A rectangle is inscribed in semicircle of ardius r with one of it

Subject

Mathematics

Class

ICSE Class 12

Pre Boards

Practice to excel and get familiar with the paper pattern and the type of questions. Check you answers with answer keys provided.

Sample Papers

Download the PDF Sample Papers Free for off line practice and view the Solutions online.
Advertisement

 Multiple Choice QuestionsShort Answer Type

11.

Using prpoperties of determinants, prove that b+ caaba + cbcca +b = 4abc


12.

Solve the following system of linear equation using matrix method.

3x + y + z = 1, 2x + 2z = 0, 5x + y + 2z = 2.


13.

If sin-1x + tan-1x = π2, prove that 2x2 + 1 = 5


14.

Write the Boolean function corrosponding to the switching circuit given below.


Advertisement
15.

Verify the conditions of Rolle's Theorem for the following function.

f(x) = log(x2 +2) - log (3) on [- 1, 1]

Find a point in the interval, where the tangent to the curve is parallel to x - axis.


16.

Find the equation of the standard ellipse, taking its axes as the coordinate axes, whose minor axis is equal to the distance between the foci and whose legth of latus rectum is 10. Also, find its eccentricity. 


17.

If log(y) = tan-1x, prove that:

(1 + x2)d2ydx2 +(2x - 1)dydx = 0


Advertisement

18.

A rectangle is inscribed in semicircle of ardius r with one of its sides on the diameter of the semicircle. Find the dimensions of the rectangle to get maximum area. Also, find the maximum area.


Let PQRS be the rectangle inscribed in the semi-circle of radius so that OR = r, where O is in centre of circle.

Let PO = OQ = x and QR = y so that sides of rectangle are of lengths 2x and 2y respectively.

Let QOR = θ°

In OQR,

xr = cosθ    x = rcosθ

yr = sinθ     y = rsinθ

Let A be area of rectangle PQRS

 A = PQ × QR

       = 2 OQ × QR

       = r2sin2θ

dAdθ = 2r2cosθ = 0

 2θ = π2 

 θ = π4 is critical point.

d2Adθ2 = - 4r2sin2θ

d2Adθ2 θ = π4 = - 4r2 < 0

 Area is maximum at θ = π4

So, sides of rectangle are 2r, 2r2

And, Area = 2r × 2r2 = rsq. units.

 


Advertisement
Advertisement
19.

Evaluate: sinx + cosx9 + 16sin2xdx


20.

Find the area of the region bound by the curves y = 6x - x2 and y - x2 - 2x.


Advertisement