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Statement-1 : tantheta+2tan2theta+4tan4theta+8tan8theta-16cot16theta=cottheta Statement-2 : cottheta-tantheta=2cot2theta.

Points A(veca), B(vecb), C(vecc) and D(vecd) are related as xveca+yvecb+zvecc+wvecd=0 and x+y+z+w=0, where x, y, z and w are scalars (sum of any two of x, y, z and w is not zero). Prove that if A, B, C and D are concyclic, then |xy||veca-vecb|^(2)=|wz||vecc-vecd|^(2).

Find X , if Y=[{:(3,2),(1,4):}]and2X+Y=[{:(1,0),(-3,2):}].

Express as a sum of a symmetric and a skew symmetric matrix:[[x,a,b],[a,y,c],[b,c,z]]

Find the values of x for which the following functions are maximum or minimum: (i)x^(3)- 3x^(2) - 9x (ii)4x^(3)-15x^(2)+12x+1 (iii)1-x-x^(2) (iv) x^(4)-8x^(3)+22x^(2)-24x +7 (v) (x^(2)-x+1)/(x^(2)+x+1)

Two dice are rolled and given that both faces are showing even numbers. The probability that their sum is more than 9 is

Find the coefficient of x^(10) in the expansion of (1+x^(2)-x^(3))^(8)

vec(r ) =(lambda-1) hat(i) + (lambda+1) hat(j) -(lambda+1) hat(k) vec(r ) =(1-mu) hat(i) + (2mu -1) hat(j) + (mu +2) hat(k).

For any three vectors overset(to)(a) , overset(to)(b) and overset(to) ( c ), prove that vectors overset(to)(a) - overset(to)(b) , overset(to)(b) - overset(to) ( c), overset(to)(c) - overset(to)(a) are coplanar.

When it is 2:01 pm Sunday afternoon in Nullepart, it is Monday in Eimissaan. When it is 1:00 pm Wednesday in Eimissaan, it is also Wednesday in Nullepart. When it is noon Friday in Nullepart, what is the possible range of times in Eimissaan?

There are seven even greeting cards each of a different colour and seven envelopes of the same seven colours. Find the number of ways in which the cards can be put in the envelopes so that exactly four of the cards go into the envelopes of the right colour.

If A={1,2,3,4},B={2,3,5,6}andC={3,4,6,7}, then

Integrate the functions (e^(5logx)-e^(4logx))/(e^(3logx)-e^(2logx))

If [(2,0,7),(0,1,0),(1,-2,1)][(-x,14x,7x),(0,1,0),(x,-4x,-2x)]=[(1,0,0),(0,1,0),(0,0,1)], then find the value of x

LetA=|(2, e^(ipi)),(-1,i^(2012))|,C=(d)/(dx)(1/x)_(x=1),D={:(1),(int),(e^(2)):}(dx)/(x). . If the sum of two roots of the equation Ax^(3)+Bx^(2)+Cx-D=0 is equal to zero, then B is equal to

If the lengths of the tangents drawn from P to the circles x^(2)+y^(2)-2x+4y-20=0 and x^(2) +y^(2)-2x-8y+1=0 are in the ratio 2:1, then the locus P is

Find x,y if (x,y) = (-3,2)

Let z_1,z_2 and z_3 be three distinct complex numbers , satisfying |z_1|=|z_2|=|z_3|=1. Which of the following is/are true :

Write the following function in the simplest form : tan^(-1)(sqrt((1-cosx)/(1+cos))),0ltxltpi

If tworootsofx^3 - 9x^2 + 14 x +24 =0are inthe ratio3:2then therootsare

Find the number of different terms in the sum (1+x)^(2009) +(1+x^(2))^(2008)+(1+x^(3))^(2007).

Evaluate : int _(1) ^(e) ( dx)/( x (1 + log x))

If the equation 3x^(2)+7xy+2y^(2)+2 gx+2 fy+2=0 represents a pair of intersecting lines and the square of the distance of their point of intersection from the origin is 2/5 , then f^2+g^2=

Let A and B be two 3xx3 matrices with integer entries . If 6AB +2A+3B=O_(3) then |det (3B+I_(3)) | is equal to ____ .

Integrate the following rational functions : int(3x+4)/(x^(2)-5x+6)dx

The set of lines ax + by + c = 0 where﻿ 3a + 2b + 4c = 0 intersect at the point

(1 +i)^(6) + (1 - i)^(3) =

If the mean deviation of the numbers 1,1+d, 1+2d, 1+3d,…1+100d from their mean is 255, then find the value of d

The vertices of a triangle are (3, 0), (3,3)﻿ and (0, 3). Then the coordinates of the﻿ circumcentre are﻿

If(1 + x)^(n) = C_(0) = C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n) , find the values of the following (sumsum)_(0leile jlen)(i +j)(C_(i)pmC_(j) )^(2)

If(1 + x)^(n) = C_(0) = C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n) , find the values of the following (sumsum)_(0leile jlen)(i +j)(C_(i)pmC_(j) )^(2)

Construct truth tables for the following and indicate which of these are tautologies ((p ^^ q) rarrp) rarr q

A plane cuts the coordinate axes X,Y,Z at A,B,C respectively such that the centroid of the triangleABC is (6,6,3). Then the equation of that plane is

The range of x for which the expansion of (1-4x^2)^(-9//4) is valid

int(e^(x)-e^(-2x))/(e^(2x)+e^(-2x))dx

Shade the feasible region for the inequations 2x+3yle6,xge0,yge0 in a rough figure.

Coefficient of x^5 in (1+x +x^2 +x^3)^10 is

If the roots of the equation x^(2)+2bx+c=0 are alpha" and "beta, " then "b^(2)-c =

For 0 lt x lt pi, sin h^(-1) (cot x)is equal to

If y= e^(3log x), then show that (dy)/(dx)= 3x^(2).

Determine P(E|F) A die is thrown three times,﻿ E : 4 appears on the third toss, F : 6 and 5 appears respectively﻿ on first two tosses﻿

The solution of the differential equation y' = (1)/(e^(-y) - x), is

If alpha, beta, gamma are roots of the equationx^(3) + qx + r = 0 and S_(r ) denotes the sum of the r the powers of the roots of the equation then 3S_(2) S_(5)=

Let f(x) and g(x) be defined by f(x) = [x] and f(x) = {{:(0, x in I),(x^(2), x in R-1):}, then (where [*] denotes the greatest integer function)

Circle x ^(2) +y^(2) + 2x - 8y - 8=0 and x ^(2) + y ^(2) + 2x - 6y - 6=0

Let A and B be two finite sets and let P(A) and P(B) respectively denote their power sets. If P(A) has112 elements more than P(B), then the number of injectivefunctions from A to B is …………….

If P(A)=(7)/(13), P(B)=(9)/(13) and P(A cap B)=(4)/(13)," find " P(A//B)

The minimum value of |z-2|+|z-3| is

A die is thrown three times, A:4 appears on the third toss, B: 6 and 5 appear respectiyely on first two tosses Determine P(A|B).

The set of lines ax + by + c = 0 where 3a + 2b + 4c = 0 intersect at the point

The vertices of a triangle are (3, 0), (3,3) and (0, 3). Then the coordinates of the circumcentre are

Determine P(E|F) A die is thrown three times, E : 4 appears on the third toss, F : 6 and 5 appears respectively on first two tosses