MODEL TEST PAPER -2 XII-20/21

         SECTION : A (80 Marks)

Question 1) (10x2= 20 Marks)

i) If A= 2     3

             4     5 , find Inverse of A.

ii) Show that the function f: R-> R, given by f(x) = | x | is neither one one or onto.

iii) Show sin⁻¹cos sin⁻¹x+ cos⁻¹sin cos⁻¹ x = π/2

iv) If y= tan⁻¹(secx+ tanx), find d²y/dx²

v) ∫ x eˣ dx    

vi) lim ₓ→₀ (log cos x)/sin²x

vii) Prove without expanding:

a - b     1     a            a      1      b

b - c     1     b    =      b      1      c

c - a     1     c            c       1      a

viii) If x > 1/2, show that the function f(x)= x(4x²-3) is strictly increasing.

ix) Solve 2ˣ⁻ʸ dx + 2ʸ ⁻ˣ dy = 0

x) A and B are two Independent events with P(A)= 2/5 and P(B)= 1/3, Evaluate P(AUB).

Question 2).                                 (4)

Prove: 1+a²-b²       2ab            - 2b

                2ab      1 - a²+b²         2a

                 2b           - 2a       1 - a² - b²

 = (1+a²+b²)³.                              

Question 3).                                   (4)

If tan⁻¹(yz/xr) + tan⁻¹(zx/yr) + tan⁻¹(xy/zr) = π/2 then Prove that, x² + y² + z² = r². 

Question 4).                                    (4)

A bag contains 5 white, 7 red and 3 black balls. If three balls are drawn one by one without replacement. Find the probability that none is red.                       

Question 5).                                 (4)

If y= (tan⁻¹x)², show that (1+x²) d²y/dx² + 2x(1+x²) dy/dx - 2= 0.  

Question 6).                                   (4)

Evaluate ∫ 2⁴ˣ sin 3x dx. 

                   OR

Evaluate:

 ∫ (cosx + x sinx)/{x(x+ cosx) dx

Question 7)                                    (4)

Find the equations of the tangent to the curve y= x² - 2x +7 which is:

a) Parallel to the line 2x - y+9= 0

b) Perpendicular to 5y-15x= 13

                       OR

Show that the maximum value of 2x + 1/2x is less than its minimum value. 

                  

Question 8).                                  (4)

Solve by matrix inversion method

x+2y+z= 7; x + 3z= 11; 2x - 3y=1

                 OR

Show that

a + b+ 2c         a                  b

      c           b+c + 2a           b

      c                 a.            c+a+2b          = 2(a+b+c)³. 

Question 9).                                   (6)

Given x+y= 3, find the maximum and minimum values of 9/x + 36/y 

                 OR

A closed right circular cylinder is has a volume of 2156cm³. What will be the radius of the base so that total surface area is minimum.

Question 10).                                  (4)

 Show that the function f in A= R - {2/3} defined as f(x)=(4x-3)/(6x-4) is one-one and onto .

Question 11).                           2+2=4

 Solve:

a) dy/dx + y secx = tan x. 

b) tan x dy/dx= 1+y² where x= π/2 and y= 1. 

Question 12).                   3+3= 6

a) Evaluate ∫ |sin x| dx at (π/2,-π/2). 

b) Prove ∫ {log(1+x)}/(1+x²) dx at (1,0) = π/8 . Log 2. 

Question 13).                      (3x2= 6)

 a) If x= sint and y= cos pt, p is constant, then find the value of (1-x²) d²y/dx² - x dy/dx.            

b) If m² = p² cos² t + q²sin²t, then show that m+ d²m/dt² = p²q²/m²

Question 14).                  3+3=6

A) It is known that 5 men out of 100 and 25 women out of 1000 are colour blind. A colour blind person is chosen at random. Assuming that males and females are in equal proportion, find the probability of the person to be male.

B) Rajiv and Robin play 12 games of chess. Rajiv wins 6 games, Robin wins 4 games and and 2 games end in a draw. They agree to play 3 more games. Calculate the probability that out of these 3 games, two games end in a draw.

    

                      Or

Evaluate:

A) ∫ x² sin⁻¹x dx

B) ∫x² eᵃˣ dx at (a,0)

            SECTION C.        (20 Marks)

Question 15).                    2+2+2

A) Given demand function x= 50- 0.5 P and cost function C=50+40x, find price for break-even price.

B) 4x+y-10= 0, 2x + 5y -14= 0 are two regression lines. Find the correlation coefficient between variables x and y.

C) The total cost C(x) of a firm is C(x)= 0.0005x³ - 0.7x² - 30x + 3000 where x is the output. Determine:

    a) average cost (AC)

    b) Marginal cost (MC)

Question 16).                               (4)

The two lines of Regression for a distribution (x,y) are 3x+2y= 7 and x+4y= 9. Find the regression coefficient X on Y and Y on X.

                         OR

Treating x as an independent variable. Find the line of best fit for the following date:

X: 15    12       11        14          13

Y: 25    28       24        22          30

Hence, predict the value of y when x= 10.

Question 17)                               (4)

The marginal cost function of manufacturing x units of a commodity is 6+10x - 6x². The total cost of producing one unit of the commodity is ₹ 12. Find the total and average cost functions.

                       OR

If c= 2x{(x+4)/(x+1)} + 6 is the total cost of production of x units of a commodity, show that marginal cost falls continuously as a x increases.

Question 18).                              (6)

A small firm manufacturers gold rings and chains. The combined number of rings and chains manufactured per day is almost 24. It takes one hour to make a ring and half an hour for a chain. The maximum number of hours available per day is 16. If the profit on a ring is ₹300 and on a chain is ₹190, how many of each should be manufactured daily so as to maximize the profit?

MODEL TEST PAPER (ICSE) (23/24)

       MODEL TEST PAPER -1

        (Marks: 80)

Question 1.                       1x 10= 10

i) What are the rate of GST on an article that was sold at a price three times its marked price ?

A) 10% B) 20% C) 100% D) 200% 

ii) Ram deposited ₹150 per month in a bank for 8 months under the Recurring deposit scheme. What will be the intrest of his deposits. If the rate of interest is 8% p.a.

A) 36 B) 360 C) 1236 D) 1536 

iii) Solve the following inequation, 2x -5 ≤ 5x+4 < 11, x belongs to R.

A) -3, -2,-1,0,1     B) 0, 1

C) {x: x belongs to R, -3≤ x ≤ 7/5

D) none 

iv) Frame the quadratic equation whose roots are 2/5, -1/2

A) x²+ x-2=0 B) x²+ x +2=0 C) 10x²+ x-2=0  D) 19x²+ x + 2=0 

v) The remainder on dividing f(x) by x-2 where f(x)= 5x² -7x +4

A) 10 B) 9 C) -9 D) -11

vi) If (2x -5):(3x+1) is the duplicate ratio of 2:3, find the value of x 

A) 6 B) 49 C) 49/6 D) 6/49

vii) Find the value of x if cos 63 sec(90- x)= 1

A) 27 B) 72 C) 18 D) 81

viii) Find the median of 31, 35, 27, 29, 32, 43,37,41, 34,28, 36, 44, 45, 42.

A) 34 B) 35.5 C) 44 D) 37

ix) Find the even number between 100 to 300 divisibile by 5.

A) 20 B) 3p C) 4p D) none

x) if the equation y= my+ c passes through the points (1,4( and (-2,-5) find the value of slope.

A) 1 B) 2/3 C) 5 D) none 

Question 2.                    2x10 = 20

i) A refrigerator is marked for sale at ₹17600 inclusive of GST. If the rate of GST is 10%, calculate

A) The list price of refrigerator.    

B) The amount of GST.                  (2)

ii) Mrs. Mathew opened a recurring deposit account in a bank with ₹500 per month for 2 years. Find the amount she will get on maturity if the intrest is paid on monthly balance at 12.5% p.a.              (2)

iii) Solve the inequation: 2x -1≥ x+ (7-x)/3 > 2. Show on the number line.   (2)                           

iv) 3x²+ 10x + 3=0.                         (2)

v) If 2x+1 is a factor of 6x³ + 5x² + ax - 2, find the value of a.              (2)

OR 

Find the equation of a straight line parallel to y-axis and passing through the point (-3,5).               (2)

vi) If X+ Y= 7       0 & X - Y= 3    0

                    2       5               0     3 then find the matrix X and Y.       (2)

vii) a) Point A(5,0) on reflection is mapped as A'(-5,0). State the equation of the mirror line.          

b) Point B(4,-3) on reflection is mapped as B'(4,3). State the equation of the mirror line.        (2) 

viii) Two right circular cylinder have equal volumes. If their heights are the ratio 4:9, find the ratio of their radii.                                            (2)

ix) A man 1.8 high stands at a distance of 3.6m from a lamp post and casts shadow of 5.4m on the ground. Find the height of the lamp.    (2)

x) Find the sum of the series 2, 7, 12, 17,...............20th term.

Question 3.                    3 X10= 30

i) If P is the solution set of - 3x+4 < 2x -3, x belongs to N, and Q is the solution set of 4x - 5< 12, x belongs to W, Find:

A) P∩Q           B) Q - P .                 

ii) 4x²- 4ax + (a² - b²)=0              

iii) Factorise completely: x³ + x² - 4x -4.                                                 

iv) If A= 1      -2 and B= 3        2

              2      -1              -2        1 find 2B - A².                                        

 OR 

Find the coordinates of the point that divides the line segment joining the point P(5,2) and Q(9,6) internally in the ratio 3:1

v) Prove: (1+ cos x)/(1- cos x)= (cosec x + cot x)².                        

vi) The radius of a sphere in 9cm. It is melted and drawn into a wire of diameter 2mm. Find the length of the wire in metres.                     

vii) The following table gives the doing wages of 50 works of a factory:

Marks: 20  70  50  60  75  90   40

F:           8   12  18   6    9    5     12 calculate the median marks.     

ix) A letter is chosen from the word TRIANGLE, what is the probability that it is vowel.                          

Or 

The nth term of an AP is 4n - 1. Find the common difference and 10th term of the AP.                             

x) Find x :

{√(2- x)+ √(2+ x)}/{√(2- x) - √(2+ x)} = 3.

OR 

If x= 6ab/(a+ b) prove (x+ 3a)/(x- 3a) + (x+ 3b)/(x- 3b) 

Question 4                         4x5= 20 

i) 'A' manufacturer motorbike at a cost price of ₹30600. Hence he sells it to a dealer B. B sells it to the dealer C. C sells it to a cust6. If the profit at each stage of selling chain is ₹1000 and the rate of GST is 12.5%, find

A) the total amount of GST paid by customer.

B) The amount which the consumer paid for the motorbike.              

OR 

On depositing ₹200, every month in a cumulative time deposit account, paying 9% p.a. intrest, a person collected ₹2517 at maturity. Find the period.                         

ii) In a flight of 2800km, an aircraft was slowed down due to bad weather. It's average speed for the trip was reduced by 100km/hr and time increased by 30 minutes. Find the original duration of flight.    

iii) If f(x)= 24x³ + px² - 5x +q has two factors 2x+1 and 3x -1, then find p and q. Also factorise completely.  

iv) The vertices of a triangle are A(10,4), B(4,-9) and C(-2,-1). Find the equation of the altitude through A.

OR 

Draw a histogram and find Mode:

Marks obtained.   No of students

24-29                             2 

29-34                             1

34-39                             5

39-44                             9

44-49.                           21

49-54                            10

54-59                             2

v) A box contains 25 cards, numbered from 1 to 25. A card is drawn from the box at random. Find the probability that the number on the card is:

a) even b) prime c) multiple of 6. 

OR 

Construct a triangle ABC in which AB= 5cm, BC= 8cm, and CA= 7cm. Draw the circumcircle 

     MODEL TEST PAPER -1
      SECTION A (Marks : 40)

Question 1)

a) Mamta has a cumulative Time Deposit Account in a Bank. She deposits ₹ 800 per month and gets ₹15198 as maturity value. If the rate of interest be 7% p.a, find the total time for which the account was held.                                           (4)

b) The vertical and slant height of a cone are 24cm and 25cm respectively. Calculate
i) curved surface
ii) volume of the cone.                 (4)

c)  The diameter of an iron sphere is 18 cm. The sphere is  melted and is drawn into a long wire of uniform cross section. If the length of the wire is 108 m,  find its diameter.  (4)

d) Solve:                                          (3)
2x/(x-4) + (2x -5)/(x-3) = 25/3.

e) The marks scored by 40 pupils of a class in a test test test in a test test test class in a test test test a test test were as:
Marks:         0    1    2     3     4    5
Frequency:  2    4   5     14  11   4
calculate the mean marks.          (3)

f) Find the sum of all the numbers between 100 and 200 which are divisible by 7.                                (3)

g) Solve 2 ≤ 2x - 3 < 5, x belongs to R and mark it on the number line. (3

h) Find the value of m if (x - m) is a factor of 3x³ + 2x² - 19x +3m.        (3

I) find x and y if
- 3    2        x       =  - 5
  0   - 5   .   2            y                       (3)

j) What must be added to each of the numbers 7, 15, 19 and 35 so that the resulting numbers numbers the resulting numbers are in the proportion?                                    (2)

k) Find the value of k so that 8k+4, 6k-2, and 2k+7 will form an A. P.  (2)

l) without using trigonometric tables, calculate:
2 tan 40°/cot 50° - cosec 61°/sec29°.                                    (1)

m) Using the remainder theorem, find the remainder when 7x³ + 5x² - 4x - 1 is divided by (x+1).          (2)

n) Find the equation of m so that the roots of the Equation (4-m)x² + (2m+4)x + (8m+1)= 0 may be equal.                                           (3)

        SECTION B(40 Marks)
    (Answer any four questions)

Question 2)
A) A shopkeeper bought a washing machine at a discount of 30% from a wholesaler, the printed price of the machine being 30000. The shopkeeper sells it to a consumer at a discount of 10% on the printed price. If the sales are intea-state and the rate of GST is 15%, find
a) the price inclusive of tax(under GST) at which the shopkeeper bought the machine.
b) the price which consumer pays for the machine.
c) the tax(under GST) Paid by the wholesaler to the State Government
d) the tax(under GST) Paid by the shopkeeper to the State Government.
e) the tax(under GST) received by the Central Government.            (4)

B) On depositing ₹200, every month in a cumulative Time Deposit Account, paying 9% p.a., interest, a person collected ₹2517 at maturity. Find the period.                                         (4)

C) Prove: (tanA + cot A) sinA . cos A = 1.                                               (2)

Question 3)

A) The point A(-3,0) on reflection in a line is a line is mapped as A'(3,0) and the point (2, -3) on reflection in the same line mapped as B'(-2,-3)
a) Name the mirror line.            (3)
b) Write the co-ordinates of the image of (-3, -4) in the mirror line.

B) Using quadratic formula, solve
4x² - 4ax + (a² - b²) = 0.             (5)

C) How many terms of the A. P 1, 4, 7,, ....are needed to give the sum 715 ?                                             (2)

Question 4)

A) The last term of an A. P 2, 5, 8, 11,....is x. The sum of the terms of the A. P is 155. Find the value of x.                                                         (3)

B) ABCD is a cyclic quadrilateral in which AB  and DC on being produced, meet at P such that PA= PD Prove that AD is parallel to BC.                                                            (3)

C) Show that X= 2   3
                             3   2 is the solution of the matrix Equation X² - 4X - 5I= O.                                                   (3)

Question 5)

A) The horizontal distance between two Towers is 140m. The angle of elevation of the top of the first Tower when seen from the top of the second Tower is 30°. If the height of the second Tower is 60 m,  find the height of the first tower.                                             (5)

B) The sum of the first fifteen terms of an arithmatic progression is 105 and the sum of the next fifteen terms is 780. Find the first three terms of the arithmatic progression.                                  (5)

Question 6)

A) The ratio of a sphere is doubled. Find the increase % in its surface area.                                              (3)

B) Find the equation of a straight line parallel to y-axis and passing through the point (-3,5).              (3)

C) Prove: (cotx +cosecx-1)/(cot x - cosec x +1)= (1+cosx)/sinx.      (4)

Question 7)

A) Find the equation of a straight line passing through (-1,2) and whose slope is 2/5.                   (3)

B) Using step deviation method, calculate the mean of the following frequency distribution:
Class           Frequency
50-60              9
60-70             11
70-80             10
80-90             14
90-100            8
100- 110         12
110- 120         11.                          (7)

Question 8)

A) On a map drawn to a  scale of 1: 250000, a triangle plot of kand has the following measurements. AB= 3 cm, BC= 4 cm and Angle ABC= 90°.
Calculate                                       (3)
a) the actual length of AB in km.
b) the area of the plot in sq.km

B) Prove: (1+cosA)/(1- cosA) = (cosec A + cotA)².                       (3)

C) Draw a histogram and hence estimate the mode for the following frequency distribution:
Class-interval    frequency
0-10                         2
10-20.                     8
20--30                    10
30-40                      5
40-50                      4
50-60.                     3                   (4)

TEST PAPER -- 1 (CLASS-XII) 20/21

                  Test Paper -1

                    -------***--------

          SECTION A.     (80 Marks)

          -------------------------------------

Question 1) Choose the Correct alternative: (MCQ).        1x10= 10

i) If sec⁻¹x= cosec⁻¹y state which of the following is the value of:  (cos⁻¹1/x+cos⁻¹1/y)

A) π   B) 2π/3    C) 3π/6   D) π/2

ii) If A= [aᵢⱼ ] is a 2x2 matrix such that aᵢⱼ= i+ 2j, then will be

A)1   3    B) 2   4   C) 3   5  D) none

    2   4         3   5        4   6  

iii) The value of ∫ e⁵ˡᵒᵍ ˣ dx is

A) (e⁵ˡᵒᵍ ˣ)/5+c  B) (e⁵ˡᵒᵍ ˣ)/(5 log x)+ c

C) x⁵/5 + c.     D) x⁶/6 + c

iv) The slope of the tangent to the rectangular hyperbola xy= c²  at the point (ct, c/t) is:

A) -1/t   B) -1/t²   C) 1/t  D) 1/t²

v) If the odds in favour of an event are 9:4, then its probability of occurrence is:

A) 9)13  B) 4/13  C) 4/9   D) 5/13

vi) the standard deviation of a binomial binomial a binomial of a binomial binomial a binomial binomial distribution with  parameters n and p is--

A) np                              B)√(np) 

C) √{np(1-p)}.               D)2√np

vii) If f(x)= [x] and g(x) = |x| then the value of f{g(8/5)} - g{f(-8/5)} is:

A) 2       B) 1       C) -1       D)-2 

viii) The value of dx/{1+√tan x} is

A) 0.    B) 1.    C) π/6.    D) π/12

ix) If A= 0    2   and KA= 0    3a

               3  -4                   2b  24 

Then K, a, b are respectively

A) 6,12,-18                 B) -4,6, 9

C) -6, -4,-9                  D) 6, -4, 9

x) If A' is the transpose of a square matrix A, then, 

A) |A| ≠ |A'|             B) |A| + |A'| 

C) |A| + |A'| =0.      D) |A| = |A'| only when A is symmetric matrix.

Question 2).     (10x2= 20 Marks)

i) A relation R is defined on the set of natural numbers N as follows: (x,y) ∈R => x+y= 12, for all x,y ∈ N. Prove that R is not transitive on N.

ii) ∫ sin x/cos 2x  dx

iii) Prove {cos(sin⁻¹x)}²= {sin(cos⁻¹)}²

iv) If A= cos t       sin t

             - sin t        cos t    prove that AA'= I

v) Find the Integrating factor of the differential equation (x+y+1) dy/dx = 1

vi) Five cards are drawn successively with replacement from a well-shuffled deck of cards. What is the probability that all the five cards are spades?

vii) Prove without expanding:

a - b     1     a          a       1        b

b - c     1     b   =     b       1        c

c - a     1     c          c       1        a

viii) lim ₓ→₀{log(1+ax)}/sin bx

ix) If y= logₓtan x, find dy/dx

x) Evaluate ∫ |sin x| dx at (π/2, -π/

Question 3).                               (4)

Show with determinants:

a² + 1      ab     ac

   ab      b²+1    bc

   ca         bc    c²+1 = 1+a²+b²+c².

Question 4).                                (4)

If sin⁻¹x + sin⁻¹y+ sin⁻¹z= π, then show that x√(1-x²) + y√(1-y²)+ z√(1-z²) = 2xyz.           OR

Prove: tan{π/4 + 1/2 cos⁻¹(a/b)} + tan{π/4 - 1/2 cos⁻¹x(a/b)}=2b/a

Question 5)                               (4)

If y= (sin⁻¹x)/√(1-x²), then prove (1-x²) dy/dx² - 3x dy/dx - y = 0

Question 6)                                 (4)

Evaluate: ∫ (x⁴+1)/(x⁶+1) dx     OR

Evaluate: ∫dx/(Cox + √3 sinx)

Question 7)                              (4)

Find the Equation of the tangent and normal to the curve:               x= a sec³ t and y= a cos³ t at t= π/4.                                       

                 OR

Find the intervals in which the function f(x)= 20 - 9y+ 6x²- x³ is.    

A) strictly increasing

B) strictly decreasing

Question 8).                            (6)

A window is in the shape of a rectangle with a semi-circular covering at the top. If the perimeter of the window is p(constant), find its maximum area.

                      OR

Of all the closed right circular cylindrical cans of volume 128π cm³, find the dimensions of the can which has minimum surface area.

Question 9)                             (4)

A dice is thrown 3 times. If getting a six is considered a success, find the probability of

A) 3 success

B) atleast two success.

          SECTION - C.   (20 Marks)

Question 10)

A)  Given that the cost function and revenue function respectively as C(x)= x+40 & R(x)= 10x - 0.2x²,  find the break even point         (2)

B) Equations of two lines of Regression are 4x+ 3y+7=0 and 3x+4y+8= 0. find the regression coefficient of y on x.               (2)

C) The demand function of a monopolist is given by p= 100 - x - x². find

i) the revenue function.          (2)

ii) marginal revenue function.   

Question 11).                          (4)

Find the equations of two lines of regression from the following observations:

(3,6),(4,5),(5,4),(6,3),(7,2).

 Find the estimate of y for x=2.5

                    OR

The lines of regression of a set of data are 8x-10y+66= 0 and 40x- 17y= 224. The variance of X is 9. Find

  i) the mean value of x and y

 ii)  Coefficient of x on y and y on x, hence find correlation coefficient.

 iii) Standard deviation of y.

Question 12)

 A profit making company wants to launch a new product. It observes that the fixed cost of the new product is Rs 35000 and the variable cost per unit is Rs 500. The revenue received on the sale of X units is given by 500x - 100x².  Find 

(i) The profit function,

ii)  break even point.                   (4)

                           OR

A Farm has the following total cost and demand functions:

  C(x)= x³/3 - 7x² + 111x + 50 and p= 100- x

 Find the profit maximizing output.

Question 13).                            (6)

Solve the following LPP by graphical method 

Minimise Z= 18c + 10y subject to 4x + y ≥ 20, 2x+ 3y≥30, x, y ≥ 0.

REVISED QUESTIONS- 12(20/21)

       INVERSE TRIGONOMETRY

                  MATRIX

               MARKS -2

1) If M= 1         2

               2         3 Find the value of k and M² - kM - I= 0

2) If A= 3       1

              7       5 , find x and y so that A²+ xI= yA.

3) If the matrix A= 6      x        2

                                 2    -1        2

                               -10    5       2 is a singular matrix, find the value of x.

4) If A= 3         -2

              4         -2 , find x such that A²= xA - 2. Hence, find inverse of A.

5) If A = x²          B= 2x          C= 7

               y²                3y               -3 and A+ 2B = 3C, then find x and y

6) If (B - 2I)(B - 3I)= 0, where. 

B =4           2

     -1           y and I is the unit matrix, find the value of y.

7) If A= 5         4. And B=  1       -2

              1         1                 1        3 with the relation AX= B. Then find the matrix X

8) If A = 2       3

               1       2 , find inverse of A

9) If A= cost          sint

             - sint          cost , show that A. A' = I

10) For what value of x is the given matrix 2x+4       4

             x+5         3 a singular matrix.             

                  MARKS -4

1) If A = [x    4       1]

B= 2         1           2

      1         0           2

      0         2          -4

C= x

      4

     -1 with the relation ABC= 0, Find x.

2) If A= 3

              1

             -2

B= [1      -5           7]

Verify (AB)' = B' A'

3) Solve: 2x - 3y =1; x+5y=7

4) If A = 3       1

               7       2 , find inverse of A, hence Solve the following Equation: 3x+7y= 4; x+2y= 1

                     Marks - 6

1) Evaluate:

3        -2      3             -1       -5        -1

2         1     -1      x     -8       -6         9

4        -3      2            -10       1         7

Hence, Solve system of equation, 3x-2y+3z=8; 2x+y-z=1; 4x-3y+2z=4.

2) Solve: x - 2y - 2z - 5= 0; -x + 3y +4= 0; - 2x + z - 4=0

3) Solve: 2/x + 3/y + 10/z= 4;

4/x - 6/y + 5/z= 1;

6/x + 9/y - 20/z= 2

4) If A= 4         2         3

              1         1         1

              3         1         -2 , find inverse of A. Hence, solve the following system of linear equations: 4x+2y+3z= 2, x+y+z= 1, 3x + y - 2z= 5.

5) If A= 2        0          -1

              5        1           0

              0        1           3 find inverse of A.

6) If A= 1       2         5

              2       3         1

             -1       1         1 , Find A⁻¹ and verify A⁻¹A = I= A A⁻¹

ANSWER: 2 Marks

1) k= 4. 2) x=8, y=8. 3) -3

4) x=1 , inverse of A= -1 1

                                       -2 3/2

5) x= -7, 3 and y= -3,-3. 6) 1

7) -3 -14

      4 17

8) - 1 2

       2 -3

10) x= 4

Marks 4:

1) x= -2 or -1. 3) x=2, y= 1

4) -2 1

     7 -3 , x= -1, y= 1

Marks 6:

1) -17; x=1, y=2, z= 3

2) -31/11, -25/11, -18/11

3) x=2, y=3, z=5

4) -3 7 -1

   1/8. 5 -17 -1

                   -2 2 2

x= 1/2, y= 3/2, z= - 1

5) 3 -1 1

  -15 6 -5

    5 -2 2

6) 2 3 -13

1/21x. -3 6 9

               5 -3 -1

                  2 Marks

                  -----------

A) Solve:

.*1) sin⁻¹cos(sin⁻¹x) = π/3.       ±1/2

.2) tan(cos⁻¹x) = 2/√5.             ± √5/3

.3) sin⁻¹(6x)+sin⁻¹(6√3x)=-π/2.  ±1/12

.4) cos(tan⁻¹x)= sin(cot⁻¹3/4).  ±3/4

5) tan⁻¹{(1-x)/(1+x)}= 1/2 tan⁻¹ x, x> 0.      1/√3

6) Cos (tan⁻¹x) = sin(cot⁻¹3/4)    3/4

7) cos⁻¹x + sin⁻¹x/2= π/6.        ±1

B) Prove:

.1) tan(2 tan⁻¹1/5) - π/4) = -7/17 

.2) sec²(tan⁻¹2) +cosec²(cot⁻¹3)= 15

.3) tan⁻¹1/4 +tan⁻¹2/9=1/2 sin⁻¹4/5

.*4) 1/2 cos⁻¹{(1-x)/(1+x)}=tan⁻¹√x

*.5) tan⁻¹{1/(1+2x)}+tan⁻¹{1/(4x+1)} = tan⁻¹(2/x²)

6) 4(2tan⁻¹1/3 + tan⁻¹1/7)= π

7) tan⁻¹1/2 + tan⁻¹1/5+ tan⁻¹1/8 = π/4

8) 2 sin⁻¹x = sin⁻¹{2x√(1-x²)}

9) (1/2)tan⁻¹x = cos⁻¹[√{(1+√(1+x²)}/{2√(1+x²)} ]

10) If sin⁻¹x +sin⁻¹y= 2π/3 then find the value of cos⁻¹x + cos⁻¹y. π/3

                      4 marks

                      ------------

A) Solve:

.1) sin⁻¹{x/√(1+x²)} - sin⁻¹{1/√(1+x²)} = sin⁻¹{(1+x)/(1+x²)}.                2

.2) Sin⁻¹5/x + sin⁻¹12/x= π/2, x≠ 0 ±13

.3) sin⁻¹x + sin⁻¹(1-x)= cos⁻¹x, x≠0. 

       1/2

4) tan(cos⁻¹x)= sin(tan⁻¹2).    ±√5/3

B) PROVE: 

1) 2tan⁻¹1/2 + tan⁻¹1/7= tan⁻¹31/17

.2) tan⁻¹1/2 = π/4 -1/2 cos⁻¹4/5

.3) tan[sin⁻¹1/√17] + cos⁻¹9/√85 =1/12

.4) sin[2tan⁻¹3/5 - sin⁻¹7/25]= 304/425

.5) 2 tan⁻¹1/5 + cos⁻¹7/5√2+ 2 tan⁻¹1/8= π/4

.6) cos⁻¹63/65 +2 tan⁻¹1/5= sin⁻¹ 3/5

7) Sin⁻¹{x/√(1+x²)} +cos⁻¹{(x+1)/√(x²+x+2)}= tan⁻¹(x²+x+1)

8) sin⁻¹12/13 + cos⁻¹4/5 + tan⁻¹63/16 =π

.9) If sin⁻¹x + tan⁻¹x= π/2 then prove 2x² + 1= √5

.10) If cos⁻¹x+ cos⁻¹y+ cos⁻¹z= π then show x²+y²+z²+2xyz= 1

11) If tan⁻¹x + tan⁻¹y+ tan⁻¹z = 0 then show x+y+z= xyz

12) cos⁻¹x/a + cos⁻¹y/b= K then show x²/a² - (2xy cos K)/ab + y²/b² = sin² K

13) 1/2 tan⁻¹x = cos⁻¹√[{1+√(1+x²)}/2√(1+x²)]