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Statement‐I: The function $f left parenthesis x right parenthesis equals not stretchy integral subscript 0 end subscript superscript x end superscript square root of 1 plus t to the power of 2 end exponent end root d t$ is an odd function and $g left parenthesis x right parenthesis equals f to the power of ’ end exponent left parenthesis x right parenthesis$ is an even function.
Statement‐II: For a differentiable function $f left parenthesis x right parenthesis$ if $f to the power of ’ end exponent left parenthesis x right parenthesis$ is an even function then $f left parenthesis x right parenthesis$ is an odd function.

Statement‐I is true, Statement‐II is true ; Statement‐II is correct explanation for Statement‐I.
Statement‐I is true, Statement‐II is true ; Statement‐II is NOT a correct explanation for statement‐I.
Statement‐I is true, Statement‐II is false.
Statement‐I is false, Statement‐II is true.

The correct answer is: Statement‐I is true, Statement‐II is false.

Related Questions to study

Statement‐I:: $not stretchy integral subscript blank superscript blank 2 to the power of t a n to the power of negative 1 end exponent x end exponent d left parenthesis c o t to the power of negative 1 end exponent x right parenthesis equals fraction numerator 2 to the power of t w to the power of negative 1 end exponent x end exponent over denominator l n 2 end fraction plus c$
Statement‐II : $fraction numerator d over denominator d x end fraction left parenthesis a to the power of x end exponent plus c right parenthesis equals a to the power of x end exponent l n$ a

Statement‐I:: $not stretchy integral subscript blank superscript blank 2 to the power of t a n to the power of negative 1 end exponent x end exponent d left parenthesis c o t to the power of negative 1 end exponent x right parenthesis equals fraction numerator 2 to the power of t w to the power of negative 1 end exponent x end exponent over denominator l n 2 end fraction plus c$
Statement‐II : $fraction numerator d over denominator d x end fraction left parenthesis a to the power of x end exponent plus c right parenthesis equals a to the power of x end exponent l n$ a

The value of $not stretchy integral subscript blank superscript blank blank$ $x. fraction numerator l n left parenthesis x plus square root of 1 plus x to the power of 2 end exponent end root right parenthesis over denominator square root of 1 plus x to the power of 2 end exponent end root end fraction d x$ equals:

The value of $not stretchy integral subscript blank superscript blank blank$ $x. fraction numerator l n left parenthesis x plus square root of 1 plus x to the power of 2 end exponent end root right parenthesis over denominator square root of 1 plus x to the power of 2 end exponent end root end fraction d x$ equals:

$not stretchy integral subscript blank superscript blank fraction numerator d x over denominator blank c o s blank x minus blank s i n blank x end fraction$ is equal to‐

$not stretchy integral subscript blank superscript blank fraction numerator d x over denominator blank c o s blank x minus blank s i n blank x end fraction$ is equal to‐

parallel

$not stretchy integral subscript blank superscript blank fraction numerator x to the power of 4 end exponent minus 4 over denominator x to the power of 2 end exponent square root of 4 plus x to the power of 2 end exponent plus x to the power of 4 end exponent end root end fraction d x$ equals‐

$not stretchy integral subscript blank superscript blank fraction numerator x to the power of 4 end exponent minus 4 over denominator x to the power of 2 end exponent square root of 4 plus x to the power of 2 end exponent plus x to the power of 4 end exponent end root end fraction d x$ equals‐

If $f left parenthesis x right parenthesis equals not stretchy integral subscript blank superscript blank fraction numerator 2 blank s i n blank x minus blank s i n blank 2 x over denominator x to the power of 3 end exponent end fraction d x$ , where $x not equal to 0$ , then Limitx $rightwards arrow 0 f to the power of ´ end exponent left parenthesis x right parenthesis$ has the value

If $f left parenthesis x right parenthesis equals not stretchy integral subscript blank superscript blank fraction numerator 2 blank s i n blank x minus blank s i n blank 2 x over denominator x to the power of 3 end exponent end fraction d x$ , where $x not equal to 0$ , then Limitx $rightwards arrow 0 f to the power of ´ end exponent left parenthesis x right parenthesis$ has the value

$not stretchy integral subscript blank superscript blank fraction numerator left parenthesis 2 x plus 1 right parenthesis over denominator left parenthesis x to the power of 2 end exponent plus 4 x plus 1 right parenthesis to the power of 3 divided by 2 end exponent end fraction d x$

$not stretchy integral subscript blank superscript blank fraction numerator left parenthesis 2 x plus 1 right parenthesis over denominator left parenthesis x to the power of 2 end exponent plus 4 x plus 1 right parenthesis to the power of 3 divided by 2 end exponent end fraction d x$

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The equation of the directrix of the parabola $y to the power of 2 end exponent plus 4 y plus 4 x plus 2 equals 0 comma$ $i s colon$

The equation of the directrix of the parabola $y to the power of 2 end exponent plus 4 y plus 4 x plus 2 equals 0 comma$ $i s colon$

The equation of the common tangent touching the circle $left parenthesis x minus 3 right parenthesis to the power of 2 end exponent plus y to the power of 2 end exponent equals 9$ and the parabola $y to the power of 2 end exponent equals 4 x$ above the $x$ ‐axis is:

The equation of the common tangent touching the circle $left parenthesis x minus 3 right parenthesis to the power of 2 end exponent plus y to the power of 2 end exponent equals 9$ and the parabola $y to the power of 2 end exponent equals 4 x$ above the $x$ ‐axis is:

If the line x‐1 =0 is the directrix of the parabola $y to the power of 2 end exponent minus k x plus 8 equals 0$ , then one of the values of $k$ is :

If the line x‐1 =0 is the directrix of the parabola $y to the power of 2 end exponent minus k x plus 8 equals 0$ , then one of the values of $k$ is :

parallel

Assertion (A): Three normals are drawn from the point $P$ ’ with slopes $m subscript 1 end subscript comma m subscript 2 end subscript comma m subscript 3 end subscript$ to the parabola $y to the power of 2 end exponent equals 4 x$ If locus of ‘ $P$ ’ with $m subscript 1 end subscript m subscript 2 end subscript equals alpha$ is a part of the parabola itself then $alpha equals 2$
Reason (R): If normals at $left parenthesis x subscript 1 end subscript comma y subscript 1 end subscript right parenthesis comma left parenthesis x subscript 2 end subscript comma y subscript 2 end subscript right parenthesis$ and $left parenthesis y subscript 3 end subscript comma y subscript 3 end subscript right parenthesis$ are concurrent then $y subscript 1 end subscript plus y subscript 2 end subscript plus y subscript 3 end subscript equals 0$

Assertion (A): Three normals are drawn from the point $P$ ’ with slopes $m subscript 1 end subscript comma m subscript 2 end subscript comma m subscript 3 end subscript$ to the parabola $y to the power of 2 end exponent equals 4 x$ If locus of ‘ $P$ ’ with $m subscript 1 end subscript m subscript 2 end subscript equals alpha$ is a part of the parabola itself then $alpha equals 2$
Reason (R): If normals at $left parenthesis x subscript 1 end subscript comma y subscript 1 end subscript right parenthesis comma left parenthesis x subscript 2 end subscript comma y subscript 2 end subscript right parenthesis$ and $left parenthesis y subscript 3 end subscript comma y subscript 3 end subscript right parenthesis$ are concurrent then $y subscript 1 end subscript plus y subscript 2 end subscript plus y subscript 3 end subscript equals 0$

ABCD and EFGC are squares and the curve $y equals k square root of x$ passes through the origin $D$ and the points $B$ and F The ratio $fraction numerator F G over denominator B C end fraction$ is:

ABCD and EFGC are squares and the curve $y equals k square root of x$ passes through the origin $D$ and the points $B$ and F The ratio $fraction numerator F G over denominator B C end fraction$ is:

Statement‐I :: With respect to a hyperbola $fraction numerator x to the power of 2 end exponent over denominator 9 end fraction minus fraction numerator y to the power of 2 end exponent over denominator 16 end fraction equals 1$ pependicular are drawn from a point (5, 0) on the lines $3 y plus-or-minus 4 x equals 0$ , then their feet lie on circle $x to the power of 2 end exponent plus y to the power of 2 end exponent equals 16.$
Statement‐II :: If from any foci of a hyperbola perpendicular are drawn on the asymptotes of the hyperbola then their feet lie on auxiliary circle.

Statement‐I :: With respect to a hyperbola $fraction numerator x to the power of 2 end exponent over denominator 9 end fraction minus fraction numerator y to the power of 2 end exponent over denominator 16 end fraction equals 1$ pependicular are drawn from a point (5, 0) on the lines $3 y plus-or-minus 4 x equals 0$ , then their feet lie on circle $x to the power of 2 end exponent plus y to the power of 2 end exponent equals 16.$
Statement‐II :: If from any foci of a hyperbola perpendicular are drawn on the asymptotes of the hyperbola then their feet lie on auxiliary circle.

parallel

A hyperbola, having the transverse axis of length $2 s i n$ , is confocal with the ellipse $3 x to the power of 2 end exponent plus 4 y to the power of 2 end exponent equals 12$ Then its equation is ‐

A hyperbola, having the transverse axis of length $2 s i n$ , is confocal with the ellipse $3 x to the power of 2 end exponent plus 4 y to the power of 2 end exponent equals 12$ Then its equation is ‐

The latus rectum of the hyperbola $16 x to the power of 2 end exponent minus 9 y to the power of 2 end exponent equals 144$ is‐

The latus rectum of the hyperbola $16 x to the power of 2 end exponent minus 9 y to the power of 2 end exponent equals 144$ is‐

Statement‐I :: If a point $open parentheses x subscript 1 end subscript comma blank y subscript 1 end subscript close parentheses$ lies in the shaded region $fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction minus fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction equals 1$ , show in the figure, then $fraction numerator x subscript 1 end subscript superscript 2 end superscript over denominator a to the power of 2 end exponent end fraction minus fraction numerator y subscript 1 end subscript superscript 2 end superscript over denominator b to the power of 2 end exponent end fraction less than 0$
Statement‐II :: $P left parenthesis x subscript 1 end subscript comma blank y subscript 1 end subscript right parenthesis$ lies outside the hyperbola $fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction minus fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction equals 1$ , then $fraction numerator x subscript 1 end subscript superscript 2 end superscript over denominator a to the power of 2 end exponent end fraction minus fraction numerator y subscript 1 end subscript superscript 2 end superscript over denominator b to the power of 2 end exponent end fraction less than 1.$

Statement‐I :: If a point $open parentheses x subscript 1 end subscript comma blank y subscript 1 end subscript close parentheses$ lies in the shaded region $fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction minus fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction equals 1$ , show in the figure, then $fraction numerator x subscript 1 end subscript superscript 2 end superscript over denominator a to the power of 2 end exponent end fraction minus fraction numerator y subscript 1 end subscript superscript 2 end superscript over denominator b to the power of 2 end exponent end fraction less than 0$
Statement‐II :: $P left parenthesis x subscript 1 end subscript comma blank y subscript 1 end subscript right parenthesis$ lies outside the hyperbola $fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction minus fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction equals 1$ , then $fraction numerator x subscript 1 end subscript superscript 2 end superscript over denominator a to the power of 2 end exponent end fraction minus fraction numerator y subscript 1 end subscript superscript 2 end superscript over denominator b to the power of 2 end exponent end fraction less than 1.$

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