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The eccentricity of the hyperbola 4x^{2} – 9y^{2} = 36 is
The length of the latus rectum of the ellipse 16x^{2}+ 25y^{2} = 400 is
Length of latus rectum =
The vertex of the parabola y2+ 6x – 2y + 13 = 0 is
(y 1)^{2}  6x  12
Vertex → (2, 1)
The coordinates of a moving point p are (2t^{2}+ 4, 4t + 6). Then its locus will be a
The equation 8x^{2}+ 12y^{2} – 4x + 4y – 1 = 0 represents
If the straight line y = mx lies outside of the circle x^{2}+ y^{2} – 20y + 90 = 0, then the value of m will satisfy
The locus of the centre of a circle which passes through two variable points (a, 0), (–a, 0) is
Centre lies on yaxis locus x = 0
The coordinates of the two points lying on x + y = 4 and at a unit distance from the straight line 4x + 3y = 10 are
The intercept on the line y = x by the circle x^{2}+ y^{2} – 2x = 0 is AB. Equation of the circle with AB as diameter is
If the coordinates of one end of a diameter of the circle x^{2}+y^{2}+4x–8y+5=0, is (2,1), the coordinates of the other end is
Centre circle (–2, 4)
If the three points A(1,6), B(3, –4) and C(x, y) are collinear then the equation satisfying by x and y is
If and θ lies in the second quadrant, then cosθ is equal to
θ in 2nd quad Cosθ < 0
The solutions set of inequation cos^{–1}x < sin^{–1}x is
cos^{–1}x < sin^{–1}x is
√5<3 No solution
If sinθ and cosθ are the roots of the equation ax^{2 }– bx + c = 0, then a, b and c satisfy the relation
If A and B are two matrices such that A+B and AB are both defined, then
Addition is defined if order of A is equal to order of B
A B
nxm nxm is defined if m = n
⇒ A, B are square matrices of same order
A = A^{T}
= Real
The equation of the locus of the point of intersection of the straight lines x sin θ + (1 – cos θ) y = a sin θ and x sin θ – (1 + cos θ) y + a sin θ = 0 is
y = a sin θ
x = a cos θ.
sin θ + cos θ = 0
⇒ tan θ = – 1
The period of the function f(x) = cos 4x + tan 3x is
If y = 2x^{3} – 2x^{2} + 3x – 5, then for x = 2 and ∆ x = 0.1 value of ∆ y is
The approximate value of correct to 4 decimal places is
y = 2 + 1/80
The general solution of the differential equation
auxilary equation m^{2} + 8m + 16 = 0 ⇒ m = – 4
Solution
The degree and order of the differential equation are respectively
The function f(x) = ax + b is strictly increasing for all real x if
f′ (x) = a
f′(x) > 0 ⇒ a > 0
The general solution of the differential equation
C_{2} → C_{2} – C_{3}
C_{3} → C_{3} + C_{2}
C_{3} → C_{3} + ωC_{1 }
C_{2} → C_{2 }– C_{1}
4 boys and 2 girls occupy seats in a row at random. Then the probability that the two girls occupy seats side by side is
A coin is tossed again and again. If tail appears on first three tosses, then the chance that head appears on fourth toss is
If A and B are coefficients of xn in the expansions of (1+ x)^{2n} and (1+x)^{2n – 1} respectively, then A/B is equal to
A = ^{2n}C_{n}.
B = ^{2n – 1}C_{n}
If n > 1 is an integer and x ≠0, then (1 + x)^{n }– nx – 1is divisible by
If ^{n}C_{4} , ^{n}C_{5} and ^{n}C_{6 }are in A.P., then n is
The number of diagonals in a polygon is 20. The number of sides of the polygon is
^{n}C_{2} –n = 20
n = 8
Let a , b, c be three real numbers such that a + 2b + 4c = 0. Then the equation ax^{2} + bx + c = 0
If the ratio of the roots of the equation px^{2} + qx + r = 0 is a : b, then
If α and β are the roots of the equation x^{2} + x + 1 = 0, then the equation whose roots are α^{19} and β^{7} is
α and β are the roots of x^{2} + x + 1 = 0
α = ω β = ω^{2}
α^{19}= ω β^{7}= ω^{2}
x^{2} – (α^{19} + β^{7})x + α^{19}β^{7}= 0
Thou,
x^{2} – (ω + ω^{2} ) x + ω . ω^{2} = 0
x^{2} + x + 1 = 0
For the real parameter t, the locus of the complex number in the complex plane is
Given
Let z = x + iy
x = 1 – t^{2}
y2 = 1 + t2
Thus, x + y^{2} = 2
y^{2} = 2 – x
y^{2} = – (x – 2)
Thus parabola
If ω ≠ 1 is a cube root of unity, then the sum of the series S = 1 + 2ω + 3ω² + .......... + 3nω^{3n – 1} is
If log_{3}x + log_{3}y = 2 + log_{3}2 and log3 (x + y) = 2, then
: log_{3}x + log_{3}y = 2 + log_{3}2
⇒ x.y = 18
log (x + y) = 2 ⇒ x + y = 9
we will get x = 3 and y = 6
If log_{7} 2 = λ, then the value of log_{49} (28) is
log_{49}28 = log_{72}4 × 7
log a . (2log a – log b)(3log a – 2 log b)
= T_{2} – T_{1 }= log a – log b
= T_{3} – T_{2} = log a – log b
If in a triangle ABC, sin A, sin B, sin C are in A.P., then
: c_{1} → c_{1} + c_{2} + c_{3}
The area enclosed between y^{2} = x and y = x is
Let f(x) = x^{3} e^{–3x}, x > 0. Then the maximum value of f(x) is
f(x) = x^{3} .e^{–3x}
= f′(x) = 3^{x2} e^{–3x} + x^{3} e^{–3x} (–3)
= x^{2}3e^{–3x}[1 – x] = 0, x = 1
Maximum at x = 1
f(1) = e^{–3}
The area bounded by y^{2} = 4x and x^{2} = 4y is
The acceleration of a particle starting from rest moving in a straight line with uniform acceleration is 8m/sec2 . The time taken by the particle to move the second metre is
Integrating Factor (I.F.) of the defferential equation
The differential equation of y = ae^{bx} (a & b are parameters) is
y = a.e^{bx} ............ (i)
y_{1} = abe^{bx}
y_{1} = by ........................... (ii)
y_{2 }= by_{1} ......................... (iii)
Dividing (ii) & (iii)
= 2x f(x)
Let f(x) = tan^{–1}x. Then f′(x) + f′′(x) is = 0, when x is equal to
f(x) = tan–1x
= π
Put A = 0.
If f(x) is continuous at x = 2, the value of λ will be
If f(x + 2y, x – 2y) = xy, then f(x, y) is equal to
The locus of the middle points of all chords of the parabola y^{2} = 4ax passing through the vertex is
2h = x, 2k = y
y2 = 4ax
k^{2} = 2ah
y^{2} = 2ax
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