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The shortest distance from (-6,0) to x^2-y^2+16=0 is

If A=[[1,2],[-2,3]]B=[[3,2],[1,4]],C=[[2,2],[1,3]]Calculate BA.

Coffiecient of x^n in e^(a +bx)is

The value of (cos"" (pi)/(2)+isin ""(pi)/(2))(cos ((pi)/(2^2))+isin((pi)/(2^2))) (cos ((pi)/(2^3))+isin((pi)/(2^3)))"………."oo is

Prove that:int_(-a)^(a) f(x) dx = {{:(2int_(0)^(a)f(x)dx, f(x) " is even "),(0, f(x) " is odd"):} and hence Evaluate int_(-t)^(t) sin^(5)(x)cos^(4)(x) dx

Solve the following D.E's (i) (dy)/(dx) = (3x - y + 7)/(x-7y - 3) (ii) (dy)/(dx) = (2x-y+1)/(x+2y-3) (iii) (dy)/(dx) = (-3x - 2y+5)/(2x+3y+5) (iv) (dy)/(dx) = (-3x -2y + 5)/(2x+3y-5) (v) (dy)/(dx) = -((12x + 5y - 9))/(5x + 2y - 4) (vi) 2(x-3y+1)dy = (4x-2y+1)dx

Find sum_(i=1)^n sum_(i=1)^n sum_(k=1)^n (ijk)

Determine order and degree (if defined) of differential equations given (y''')^(2) + (y'')^(3) + (y')^(4) + y^(5) = 0

A straight line passing through the point A(-2,-3) cuts lines x +3y = 9 and x +y +1 = 0 at B and C, respectively. If AB. AC = 20, then equation of the possible line is

Draw the graph of y = tan^(-1)(tan x)

Out of (2n+1) tickets consecutively numbered , three are drawn at random . The chance that the numbers on them are in A.P is

Check the validity of r:60 is a multiple of 3 or 5.

The radius of the circle having centre at (2,1) whose one of the chords is diameter of the circle x^2 + y^2- 2x - 6y + 6 = 0 is

Out of 30 observations 10 are equal to 70 -2alphaeach, 10 are equal to 7 each and the remaining 10 are equal to 7+ 2alphaeach. If standard deviations of the data is equal to5sqrt((2)/(3)) then |alpha| is equal to _____________

Find the angle between tangents drawn from P(2, 3) to the parabola y^(2) = 4x

If from a point on the line (x -1) =0, tangents are feawn to the parabola y ^(2) - 2y - 4x + 9=0 such that the pair of tangents and the chord of contact from a triangle, then the minimum area of the triangle is

A binary operation ""^(ast) defined on N, is given by a^(ast)b=H.C.F. (a, b), foralla, b in N. Check the commutativity and associativity.

Two pillars of height a and b subtend the same angle alpha at a point on the line joining their feet. If the pillars subtend angles beta and gamma at another point in the horizontal plane at which the line joining their feet subtends a right angle then,

If alpha and beta are two points on the hyperbola x^(2)/a^(2)-y^(2)/b^(2)=1 and the chord joining these two points passes through the focus (ae, 0) then e cos ""(alpha-beta)/(2)=

Using elementary transformations, find the inverseof the matrices [(1,3),(2,7)]

If 0 lt A lt pi //4 and cos A = 4//5,thenfind the values ofsin 2 A and cos2A

(x-12)/(-9) =(y-1)/(4)=(z-5)/(2) " and " (x-23)/(-6) =(y-19)/(-4) =(z-25)/(3)

Find the area of the region enclosed by the curves y^(2)=3x and x=3

If 1/(x^(4)+x^(2)+1)=(Ax+B)/(x^(2)+x+1)+(Cx+D)/(x^(2)-x+1) " then "C+D=

Let P be the point of intersection of two lines L_(1) : (x + 10)/(1) = (y - 21)/(7) = (z + 11)/(5) and L_(2) : (x - 1)/( 5) = (y - 46)/( 9) = (z)/(3) . If Q be the point (-10, 21, -11) , then PQ is equal to :

Solve the following systems of linear inequalities graphically : 2x - y ge 0 , x - 2y le 0 , x le 2 , y le 2.

The point P( (pi )/(4))lie on the ellipse(x^(2))/( 4) +( y^(2))/(2) =1 whose foci S and S^(-1) .The equation of the external angular bisector ofSPS^(-1) of Delta ^(le) SPS^(1)is

Which of the given values of x and y make the following pair of matrices equal [[3x+7, 5],[y+1, 2-3x]], [[0, y-2],[8, 4]]

Find the roots of x^4-16x^3+86x^2-176x+105=0

If the circles x^(2) +y^(2) - 6x – 8y +c = 0 and x^(2) + y^(2) =9 have three common tangent then c=

If [{:(2x+y,4x),(5x-7,4x):}]=[{:(7,7y+13),(y,x+6):}], then the value of x+y=.......

A ( a e, 0), B( - ae, 0)aretwopoints.Theequationto thelocusofPsuchthatPA - PB = 2ais

C_0 + 2.C_1 + 4.C_2 + …….+C_n.2^n = 243 , then n =

If m and n are the order and degree of the differential equation of the family of parabolas with focus at the origin and X-axis as its axis, then mn -m + n =

Greg is making a triangular sail for a boat , shaped like a right triangle and shown below . To determine how much trim to buy for the sail, Greg calculated the sail,s perimeter . What is the sail's perimeter, in feet ?

int_(0)^(1) ((1-x^(2))dx)/(x^(4)+x^(2)+1)=

Integrate the following functions with respect to x. sqrt(1-sin2x)x in((pi)/(4),(5pi)/(4))

Find the equation of the auxiliarly circle of the hyperbola(x^(2))/(6)-(y^(2))/(4) = 1

A regular polygon of 10 sides is constructed. Triangles are formed joining vertices of the polygon. Find the number of triangles (i) if two sides of trinangle coincide with the sides of polygon. (ii) if only one side of triangle coincide with the side of polygon.

Find the length of the latus rectum of the ellipse5x^(2) + 3y^(2) = 15.

If the distance between two latus rectum of a ellipse is 10 unit and length of major axis 12unit then find its eccentricity.

Find the length of mejor and minor axis oftheellipse25(x +1) ^(2) + 9 (y + 2)^(2) = 225.

Two balls are projected simultaneously from the top of a tall building. One vertically upward and other vertical downwards with same speed of 60 m/s.Then time interval between the balls striking the ground is :-

The sum of the integers from 1 to 100 that are divisible by 2 or 5 is :

{:("Column A","|a| is the distance point a is from the origin on the number line" x != 0,"Column B"),(|x| + |-2|,,|x-2|):}

Solve the following linear programming problem graphically: Maximize : z=x+2y Subject to: 2x+yge3 x+2yge6 xge0 yge0 Show that z is minimum at two points.

Evaluate : (i) int_(0)^(1) sin^(-1)((2x)/(1+x^(2)))dx , (ii) int_(0)^(1) (xtan^(-1)x)/((1+x^(2))^(3//2))dx (iii) int_(a)^(b) sqrt((x-a)(b-x))dx, a gt b (iv) int_(0)^(sqrt(3)) tan^(-1)((2x)/(1-x^(2)))

If bar(OA) = bar(i) + bar(j)+ bar(k), bar(AB) = 3bar(i)-2bar(j)+bar(k), bar(BC) = bar(i)+2bar(j)-2bar(k), bar(CD) = 2bar(i)+bar(j)+3bar(k) then find the vector bar(OD).

If S=Sigma_(n=2)^(oo)""^(n)C_(2) (3^(n-2))/(n!) then S equals