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Solve the following differential equations. (dy)/(dx)=(4x+6y+5)/(3y+2x+4)

Evaluate the following determinants. [[-6,0,0],[3,-5,7],[2,8,11]]

If the system of equations : 2x+3y-z=0,3x+2y+kz=0,4x+y+z=0 have a set of non-zero integral solutions then,find the smallest positive value of z

Pentagons have 5 diagonals, as illustrated below. How many diagonals does the octagon below have?

Draw the graph of y=2cosx + sin2x

Let f(x) be a quadratic expression such that f(x) lt 0 AA x in R . If g(x)f'(x) + f''(x)then for x in R.

Evaluate the following integrals int x " tanh"^(-1)xdx, |x| lt1

IF bar(a)=2bar(i)+5bar(j)+bar(k),bar(b)=4bar(i)+mbar(j)+nbar(k)are collinear vectors then find m+n

Fill in the gaps with correct answer choosing from the brackets.The correct expression is _____.(sin 1^0 > sin1,sin1^0 < sin1, sin1^0 =sin1)

How many positive integers less than 1000 have the property that the sum of the digits of each such number is divisible by 7 and the number itself is divisible by 3?

If A,B and C are angles of a triangle, then the determinant |{:(-1,cosC,cosB),(cosC,-1,cosA),(cosB,cosA,-1):}| is equal to"........."

State true or False .The planes 2x +4y -z + 1 = 0 and x - 2y - 6z + 3 = 0 are perpendicular to each other.

Evaluate the determinants in Exercises 1 and 2.{:|( 2,4),( -5,-1) |:}

Match the points on the curve 2y^2=x+1 with the slope of normals at those points {:("I.",(7,2),"a",-4sqrt2),("II".,(0.1//sqrt2),b,-8),("III.",(1,-1),c,4),("IV.",(3,sqrt2),d,0),("","",e,-2sqrt2):}

If Delta_r= |[2r-1, mC_r,1],[m^2-1,2^m,m+1],[m^2+m+1,2^m+1,m+2]|, then sum_(r=0)^m Delta_r=

The median of 100 observation grouped in classes of equal width is 55 if the median class is 50-60 and the number of observation less than 50 is 45, then frequency of the median class is _______

By vector method prove that the angle in a semi circle is a right angle.

If a and b are two unit vectors inclined to X-axis at angles 30^(@) and 120^(@), then |a+b| is equal to

If x+ky-5=0 is a tangent to the ellipse 4x^(2)+9y^(2)=20 then k =

Integrate the following functions : intx(4+x)^(1/4)dx

If each of the obsservationsx_(1) , x_(2) …..x_(n)is o increased by k , is a positive or negative number , then show that the variance ramains unchanged .

If A=[{:(1,0,2),(0,2,1),(2,0,3):}]then , prove that A^(3)-6A^(2)+7A+2I=O.

Find the number of ways of selecting cricket eleven from 20 players such that at least one of Sachin and Dravid must be excluded

The value of int_(-(pi)/(2))^((pi)/(2)) sqrt((1-cos 2x)/(2x)) dxis

Find a vector of magnitude sqrt(51) and makes equal angle with the vectors vec(a)=(1)/(3)(hati-2hatj+2hatk),vec(b)=(1)/(5)(-4hati-3hatk) and vec( c )=hatj.

If the coefficients of rth and (r+1)th terms in the expansion of (1+x)^(24) are in the ratio 12 : 13, then r is the root of the quadratic equation

If a=1,+x+x^2... and b=1+y+y^2+...[x|lt1and]y|lt1, then prove that 1-xy+x^2y^2+x^3y^3+....=(ab)/(a+b-1)

If a random variable X has a binomial distribution B (8,1/2) then find X for which the outcome is the most likely.

Find the number of ways of arranging 7 gents and 4 ladies around a circular table if no two ladies wish to sit together.

The solution of (x^(2) + 1) (dy)/(dx) + 4xy = (1)/(x^(2) + 1) is

Three persons A, B and C, fire at a target in turn starting with A. Their probability of hitting the target are 0.4, 0.3 and 0.2 respectively. The probability of two hits is …………

Find the rank of the following matrices by row reduction method.[[3,-8,5,2],[2,-5,1,4],[-1,2,3,-2]]

If Q is the image of the point P(2,3,4)under the reflection in the plane x-2y+5z =6 , then the equation of the line PQ is

Construct truth tables for the following and indicate which of these are tautologiesp^^(porq)rarr q.

If in a triangleABC, r_1=2r_2=3r_3, then b : c =

Form the differential equation representing the given family of curves y= asin(x+b) where a,b are arbitrary constants.

If x = alpha, y = beta, z = gamma is the solution, for the system of equations 2x - y + 8z = 13 3x + 4y + 5z = 18 5x - 2y + 7z = 20 then alpha beta + beta gamma + gamma alpha =

The volume of a cube is increasing at the rate of 8 cm^(3) /s. How fast is the surface area increasing when the length of an edge is 12 cm?

Find the domains of defination of the following functions: (a) f(x)=sqrt(x^(3)-x^(2)), (b f(x)=sqrt(sin sqrtx) (c) fx)=sqrt(-sin^(2) pix), (d) f(x)=(1)/sqrt(|x|-x|) and g(x)=1/sqrt(x-|x|) (e) f(x)=arc sin (|x|x-3), (f) f(x)=arc cos ""1/(sin x)

Prove thatsum_(k=0)^(n) (-1)^(k).""^(3n)C_(k)= (-1)^(n). ""^(3n-1)C_(n)

Let X ={x,y} and Y ={u,v}. Write down all the functions that can be defined from X To Y. How many of these are (i) one-one (ii) onto and (iii)one-one and onto ?

Consider the unction f(x)=int_(0)^(x)(5ln(1+t^(2))-10t tan^(-1)t+16sint)dt Which is not true for int_(0)^(x)f(t)dt gt?

int(x^(2))/((x^(2)+a^(2))(x^(2)+b^(2)))dx

The magnitude of the projection of vector (4, 1, 3) and (1, -2, 3) is …………….

Hannah is 5 years younger than Nora , who is x years old, Which of the following is an expression for Hannah's age in 2 years ?

If f(x) is a function of x such that (1)/((1+x)(1+x^(2)))=(A)/(1+x)+(f(x))/((1+x^(2)) for all x in R then f(x) is

Consider the non-empty set consisting of children in a family and a relation R defined as aRb, if a is brother of b. Then, R is

Find the relationship between a and b so that the function f defined by f(x)={{:(ax+1," if "x le 3),(bx+3," if "x gt 3):} is continuous at x=3.