 5 rows, then what
type of matrix is a row of A.
Q3
(i) If the matrix has 5 elements, write all the
possible orders it can have ?
(ii) If a matrix has 8 elements, what are the
possible order it can have ?
(iii) If a matrix has 18 elements, what are the
possible order it can have ?
(iv) If a matrix has 24 elements, what are the
possible order it can have ?
Q4
(i) For 2 × 2 matrix, A = [a i j] whose elements are
given by 𝑎 𝑖 𝑗 =𝑖
𝑗, write the value of a12 .
(ii) If A is a 3 × 3 matrix whose elements are given
by 𝑎 𝑖 𝑗 =1
3[−3 𝑖 + 𝑗] write the value of a23.
(iii) Construct a 2 × 2 matrix A = [a i j] whose
elements a i j are given by a i j = i + 2 j.
(iv) Construct a 2 × 2 matrix A = [a i j] whose
elements a i j are given by
a) a i j = 2 i j b) a i j = (i 2j)2
c) a i j = |2i 3j|
(v) Constant a 2 × 3 matrix B = [b i j] whose
elements b i j are given by
a) b i j = i 3j
b) b i j = (i + 2j)2
Q5 Find the value of x , y.
(i) 𝑥 + 3 4𝑦 − 4 𝑥 + 𝑦
= 5 43 9
(ii) 𝑥 + 2𝑦 −𝑦
3𝑥 4 =
−4 36 4
(iii) 𝑥 + 3𝑦 𝑦7 − 𝑥 4
= 4 −10 4
(iv) 𝑥 − 𝑦 2
𝑥 5 =
2 23 5
(v) 3𝑥 + 𝑦 −𝑦2𝑦 − 𝑥 3
= 1 2
−5 3
(vi) 2𝑥 − 𝑦 3
3 𝑦 =
6 33 −2
(vii) 2𝑥 + 𝑦 4𝑥5𝑥 − 7 4𝑥
= 7 7𝑦 − 13𝑦 𝑥 + 6
Q6 Write the value of x y + z from the equation
𝑥 + 𝑦 + 𝑧
𝑥 + 𝑧𝑦 + 𝑧
= 957
Q7 If 𝑥𝑦 4
𝑧 + 6 𝑥 + 𝑦 =
8 𝑤0 6
, find the value of
x , y , z , w.
Q8 What is the number of all possible matrix of
order 3 × 3 with each entry 0 or 1.
Q9 If
𝑥 + 3 𝑧 + 4 2𝑦 − 7
4𝑥 + 6 𝑎 − 1 0𝑏 − 3 3𝑏 𝑧 + 2𝑐
=
0 6 3𝑦 − 2
2𝑥 −3 2𝑐 + 22𝑏 + 4 −21 0
Find the value of a, b, c, x, y, z.
Ex 2
Q1 Find the value of k, a non – zero scalar, if
2 1 2 3
−1 −3 2 + 𝑘
1 0 23 4 5
=
4 4 104 2 14
Q2 Solve for x and y
2𝑥 + 3𝑦 = 2 34 0
3𝑥 + 2𝑦 = −2 21 −5
Q3 If 𝐴 = 2 43 2
, 𝐵 = 1 3
−2 5 , 𝐶 = −2 5
3 4
9
Find the following
(i) A + B
(ii) A B
(iii) 3A C
(iv) 2A 3B
(v) 2A B
Q4 (i) if 𝐵 = −1 50 3
and 𝐴 − 2𝐵 = 0 4
−7 5
Find the matrix A.
(ii) If 9 −1 4
−2 1 3 = 𝐴 +
1 2 −10 4 9
then find the matrix A.
Q5 If A = diagonal (1, 2,5) , B = diagonal (3,0, 4)
and c = diagonal (2, 7, 0) then find
(i) 3A 2B (ii) A + 2B 3c
Q6 Find x , y , a , b , c , k .
(i) 𝐴 = 2 −35 0
and 𝑘𝐴 = 8 3𝑎
−2𝑏 𝑐
(ii) 𝑥 23 + 𝑦
−11
= 105
(iii) 𝑥2
𝑦2 + 2 2𝑥3𝑦
= 3 7
−3
(iv) 2 1 30 𝑥
+ 𝑦 01 2
= 5 61 8
(v) 2 𝑥 57 𝑦 − 3
+ 3 −41 2
= 7 6
15 14
(vi) 3 𝑎 𝑏𝑐 𝑑
= 𝑎 6
−1 2𝑑 +
4 𝑎 + 𝑏𝑐 + 𝑑 3
Q7 Find X and Y , if
(i) 𝑌 = 3 21 4
and 2X + Y = 1 0
−3 2
(ii) 𝑋 + 𝑌 = 5 20 9
and 𝑋 − 𝑌 = 3 60 −1
(iii) 2X Y = 6 −6 0
−4 2 1 and X + 2 Y =
3 2 5−2 1 −7
(iv) If A = −1 23 4
and B = 3 −21 5
and 2A + B + X = 0
(v) Find X if 3A 3B + X = 0 where 𝐴 = 4 21 3
and 𝐵 = −2 13 2
(vi) 𝐴 = 8 04 −23 6
and 𝐵 = 2 −24 2
−5 1
Find X if 3A + 2X = 5B.
Ex 3
Q1 (i) Write the order of the product of matrix
123 3 3 4
(ii) Write the order of AB and BA if A = [1 2 5]
and 𝐵 = 2
−17
(iii) Write the order of AB and BA if
𝐴 = 2 1 44 1 5
and 𝐵 = 3 −12 21 3
Q2 If 𝐴 = 0 −10 2
and 𝐵 = 3 50 0
Find AB.
Q3 (i) If 3 25 7
1 −3
−2 4 =
−1 −1−9 𝑥
Find x.
(ii) Find x + y + z if 1 0 00 1 00 0 1
𝑥𝑦𝑧 =
1−10
Q4 If 𝐴 = 1 00 −1
and 𝐵 = 0 11 0
Find AB and BA.
Q5 (i) Give an example of two non – zero 2 × 2
matrix A and B such that AB = 0.
Q6 Find the Product of
𝑥 𝑦 𝑧
𝑎 𝑔 𝑏 𝑓𝑔 𝑓 𝑐
𝑥𝑦𝑧
Q7 If 𝐴 = 0 0
−1 0 find A6.
Q8 If 𝐴 = 𝑥 𝑦𝑧 −𝑥
and A2 = I.
Find the value of x2 + yz
Q9 If 𝐴 = 1 22 1
then show that A2 = 2A + 3I
Q10 If A is a square matrix such that A2 = A then
show that (I + A)3 = 7A + I.
10
Q11 Simply 1 −2 3 2 −1 50 2 47 5 0
−
2 −5 7
Q12 If 𝐴 = 2 −13 2
and 𝐵 = 0 4
−1 7
Find 3A2 3B + I.
Q13 Solve for x and y
(i) 2 −31 1
𝑥𝑦 =
13
(ii) 𝑥 𝑦
3𝑦 𝑥 12 =
35
Q14 If 𝐴 = cos 𝛼 sin 𝛼
− sin 𝛼 cos 𝛼 show that
𝐴2 = 𝑐𝑜𝑠2 𝛼 𝑠𝑖𝑛2𝛼−𝑠𝑖𝑛2𝛼 𝑐𝑜𝑠2𝛼
Q15 If 𝐴 = 3 −5−4 2
show that A2 5A 14I
= 0
Q16 If 𝐴 = 4 2
−1 1 prove that (A 2I) (A 3I) = 0
Q17 Find K if A2 = KA 2I, 𝐴 = 3 −24 −2
Q18 (i) If 𝐴 = 1 22 1
show that
f (A) = 0 where f (x) = x2 2x 3
(ii) If = −1 23 1
, find f (A), where f (x) = x2
2x + 3
Q19 If 𝐴 = 2 31 2
, and 𝐼 = 1 00 1
(i) Find , so that A2 = A + T
(ii) Prove that A3 4A2 + A = 0
Q20 Find x if
(i) 1 𝑥 1 1 3 22 5 1
15 3 2
12𝑥 = 0
(ii) 1 2 1 1 2 02 0 11 0 2
02𝑥 = 0
Q21 If 𝐴 = 2 3
−1 2 show that
A2 4A + 7I = 0, Hence find A5.
Q22 If 𝐴 = 0 02 0
find A10
Q23 (i) If 𝐴 = 𝑎 10 𝑎
prove that 𝐴𝑛 = 𝑎𝑛 𝑛𝑎𝑛−1
0 𝑎𝑛
n N
(ii) If 𝐴 = 3 −41 −1
prove that
𝐴𝑛 = 1 + 2𝑛 −4𝑛
𝑛 1 − 2𝑛
n N
(iii) If 𝐴 = 1 11 1
prove that for n N
An = 2𝑛−1 2𝑛−1
2𝑛−1 2𝑛−1
Q24 Find the matrix A su that
(i) 𝐴 1 −21 4
= 6𝐼2
(ii) 𝐴 3 −4
−1 2 = 𝐼2
(iii) 1 10 1
𝐴 = 3 3 51 0 1
Ex 4
Find the inverse of the following matrix
(i) 2 35 7
(ii) 1 32 7
(iii) 3 102 7
(iv) 1 −12 3
(v) 10 −2−5 1
(vi) 3 0 −12 3 00 4 1
(vii) 1 2 32 5 7
−2 −4 −5 (viii)
2 −1 44 0 23 −2 7
(ix) −1 1 21 2 33 1 1
(x) 1 3 −2
−3 0 −52 5 0
Ex 5
Q1 If 𝐴 = 2 −3 0
−1 4 5 then find (3A)T
Q2 If 𝐴 = 2 −1 54 0 3
and 𝐵 = −2 3 1−1 2 −3
Find AT + BT
Q3 If = cos 𝑥 sin 𝑥
− sin 𝑥 cos 𝑥 , 0 < x < π / 2
And A + AT = I. find x.
Q4 Find x , y, z if
(i) 0 6 − 5𝑥𝑥2 𝑥 + 3
is symmetric
(ii) −2 𝑥 − 𝑦 51 0 4
𝑥 + 𝑦 𝑧 7 is symmetric
11
(iii) 0 −1 −2
−1 0 3𝑥 −3 0
is skew symmetric
(iv) 0 𝑎 32 𝑏 −1𝑐 1 0
is skew symmetric.
Q5 (i) If A is square matrix prove that AT A is
symmetric
(ii) If A , B are symmetric matrix and AB = BA .
Show that AB is symmetric Matrix.
(iii) If A , B are square matrix of equal order, B is
skew symmatrix then check ABAT is symmetric
or skew symmetric Matrix.
(iv) If A, B are square matrix of equal order and
B is symmetric then show that ATBA is also
symmetric Matrix.
(v) If A, B are skew symmetric matrix and AB =
BA then show that AB is symmetric matrix.
(vi) If a matrix is both symmetric and skew
symmetric, then show that it is a null matrix.
Q6 If 𝐴 = 2 31 0
= P + Q. where P is symmetric
and Q is skew symmetric then find the matrix P.
Q7 If 𝐴 = 3 3 24 2 0
and 𝐵 = 2 −1 21 2 4
Then verify that
(i) (AT)T = A
(ii) (A + B)T = A T + B T
(iii) (kB) T = kB T where k is any real number
(iv) find (A + 2B) T
Q8 If 𝐴 = 2 4 03 9 6
and 𝐵 = 1 42 81 3
Verify that (AB) T = B T A T.
Q9 If 𝐴 = 3 2
−1 1 and 𝐵 =
−1 02 53 4
Find (BA)T
Q10 Find x if
𝑥 4 −1 2 1 −11 0 02 2 4
𝑥 4 −1 𝑡 = 0
Answer Key
Ex 1
Q1 1 × q Q2 5 × 1
Q3 (i) 1 × 5, 5 × 1
(ii) 1 × 8, 8 × 1, 2 × 4, 4 × 2
(iii) 1 × 18, 18 × 1, 2 × 9, 9 × 2, 3 × 6, 6 × 3
(iv) 1 × 24, 24 × 1, 2 × 12, 12 × 2, 3 × 8, 8 × 3, 4 × 6, 6 ×
4
Q4 (i) a12 = 1/2 (iii) 𝐴 = 3 54 6
(ii) a23 = 1
(iv) (a) 𝐴 = 1 03 2
(b) 𝐴 = 1/2 9/2
0 2
(c) 𝐴 = 1 41 2
(v) (a) 𝐴 = −2 −5 −8−1 −4 −7
(b) 𝐴 = 9 25 4916 36 64
Q5 (i) x = 2 , y = 7 (ii) x = 2 , y = 3
(iii) x = 7 , y = 1 (iv) x = 3 , y = 1
(v) x = 1 , y = 2 (vi) x = 2 , y = 2
(vii) x = 2 , y = 3
Q6 x = 2 , y = 4 , z = 3
Q7 x = 2 , y = 4, z = 6, w = 4
or
x = 4 , y = 2, z = 6, w = 4
Q8 29 = 512
Q9 a = 2, b = 7, c = 1, x = 3, y = 5, z = 2
Ex 2
Q1 k = 2
Q2 𝑋 = −2 0−1 −3
𝑌 = 2 12 2
Q3 (1) 3 71 7
(2) 1 15 −3
(3) 8 76 2
(4) 1 −1
12 −11
12
(5) 3 58 −1
Q4 (i) −2 14−7 11
(ii) 8 −3 5
−2 −3 −6
Q5 (i) diagonal [3, 6, 23]
(ii) diagonal [13, 23, 3]
Q6 (i) k = 4, a = 4, b = 10, c = 0
(ii) x = 3, y = 4
(iii) x = 3, y = 3, or x = 7, y = 4
(iv) x = 3, y = 3
(v) x = 2, y = 9
(vi) a = 2, b = 4, c = 1, d = 3
Q7 (i) −1 −1−2 −1
(ii) 𝑋 = 4 40 4
𝑌 = 1 −20 5
(iii) 𝑋 = 3 −2 −1
−2 1 −1 𝑌 =
0 2 20 0 −3
(iv) 𝑋 = −1 −2−7 −13
(v) 𝑋 = −16 −4
3 −5
(vi) −2 −10/34 14/3
−31/3 −7/3
Ex 3
Q1 (i) 3 × 3 (ii) 1 × 1, 3 × 3
(iii) 2 × 2, 3 × 3
Q2 0 00 0
Q3 (i) 13 (ii) 0
Q4 𝐴𝐵 0 1
−1 0 , 𝐴𝐵 =
0 −11 0
Q5 𝐴 = 1 00 0
, 𝐵 = 0 00 1
Q6 [ax2 + by2 + cz2 + 2hxy + 2fyz + 2gzx] 1 × 1
Q7 0 00 0
Q8 1
Q9 verify
Q10 verify
Q11 21 15 −10
Q12 4 −101 13
Q13 (i) x = 2, y = 1 (ii) x = 1, y = 1
Q14 verify
Q17 k = 1
Q18 (ii) 𝑓 𝐴 = 12 −4−6 8
Q19 (i) = 4, x = 1
Q20 (i) 2, 14 (ii) 1
Q21 −118 −93
31 −118
Q22 0 00 0
Q24 (i) 4 2
−1 1
(ii) 1 2
1/2 3/2
(iii) 2 3 41 0 1
Ex 4
Q1 (i) −7 3−5 2
(ii) 7 −3
−2 1
(iii) 7 −10
−2 3 (iv)
3/5 1/5−2/5 1/5
(v) does not exist (vi)
3 −4 3
−2 3 −28 −12 9
(vii) 3 −2 −1
−4 1 −12 0 1
(viii) −2 1/2 111 −1 −64 −1/2 −2
(ix) 1 −1 1
−8 7 −55 −4 3
(x) 1 −2 −3
−2 4 7−3 5 9
Ex 5
Q1 6 −3
−9 120 15
Q2 0 32 26 0
Q3 π / 3
13
Q4 (i) 1, 6
(ii) x = 3, y = 2, z = 4
(iii) 2
(iv) a = 2, b = 0, c = 3
Q6 2 22 0
Q9 −3 −21 95 10
𝑇
= −3 1 5−2 9 10
DETERMINANTS
Ex – 1
Q1 Evaluate the determinants
(i) cos 15 sin 15sin 75 cos 75
Ans. = 0
(ii) 0 2 02 3 44 5 6
Ans. = 8
(iii) cos 90 − cos 45o
sin 90 sin 45o Ans. = +1
2
(iv) 2 cos θ −2 sin θsin θ cos θ
Ans. = 2
(v) sin 30 cos 30
− sin 60 cos 60 Ans. = 1
Q2 Find the value of x if
(i) 𝑥 − 2 −3
3𝑥 2𝑥 Ans. = 𝑥 =
1
2 , –
3
(ii) 2𝑥 35 𝑥
= 16 35 2
Ans. x =
4 , –4
(iii) 3 𝑦𝑥 1
= 2 24 1
Ans. x =
4 or 8
(iv) 1 −2 52 𝑥 −10 4 2𝑥
= 86 Ans. –4
Find the sum of roots. In part (iv)
(v) 2𝑥 + 5 35𝑥 + 2 9
= 0 Ans. = –
13
(vi) 𝑥 + 2 3𝑥 + 5 4
= 3 Ans. =
130
(vii) 2𝑥 46 𝑥
= 2 45 1
Ans. = ± 3
(viii) 𝑥 3
2𝑥 5 =
2 34 5
Ans. = 2
(ix) 𝑥 + 1 𝑥 − 1𝑥 − 3 𝑥 + 2
= 4 −11 3
Ans. = 2
(x) 6 𝑥
20 24 =
6 25 2
Ans. =
5
(xi) x N and 𝑥 34 𝑥
= 4 −30 1
Ans. = ±
4
(xii) 𝑥 𝑥1 𝑥
= 3 41 2
Ans. 22
– 1
(xiii) x I 2𝑥 3−1 𝑥
= 3 1𝑥 3
Ans. = –2
(xiv) 2 sin 𝑥 −1
1 sin 𝑥 =
3 0−4 sin 𝑥
Ans. = 𝜋
6 ,
𝜇
2
x N of 0 ≤ x ≤ 𝜋
2
(xv) 𝑥2 𝑥 10 2 13 1 4
= 28 Ans. = 2
Q3 Show that the following determinants are
independent of x
(i) 𝑎 sin 𝑥 cos 𝑥
− sin 𝑥 −𝑎 1cos 𝑥 1 𝑎
(ii) 0 tan 𝑥 11 − sec 𝑥 0
sec 𝑥 0 tan 𝑥
Q4 If 𝐴 = 2 −13 2
& 𝐵 = 0 4
−1 7
Find 3𝐴2 − 2𝐵 Ans. 727
Q5 (i) If 𝐴 = 1 0 10 1 20 0 4
Then show | 3 A | = 27 | A |
(ii) If 𝐴 = 1 24 2
Then show | 2 A | = 4 | A |
Q6 Prove that
14
1 𝑎 𝑏
−𝑎 1 𝑐−𝑏 −𝑐 1
= −𝑎2 + 𝑏2 + 𝑐2
Ex – 2
Q1 With expending, Evaluate
(i) 49 1 639 7 426 2 3
Ans. = 0
(ii) 1 𝑎 𝑏 + 𝑐1 𝑏 𝑐 + 𝑎1 𝑐 𝑎 + 𝑏
Ans. = 0
(iii) 2 3 45 6 8
6𝑥 9𝑥 12𝑥
Ans. = 0
(iv) 𝑎 − 𝑏 𝑏 − 𝑐 𝑐 − 𝑎𝑏 − 𝑐 𝑐 − 𝑎 𝑎 − 𝑏𝑐 − 𝑎 𝑎 − 𝑏 𝑏 − 𝑐
Ans. = 0
(v) 0 𝑏 − 𝑎 𝑐 − 𝑎
𝑎 − 𝑏 0 𝑐 − 𝑏𝑎 − 𝑐 𝑏 − 𝑐 0
Ans. = 0
(vi) cosec2 𝑥 cot2 𝑥 1
cot2 𝑥 cosec2 𝑥 −142 40 2
Ans. = 0
(vii) 𝑏2 − 𝑎𝑏 𝑏 − 𝑐 𝑏𝑐 − 𝑎𝑐𝑎𝑏 − 𝑎2 𝑎 − 𝑏 𝑏2 − 𝑎𝑏𝑏𝑐 − 𝑎𝑐 𝑐 − 𝑎 𝑎𝑏 − 𝑎2
Ans. = 0
Q2 Show that the determinant is independent of x.
cos 𝑥 + 𝑦 − sin 𝑥 + 𝑦 cos 2𝑦
sin 𝑥 cos 𝑥 sin 𝑦− cos 𝑥 sin 𝑥 cos 𝑦
Ans. =
(1+ cos 2y)
Q3 Prove that
(i) 𝑏2𝑐2 𝑏𝑐 𝑏 + 𝑐𝑐2𝑎2 𝑐𝑎 𝑐 + 𝑎𝑎2𝑏2 𝑎𝑏 𝑎 + 𝑏
= 0
(ii) 1 𝑎 𝑎2 − 𝑏𝑐1 𝑏 𝑏2 − 𝑐𝑎1 𝑐 𝑐2 − 𝑎𝑏
= 0
(iii) 0 𝑏2𝑎 𝑐2𝑎
𝑎2𝑏 0 𝑐2𝑏𝑎2𝑐 𝑏2𝑐 0
= 2𝑎3𝑏3𝑐3
Q4 If a , b , c are in AP. Prove that
𝑥 + 1 𝑥 + 2 𝑥 + 𝑎𝑥 + 2 𝑥 + 3 𝑥 + 𝑏𝑥 + 3 𝑥 + 4 𝑥 + 𝑐
= 0
Q5 Prove that
(i)
𝑥 + 𝑦 𝑥 𝑥5𝑥 + 4𝑦 4𝑥 2𝑥
10𝑥 + 8𝑦 8𝑥 3𝑥 = 𝑥3
(ii) 𝑎 𝑎 + 𝑏 𝑎 + 𝑏 + 𝑐
2𝑎 3𝑎 + 2𝑏 4𝑎 + 3𝑏 + 2𝑐3𝑎 6𝑎 + 3𝑏 10𝑎 + 6𝑏 + 3𝑐
= 𝑎3
(iii) 𝑥 + 𝑦 𝑦 + 𝑧 𝑧 + 𝑥𝑦 + 𝑧 𝑧 + 𝑥 𝑥 + 𝑦𝑧 + 𝑥 𝑥 + 𝑦 𝑦 + 𝑧
= 2
𝑥 𝑦 𝑧𝑦 𝑧 𝑥𝑧 𝑥 𝑦
(iv) 𝑎 − 𝑏 − 𝑐 2𝑎 2𝑎
2𝑏 𝑏 − 𝑐 − 𝑎 2𝑏2𝑐 2𝑐 𝑐 − 𝑎 − 𝑏
=
𝑎 + 𝑏 = 𝑐 3
(v) 𝑏 + 𝑐 𝑐 + 𝑎 𝑎 + 𝑏𝑐 + 𝑎 𝑎 + 𝑏 𝑏 + 𝑐𝑎 + 𝑏 𝑏 + 𝑐 𝑐 + 𝑎
= 2 𝑎 + 𝑏 + 𝑐 𝑎𝑏 + 𝑏𝑐 + 𝑐𝑎 − 𝑎2 − 𝑏2 − 𝑐2
(vi) 𝑎 𝑏 𝑐
𝑎 − 𝑏 𝑏 − 𝑐 𝑐 − 𝑎𝑏 + 𝑐 𝑐 + 𝑎 𝑎 + 𝑏
= 𝑎3 + 𝑏3 + 𝑐3 −
3𝑎𝑏𝑐
(vii) 𝑏 + 𝑐 𝑎 𝑎
𝑏 𝑐 + 𝑎 𝑏𝑐 𝑐 𝑎 + 𝑏
= 4𝑎𝑏𝑐
(viii) 1 𝑏 + 𝑐 𝑏2 + 𝑐2
1 𝑐 + 𝑎 𝑐2 + 𝑎2
1 𝑎 + 𝑏 𝑎2 + 𝑏2
= 𝑎 − 𝑏 𝑏 −
𝑐(𝑐−𝑎)
Q6 Find the value of x using properties of
determinant
(i) 𝑎 + 𝑥 𝑎 − 𝑥 𝑎 − 𝑥𝑎 − 𝑥 𝑎 + 𝑥 𝑎 − 𝑥𝑎 − 𝑥 𝑎 − 𝑥 𝑎 + 𝑥
= 0 Ans. x = 0, 0, +
3a
(ii) 𝑥 − 2 2𝑥 − 3 3𝑥 − 4𝑥 − 4 2𝑥 − 9 3𝑥 − 16𝑥 − 8 2𝑥 − 27 3𝑥 − 64
= 0
Ans. = 4
Q7 Prove that, using properties.
15
(i)
𝑥 𝑝 𝑞𝑝 𝑥 𝑞𝑞 𝑞 𝑥
= 𝑥 − 𝑝 (𝑥2 + 𝑝𝑥 − 2𝑞2)
(ii)
𝑎 𝑏 𝑎𝑥 + 𝑏𝑦𝑏 𝑐 𝑏𝑥 + 𝑐𝑦
𝑎𝑥 + 𝑏𝑦 𝑏𝑥 + 𝑐𝑦 0
= 𝑏2 − 𝑎𝑐 (𝑎𝑥2 + 2𝑏𝑥𝑦 + 𝑐𝑦2)
(iii) −𝑏𝑐 𝑏2 + 𝑏𝑐 𝑐2 + 𝑏𝑐
𝑎2 + 𝑎𝑐 −𝑎𝑐 𝑐2 + 𝑎𝑐𝑎2 + 𝑎𝑏 𝑏2 + 𝑎𝑏 −𝑎𝑏
=
𝑎𝑏 + 𝑏𝑐 + 𝑐𝑎 3
(iv) 𝑎 𝑏 − 𝑐 𝑐 + 𝑏
𝑎 + 𝑐 𝑏 𝑐 − 𝑎𝑎 − 𝑏 𝑏 + 𝑎 𝑐
= 𝑎 + 𝑏 + 𝑐 (𝑎2 +
𝑏2 + 𝑐2)
(v) 1 + sin2 𝑥 cos2 𝑥 4 sin 2𝑥
sin2 𝑥 1 + cos2 𝑥 4 sin 2𝑥sin2 𝑥 cos2 𝑥 1 + 4 sin 2𝑥
=
2 + 4 sin2 𝑥
Q8 In triangle ABC if
1 1 1
1 + sin A 1 + sin B 1 + sin Csin A + sin2 A sin B + sin2 B sin C + sin2 C
=
0
Show ABC is isosceles triangle.
Q9 if p , q , r are not in GP. And
1𝑞
𝑝𝛼 +
𝑞
𝑝
1𝑟
𝑞𝛼 +
𝑟
𝑞
𝑝𝛼 + 𝑞 𝑞𝛼 + 𝑟 0
= 0
Show p 2 + 2 q + r = 0
Q10 If A & B are square matrix of order 3 such that |
A | = –1 , | B | = 3 find | 3 AB |.
Ans. – 81
Q11 If A is a square matrix and | A | = 2 find the value
of | A AT |
Ans. = 4
Q12 (i) If A is a square matrix of order 2 and | A | = –5
find | 3 A |
(ii) If A is a square matrix of order 3 and | A | = 4
find | 2 A |
(iii) If A is a square matrix of order 3 and | A | = –2
find | –5 A |
Ans. (i) –45 (ii) 32 (iii) 250
Q13 Find the value of
2𝑥 + 2−𝑥 2 2𝑥 − 2−𝑥 2 1
3𝑥 + 3−𝑥 2 3𝑥 − 3−𝑥 2 1 4𝑥 + 4−𝑥 2 4𝑥 − 4−𝑥 2 1
Q14 Verify that | AB | = | A | . | B |
𝐴 = 3 4 0
−1 5 62 1 1
& 𝐵 = 1 2 52 −3 −14 6 0
Ex – 3
Q1 Show that the points (a + 5 , a – 4) (a – 2 , a + 3)
& (a , a ) are not lie on a straight line for any
value of a.
Q2 Find the value of if the points are collinear ( ,
7) , (1 , – 5) , (–4 , 5).
Ans. –5
Q3 Find the equation of line passing through (–1 , 3)
and (0 , 2).
Ans. = x + y – 2 = 0
Q4 If area of triangle is 9 sq. unit find the value of k.
(–3 , 0) (3 , 0) & (0 , k) . Ans. = 3 , –
3
Q5 If the points are collinear then show that a + b =
ab.
(a , 0) , (0 , b) and (1 , 1).
Ex – 4
Q1 If A is singular matrix then find the value of A.
1 −2 31 2 1𝑥 2 −3
Ans. x = – 1
Q2 if | A | = 7 and A is a sq. matrix of order 3 find |
adj A | Ans. 49.
Q3 if 𝐴 = 𝑎 𝑏𝑐 𝑑
verify A (Adj A) = | A | I2.
Q4 Find the inverse of 𝐴 = cos 𝜃 sin 𝜃
− sin 𝜃 cos 𝜃
Ans. A–1 = cos 𝜃 − sin 𝜃sin 𝜃 cos 𝜃
Q5 If 𝐴 = 2 35 −2
write A–1 in the terms of A
16
Ans. A–1 = 1
19 A
Q6 If 𝐴 = 2 41 −3
verify (A–1)T = (AT)–1
Q7 If 𝐴 = 3 17 5
Find x & y such that A2 + x I = y A.
Hence find A–1
Ans. x = y = 8
Q8 𝐴 = −1 2 0−1 1 10 1 1
show A–1 = A2
Q9 Find the matrix A such that
2 13 2
𝐴 −3 25 −3
= 1 00 1
Ans.
𝐴 = 1 11 0
Q10 If 0 < x < π and Matrix 2 sin 𝑥 3
1 2 sin 𝑥 is
singular. Find x
Q11 Find the value of k if matrix 2 𝑘3 5
has no
inverse
Q12 If 𝐴 = 3 −14 5
find a d j (a d j A )
Q13 For the matrix 𝐴 = 5 2−3 −1
verify (a d j A )T = a
d j (AT)
Q14 Find the matrix P such that
2 13 2
𝑃 −3 25 −3
= 1 22 −1
Q15 If 𝐹 𝑥 = cos 𝑥 − sin 𝑥 0sin 𝑥 cos 𝑥 0
0 0 1 verify [F(x)]–1 = F
(–x)
Q16 If 𝐴 = 3 27 5
& 𝐵 = 6 78 9
find (AB)–1
Q17 If 𝐴 = 5 0 42 3 21 2 1
& 𝐵 = 1 3 31 4 31 3 4
find
(AB)–1
Ex – 5
Q1 Solve using matrix method.
(i) 4x + 3y + 2z = 60
x + 2y + 3z = 45
6x + 2y + 3z = 70 Ans. (5,
8 , 8)
(ii) 2/x + 3/y + 10/z = 4
4/x – 6/y + 5/z = 1
6/x + 9/y – 20/z = 2 Ans. (2 ,
3 , 5)
(iii) 2x – y + z = 3
–x + 2y – z = –4
x – y + 2z = 1
Q2 If 𝐴 = 1 −1 02 3 40 1 2
& 𝐵 =
2 2 −4
−4 2 −42 −1 5
Find AB and use it to solve the system
x – y = 3
2x + 3y + 4z = 17
Y + 2 z = 7
Ans. (2 , –1 , 4)
Q3 If 𝐴 = 1 2 0
−2 −1 −20 −1 1
. find A–1 and use it to
solve the system
x – 2y = 10
2x – y – z = 8
–2y + z = 7 Ans. 0, –5, –
3
Q4 The management committee of a residential
colony decided to award some of its members
(say x) for honesty, same (say y) for helping
others and same other (say z) for supervising the
workers to keep the colony neat and clean. The
sum of all the awards is 12. Three times of
awardees of sum of helping others and
supervision is added to two times the number of
awardees for honesty is 33. If the sum I the
number of awardees for honesty and supervision
is twice the number of awardees for helping
others. Using matrix method, find the number of
awardees of other category. Ans. 3 , 4 ,
5
17
Q5 10 Students were selected from a school on the
basis of values for giving award and were divided
into three groups. The first group comprises hard
workers, the second group has honest and law
abiding students and the third group contains
vigilant and obedient students. Double the
number of students of first group added to the
number of second group gives 13. While the
combined strength of first and second group is
four times that of the third group. Using matrix
method, find the number of students in each
group. Ans. 5 , 3 2
Q6 A school wants to award its student for the value
of Honesty, Regularity and hard work. With a
total cash award I Rs 6000. Three times the
award money for hard work added to that given
for honesty amount to Rs 11000. The award
money given for Honesty and Hard work
together is double the one given for Regularity.
Represent the above situation algebraically and
find the award money for each value using
matrix method. Ans. 500, 2000 ,
3500
Q7 Two cricket teams honored their players for
three values excellent batting, to the point
bowling and unparalleled fielding by giving Rs. x ,
Rs. y and Rs. z per player resp. The first team
paid resp. 2 , 2 , and 1 player for the above
values with a total prize Rs 11 lakh. While the
second team paid resp. 1 , 2 and 2 player for
these values with a total of prize 9 lakh. If the
total award money for one person each for these
value amount to Rs. 6 lak. Then express the
above situation as a matrix method. Find the
award money per person for each value.
Ans. 3 lakh, 2 lakh, & 1 lakh
INVERSE TRIGNOMETRY
Q1. Find the principal values of
(i) sin−1 3
2 (Ans.
–𝜋
3 )
(ii) sec−1(−2) (Ans. 2𝜋
3 )
(iii) cos−1 cos 7𝜋
6 (Ans.
5𝜋
6 )
(iv) tan−1 tan 7𝜋
6 (Ans.
𝜋
6 )
(v) tan−1 3 − sec−1(−2) (Ans.
–𝜋
3 )
(vi) sin 𝜋
3− sin−1 −
1
2 (Ans. 1 )
(vii)
Q2. Evaluate
(i) sin 2 cot−1 −5
12 (Ans. −
120
169 )
(ii) tan 1
2cos−1 5
3 (Ans.
3− 5
2 )
(iii) sin 2 sin−1 3
5 (Ans.
24
25 )
Q3. Find the principal value.
(i) tan−1 −1 (Ans.
− 𝜋4 )
(ii) tan−1 tan 9𝜋8 (Ans.
𝜋8 )
(iii) 𝑐𝑜𝑠𝑒𝑐−1 𝑐𝑜𝑠𝑒𝑐 13𝜋
4 (Ans. − 𝜋
4 )
(iv) sec−1 −2
3 (Ans.
5𝜋
6 )
(v) cos−1 cos 2𝜋
3 + sin−1 sin
2𝜋
3 (Ans.
2𝜋
3
)
(vi) cos−1 1
2 − 2 sin−1 −
1
2 (Ans.
2𝜋
3 )
(vii) tan−1 3 − cot−1(− 3) (Ans.
–𝜋
2 )
18
(viii) tan−1 1 + cos−1 −1
2 (Ans.
11𝜋
2 )
(ix) tan−1 2 cos 2 sin−1 1
2 (Ans.
𝜋
4 )
(x) tan−1 2 sin 2 cos−1 3
2 (Ans.
𝜋
3 )
Ex – 2
Q1. Find the value of
(i) tan 2 tan−1 1
5
(ii) If x y < 1 and tan−1 𝑥 + tan−1 𝑦 =𝜋
4 find x + y
+ xy
(Ans. (i) 5
12 (ii) 1)
Q2. Convert in simplest form
(i) tan−1 cos 𝑥
1+sin 𝑥 (Ans.
𝜋
4−
𝑥
2 )
(ii) cot−1 1+sin 𝑥+ 1−sin 𝑥
1+sin 𝑥+ 1−sin 𝑥 (Ans.
𝑥
2 )
(iii) tan−1 1+𝑥2+ 1−𝑥2
1+𝑥2+ 1−𝑥2
(Ans. 𝜋
4+
1
2cos−1 𝑥2 )
(iv) cos−1 3
5cos 𝑥 +
4
5sin 𝑥 (Ans.
cos−1 3
5 )
Q3. Prove that
(i) 2 tan−1 1
3tan
𝑥
2 = cos−1
1+2 cos 𝑥
2+cos 𝑥
(ii) sin−1 𝑥
1+𝑥 = tan−1 𝑥
(iii) cot−1 1 + 𝑥2 − 𝑥 =𝜋
4+
1
2tan−1 𝑥
(iv) sin cot−1 tan−1 𝑥 = 1+𝑥2
2+𝑥2
(v) tan−1 1 + tan−1 1
2 + tan−1
1
3 =
𝜋
2
(vi) cot−1 1 + cot−1 2 + cot−1 3 =𝜋
2
(vii) tan−1 1 + tan−1 2 + tan−1(3) = 𝜋
(viii) sin−1 1
5 + sin−1
2
5 =
𝜋
2
(ix) cos−1 5
41 + cot−1
4
5 =
𝜋
2
(x) sin−1 1
17 + cos−1
9
85
(xi) cos sin−1 3
5 + sin−1
5
13 =
33
65
(xii) cos−1 4
5 + sin−1
2
13 = tan−1
17
6
Q4. Solve for x.
(i) tan−1 2 + 𝑥 + tan−1 2 − 𝑥 = tan−1 2
3
(Ans. 3, –3)
(ii) tan−1 1 + 𝑥 + tan−1 𝑥 − 1 = tan−1 6
17
(Ans. 𝑥 =1
3)
(iii) cos 2 𝑠𝑖𝑛 −1 𝑥 =1
9 (Ans.
𝑥 =2
3 , −
2
3 )
(iv) sin−1 8
𝑥 + sin−1
15
𝑥 =
𝜋
2 (Ans.
𝑥 = ±17 )
Q5. If tan−1 𝑥 + tan−1 𝑦 + tan−1 𝑧 = 𝜋
Prove that x + y + z = xyz
Q6. If tan−1 𝑥 + tan−1 𝑦 + tan−1 𝑧 =𝜋
2
Prove that x y + y z + z x = 1
Q7. If cos−1 𝑥
2+ cos−1 𝑦
3= A
Prove that 9x2 – 12 xy cos A + 4y2 = 36
Sin2 A.
Q8. Find the principal value
a) tan−1 1
3
b) sec−1 − 2
c) cot−1(− 3)
d) 𝑐𝑜𝑠𝑒 𝑐−1 2
3
e) sec−1(2)
f) 𝑐𝑜𝑠𝑒 𝑐−1 −2
3
Q9. Evaluate
19
a) tan−1 3 − sec−1 − 2 + 𝑐𝑜𝑠𝑒 𝑐−1 2
3
b) cos−1 cos 7𝜋
6
c) tan−1 cos 𝑥 − sin 𝑥
cos 𝑥+ sin 𝑥
d) tan−1 cos 𝑥
1−sin 𝑥
e) tan−1 1 – cos 𝑥
1+cos 𝑥
Q10. Prove that
a) cos−1 4
5 + cos−1
12
13 = cos−1
33
65
b) cot−1 1+sin 𝑥+ 1−sin 𝑥
1+sin 𝑥 − 1−sin 𝑥 =
𝑥
2
c) tan−1 1+𝑥2+ 1−𝑥2
1+𝑥2 – 1−𝑥2 =
𝜋
4+
1
2cos−1 𝑥2
Q11. Simplify
a) cos−1 3
5cos 𝑥 +
4
5sin 𝑥
b) sin−1 5
13cos 𝑥 +
12
13sin 𝑥
c) sin−1 sin 𝑥 + cos 𝑥
2
Q12. Prove that
a) cos 𝑡𝑎𝑛 −1 𝑠𝑖𝑛 𝑐𝑜𝑡 −1 𝑥 = 𝑥2+1
𝑥2+2
b) sin−1 4
5 + sin−1
5
13 + sin−1
16
65 =
𝜋
2
c) sin−1 3
5 + sin−1
8
17 = cos−1
36
65
Q13. Solve for x.
a) tan−1 𝑥
2 + tan−1
𝑥
3 =
𝜋
4
b) tan−1 𝑥 + 2 + tan−1 𝑥 − 2 = tan−1 8
79
c) cos−1 𝑥 + sin−1 𝑥
2=
𝜋
6
d) tan−1 1
4 + 2 tan−1
1
5 + tan−1
1
6 +
tan−11=4
e) cot−1 7 + cot−1 8 + cot−1 18 = cot−1 3
f) cot−1 3 + cot−1(5) + cot−1(7) +
cot−1(8) =𝜋
4
Ex 1
Q1 If A is a matrix of type p × q and R is a row of A,
then what is the type of R as a matrix ?
Q2 If A is a column matrix with 5 rows, then what
type of matrix is a row of A.
Q3
(v) If the matrix has 5 elements, write all the
possible orders it can have ?
(vi) If a matrix has 8 elements, what are the
possible order it can have ?
(vii) If a matrix has 18 elements, what are the
possible order it can have ?
(viii) If a matrix has 24 elements, what are the
possible order it can have ?
Q4
(vi) For 2 × 2 matrix, A = [a i j] whose elements are
given by 𝑎 𝑖 𝑗 =𝑖
𝑗, write the value of a12 .
(vii) If A is a 3 × 3 matrix whose elements are given
by 𝑎 𝑖 𝑗 =1
3[−3 𝑖 + 𝑗 ] write the value of a23.
(viii) Construct a 2 × 2 matrix A = [a i j] whose
elements a i j are given by a i j = i + 2 j.
(ix) Construct a 2 × 2 matrix A = [a i j] whose
elements a i j are given by
d) a i j = 2 i j e) a i j = (i 2j)2
f) a i j = |2i 3j|
(x) Constant a 2 × 3 matrix B = [b i j] whose
elements b i j are given by
a) b i j = i 3j
b) b i j = (i + 2j)2
Q5 Find the value of x , y.
(i) 𝑥 + 3 4𝑦 − 4 𝑥 + 𝑦
= 5 43 9
(ii) 𝑥 + 2𝑦 −𝑦
3𝑥 4 =
−4 36 4
(iii) 𝑥 + 3𝑦 𝑦7 − 𝑥 4
= 4 −10 4
(iv) 𝑥 − 𝑦 2
𝑥 5 =
2 23 5
20
(v) 3𝑥 + 𝑦 −𝑦2𝑦 − 𝑥 3
= 1 2
−5 3
(vi) 2𝑥 − 𝑦 3
3 𝑦 =
6 33 −2
(vii) 2𝑥 + 𝑦 4𝑥5𝑥 − 7 4𝑥
= 7 7𝑦 − 13𝑦 𝑥 + 6
Q6 Write the value of x y + z from the equation
𝑥 + 𝑦 + 𝑧
𝑥 + 𝑧𝑦 + 𝑧
= 957
Q7 If 𝑥𝑦 4
𝑧 + 6 𝑥 + 𝑦 =
8 𝑤0 6
, find the value of x
, y , z , w.
Q8 What is the number of all possible matrix of
order 3 × 3 with each entry 0 or 1.
Q9 If
𝑥 + 3 𝑧 + 4 2𝑦 − 7
4𝑥 + 6 𝑎 − 1 0𝑏 − 3 3𝑏 𝑧 + 2𝑐
=
0 6 3𝑦 − 2
2𝑥 −3 2𝑐 + 22𝑏 + 4 −21 0
Find the value of a, b, c, x, y, z.
Ex 2
Q1 Find the value of k, a non – zero scalar, if
2 1 2 3
−1 −3 2 + 𝑘
1 0 23 4 5
= 4 4 104 2 14
Q2 Solve for x and y
2𝑥 + 3𝑦 = 2 34 0
3𝑥 + 2𝑦 = −2 21 −5
Q3 If 𝐴 = 2 43 2
, 𝐵 = 1 3
−2 5 , 𝐶 =
−2 53 4
Find the following
(i) A + B
(ii) A B
(iii) 3A C
(iv) 2A 3B
(v) 2A B
Q4 (i) if 𝐵= −1 50 3
and 𝐴− 2𝐵= 0 4
−7 5
Find the matrix A.
(ii) If 9 −1 4
−2 1 3 = 𝐴+
1 2 −10 4 9
then find the matrix A.
Q5 If A = diagonal (1, 2,5) , B = diagonal (3,0, 4)
and c = diagonal (2, 7, 0) then find
(i) 3A 2B (ii) A + 2B 3c
Q6 Find x , y , a , b , c , k .
(i) 𝐴 = 2 −35 0
and 𝑘𝐴 = 8 3𝑎
−2𝑏 𝑐
(ii) 𝑥 23 + 𝑦
−11
= 105
(iii) 𝑥2
𝑦2 + 2 2𝑥3𝑦
= 3 7
−3
(iv) 2 1 30 𝑥
+ 𝑦 01 2
= 5 61 8
(v) 2 𝑥 57 𝑦− 3
+ 3 −41 2
= 7 6
15 14
(vi) 3 𝑎 𝑏𝑐 𝑑
= 𝑎 6
−1 2𝑑 +
4 𝑎 + 𝑏𝑐 + 𝑑 3
Q7 Find X and Y , if
(i) 𝑌 = 3 21 4
and 2X + Y = 1 0
−3 2
(ii) 𝑋+ 𝑌 = 5 20 9
and 𝑋− 𝑌 = 3 60 −1
(iii) 2X Y = 6 −6 0
−4 2 1 and X + 2 Y =
3 2 5
−2 1 −7
(iv) If A = −1 23 4
and B = 3 −21 5
and 2A + B + X = 0
(v) Find X if 3A 3B + X = 0 where 𝐴 = 4 21 3
and 𝐵= −2 13 2
(vi) 𝐴 = 8 04 −23 6
and 𝐵= 2 −24 2
−5 1
Find X if 3A + 2X = 5B.
Ex 3
Q1 (i) Write the order of the product of matrix
123 3 3 4
(ii) Write the order of AB and BA if A = [1 2 5]
and 𝐵 = 2
−17
21
(iii) Write the order of AB and BA if
𝐴 = 2 1 44 1 5
and 𝐵= 3 −12 21 3
Q2 If 𝐴 = 0 −10 2
and 𝐵= 3 50 0
Find AB.
Q3 (i) If 3 25 7
1 −3
−2 4 =
−1 −1−9 𝑥
Find x.
(ii) Find x + y + z if 1 0 00 1 00 0 1
𝑥𝑦𝑧 =
1−10
Q4 If 𝐴 = 1 00 −1
and 𝐵= 0 11 0
Find AB and BA.
Q5 (i) Give an example of two non – zero 2 × 2
matrix A and B such that AB = 0.
Q6 Find the Product of
𝑥 𝑦 𝑧 𝑎 𝑔 𝑏 𝑓𝑔 𝑓 𝑐
𝑥𝑦𝑧
Q7 If 𝐴 = 0 0
−1 0 find A6.
Q8 If 𝐴 = 𝑥 𝑦𝑧 −𝑥
and A2 = I.
Find the value of x2 + yz
Q9 If 𝐴 = 1 22 1
then show that A2 = 2A + 3I
Q10 If A is a square matrix such that A2 = A then
show that (I + A)3 = 7A + I.
Q11 Simply 1 −2 3 2 −1 50 2 47 5 0
−
2 −5 7
Q12 If 𝐴 = 2 −13 2
and 𝐵= 0 4
−1 7
Find 3A2 3B + I.
Q13 Solve for x and y
(i) 2 −31 1
𝑥𝑦 =
13
(ii) 𝑥 𝑦
3𝑦 𝑥
12 =
35
Q14 If 𝐴 = cos 𝛼 sin 𝛼
− sin 𝛼 cos 𝛼 show that
𝐴2 = 𝑐𝑜𝑠 2 𝛼 𝑠𝑖𝑛 2𝛼−𝑠𝑖𝑛 2𝛼 𝑐𝑜𝑠 2𝛼
Q15 If 𝐴 = 3 −5
−4 2 show that A2 5A 14I
= 0
Q16 If 𝐴 = 4 2
−1 1 prove that (A 2I) (A 3I) = 0
Q17 Find K if A2 = KA 2I, 𝐴 = 3 −24 −2
Q18 (i) If 𝐴 = 1 22 1
show that
f (A) = 0 where f (x) = x2 2x 3
(ii) If = −1 23 1
, find f (A), where f (x) = x2
2x + 3
Q19 If 𝐴 = 2 31 2
, and 𝐼 = 1 00 1
(i) Find , so that A2 = A + T
(ii) Prove that A3 4A2 + A = 0
Q20 Find x if
(i) 1 𝑥 1 1 3 22 5 1
15 3 2
12𝑥 = 0
(ii) 1 2 1 1 2 02 0 11 0 2
02𝑥 = 0
Q21 If 𝐴 = 2 3
−1 2 show that
A2 4A + 7I = 0, Hence find A5.
Q22 If 𝐴 = 0 02 0
find A10
Q23 (i) If 𝐴 = 𝑎 10 𝑎
prove that 𝐴𝑛 = 𝑎𝑛 𝑛𝑎𝑛−1
0 𝑎𝑛
n N
(ii) If 𝐴 = 3 −41 −1
prove that
𝐴𝑛 = 1 + 2𝑛 −4𝑛
𝑛 1 − 2𝑛
n N
(iii) If 𝐴 = 1 11 1
prove that for n N
An = 2𝑛−1 2𝑛−1
2𝑛−1 2𝑛−1
Q24 Find the matrix A su that
(i) 𝐴 1 −21 4
= 6𝐼 2
(ii) 𝐴 3 −4
−1 2 = 𝐼 2
(iii) 1 10 1
𝐴 = 3 3 51 0 1
Ex 4
22
Find the inverse of the following matrix
(i) 2 35 7
(ii) 1 32 7
(iii) 3 102 7
(iv) 1 −12 3
(v) 10 −2−5 1
(vi) 3 0 −12 3 00 4 1
(vii) 1 2 32 5 7
−2 −4 −5 (viii)
2 −1 44 0 23 −2 7
(ix) −1 1 21 2 33 1 1
(x) 1 3 −2
−3 0 −52 5 0
Ex 5
Q1 If 𝐴 = 2 −3 0
−1 4 5 then find (3A)T
Q2 If 𝐴 = 2 −1 54 0 3
and 𝐵= −2 3 1−1 2 −3
Find AT + BT
Q3 If = cos 𝑥 sin 𝑥
− sin 𝑥 cos 𝑥 , 0 < x < π / 2
And A + AT = I. find x.
Q4 Find x , y, z if
(i) 0 6 − 5𝑥𝑥2 𝑥 + 3
is symmetric
(ii) −2 𝑥 − 𝑦 51 0 4
𝑥 + 𝑦 𝑧 7 is symmetric
(iii) 0 −1 −2
−1 0 3𝑥 −3 0
is skew symmetric
(iv) 0 𝑎 32 𝑏 −1𝑐 1 0
is skew symmetric.
Q5 (i) If A is square matrix prove that AT A is
symmetric
(ii) If A , B are symmetric matrix and AB = BA .
Show that AB is symmetric Matrix.
(iii) If A , B are square matrix of equal order, B is
skew symmatrix then check ABAT is symmetric
or skew symmetric Matrix.
(iv) If A, B are square matrix of equal order and
B is symmetric then show that ATBA is also
symmetric Matrix.
(v) If A, B are skew symmetric matrix and AB =
BA then show that AB is symmetric matrix.
(vi) If a matrix is both symmetric and skew
symmetric, then show that it is a null matrix.
Q6 If 𝐴 = 2 31 0
= P + Q. where P is symmetric and
Q is skew symmetric then find the matrix P.
Q7 If 𝐴 = 3 3 24 2 0
and 𝐵= 2 −1 21 2 4
Then verify that
(i) (AT)T = A
(ii) (A + B)T = A T + B T
(iii) (kB) T = kB T where k is any real number
(iv) find (A + 2B) T
Q8 If 𝐴 = 2 4 03 9 6
and 𝐵= 1 42 81 3
Verify that (AB) T = B T A T.
Q9 If 𝐴 = 3 2
−1 1 and 𝐵=
−1 02 53 4
Find (BA)T
Q10 Find x if
𝑥 4 −1 2 1 −11 0 02 2 4
𝑥 4 −1 𝑡 = 0
Answer Key
Ex 1
Q1 1 × q Q2 5 × 1
Q3 (i) 1 × 5, 5 × 1
(ii) 1 × 8, 8 × 1, 2 × 4, 4 × 2
(iii) 1 × 18, 18 × 1, 2 × 9, 9 × 2, 3 × 6, 6 × 3
(iv) 1 × 24, 24 × 1, 2 × 12, 12 × 2, 3 × 8, 8 × 3, 4 × 6, 6 ×
4
Q4 (i) a12 = 1/2 (iii) 𝐴 = 3 54 6
(ii) a23 = 1
(iv) (a) 𝐴 = 1 03 2
(b) 𝐴 = 1/2 9/2
0 2
(c) 𝐴 = 1 41 2
(v) (a) 𝐴 = −2 −5 −8−1 −4 −7
23
(b) 𝐴 = 9 25 49
16 36 64
Q5 (i) x = 2 , y = 7 (ii) x = 2 , y = 3
(iii) x = 7 , y = 1 (iv) x = 3 , y = 1
(v) x = 1 , y = 2 (vi) x = 2 , y = 2
(vii) x = 2 , y = 3
Q6 x = 2 , y = 4 , z = 3
Q7 x = 2 , y = 4, z = 6, w = 4
or
x = 4 , y = 2, z = 6, w = 4
Q8 29 = 512
Q9 a = 2, b = 7, c = 1, x = 3, y = 5, z = 2
Ex 2
Q1 k = 2
Q2 𝑋 = −2 0−1 −3
𝑌 = 2 12 2
Q3 (1) 3 71 7
(2) 1 15 −3
(3) 8 76 2
(4) 1 −1
12 −11
(5) 3 58 −1
Q4 (i) −2 14−7 11
(ii) 8 −3 5
−2 −3 −6
Q5 (i) diagonal [3, 6, 23]
(ii) diagonal [13, 23, 3]
Q6 (i) k = 4, a = 4, b = 10, c = 0
(ii) x = 3, y = 4
(iii) x = 3, y = 3, or x = 7, y = 4
(iv) x = 3, y = 3
(v) x = 2, y = 9
(vi) a = 2, b = 4, c = 1, d = 3
Q7 (i) −1 −1−2 −1
(ii) 𝑋= 4 40 4
𝑌 = 1 −20 5
(iii) 𝑋= 3 −2 −1
−2 1 −1 𝑌 =
0 2 20 0 −3
(iv) 𝑋= −1 −2−7 −13
(v) 𝑋= −16 −4
3 −5
(vi) −2 −10/34 14/3
−31/3 −7/3
Ex 3
Q1 (i) 3 × 3 (ii) 1 × 1, 3 × 3
(iii) 2 × 2, 3 × 3
Q2 0 00 0
Q3 (i) 13 (ii) 0
Q4 𝐴𝐵 0 1
−1 0 , 𝐴𝐵 =
0 −11 0
Q5 𝐴 = 1 00 0
, 𝐵= 0 00 1
Q6 [ax2 + by2 + cz2 + 2hxy + 2fyz + 2gzx] 1 × 1
Q7 0 00 0
Q8 1
Q9 verify
Q10 verify
Q11 21 15 −10
Q12 4 −101 13
Q13 (i) x = 2, y = 1 (ii) x = 1, y = 1
Q14 verify
Q17 k = 1
Q18 (ii) 𝑓 𝐴 = 12 −4−6 8
Q19 (i) = 4, x = 1
Q20 (i) 2, 14 (ii) 1
Q21 −118 −93
31 −118
Q22 0 00 0
Q24 (i) 4 2
−1 1
(ii) 1 2
1/2 3/2
(iii) 2 3 41 0 1
Ex 4
Q1 (i) −7 3−5 2
(ii) 7 −3
−2 1
24
(iii) 7 −10
−2 3 (iv)
3/5 1/5−2/5 1/5
(v) does not exist (vi)
3 −4 3
−2 3 −28 −12 9
(vii) 3 −2 −1
−4 1 −12 0 1
(viii) −2 1/2 111 −1 −64 −1/2 −2
(ix) 1 −1 1
−8 7 −55 −4 3
(x) 1 −2 −3
−2 4 7−3 5 9
Ex 5
Q1 6 −3
−9 120 15
Q2 0 32 26 0
Q3 π / 3
Q4 (i) 1, 6
(ii) x = 3, y = 2, z = 4
(iii) 2
(iv) a = 2, b = 0, c = 3
Q6 2 22 0
Q9 −3 −21 95 10
𝑇
= −3 1 5−2 9 10
25
INFORMATICS PRACTICES
worksheet 1
Table: CLUB
COACH ID
NAME AGE SPORTS DATEOF APP
PAY COMM SEX
1 KUKREJA 35 KARATE 27/3/1996 10000 100 M
2 RAVEENA 34 KARATE 20/1/1998 12000 200 F
3 KARAN 34 SQUASH 19/2/1998 20000 M
4 TARUN 33 BASKETBALL 1/1/1998 15000 600 M
5 ZUBIN 36 SWIMMING 12/1/1998 7500 M
6 KETAKI 36 SWIMMING 24/2/1998 800 F
7 ANKITA 39 SQUASH 20/2/1998 2200 0 F
8 ZAREEN 37 KARATE 22/2/1998 1100 0 F
9 KUSH 41 SWIMMING 13/1/1998 900 200 M
10 SHAILYA 37 BASKETBALL 19/2/1998 1700 M
1. Write the command to create the above table
2. display the information about CLUB
3. show all the information about SWIMMING COACHES in the CLUB
4. list the name,age and pay of all the female coaches
5. display the records in ascending order of age
6. display name,sports and age for all Coaches with their date of appointment in descending order
7. display the ID,name and sports of those coaches whose age is more than 30 but less than 36
8. display the coach name,ID along with the salary who are earning salary between 1000-5000
9. display the records of all female KARATE coaches
10. display the name,ID,AGE,and annual salary for all the coaches
11. display the name,salary of coaches who are getting commission
12. display the name,salary of coaches who are not getting commission
13. write the command to provide an increment of Rs 1000 to all the coaches
14. write the command to provide an increment of 10% to all the male coaches
15. write the command to find out the total salary to be paid to all the coaches
16. write the command to find out the maximum salary paid
17. write the command to find out the minimum commission paid
18. write command to find out the average salary paid to all the basketball coaches
19. delete the records of cricket coaches
20. display a report showing coach name,pay,age & bonus{15% of pay} for all the coaches
21. change the sports from swimming to mountaining
22. count the coaches having sports as KARATE and getting the pay more than 1000
23. drop the column age from the table
26
24. truncate the above table
25. drop the the table
WRITE THE OUTPUT OF THE FOLLOWING ON THE BASIS OF ABOVE TABLE:-
A. SELECT LCASE(SPORTS)FROM CLUB;
B. SELECT MOD(AGE,5)FROM CLUB WHERE SEX =’F’;
C. SELECT POWER(3,2)FROM CLUB WHERE SPORTS=’KARATE’;
D. SELECT SUBSTR(COACHNAME,1,2)FROM CLUB WHERE DATEOF APP>’1998-O1-31’;
E. SELECT CONCAT(NAME,SPORTS)FROM CLUB;
F. SELECT CONCAT(NAME,AGE)FROM CLUB WHERE AGE>35;
G. SELECT ROUND( PAY/30)”PAY PER DAY FROM CLUB;
H. SELECT LEFT(SPORTS,3)FROM CLUB;
I. SELECT MID(SPORTS,2,3)FROM CLUB;
J. SELECT MONTH(DATEOF APP)FROM CLUB WHERE SEX=’F’;
K. SELECT EAR(DATEOF APP)FROM CLUB WHERE SEX=’F’;
L. SELECT NOW( );
M. SELECR RIGHT(COACH NAME,3)WHERE SEX=’M’;
N. SELECT COUNT(*) FROM SPORTS;
O. SELECT COUNT(DISTINCT SPORTS)FROM CLUB;
P. SELECT MIN(PAY)FROM CLUB;
Q. SELECT MIN(COMM)FROM CLUB;
R. SELECT AVG(PAY)FROM CLUB;
WORKSHEET 2
Q2. Commands Based on EMPLOYEE And DEPT table
1. create a database named as EMPLOYEE of the following structure using SQL command
2.a)Create a table EMP with the following details:
Field name Field type Constraint
EMPNO Int(40) NOT NULL PRIMARY KEY
ENAME Varchar(20)
JOB Varchar(9)
MGR Int(4)
HIREDATE Date
SAL Float(7,2) NOT LESS THAN 10000
COMM Float(7,2)
DEPTNO Int(2) DEFAULT IS 10
b. create another tabl named as DEPTM with the following structure:-
27
Field name Field type Constraint
DEPTNO Int(2) NOT NULL PRIMARY KEY
DNAME Varchar(14)
LOC Varchar(13)
C.Insert the following records in the DEPARTMENT (DEPT)TABLE:-
Deptno Dname Location
10 ACCOUNTING DELHI
20 RESEARCHING KOLKATA
30 SALES MUMBAI
40 OPERATIONS BANGALORE
EMP TABLE Contains the following data:-
ENO ENAME JOB MGR HIREDATE SAL COMM DEPT
7369 Sunita Sharma
CLERK 7902 17/12/1980 2800 20
7499 Ashok singhal
SALESMAN 7698 20/2/1981 3600 300 30
7521 Rohit rana
SALESMAN 7698 22/2/1981 5250 500 30
7566 Jyoti lamba
MANAGER 7839 02/04/1981 4975 20
7654 Martin s. SALESMAN 7698 28/09/1981 6250 1400 30
7698 Binod goel
MANAGER 7839 01/05/1981 5850 30
7782 Chetan gupta
MANAGER 7839 09/06/1981 2450 10
7788 Sudhir rawat
ANALYST 7566 19/04/1981 5000 20
7839 Kavita sharma
PRESIDENT 17/11/1981 5000 10
7844 Tushar tiwari
SALESMAN 7698 08/09/1981 4500 0 30
7876 Anand rathi
CLERK 7788 23/01/1987 6100 20
7900 Jagdeep rana
CLERK 7698 03/12/1981 4950 30
7902 Sumit vats
ANALYST 7566 03/12/1981 3500 3600 20
7934 Manoj kaushik
CLERK 7782 23/01/1982 5300 10
28
2. list the name and employee number from emp table
3. list all names,hiredate&salary of all the employees
4. display the employee name& incremented value of sal as sal+300;
5. display the employee name& hid annual salary(annual salary=12*sal+100)
6. display the employee name&salary where comm. is NULL,.
7. list the distinct department no.from the table.
8. list the unique jobs from the table.
9. list the salary where salary is less than is commission .
10. list the ename,job,sal and hiredate for the employees whose salary is between 3000-4000
11. list the names & mgr which are in 7902,7566,7788.
12. list the ename starting with”S”
13. display the records of thode employees whose salary is more than 4100
14. list all the columns in ascending order of hiredate
15. list all the columns in ascending order of deptno & descending order of salary
16. display the employee name & job of employees hired between feb-20,1981 & may-1,1981
17. display employee name & their dept no,dept name
18. display the total salary of employees in each department
19. count the number of employees in each department & display the number of employees,department
no.& department name & location
20. display the ename & deptno of all the employees in the department 20 & 30 in alphabetical order by
name
21. list the name & salary of all the employees who earn commission
22. create a view vmp containing deptno,dname,sal,hiredate from emp table & dept table
23. add a new column in the table as Contact_number in employee table
24. provide an increment of 5% in sal to those employees who are not getting commissions
25. delete the records of those employees who are working in deptno 10
26. provide Rs 500 as commission to those employees who are working in either deptno 10,20 or 40
27. delete the column hiredate from employee table
28. delete the records of department table
29
PHYSICAL EDUCATION
Write short notes on the following topics
1. River Rafting
2. Camping
3. Trekking
Project Work:
“Adventure sports”