MISCELLANEOUS QUESTIONS -XII(2023/24)

RELATION AND MAPPING

1) For all x∈ z, the function f: z --> is defined by f(x)= 3x+4, Find the function goz --> z  such that (gof=I.         g(x)= (x -4)/3 where x∈z

2) Find the number of equivalence relation on set A={a,b,c} containing elements (b,c) and (c,b).           2

3) If f(x)= sin x, g(x)= x² and h(x)= log x, find the composite function [ho(gof)](x).    2 log(sinx)

4) Let N be the set of all natural numbers and R be the relation on N x N defined by (x,y) R(z,t) => xt(y + z)= yz(x +t). 

 Check whether R is anequivalence relation on N x N.          Yes

5) Let R be the set of real numbers and X={x ∈ R : -1 < x < 1}= Y. Is the mapping f: x --> R defined by f(x)= (2x -1){1 - |2x -1|} bijective ?     Yes

TRIGONOMETRIC FUNCTION

1) PROVE:

a) tan⁻¹{(2 sin2x)/(1+ 2 cos2x)} -1/2  sin⁻¹{(3 sin2x)/(5+ 4 cos2x)}= x.

b) tan⁻¹{(3 sin2x)/(5+ 3 cos2x)} + tan⁻¹{(1/4  tanx)} = x.

c) 2 sin⁻¹(2/√13) + 1/2 cos⁻¹(7/25)+ tan⁻¹(63/16)=π.

2) If cos⁻¹x + cos⁻¹y + cos⁻¹z =π and x + y + z =3/2, then show that x= y= z.

3) If tan⁻¹a + tan⁻¹(1/b)= tan⁻¹3, then find the positive integral values of a and b.    1,2 or 2,7

4) If tan⁻¹[{√(1+ x²) - √(1- x²)}/{√(1+ x² + √(1- x²)}] = θ  ; then show that sin2θ  = x².

5) If (n tanθ)/cos²(α -θ) = {m tan²(α -θ}/cos²θ} then show that 2θ= α - tan⁻¹{(n - m)/(n + m)  tanα}.

6) If φ = tan⁻¹{x √3/(2k - x)} and θ = tan⁻¹{(2x - k)/k √3}, then show that one value of φ - θ is 30°

7) If sin⁻¹(x/a) + sin⁻¹(y/b) = sin⁻¹(c²/ab),  then show that b²x² + 2xy √(a²b² - c⁴) + a²y² = c⁴.

8) Solve: cos⁻¹x - sin⁻¹x = cos⁻¹(x √3).         0, ±1/2

DETERMINANT

By using properties of determinant. Prove that:

1) (x +4)        2x            2x

        2x         x +4          2x  = (5x +4)(x -4)²

        2x          2x           x+4

2) 1        1+ x           1+ x + y

     2        3+2x         1+3x +2y     =1

     3        6+3x         1+6x+3y

3) a         a²        1+ xa³ 

     b         b²        1+ xb³  

     c         c²         1+ xc³

= (1+ abcx)(a - b)(b - c)(c - a)

4) (y + z)²       x²          x² 

         y²       (z + x)²      y²  = 2xyz(x + y+ z)³

         z²            z²     (x + y)²

5)  - yz        y²+ yz        z²+ yz 

   x²+ zx         -zx           z²+ zx 

   x²+ xy      y²+ xy           -xy

= (xy + yz + zx)³

6) (y + z)²        z²           y² 

         z²        (z + x)²       x² 

         y²             x²      (x + y)²

= 2(xy+ yz + zx)³

7) ax - by - cz        ay+ bx              cx+ az

      ay + bx          by - cz- ax          bz+ cy

      cx+ az              bz+ cy          cz - ax- by

= (a²+ b²+ c²)(x² + y²+ z²)(ax + by + cz)

METRICES

1) If A=  -4       4       3  & B= 1     -1      1

               -7       1       3           1     -2     -2

                5      -3      -1           2      1      3 find AB and use this to solve the following system of equation: x - y + z=4; xx - 2y - 2z=0 ; 2x + y + 3z= 1.       3,-2,-1

2) Let X= 3       2      5

                 4       1      3

                 0       6      7 express X as sum of two matrices such that one is symmetric and other is skew-symmertric.

LIMIT, CONTINUITY AND DIFFERENTIABLE

1) lim ₓ→∞ √x{√(x +3) - √x}.                 3/2

2) lim ₓ→₀ {(x -1 + cosx)/x}¹⁾ˣ .             e⁻¹⁾²

3) lim ₓ→∞ {(x +5)/(x +1)}ˣ.                 e⁴

4) lim ₓ→π/2    (1+ cosx)³ˢᵉᶜˣ.             e³

5) lim ₓ→₀ (1+ 3x)^(x+3)/x.                 e³

6) lim ₓ→₀ {log(x²+ x+1)+ log(x²- x+1)}/(secx - cosx).       1

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

8) lim ₓ→₀ {tan(π/4  + x)}¹⁾ˣ.               e²

9) lim ₓ→₀ (sinx + cosx)¹⁾ˣ.                 e

10) lim ₓ→₀ {sin log(1+ x)}/log(sinx +1).        1

1) Prove that the function f(x)= sinπ |x| is continuous at x =0 but not differentiable at the same point.     

2) If g(x) is the universe of f(x) and f'(x)= (1+ x³)⁻¹, show that g'(x)= 1+ {g(x)}³.

3) A function f(x) is defined as follows:

                 - 2sinx,        when -π ≤ x ≤ -π/2

 f(x)=          a sinx + b, when -π/2< x < π/2

                     cosx,        when  π/2≤ x ≤ π

If f(x) is continuous in the interval -π≤ x ≤π. find the value of a and b.     -1,1

DIFFERENTIATION

1) Find dy/dx when

a) xˢᶦⁿ ʸ + yˢᶦⁿˣ =1.          

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

c) (√x)ˣ + (x)^√x.

d) y= tan⁻¹(√x - ⁴√x)/(1+ ⁴√x³).

e) xˡᵒᵍˣ + (sinx)ˣ+ 15x.

2) Find d²y/dx²

a) x= e⁻ᵗ and y = te⁻ᵗ.

b) x = a sin³t and y= a cos³t at t=π/4.

c) x= a(t - sint) and y= a(1+ cost) at t=π/2

d) x=√3(3 sint + sin3t) and y= √3(3cost + cos3t) at t=π/3.

e) if 3px²= y²(p - x⁶), show dy/dx = y³/x³ - 2y/x.

f) If y²(1- x²)= x²+1, show that (1- x⁴)(dy/dx)²= y⁴-1.

g) If √(1- x⁴) + √(1- y⁴)= k(x²- y²), show dy/dx= x√(1- y⁴)/y√(1- x⁴).

h) If f(x)= sin(logx) and y= f{(2x+3)/(3- 2x)} show that dy/dx= 12/(9- 4x²)  cos{log{(2x+3)/(3- 2x)}}.

i) If x= sect - cost, y= secⁿt - cosⁿt, then show that (x²+4)(dy/dx)²= n²(y²+4).

j) 

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