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agt0,int_(-pi)^(pi)(sin^(2)x)/(1+a^(x))dx=

Show that the three lines with direction cosines (12)/(13),(-3)/(13),(-4)/(13),(12)/(13),(3)/(13),(-4)/(13),(12)/(13) are mutually perpendicular.

The product of the perpendicular from the foci on any tangent to the hyperbolax^(2) //a^(2) -y^(2) //b^(2) =1is

If P(A)=3/5 and P(B)=1/5 find P(A cap B), where A and B are independent events.

The probability that a boy will get a scholarship is 0.7 and that another boy will get is 0.8. What is the probability that atleast one them will get scholarship.

If alpha, beta are the eccentric angles of the extremities of a focal chord of the ellipse x^(2)/16+y^(2)/9, " then "tan""alpha/2tan""beta/2=

Let AP(a:d) denote the set of the all terms of an inginate arithmetic progression with first term a and common difference d gt 0 if AP (I,3)cap AP(2,5)capAP(3,7)=AP(a,d) then a+d equals...........

Two objects at the points P and Q subtend an angle of 30^(@) at a point A. Lengths AR = 20 m and AS = 10 m are measured from A at right angles to AP and AQ respectively. If PQ subtends equal angle of 30^(@), at R and S, then length of PQ is

Solve the inequality 3(x-145) le 2(x-3)

Let bar z+b bar z= c, b ne 0 be a line in the complex plane where bar b

Find the number of 4 - digit telephone numbers that can be formed using the digits 1,2,3,4,5,6 with atleast one digit repeated.

On Z ** defined by a**b = a+b+1 . Is ** associative ? Find identity element and inverse if it exists.

If A and B are mutually exclusive and exhaustive events with P(A)=(2)/(3)P(B) then odds in favour of B are

Assume that in a family, each child is equally likely to be a boy or a girl. A family with three children is chosen at random. The probability that the eldest child is a girl given that the family has atleast one girl is ………..

Find the number of terms in the expansion of (4x - 7y)^49 + (4x + 7y)^49

In what ratio should a given line be divided into 2 parts so that the rectangle contained by them is maximum ?

If (x_(1)-x_(2))^(2) + (y_(1)-y_(2))^(2) =a^(2), (x_(2)-x_(3))^(2) + (y_(2)-y_(3))^(2) = b^(2), (x_(3)-x_(1))^(2)(y_(3)-y_(1))^(2) =c^(2) and |{:(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1):}|^(2) = (a+b+c)(b+c-a)(c+a-b)(a+b-c) then the value of lambda is:

Probability of happening of an event in an experiment is 0.4 . The probability of happening of the event atleast once , if the experiment is repeated 3 times under similar condition is

Iff(x)[ log_(e)x+ sqrt({log_(e)x}), x lt 1, where [.]and f{.} denotethefreatestfunction and the fractional part function respectively , then

Matrices of any order can be added.

An unbiased die is tossed 6 times. The mean of number odd numbers is

Prove the following overset(1)underset(0)int sin^(-1)x dx=(pi)/(2)-1

Find (dy)/(dx) when x and y are connected by the relation given: (x^(2) + y^(2))^(2)= xy

If the sum of the surface areas of cube and a sphere is constant, what is the ratio of an edge of the cube to the diameter of the sphere, when the sum of their volumes is minimum ?

In the matrix A=[(2,5,19,-7),(35,-2,(5)/(2),12),(sqrt(3),1,-5,17)], write: (i) The order of the matrix, (ii) The number of elements, (iii) Write the elements a_(13), a_(21), a_(33), a_(24), a_(23).

inte^x(tanx+Insecx)dx

Evaluate : int_((pi)/(5))^((3pi)/(10))(dx)/(tan^(n) x +1)

Find the shortest distance between the lines (x)/(5)= (y-2)/(2) =(3-z)/(-3) and (x+3)/(5)=(1-y)/(-2)=(z+4)/(3)

lim_(x rarr oo) (sqrt(a^(2) x^(2) + bx + c) - ax) =

int_(0)^(oo)(logx)/(1+x^(2))dx=

Let 'P' be an interior point of Delta ABC. If angle A=45^(@), angle B=60^(@) and angle C=75^(@). If X=area of Delta PBC,Y= area of Delta PAC and Z = area of Delta PAB, then which of the following ratios is/are true ?

Which of the following statements is not true about f(x) = 1+x+ x^(2)/(2!) +...+ x^(2n)/ (2n!)(n gt 1)

A variable line L is drawn through O(0,0) to meet the line L_(1) " and " L_(2) given by y-x-10 =0 and y-x-20=0 at Points A and B, respectively. Locus of P, if OP^(2) = OA xx OB, is

With usual notions, prove that in a triangle ABC, cot (A)/(2)+cot (B)/(2)+cot(C )/(2)= (s)/(r ).

cot{ (2019pi)/(2) - ( cosec^(-1) (5)/(3) + tan^(-1)"" (2)/(3) ) } = .....

Resolve (3x^(3)-2x^(2)-1)/(x^(4)+x^(2)+1) into partial fractions.

The vector equations of two lines L_(1) and L_(2) are respectively r=17i-9k+lamda(3i+j+5k) and r=15i-8j+k+mu(4i+3j). Then the correct statement(s) is/are:

Statement I The triangle so obtained is an equilateral triangle. Statement II If roots of the equations be tan A, tan B and tanC then tan A + tanB+tanC=3sqrt (3)

For theta in[ 0, 2pi] , consider the followingstatement : p : sqrt(1 - cos (2 theta)) = sqrt(2) | sin theta| q: sqrt(1 - cos (2 theta)) + sqrt(2) sin theta = 0 " if " pi le theta le 2 pi Then truth values of p and q are respectively:

Evaluate the definite integrals int_(0)^(pi/4)tanxdx

If bar(a),bar(b) and bar( c ) are not coplanar then (bar(a)+bar(b)+bar( c ))*[(bar(a)+bar(b)) times (bar( a )+bar(c))] = ……………

Let A = { 1, 2, 3, …….n} , if a_(i) is the minimum element of the set A, (where A , denotes the subset of A containingexactly three elements) and X denotes the set of A_(i)'s , then evalute sum_(A_(i)inX) a

Three point A, B and C with position vectors a_(1)=3hati-2hatj-hatk, a_(2)=hati+3hatj+4hatk and a_(3)=2hati+hatj-2hatk relative to an origin O. The distance of A from the plane OBC is (magnitude)

Using the binomial theorem show that 1^(99) + 2^(99) +3^(99) + 4^(99) + 5^(99) is divisible by 3 and 5 so that it is actually divisible by 15.

The statement ~p vv q is equivalent is

Fleshy buds produced in the axil of leaves, which grow to form new plants when shed and fall on ground, are called

On Z defined by ab = a+b+1 . Is ** associative ? Find identity element and inverse if it exists.