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The area common to the parabola y = 2x^{2} and y = x^{2} + 4 is
The equation of the line parallel to the tangent to the circle x^{2} + y^{2} = r^{2} at the point (x₁, y₁) and passing thro' origin is
Correct Answer : D
Explanation : mm_{1} = 1
(y_{1}/x_{1})*m_{1} = 1
m_{1} =  x_{1}/y_{1}
So, equation of line passing through (x₁, y₁)
y = m_{1}x
y = (x_{1}/y_{1})*x
yy_{1}+ xx_{1} = 0
The lines 2x3y=5 and 3x4y=7 are diameters of a circle of area 154 sq. units. Then the equation of this circle is :
The solution of the equation cosx cosy (dy/dx)=sinx siny is
The differential equation with respect to the curve y=e^{mx} is
ANSWER : c
Solution : ⇒z = [(1−i(3)^1/2)^2 * (1+i(3)^1/2)^2]/[(1+i(3)^1/2)^2 * (1−i(3)^1/2)^2]
⇒z = [(1−i(3)^1/2)^2]/(1+3)
⇒z = [1−3−2i(3)^½]/4
⇒z = − 1/2 − [i(3)^½]/2
r = [(½)^2 + ((3)^½ /2)^2]^½ =1
Comparing the above equation with z=rcosα+irsinα
rcosα = − 1/2 ⇒cosα = −1/2
rsinα = − (3)^1/2/2 ⇒sinα = −(3)^1/2/2
Since sinα and cosα, both are negative, thus the argument will be in IIIrd
quadrant.
α=180°+60°=240°(∵sin60° = (3)^1/2/2 & cos60° = 1/2)
Hence argument of given complex is 240°.
At a point a metres high above a lake the angle of elevation of a cloud is α and the angle of depression of its image is β, the height of the cloud is
The tangents to the hyperbola x^{2}  y^{2} = 3 are parallel to the st. line 2x + y + 8 = 0 at the following points
The differential equation (d^{2}y/dx^{2})^{2/3} = (y + (dy/dx))^{1/2} is of
Order = 2, Degree = 4
The solution of the differential equation (dy/dx) = (y/x) + (φ (y/x)/φ' (y/x)) is
The function f(x) = cot^{⁻1} x + x increases in the interval
For a skew symmetric odd ordered matrix A of integers, which of the following will hold true:
The sum of the products of the elements of any row of a determinant A with the same row is always equal to
The point on the curve y = x^{2} + 4x + 3 which is closest to the line y = 3x + 2 is
Which of the following statements are true ?
(1) The amplitude of the product of complex numbers is equal to the product of their amplitudes.
(2) For any polynomial f(x) =0 with real coefficients, imaginary roots occurs in conjugate paris.
(3) Order relation exists in complex numbers whereas it does not exist in real numbers.
(4) The value of ω used as a cube root of unity and as a fourth root of unity are different.
The acute angle between the lines joining origin to the intersection points of the curve x^{2}+y^{2}2x1=0 and line x+y=1 is
The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of there parallel lines is :
ANSWER : b
Solution : To form parallelogram we required a pair of line from a set of 4 lines and another pair of line from another set of 3 lines.
Required number of parallelograms = 4C2 x 3C2 = 6×3 = 18
A group of 7 is to be formed from 6 boys and 4 girls. In many ways can this be done if the boys are in majority?
In eight throws of 1 die or 3 is considered a success. Then the standard deviation of the success is
In a competitions A, B and C are participating. The probability that A wins is twice that of B, the probability that B wins is twice that of C. The probability that A loses is
In a triangle ABC, A = 30^{o}, b = 8, a = 6, then B = sin^{⁻1} x where x =
x^{2}+x+1+2k(x^{2}x1)=0 is a perfect square for how many values of k?
If the arithmetic and geometric means of two distinct, positive numbers are A and G respectively, then their harmonic mean is
a, b, c are three unequal numbers such that a, b, c are in A.P. ; b  a, c  b, a are in G.P., then a : b : c : :
Since a, b, c are in A.P.,
∴ b  a = c  b
Also b  a, c  b, a are in G.P
∴ (c  b)^{2} = (b  a)a
⇒ (b  a)^{2} = (b  a)a
⇒ b  a = a
⇒ b = 2a
Also c = 2b  a = 2(2a)  a = 3a
∴ a : b : c = a : 2a : 3a = 1 : 2 : 3
If A and B are two sets such that n(A) = 70, n(B) = 60 and n(A ∪ B)= 110, then n (A ∩ B) is equal to
ANSWER : c
Solution : y= sin^{6}(x) + cos^{6}(x).
y =(sin^{2}x)^{3} + (cos^{2}x)^{3}
y=(sin^{2}x+cos^{2}x)(sin^{4}x+cos^{4}xsin^{2}x.cos^{2}x)
y=(1).[ (sin^{2}x+cos^{2}x)^{2}–3.sin^{2}x.cos^{2}x]
y= 1  3 sin^{2}x.cos^{2}x
y = 1 3(sin x.cos x)^{2}
y= 1 3(1/2.sin2x)^{2}
y = 1  (3/4).(sin 2x)^{2}
For minimum value of y , sin2x should be maximum , maximum value of sin 2x =1 or
x=45°.
Minimum value of y= 1 (3/4).(1)^{2}
= 1  3/4 = 1/4
The equation of a line passing through the intersection of lines 3x2y1=0 and x4y+3=0 and point (π,0) is
If y = 4x  5 is a tangent to the curve y^{2} = ax^{3} + b at (2, 3), then
The smallest radius of the sphere passing thro'(1,0,0),(0,1,0) and (0,0,1) is
The equation of the sphere concentric with the sphere x^{2}+y^{2}+z^{2}4x6y8z=0 and which passes thro' (0.1,0) is
If α is a root of 25cos^{2}θ + 5cosθ  12 = 0, (π/2)<α<π, then sin2α is equal to
If 0≤x≤π and 81^{sin2x} + 81 ^{cos2x} = 30, then x is equal to
Given that θ is a acute and that sin θ =3/5. Let x,y be positive real numbers such that 3(xy) =1, then one set of solutions for x and y expressed in terms of θ is given by
The number of vectors of unit length perpendicular to vectors and is
A force acts at a point A whose position vector is If point of application of moves from A to the point B with position vector then work done by is
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