Let $A comma$ $B comma$ $C$ be three mutually independent events Consider the Two Statement $S subscript 1 end subscript$ and $S subscript 2 end subscript$ $S subscript 1 end subscript$ : $A$ and $8 union C$ are independent $S subscript 2 end subscript colon$ $A$ and $B intersection C$ are independent Then

The equations of tangents to the ellipse $x to the power of 2 end exponent plus 4 y to the power of 2 end exponent equals 4$ , which are inclined at 60° to x-axis, are

The area of the triangle formed by the lines joining the vertex of the parabola $x to the power of 2 end exponent equals 12 y$ to the ends of its latus rectum is

The length of chord of contact of the tangents drawn from the point (2, 5) to the parabola $y to the power of 2 end exponent equals 8 x comma text is end text$

Area bounded by curve $x y equals c comma x$ ‐axis between $x equals 1$ and $x equals 4$ is

For such questions, we should know different formulas of integrals.

Area bounded by curve $x y equals c comma x$ ‐axis between $x equals 1$ and $x equals 4$ is

For such questions, we should know different formulas of integrals.

If $2 not stretchy integral subscript 0 end subscript superscript 1 end superscript t a n to the power of negative 1 end exponent x d x equals not stretchy integral subscript 0 end subscript superscript 1 end superscript c o t to the power of negative 1 end exponent left parenthesis 1 minus x plus x to the power of 2 end exponent right parenthesis d x$ , then $not stretchy integral subscript 0 end subscript superscript 1 end superscript t a n to the power of negative 1 end exponent left parenthesis 1 minus x plus x to the power of 2 end exponent right parenthesis d x$ is equal to

The integral $not stretchy integral subscript 2 end subscript superscript 4 end superscript fraction numerator blank l o g blank x to the power of 2 end exponent over denominator I o g x to the power of 2 end exponent plus blank l o g blank left parenthesis 36 minus 12 x plus x to the power of 2 end exponent right parenthesis end fraction d x$ is equal to

The solution for $x$ of the equation $not stretchy integral subscript square root of 2 end subscript superscript x end superscript fraction numerator d r over denominator r square root of t to the power of 2 end exponent minus 1 end root end fraction equals fraction numerator pi over denominator 2 end fraction$ is

If the differential equation representing the family of all circles touching $x$ ‐axis at the origin is $left parenthesis x to the power of 2 end exponent minus y to the power of 2 end exponent right parenthesis fraction numerator d y over denominator d x end fraction equals g left parenthesis x right parenthesis y comma t h e n g left parenthesis x right parenthesis$ equals

Which one of the following is a differential equation of the family of curves $y equals A e to the power of 2 x end exponent plus B e to the power of negative 2 x end exponent$

A function $y equals f left parenthesis x right parenthesis$ has a second‐order derivatives $f to the power of ´ ´ end exponent left parenthesis x right parenthesis equals 6 left parenthesis x minus$ 1) If its graph passes through the point $left parenthesis 2 comma blank 1 right parenthesis$ and at that point the tangent to the grraph $i s y equals 3 x minus 5$ , then the function is

For such questions, the important part is integration. We should know the methods to integrate the derivate. We should know the properties of a tangent.

A function $y equals f left parenthesis x right parenthesis$ has a second‐order derivatives $f to the power of ´ ´ end exponent left parenthesis x right parenthesis equals 6 left parenthesis x minus$ 1) If its graph passes through the point $left parenthesis 2 comma blank 1 right parenthesis$ and at that point the tangent to the grraph $i s y equals 3 x minus 5$ , then the function is

For such questions, the important part is integration. We should know the methods to integrate the derivate. We should know the properties of a tangent.

The differential equation $y fraction numerator d y over denominator d x end fraction plus x equals o$ ( $0$ is any constant) represents

For such questions, we know different methods to solve differential equations. We should also know the formulas of different shapes.

The differential equation $y fraction numerator d y over denominator d x end fraction plus x equals o$ ( $0$ is any constant) represents

For such questions, we know different methods to solve differential equations. We should also know the formulas of different shapes.

An integrating factor of the differential equation $x fraction numerator d y over denominator d x end fraction plus y to the power of 1 end exponent o g x equals x e to the power of x end exponent x to the power of negative fraction numerator 1 over denominator 2 end fraction 1 o g x end exponent comma$ $left parenthesis x greater than 0 right parenthesis$ is

The solution of the differential equation $fraction numerator d y over denominator d x end fraction plus fraction numerator 3 x to the power of 2 end exponent over denominator 1 plus x to the power of 3 end exponent end fraction y equals fraction numerator s i n to the power of 2 end exponent x over denominator 1 plus x to the power of 3 end exponent end fraction$ is

For such questions, we should know different methods to solve differential equation.

The solution of the differential equation $fraction numerator d y over denominator d x end fraction plus fraction numerator 3 x to the power of 2 end exponent over denominator 1 plus x to the power of 3 end exponent end fraction y equals fraction numerator s i n to the power of 2 end exponent x over denominator 1 plus x to the power of 3 end exponent end fraction$ is

For such questions, we should know different methods to solve differential equation.

The solution of $left parenthesis x minus y to the power of 3 end exponent right parenthesis d x plus 3 x y to the power of 2 end exponent d y equals 0$ is