Mathematics Test - 24

2. Main Concept :

Properties of Determinant P - 1: The value of determinant remains unchanged, if the rows and the columns are interchanged.
Since the determinant remains unchanged when rows and columns are interchanged, it is obvious that any theorem which is true for 'rows' must also be true for 'columns.'
P - 2: If any two rows (or columns) of a determinant be interchanged, the determinant is unaltered in numerical value but is changed in sign only.
P - 3: If a determinant has two rows (or columns) identical, then its value is zero.
P - 4: If all the elements of any two (or column) be multiplied by the same number, then the value of determinant is multiplied by that number.
P - 5: If each element of any row (or column) can be expressed as a sum of two terms, then the determinant can be expressed as the sum of the determinants.

P - 6: The value of a determinant is not altered by adding to the elements of any row (or column) the same multiples of the corresponding elements of any other row (or column)

Then b.
P - 7: If all elements below leading diagonal or above leading diagonal or except leading diagonal elements are zero then the value of the determinant equal to Product of all leading diagonal elements.
P - 8: If a determinant D becomes zero on putting x = a, x = a, then we say that (x - a) is factor of determinant.

P - 9: It should be noted that while applying operations on determinants then at least one row (or column) must remain unchanged or, Maximum number of operations = order of determinant - 1
P - 10: It should be noted that if any row (or column) which is changed by multiplied a non - zero number, then the determinant will be divided by that number.

Given, $y^2=x$...(i)

$\Rightarrow 2y\frac{du}{dx}=1$

$\Rightarrow \frac{dy}{dx}=\frac{1}{2y}(\text{ slope of tangent })$

Also, slope of tangent=tan⁡45°=1

Hence $\frac{1}{2y}=1$$\Rightarrow y=\frac{1}{2}$

From Eq. (i), if $y=\frac{1}{2}$, then $x=\frac{1}{4}$

Main Concept :
Basics of Ellipse(1)Focus : The fixed point is called the focus of the ellipse.
(2) Directrix : The fixed straight line is called the directrix of the ellipse.
(3) Eccentricity : The constant ratio is called the eccentricity of the ellipse and is denoted by e.
(4) Axis : The straight line passing through the focus and perpendicular to the directrix is called the axis of the ellipse.
(5) Vertex : The points of intersection of the ellipse and the axis are called vertices of ellipse
(6) Centre : The point which bisects every chord of the ellipse passing through it, is called the centre of ellipse.
(7) Latus - rectum : The latus - rectum of a ellipse is the chord passing through the focus and perpendicular to the axis.
(8) Double ordinate : The double ordinate of a ellipse is a chord perpendicular to the axis.
(9) Focal chord : A chord passing through the focus of the ellipse is called a focal chord.
(10) Focal distance : The distance of any point on the ellipse from the focus is called the focal distance of the point.