The maximum coefficients in the expansion of (1+x)^{(2n+2)} is :
The angle between the tangents drawn from (0,0) to the circle x^{2} + y^{2} + 4x  6y + 4 = 0 is
If a, b, c are different and
The solution of the equation cosylog(secx+tanx)dx=cosxlog(secy+tany)dy is
Let y=x^{2}e^{x}, then period in which y is increasing is
The centre of the ellipse
The line p = x cos α + y sin α. becomes tangent to x^{2}/a^{2}  y^{2}/b^{2} = 1 is
The equation of the ellipse whose focus is at (4, 0) and whose eccentricity is 4/5 is
If cot⁻^{1}[(cos α)^{1}∕^{2}]  tan⁻^{1}[(cot α)^{1}∕^{2}] = x, then sin x =
If the equation x  sin x = k has a unique root in , , then the range of values of k are
Consider the function
The equation of the parabola whose focus is the point (0,0) and the tangent at the vertex is xy+1=0 is
Out of 15 students studying in a class, 7 are from Maharashtra, 5 are from karnataka and 3 are from Goa. Four students are to be selected at random. What are the chances that at least one is from Karantaka?
No. of ways of selecting 4 students out of 15 students
= ^{15}C_{4} = 15 x 14 x 13 x 12 /1 x 2 x 3 x 4 = 1365
The no of ways of selecting 4 students in which no student belongs to Karnataka = ^{10}C_{4}
∴ No of ways of selecting at least one student from Karnataka = ^{15}C_{4}  ^{10}C_{4} = 1155
∴ Required probability = 1155 1365 = 77 91 = 11 13
The line y=mx+c touches the parabola x^{2}=4ay if
From a box containing 10 cards, numbered 1, 2,3,......,10. Four cards are drawn together, what is the probability of their sum is even ?
A and B play 12 games of chess of which 6 are won by A. 4 are won by B, and 2 end in a tie. They agree to play a tournament consisting of 3 games. The probability that A and B win alternately is,
If b,c and sinB are given such that ∠B is acute and b<c sinB, then
When two sides and angle opposite to one of then be given. In this case, the triangle is not always uniquely determined. It is quite possible to have no triangle, one triangle and two triangles with this type of data.
So, it is called an ambiguous case.
Let b, c and B are given parts.
When B is acute and b < c sin B, then there is no triangle
By Cosine Law
b^{2} = c^{2} + a^{2}  2ac cos B
⇒ a^{2}  2ac cos B = b^{2}  c^{2}
⇒ a^{2}  2ac cos B + c^{2} cos^{2} B
= b^{2}  c^{2} + c^{2} cos^{2} B
⇒ (a  c cos B)^{2} = b^{2}  c^{2} (1  cos^{2} B)
= b^{2}  c^{2} sin^{2} B
This equation helps us to determine a, when b, c and B are being given. We observe that:
When b < c sin B, then a is imaginary and so there is no solution.
In Δ A B C , r
For what value of p, the difference of the roots of the equation x^{2}  px + 8 = 0 is 2 ?
Let α and β be the roots of the equation x^{2}+x+1=0, then the equation whose roots are α^{19},β^{7} is
If f : R → R is continuous and differentiable function such that
Then, value of f'(4) is
if A, G and H are respectively the A.M., the G.M and the H.M. between two positive numbers 'a' and 'b', then the correct relationship is
If x, y, z are positive real numbers, then (x^{3}/z) < (x^{3} + y^{3} + z^{3})/(x + y + z) < (z^{3}/x) if
If A = {x : x^{2 } 5x + 6 = 0}, B = {2, 4}, C = {4, 5} then A x (B ∩ C) is
The obtuse angle between lines y=2 and y=x+2 is
The condition for the roots of the equation (c^{2}ab)x^{2}2(a^{2}bc)x+(b^{2}ac)=d are equal, is
The sum to infinity of the series
The equation of the curves for which the tangent is of constant length, is
The remainder left out when 8^{2n} − (62)^{2n }+ 1 is divided by 9 is:
Let f x be an odd function in the interval with a period T, then is
If f x = x^{5} − 20^{x3} + 240 x , then f x satisfies which of the following?
If , then value of x, y, z are respectively (where m , n , r ∈ I )
The shortest distance between the lines
The area of the region bounded by the parabola (y − 2 )^{2} = x − 1 , the tangent to the parabola at the point (2, 3) and the xaxis is:
Two of the lines represented by are perpendicular for
Let the line lie in the plane . Then α , β equals:
The direction ratios of two lines are a, b, c and . The lines are
The number of functions f from the set A = {0, 1, 2} in to the set B = {0, 1, 2, 3, 4, 5, 6, 7} such that f(i) ≤ f(j) for i < j and i, j ∈ A is
A fair coin is tossed 9 times the probability that at least 5 consecutive heads occurs is
The value of
Let f(x) then for all x
If in a Δ A B C , a = 6, b = 3 and cos (A − B )= 4/ 5 then
Let f(x) is a quadratic expression with positive integral coefficients such that for every
Let d has local extrema at x = α and β such that α , β < 0, f α , f β > 0 ; Then the equation f(x)=0
The equation x^{2 } − 6 x + 8 + λ (x^{2} − 4 x + 3) = 0 , λ ∈ R has
If has the value equal to
is possible, if
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