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What is the area under the curve y = x +  x  1 between x = 0 and x = 1 ?
The greatest term in the expansion of (3 + 2x)^{9}, when x = 1, is
The coordinates of the pole of the line lx+my+n=0 with respect to the circle x^{2}+y^{2}=1 are
ANSWER : d
Solution : Let (x1,y1) be the pole of the line lx+my+n=0 with respect to the hyperbola x2a2−y2b2=1. Then, the equation of the polar is
xx1/ + yy1/ = 1………(i)
Since, (x1,y1) is the pole of the line lx+my+n=0. So, the polar of (x1,y1) is also the line
lx+my+n=0...(ii)
clearly, (i) and (ii) represent the same line. Therefore,
x1(l) = y1(m) = 1/(−n)
⇒x1 = −l/n, y1 = m/n.
Hence, the pole of the given line with respect to the given hyperbola is
(−l/n, m/n)
If the line 2x  y + k = 0 is a diameter of the circle x^{2} + y^{2} + 6x 6y + 5 =0, then k is equal to
The differential equation of the family of lines passing through the origin is
The equation of line passing through the origin is y = mx , when m is constant
Diffrence w.r.t x
Which of the following is a solution of the differential equation
dy/dx= x−y/x+y
Put,y=vx
⟹dy/dx=v+x(dv/dx)
⟹v+xdv/dx=1−v/1+v
⟹xdv/dx = 1−2v−v^2/(1+v)
⟹∫v+1/(v+1)^2−2dv=−∫1/xdx
⟹1/2ln[(v+1)^2−2] = 2lnc1/x
⟹x^2(v^2+2v−1)=C
Where C = 2lnc1/x
Since,v=y/x, we get
⟹ y^{2}+2xy−x^{2}=C
Value of 1 + log x + (log x)^{2}/2! + (log x)^{3}/3! + ..... ∞ is
The angle of elevation of a cloud from a point h mt above the surface of a lake is θ and the angle of depression of its reflection in the lake is φ . The height of the cloud is
The eccentricity of the conjugate hyperbola of the hyperbola x^{2}  3y^{2} = 1 is
Which of the following functions is a solution of the differential equation (dy/dx)^{2}  x (dy/dx) + y = 0?
The solution of the differential equation (dy/dx) = (y/x) + (φ (y/x)/φ' (y/x)) is
ANSWER : c
Solution : 23n−7n−1=8n−7n−1
=(7+1)n−7n−1
=C(n,0)7n+C(n,1)7n−1+...+C(n,n−2)72+C(n,n−1)71+C(n,n)70−7n−1
Now, C(n,0)7n+C(n,1)7n−1+...+C(n,n−2)72 is clearly divisible by 49.
So, we can write it as 49k.
So, our expression becomes,
=49k+C(n,n−1)71+C(n,n)70−7n−1
=49k+7n+1−7n−1
=49k
∴23n−7n−1=49k
So, clearly 23n−7n−1 is divisible by 49.
If A, B are two square matrices such that AB = A and BA = B, then
For a square matrix A, it is given that AA' = I, then A is a
The real value of α for which the expression 1i sin α/1+2 i sin α is purely real is
For xy = 0
The lines are: x = 0 & y = 0 which are the Y and X axis respectively which are perpendicular.
The equation of the normal to the curve x^{2} = 4y at (1, 2) is
Two finite sets have m and n elements, the total number of subsets of the first set is 56 more than the total number of subsets of the second. The value of m and n are respectively
Let A denote the first set and B denote the second set
We have, n(A) = 2^{m} and n(B) = 2^{n}
As per the question, we have
n(A) = 56 + n(B)
⇒ n(A)  n(B) = 56
⇒ 2^{m}  2^{n} = 56
⇒ 2^{n} (2^{m  n}  1)
⇒ 2^{n} (2^{m  n}  1) = 8 x 7
⇒ 2^{n} = 8 = 2^{3} or (2^{m  n}  1) = 7
⇒ n = 3 or 2^{m  n} = 8 = 2^{3} = 2^{6  3}
⇒ n = 3 or m  n = 3
⇒ n = 3 or m = 6
Hence, the required values of m and n are 6 and 3 respectively
In how many ways can the letters of the word ARRANGE be arranged so that R's are never together?
Reqd. ways = = 1260  360 = 900
A and B are events such that P(A ∪ B) = 3/4, P(A ∩ B) = 1/4, P(A̅)= 2/3, then P(A̅ ∩ B) is
The probability that a number selected at random from the set of numbers {1,2,3,....,100} is a cube is
A = B = C = 60^{o}
r : R : r_{1} = 4R sin (A/2) sin (B/2) sin (C/2) : R : 4R sin (A/2) cos (B/2) cos (C/2)
= 4 (1/2) (1/2) (1/2) : 1 : 4 (1/2) (√3 /2) (√3 /2) = (1/2) : 1 : (3/2) = 1 : 2 : 3
The perimeter of a triangle is 16cm. One of the sides is of length 6cm. If the area of the triangle is 12sq.cm, then the triangle is
If the sum of first n terms of an A.P. be 3n^{2}  n and its common difference is 6, then its first term is
In a town of 10,000 families it was found that 40% family buy newspaper A, 20% buy newspaper B, and 10% families buy newspaper C, 5% families buy A and B, 3% buy B and C and 4% buy A and C. If 2% families buy all the three newspapers, then number of families which buy A only is
If f : N x N → N is such that f (m,n) = m + n where N is the set of natural numbers, then which of the following is true?
The ortho centre of triangle whose vertices are (0,0), (3,0) and (0,4) is
The angle between the curves y^{2}=x at x^{2}=y at (1,1) is
The acute angle between the planes 2xy+z=6 and x+y+2z=3 is
Max. value will occur when cosx=cosy=1 and cosz=0
If 3i+4j and 5i+7j represent the sides of a triangle, then its area is
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