Class 12

Math

Calculus

Application of Derivatives

Find the equation of the normals to the curve $y=x_{3}+2x+6$which are parallel to the line $x+14y+4=0$.

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Determine $p$ such that the length of the such-tangent and sub-normal is equal for the curve $y=e_{px}+px$ at the point $(0,1)˙$

If the tangent at $(1,1)$ on $y_{2}=x(2−x)_{2}$ meets the curve again at $P,$ then find coordinates of $P˙$

A curve is defined parametrically be equations $x=t_{2}andy=t_{3}$ . A variable pair of perpendicular lines through the origin $O$ meet the curve of $PandQ$ . If the locus of the point of intersection of the tangents at $PandQ$ is $ay_{2}=bx−1,$ then the value of $(a+b)$ is____

Draw the graph of $f(x)=∣∣ x_{2}−1x_{2}−2 ∣∣ $ .

A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle. Show that the maximum length of the hypotenuse is $(a_{32}+b_{32})_{23}$ .

Find the value of $a$ if $x_{3}−3x+a=0$ has three distinct real roots.

A running track of 440 ft is to be laid out enclosing a football field, the shape of which is a rectangle with a semi-circle at each end. If the area of the rectangular portion is to be maximum, then find the length of its sides.

Two towns $AandB$ are 60 km apart. A school is to be built to serve 150 students in town $Aand50$ students in town $B˙$ If the total distance to be travelled by 200 students is to be as small as possible, then the school should be built at (a)town B (b)town A (c)45 km from town A (d)45 km from town B