The coefficient of x^{3} in ((√x^{5}) + (3/√x^{3}))^{6} is
If the coefficient of x^{7} in [ax^{2}+(1/bx)]^{11} equals the coefficient of x^{7} in [ax(1/bx^{2})]^{11}, then a and b satisfy the relation
The third term in the expansion of ((x^{2})  (1/x^{3}))^{n} is independent of x, when n is equal to
The equation of the circle passing through intersection of circles x^{2}+y^{2}1=0 and x^{2}+y^{2}2x4y+1=0 and touching x+2y=0 is
The square root of the number 5 + 12i is
The equation of circle which passes through (4,5) and whose centre is (2,2) is
The volume of a solid which is obtained by revolving area bounded by an ellipse x^{2}+9y^{2}=9 and straight line x+3y=3 about yaxis is
If
The degree of the differential equation d^{2}y/dx^{2}+(dy/dx)^{3}+6y=0 is
The roots of the equation =0 are
Differential of x^{6} w.r. to x^{3} is equal to
If x = a cos^{3} θ, y = a sin^{3} θ, then
√[1 + (dy/dx)^{2}] =
The eccentricity of the conic 9x^{2} + 25y^{2} = 225 is
The graph represented by the equations x = sin^{2}t,y = 2cost is
If the major axis of an ellipse is thrice the minor axis, then its eccentricity is equal to
The curve represented by x = a (coshθ + sinhθ) , y = b(coshθ − sinhθ) is
The least possible value of k for which the function f(x) = x^{2} + kx + 1 may be increasing on [1,2] is
We have , f x = x 2 + k x + 1
⇒ f ′ x = 2 x + k . Also , f ″ x = 2
Now , f ″ x = 2, ∀ x ∈ [ 1,2 ]
⇒ f ″ x > 0, ∀ x ∈ [ 1,2 ]
⇒ f ′ x is an increasing function in the interval [ 1,2 ] .
⇒ f ′ 1 is the least value of f′(x) on [ 1,2 ]
But f ′ x > 0 ∀ x ∈ [ 1,2 ]
[ ∵ f x is increasing on [1,2] ]
∴ f ′ 1 > 0, ∀ x ∈ [ 1,2 ] ⇒ k > − 2
Thus, the least value of k is 2
If z=1+i,then the multiplicative inverse of z^{2} is
The equation of the parabola whose vertex is at (2,1) and focus at (2,3) is
sin[(1/2)cos⁻^{1}(4/5)] is equal to
The pole of the line 2x + 3y − 4 = 0 with respect to the parabola y^{2 } = 4x is
How many total words can be formed from the letters of the word INSURANCE in which vowels are always together?
The probability of drawing a card which is atleast a spade or a king from a well shuffled pack of cards is
Let A be the event that the card drawn is spade,
B be the event that card drawn is king and S be the sample space,
When a card is drawn from a pack of cards.
Now n(S) = ^{52}C_{1} = 52, n(A) = ^{13}C_{1} = 13, n(B) = ^{4}C_{1} = 4, n(A ∩ B) = ^{1}C_{1} = 1
In (in terms of x , y , z only) is
In
If the equation (λ1)x^{2} + (λ+1)x+(λ1)=0 has real roots, then λ can have any value in the interval ,
Given that f(x) is continuously differentiable on a ≤ x ≤ b where a < b , f a < 0 and f b > 0, which of the following are always true ?
(i) f(x) is bounded on a ≤ x ≤ b .
(ii) The equation f(x) = 0 has at least one solution in a < x < b .
(iii) The maximum and minimum values of f(x) on a ≤ x ≤ b occur at points where f ′ c = 0
(iv) There is at least one point c with a < c < b where f ′ c > 0.
(v) There is at least one point d with a < d < b where f ′ c < 0.
(i) This statement is true, every continuous function is bounded on a closed interval.
(ii) True again, by Intermediate Value Theorem.
(iii) Not ture, because maximum and/or minimum value could also occur at a or b, without the derivatives being O.
(iv) True By the Mean Value Theorem, there exists a point betweem a and b, where the derivative is exactly a clearly positive value.
(v) Not always true, for example, the function might be strictly increasing guaranteeing the derivative to be always positive.
Thus, the true statements are (i), (ii) and (iv).
If f (x) = {2x  3, x ≤ 2} {x, x < 2} then f (1) is equal to
If a, b, c, d, e, f are in A.P., then e  c is equal to
Given a, b, c, d, e, f are in A.P.
Let p be the common difference.
b = a+p
c = a+2p
d = a+3p
e = a+4p
f = a+5p
dc = a+3p – a2p = p
ec = a+4p(a+2p) = 2p
= 2(dc)
A line passing through point A(5,4) meet other three lines x + 3y + 2 = 0, 2x + y + 4 = 0 and x  y  5 = 0 at B, C and D respectively. If ( 15 AB )^{2} + ( 10 AC )^{2} = ( 6 AD )^{2}, then the equation of line is
Equation of any line through A(5, 4) is
= r(say)
then the coordinates of any point on this line at a distance r from A are
(r cosθ  5, r sinθ  4)
If AB = r_{1}, AC = r_{2}, AD = r_{3}
then (r_{1} cosθ  5, r_{1} sinθ  4) lies on x + 3y + 2 = 0
⇒ r_{1} cosθ  5 + 3 (r_{1} sinθ  4) + 2 = 0
⇒ r_{1} (cosθ + 3 sinθ) = 15
⇒ r_{1} =
Therefore according to the given condition
(cosθ + 3 sinθ)^{2} + (2 cosθ + sinθ)^{2}
= (cosθ  sinθ)^{2}
⇒ 4 cos^{2}θ + 9 sin^{2}θ + 12 sinθ cosθ = 0
⇒ (2 cosθ + 3 sinθ)^{2} = 0
⇒ 2 + 3 tanθ = 0
⇒ θ = 
Hence the required equation of the line is
y + 4 =  (x + 5)
or 2x + 3y + 22 = 0
If α_{1} , α_{2} … α_{n} be the roots of the equation is equal to_____
If a is any real number, the number roots of cot x − tan x = a in the first quadrant is
Evaluate the differentiation of
Explanation:
If the normals at (x_{i}, y_{i}), i = 1,2,3,4 on the rectangular hyperbola xy = c^{2} meet the point α , β then
Let = 1 and a x + b y = 1 be two variable lines and a, b be parameters such that a^{2} + b^{2} = ab then the locus of point of intersection of the lines is
The determinant is equal to zero if
The locus of the point of intersection of the lines (t is a parameter) is a hyperbola with eccentricity
{x} and [x] represent fractional and integral part of x, then is equal to
If 0 < x < 1, then [{x cos (cot^{1} x) + sin (cot^{1} x)}^{2}  1]^{1/2} =
If p, q, r are positive integers and ω be an imaginary cube roots of unity and
The sum of the infinite terms of the series tan^{−1} + tan^{−1} + tan^{−1} + … is equal to
If are noncoplanar nonzero vectors and is any vector in space then,
is equal to
(ο) is a binary function such that aοb = ab+1 where a,b∈z then the given binary function is
If a ∗ b = a+b2 and if x ∗ 3 = 7 then what is the value of x^{1}?
The system of confocal conics
being an arbitary constant
If be any point on a line then the range of values of t for which the point P lies between the parallel lines x + 2y = 1 and 2x + 4y = 15 is:
If , then the value of x is
Which of the following expression are meaningful ?
The polynomial expression (x^{3} + ax^{2} + bx + c) is divisible by (x+ 1)^{2}, then
If y = f(x) be the equation of a parabola (the axis of the parabola being parallel to the yaxis) which is touched by the line y = x at x = 1, then
If are any two vectors then the vector bisecting the angle between
If 8 sinθ + 7 cosθ = 8 , then the value of 7 sin θ − 8 cosθ is equal to
Let be vector parallel to line of intersection of planes P_{1} and P_{2}. Plane P_{1} is parallel tp the vectors then the angle between vector and a given vector
In Δ ABC, if tan A and tan B are the roots of the equation ab(x^{2} + 1) = c^{2}x where a, b, c are the sides of the triangle, then
If 3 unequal numbers a, b, c are in H.P. and their squares are in A.P., then a : b : c is equal to
The function
has a local minimum at x
does not exist when
Where [x] denotes step up function and {x} fractional part function
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