holidays homework(2015-16) class xii english · holidays homework(2015-16) class xii english q1....
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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.
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PHYSICS
Q1. A spherical conducting shell of inner radius r1 and outer r2 has a charge ‘Q’. A charge ‘q’ is placed at
The center of the shell.
a) What is the surface charge density on the i) inner Surface, ii) outer surface of the shell?
b) What is the impression for electric field at a point x>r2 from the center of the shell.
Q2. Two points charges q1 =+0.2C and q2 =+0.4C are placed 0.1m apart. Calculate the electric field at a) The midpoint between the charges b) a point on the line joining q1 and q2 such that it is 0.05m Away from q2 and 0.15 m away from q1. Q.3) A Cylindrical wire is stretched to increase its length by 10%. Calculate the percentage increase its Length by 10%.Calculate the percentage increase in resistance. Q.4) Two cells of emfs 1.5V and 2V and internal resistance 1Ω and 2Ω respectively are connected in Parallel so as to send current in the same direction through an external resistance of 5Ω.
a) Draw the circuit diagram. b) Using Kirchhoff’s Laws , calculate
i. Current through each branch of the circuit ii. Potential difference across 5Ω resistance.
Q.5) Calculate the equivalent resistance of the given electrical network between A & B.
Also calculate the current through CD & ACB, if a 20V dc source is connected between A & B and the Value of R is assumed as 2Ω. Q.6) Charges of magnitudes 2Q and –Q are located at points (a, 0, 0)and (4a, 0, 0). Find the ratio of the These of electric field, due to these charges, through concentric spheres of radii 2a and 8a centered At the origin. Q.7) The electric field components due to a charge inside the cube of side 0.1m are shown in fig.
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Ex = αx, where α = 500 N/C-m, Ey =0, Ez =0
Calculate i) the flux through the cube. ii) The charge inside the cube. Q.8) Calculate the value of resistance R in the circuit shown in fig. so that the current in the circuit is 0.2 A. What would be the potential difference between points A and B.
Q.9) The voltage current variation of two metallic wires X and Y at constant temperature are shown in Fig. Assuming that the wires have the same length and the same diameter, explain which of the Two wires will have larger resistivity.
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Q.10) ABCD is a square of side 5m charges of +50C, -50C and +50C are placed at A, C and D respectively. Find the resultant electric filed at B.
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CHEMISTRY
SOLID STATE
1. What makes a glass different from a solid such as quartz under what conditions could quartz be converted into glass.
2. Why the window glasses of the old building looks milky.
3. Refractive index of a solid is observed to have the same value along all directions. Comment on the nature of this
solid.Would it show cleavage property.
4. Classify each of the following solids as ionic, metallic, molecular, covalent or amorphous.
(i)tetraphosphorousdecaoxide(P4O10) (ii) Ammonium phosphate (iii) SiC (iv) I2 (v) P4 (vi) Plastic (vii) Graphite (viii) Brass
(ix) Rb (x) LiBr (xi) Si
5. Solid A is very hard electrical insulator in solid as well as in molten state and melts at extremely high temperature.
Why type of solid is it.
6. A compound forms hexagonal close packed structure. What is the total number of voids in 0.5 mol of it. How many of
these are tetrahedral voids.
7. What type of defect can arise when a solid is heated.Which physical peoperty is affected by it and in what way.
8. On heating crystal of KCl in potassium vapours, the crystals start exhibiting a violet colour, why.
9. Non stoichiometric cuprous oxide (Cu2O) can be prepared in laboratory. In this oxide copper to oxygen ratio is slightly
less than 2 : 1 , can you account for the fact that this substance is a p type semiconductor.
10. Analysis shows that nikel oxide has formula Ni0.98O1.00 What fraction of nickel exists as Ni2+ and Ni3+ ions.
11. If NaCl is doped with 10-3 mol % of SrCl2, What is the concentration of cation vacancies.
12. Explain the nature of crystal defect produced when NaCl is doped with AlCl3, assuming AlCl3 to be ionic compound.
13. What is a semiconductor. Describe the two main types of semiconductors and contrast their mechanism.
14. Pure silicon is an insulator, silicon doped with phosphorous is a semiconductor. Silicon doped with Ga is also a
semiconductor. What is the difference between the two doped silicon semiconductors.
15. A group 14 element is to be converted into n-type semiconductor by doping it with a suitable impurity. To which
group should this impurity belong.
16. An alloy of gold and cadmium crystallises with cubic structure in which gold atoms occupy the corners and cadmium
atoms fit into the face centres. What formula would you assign to the allow.
17. Calculate the density of silver which crystallizes in the face centred cubic structure. The distance between the
nearest silver atoms in the structure is 287 pm. (atomic mass of silver is 108 g mol-1).
18. A density of CsCl (bcc) structure is 4.4 g cm-3. The unit cell edge length is 400 pm. Calculate the interionic distance in
crystal of CsBr (NA= 6.023 x1023 mol-1 , atomic masses : Cs= 133, Br =80 )
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19. Given for Fe a/pm =286, ρ /gcm-3 = 7.86,find the type of cubic lattice to which the crystal belongs . Also calculate the
radius of Fe (Given that at wt of Fe= 55.85 )
20. Metallic copper crystallises in b.c.c. lattice. If the length of cubic unit cell is 362 pm, then calculate the closest
distance between two copper aloms. Also calculate the density of crystalline copper .
SOLUTION
1.2g sodium chloride is present in 18g water,determine the percentage of solute by mass.
2. 3.42g sucrose is present in 100ml of solution, determine the molarity of the solution.
3. A solution of sulphuric acid is 15% by mass, if density of solution is 1.01gml-1, then determine the molarity of the
solution.
4. How much volume of 10M HCl should be diluted with water to prepare 2L of 5M HCl solution.
5. A solution of nitric acid is 20% by mass, determine the molality of the solution.
6. The density of 3M solution of NaCl is 1.25 gml-1. Calculate the molality of the solution.
7. Calculate the strength of 2M H2SO4 solution.
8. A sample of rectified spirit is 46% by mass, determine the mole fraction of ethanol and water in it.
9. A sample of drinking water was found to be severely contaminated with chloroform (CHCl3), supposed to be
carcinogenic in nature. The level of contamination was 15ppm (by mass).(i) Express this in percentage by mass(ii)
Determine the molality of chloroform in water
10. Henry’s law constant for CO2 in water is 1.67x108 Pa at 298K. Calculate the quantity of CO2 in 500ml of soda water
when packed under 2.5 atm CO2 pressure at 298K.
11. At higher altitudes, people suffer from a disease called anoxia. In this disease, they become weak and can not think
clearly. Give reason.
12. Two liquids X and Y on mixing form an ideal solution. At 30oC the vapour pressure of the solution containing 3 moles
of X and 1 mole of Y is 550mm of Hg. But when 4 moles of X and 1 mole of Y are mixed the vapour pressure of the
solution formed is 560 mm of Hg. What would be the vapour pressure of pure X and Y at this temperature.
13. The vapour pressure of pure benzene at a certain temperature is 0.850 bar. A non-volatile, non-electrolyte solute
weighing o.5g is added to 39g of benzene. The vapour pressure of the solution then is 0.845 bar. What is the molecular
mass of the solid substance.
14. The boiling point of benzene in 353.23K, when 1.80g of a non-volatile solute was dissolved in 90g of benzene, the
boiling point in raised to 354.11K. Calculate the molar mass of the solute. Kb for benzene is 2.53Kkgmol-1.
15. The freezing point of cyclohexane is 279.65K. A solution of 14.75g of a solute in 500g cyclohexane has a freezing
point of 277.33K. Calculate the molar of the solute. (Given Kf = 20.2Kkgmol-1)
16. Calculate the osmotic pressure of a solution containing 17.1g of cane sugar (molecular mass = 342) in 500g water
at 300K. (R = 0.0821L atmK-1mol-1, density of solution as 1.034gcm-3)
17. Given below is the sketch of a plant for carrying out a process.
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(i)Name the process occurring in the above plant.(ii)To which container does the net flow of solvent take place. (iii)Name
the SPM which can be used in this plant.(iv) Give one practical use of the plant.
18. What is the Van’t Hoff factor for a compound which undergoes tetramerization in an organic solvent.
19. Determine the osmotic pressure of a solution prepared by dissolving 25mg of K2SO4 in 2 litre of water at 25oC,
assuming that it is completely dissociated.
20. A decimolar solution of K4[Fe(CN)6] is 50% dissociated at 300K. Calculate osmotic pressure of solution.
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BIOLOGY
Q1. Revise whatsoever has been done in the class.
Q2. Make a project on any topic related to biology.
(a) Any disease with its cause, diagnosis & treatment.
(b) Any mechanism on Physiology (plant/ animal)
(c) Any recent technique ( in relevance to biology)
Conceptual Questions
Q3. Write the process of spermatogenesis in detail.
Q4. Write the process of oogenesis in detail.
Q5. What is ART? How is it useful? Explainthe techniques.
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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
10
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.
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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
12
(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
13
(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
14
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
15
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.
16
(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
17
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
18
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 )
19
(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
20
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
21
(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
22
(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
23
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
24
(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
25
(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
26
COMPUTER SCIENCE
1. To Create an array and display the elements which are divisible by 2 and 5 both
2. Write a program to find the length of string, compare two strings and concatenate two strings
3. To read an integer array, sort it using bubble sort and search an element using binary search.
4. To create a two dimensional array and find out the sum of both its diagonal.
5. To create an array and display in sorted order using selection sort
6. program to find the sum of rows and column of an array
7. To create a two dimensional array and find out its transpose
8. To find the product of each row and column in an array
9. Write a function in C++ to merge the contents of two sorted array A and B in a third array C.Assuming array A is sorted in ascending order and array b is stored in descending order, the resultant array is required to be in ascending order.
10. Write a c++ program to create a function that searches a given item using linear search technique. If the item is found, it should return its position otherwise it should return -1
11. Define a string class having a member function upit() that converts the string to upper case. toupper() library function van also be used.
12. Write a program to prepare invoice from the following data: Customer number, customer name, customer address, date of sale, itemno, item description, item sold, unit price of item, discount percentage, sales tax percentage.
13. To create a text file , count the number of vowels, digits and words and replace all the blank spaces with #
14. To create a class travel plan in C++ with the following description:
Private members: PlanCode long Place string Number_of_travellers integer Number_of_buses integer Public Members: A constructor to assign initial value of Plan code as 1001, Place as “Agra”, number_ of_ travelers as 5 and number_of _Buses as 1 A function Newplan() which allows user to enter Plan code, Place and number_of_travellers. Also assign the number of buses as per the following conditions:
27
Number_of_TravellersNumber_of_Buses Less than 20 1 >=20 and <40 2 >=40 3
15. To demonstrate the working of constructor and destructor.
16. To declare a class, append record in file and to delete record of a given book number.
17. To declare a class, append record in a file and to delete record of a given student number.
18. To implement stack using linear array(push, pop and display)
19. To create a double ended queue.
20. program to calculate occurrence of character in string
21. Suppose 7 names are stored in an array of pointers names[] as shown below:-
Char * names[]= “kiran”, “kunal”, “suman”, “kriti”, “sonam”; Write a program to reverse the order of these names.
22. To declare a class date and a function which takes two date as argument and display the greater date.
23. To create a linked implementation of stack and the body for push, pop, and traverse function.
24. To create a stack containing pin code end name of city and perform push and pop function.
25. To create two single dimensional arrays, perform their merging and stored them in another array
26. To create a linked implementation of queue and perform insert, delete and traverse functions.
27. Write a program that displays the size of a file in bytes.
28. Given a binary file SPORTS.DAT, containing records of the following structure type:
struct Sports char Event[20];
28
Char participant[10][30]; ; Write a program to create the complete structure with a function that would read the contents from the existing file SPORTS.DAT and creates a file named ATHELETIC.DAT copying only those records from SPORTS.DAT where the Event name is “Athletics”.
29. Create the following structure to access the records of a file “PRODUCT.DAT”
Struct PRODUCT Char p_code[10]; Char p_description[10]; int stock; ; Write a function in C++ to update the file with a new value of stock. The stock and the product code , whose stock to be updated are read during the execution of the program.
30. To create a queue containing pin code end name of city and perform push and pop function.
29
PHYSICAL EDUCATION
Write short notes on the following topics
1. River Rafting
2. Camping
3. Trekking
Project Work:
“Adventure sports”