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In the expansion of (1+x)^{(2n+2)} the maximum coefficient is :
The length of tangents drawn from the point (5,1) is to the circle x^{2}+y^{2}+6x4y3=0
If ω is cube root of unity, then (1 + ω^{3})  (1 + ω^{2})^{3} =
Let f(x) be a function satisfying f ′(x)= f x with f(0) = 1 and g(x) be a function that satisfies f(x) + g(x) = x^{2}, then value of integral
is equal to
A singular solution of the differential equation y^{2}[1+(dy/dx)^{2}]=R^{2} is :
The order and degree of the differential equation d^{2}y/dx^{2} + (dy/dx)^{1}/^{3} + x^{1}=0 are respectively
The eccentricity of the ellipse 9x^{2} + 5y^{2} − 30y = 0 is
The product of the perpendicular, drawn from any point on a hyperbola to its asymptotes is
A and B are square matrices of order n x n, then (A  B)^{2} is equal to
(A  B) x (A  B)
(A  B) x A  (A  B) x B
= A^{2}  AB  BA + B^{2}
The coordinates of a point of the parabola y=x^{2}+7x+2 which is the closest to the straight line y=3x3 is
Let a + b = 4, a < 2 and g(x) be a monotonically increasing function of x.
Then,
We have a + b = 4 ⇒ b = 4 − a and b − a = 4 − 2 a = t (say)
Thus, f(a) is an increasing function of t. Hence, the given expression increases with increase in (ba).
The length of the latus rectum of the parabola x^{2}4x8y+12=0 is
The length of the latus rectum of the parabola 4y^{2}+2x20y+17=0 is
Five digit number divisible by 3 is formed using the digits 0, 1 , 2, 3, 4 and 5 without repetition. Total number of such numbers is
A number is divisible by 3 if and only if the sum of its digits are divisible by 3
Notice that 1 + 2 + 3 + 4 + 5 = 15, which is divisible by 3
The only other way we can have a sum of 5 digits divisible by 3 is to replace the 3 by the 0 making the sum 3 less:
1 + 2 + 0 + 4 + 5 = 12, which is divisible by 3
No other choice of 5 digits can have a sum divisible by 3, because there is no other way to make the sum 12 or 15, and we certainly can't have a sum of 9 or 18
So the number of 5digit numbers that can be formed from the digits {1,2,3,4,5} is
Number of ways = 1 x 2 x 3 x 4 x 5 = 120
And the number of 5digit numbers that can be formed from the digits {1, 2, 0, 4, 5} is figured this way
Number of ways = 1 x 2 x 3 x 4 x 4 = 96
Total Number of ways = 120 + 96 = 216
cos A + cos B + cos C = 1 + 4 sin
Σ(b + c) tan A 2 tan ( B  C 2 ) = Σ(b + c) tan
If f(x) is continuous and differentiable over [−2,5] and −4 ≤ f′ (x) ≤ 3 for all x in (−2,5) then the greatest possible value of f(5) − f(−2), is
The sum of the series 1.3^{2} + 2.5^{2} + 3.7^{2} +.....upto 20 terms is
If a,b,c are in G.P.,then equations ax² + 2bx + c = 0 and dx² + 2ex + f = 0 have a common root if d/a,e/b,f/c are in
The sum of n terms of two A.P.'s are in the ratio of (7n + 1) : (4n + 27). The ratio of their 11 terms is
Hence the ratio of the 11th terms is 148 : 111
The condition that the cubic equation x^{3}  px^{2} + qx  r = 0 should have roots in G.P. is given by____
The roots of the cubic equation
The equation 4 sin^{2}x + 4 sin x + a^{2} − 3 = 0 possesses a solution if a belongs to the interval
If p , x_{1} , x_{2} , … x_{i} … and q , y_{1} , y_{2} , … y_{i} … are in A.P., with common difference a and b respectively, then the centre of mean position of the points A_{i}(x_{i}, y_{i}) where i = 1, 2, ..., n lies on the line
If the roots of the equation bx^{2} + cx + a = 0 be imaginary, then for all real values of x. The expression
3b^{2}x^{2} + 6bc x + 2c^{2} is
The locus of the orthocentre of the triangle formed by the lines
(1 + p)x  py + p(1 + p) = 0,
(1 + q)x  qy + q(1 + q) = 0,
and y = 0, where p ≠ q , is
The differential equation determines a family of circles with
The area of the region between the curves bounded by the lines x = 0 and x = is
Let A = N x N, and let ' ∗ ' be a binary operation on A defined by (a,b) ∗ (c,d) = (ad+bc,bd) for all (a,b), (c,d) ∈ N x N. Then find identity element in A.
Let A be a nonempty set and S be the set of functions from A to itself. The composition of function 'O' is a.
Let A_{1} , A_{2} , A_{3} , … … , A_{n} be n skewsymmetric matrix of same order then will be
Let a,b,c be any real numbers. Suppose that there are real numbers x,y,z not all zero such that x=cy+bz, y=az+cx and z=bx+ay. Then a^{2}+b^{2}+c^{2}+2abc is equal to
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
Given that converges, what is the value to which it converges?
Circles are drawn on chords of the rectangular hyperbola xy = c^{2} parallel to the line y = x as diameters. All such circles pass through two fixed points whose coordinates are
If sinβ is the G.M. between sinα and cos α , then cos2 β is equal to
Z_{1} , Z_{2} , Z_{3 }correspond to the vertices of an equilateral triangle and Z_{1} − 1 = Z_{2} − 1 = Z_{3} − 1 . Then
In a certain culture of bacteria, the rate of increase is proportional to the number present. It if be known that the number doubles in 4 hours, then
The polynomial 1 − x + x^{2} − x^{3} + . . . + x^{16} − x^{17} may be written in the form where y = x + 1 and are constants . Then a_{2} equals
The set of discontinuities of the function f (x) = ( 1/2 − cos2x ) contains the set
If p^{th}, 2p^{th}, 4p^{th} terms of an A.P. are in G.P. then common ratio of G.P. is
An integral solution of the equation tan^{−1 } x + tan^{−1} = tan^{−1} 3 is
If represent the equations of three planes, then point of intersection of these planes is
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