Revised Questions (IX)

        CUBE AND CUBOID

           -------------------------

1) A pit 7.5 metre long, 6 metres wide and 1.5m deep is a dug in a field in a field. find the volume of soil removed.                        67.5m³

2) Find the length of the longest pole that can be placed in an Indoor Stadium 24 metre long. 18 metre wide and 16m high.                 34m

‌3) The Length, breadth and the height of a room are in the ratio of 3:2:1. If volume be 1296m³, find its breadth.                                    12m

4) The whole surface of a rectangular block is 8788 square cm. If Length, breadth and height are in the ratio of 4:3:2, find its length.                                  52cm

5) Three metal cubes with edges 6cm, 8cm and 10cm respectively are melted together and formed into a single cube. Find the side of the resulting cube.               12cm

6) Find the number of bricks, each measuring 25cm x 12.5cm x 7.5cm, required to construct a wall 12m long, 5m high and 0.25m thick, while the sand and cement mixture occupies 5% of the total volume of wall.                                      6080

7) Seven equal cubes each of side 5cm are joined end to end. Find the surface area of the resulting cuboid.                                750cm²

 

8) In a swimming pool measuring 90m by 40m, 150 men take a dip. If the average displacement of water by a man is 8 cubic metres, what will be the rise in water level?

33.33cm 

9) A closed wooden box measures externally 10cm long, 8 cm broad and 6cm high. Thickness of wood is 0.5 cm. find the volume of wood used.                                           165

10) A cuboid of dimension 24cm x 9cm x 8cm is melted and smaller cubes are of side 3cm is formed. Find how many such cubes can be formed?                                      64

11) Three cubes each of volume of 216m³ are joined end to end. find the surface area of the resulting figure.                                   504m²

12) The area of of three adjacent faces of a cuboid are are x,y, z. if the volume is V, then V² will be equals to.                                     xyz

    

          

MODEL TEST PAPER-1 For (XII)

Question 1).                   2x6= 12

a) Find the maximum value of 

1        1           1

1    1+ sinx     1 

1        1     1+ cosx

b) Solve the equation for x:

cos(tan⁻¹x) = sin(cot⁻¹3/4)

c) Evaluate lim ₓ→₀ (sinx-x)/x³

d) ∫ x sinx/(1+ sinx)dx at (π, 0)

e) For what value of x is the given matrix 

2x+4        4

 x+5         3 a singular matrix

f) if Y= x ʸ , prove that 

x dy/dx = y²/(1 - y logx)

2) Using property of                     (4) determinants, prove that 

1+ a            1          1 

1                1+b       1 

1                  1       1+c

= abc(1 + 1/a +1/b+ 1/c)         

3) If cos⁻¹(x/a)+ cos⁻¹(y/b) = k

prove that x²/a² - (2xy cos k)/ab + y²/b² = sin² k.                                (4)

4) If y= {x+√(1+x²)ⁿ,                      (4)

then show that 

(1+x²) d²y/dx² + x dy/dx = n²y

5) ∫(3x+1)/√(5- 2x - x²) dx.            (4)

6) Find the equations of the tangent to the curve y²=x²-2x+7 which is 

i) parallel to the line 2x- y+9=0

ii) perpendicular to the line 5y -15x = 13.                                                (4)

                         OR

Find the intervals in which the function f given by f(x)= sinx - cosx, 0≤ x ≤2π is strictly increasing or strictly decreasing.

7) Solve:

A) dy/dx =(xy+y)/(xy+x).               (2)

B) dy/dx + 1= eˣʸ.                           (2)

8) Using metrics, solve the following equations:

5x+3y+z= 16, 2x+y+3z= 19, x+2y+4z= 25. (6)

                      OR

 If A = 1     -1       0 

           2      5       3

           0      2       1 , 

find inverse of A, 

9) Prove that the area of right angle triangle triangle triangle of right angle triangle triangle triangle of given hypotenuse is a maximum, when the triangle is isosceles.    (6)     

                          OR 

show that of all the rectangles inscribed in a given fixed circle, the square has the maximum the maximum has the maximum area.

10) Evaluate:

A) ∫ x² sin⁻¹x dx.                          (3)

B) ∫ x/(x² + 4x +3) dx.                 (3)

11) The fixed cost of the product is Rs18000 and the variable cost per unit is Rs550. If the demand function is p(x)= 4000 - 150x, find the break even even values.       (2)

12) Given x+ 4y = 4 and 3x+y=16/3

are regression lines. find the line of regression of x on y.                     (2)

13) the cost function for a commodity is commodity is 

C(x)=₹(200+20x - x²/2) 

A) Find the marginal cost(MC)

B) calculate the marginal cost when x= 4 and interpret it.                    (2)

14) Two regression lines are represented by 2 X + 3 Y - 10= 0 and 4X + Y - 5= 0. Find the line of regression of Y on X.                    (4)

                          OR

15) Fit a straight line line to the following data, treating y as the dependent variable.

 X: 1        2          3         4         5 

 Y: 7        6          5         4         3

Hence, estimate the value of y when x= 3.5

16) The marginal cost function of a firm is MC= 33 log x. Find the total cost function when the cost of producing one producing one cost of producing one producing one unit is ₹11.                                    (4)

                          OR

If the marginal cost of a commodity is equals to to half its average average cost, show that fixed cost is zero. If the cost of producing 9 units of the commodity is ₹60. find the cost function.

17) A manufacturer produces two products A and B. Both the products are are processed on two different machines. The available capacity of the first first machine is 12 hours and that of the second second machine the second machine is 9 hours per day. Each unit of a product A requires 3 hours on both machines and each unit of product B requires 2 hours on the first machine and 1 hour on the second second machine. Each unit of product A is sold at profit of ₹7 and that of B at a profit of ₹4. Find graphically the production level per day for maximum profit.               (6)

FULL SYLLABUS FOR MATHS (XII) 20/21

              INVERSE TRIGO

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

1) Simplify :

a) tan⁻¹[2 sin(2cos⁻¹√3/2)].    π/3

2) Solve:
a) tan⁻¹{(1-x)/(1+x)}=(1/2)(tan⁻¹x)
                                    Ans. x= 1/√3

b) tan⁻¹{(x-2)/(x-4)} +tan⁻¹{(x+2)/(x+4)} =π/4.      ±√2      

c) cos⁻¹x + sin⁻¹x/2 = π/6          1

d) sin (sin⁻¹1/5 +cos⁻¹x) = 1.   1/5

3) Prove:
a) cot⁻¹7+ cot⁻¹8 +cot⁻¹18= cot⁻¹3

b)sin⁻¹8/17+sin⁻¹3/5=cos⁻¹36/85

c)tan⁻¹√x=(1/2) cos⁻¹{(1-x)/(1+x)}

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

e) tan⁻¹3/4 +tan⁻¹3/5- tan⁻¹8/19
= π/4

f) tan(2tan⁻¹1/2 - cot⁻¹3) = 9/13  

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

       L'HOSPITAL THEOREM

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

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

2) lim ₓ→₀ (x - tanx)/x³.           -1/3

3) lim ₓ→₀ (1- tanx)/cos2x.        1

4) lim ₓ→₁ (1- logx - x)/(1--2x+x²)

                                                     -1/2

5) lim ₓ→₀ (x - tan⁻¹x)/(x- sinx).  2

6) lim ₓ→π/2.  tan 5x/tanx       1/5

7) lim ₓ→₀ (cosecx  - 1/x).         0

8) lim ₓ→π/2  (xtanx - π/2 secx)   -2

9) lim ₓ→π/2  (cosx logtanx).       0

10) lim ₓ→₀  (1+ sinx) ᶜᵒᵗ ˣ            e

11) lim ₓ→₀  (1- cosx)/x².          1/2

12) lim ₓ→₀ (sinx -x + x³/6)/x³.     0

13) lim ₓ→₀ logx/ cotx.                  0

      INCREASING-DECREASING 

                  FUNCTION

                 --------------------

Find the interval innovate the 1) Find the intervals in which function f(x)= 2x³+9x²+12x+20 is A) increasing          B) decreasing
   Increasing (-∞,-2)and [-1,∞)
   Decreasing [-2, -1]

2) Find the interval in which the function f(x)= 3x⁴- 4x³-12x²+5 is
A) strictly increasing
B) strictly decreasing.
                    A) (-1,0) and (2, ∞)
                    B) (-∞, -1) and (0,2)

3) Find the interval in which the function f given by
f(x)= (x-1)(x+2)² is
A) strictly increasing
B) strictly decreasing
                           A) (∞,-2)∪(0, ∞)
                           B) (-2,0)

4) Find the interval in which the function given by
f(x)= sinx + cosx, 0 ≤ x ≤ 2π is
A) increasing           B) decreasing
               Decreasing in[π/4, 5π/4]

        TANGENT AND NORMAL.     

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

1) At what point on the curve
y= x² does the tangent make an angle of 45° with the x-axis?
                                               1/2,1/4

2) Show that the line x/a + y/b= 1 touches the curve y= be⁻ˣ⁾ᵃ at the point when the curve intersects the axis of y.

3) Find the point (S) on the curve x²/9 + y²/4 = 1, where the tangent is parallel to the y-axis.            (±3,0)

4) Find the Equation of the tangent and normal to the curves x²/a² + y²/b² = 1 at the point (√2a, b)                                            √2 bx- ay-ab=0,
                   ax+√2 by - √(a²+b²)= 0

5) Find the equation of the tangent to the curve y=x²-2x+7  which is
A) parallel to the line 2x- y +9= 0
B) perpendicular to the line 5y-15y = 13.    (2x - y+3=0, 36y+ 12x - 227= 0)

6) Find the equation of the tangent to the curve 4x²+9y²=36 at the point (3 cost,2sint).     2xcost+3ysint-6=0

7) Find the slope of the tangent to the curve y= 3x² - 4x at the point whose x-co-ordinate is 2.             8
                        

8) Find the points on the curve y²= x³ - 11x +5 at which the equation of the tangent is y=x-11   (2,-9)&(-2,19)

9) Find the equation of the tangent to the curve x²+3y= 3, which is parallel to the line y-4x+5=0.                                                     4x - y+13 = 0           

10) Find the equations of the normal to the curve y= x³+2x+6,  which is parallel to the line, x+14y+4= 0.                                                  x+ 14y= 254 and x+ 14y= - 86

11) Find the equations of the tangent and normal to the curve x= 1 - cos k, y= k - sin k at k=π/4
          4√2 x + (8 -4√2)y =π(2 - √2)