LAST REVISED MIXED QUESTION - XII

1) If A= 3 &  B= (1  -5  7), 
              1
             -2
Verify that (AB)' = B'A'.

2) Find a 2 x 2 Matrix B such that BA = A where
A= 1    -2  = 6     0                        4       2
      1     4     0     6                        -1      1

3) Find the equation of the tangent to the curve √x + √y = a, at the point (a²/4 , a²/4).      2y + 2x - a²=0

4) Discuss the continuity of the function f(x) at x=0, where 
f(x)=   x, if x >0
           1, if x=0
          -x, if x <0                 discontinue 

5) Evaluate : ∫ log(1+ tanx)dx at (π/4,0).        π/8    log2

6) Evaluate: ∫xeˣ/(1+ x)² dx.         eˣ/(1+ x)  + C

7) Evaluate: ∫ dx/{(x+1)(x+2)(x+3)}.       1/2 log|x +1| -  log|x +2| + 1/2 log|x +3| + C

8) Evaluate: ∫ eᵃˣ cos(bx+ c) dx.        eᵃˣ/(a²+ b²)   [a sin(bx+ c) - b cos(bx +c)] + C

8) One card is drawn from a set of 17 cards in numbered 1 to 17. Find the probability that the number drawn is divisibility by 3 or 7.         7/17

9) Cards are numbered 1 to 25. Two cards are drawn one after another . Find the probability that the number on one card is a multiple of 7 and on the other it is a multiple of 11.      1/50

10) Out of or 9 outstanding students in a college, there are 4 boys and 5 girls. A team of 4 students is to be selected for a quiz programme . Find the probability that two are boys and two girls.         10/21

11) Show that the differential equation of which y= 2(x²-1)+ ₑ-x² is a solution, is dy/dx + 2xy = 4x³.

12) Solve the following system equations, using Cramer's rule: 2x - 3y + 4z =-9, -3x +4y + 2z =-12, 4x - 2y -3z =-3.          x= -342/53, y= -321/53, z= -189/53

13)  Show that A= 5      3  satisfy the equation x² - 3x - 7 = 0 . Thus , find A⁻¹.    2/7   3/7
                           -1     -2                                                           -1/7  -5/7

14) Find the intervals in which f(x)= x³- 6x²- 36x +2 is increasing or decreasing.

15) Differentiate: sin⁻¹(3x - 4x³) with the respect to x.                3/√(1- x²)

16) The surface area of a balloon being inflated, changes at a rate proportional to time t. If initially its radius is 1 unit and after 3 seconds, it is 2 units; find the radius of after time t.        r= √(t²/3 + 4) units 

17) Solve the differential equation: (x²+ xy)dy = (x²+ y²)dx.      x/(x - y)²= C ₑy/x

18) Evaluate ²₀∫ (x²+1)dx.          14/3

19) Find the area bounded by the curve, y²= 4a²(x -1) and the lines x=1 and y = 4a.     16a/3

20) sketch the graph, y=|x -5|. Evaluate ¹₀∫|x - 5| dx. What does the value of this integral represent on the graph?    9/2

21) A bag contains 4 white and 5 black balls and another contains 3 white and 4 black balls. A ball is taken out from the first bag and without seeing its colour is put in the second bag. A ball is then taken out from the lattter. Find the probability that the ball drawn is white.     31/72

22) For 5 observations of pairs (x,y) of variable x and y, the following results are obtained:
∑x= 15, ∑y= 25, ∑x²= 55, ∑y²= 135, ∑xy = 83.
 Find the equations of the lines of regression and estimate the value of x when y = 12 and the value of y when x= 8.       Ty= 4x +13, 5x = 4y - 5. When x=8, then y=9 and y=12, then x= 43/5

23) Solve the following system are equations , using Matrix method: 6x - 12y + 25z =4; 4x - 15y - 20z =3; 2x +18y + 15z =10.      1/2,1/3,1/5

24) Find the equation of the plane passing through the point (3, 4, 2) and (7,0,6) and perpendicular to the plane 2x - 5y= 15.      5x+ 2y - 3z =17

25) Using integration, find the area of triangle ABC, coordinates of whose vertices are A(2,0), B(4,5) and C(6,3).        7 sq.units

26) The combined resistance R of two resistors R₁ and R₂ (R₁ , R₂ >0) is given 1/R = 1/R₁ + 1/R₂ . If R₁ + R₂ = C(constant), find R₁ and R₂ so that R is maximum.       C/2, C/2

27) Find the magnitude of the vector a x b, if a= 3i + 4j and b= 5j + 12k.    15√17

28) Find the centre and radius of the sphere 2(x -5)(x+1)+ 2(y +5)(y -1)+ 2(z -2)(z +2) -7=0.     (2,-2,0),√(51/2)

29) If a= 2j - 3j + 4k,  3i + 2j - 4k =b, c= 4i - 3j + 5k, which of the following are meaningful and evaluate any one of those that are meaningful: (a. b) x c, a x (b x c), a.(b x c).       a x (b x c)= 175i+ 26j - 68k, a. (b x c)=21

39) Show by vector method that the line segment joining the midpoints of any two sides of a triangle is parallel to the third side and is half its length.

31) Find the shortest distance between the lines r= i - j + λ(2i + k) and r= 2i - j + μ(i +j -k).      1/√14

32) Find the derivative of tan√x.          Sec²√x/(2√x)

33) Evaluate : ∫ sinx/(1+ cos²x)  dx at (π/2,0).           π/4

34) Solve : (x -1) dy/dx = 2xy, given that y(2)= 1.            2x + 2log(x -1)= logy +4

35) Solve: x dy/dx = x + y.           y= x logx + cx

36) Using vectors show that the diagonas of a parallelogram bisect each other.

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

38) Evaluate: ∫ dx/(6- cosx) at (π,0).        π/√35

39) Prove a     a²    bc 
                  b    b²    ca 
                  c    c²     ab
 =(a - b)(b - c)(c - a)(ab + bc+ ca).



40) For the function f(x)= 6+ 12x + 3x²- 2x³, find:
a) the interval where it is increasing.       ]-1,2[
b) the intervas where it is decreasing.     ]2, ∞[ , ]-∞, -1[

41)  Find dy/dx : (x²+1)/x w.r.t.x.       (x²-1)/x²

42) Solve: dy/dx = y sin2x, given that y(0)=1.        y=ₑsin²x

53) Using determinants, find the area of the triangle with vertices (-3,5),(3,-6) and (7,2).    46 sq.units

44) Solve: x dy/dx = x+ y.       y= x logx + Cx

45) Find the inverse of the matrix 
A= 5     2                 1      -2
      2     1                -2       5 

46) Using vectors show that the line segment joining the midpoints of two sides of a triangle is parallel to the third side.

47) If a= (i - 2j + 3k) and b=(2i+3j- 5k), then find a x b. verify that a and a x b are perpendicular to each other.       i+11j + 7k

48) Evaluate: ¹₀∫ eˣ/(1+ e²ˣ) dx.     (tan⁻¹e - π/4)

49) Evaluate: ∫ dx/√(2+ 2x - x²).        sin⁻¹{(x -1)/√3}

50) In a group there are 3 men and 2 women. Three persons are selected at random from this group. Find the probability that one man and two women or two men and one woman are selected.             9/10

51) One card is drawn from a well shuffled pack of 52 cards. If E is the event . 'the card drawn is the king or queen' and F is the event, 'the card drawn is a Queen or an ace',  then find the probability of the conditional event (E/F).      1/2

52) A die is thrown from 7 times. If getting an even number is success. Find the probability of getting at least 6 success .    1/16


53) Find the radius of the circular section of the sphere x²+ y²+ z²= 49 by the plane 2x + 3y - z - 5 √14=0.     2√6

54) Find dy/dx, when y= xˢᶦⁿˣ ⁺ ᶜᵒˢˣ + (x²-1)/(x²+1).      xˢᶦⁿˣ ⁺ ᶜᵒˢˣ{(sinx - cosx)/x  + logx(sinx + cosx)} + 4x/(x²+1)²

55) A  box contains 16 bulbs out of which 4 bulbs are defective. Three bulbs are drawn one by one from the box without replacement. Find the probability distribution of the number of defective bulbs drawn.       

56)  Prove that : [(b + c) x (c + a)] . (a + b)= 2[a  b  c].

57) Define the line of the shortest distance between two skew lines given. Find the shortest distance and the vector equation of the line of shortest distance between the lines given by:
r= (3i + 8j+ 3k)+ λ(3i - j+k) and
r= (3i -7j+ 6k) + μ(-3i + 2j+ 4k).      3√30

58) Find the Karl Pearson's coefficient or correlation between x and y for the following:
 X: 10    7   12    15      9    15      8
Y:   6     4     7     10     11    8      10
Also, state whether y increases or decreases with increase in x.    0.17

59) If  A = 1     - 2       0
                  2      1        3
                  0    - 2        1  find A⁻¹. Using A⁻¹, solve the system of linear equation:  x - 2y = 10;  2x + y + 3z = 8, -2y + z =7.         4,-3,1

60)  Find the point on the curve y² = 4 x which is nearest to the point (2, - 8).     (4,4),(4,-4)

61) Draw a rough sketch of the region [(x, y): y² ≤ 4x, 4x²+ 4y² ≤ 9] and find the area  enclosed region using method of integration.         (√2/6+ 9π/8 - 9/4  sin⁻¹(1/3)) sq.units

62) A is a given Matrix 3x2. If the order of matrix AB be 3x3, what will be the order of B?
a) 1x3 b) 2x3 c) 3x3 d) 2x2

63) if d/dx{(2x³+ 5)¹⁰}= 60x²f(x), then f(x)=___

64) Which of the following will be the domain of the differential coefficient of sin⁻¹x with respect to x?
a) -1≤x ≤1 b) -1<x <1 c) -1≤x <1 d) -1<x ≤ 1



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