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If the direction ratios of two lines are given by l+m+n=0, mn-2ln +lm=0 then the angle between the lines is

If A and B are two independent events then P(A cap B) = P(A)*P(B).

Consider the exponential equation y=p^((x+1))/K, where K and p are positive real constants and x is a positive real number. The value of y decreases as the value of x increases if and only if which of the following statements about p is true ?

(1+cos theta+i sintheta)^n=

If xy ne 0, x + y ne 0 and x^(m)y^(n) = (x + y)^(m+n), where, m, n ne N, then (dy)/(dx) is equal to

A balloon, which always remains spherical, has a variable diameter (3)/(2)(2x+1). Find he rate of change of its volume with respect to x.

Integrate the following functions sqrt(4-x^2)

The number of ways in which 9 things can be divided into 3 equal groups in

Integrate the following functions : int(tanx)/(secx+cosx)dx

Letf(x) = |x-1| , thenwhichof thefollowingis true

Let P be a point in first quadrant on parabola y^(2) = 2x whose focal distance is 5. If PF is perpendicular to SP (S is focus) which intersect x axis at point F, then the length of SF is :

Area of the region bounded by the curve y = cos x, x = 0 and x = pi is

One of the following is perpendicular to 2hati + 2hatj - hatk

If a,b,c are the lengths of tangents from (0,0) to the circles x^(2)+y^(2)-3x-4y+1=0, x^(2)+y^(2)+4x-6y+4=0, x^(2)+y^(2)-6x-12y+9=0 then the ascending order of a,b,c is

Find the points at which the function f given by f(x)=(x-2)^(4)(x+1)^(3) haslocal minima,

Find the period for each of the following functions: (a) f(x)=arc tan (tanx) (b) f(x)=2 cos ""(x-pi)/(3)

intdx/(sqrt(x+1)+sqrt(x+2))

If either a=0,b=0, then a.b=0. But the converse need not to be true . Justify your answer with an example.

|((b+c)^(2),a^(2),a^(2)),(b^(2),(c+a)^(2),b^(2)),(c^(2),c^(2),(a+b)^(2))|=

Without expanding the determinant prove that |{:(a,a^2,bc),(b,b^2,ca),(c,c^2,ab):}|=|{:(1,a^2,a^3),(1,b^2,b^3),(1,c^2,c^3):}|

If log_(3)^(2),log_(3)(2^(x) - 5) and log(2^(x) - 7/2)are in A.P then the value is x is

If the symbolic form is (p ^^ r) vv (~q ^^ ~r) vv (~q ^^ ~r),then switching circuit is

For all positive intergral values of n, 3 ^(2n)-2n +1 is divisible by

Evaluate (iii) int_(0)^(pi)(x sin^(3)x)/(1+cos^(2)x)dx

If the pairs of straight lines x^(2) - 2pxy-y^(2)=0﻿ and x^(2) – 2qxy - y^(2) = 0 be such that each﻿ pair bisects the angle between the other﻿ pair, then﻿

The locus of the point of intersection of the perpendicular tangents to the ellipse x^(2)//a^(2)+y^(2)//b^(2)=1 is

The solution of (dy)/(dx)=(px+q)/(ry+s) represents a parabola when

The probability of getting atmost 4 heads when tossing 7 coins is

Prove that : If |x| is so small that x^(2) and higher powers of x may be neglected, then find an approximate value of

If ""^(n+1)C_(3)=4(""^(n)C_(2)), then n is equal to

Find the Principle values of the following : cosec^(-1)(2)

Show that the volume of the greatest cylinder which can be inscribed in a cone of height hand semi - vertical angle alpha is (4)/(27)pi h^(3)tan^(2)alpha . Also show that the height of the cylinder is(h)/(3) .

The term independent of x in the expansion of (x + 1/x^())^(6) is

A biased coin is tossed 10 times. The head is 2 times more likely to appear than the tail. The probability that 2^("nd") tail and 4^("th") tail occur at 4^("th") and 10^("th") tosses respectively is

For an ellipse with eccentricity 1/2 the centre is at the origin, if one directrix is x=4, then the equation of the ellipse is

The vlaue of cosec^(2)""(pi)/(7)+cosec^(2)""(2pi)/(7)+cosec^(2)""(3pi)/(7), is

If a line makes angle 90^@, 60^@" and "30^@ with the positive direction of x, y and z-axis respectively, find its direction cosines.

Find the maximum value of absz when abs(z-3/z)=2,z being a complex number.

For positive integers m and n, let god(m, n) denote the largest integer that is a factor of both m and n. Find the sum of all possible values of gcd (a - 1, a^2 + a + 1) where a is a positive integer.

Integrate the function (3x^(2))/(x^(6)+1)

Examine the continuity of the following functions at indicated points. f(x)={((x^2-1)/(x-1)ifxne1 at x=1),(2if x=1):}

If the system of linear equations ax+(a+1)y+(a-1)z=0 (a-1)x+(a+2)y+az=0 (a+1)x+ay+(a+2)z=0 has a nontrivial solution then sum of possible values of |a| is _________

Integrate the following functions with respect to x. (x^(2))/(sqrt(1+x^(2))(1+sqrt(1+x^(2))))

(1 - omega + omega^(2)) ( 1 - omega^(2) + omega^(4)) ( 1- omega^(4) + omega^(8)) to 2n factors =

If n is six more than two thirds of twelve, what is the value n?

If the Rolle's theorem is applicable to the function f defined by f(x)={{:(ax^(2)+b",", |x|le1),(1",", |x|=1),((c)/(|c|)",",|x|gt1):} in the interval [-3, 3], then which of the following alternative(s) is/are correct?

If x " log " x (dy)/(dx) + y = log x^2 " and " y ( e) = 0, then y(e^2)=

If A=[(3,-3,4),(2,-3,4),(0,-1,1)], then A^(-1)=

The value of |[overline(a)*overline(a), overline(a)*overline(b), overline(a)*overline(c)], [overline(b)*overline(a), overline(b)*overline(b), overline(b)*overline(c)], [overline(c)*overline(a), overline(c)*overline(b), overline(c)*overline(c)]|=

If the pairs of straight lines x^(2) - 2pxy-y^(2)=0 and x^(2) – 2qxy - y^(2) = 0 be such that each pair bisects the angle between the other pair, then

The value of |[overline(a)overline(a), overline(a)overline(b), overline(a)overline(c)], [overline(b)overline(a), overline(b)overline(b), overline(b)overline(c)], [overline(c)overline(a), overline(c)overline(b), overline(c)*overline(c)]|=