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The number of tangents to the curve `x^(3/2)+ y^(3/2) = 2a^(3/2) ,a > 0` , which are equally inclined to the axes, is

If m is the slope of a tangent to the curve `e^y =1+ x^2`, then

The slope of the tangent to the curve `y =sqrt(4-x^2)` at the point where the ordinate and the abscissa are equal is

Let `f(x)= {(-x^2, text(for) x<0),(x^2+8 ,text(for) x>=0):}` . Then x-intercept of the line, that is, the tangent to the graph of f(x) is

If x+4y=14 is a normal to the curve `y^2=alphax^3-beta` at (2,3), thenthe value of `alpha+beta` is

In the curve represented parametrically by the equations x=2 ln cot t +1 and y = tan t + cot t,

Let C be the curve `y=x^3`(where x talks all real values). The tangent at A meets the curve again at B. If the gradient at B is K times the gradient at A, then K is equal to

A curve is represented by the equations `x =sec^2 t` and `y=cot t,` where t is a parameter. If the tangent at the point P on the curve where `t=pi/4 `meets the curve again at the point Q, then |PQl is equal to

If the length of sub-normal is equal to the length of subtangent at any point (3, 4) on the curve `y=f(x)` and the tangent at (3, 4) to `y = f(x)` meets the coordinate axes at A and B, then the maximum area of the triangle OAB, where O is origin, is '

Tangent of acute angle between the curve `y=|x^2-1|` and `y=sqrt(7-x^2)` at their points of intersection is

The lines tangent to the curves `y^3-x^2y+5y-2=0` and `x^4-x^3y^2+5x+2y=0` at the origin intersect at an angle `theta` equal to

The two curves `x = y^2 , xy = a^3` cut orthogonally at a point. Then `a^2` is equal to

A point on the parabola `y^2 =18x` at which the ordinate increase at twice the rate of the abscissa is

A lamp of negligible height is placed on the ground `l_1` ,away from a wall. A man `l_2` m tall is walking at a speed of `l_1/10` m/s , from the lamp to the nearest point on the wall. When he is midway between the lamp and the wall, the rate of change in the lengh of this shadow on the wall is

A man is moving away from a tower 41.6 m high at a rate of 2 m/s. If the eye level of the man is 1.6 m above the ground, then the rate at which the angle of elevation of the top of the tower changes, when he is at a distance of 30 m from the foot of the tower, is

The coordinates of the point n the parabola `y^2 = 8x` whose distance from the circle `x^2 +(y+6)^2 =1` is minimum is

At the point `P(a, a^n)` on the graph of `y = x^n, (n in N)`, in the first quadrant, a normal is drawn.
The normal intersects the y-axis at the point (0, b). If `lim_(a->0)b=1/2` then n equals

The tangent to the curve `y=e^(kx)` at a point (0, 1) meets the x-axis at (a,0), where a `in` [-2,-1]. Then `k in .`

Given g(x)=`(x+2)/(x+1)` and the line `3x+y-10=0`. Then the line is

The radius of the base of a cone is increasing at the rate of 3 cm/min and the altitude is decreasing at the rate of 4 cm/min. The rate of change of lateral surface when the radius is 7 cm and altitude is 24 cm is

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