The locus of the middle point of chords of the ellipse x2/a2+y2/b2=1 whose pole lies on the auxiliary circle is
(x2/a2+y2/b2)2= (x2+y2)/a2
(x2/a2+y2/b2)2= (x2-y2)/a2
(x2/a2-y2/b2)2= (x2+y2)/a2
(x2/a2-y2/b2)2= (x2-y2)/a2
The angle between the two line having slopes 3/2 and -2/3
π/4
π/6
π/2
π/3
Orthocentre of thele whose vertices are (2,-1/2), (1/2,-1/2), (2,(√3-1)/2) is
(2,-1/2)
(1/2,-1/2)
(2,(√3-1)/2)
(1/2,(√3-1)/2)
If X is a poisson variate with P(X = 0) = 0.8, then the variance of X is :
loge20
log1020
loge5/4
0
(√2+1)6+(√2-1)6=
198
992
99
If f : R → R is defined by f(x) = [2x] - 2[x] for x ε R, where [x] is the greatest integer not exceeding x, then the range of f is :
{ x ε R : 0 ≤ x ≤ 1}
{ 0, 1}
{x ε R : x > 0 }
{x ε R : x ≤ 0 }
cot(A+150)- tan(A- 150)=
4cos 2A/1+2 cos 2A
4cos 2A/1-2 sin 2A
4cos 2A/1+2 sin 2A
4cos 2A/1-2 cos 2A
There are 25 railway stations between Nellore and Hyderabad.The number of different kinds of single second class tickets to be printed so as to enable a passenger to travel from the station to another is
28P2
27P2
26P2
25P2
The roots of x5-5x4+9x3-9x2+5x-1=0 are
1,1,-2,-1/2
1,1±√3i/2, 3±√5/2
1,-1,-2,-1/2,3,1/3
2,1/2,3,1/3
If ax2+bx+c=0 and bx2+cx+a=0 have a common root and a≠0 then a3+b3+c3/abc=
1
2
3
9
The co-efficient of x4 in the expansion of (1-3x)2/(1-2x) is equal to:
4
The equation of the circle passing through the points of intersection of the circle x2+y2-2x+4y-20=0, the line 4x-3y-10=0 and the point (3, 1) is
x2+y2-50x+40y+100=0
2x2+3y2+100x+40y+100=0
x2+y2+50x-40y+100=0
3x2+4y2+50x+20y+100=0
The two curves y=x-3, y=e3(1-x) at (1, 1)
Touch each other
Cut orthogonally
Cut at an angle of 450
None
The equation of the tangent to the curve y2=4ax at (at2, 2at) is
x+yt-at2=0
xt-y=2at+at3
xt+y=2at+at3
x-yt+at2=0
The equation of the line joining the points represented by 2-3i and -3+4i in the Argand plane is
7x-5y-1=0
7x+5y-1=0
7x+5y+1=0
7x-5y+1=0
If 2x-3y=5 and 3x-4y=7 are the equation of two diameters of a circle whose area is 154sq units, then the equation of the circle is
x2+y2+2x-2y-47=0
x2+y2-2x+2y-49=0
x2+y2-2x+2y+47=0
x2+y2-2x+2y-47=0
The locus of the point of intersection of perpendicular tangents to the circle x2 + y2 = 16 is a circle whose diameter is
4√2
8√2
2√2
8
The ratio in which ys-plane divides the line segment joining (-3, 4, - 2) and (2,1, 3) is
-4 :1
3 :2
-2 :3
1: 4
A=(cosθ, sinθ) and B==(sinθ, -cosθ) are two points. The locus of the centroid of ΔOAB where O is the origin is
x2+y2=3
9x2+9y2=2
2x2+2y2=9
3x2+3y2=2
C1+2.C2+ 3.C3+…….+n.Cn=
(n +1)2n-1
(n +1)2n
n.2n-1
n.2n
If the number of common tangents of the circles x2+y2+8x+6y+21=0, x2+y2+2y-15=0 are 2,then the point of their intersection is
(8,5)
(8,-5)
(-8,-5)
(-4,-3)
The equation of the circle passing through the points of intersection of the circlesx2+y2=5, x2+y2+12x+8y-33=0 and touching x-axis is
x2+y2-6x-4y+9=0, 9x2+9y2-30x-20y+25=0
3 x2+3y2-16x-40y+29=0, 9x2+9y2-30x-20y+25=0
x2+y2-6x-4y+9=0, 9x2+9y2-30x-20y-25=0
x2+y2-6x-4y+9=0, x2+y2-10x+50y-25=0
If (a+ib)2= x+iy then x2+y2=
(a2+b2)2
(a2-b2)2
tan 100. tan 200. tan 300. tan 400. tan 500. tan 600. tan 700. tan 800 =
I: In a ΔABC, if 4s(s-a) (s-b) (s-c) =a2b2 then it is right angled triangle II: In a ΔABC, if sin A+ sin B +sin C maximum then triangle is equilateral
only I is true
only II is true
both I,II are true
neither I and II is true
Let A and B be two fixed points, If a perpendicular p is drawn from A to the polar of B with respect to the circle x2+y2=a2 and perpendicular q is drawn from B to the polar of A then
p=q
pOA=qOB
pOB=qOA
p2=q2
If [(x2+x+1)/(x2+2x+1)]=A+[B/(x+1)]+[C/(x+1)2] then A-B=
4C
4C+1
3C
2C
A(3x1, 3y1), B(3x2,3y2),C(3x3,3y3) are vertices of a triangle with orthocenter H at (x1+x2+x3,y1+y2+y3) then the
y = Aex + Be2x + Ce3x satisfies the differential equation :
y''' - 6y'' + 11y' - 6y = 0
y''' + 6y'' + 11y' + 6y = 0
y''' - 6y'' - 11y' + 6y = 0
The solution of (x2+x)(dy/dx)=1+2x is
ey=c(x2+x)
y=x(x+1)+c
y=(1+2x)+c
xy=x2+x+c
In ΔABC, if r1 =3, r2= 10, r3= 15, then c=
5
12
13
13/2
If f: R→R,, g: R→R, are defined by f(x)=4x-1, g(x)=x3+2, then gof(a+1/4)=
43
345
a3+2
a2-1
The equation of one tangent to the circle with Centre(2,-1) from the origin is 3x + y = 0, then the equation of the other tangent through the origin is
3x-y=0
x+3y=0
x-3y=0
x+2y=0
√3 cosec 200- sec 200 =
If Q is the radical centre of the three circles x2+y2=a2, (x-g)2+y2=a2and x2+(y-f)2=a2,then Qx+Qy=
–(g+f)/2
2g+2f
(g+f)/2
g+f
If 4 sin(600+ θ) sin(600-θ)-1= kcos 2θ, the value of k is
none
The locus of the centre of the circles which touch the lines 6x – 8y + 5=0 and 6x – 8y + 13=0 is 6x – 8y+k=0 then k is
18
10
The number of positive divisors of 253673 is
14
167
168
166
The locus of the midpoint of the chord of the circle x2+y2=25 which subtends a right angle at (2,-3) is
x2+y2+2x-3y-6=0
x2+y2-2x-3y-12=0
x2+y2-2x+3y-6=0
x2+y2+2x+3y-24=0
If x ≥ y and y > 1, then the value of the expression logx (x/y) + logy (y/x) can never be
-1
-0.5
The unit vector orthogonal to a=2i+2j+k, b=3i+4j-12k and forming a right handed system with a and b is
28i-27j-2k
-28i+27j+2k
28i-27j-2k/√1517
-28i+27j+2k/√1517
cos A+sin(2700+A)-sin(2700-A)+cos(1800-A)=
sin θ
cos θ
The point of the curve y=x4-4x3+4x2+1 at which the tangent is parallel to x-axis is
(0, 1), (1, 2), (2, 1)
(0, -1), (-1, 2), (2, 1)
(0, 1), (1, -2), (2, -1)
(0, -1), (-1, 2), (-2, 1)
The locus of the midpoints of the chords of the circle x2+y2-2x+2y-2=0 parallel to the line y=x+5 is the line which passes through the point is
(2,-1)
(0,0)
(1,1)
(-1,-1)
The pole of the line y=x+2 e with respect to the ellipse x2+4y2-2x-6y-10=0 is
(9/4, -5/24)
(6/7, -17/7)
(-26, 35/4)
The line x cosα+y sinα=p touches the circle x2+y2-2axcosα-2aysinα=0, then p=
a
2a
–a
a/2
For any integer n ≥ 1, the number of positive divisors of n is denoted by d(n). Then for a prime P, d(d(d(P7}}) =
P
If the parabola y2=-4ax passes through (-3,2) then the length of its latusrectum is
4/3
1/3
2/3
2+3+5+6+8+9+…..2n terms=
3n2+2n
4n2+2n
4n2
f(x)= x-1/x is
Increasing in R
Decreasing in R+
Not decreasing
Not increasing
If A,B,C are the remainders of x3-3x2-x+5,3x4-x3+2x2-2x-4,2x5-3x4+5x3-7x2+3x-4 when divided by x+1,x+2,x-2 respectively then the ascending order of A,B,C is
A,B,C
B,C,A
A,C,B
B,A,C
For all values of a and b(a + 2b)x + (a- b)y + (a + 5b) = 0 passes through the point:
(-1, 2)
(2, -1)
(-2, 1)
(1, -2)
I: If the vectors a=(1, x, -2), b=(x, 3, -4) are mutually perpendicular,then x=2 II: If a=i+2j+3k, b=-i+2j+k, c=3i+j and a+tb is perpendicular to c then t=5
Only I is true
Only II is true
both I and II are true
neither I nor II are true
cos θ+ cos (2400 + θ)- sin (2400- θ)=
1/4
3/4
If 3-√2 is a root of x4-8x3+21x2-26x+14=0 then the roots are
1±√2,1±i
-1±√2,-1±i
3±√2,1±i
-3±√2,-1±i
If (x+1)/(2x-1)(3x+1)=A/(2x-1)+B/(3x+1), then 16A+9B is equal to :
6
The length of the latus rectum of the hyperbola 9x2-16y2+72x-32y-16=0 is
9/2
32/3
11/5
21/5
Angle between the tangents to the curve y=x2-5x+6 at the points (2, 0) and (3, 0) is
If the equation x2+2(k+1)x+9k-5=0 has only negative roots,then
k
k>=0
k>=6
The angle between the pair of lines 2x2+5xy+2y2+3x+3y+1=0,is:
cos-1(4/5)
tan-1(4/5)
. I: The equation to the pair of lines passing through the point (2,-1) and parallel to the pair of lines 3x2-5xy+2y2-17x+14y+24=0. II: The equation to the pair of lines passing through (1,-1) and perpendicular to the pair of lines x2-xy-2y2=0 is 2x2-xy-y2-5x-y+2=0.
The angle between the curves y2=8, x2=4y-12 at (2, 4) is
If y=(x+√x2-1)m then (x2-1)y2+xy1=
m2y
m2
y
B and C are two points on the circle x2+y2=a2. From a point A(b, c) on that circle AB=AC=d. The equation to Bc is
bx+ay=a2-d2
bx+ay=d2-a2
bx+ay=2a2-d2
2(bx+ay)=2a2-d2
The solution of (dy/dx)=ey-x is
ey+ex=c
e-x=e-y+c
ey-x=c
ey+x=c
The minimum value of 2x2+x-1 is
3/2
-9/8
9/4
The area of the plane region bounded by the curve x + 2y2 = 0 and 3y2 = 1 is equal to
5/3
If (a1+ib1)(a2+ib2)……(an+ibn)=A+iB,then (a12+b12) (a22+b22)……. (an2+bn2) =
A2+B2
A2-B2
A3+B3
A3-B3
The angle between the circles x2+y2-4x-6y-3=0 and x2+y2+8x-4y+11=0 is
If an error of 0.01 cm is made while measuring the radius 10cm of a circle, then the relative error in the area is
0.02π sq.cm
4.4sq.cm
0.4π cm
0.6π cm
The length of the chord of the circle x2+y2+4x-7y+12=0 along the y-axis is
1/2
The centre of the circle x2+y2-4x-2y-4=0 is
(2, 1)
(0, 0)
(-2, -1)
(1, -1)
If x2+y2=a2 then dy/dx=
x/y
–y/x
–x/y
y/x
The intercepts of line joining the points (4,-7),(1, -5) are
5, 5/3
7/5, -7/3
5, 7/2
2, 4/3
d/dx{Tan-1(3a2x-x3/a3-3ax2)}=
2a/2(a+x2)
3a/√a+x2
3a/a2+x2
2a/√(a2+x2)
If α, β, γ are the roots of x3+px2+qx+r=0 then (β+γ-3α)(γ+α-3β)(α+β-3γ) =
3p3+16pq
3p3-16pq
3p3-16pq+64r
3p3+16pq+64r
I: The equation of the line parallel to 2x+3y-5=0 and passing through the point (3, -4) is 2x+3y-13=0 II: The equation of the line perpendicular to 2x+3y-5=0 and passing through the point (3, -4) is 3x-2y- 17=0
Both I and II are true
Neither I nor II are true
A tower 51 m high has a mark at a height of 25m from the ground. If the two parts subtend equal angles to an eye at the height of 15 m from the ground, the distance of the tower from the observe is
160 m
150 m
140 m
The angle between the lines whose direction cosines satisfy the equations l + m+ n =0, l2 + m2 – n2 = 0 is
If ( x - 2 ) is a common factor of the expressions x2 + ax + b and x2 +cx+ d, then b-d/c-a is equal to :
-2
∑((12+22+32+….+n2)/1+2+3+….+n)=
(n2+2n)/3
n2-2n/6
n2+11/12n
The equation of the circle passing the origin having its centre on the line x+y=4 and cutting the circle x2+y2-4x+2y+4=0 orthogonally is
x2+y2-2x-6y=0
x2+y2-4x-4y=0
x2+y2-6x-3y=0
A bag contains n white and n black balls. Pairs of balls are drawn at random without replacement successively , until the bag is empty , if the number of ways in which each pair consists of one white and one black ball is 14,400 then n=
If the line lx+my = 1 is a normal to the hyperbola (x2 /a2 ) - (y2 / b2) =1 then (a2/l2) - (b2/m2) is equal to
a2 - b2
a2 + b2
(a2 + b2 )2
(a2 - b2 )2
Observe the following statements A: f'(x) = 2x3 - 9x2 + 12x - 3 is increasing outside the interval (1, 2)R: f'(x) < 0 for x belongs to (1,2).Then which of the following is true
Both A and R are true, and R is not the correct reason for A
Both A and R are true, and R is the correct reason for A
A is true but R is false
A is false but R is true
The tangents are drawn to the ellipse x2/a2+y2/b2=1 at point where it is intersected by the line lx+my+n=0. The point of intersection of tangents at these points is
(a2l/n, b2m/n)
(-a2l/n, b2m/n)
(a2l/n, -b2m/n)
(-a2l/n, -b2m/n)
If tan θ=-4/3 and θ is not in the fourth quadrant , then the value of 5 sin θ+10cos θ+ 9 secθ+16 cosec θ – 4 cot θ=
A line segment of length 10 cm is divided into two parts and a rectangle is formed with these as adjacent sides, then the dimensions of the rectangle in order that its area is maximum is
4, 6
5, 5
2, 8
The distance of the point (1,2) from the common chord of the circles x2+y2-2x-6y-6=0 and x2+y2+6x-16=0 is
1/5
If |z-1/z+a|=1 where Re(a)≠0 then the locus of z=x+iy is
y=0
x=0
x2+y2+2x-4y=0 such that y1
x2+y2+2x-4y=0 such that 2x-y+4>0
If the normals from any point to the parabola x2 = 4y cuts the line y = 2 in points whose abscissa are in A.P, then the slopes of the tangents at the 3 conormal points are in
Ap
Gp
Hp
The equation of the normal to the curve 3y2=4x+1 at (1, 2) is
3x+y+5=0
3x+y-5=0
3x-y+5=0
3x-y-5=0
The centre of the circumscribing the quadrilateral whose sides are 3x+y=22, x-3y=14 and 3x+ y=62 is
(3/2, 27/2)
(27/2, 3/2)
(27, 3)
(1, 2/3)
The number of common tangents to the circles x2+y2+2x+8y-23=0, x2+y2-4x-10y+19=0 is
(1+sec 200)( 1+sec 400) (1+sec 800)=
The sum of the slopes of the lines represented by 6x2-5xy+y2=0 is
7
In a ΔABC, cos[(B+2C+3A)/2]+cos[(A-B)/2] is equal to
In a class there are 60 boys and 20 girls. In it, half of the boys and half of the girls know cricket. The probability of a person selected from the class is either a boy or a person who knows ticket is
7/8
8/7
5/7
7/5
If u=3(lx+my+nz)2-(x2+y2+z2) and l2+m2+n2=1 then uxx+uyy+uzz=
u
2u
3√1003 -3√997
0.01
0.02
0.03
0.04