The first three terms in the expansion of (1 + ax)^{n} (n ≠ 0) are 1, 6x and 16x ^{2}. Then the values of a and n are respectively
In the expansion of (1+x)^{(2n+2)} the maximum coefficient is :
The limiting point of the system of coaxial circles x^{2}+y^{2}6x6y+4=0, x^{2}+y^{2}2x4y+3=0 is
Find the value of
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
The eccentricity of the conic 9x^{2} + 25y^{2} = 225 is
The line y = 4x + c touches the hyperbola x^{2}  y^{2} = 1 if
We know that the line y = mx + c touches the hyperbola x 2 a 2  y 2 b 2 = 1, then
c^{2} = a^{2}m^{2}  b^{2}
Here the hyperbola is x^{2}  y^{2} = 1
ie here a^{2} = b^{2} = 1
and comparing y = 4x + c with y = mx + c,
we get
m = 4
∴ c^{2} = 16  1 = 15
The st. line lx + my + n = 0 touches the hyperbola x^{2}/a^{2}  y^{2}2/b^{2} = 1 if
The equation x + e^{x} = 0 has
Let f x = x + e x = 0.
Since f − ∞ = − ∞ and f + ∞ = ∞ ,
∴ f x = 0 has a real root.
Let the real root be α . Then f( α ) = 0.
Now , f ′ x = 1 + e x > 0, ∀ x ∈ R
∴ f x is an increasing function ∀ x ∈ R .
∴ for any other real number β ,
f β > f α or f β < f α .
But f a = 0 ; so , f β ≠ 0.
∴ f x = 0 has no other real root.
Hence, the equation has only one real root.
tan⁻^{1}(1/7)+2tan⁻^{1}(1/3)=
The line y=mx+c touches the parabola x^{2}=4ay if
If z₁=z₂ and amp. z₁+amp.z₂=0, then
The normals to the parabola y^{2}=4ax from the point (5a,2a) are
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
If two angles of a Δ A B C are 45 º and 60 º then the ratio of the smallest and the greatest sides are
If the equation x^{3}  ax^{2} + bx  a = 0 has real roots, then it must be the case that :
If α,β are the roots of the equation x^{2}+x+1 = 0 and α/β, β/α are roots of the equation x^{2}+px+q = 0, then p equals
If cos2θ=(√2+1)(cosθ(1/√2)), the value of θ is
Let f be real valued function satisfying and φ x = ∫ x x + 8 f t d t , then φ ′ x is
If α , β and γ are the altitudes of the Δ A B C from the vertices A, B and C respectively, then the value of is
If f(x) is an even and differentiable function, then the value of
If the line x cos θ + y sin θ = 2 is the equation of a transverse common tangent to the circles x 2 + y 2 = 16 and , then θ equals
If two events A and B are such that P(A^{c}) = 0.3, P(B) = 0.4 and , then is equal to
Area bounded by , xaxis and ordinates x = 0 and x = 3 2 is
The distance between the chords of contact of the tangents to x ^{2} + y^{ 2} + 2 g x + 2 f y + c = 0 from (0, 0) and (g, f) is
Let F denote the set of all onto functions from A = { a 1 , a 2 , a 3 , a 4 } to B = { x , y , z } . A function f is chosen at random from F. The probability that f − 1 consists of exactly one element is
For any two complex numbers z_{1}, z_{2} and a, b ∈ R,
Let a and b be nonzero real numbers. Then, the equation (ax^{2} + by^{2} + c) (x^{2}  5xy + 6y^{2}) = 0 represents
The values of a, b, c for which the equation has infinitely many solutions are
The function defined by , Where [ ] denotes greatest integer function satisfies
The number of real roots of the equation
Tangents drawn from the point P(1,8) to the circle
x^{ 2} + y^{ 2} − 6 x − 4 y − 11 = 0
touch the circle at the points A and B. The equation of the circumcircle of the triangle PAB is
The equation sin x = [ 1 + sin x ] + [ 1 − cos x ] has
{where [x] is the greatest integer less then or equal to x }
If n is the number of positive integral solutions of X_{1} X_{2} X_{3} X_{4} = 210 then
Let a, b, c, d real numbers. Suppose that all the roots of the equation z^{4} + az^{3} + bz^{2} + cz + d = 0 are complex numbers lying on the circle z = 1 in the complex plane. The sum of the reciprocals of the roots is necessarily
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
Two lines 0 where a , b ∈ C − { 0 } and c , d ∈ R , are:
All points lying inside the triangle formed by the points (1, 3), (5, 0) and (1, 2) satisfy
Let A(z_{1}), B(z_{2}), C(z_{3}) be the vertices of a triangle in the Argand plane. then which of the following statements is correct?
Let f x = max { 1 + sin x , 1 , 1 − cos x } , x ∈ [ 0 , 2 π ] and g x = max { 1 ,  x − 1  } ∀ x ∈ R then,
If P(1, 2), Q(4, 6), R(5, 7) and S(a, b) are the vertices of a parallelogram PQRS, then
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