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Which of the following represents direction cosines of the line :

(a)`0,1/sqrt2,1/2`

(b)`0,-sqrt3/2,1/sqrt2`

(c)`0,sqrt3/2,1/2`

(d)`1/2,1/2,1/2`

`A=[[1,2],[3,4]]` ans A(Adj A)=KI, then the value of 'K' is

(a) 2

(b) -2

(c) 10

(d) -10

The general solution of the trigonometric equation tan^{2} θ = 1 is..........................

(a)`theta =npi+-(pi/3),n in z`

(b)`theta =npi+-pi/6, n in z`

(c)`theta=npi+-pi/4, n in z`

(d) `0=npi, n in z`

If `bara, barb, bar c` are the position vectors of the points A, B, C respectively and ` 2bara + 3barb - 5barc = 0` , then find the ratio in which the point C divides line segment AB.

The Cartestation equation of line is `(x-6)/2=(y+4)/7=(z-5)/3` find its vector equation.

Equation of a plane is vecr (3hati-4hatj+12hatk)=8. Find the length of the perpendicular from the origin to the plane.

Find the acute angle between the lines whose direction ratios are 5, 12, -13 and 3, - 4, 5.

Write the dual of the following statements:

(l) (p ∨ q) ∧ T

(2) Madhuri has curly hair and brown eyes .

If the lines `(x-1)/2=(y+1)/3=(z-1)/4 ` and `(x-3)/1=(y-k)/2=z/1` intersect each other then find value of k

Prove that three vectors `bara, barb and barc ` are coplanar, if and only if, there exists a non-zero linear combination `xbara+ybarb +z barc=0`

Using truth table prove that :

`~p^^q-=(p vv q)^^~p`

In any ΔABC, with usual notations, prove that b^2=c^2+a^2-2ca cosB.

Show that the equation `x^2-6xy+5y^2+10x-14y+9=0 ` represents a pair of lines. Find the acute angle between them. Also find the point of intersection of the lines.

Express the following equations in the matrix form and solve them by method of reduction :

2x- y + z = 1, x + 2y + 3z = 8, 3x + y - 4z =1

how that every homogeneous equation of degree two in x and y, i.e., ax^{2} + 2hxy + by^{2} = 0 represents a pair of lines passing through origin if h^{2}−ab≥0.

find the symbolic fom of the following switching circuit, construct its switching table and interpret it.

if A, B, C, D are (1, i, I), (2, l ,3), (3; 2, 2) and (3, 3, 4) respetivly., then find the volume of the parallepiped with AB, AC and AD as concurrent edges

Find the equation of the plane passing through the line of intersection of planes 2x - y + z = 3 and 4x- 3y + 5z + 9 = 0 and parallel to the line

` (x+1)/2=(y+3)/4=(z-3)/5`

Minimize :Z=6x+4y

Subject to : 3x+2y ≥12

x+y ≥5

0 ≤x ≤4

0 ≤ y ≤ 4

Show that:

`cos^(-1)(4/5)+cos^(-1)(12/13)=cos^(-1)(33/65)`

If y =1-cosθ , x = 1-sinθ , then ` dy/dx at " "0 =pi/4` is ............

The integrating factor of linear differential equation `dy/dx+ysecx=tanx` is

(a)secx- tan x

(b) sec x · tan x

(c)sex+tanx

(d) secx.cotx

The equation of tangent to the curve y = 3x^{2} - x + 1 at the point (1, 3) is

(a) y=5x+2

(b)y=5x-2

(c)y=1/5x+2

(d)y=1/5x-2

Examine the continuity of the function

f(x) =sin x- cos x, for x ≠ 0

=- 1 ,forx=0

at the poinl x = 0

Verify Rolle's theorem for the function

f(x)=x^{2}-5x+9 on [1,4]

Evaluate : `intsec^nxtanxdx`

The probability mass function (p.m.f.) of X is given below:

X=x | 1 | 2 | 3 |

P (X= x) | 1/5 | 2/5 | 2/5 |

find E(X^{2})

Given that X~ B(n = 10, p), if E(X) = 8. find the value of p.

Ify y=f(u) is a differentiable function of u and u = g(x) is a differentiable function of x then prove that y = f (g(x)) is a differentiable function of x and

`(dy)/(dx)=(dy)/(du)*(du)/(dx)`

Obtain the differential equation by eliminating arbitrary constants A, B from the equation -

y = A cos (log x) + B sin (log x)

Evaluate : `int x^2/((x^2+2)(2x^2+1))dx`

An open box is to be made out of a piece of a square card board of sides 18 cms. by cutting off equal squares from the comers and tumi11g up the sides. Find the maximum volume of the box.

Prove that :

`int_0^(2a)f(x)dx=int_0^af(x)dx+int_0^af(2a-x)dx`

If the function f (x) is continuous in the interval [-2, 2],find the values of a and b where

`f(x)=(sinax)/x-2, for-2<=x<=0`

`=2x+1, for 0<=x<=1`

`=2bsqrt(x^2+3)-1, for 1<x<=2`

Solve the differential equation `dy/dx=(y+sqrt(x^2+y^2))/x`

A fair coin is tossed 8 times. Find the probability that it shows heads at least once

If x^{p}y^{q}=(x+y)^{p+q} then Prove that ` dy/dx=y/x`

Find the area of the sector of a circle bounded by the circle x^{2} + y^{2} = 16 and the line y = x in the ftrst quadrant.

Prove that `int sqrt(x^2-a^2)dx=x/2sqrt(x^2-a^2)-a^2/2log|x+sqrt(x^2-a^2)|+c`

A random variable X has the following probability distribution :

X=x | 0 | 1 | 2 | 3 | 4 | 5 | 6 |

P[X=x] | k | 3k | 5k | 7k | 9k | 11k | 13k |

(a) Find k

(b) find P(O <X< 4)

(c) Obtain cumulative distribution function (c. d. f.) of X.