A to Z Exams

If a, b, c form a geometric progression with common ratio r,then the sum of the ordinates of the Points of intersection of the line ax + by + c = 0 and the curve x + 2y2=0

If f : R → R is continuous such that f(x + y) = f(x) + f(y), ∀x∈R, y∈R, and f(1) = 2 then f(100) =

If f(x)=x3-x, g(x)=sin 2x, then

The area (in square unit) of the region enclosed by the curves y=x2 and y=x3 is

The range of 5 Co-1(3x) is

If au+b=a2x+y then uxuy=

The orthocentre of the triangle formed by the points(2,1,5), (3,2,3), (4,0,4) is

If sin-1 (3/5)+sin-1(5/13)= sin-1 x, then x=

The equation of the normal to the hyperbola x2-4y2= 5  at (3,-1) is

If the orthocenter of the angle formed by the lines 2x+3y-1=0, x+2y-1=0, ax+by-1=0  is at the origin, then (a, b) is given by

nC0+nC1+nC2+………+nCn =

12+32+52+…+(2n-1)2=

The length of the tangent of the curves x=a cos 3θ, y= a sin3θ (a>0) is

Eight chairs are numbered 1 to 8. Two women and three men wish to occupy one chair each. First the women choose the chairs from amongst the chairs marked 1 to 4 and then the men select the chairs from amongst the remaining. The number of possible arrangements is

In a ∆ABC, a(cos2 B + cos2 C) + cos A (c cos C + b cos B) is equal to

The equation of conjugate axis of the  hyperbola 5x2-4y2-30x-8y-30=0  is

If (2, -2),(-1, 2),(3, 5) are the vertices of a triangle then the equation of the side not passing through (2, -2) is

If one root of px2-14x+8=0 is 6 times the other,then p=

Solution of (dy/dx)=(y+2)/(x-1) is

The area bounded by x=0,x=6+5y-y2 is

The in-centre and ex centre of the triangle formed by the lines 3x+4y=0, 5x-12y=0, y=15 are

The circumcircle of a triangle is given by x2+y2-4x+6y-3=0. The radius of the nine point circle of the triangle is

The curve whose subtangent is twice the abscissa of the point of contact and passing through (1,2) is

A circle C of radius 2 units rolls inside the rim of the circle x2+y2+8x-2y-19=0.Then the locus of the centre of C is

If ω is a complex cube root of unity, then sin[(ω10+ω23)π-π/4]=

The angle A ofABC is found by measurement to be 630 an the area is calculated by the formula 1/2bc sin A. the percentage error in the calculated value of the area due to an error of 15 minutes in the measured value of A is

The period of sin x sin(1200+x) sin(1200-x)is

The letters of the word SUCCESS are arranged in a row at random. The probability that all S’s come together is

The length of the chord of the circle x2+y2+4x-7y+12=0 along the y-axis is

(1+ω-ω2) (1-ω+ω2)=

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

If the equation of the pair of tangents drawn from (1, 2) on the ellipse x2+2y2=2 is 3x2-4xy-y2+ax+by+c=0 then the ascending order of a, b, c is

If f : R→ C is defined by f(x) = e2ix  for x є R, then f is (where C denotes the set of all complex numbers)

The angle subtended by the double double ordinate of length 16 of the parabola y2=8x at its vertex is

The points 2a-b+3c, -a+2b-4c, 12a-b-3c, 6a+2b-c lie in a plane. Then the vector equation of the plane is

The tangent of the difference of the angles made by the lines 4x2-24xy+11y2=0 with x-axis is

The equation of the line passing through (-4, 3) and having intercepts whose ratio is 5:3 is

If α,β,γ are the roots of the equation x3-px2+qx-r=0,then α3β3γ3=

If A=cos 150- cos 750,B= tan 150+ tan 750, C= cos2450 – sin2 150 then ascending order is

The number of circles that touch all the straight lines x + y = 4, x - y = -2 and y = 2 is:

If two angles of ∆ ABC are 45o and 60o, then the ratio of the smallest and the greatest sides are

Let A(-2,3), B (-7,5), C (3,-5) are three vertices of a triangle and area of the triangle ABC is P and Area of the triangle formed by the midpoints of AB, BC, CA is Q and area of the triangle formed by two vertices and centroid is R then decreasing order of P, Q,R is

If A=diag(3,3,3) then A4 =

The excentre of the triangle formed by the points (1,2), (1,5), (5,2) which is opposite to (1,2) is

The difference of focal distance of any point on the hyperbola [(x2/36)-(y2-9)]=1 is

The equations of the tangents drawn from the origin x2+y2+2gx+2fy+f2=0 is

If the orthocentre and circumcentre of a triangle  are (2,-3), (5,6) then the centroid is

The equation of the line dividing the line segment joining the points (2, 5), (6, 3) in the ratio 3:4 and externally and parallel to x+2y+7=0.  Is

The intersection of the sphere x2+y2+z2+7x – 2y –z =13 and x2+y2+z2-3x+3y+4z=8 is the same as the intersection of one of the sphere and the plane

The vertex of a parabola is the point (a,b) and latusrectum is of length 1. If the axis of the parabola is along the positive direction of y – axis, then its equation is

The parametric equations of circle (x-3)2+(y-2)2=100 are

If x+3y=16 is the perpendicular bisector of AB and A (5, 7), then B is

If α,β are the roots of ax2+bx+c=0 then (aα+b)-2+(aβ+b)-2=

The ordinate of  the centroid of the triangle formed by conormal points on the parabola y2=4ax is

If the product of the roots of the equation 5x2-4x+2+k(4x2-2x-1)=0 is 2,then k=

If the distance the points(5,-1,7) and (c,5,1) is 9 then c=

The vector r satisfying the conditions that i) it is perpendicular to 3i+2i+2k and 18i-22j-5k ii) it makes an obtuse angle with y-axis, iii)|r|=14 is

The equation of the circle with centre at (-3, 4) and touching y-axis is

2 tan h -1 1/2  is equal to

From 101 to 1000 natural numbers a number is taken at random. The probability that the number is divisible by 17 is:

1+ n/3+n(n+1)/3.6+n(n+1)(n+2)/3.6.9+............∞=

If α,β,γ are the roots of x3-x-1=0 then the transformed equation having the roots 1+α/1-α,1+β/1-β,1+γ/1-γ is obtained by taking x=

If the algebraic sum of the perpendicular distances from the points(2,0),(0,2),(4,4) to a variable line is equal to zero. Then the line passes through the point

4 cos 6θ cos 4θ cos 2θ=

(2-)5 + (2+)5 =

A straight line through the point (2, 2) intersects the lines √3x-y=0 at the points A and B. the equation to the line AB so that the triangle OAB is equilateral is

d/dx{Tan-1(x/1+√1-x2)}=

The product of the perpendicular distances from the origin on the pair of straight lines12x2 + 25xy + 12y2 + 10x + 11y + 2 = 0, is

If the tangents and normals at the extremities of a focal chord of a parabola intersect at (x1,y1) and (x2,y2) respectively, then

The shortest distance between the straight line passing through the point A=(6, 2, 2) and parallel to the vector(1, -2, 2) and the straight line passing through A’=(-4, 0, 1) and parallel to the vector (3, -2, -2) is

32cos4 θ.sin2 θ=

The number of positive divisors of 253673 is

If the lines joining the origin to the points of intersection of x+2y=k, 2x2-2xy+3y2+2x-y-1=0 are at right angles, then k=

From a well shuffled pack of 52 playing cards two cards are drawn at random, one after another without replacement. The probability that 1st one is a king and second one is queen is

The period of sec(2x+5) is

If a=2i-3j-k, b=i+4j-2k then (a+b)x(a-b)=

If 3p2=5p+2 and 3q2=5q+2 where p≠q,then the equation whose roots are 3p-2q and 3q-2p is

In a class of 125 students 70 passed in Mathematics and 55 in statistics and 30 in both. The probability that a student, selected at random from the class, has passed in only one subject is

If α,β,γ,δ are the roots of x4+px3+qx2+rx+s=0 then Σα2βγ

In measuring the vertical angle  of the sector of acircle of radius  30cms, an error of  10 is made. The error in the area of the sector is

The lines represented by the equation 3x2-5xy+2y2=0 are

If P is a point on the parabola y2 = 4ax such that the subtangent and subnormal at P are equal, then the coridinate of P are

If x4-5x3+9x2-7x+2=0 has a multiple root of order 3 then the roots are

If the circle x2+y2=a2 intersects the hyperbola xy=c2 in four points (xi, yi), for i=1,2,3 and 4 then y1+y2+y3+y4=

If the sum of odd terms and the sum of even terms in the expansion of (x+a)n are p and q respectively then p2+q2=

The condition for f(x)= x3+px2+qx+r(x R) to have no extreme value, is

2 sec2 θ- sec4 θ- 2 cosec2 θ+ cosec4θ=

If x ≥ y and y > 1, then the value of the expression logx (x/y) + logy (y/x) can never be

The value of (i)i is equal to

If θ is the angle between the curves xy=2, x2 +4y=0 then tan θ=

If the 2nd term in the expansion (13√a+a/√a-1) is 14a5/2, then the value of nC3/nC2 is

The number of common tangents that can be drawn to the circles x2+y2=1 and  x2+y2-2x-6y+6=0 is

sin 3π/5+sin 4π/5+ sin 6π/5+sin 7π/5=

The locus  of the  midpoint its of chords of x2/a2-y2/b2=1 which pass through the focus (ae, 0) is

If cos6 θ+sin6 θ+ k sin2 2θ=1.then k=

If f : R --->R  and g  :  R ---> R are defined by  f(x) = x -[x]  and g(x) = x- [x] for x belongs to R where [x] is the greatest integer not exceeding x then for every x belongs to R f(g(x)) is equal to

Cos (sin-13/5+Sin-1 5/13)=

The length of the intercept made by the normal at (1,6) of the circle x2+y2-4x-6y+3=0 between the coordinate axes is

If the normal at (1,2) on the parabola again at the point (l2,2t), then the value of t is

The angle of elevation of the summit of a mountain at a point A is 450. After walking 200 mt from A towards the mountain along a road included at 150, it is observed that the angle of elevation of the summit is 600. The height of the mountain is