REVISION - XII

DETERMINANT 

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

a) 0 b) 2 c) -1 d) 1

2) If D= 0     a    b

             -a     0    c

             -b    -c    0 then D is 

a) 0 b) 1 c) a d) b

3) a+b    a+2b    a+3b

    a+2b  a+3b    a+4b= 0

    a+4b  a+5b     a+6b

4) 1+ a      1        1

      1       1+ b     1 

      1         1      1+ c

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

5) loga       p      1

    logb       q       1 = 0 (a,b,c> 0)

    logc       r        1

6) x    b      c

    a    y       c

    a    b      z

= (x - a)(y - b)(z - c){x/(x -a) + y/(y- b) + z/(z - c)  -2}

7) a   a²   bc     1    a²     a³

    b   b²   ca=   1    b²     b³

    c   c²   ab     1    c²     c³

8) 2a       a- b - c          2a

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

  c- a-b         2c             2c

9) 1+ a²- b²     2ab       -2b

         2ab     1-a²+ b²    2a= (1+ a²+ b²)³

         2b          -2a    1- a²-b²

10) (a²+b²)/c        c           c

           a         (b²+c²/a       a         =4abc

           b                 b     (c²+a²)/b

11) b²c²       bc      b+ c

      c²a²        ca      c+a = 0

      a²b²        ab     a+ b

12) x        y         z 

      x²       y²        z²

    y+z    z+a      x+ y

= (x - y)(y-z)(z -x)(x+ y+ z)

13) x     x²        1+x³

       y     y²        1+ y³=0

       z     z²        1+ z³ show 1+ xyz=0

14) Find the value of 

    1                1               1

 (eˣ+ e⁻ˣ)²  (πˣ+ π⁻ˣ)²    2

 (eˣ -  e⁻ˣ)²  (πˣ- π⁻ˣ)²   -2

15) y+z    z+x   x+y   2x  2y  2z

      z+ x    x+y  y+z = 2y  2z  2x

      x+y     y+z  z+ x.  2z  2x  2y

= -(x³+ y³+ z³- 3xyz)

16) a      b       c 

    a- b  b- c   c - a = a³+b³+c³-3abc

    a+c  c+a   a+b 

17) 1    bc    a(b+c)

       1    ca    b(c+a)

       1    ab    c(a+b) simplify 

18) (a+b)²     ca      bc

          ca      (b+ c)² ab= 2abc(a+b+c)²

          bc        ab   (c+a)²

19) a-b-c     2a      2a

          2b    b-c-a    2b= (a+b+c)³

          2c      2c    c-a-b

20) 1    ab    1/a+ 1/b

       1    bc    1/b+ 1/c=0

       1    ca    1/c+ 1/a

21) (y+b)²     a²        a²

          b²     (a+y)²     b²= 2aby(a+b+y)

          y²         y²     (a+b)²

22) p     q-y    r-z

     p-x     q     r-z=0

     p-x    q-y    r 

Show that p/x + q/y + r/z =2.

23) Value of 

4²    4³     4⁴

4³    4⁴     4⁵

4⁴     4⁶     4⁷ is 

a) 4² b) 4⁵ c) 0 d) 4⁴

24) a          b       ax+ by

       b         c       bx+ cy

    ax+by  bx+ay    0

= (b²- ac)(ax²+ 2bxy + cy²)

25) 2   7      65

       3   8      75 =0

       5   9      86

26) -1   b      c

        a   1      c

        a  -b     -1

= (a+1)(b+1)(c+1){a/(a+1) + b/(b+1) + c/(c+1)  -1}

27) p    b     c 

       a    q     c=0

       a    b     r then show p/p-a) + q/(q - b) + r/(r - c) -2

28) 0        2016      -2017

   -2016       0           2018 = 0

    2017    -2018         0

29) a²        bc       c²+ ca

     a²+ab    b²         ca

       ab     b²+bc      c²    simplify 

30) 1        q/p      a+ q/p

       1        r/p       a+ r/q = 0

    pa+ q   qa+r        0 then show that pa²+ 2qa + r=0

31) a²+1     ab        ac

        ab      b²+1     bc= 1+ a²+b²+ c²

        ca        cb     c²+1

32) If A= 1     0

               1/2  1 then find A⁵⁰

a) 1      0 b) 1   0 c) 1   0 d) none

   1/10 0     50  1     25 1

33) a²     bc      ac+c²

    a²+ab  b²       ac = 4a²b²c²

       ab   b²+bc    c²

34) 9     9     12

       1    -3    -4 = 0

       1     9    12

Solve:

1) 3x + y+ z= 10

, x + y+ z= 0, 5x -9y1= 1

2) x-2     2x -3       3x-4

    x-4     2x-9        3x-16=0

    x-8     2x-27     3x-64

3) 3+x    3- x     3- x

     3-x    3+ x    3 - x =0

     3- x   3- x     3+x

4) x -1       1       1

      1       x-1      -1 = 0

      -1         1    x+1

5) x+1  x+2  x+a

    x+2   x+3  x+b= 0

    x+3   x+4  x+c

6) If x+2    3 

        x+5    4 =3 find x.

7) a - x      c        b

       c      b- x       a = 0

       b         a     c-x 

And a+ b+ c=0, then find the value of x.

8) x     c+ x     b+x

   c+x     x       a+ x= 0

   b+x   a+x      x

9) 

DIFFERENTIATION 

Short Question 

1) y= sin⁻¹{2x/(1+ x²)}

2)  If y= tan⁻¹{(5-x)/(1+ 5x)} then dy/dx is 

a) -1/(1+ x²) b) 1/(1+ x²) c) 5 d) 5/(1+ x²) 

3) The derivative of sin⁻¹{2x/(1+ x²)} w.r.t. tan⁻¹x is 

a) 1/2 b) 2 c) -1/2 d) 1

4) d/dx (xˣ)=? 

a) xˣ(1- logx) b) xˣ(1+ logx) c) x.xˣ⁻¹ d) xˣlogx

5) If y= cosx° then dy/dx is 

a) (-π/180) sinx° b) (π/180) sinx° c)  sinx° b d) (180/π) sinx° 

6) If f(x)= log{tan(x/2)}, then the value of f'(x) is 

a) sinx b) sec(x/2) cosec(x/2) c) - cosecx d) cosecx

7) The function u= eˣ sinx and v= eˣ cosx the value of u du/dx - v dv/dx is 

a) u²+ v² b) u+ v c) u²- v² d) u - v

8) y= log₁₀x then dy/dx is 

a) (1/x) log₁₀e b) (1/x) logₑ10 c) 1/(xlog₁₀x) d) 1/10x

9) If x²y³= (x + y)⁵, then dy/dx is 

a) x/y b) -x/y c) y/x d) 1

10) y= x + x³/3! + x⁵/5!+ ....∞ then dy/dx =

a) 1/(1- x) b) 1/(1+ x) c) 1/(1- x²) d) 1/(1+ x²)

11) The derivative of f{g(x)} with respect to g(x) is

a) f'(g(x)) b) f'(g(x)) g'(x) c) f'(g(x))/g'(x) d) f'(g'(x))

12) if y= cos²x then [d²y/dx²]ₓ₌π/4 is

a) -1 b) 0 c) 1 d) -2

13) If xy = 1 then the value of d²y/dx² is 

a) 0 b) 3 c) -1 d) 2

Max-Min Question 

1) If cosy= x cos(a+ y), (a≠0) then dy/dx = cos²(a+y)/sins.

2) If x= sint, y= sinkt (k≠ 0) then (1- x²) d²y/dx² - x dy/dx + k²y =0.

3) If y= (1+ sinx+ cosx)/(1+ sinx - cosx) then show dy/dx + 1/(1- cosx) = 0

4) If 2x = y¹⁵ + y⁻¹⁾⁵ then show (x²- 1)y₂ + xy₁ = 25y.

5) If y= 2 tan⁻¹√{(x-a)/(b -x)} then show (dy/dx)²+ 1/{(x - a)(x - b)}= 0

6) y= sin⁻¹[{√(1+ sinx)+ √(1- sinx)}/{√(1+ sinx) - √(1- sinx)}] 

7) y= cos(msin⁻¹x) then show (1- x²)y₂ - xy₁+ m²y = 0.

8) If p= a(1- cosx), q= a(x + sinx) and x=π/2 then find dy/dx.

9) If y= tan⁻¹{4x/(1+ 5x²)+ tan⁻¹{(2+ 3x)/(3- 2x)} Then show dy/dx = 5/(1+ 25x²).

10 If y= ₑsin⁻¹x then o(1- x²)y₂ - xy₁ = m²y.

11) y= xˣ+ x²

12) If √(1- x²)+ √(1- y²)= a(x - y) then show dy/dx = √(1- y²)/√(1- x²).

13) If y=√(x+1) - √(x -1) then show (x²-1)y₂ + xy₁ = y/4.

14) If y= tan⁻¹[x/{1+ √(1- x²)}] + sin[2 tan⁻¹ √{(1- x)/(1+ x)}] then show dy/dx = (1- 2x){√(1- x²)}

15) If y= f(x) and x = 1/z then show d²f/dx²= 2z³ dy/dz + z⁴ d²y/dz².

16) sin⁻¹{2x/(1+ x²)} w,r,t tan⁻¹{2x/(1+ x²)}.

17) if y= sin(sinx) then show y₂ + y₁ tanx + y cos²x = 0.

18) If y= f{f(x)} and f(0) = 0, f'(0)= 5 then find [dy/dx]ₓ₌₀.

19) If y= log{x + √(x²- a²)}.

20) If v= A/r + b, then show d²v/dr² + 2dv/rdr = 0. (A, B - constant)

21) If y= sin⁻¹x then show (1- x²)y₂ - xy₁ = 0.

22) y= (sinx)ᶜᵒˢˣ + (cosx)ˢᶦⁿˣ.

23) y= (sin⁻¹x)²+ (cos⁻¹x)² then show that (1- x²)y₂ - x y₁  = 4.

24) f(x)= {(x +3)/(x+1)³ˣ⁺² then f'(0) is 

25) xʸ = yˣ find dy/dx

26) If y= eᵗ cost and x= eᵗ sint, then show dy/dx . dx/dy = 1.

27) If x²+ xy + y²= a², then show d²y/dx²+ 6a²/(x + 2y)²= 0.

28) y= cos⁻¹(4x³- 3x), z= tan⁻¹{√(1- x²)/x} then dy/dx is 

29) If y= (tan⁻¹x)² then show (1+ x²)y₂ + 2x(1+ x²)y₁ = 2.

30) y= cos⁻¹(8x⁴- 8x²+1).

31) If x√(1+ y)+ y√(1+ x)= 0 then show dy/dx = -1/(x+1)².

32) y= (cos2x - cosx)/(cosx -1).

33) y=√[x + √{x + √(x +..........∞

34) If y= sin(m sin⁻¹x) then show that (1- x²)y₂ -xy₁+ m²y = 0. (y₁≠ 0).

35) If xᵐ yⁿ = (x + y)ᵐ⁺ⁿ then ody/dx = y/x.

36) If x= (1- ₑ-t²)/2t², show that t dx/dt + 2x = (1/2t)+ ₑ-t²

37) If xʸ = eˣ⁻ʸ then show dy/dx = logx/(logex)²

38) y= xⁿ⁻¹logx then show x² d²y/dx² +(3 - 2n)x dy/dx + (n -1)²y=0

39) y= sin⁻¹{(a + b cosx)/(b + a cosx)} then show dy/dx =√(b²- a²)/(b + a cosx).

40) xʸ + yˣ =1.

41) If eʸ = x + √(a²+ x²) then show that (a²+ x²) d²y/dx²+ x dy/dx = 0.

42) If f(x)= {(a+ x)/(b+ x)}ᵃ⁺ᵇ⁺²ˣ, then show f'(0)= [2log(a/b) + (b²- a²)/ab](a/b)ᵃ⁺ᵇ

43) If y= sin(sin⁻¹x), then show (1- x²)y₂ -xy₁+ 4y = 0. 

44) If x³y¹⁰= (x + y)¹³ then show that d²y/dx²= 0

45) If sin⁻¹{(x²- y²)/(x²+ y²)} = k (k is constant) then show dy/dx = y/x.

46) If f(x)= log tan(π/4 + x/2) then show f'(x)= sex.

47) If f(x)= (x +1)ˣ+ xˣ⁺¹ then f'(1).

48) If y= a cos(ln x)+ b sin(ln x) then show x² d²y/dx² + dy/dx + y= 0.

49) If y= (ae⁻ᵏˣ - be⁻ᵏˣ) then show d²y/dx²= k²y.

50) If y= tan⁻¹(sex + tanx), find y₂ at x=π/4.

51) If sin(logx), show x² d²y/dx² + x dy/dx + y= 0.

52) If y= 1+ cos2x then d²y/dx² + 4y= ?

53) y= x logₑ{x/(a+ bx)} then show x³ d²y/dx² = (y - x dy/dx)².

54) If x= √ₐsin⁻¹(-1/t) and y= √ₐcos⁻¹(-1/t), show dy/dx = -y/x.

TANGENTS AND NORMAL 

1) At what point on the curve y= x² does the tangent make an angle 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 interesects the axis of y.         x/a + y/b =1

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 ratio of the tangent and normal to the curves x²/a²- y²/b²= 1 at the point (√2 a, b).               ax + √2 by - √2(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.   2x - y+3=0

b) perpendicular to the line 5y - 15x =13.       36y+ 12x -227=0.

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

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

8) Find the points on the curve y= x³- 11x +5 at which the equation of the tangent is y= x -11.         (2,-9) And (-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 equation 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 equation of the tangent and normal to the curve x= 1 - cos t, y= t - sin t at t=π/4.         4√2x + (8- 4√2)y = π(2- √2)

12) Show that the question of the normal at any points t on the curve x = 3 cos t - cos³t and y= 3 sin t - sin³t is 4(y cos³t - x sin³t)= 3 sin4t.

13) The equation of the tangent at (2,3) on the curve y²= ax + b is y= 4x -5. Find the value of a and b.      2,-7

14) Find the points on the curve y= x³ at which the slope of the tengent is equal to the y-ordinate to the point.      (0,0) And (3,27)

15) Find the equation of the tangent and the normal the curve y²= 4ax at the point (at², 2at).      tx + y = 2at + at³.

16) At what points on the curve x²+ y²- 2x - 4y +1=0 the tangent are parallel to the y-axis?         (3,2) And (-1,2)

17) Find the equation of the tangent line to the curve y= √(5x -3) -5, which is parallel to the line 4x - 2y +5=0.        80x - 40y -223=0

18) Find the equation of tangents to the curve y= cos(x + y), -2π≤ x ≤ 2π that are parallel to the line x+ 2y= 0.       4y + 2x +3π=0

19) Find the equation of the normal to the curve ay²= x³ at the point whose x coordinate is am².         2x + 3my - am²(3m²+2)=0

20) Find the equation of the tangent and normal to the curve x= a sin³t and y= a cos³t at t=π/4.     x - y =0

21) At what points will the tangent to the curves y²= 2x³- 15x²+ 36x -21 be parallel to the x-axis ? Also find the equations of the tangents to the curve at these points.     (2,7) and (3,6), y-7=0, y-6=0

22) Show that the curves 2x = y² and 2xy = k cut at right angles if k²= 8.

RATE MEASURE, INCREASING - DECREASING FUNCTIONS

1) Find the rate of change of the area of a circle with respect to its radius. How fast is the changing with respect to the radius when the radius 3 cm ?     6π

2) The amount of pollution content added in air in a city due to x diesel vehicle is given by P(x)= 0.005x⅔+ 0.02x²+ 30x.

 Find the marginal increase in pollution content when 3 diesel vehicles are added and write which value is indicated in the above question.

               Pollution control in the city increases with the increase in number of diesel vehicles.

3) The total revenue received from the sale of x unit of a product is given by  R(x)= 3x²+ 36x +5. Find the marginal revenue when x= 5, where, by marginal revenue we mean the rate of change of total revenue with respect to the number of items sold at an instant.        Rs66

4) Find the rate of change of volume with respect to its surface area when the radius is 2cm.          1

5) The radius of a circle is increasing uniformly at the rate of 4 cm/s. Find the rate at which the area of the circle is increasing when the radius is 8cm.     64π cm²/s

6) The radius of a balloon is increasing at the rate of 10 cm/s. At what rate is the surface area of the balloon increasing when the radius is 15 cm?      1200π cm²/s

7) Find the intervals in with the function f(x)= 2x³+ 9x²+ 12x +20 is

a) increasing.      (-∞,-2) And [-1, ∞)

b) decreasing .       [-2,-1]

8) The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing, when the side of the triangle is 20 cm ?      20√3 cm²/s

9) Find the intervals in which the function f(x)= 3x⁴- 4x³- 12x²+ 5 is

a) strictly increasing.     (-1,0) and (2, ∞)

b) strictly decreasing.      (-∞,-1) and (0,2)

10) Find the intervals in which the function given by f(x)= sinx + cosx, 0≤ x <2π is

a) increasing.       [0,π/4] and [5π/4,2π]

b) decreasing.          [π/4,5π/4]

11) The area of a circle of radius r Increase at the rate of 5 cm²/s

a) Find the rate at which the radius increase.     5.2πr cm/s

b) Find the value of this rate when the circumference is 10cm.      0.5cm/s

12) Find the intervals in which the function f(x) is strictly increasing where f(x)= 10 -6x -2x².     (-∞,-3/2)

13) The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20cm ?     20√3 cm²/s

14) Find the value/s of x for which y= {x(x -2)}² is an increasing function.      [0,1] U [2, ∞)

15) Find the intervals in which the function f given by f(x)= (x -1)(x +2)² is strictly increasing or strictly decreasing.     I: (-∞,-2) U(0, ∞), d: (-2,0)

16) The area of a circle of radius r increases at the rate of 5 cm²/s. Find the rate at which the radius increases. Also, find the value of this rate when the circumference is 10cm.          5/2πr cm/s,   0.5 cm/s

17) Find the intervals in which the function f(x)= 20- 9x + 6x¹- x³ is 

a) strictly increasing.       (1,3)

b) strictly decreasing.         (-∞, 1) and (3, ∞)

18) Find the intervals in which the following function is strictly increasing or strictly decreasing: f(x)= (x -1)³(x +2)².          (-∞,-2) and (-4/5, ∞) d: (-2,-4/5)

19) A balloon which always remains spherical has a variable diameter of (3/2)(2x +1). Find the rate of change of its volume with respect to x.     dv/dx = (27/8) π(2x +1)²

20) Prove that the function defined by f(x)= log sin x is strictly increasing on (0,π/2) and strictly decreasing on (π/2,π)

21) Find the intervals in which the function f given by f(x)= 2x³- 9x²+ 12x +15 is strictly increasing or strictly decreasing.       I: (-∞,1) U (2, ∞) d: (1,2)

22) Find the interval for which the function f(x)= cot⁻¹x + x increases.     (-∞,∞)

23) The area of an expanding rectangle is increasing at the rate of 48 cm²/s. The length of the rectangle is always equal to square of the breadth. At what rate is the length increasing at the instant when breadth is 4.5 cm ?      7.11 cm/s

24) Show that the funrf given by f(x)= tan⁻¹(sinx + cosx), x> 0 is always a strictly increasing function in (0,π/4).

INVERSE TRIGONOMETRIC FUNCTION 

1) Solve: 

a) sin(sin⁻¹(1/5)+ cos⁻¹x)= 1 find x.       1/5

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

c) sin[cot⁻¹(x+1)] = cos(tan⁻¹x).                -1/2

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

e) cos⁻¹x + sin⁻¹(x/2)= π/6.            1

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

g) cos⁻¹[sin(cos⁻¹x] = π/6.       ±1/2

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

2) Show 

a) cot⁻¹7 + cot⁻¹8 + cot⁻¹18 = cot⁻¹3.

b) sin⁻¹(8/17)+ sin⁻¹(3/5)= cos⁻¹(36/85).

c) tan⁻¹√x= (1/2) cos⁻¹{(1- x)/(1+ x)}, x ∈ {0,1}.     

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

e) tan⁻¹(3/4)+ tan⁻¹(3/5) - tan⁻¹(8/19)= π/4.

f) sin⁻¹(8/17) + sin⁻¹(3/5)= tan⁻¹(77/36).

g) tan⁻¹(1/4)+ tan⁻¹(2/9) = (1/2) sin⁻¹(4/5)

h) tan⁻¹(1/2)= π/4 - (1/2) cos⁻¹(4/5).        

i) tan[sin⁻¹(1/√17) + cos⁻¹(9/√85)]= 1/2.       

j) sin[2tan⁻¹(3/5) - sin⁻¹(7/25)]= 304/425.

k) sec²(tan⁻¹2) + cosec²(cot⁻¹3)= 15.

l) tan⁻¹(1/4) + tan⁻¹(2/9) = (1/2) sin⁻¹(4/5).

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

n) sin⁻¹{x/√(1+ x²)} + cos⁻¹{(x +1)/√(x²+ 2x +2)}= tan⁻¹(x²+ x +1).

3) Find the value of cot(π/2 - 2cot⁻¹√3).          √3

4) Value of tan⁻¹[2sin(2cos⁻¹(√3/2)}.              π/3

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

6) Find the principal value of tan⁻¹(√3) - sec⁻¹(-2).     -π.3

7) Evaluate: tan[2tan⁻¹(1/2) - cot⁻¹3]        9/13

8) Evaluate: tan[2tan⁻¹(1/5) - π/4].       -7/17

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

10) If tan⁻¹x  + tan⁻¹y + tan⁻¹z = 0, show that x+ y + z = xyz.

11) If cos⁻¹(x/a) + cos⁻¹(y/b) = k, show x²/a²- (2xy cos k)/ab + y²/b² = sin²k.