x^{2} + y^{2 }+ 2(2K+3)x  2Ky +(2K+3)^{2} + K^{2}  r^{2} = 0 represents the family of circles with centres on the line
If the equation [(k(x+1)^{2/}3)]+[(y+2)^{2}/4]=1 represents a circle, then k=
Let (1 + x)^{n} =
C_{0} + C_{1}x + C_{2}x^{2} + ... + C_{n}x^{n} and (C_{1}/C_{0}) + (2 C_{2}/C_{1}) + (3 C_{3}/C_{2}) + .... + ((n C_{n})/(C_{n}  1)) = (1/k)n(n + 1), then the value of k is
The solution of
Put x + y + 1 = z
⇒ 1 + dy /dx = dz /dx ⇒ dy /dx = dz /dx  1
(x + y + 1) dy dx = 1
⇒ z( dz /dx  1) = 1 ⇒ dz /dx = 1 + 1 z ⇒ z z + 1 dz = dx ⇒ ∫(1  1 z + 1 )dz = ∫dx
⇒ z  log(z + 1) = x + c ⇒ x + y + 1 = log(x + y + 2) + x + c ⇒ y = log(x + y + 2) + log c
⇒ e^{y} = (x + y + z)c ⇒ x + y + 2 = ce^{y}
Solution of the differential equation tan y sec^{2} x dx + tan x sec^{2} y dy = 0 is
If y=cot⁻^{1}[(1+x)/(1x)], (dy/dx)=
The latus rectum of the conic 3x^{2} + 4y^{2}  6x + 8y  5 = 0 is
If the length of the major axis of an ellipse is three times the length of its minor axis, then it's eccentricity is
The eccentricity of the conic 9x^{2}  16y^{2} = 144 is
In the interval ( 3,3) the function
The value of
The maximum area of the rectangle that can be inscribed in a circle of radius r is
If z^{2 }= i, then z =
The equation of the normal to the curve x^{2} = 4y at (1, 2) is
The length of the latus rectum of the parabola x^{2}4x8y+12=0 is
If ^{n}P_{r}=840, ^{n}C_{r}=35, then n=
Four numbers are chosen at random from {1, 2, 3, . . . , 40}. The probability that they are not consecutive is
A purse contains 4 copper coins and 3 silver coins, the second purse contains 6 copper coins and 2 silver coins. A coin is taken out from any purse. The probability that it is a copper coin is
If two equations x^{2}+a^{2}=12ax and x^{2}+b^{2}=12bx have only one common root, then
If 2a + 3b + 6c = 0, then at least one root of the equation ax^{2}+bx+c = 0 lies in the interval
The fourth ,seventh and tenth of a G.P. are p,q,r respectively then
The equation of lines passing through the intersection of 4x3y1=0 and 2x5y+3=0 which are equally inclined with axes, are
The number of roots of the equation x^{2}7x+12=0 is
If α , β are different values of x satisfying a cos x + b sin x = c then tan (α + β) /2 =
In how many different ways can the letters of the word DETAIL be arranged in such a way that the vowels occupy only the odd positions?
Let d _{1} , d _{2} , d _{3} , … …, d k be all the divisors of a positive integer n including 1 and n.
Suppose d _{1} + d_{ 2} + d _{3} + … … + d k = 72 , then the
value of 1 d _{1} + 1 d_{ 2} + 1 d _{3} + … … + 1 d k is
For all x ∈ (0,1):
The value of tan^{− 1} e (i θ)is equal to
The value of cot
For a positive integer n, let
Statement1: For every natural number
Statement2: For every natural number
Let be two unit vectors and α the angle between them. Vector → will be a unit vector is α is equal to
The quadratic equations x^{2}6x+a=0 and x^{2}cx+6=0 have one root in common. The other roots of the first and second equations are integers in the ratio 4:3. Then the common root is
Let be three nonzero vectors. Then if and only if
The value of p for which the function:
f
Given P x = x ^{4} + a x^{ 3} + b x^{ 2} + c x + d such that x=0 is the only real root of P ′ x = 0. If P − 1 < P 1 , then in the interval [1, 1]:
The number of seven digit integers, with sum of the digits equal to 10 and formed by using the digits 1,2 and 3 only, is
For which of the values of m the area of the region bounded by the curve y = x − x^{2} and y = mx equal 9 2 .
be a continuous function, then λ is equal to
Let a, b, c, d be real numbers such that
(a ^{2} + b^{ 2} − 1 )(c^{ 2} + d^{ 2} − 1 )> (a c + b d − 1 )^{2}
then which of the following statements is correct?
If equals
For a positive interger n, let
Let D be the domain of the function,
Let [x] = the greatest integer less than or equal to x. Then equation sinx = [1+sin x] +[1cos x] has
If 1, log_{y}x, log_{z}y, 15log_{x}z are in A.P., then
Let a 1, a 2, a 3, … … , a n n > 2 be real numbers such that a i = − a _{n − i + 1} for 1 ≤ i ≤ n and k = ∑ 1 ≤ i < j < k ≤ n ∑ ∑ ^{x i x j x k} , then which of the following is not true?
The solutions of are given by (where p = dy/dx and k is constant)
If A = [ α β 0 α ] is the n^{th} root of I_{2}, then choose the correct statement
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