A to Z Exams

(1+ tan 320)(1+ tan130)/ (1+tan 230)(1+ tan220)=

In a ?ABC, AB =6, BC =5, CA =4 and AP bisects angle A. If P lies on BC then BP =

The distance from a fixed point O of a particle P moving in a straight line from O is given by s=16+48t-t3. The direction of motion of the particle after t=4 sec is

The work done by force F=ai+j+k in moving a particle from (1, 1, 1) to (2, 2, 2) along a straight line is 5 unit. Then a=

A, B, C are three routes from the house to the office. On any day, the route selected by the officer is independent of the climate. On a rainy day, the probabilities of reaching the office late, through these routes are 1/25, 1/10, 1/4respectively. If a rainy day the officer is late to the office then the probability that the route to be B is

If x2+xy+y2=a2, then y2=

The domain of 1/√[x]2-[x]-6 is

The equation of the sphere on the join of (3, 4, -1), (-2, -1, 0) as diameter is

The length and the midpoint of the chord 2x+y-5=0 w.r.t the circle x2+y2=9 is

The circum centre of the triangle passing through (1, √3), (1, -√3), (3, -√3) is

The area bounded by y = x2 + 2 , x - axis, x = 1 and x = 2 is :

The vectors i-2j+3k, 2i-3j+4k, i-3j+5k are

If α,β,γ are the roots of the equation x3-7x+7=0,then the value of α-4+β-4+γ-4 is

PN is the ordinate of any point P on the hyperbola x2/a2 – y2/b2 =1. If Q divides AP in the ratio a2:b2 then NQ is

The area (in sq unit ) of the region bounded by the curves 2x = y2-1 and x = 0 is

If the lines 2x+3y+1=0, 3x+2y-1=0 intersect the coordinate axes in four concyclic points then the equation of the circle passing through these four points is

If  f(x)=αx+β and f={(1, 1), (2,3), (3,5), (4, 7)} then the values of α, β are

If tan (πcos x) = cot (π sinx) then cos(x-π/4) =

The centre of the circle r2 - 4r (cosθ + sin θ) - 4 = 0 in cartesian coordinates is :

At a given instant the legs of a right angled triangle are 8 inch and 6 inch respectively. The first leg decreases at 1 inch per minute and second increases at 2 inch per minute. The rate of increasing of the area after 2 minutes is

The ratio in which the line joining the points A(-1, -1) and B(2, 1) divides the line joining C(3, 4) and D(1, 2) is

If the normal to the curve x3-y2 =0 at (m2, -m3) is y=mx-2m3, then the value of m2  is

The radical centre of the circle x2+y2+arx+br y+c=0, r=1, 2, 3 is

The number of ways that 12 prizes can be divided among 4 students so that each may have 3 prizes is

The locus of the point of intersection of perpendicular tangents to the circle x2 + y2 = 16 is a circle whose diameter is

The conic represented by x2-4x+3y-1=0 is

If the plane 2ax-3ay+4az+6=0 passes through the midpoint of the line joining the centres of the spheres x2+y2+z2+6x-8y-2z=13 and x2+y2+z2-10x+4y-2z=8 then a=

6 boys and 4 girls sit around a round table at random. The probability that the no two girls sit together is

The locus of poles of tangents to the hyperbola x2-y2=a2 w. r. t the parabola y2=4ax is

The lines (a+b) x+(a-b)y=2ab, (a-b)x+(a+b)y=2ab,x+y=0 form an isosceles triangle whose vertical angle is

The solution of dy/dx +1  = e x+y is

If y= log sin x then y2=

If tan x+ tan 4x+tan 7x=tan x tan 4x tan 7x, then x=

A boat is to be manned by 9 crew with 4 on the stroke side, 4 on the row side and one to steer. There are 11 crew of which 2 can stroke only, 1 can row only while 3 can steer only. In how many ways the crew can be arranged for the boat?

If the lengths of the tangent from P(h,k) to the circles x2+y2-4x-5=0 and x2+y2+6x-2y+6=0 are equal then

The equation of the line whose y-intercepts is -3/4 and which is parallel to 5x+3y-7=0 is

A man observes that angle of elevation of the top of a tower from a point P on the ground is θ. He moves a certain distance towards the foot of the tower and finds that the angle of elevation of the top has doubled. He further moves a distance 3/4 of the previous and finds that the angle of elevation is three times that at P. Then angle θ is given by

Let C be the circle with centre (0, 0) and radius 3 units. The equation of the locus of the midpoints of the chords of the circle C that subtend an angle of 2π/3 at its centre is

5 digit numbers can be formed from the digits 0, 2,2,4,5. One number is selected at random.the probability that it is divisible by 5 is

The sum of all possible numbers greater than 2000 formed by using the digits 2,3,4,5 is

The two circles x2+y2+2ax+2by+c=0 and x2+y2+2bx+2ay+c=0 have three real common tangents, then

If sinA + sinB = l, cosA - cosB = m, then the value of cos(A - B) =

If the slope of one line of 8x2+2hxy+by2=0 is double the other, then h2=

A rectangular sheet of dimensions 30 cm * 80 cm four equal squares of side x cm are removed at the corners, and the sides are then turned up so as to form an open rectangle box. The value of x, so that the value of the box is the greatest is

If Z=1n(1+xr+yr) then Zxy=

Consider the circles x2+(y-1)2=9, (x-1)2+y2=25. They are such that

The lines joining the origin to the points of intersection of the line x-y=2 with the curve 5x2+12xy-8y2+8x-4y+12=0 are equally inclined to

The centroid of the triangle formed by the pair of straight lines I2x2 - 20xy + 7y2 = O and the line 2x- 3y + 4 = 0 is:

The two curves x2+y2=25, 2x2-9y+18=0

If |x|

A point is moving on y = 4-2x2. The x-co-ordinate of the point is decreasing at the rate of 5 units per second. Then the rate at which y co-ordinate of the point is changing when the point isat (1, 2) is :

The lines joining the origin to the points of intersection of the line y=6x+8 with the curve 3x2+4xy-4y2-11x+2y+6=0 are equally inclined to

If √(x2+4ax+5)+√(x2+4bx+5)=2(a-b) then x=

If the expression x2-(5m-2)x+(4m2+10m+25)=0 can be expressed as a perfect square,then m=

The complex numbers sin x+ i cos 2x- i sin 2x are conjugate to each other for

A five digit number is formed by using the digits 0, 1,2,3,4 and 5 without repetetion.The probability that the number is divisible by 6 is

cos θ+ cos (θ + 1200)- cos (1200- θ)=

The equation of the circle cutting orthogonally circles x2+y2+2x+8=0, x2+y2-8x+8=0 and which touches the line x-y+4=0 is

The angle between the circles x2+y2-4x-6y-3=0 and x2+y2+8x-4y+11=0 is

The area (in square unit) of the triangle formed by   the   points   with   polar   coordinates (1,0) , (2 , π/3)and (3, 2π/3)

cot2θ(sec θ-1/ (1+sin θ))+ sec2 θ (sin θ-1/ (1+sec θ))=

One focus of an ellipse is (1,0) and (0,0). If the length of major axis is 6 its e=

2 cosh 3 cosh 5=

(2 cos23θ-1) cos 5θ=

If the lines x+ky+3=0 and 2x-5y+7=0 intersects the coordinates axes in concyclic points then k =

Equation of the latusrectum of the parabola x2 + 8x + 12y + 4=0 is

If A+B+C+D= 2π, then -4 cos (A+B/2) sin (A+C/2) cos (A-D/2)=

The diameter x of a circle is found by measurements to be 5 cm with maximum error of 0.05 cm. the relevant error in the area is

The equation of the plane passing through the points (3, -5, -1), (-1, 5, 7) and parallel to the vector (3, -1, 7) is

The vectors (1, -1, 1), (0, 1, 1), (0, 0, 2) are

If (3,2 )is limiting point of the coaxal system of circles whose common radical axis is 4x+2y=11, then the other limiting point is

If α,β are the roots of 3x2+5x-7=0,then the value of (1/3α+5)2+(1/3β+5)2 is

If x is real, then the minimum value of [(x2-x+1)/(x2+x+1)], is

The equations to the direct common tangents to the circles x2+y2+22x-4y-100=0, x2+y2-22x+4y+100=0is

The equation of the circle with centre at (2,3) and touching x-axis is

13+12+1+ 23+22+2+ 33+32+3+…3n terms=

A particle moves along the curve y = x2 + 2x. Then the point on the curve such that x and y co-ordinates of the particle change with the same rate is :

The domain of √1-3x+Cos-1 3x-1/2 is

Statement I : If f:A→B, g:B→C are such that gof is an injection, then f is an injection. Statement II : If f:A→B, g:B→C are such that gof is an injection, then g is an injection. The correct statement is

The value of k. if (1, 2), (k, - 1) are conjugate points with respect to the ellipse 2x2+ 3y2 = 6 is

The function f(x) = tan x has

The odds against A solving a problem are 3 to 2 and the odds in favour of B solving the same problem ae5 to 4.Thenthe probability that the problem will be solved if both of them try the problem is

The sum of divisors of 253453 is

If the area of the triangle with vertices (2a,a), (a,a), (a,2a) is 18sq.unit5, then the circumcentre of the triangle is

If sin 7θ+sin 4θ+ sin θ=0, 0≤θ≤ π/2 then θ=

The circumcentre of the triangle formed  by(2,-5), (2,7), (4,7) is

The equation to hyperbola whose centre is (0,0) distance between the foci is 18 and between the directrices is 8 is

A= cos 200- sin 200, B= cos 1000+ sin 1000, C= cos 5π/6+ sin 2π/3 then the ascending order is

In ΔABC, if cos A cos B +sin A sin B sin C =1, then a:b:c =

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

In a triangle ABC, (a2-b2-c2) tan A+ (a2-b2+c2) tan B is equal to

The vertices of a triangle are (6, 6), (0, 6) and (6, 0) the distance between its circumcentre and centroid is:

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

1+ n/2+n(n-1)/2.4+n(n-1)(n-2)/2.4.6+............…∞

x2+y2=t+(1/t),x4+y4=t2+(1/t2)⇒x3y(dy/dx)=

The equation of the circle having a radius 2 and passing through the limiting points of the coaxal system x2+y2-6-2λ(x+y-4)=0 is

If the centroid of an equilateral triangle is (1,1) and its one vertex is (-1, 2) then the equation of the circum circle is

Observe the following statements :A : Integrating factor of (dy/dx) + y = x2 is exR :Integrating factor of (dy/dx) + P(x) y = Q(x) is e∫p(x)dx. Then the true statement among the following is

3 integers are chosen at random without replacement from the first 20 integers. The probability that the product is odd is

The angle between a pair-of tangents drawn from a point P to the circle x2+y2+4x-6y+9sin2α+13cos2 α=0 is 2α.The  equation of the locus of the point P is