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Points A(2,2), B(4,4), C(5,8) are vertices of ∆ ABC the length of the median through C is
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
If f (x) = cos [π^{2}] x + cos [− π^{2}] x where [x] is the step function, then
The sum of the focal distances from any point on the ellipse 9x^{2} + 16y^{2} = 144 is
In the following question, a Statement of Assertion (A) is given followed by a corresponding Reason (R) just below it. Read the Statements carefully and mark the correct answer
Assertion(A): f (x) = log x^{3} and g (x) = 3 log x are equal.
Reason(R) : Two functions f and g are said to be equal if their domains, ranges are equal and f (x) = g (x)∀ x in the domain .
If is continuous at x = x_{0} , then f ′ (x_{0}) is equal to
In the following question, a Statement1 is given followed by a corresponding Statement2 just below it. Read the statements carefully and mark the correct answer
Tangents are drawn from the point (17,7) to the circle x^{2}+y^{2}=169.
Statement1:
The tangents are mutually perpendicular.
Statement2:
The locus of the points from which mutually perpendicular tangents can be drawn to the given circle is x^{2}+y^{2}=338.
The sum of the digits in the unit place of all the numbers formed with the help of 3,4,5,6 taken all at a time is
If θ + Φ = π/3 then sin θ . sinΦ has a maximum value at θ =
Here, y = sin θ · sinΦ = sinθ · sin
An unbiased coin is tossed to get 2 points for turning up a head and one point for the tail. If three unbiased coins are tossed simultaneously, then the probability of getting a total of odd number of points is
The probability of A = probability of B = probability of C = 1/4, P(A∩B) = P(C∩B) = 0 and P(A∩C) = 1/8, then P(A∪B∪C) is equal to:
If the equations x^{2}+ ba + a = 0 and x^{2} + ax + b = 0 have a common root, then a + b =
If a, b, c, d, e, f are in A.P., then e  c is equal to
If a line passes through points (4,3) and (2,λ) and perpendicular to y=2x+3, then λ=
The median of a set of 9 distinct observations is 20.5. If each of the largest 4 observations of the set is increased by 2, then the median of the new set
If a, b are odd integers then number of integral root, of equation x^{10} + ax^{9} + b = 0 is equal to :
Let P is a root (∈I)
CaseI : p is odd
p^{10} + ap^{9} + b ≠ 0
Because LHS odd
CaseII : p is even
p^{10} + ap^{9} + b ≠ 0
because, LHs is odd
If the number of distinct positive rational numbers p/q smaller than 1, where p, q ∈ {1, 2, 3 ....., 6} is k then k is :
Out of numbers will result only 3 distinct rational numbers.
⇒ Total numbers = ^{6}C_{2} – 7 + 3 = 11
If two distinct chords of a parabola y^{2} = 4ax passing through (a, 2a) are bisected on the line x + y = 1, then the sum of integral values of the length of possible latus rectums is equal to :
Any point on x + y = 1 can be taken as (t, 1–t)
The equation of chord with this as midpoint is y(1–t) –2a (x + t) = (1 – t)^{2} – 4at
It passes through (a, 2a)
So, t^{2} – 2t + 2a^{2} – 2a + 1 = 0
This should have two distinct real roots. So D > 0
⇒ a^{2} – a < 0
⇒ 0 < a < 1
So, length of latus rectum < 4 and 0 < a < 1
⇒ LR = 1, 2, 3
If be three vectors of magnitude √3, 1, 2, such that is the angle between is equal to:
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