QUICK REVISION (XII)

REVISION

LINEAR PROGRAMMING

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

SOLVE GRAPHICALLY:

1) 3y - 2x < 4, x+ 3y > 3 and x + y≤ 5.

2) 2x + y- 3 ≤ 0, 2x+ y- 6 > 0.

DETERMINE THE MAXIMUM AND MINIMUM VALUES:

1) f(x,y)= 3x + 5y, vertices at (4,8), (2,4), (1,1),(5,2).     
          Max 52 at (4,8), Min 8 at (1,1)

2) f(x,y)= x + 4y, vertices at (0,7), (0,0), (5,4),(6,2).     
          Max 28 at (0,7), Min 0 at (0,0)

3) Z= x+ 3y subject to x,y ≥0, 5x + 2y≤ 20, 2y ≥ x.   
        Max 30 at (0,10), Min 0 at (0,0)

4) f(x,y)= 10x + 12y, subject to x, y ≥ 0, 2x + 5y ≥22, 4x + 3y ≥ 28, 2x + 2y ≤ 17.     
    Max 97 at (5/2,6), Min 562/7 at (37/7,16/7)

5) 5x + y≥ 10, x + y ≥ 6, x + 4y≥ 12, x≥ 0, y ≥0. Z= 3x + 2y         min 13 at (1,5)

6) x - y≤ - 1, - x + y ≤ 0, x≥ 0, y ≥0. Z= 3x + 4y                  no solution exists

7) A shopkeeper deals in two items - wall hangings and artificial plants. He has ₹15000 to invest and a space to store at most 80 pieces. A wall had hanging costs him ₹300 and an artificial plants ₹150. He can sell a wall hanging at a profit of ₹50 and an artificial plant at a profit of ₹ 18. Assuming that he can sell all the items that he buys, formulate a linear programming problem in order to maximise his profit.      

8) A housewife wishes to mix together two kinds of food X and Y in such a way that the mixture contains at least 10 units of vitamins A.12 units of Vitamin B and 8 units of Vitamin C. The vitamin contents of one kg of foods X and Y are as below:
                Vit A       Vit B        Vit C
Food X     1              2              3
Food Y     2              2              1
One kg of food X costs ₹6 and 1 kg of food Y costs ₹10. Formulate the above problem as a linear programming problem, and use iso- cost method to find the least cost of the mixture which will to produce the diet.                                Min ₹52

9) A man has ₹1500 for purchase of rice and wheat. A bag of rice and a bag of wheat cost ₹180 and ₹120, respectively. He has storage capacity of 10 bags only. He earns a profit of ₹ 11 and ₹9 per bag rice and wheat respectively. Formulate an LPP to maximize the profit and solve it.                  5 rice bags and 5 wheat bags, Max. Profit=₹100

1p) A manufacturer makes two types of the tea-cups, say, A and B. Three machines are needed for their manufacturing and the time (in minutes) required for each Cup on the Machine is given below:
Cup                Machine
                I              II                III
A            12           18               6
B              6             0                9
Each machine is available for a maximum of 6 hours per day. If the profit on each Cup A is 75 paise and on each cup B is 50 paise, show that 15 tea- cups of  type A and 30 of type of B should be manufactured in a day to get maximum profit.    

11) A factory owner purchases two types of machines. A and B, for his factory. The requirements and  limitations for the machines are as follows:
Machine   Area       labour   daily
             occupied     force    output
                 by the         each      on
              machine   machine  units
A           1000 sq.m   12 men    60
B           1200 sq.m    8 men     40
He has an area 9000 sq.m available and 72 skilled men who operate the Machines. How many Machines of each type should he buy to maximize the daily outputs ?        4 type A, 3 type B or 6 type A, no machine of type B.

12) A dietician wishes to mix two types of food in such a way that the vitamin contents of the mixture contain atleast 8 units of Vitamin A, and 10 units of Vitamin C. Food I contains 2 units/kg of Vitamin A and 1 unit/kg of vitamin C. Food II contains 1 unit/kg of Vitamin A and 2 unit/kg of Vitamin C. It costs ₹5  per/kg to purchase food I and ₹7 per/kg to purchase Food II. Determine the minimum cost of such a mixture.                 2 kg food I, 4 kg of Food II. Min cost= 38

13) A factory manufacturer produces two types of screws. A and B each types  requiring the use of two machines - an automatic and a hand operated. it takes 4 minutes on the automatic and 6 minutes on the hand operated machine to manufacture a package of screws A. while it takes 6 minutes on the automatic and 3 minutes on the hand operated machine to manufacture a package of screws B, each machine is available for atmost 4 hours on any day. The manufacturer can sell a package of screws A at a profit of ₹7 and of screws B at a profit of ₹10. Assuming that he can sale all the screws he can manufacture, how many package of each type should the factory owner produce in a day in order to maximize profit ? Determine the maximum profit.                      Max profit ₹410 at (30,20)

14) A brick manufacture has two depot A and B, with stocks of 30000 in 20000 bricks respectively. He receives a orders from three builders P, Q and R for 15000, 20000 and 15000 respectively. The cost (in Rs) of transporting 1000 bricks to the builders from the depot are given below:
To/from   Transport cost per
                  1000 bricks (in Rs)
               P        Q.        R
A           40      20        20
B           20      60        40
The manufacturer wishes to find how to fulfill the the order so that transportation cost is minimum.
Formulate L. P. P.     

15) A company has two factories located at P and Q and has three depots situated at A, B and C. The weekly requirement of the depot at A, B, C is respectively 5, 5 and 4 units, while the production capacity of the factories at P and Q are respectively 8 and 6 units. The cost of transportation per unit is given below:
To/from.        Cost(in Rs)
                    A           B           C
P                16         10         14
Q.               10         12         10
How many units should be transported from each factory to each depot in order that the transportation cost is minimum.    

REVISION

MEAN THEOREM & TANGENT AND NORMAL

_______________&&&&_____________

1) Use lagrange's mean value theorem to determine a point P on the curve y= √(x² - 4) defined in the interval [2,4] where the tangent is parallel to the chord joining the end-points on the curve. (√6, √2)

2) Verify Rolle's theorem for the function f(x)= e ²ˣ(sin2x - cos 2x) defined in the interval [π/8, 5π/8].  

3) Given f(x)= (x - 3)log x prove that there is atleast one value of x in the interval [1, 3] which satisfies the equation x log x = 3 - x

4) Examine the validity and conclusion of Rolle's theorem for the function

f(x)= eˣ sin x, ∀ x ∈ [0, a] 

5) Examine the validity and conclusion of lagrange's mean value theorem for the function f(x)= x(x - 1)(x - 2) for every x ∈ (0, 1/2). Not acceptable

6) Verify Rolle's theorem for the function f(x)= log{(x² + ab)/(ax+ bx)} x ∈ [a, b] and 0 not belongs to [a,b]

7) Verify Rolle's theorem for the function f(x)= sinx + cos x - 1 in [0, a/2] and find the point where the derivation vanishes. π/4

8) Show that the function f(x)= x² - 6x + 1 satisfies the Lagrange's mean value theorem. Also find the coordinates of a point at which the tangent to the curve represent by the above function is parallel to the chord A(1, -4) and B(3, -8). B b(2, -7)

9) Verify lagrange's mean value theorem for f(x)= sinx - sin2x in [0,π].    

10) Verify Rolle's theorem for f(x)= x(x +3) e⁻ˣ⁾² [-3, 0].

11) Verify Lagrange's mean value theorem for the following function f(x)= 2x - x², 0≤x≤ 1

12) Check if Lagrange's mean value theorem is applicable to f(x)= 4 - (6 - x)²⁾³ in [5, 7].

13) Find c of the Lagrange's mean value theorem for the function f(x)= x(x - 2) in the interval to [1,2]. 3/2

Tangent and Normal

1) Find the point on the curve 9y²= x³, where the normal to the curve makes equal intercepts on the axes.

2) Find the Equations of the tangent to the curve x= sin 3t, 

y= cos 2t at t= π/4

3) Find the slope of the tangent to the curve y= 3x² - 6 at the point on it whose x-ordinate is 2.

4) Find the Equation of the tangent to the curve y= x⁴-6x³+13x²-10x+5 at the points x=2, y=0

5) The equation of the tangent at

(2,3) on the curve y²=ax³+b is.  y=4x-5. Find the values of  a and b.

6) Find the equation of tangents to the curve y=cos(x+y), - 2π ≤ x ≤ 2π that are parallel to the line x+2y=0 

7) If the tangent to the curve y= x³+ax+b at (1, - 6) is parallel to the line x - y + 5=0 find a, b.

8) Find the Equation of the tangent to the curve x= sin 3t, y= cos 2t at t=π/4. Show that the curves 2x= y² and 2xy = k cut right angles If k² = 8.

9) Find the Equation of tangent lines to the curve y= 4x³ - 3x +5 which are perpendicular to the line 9y+x+3=0.

                         

     

1) Solve: cos(sin⁻¹x)= 1/7.    ±4√3/7

2) Use Matrix rule to solve 2x+ 3y= 10 and x+ 6y= 4.               16/3, -2/9

3) Find dy/dx if y+ sin y= x².           x sec²(y/2)

4) Let f: X --> be a function defined on a relation R on X is given by R={(a,b): f(a)= f(b). Show that R is an equivalence relation on X.   

5) If f: R--> R is defined by f(x)= (3 - x³)¹⁾³. then find f of(x).                   x

6) consider f: R₊ -->[-9, ∞] given by f(x)= 5x²+6x-9, prove that f is invertible with f⁻¹(y)={√(54+5y) -3}/5.    

7) Value of tan(2 tan⁻¹ 1/5).     5/12

                                       √x...∞

                                  √x

8) Differentiate: √x     with respect of x .                         y²/{x(2- log x)}

9) solve for x if x²     x     1 

                           0      2     1 = 28

                           3      1     4    -17/7, 2

10) Inverse of:

   -2     5                     -4/23       5/23

    3     4                      3/23       2/23

11) Using Matrix rule to solve the system of equation: 5x+ 7y =-2 and 4x + 6y =-3.                        9/2, -7/2

12) solve x: cos⁻¹x+ sin⁻¹(x/2)=π/6.        ±1

13) dy/dx of cos⁻¹{√(1-cos x)/2}with respect to x.               -1/2

14) If f is an invertible function defined as f(x)= (3x-4)/5, then find f⁻¹(x).                                    (5x+4)/3

15) If y=√{1-cos 2x/(1+cos 2x)}, find dy/dx.                                Sec²x

16) A company wants to launch a new product. it invested ₹37500 as fixed cost of production. The revenue function for the sale of x units is given by 4825x - 125x². Find break even points.                   12,25

17) Does the Lagrange's mean value theorem apply to f(x)= ³√x; -1≤ x ≤ 1 ?                                            No

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

19) If y= tan⁻¹[{√(1+x²)-1}/x] then find dy/dx.                        1/{2(1+x²)}

20) Using determinant, Find the value of x, such that points (02), (1,x) and (3,1) are collinear.         5/3

21) if f(x)= 27x³ and g(x)= ³√x, then find g o f(x).                                    3x

22) If f: R--> R defined by f(x)=(3x+5)/2 is an invertible function, then find f⁻¹(x).                     (2x-5)/3

23) Without expanding the determinant show that

1     a        b+c 

1     b        c+a = 0.

1     c        a+b 

24) Using determinants show that the points (11,7), (5,5), (-1,3) are collinear.

25) The regression equation of Y on X is 3x- 5y= 13 and the regression equation of X on Y is 2x- y= 7. Estimate the value of x when y = 10.

  8.5

26) If eˣ + eʸ = eˣ⁺ʸ, prove that dy/dx + eʸ⁻ ˣ = 0.

27) Differentiate xˣ w.r.t.x.       xˣ(log x+1)

28) Find the local maxima and minima for the function x³ -12x.

            Min. at x=2 value: -16. Max at x=-2, value 16

29) If y= x + tan x, prove that cos²x d²y/dx² - 2y + 2x = 0.

30) lim ₓ→₃ (x⁴-81)/(x-3).           108

31) find dy/dx if y tan x - y² cos x + 2x = 0.                   (y sec²x + y² sin x +2)/(2y cos x - tan x).

31) Out of the following two regression line of regression of y on x; 3x+ 12y= 9, 3y+ 9x= 46.          3x + 13y= 9

32) Solve: 7x= 6y-8 and 13y= 9 + 5x by find Matrix method.    -7/9, 23/54

33) If A= 5       3

               -1      -2 find A⁻¹.   2/7   3/7

                                             -1/7  -5/7

34) verify rolle's theorem for the function f(x)= sin x in the interval (π/4, 3π/4).         

35) find the inverse of the matrix

1      2                          7      -2

3      7                         -3       1

36) The function f(x)= x⁴ - 62x² + Kx +9 attains its maximum value on the interval [0, 2] at x= 1. find the value of k.                                120

37) If y= 2 sin x + 3 cos x, find the value of d²y/dx² + y.                    0

38) Using metrix method, solve the following system of equations: 6x+ y - 3z= 5; x+ y - 2z= 5; 2x+ y + 4z= 8;            1, 2, 1

39) without expanding the determinant prove that

x+ y     y+ z       z+x

  z           x            y = 0

  1           1           1

40) If A= x 4 1 & B= 2     1     2

                                  1      0     2

                                  0      2    -4 and C= x

      4

     -1 then ABC= 0 then x is.       -2,-1

41) differentiate cos⁻¹{(1-x²)/(1+x²) with respect to x.                 2/(1+x²)

42) Using determinants prove that (11,7),(5,5) and (-1,3) are collinear.

43) If A=3 and B= 1    -5     7

               1

              -2 then verify (AB)'= B' A'

44) Find the coefficient of correlation between x and x, when Cor(x,y)= -16.5,.Var(x)=2.89 and Var(y)= 100.                                -0.97

45) The fixed cost of a new product is ₹18000 and the variable cost per unit is 550. If the demand function is p(r)= 4000 - 15x, find the break-even points.                              8,15

46) If y= √x + 1/√x, show that 2x dy/dx + y= 2 √x.

47) Among all points a positive numbers with sum 24. Find those whose product is maximum.   12,12

48) If x √(1+y) + y√(1+x)= 0, prove that dy/dx= -1/(1+x)²

49) If y√(1-x²) + x√(1- y²) = 1, prove that dy/dx= √{(1-y²)/(1-x²)}.

50) If sin y= x sin(a+y), prove that dy/dx= {sin²(a+y)}/sin a.

51) If ax² + 2hxy + by²+ 2gx + 2fy + c= 0, Find dy/dx.                               -{(ax+by+g)/(hx+ by+f)}

52) The demand function of a monopolist is given by P= 1500 - 2x - x². Find the marginal revenue for any level of output x. Also, find marginal revenue(MR) when x= 10.

           MR= 15004x-3x², .R= 1160

53) Find regression coefficients bᵧₓ and b ₓᵧ if ∑x=30, ∑y= 42, ∑xy= 199, ∑x²= 184, ∑y²= 318 and n= 6. Also , find p(X,Y).       -0.323, 0.458, -0.384

54) Show 1+ x      1        1 

                  1        1+ x      1 =x²(x+3)   

                  1          1       1+x  

55) Using Matrix method solve: 2x- 3y= 1, x+ 5y= 7.                          2, 1

56) find the interval in which the function f(x)= x³ - 12x² +36x +17 is an increasing or decreasing function.          Inc(-∞,2) and (6, ∞) and Dec [2,6]

57) Find the point of the curve 9y² = x³, where the normal to the curve makes equal intercepts on the axis.            (4, 8/3) & (4, -8/3)

58) find the equation of the tangent to the curve x= sin 3t, y= cos 2t at t= π/4.                            6√2 x - 8y-2= 0

59) Find the slope of the tangent to the curve y= x⁴ - 6x³ + 13x² - 10x+5 at the points x=1, y=0.          2x-y= 2

60) It is given that for the function f(x)= x³ + bx² + ax +5 on [1,3] , Rolle's theorem holds with C= 2 + 1/√3. find the values of a and b.     11, -6

61) If y= sin[2tan⁻¹√{(1-x)/(1+x)}], prove that dy/dx= - x/√(2- x²).      

62) differentiate Cos⁻¹{x - x⁻¹)/(x + x⁻¹)} w r.t.x. -2/(1+x²)

63) Find the maximum value of

1              1             1

1          1 + sin x     1

1             1          1+ cos x.           1/2

64) Matrix A= 0      2b     -2 

                         3       1       3 

                        3a      3      -2 is given to be symmetric, find the values of a, b.           -2/3, 3/2

65) If A= 3    1

                7    2 find A⁻¹ and hence, Solve the following equestions: 3x+7y= 4, x + 2y= 1.    -2   1

                                       7   -3,  -1,1

66) find inverse of the matrix of:

Cos x         Sin x           cos x  - sin x

- sin x         cos x           sin x    cos x

67) Two lines of regression are given by x+ 2y= 5, 2x+ 3y= 8. calculate mean of x and y. regression coefficients of x on y and y on x.        1, 2, -1/2, -3/2, -√3/2

68) If y = tan⁻¹{4x/(1+5x²) + tan⁻¹{(2+3x)/(1+5x²), find dy/dx.

     5/(1+ 25x³)

69) If cos⁻¹x + cos⁻¹y+ cos⁻¹z=π, prove that x²+ y²+ z² +2xyz= 1.

70) Prove sin⁻¹12/13 + cos⁻¹4/5+ tan⁻¹63/16=π. 

** Evaluate:

71) lim ₓ→₀ (sin x - x + x³/6)/x³.     0

72) lim ₓ→₀(xeˣ - log(1+x))/x².    3/2

73) lim ₓ→₀ {log(1+x³)}/sin³x.         1

74) lim ₓ→₀(tan⁻¹x - x)/(sin x - x).   2

75) lim ₓ→₀ (x - sin x)/x³.             1/6

76) lim ₓ→₀{1+ sin x - cos x + log(1-x)}/(x tan²x).                            -1/2

77) Find the value of K if A=1     2 

                                                 2     3 and A² - KA - I= 0.                            4

78) Solve: cos⁻¹sin(cos⁻¹x)π/6.    ±1/2

MATH TEST PAPER (XII) -3

1) If two rows or two columns of a determinant are identical then value of the determinant is:

A) 0        B) 2       C) -1         D) 1 

2) The value of tan(π/2 - tan⁻¹1/3) is equal to 

A) 1/3     B) 3      C) 2/3       D) 3/2 

3) The domain for which the functions f(x)= 3x² - 2x and g(x)= 3(3x -2) are equal, will be..

A){1,2/3} B) {1,3} C) {2/3,3} D) {2/3,0} 

4) If y= tan⁻¹{(5-x)/(1+5x)}, then value of dy/dx=? 

A) -1/(1+x²) B) 1/(1+x²) C) 5 D) 5/(1+x²).

5) Solve: 2 sin⁻¹x= cos⁻¹x, 0<x<1.

A) 1/2 B) 1/3 C) 1 D) none

6) Let A={1,2,3}. Define a relation (on A) which is reflexive and symmetric but not transitive. 

A) (2,2),(3,3),(2,3),(1,2),(2,1)

B) (1,1),(2,2),(3,3),(2,3),(1,2),(2,1)

C) (1,1),(2,2),(3,3),(2,3),(1,2)

D) (1,1),(2,2),(3,3),(1,2)

7) If A= 8      0   &  B= 2      -2

              4     -2            -5       1 find another matrix X where 2A + 3X = 5B.

A) 2    10/3       B) -2       10/3

    11        3            -11         3 

C) -2    -10/3     D) -2        -10/3

    -11        3            11          -3 

8) If 2      3 =  x      3

        4      5    2x     5. Find the value of x.

A) 0       B) 1     C) -2       D) 2 

9) If y= sin⁻¹{2x/(1+x²) then dy/dx

A) 1/(1+x²) B) -1/(1+x²) C) 2/(1+x²) D) -2/(1+x²)

10) f(x)= 5 - | x - 1 |. Find the maximum value of f(x). Also find the value of x for which f(x) is maximum.

A) -5, -1  B) 5, 1  C) 1, 5  D) -1,-5 

11) If R₁ and R₂ are two equivalence relation defined on set A(≠ 0), then R₁∩R₂ is an

A) one-one function

B) onto function

C) equivalence relation

D) none 

12) If tan⁻¹x + tan⁻¹y + tan⁻¹z=π/2 and x+y+ z =√3, then.

A) x=y≠z B) x≠y≠z C) x≠y=z D) x= y=z.

13)     1      2       2

If A=   2     1        2

           2     2        1 then A²-4A= ?

A) 5      B) 4       C) 3         D) 1 

14) Solve: 3x+y+z=20; 3x+y-z= 0; 5x- 9y= 1;

A) 1,2,3  B) 2,1,3  C) 3,2,1  D) 1,3,2

15) If cos y= x cos(a+y), (a≠0), then dy/dx is

A) cos{(a+y)}/sin a 

B) cos a/sin (a +y)

C) cos²{(a+y)}/sin a 

D) sin²{(a+y)}/cos a 

16) If x= sin t, y= sin kt (k≠0, constant) then (1-x²) d²y/dx² - x dy/dx =?

A) - ky    B) -ky²  C) -k²y   D)- k²y² 

17) Find maximum and minimum values of (x²-x+1)/(x²+x+1).

A) -3, 1/3 B) -3, -1/3 C) 3, -1/3 D) 3, 1/3

            

TEST PAPER-2(State Board) For NOVEMBER

1) For what value of x is the given matrix 2x+4       4
              x+ 5       3  a singular matrix 
A) 3           B) 4.           C) 5        D) 6

2) If y= xʸ then value of x dy/dx
A) y/(1-y logx)
B) y²/(1- y logx).
C) -y²/(1- y logx)
D) -y/(1- y logx) 

3) Apply Rolle's theorem to find a point (or points) on the curve y= -1 + cos x where the tangent is parallel to the x-axis in [0,2π].
A) (π,0) B) (π,2) C) (π, -2). D) (-π,0)

4) if the following function is differentiable at x= 2, then find the values of a and b.
   f(x)=  x²,       If x≤ 2
           ax+ b,  If x > 2
A) 4,4     B) 4,-4.   C) -4,4     D) -4,-4

5) If y= (x+ √(1+x²)))ⁿ, then find the value of (1+x²) d²y/dx² + x dy/dx

A) ny     B) n²y.    C) ny²        D) n²y²

6) Find the equation of the tangent to the curve y= x² - 2x+7 which is parallel to the line 2x- y+9= 0
A) y-2x-3= 0.           B) y+2x-3= 0
C) y-2x+3= 0           D) y+2x+3= 0
          OR

Find the equation of the tangent to the curve y= x² - 2x+7 which is parpenducular to the line 15x- 5y -13 = 0
A) 36y-12x-227= 0.          
B) y+2x-3= 0
C) y-2x+3= 0          
D) 36y+12x -227= 0.

7) If f(x)= x |x| then dy/dx is 

A) 2x B) -2x C) 2|x| D) none

8) Let f: N --> N be a function defined as f(x)= 4x² + 12x+ 15. Find f⁻¹(31)
A) 1.       B) 4       C) -4          D) -1

9) The area of a right-angled triangle of given hypotenuse is maximum, when the triangle is
A) isosceles triangle.
B) equilateral triangle
C) scalene triangle
D) Isosceles right angled triangle

OR
All the rectangle inscribed in a given fixed circle, the ____ has the maximum area.
A) triangle                    B) square.
C) equilateral triangle
D) parallelogram

10) sin cos⁻¹(x) w.r.t. cos⁻¹x is 

A) cosx  B) - x C) x D) none

11) R is a relation in N x N  defined by (a, b) R(c, d) if and only if ad= bc. Then R is
A) equivalence relation.
B) Reflexive relation
C) Transitive relation
D) Identity relation

12) find the value of K if A²= 8A + KI, where A= 1      0
                      -1     7
A) 7     B) -7        C) O            D) I

13) If y= log{√(x+1)+√(x-1)}/{√(x+1) - √(x-1)}, then dy/dx is..
A) 1/(x²-1)              B) 1/√(x²-1).
C) 1/√(x² +1)         D) 1/(x²-1) 

14) If y= x sin 2x then find the value of x² d²y/dx² - 2x dy/dx is
A) 0    B) 2y    C) 4x²y D) -(2y+ 4x²y)

15) The value of c of Lagrange's mean value theorem if
f(x)= x(x-1)(x-2); a= 0, b= 1/2, i.e., for every x belongs to [0,1/2]
A) 0.24.   B) 2.4      C) 24      D) 42

16)         3x - 2,    0< x ≤ 1
If f(x)=2x² - x,    1< x ≤ 2
            5x - 4,         x> 2  then
A) Differentiable at x= 2, continuous at x= 2
B) not Differentiable at x= 2, but continuous at x= 2
C) Differentiable at x=2 , but not continuous at x= 2
D) Neither Differentiable nor continuous at x= 2

17) The curves 2x= y² and 2xy= k cut at right angles if k²=?
A) 2 B) 4 C) 6 D) 8.

18) If f(x)= (4x+3)/(6x-4), x≠ 2/3, then (f o f)(x)= ? Also find f⁻¹
A) x for all , f⁻¹= Intwger
B) x for all x= 2/3, f⁻¹= N
C) x for all x≠2/3, f⁻¹= f.
D) x for all irrational, f⁻¹= R

19) sum of the surface area of a rectangular parpalloid with sides x, 2x and x/3  and a sphere is given to be constant. Then the sum of their volume is minimum if x equal to
A) three times the radius of the sphere.
B) 2 times the radius of the sphere.
C) equal to the radius of the sphere
D) 4 times the radius of the sphere

20) For a certain establishment, the total revenue function R and the total cost function C are given by R= 83x - 4x² - 21 and C= x³ - 12x² + 48x + 11, where x= output. Obtain the output for with the profit is maximum.
A) 5 B) 6 V) 7. D) 8

21) the cost function of a Firm is given by C = x³/3 - 5x² + 30x + 10 where C is the total cost for x items. Determine x at which the marginal cost is minimum.
A) 5. B) 6 V) 7 D) 8

TEST PAPER -2 (ISC) For November

1) Solve for x: cos(tan⁻¹x)= sin(cot⁻¹3/4)

A) 1/4 B) 3/4. C) 5/4 D) 6/5

2) Evaluate: lim ₓ→₀{sin x -x)/x³}

A) 1/6 B) -1/6. C) 5/6 D) -5/6

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

              x+ 5 3 a singular matrix ?

A) 3 B) 4. C) 5 D) 6

4) If y= xʸ then value of x dy/dx

A) y/(1-y logx)

B) y²/(1- y logx).

C) -y²/(1- y logx)

D) -y/(1- y logx)

5) using properties of determinants evaluate

1+ a 1 1

   1 1+ b 1

   1 1 1+ c

A) abc

B) 1/a + 1/b +1/c 

C) abc(1/a + 1/b +1/c)

D) abc(1 + 1/a + 1/b +1/c).

6) Apply Rolle's theorem to find a point (or points) on the curve y= -1 + cos x where the tangent is parallel to the x-axis in [0,2π].

A) (π,0) B) (π,2) C) (π, -2). D) (-π,0)

7) if the following function is differentiable at x= 2, then find the values of a and b.

   f(x)= x², If x≤ 2

           ax+ b, If x > 2

A) 4,4 B) 4,-4. C) -4,4 D) -4,-4

8) If cos⁻¹x/a + cos⁻¹y/b = k, then the value of x²/k + y²/b² - sin²k is

A) 2xy B) 2xy/ab C) 2xy cos k D) (2xycos k)/ab.

9) If y= (x+ √(1+x²)))ⁿ, then find the value of (1+x²) d²y/dx² + x dy/dx

A) ny B) n²y. C) ny² D) n²y²

10) Find the equation of the tangent to the curve y= x² - 2x+7 which is parallel to the line 2x- y+9= 0

A) y-2x-3= 0. B) y+2x-3= 0

C) y-2x+3= 0 D) y+2x+3= 0

          OR

 Find the equation of the tangent to the curve y= x² - 2x+7 which is parpenducular to the line 15x- 5y -13 = 0

A) 36y-12x-227= 0.           

B) y+2x-3= 0

C) y-2x+3= 0           

D) 36y+12x -227= 0.

11) Find the interval in which the function f given by f(x)= sin x - cos x, 0 ≤ x ≤ 2π is

A) strictly increasing in (0,3π/4).

B) strictly decreasing in (3π/4,π/4)

C) Increasing at (0, π)

D) Decreasing at (0,-π)

12) Let f: N --> N be a function defined as f(x)= 4x² + 12x+ 15. Find f⁻¹(31)

A) 1. B) 4 C) -4 D) -1

13) The area of a right-angled triangle of given hypotenuse is maximum, when the triangle is

A) isosceles triangle.

B) equilateral triangle

C) scalene triangle

D) Isosceles right angled triangle

OR

All the rectangle inscribed in a given fixed circle, the ____ has the maximum area.

A) triangle B) square.

C) equilateral triangle

D) parallelogram

14) the fixed cost of a product is ₹ 18000 and the variable cost per unit is ₹550. If the demand function is p(x)= 400 - 150x, find the breakable values.

A) 8, 10 B) 8, 15. C) 10,15 D) 1,15

15) The cost function for a commodity is C(x)= ₹(200 + 20x - x²/2) find the marginal cost (MC). calculate also the marginal cost when x= 4 and interpret it.

A) 20+x, 10 B) 20- x, 10  

C) 20+x, 16 D) 20-x, 16

16) 2 sin⁻¹x = ?

A) sin⁻¹(x √(1+x²))

B) sin⁻¹(2x √(1+x²))

C) sin⁻¹(x √(1- x²))

D) sin⁻¹(2x √(1- x²)).

17) R is a relation in N x N defined by (a, b) R(c, d) if and only if ad= bc. Then R is 

A) equivalence relation.

B) Reflexive relation

C) Transitive relation

D) Identity relation

18) find the value of K if A²= 8A + KI, where A= 1       0

                       -1      7

A) 7 B) -7 C) O D) I

19) If y= log{√(x+1)+√(x-1)}/{√(x+1) - √(x-1)}, then dy/dx is..

A) 1/(x²-1) B) 1/√(x²-1).

C) 1/√(x² +1) D) 1/(x²-1)

20) If x         y          z

          x²        y²        z² = 0

          yz      zx        xy then find the value of (yz + zx + xy)

A) y- z B) z - x C) x - y D) 0

21) If y= x sin 2x then find the value of x² d²y/dx² - 2x dy/dx is

A) 0 B) 2y C) 4x²y D) -(2y+ 4x²y)

22) The value of tan⁻¹√x is

A) cos⁻¹{(1-x)/(1+x)}

B) cos⁻¹{(1+ x)/(1-x)}

C) 1/2 cos⁻¹{(1-x)/(1+x)}.

D) 1/2 cos⁻¹{(1+ x)/(1-x)}

23) The value of c of Lagrange's mean value theorem if 

f(x)= x(x-1)(x-2); a= 0, b= 1/2, i.e., for every x belongs to [0,1/2]

A) 0.24. B) 2.4 C) 24 D) 42

24) 3x - 2, 0< x ≤ 1

 If f(x)=2x² - x, 1< x ≤ 2

            5x -4, x> 2 then

A) Differentiable at x= 2, continuous at x= 2

B) not Differentiable at x= 2, but continuous at x= 2

C) Differentiable at x=2 , but not continuous at x= 2

D) Neither Differentiable nor continuous at x= 2

25) The curves 2x= y² and 2xy= k cut at right angles if k²=?

A) 2 B) 4 C) 6 D) 8.

26) If f(x)= (4x+3)/(6x-4), x≠ 2/3, then (f o f)(x)= ? Also find f⁻¹

A) x for all , f⁻¹= Intwger

B) x for all x= 2/3, f⁻¹= N

C) x for all x≠2/3, f⁻¹= f.

D) x for all irrational, f⁻¹= R

27) sum of the surface area of a rectangular parpalloid with sides x, 2x and x/3 and a sphere is given to be constant. Then the sum of their volume is minimum if x equal to

A) three times the radius of the sphere.

B) 2 times the radius of the sphere.

C) equal to the radius of the sphere

D) 4 times the radius of the sphere

28) the cost of producing x items per day is given in Rupees as C(x)= 2000 + 100√x. if each item can be sold for ₹10, then the break even point is .

A) 400 B) 500 C) 600 D) 700

29) the marginal cost C is given to be a constant multiple of number of units (x) produced. Find the total cost and the average cost function if the fixed cost is ₹1000 and cost of producing 30 unit is ₹2800

A) 2x²+1000 , 2x+ 1000/x.

B) 2x+1000/x , 2x²+ 1000

C) 2x²+1000x , 2x+ 1000

D) 2x+1000 , 2+ 1000/x

30) For a certain establishment, the total revenue function R and the total cost function C are given by R= 83x - 4x² - 21 and C= x³ - 12x² + 48x + 11, where x= output. Obtain the output for with the profit is maximum.

A) 5 B) 6 V) 7. D) 8

31) the cost function of a Firm is given by C = x³/3 - 5x² + 30x + 10 where C is the total cost for x items. Determine x at which the marginal cost is minimum.

A) 5. B) 6 V) 7 D) 8

TEST PAPER -1 CLASS- XII (2021-22)

1) Fill the gap: The value of the determinant  3    1975     1978
                        4    1982     1986
                        5    1995     2000  is_____
OR
State whether the following statement is true or false :
" the product of two non zero matrices must be a non-zero metrix."

2) If y= log log x, x> 1 which one of the following answer is true?
A) x dy/dx= 1  B)(x logx)dy/dx= 1
C) (logx)dy/dx= 1
D) (logx)dy/dx= x
OR
* If x= a(t - sint) and y= a(1+cos t) then which one of the following is the value of dy/dx.
A) - cot(t/2)        B) cot t
C) - tan(t/2)        D) cot(t/2)

3) If y= cos²x, then Which one of the following is the value of d²y/dx² ?
A) - 2 cos 2x    B) 2 cos 2x
C) -2 sin 2x      D) cos 2x

4) State whether the following statement is true or false:
the gradient of the tangent at the point (8,-4) to the parabola y²= 8(x-6) is - 1.
OR
* State whether the following is true or false:
f(x)= (x-1)(3-x) had an extreme value at the point x= 2.

5) The equation of the normal to the ellipse x²+ 4y²= 4 which is parallel to the line 8x+ 3y= 0 be..
A) 8x+ 3y= ±12   B) 16x+ 6y= ±15
C) 40x+ 15y= ±36  D) 24x+ 9y= ±15

6) The difference between the maximum and minimum values of the function f(x)= x³/3 - 2x² + 3x +1 is...
A) 4    B) 2     C) 1    D) 4/3.

7) The minimum value of 1/2 (7 - cos 2x) is...
A) 7/2     B) 4     C) 5/2      D) 3.

8) The length of the rectangle of maximum area that can be inscribed in a semicircle of radius 1 unit, so that two vertices lie on the diameter, is..
A) √2. B) 2  C) √2/3 D) √3 unit

9) If the tangent at the point P on the circle x²+ y²+ 6x + 6y= 2 meets the straight line 5x - 2y+6= 0 at the point Q on the y-axis, then the length of PQ is...
A) 4      B) 2√5   C) 5.     D) 3√5

10) Equations of the tangent and normal drawn at the point (6.0) on the ellipse x²/36+ y²/9= 0 respectively are...
A) x= 6, y= 0.  B) x+y= 6, y-x+6= 0
C) x= 0, y= 3  D) x= -6, y= 0

11) If the function f(x)= x²(x-2)² is an increasing function of 3, then
A) 1< x<2      B) x<0 
C)0< x<1 or x>2.  D)1< x<2 or 0< x

12) If f(x)=kx³ - 9x²+ 9x+3 is an increasing function then...
A) k< 3 B) k≤3 C) k>3 D) k is indeterminate.

13) The function f(x)= 1- x³ - x⁵ is decreasing for.....
A) 1≤ x≤5 B) all real values of x
C) x≤ 3.   D) x≥ 5

14) If v= 4πr³/3, then the rate (in cubic unit) at which v is increasing when r= 10 and dr/dt= 0.01, is ..
A) 4π.   B) π  C) 40π.   D) 4π/3

15) If the time rate of change of the radius of a sphere is 1/2π, then the rate of change of its surface area (in sq cm), when the radius is 5cm is..
A) 20      B) 10.    C) 4     D) 5

16) If f(x)= x      when 0≤ x ≤ 1
               2x -1 when x> 1 then
A) f(x) is discontinues at x= 1
B) f(x) is discontinues but not differentiable at x= 1
C) f(x) is differentiable at x= 1
D) none of these

17) Which of the following statements is not true?
A) a polynomial function is always continuous
B) a differentiable function is always continuous
C) a continuous function is always differentiable
D) log x is continuous for all x> 0

18) If the function
f(x)= (x²-9)/(x-3) when x≠ 3
          2x+a when x= 3 is continuous at x=3, then the value of a is...
A) 3      B) 6      C) 0        D) 4

19) If y= √(x+1) - √(x-1), then the value of (x²-1)d²y/dx² + x dy/dx is
A) 2y      B) -2y    C) y/4    D) y/2

20) In the mean value theorem f(b) - f(a) = (b - a) f'(c), (a<c<b), if f(x)= x³ - 3x -1, a= -11/7, b= 13/7 then the value of c is...
A) 0      B) 1      C) -1        D) ±1

21) The derivative of the function
tan⁻¹[{2x√(1-x²)}/(1-2x²)] w.r.t. the function tan⁻¹[{√(1+x²) - 1}/x] at x= 0 is...
A) 1     B) 2      C) 4           D) 8

22) If f(x)g(x)= k(a constant) and g"(x)/g'(x)= f"(x)/f'(x) + a.f'(x)/f(x) , then the value of a is...
A) 4      B) -4   C) 2    D) -2

23) If 2x= y¹⁾⁵ + y⁻¹⁾⁵ and (x²-1)y₂ + xy₁= ky, then the value of k is..
A) 5      B) -5       C) 25     D) -25

24) lim ₓ→∞{1 - 4/(x-1)}³ˣ⁻¹=
A) e⁴   B) e³    C) e¹²    D)1/e¹²

25) If the function f(x)= 4x³ + ax² + bx -1 satisfies all the conditions of Rolle's theorem in -1/4 ≤x ≤ 1 and f'(1/2)=0, then the values of a and bare
A) a=2, b =-3       B) a=1, b =-4
C) a=-1, b =4       D)  a=-4, b =-1

26) lim ₓ→∞ {(n-3)/(n+2)ⁿ is..
A) 1/e⁵ B) 1/e⁴ C) 1/e² D) 1/e

27) The value of c in Rolle's theorem f(x)= 2x³ - 5x² - 4x +3, x belongs [1/2, 3] is...
A) -1/3  B) 2/3    C) 2       D) -2

28) If sin⁻¹(x/5) + cosec⁻¹(5/4)= π/2, then the value of x is..
A) 1     B) 2      C) 3          D) 4

29) The equation sin⁻¹x - cos⁻¹x = cos⁻¹(√3/2) has
A) unique solution
B) two solution
C) no solution
D) infinite number of solutions

30) If x+y+z= xyz, then the value of (tan⁻¹x + tan⁻¹y + tan⁻¹z) is equal to
A) 3π/2   B) π    C) π/2     D) 2π

31) The value of cos⁻¹{(3+ 5 cosx)/(5+ 3cosx)} is...
A) 1/2 tan⁻¹(2 tan(x/2))
B) 2 tan⁻¹(1/2 tan(x/2))
C) tan⁻¹(1/2 tan x)
D) 2 tan⁻¹(2 tan(x/2))

32) If the determinant of the matrix a₁       b₁        c₁
            a₂        b₂        c₂
            a₃        b₃        c₃ is denoted by D, then the determinant of the matrix  a₁+3b₁- 4c₁    b₁       4c₁
              a₂ +3₂- 4c₂     b₂       4c₂
              a₃ +3b₃-4c₃    b₃       4c₃ will be
A) D B) 2D C) 3D D) 4D

33) If A= 3    - 5
               -4      2 , then the value of A² - 5A is equal to
A) I     B) 14I     C) O     D) none

34) If A= 1     2    and B= 1      2
                 2     3                 2      1
                 3     4
Then,
A) both AB and BA exist
B) neither AB nor BA exist
C) AB exists but BA does not exist
D) AB does not exist but BA exist

35) The value of the determinant b²c²         bc         b+ c
c²a²         ca         c+ a
b²a²         ab         a+ b
A) abc(a²+b²+c²)         B) 0
C) abc(bc+ca+ab)
D) (a+b+c)(a²+b²+c²) (ab+bc+ca)

36) The maximum value of the function 3cosx - 4 sinx -2 is..
A) 0   B)1   C) 4       D) 3

37) If y= cot⁻¹{(b- ax)/(a+ bx)}, then the value of dy/dx is...
A) 1  B) 1/(1+x²) C) -1 D)-1/(1+x²)

38) If y= cot⁻¹{8x⁴ - 8x²+1}, then the value of dy/dx.
A) 4/√(1-x²)       B) -4/√(1-x²)
C) 4/(1+x²)        D) - 4/(1+x²)

39) State whether sin(sin⁻¹(√2))= √2 is true or false
OR
Find y in terms of x where tan⁻¹{x/√(1- x²)}= sin⁻¹y.

40) tan⁻¹x + tan⁻¹y=π/4, then...
A) x+y+z+1=0     B) x+y+xy-1=0
C) x+y- xy+1=0   D) x+y- xy-1=0

REVISED CMA

16/8/21

1) If the 3rd and 6th terms of an AP are 7 and 13 respectively, find the first term and the common difference.                              3 and 2

2) find the sum of all natural numbers between 100 and 1000 which are multiple of 5.      98450

3) how many terms of the AP -6, -11/2, -5,.... are needed to give the sum -25 ?                            5 or 20.

4) Determine the sum of the first 35 terms of an AP if a₂ = 2 and a₇ = 22.                        2310

5) If the first term of an AP is 2 and the sum of first five terms is equal to one fourth of the sum of the next five terms, show that the 20th term is --112

6) Insert 3 arithmetic mean between 2 and 10.                  4,6,8

7) The sum of three decreasing numbers in AP is 27. If -1, -1, 3 are added to them respectively, the resulting series is in GP. The numbers are 

A) 5,8,13 B)15,9,3 C)13,9,5 D) 17,9,1

8) The sum of all odd numbers between 1 and 100 which are divisible by 3, is..

A) 83667 B) 90000 C) 83660 D) n 

9) If 7th and 13th terms of an AP be 34 and 64 respectively, then its 18th term is.

A) 87   B) 88   C) 89   D) 90 

10) If the sum of p terms of an AP is q and the sum of q terms is q, then the sum of the p + q terms will be..

A) 0    B) p-q   C) p+q   D) -(p-q)

11) If the sum of n terms of AP be n² - n and its common difference is 6, then its first term is..

A) 2    B) 3      C) 1     D) 4 

12) Sum of all two digit numbers which when divided by 4 yield Unity as reminder is..

A) 1200   B)1210.  C)1250.  D) n

13) In n AM's introduced between 3 and 17 such that the ratio of the last mean to the first mean is 3:1, then the value of n is..

A) 6      B) 8      C) 4     D) n 

14) The 1st and last terms of an AP are 1 and 11. If the sum of its terms is 36, then the number of terms will be.

A) 5     B) 6      C) 7         D) 8 

15) Find the sum of all odd integers from 1 to 1001.                      251001

16) If the ratio between the sums of n terms of two AP is (7n+1):(4n+27) find the ratio of their 11th term.   148: 111

17) If the sum of m terms of an AP be n and the sum of n terms be m, show that the sum of m+n terms is -(m+n).

18) If the sum of n terms of an AP is (pn+ qn²), where p and q are constants, find the common difference.                                   2q

19) In an AP, the first term is 2 and the sum of first five terms is one-fourth of the sum of next terms. Show that the 20th term is - 112 and the sum of first 20 term is -1100.

21) If the sum of n terms of an AP is given by (3n²+ 4n), find its rth term.                                        6r +1

22) The digits of a three-digit numbers are in AP and their sum is 15. The number obtained by reversing the digits is 594 less than the original number. Find the number.                                    852

23) Between 1 and 31, m numbers have been inserted in such a way that the ratio of 7th and (m-1)th numbers is 5:9. Find the value of m.                                 14

24) In the arithmetic progression whose common difference is non zero, the sum of the first 3n terms is equal to the sum of next n terms. Then the ratio of the sum of the first 2n terms the next to 2n terms is 

A) 1/5. B) 2/3  C) 3/4 D) none

25) If four numbers in AP are such that their sum is 50 and the greatest number is 4 times the least, then the numbers are:

A) 5,10,15,20     B) 4,10,16,22

C) 3,7,11,15       D) none

26) The first and the last term of an AP are a and l respectively. if S is the sum of all the terms of the AP. and the common difference is given by (l²-a²)/{k -(l+a)}, then k is

A) S   B) 2S      C) 3S    D) none

27) If the sum of the first n even natural number is equal to K times the sum of the first n odd natural numbers, then k is..

A) 1/n B) (n-1)/n  C)(n+1)/2n D)(n+1)/n  

28) If the first, second and last term of an AP are a,b and 2a respectively, then its sum is 

A) ab/{2(b-a)}       B) ab/(b-a)

C) 3ab/{2(b-a)}     D) none

29) If x is the sum of an arithmetic progression of n odd number of terms and y the sum of the terms of the series in odd places, then x/y is

A) 2n/(n+1)               B) n/(n+1)

C) (n+1)/2n               D) (n+1)/n 

30) If the first term of an AP is 2 and common difference is 4, then the sum of its 40 terms is

A) 3200 B) 1600 C) 200 D) 2800

31) The number of terms of the AP 3, 7, 11, 15, ... to be so that the sum is 406 is...

A) 5 B) 10 C) 12  D) 14   E) 20

32) If a(1/b+ 1/c), b(1/c + 1/a), c(1/a + 1/b) are in AP , then

A) a, b, c are in AP

B) 1/a, 1/b, 1/c are in AP

C) a, b, c are in HP

D) 1/a, 1/b, 1/c are in GP. 

33) If the sum of the three numbers in AP be 18 then what is the middle term ?                                       6

34) The fifth term and the 11th term of an AP are 41 and 20 respectively. Find the first term. What will be the sum of first 11 terms of the AP. ?          425/2

35) The n-th term of an AP is p. Show that sum of first (2n-1) terms is (2n-1)p.

36) The middle term, of an AP having 11th term is 12. Find the sum of the 11 terms of that progression.                               132

37) There are n arithmetic means between 4 and 31. If the second mean : last mean=5: 14 then find the value of n.                         8

38) If the sum of the first P terms of an AP be equal to the sum of the first Q terms then show that the sum of the first P +Q terms is zero.

) Find the sum upto n terms of the series 1²- 2²+ 3²- 4²+ 5²- 6²+.. ..          -n/2 (n+1) (n= 2r)

39) if the sum of p terms of an AP is to the sum of q terms as p²:q², show that (pth term)/(qth term)= (2p-1)/(2q-1).

40) The first term of an AP is a, the second term is b and the last term is c. Show that the sum is {(a+c)(b+c-2a)}/{2(b-a)}.

41) The sides of a right angled triangle are in AP. if the smallest side is 5cm then find the largest side.                                  25/3

42) find the sum of natural numbers from 1 to 200 excluding those divisible by 5.                           16000 

43) Show that the sum of all odd numbers between 2 and 1000 which are divisible by 3 is 83667 and of those not divisible by 3 is 166332.

44) Find the 14 A. M which can be inserted between 5 and 8 and show that their sum is 14 times the Arithmetic mean between 5 and 8.

45) Divide 25/2 into five parts in AP, such that the first and the last parts are in the ratio 2: 3.     2,9/4,5/2, 11/3, 3.

46) For what value of m, the sequence 2(4m+7), 6m + 1/2, 12m-7 forms an AP.               -3/4 

47) Find the 20th term of the AP 80, 75, 70,... Calculate the number of terms required to make the sum equal to zero.                         35 

48) Prove that if unity is added to the sum of any number of terms of the AP 3, 5,7,9...the resulting sum is a perfect square.

49) The sum of n terms of the series 25, 22, 19, 16,.. is 116. Find the number of terms and the last term. The given series is AP.  18405

50) Find the sum of all natural numbers from 100 to 300:

a) which is divisible by 4.       10200

b) excluding those which are divisible by 4.                        30000

c) which are exactly divisible by 5. 

d) which are exactly divisible by 4 and 5.               8200, 2200

e) which are exactly divisible by 4 or 5.                                        16200

Profit and Loss(Basic)

A) Find Selling Price:

1) C. P= ₹78, Profit= ₹30.

2) C. P= ₹121, Profit= ₹13.

3) C. P= ₹1028, Profit=₹ 329

4) C. P=₹ 998.50, Profit= ₹37.25.

5) C. P= ₹1937, Profit=₹ 789.

6) C. P= ₹12000, Profit= ₹3000.

7) C. P= ₹178.30, Profit=₹ 52.13.

8) C. P= ₹2.35, Profit= ₹5.75.

9) C. P=₹ 1999.99, Profit= ₹22.34

10) C. P=₹ 841, Profit= ₹329.

11) C. P= ₹178, loss= ₹49.

12) C. P= ₹782, loss=₹ 93.

13) C. P= ₹849.60, loss= ₹90.10.

14) C. P= ₹1009.11, loss= ₹39.21.

15) C. P= ₹896.79, loss= ₹321.

16) C. P= ₹8920, loss= ₹651.

B) Find the Cost Price:

1) S.P= ₹79, loss= ₹9.

2) S.P= ₹892, loss= ₹19.

3) S.P= ₹892, loss= ₹19.

4) S.P= ₹9927, Profit= ₹342.

5) S.P= ₹21.32, loss= ₹3.96.

6) S.P= ₹873, loss= ₹415.

7) S.P= ₹3975, Profit= ₹653.

8) S.P= ₹4321, Profit= ₹19.

9) S.P= ₹653, loss= ₹213.50.

10) S.P= ₹5641, loss= ₹327.

C) Find the Profit or Loss, when

1) S.P= ₹55, C. P= ₹72.60

2) C.P= ₹55, S. P= 72.60

3) C.P= ₹490, S. P= 416.50

4) C.P= ₹4000, S. P= 4160.

5) C.P= ₹567.77 S. P= 526.50

6) C.P= ₹8769, S. P= 9887.

7) C.P= ₹490, S. P= 416.50