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

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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.     

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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)