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Physics-

General

Easy

Question

The system of the wedge and the block connected by a massless spring as shown in the figure is released with the spring in its natural length. Friction is absent. maximum elongation in the spring will be

$\frac{3 M g}{5 k}$
$\frac{6 M g}{5 k}$
$\frac{4 M g}{5 k}$
$\frac{8 M g}{5 k}$

The correct answer is: $fraction numerator 6 M g over denominator 5 k end fraction$

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A bead is free to slide down a smooth wire tightly stretched between points A and B on a vertical circle. If the bead starts from rest at A, the highest point on the circle

A bead is free to slide down a smooth wire tightly stretched between points A and B on a vertical circle. If the bead starts from rest at A, the highest point on the circle

physics-General

Balls are thrown vertically upward in such a way that the next ball is thrown when the previous one is at the maximum height. If the maximum height is 5m, the number of balls thrown per minute will be

Balls are thrown vertically upward in such a way that the next ball is thrown when the previous one is at the maximum height. If the maximum height is 5m, the number of balls thrown per minute will be

physics-General

The position vector of a point P is $bold r with rightwards arrow on top equals x ı with not stretchy hat on top plus y ȷ with not stretchy hat on top plus z k with not stretchy hat on top$ where $straight x comma straight y comma straight z element of straight N$ and $times a with not stretchy bar on top equals ı with not stretchy hat on top plus 2 ȷ with not stretchy hat on top plus k with not stretchy hat on top times ∣ f times r with not stretchy rightwards arrow on top times a with not stretchy rightwards arrow on top equals 20$ . If , the number of possible position of P is

The position vector of a point P is $bold r with rightwards arrow on top equals x ı with not stretchy hat on top plus y ȷ with not stretchy hat on top plus z k with not stretchy hat on top$ where $straight x comma straight y comma straight z element of straight N$ and $times a with not stretchy bar on top equals ı with not stretchy hat on top plus 2 ȷ with not stretchy hat on top plus k with not stretchy hat on top times ∣ f times r with not stretchy rightwards arrow on top times a with not stretchy rightwards arrow on top equals 20$ . If , the number of possible position of P is

parallel

$text Let end text f left parenthesis x right parenthesis equals x plus s i n invisible function application x text and end text g left parenthesis x right parenthesis text be the inverse of end text f left parenthesis x right parenthesis text . Then area bounded by end text g left parenthesis x right parenthesis text and end text text the ordinates end text x equals 0 comma x equals pi text is end text$

$text Let end text f left parenthesis x right parenthesis equals x plus s i n invisible function application x text and end text g left parenthesis x right parenthesis text be the inverse of end text f left parenthesis x right parenthesis text . Then area bounded by end text g left parenthesis x right parenthesis text and end text text the ordinates end text x equals 0 comma x equals pi text is end text$

A function f is continuous on the interval $left square bracket 0 comma pi right square bracket. text If end text I subscript 1 end subscript equals stretchy integral subscript 0 end subscript superscript pi end superscript f left parenthesis s i n invisible function application x right parenthesis d x text and end text I subscript 2 end subscript equals stretchy integral subscript 0 end subscript superscript pi end superscript x f left parenthesis s i n invisible function application x right parenthesis d x$ then evaluate ratio $fraction numerator I subscript 2 end subscript over denominator I subscript 1 end subscript end fraction$

A function f is continuous on the interval $left square bracket 0 comma pi right square bracket. text If end text I subscript 1 end subscript equals stretchy integral subscript 0 end subscript superscript pi end superscript f left parenthesis s i n invisible function application x right parenthesis d x text and end text I subscript 2 end subscript equals stretchy integral subscript 0 end subscript superscript pi end superscript x f left parenthesis s i n invisible function application x right parenthesis d x$ then evaluate ratio $fraction numerator I subscript 2 end subscript over denominator I subscript 1 end subscript end fraction$

Suppose A and B are two nonsingular matrices such that $A B equals B A to the power of 2 end exponent text and end text B to the power of 5 end exponent equals I comma text then end text$

Suppose A and B are two nonsingular matrices such that $A B equals B A to the power of 2 end exponent text and end text B to the power of 5 end exponent equals I comma text then end text$

parallel

If A is the area between the curve $y equals x to the power of 6 end exponent left parenthesis 1 minus x right parenthesis to the power of 7 end exponent$ and x - axis then the number of prime factors of A $blank to the power of negative 1 end exponent text is end text$

If A is the area between the curve $y equals x to the power of 6 end exponent left parenthesis 1 minus x right parenthesis to the power of 7 end exponent$ and x - axis then the number of prime factors of A $blank to the power of negative 1 end exponent text is end text$

$stretchy integral subscript 0 end subscript superscript pi end superscript fraction numerator d x over denominator left parenthesis 5 plus 4 c o s invisible function application x right parenthesis to the power of 2 end exponent end fraction equals$

$stretchy integral subscript 0 end subscript superscript pi end superscript fraction numerator d x over denominator left parenthesis 5 plus 4 c o s invisible function application x right parenthesis to the power of 2 end exponent end fraction equals$

The value of the definite Integral $text . end text stretchy integral subscript negative 2008 end subscript superscript 2008 end superscript fraction numerator f to the power of ´ end exponent left parenthesis x right parenthesis plus f to the power of ´ end exponent left parenthesis negative x right parenthesis over denominator left parenthesis 2008 right parenthesis to the power of x end exponent plus 1 end fraction d x text equals end text$

The value of the definite Integral $text . end text stretchy integral subscript negative 2008 end subscript superscript 2008 end superscript fraction numerator f to the power of ´ end exponent left parenthesis x right parenthesis plus f to the power of ´ end exponent left parenthesis negative x right parenthesis over denominator left parenthesis 2008 right parenthesis to the power of x end exponent plus 1 end fraction d x text equals end text$

parallel

The integral $integral s i n invisible function application 2 x open parentheses 1 minus fraction numerator 3 over denominator 2 end fraction c o s invisible function application x close parentheses e to the power of s i n to the power of 2 end exponent invisible function application x plus c o s to the power of 3 end exponent invisible function application x end exponent d x text end text$ is equal to

The integral $integral s i n invisible function application 2 x open parentheses 1 minus fraction numerator 3 over denominator 2 end fraction c o s invisible function application x close parentheses e to the power of s i n to the power of 2 end exponent invisible function application x plus c o s to the power of 3 end exponent invisible function application x end exponent d x text end text$ is equal to

A point P moves in xy – plane in such a way that $left square bracket vertical line x vertical line right square bracket plus left square bracket vertical line y vertical line right square bracket equals 1 comma text where end text left square bracket. right square bracket$ denotes the greatest integer function. Area of the region representing all possible positions of the point P is equal to

A point P moves in xy – plane in such a way that $left square bracket vertical line x vertical line right square bracket plus left square bracket vertical line y vertical line right square bracket equals 1 comma text where end text left square bracket. right square bracket$ denotes the greatest integer function. Area of the region representing all possible positions of the point P is equal to

The value of $stretchy integral subscript 0 end subscript superscript 1 end superscript fraction numerator l o g subscript e end subscript invisible function application left parenthesis x plus 1 right parenthesis over denominator 1 plus x to the power of 2 end exponent end fraction d x text is end text$

The value of $stretchy integral subscript 0 end subscript superscript 1 end superscript fraction numerator l o g subscript e end subscript invisible function application left parenthesis x plus 1 right parenthesis over denominator 1 plus x to the power of 2 end exponent end fraction d x text is end text$

parallel

If function f(x) $equals c o s invisible function application x minus stretchy integral subscript 0 end subscript superscript x end superscript left parenthesis x minus 1 right parenthesis f left parenthesis t right parenthesis d t comma text then end text f to the power of ´ ´ end exponent left parenthesis x right parenthesis plus f left parenthesis x right parenthesis text end text$ is equal to

If function f(x) $equals c o s invisible function application x minus stretchy integral subscript 0 end subscript superscript x end superscript left parenthesis x minus 1 right parenthesis f left parenthesis t right parenthesis d t comma text then end text f to the power of ´ ´ end exponent left parenthesis x right parenthesis plus f left parenthesis x right parenthesis text end text$ is equal to

If $I equals stretchy integral subscript 0 end subscript superscript 12 end superscript fraction numerator d x over denominator square root of 1 minus x to the power of 2 n end exponent end root end fraction comma n element of N comma text end text$ then

If $I equals stretchy integral subscript 0 end subscript superscript 12 end superscript fraction numerator d x over denominator square root of 1 minus x to the power of 2 n end exponent end root end fraction comma n element of N comma text end text$ then

Statement - I : $ϕ left parenthesis x right parenthesis equals stretchy integral subscript 0 end subscript superscript x end superscript left parenthesis 3 s i n invisible function application t plus 4 c o s invisible function application t right parenthesis d t comma open square brackets fraction numerator pi over denominator 6 end fraction comma fraction numerator pi over denominator 3 end fraction close square brackets ϕ left parenthesis x right parenthesis text end text$ attain its maximum value $text at end text x equals fraction numerator pi over denominator 3 end fraction$
Statement - 2: $ϕ left parenthesis x right parenthesis equals stretchy integral subscript 0 end subscript superscript x end superscript left parenthesis 3 s i n invisible function application t plus 4 c o s invisible function application t right parenthesis d t comma ϕ left parenthesis x right parenthesis text end text$ increasing function in $open square brackets fraction numerator pi over denominator 6 end fraction comma fraction numerator pi over denominator 3 end fraction close square brackets text . end text$

Statement - I : $ϕ left parenthesis x right parenthesis equals stretchy integral subscript 0 end subscript superscript x end superscript left parenthesis 3 s i n invisible function application t plus 4 c o s invisible function application t right parenthesis d t comma open square brackets fraction numerator pi over denominator 6 end fraction comma fraction numerator pi over denominator 3 end fraction close square brackets ϕ left parenthesis x right parenthesis text end text$ attain its maximum value $text at end text x equals fraction numerator pi over denominator 3 end fraction$
Statement - 2: $ϕ left parenthesis x right parenthesis equals stretchy integral subscript 0 end subscript superscript x end superscript left parenthesis 3 s i n invisible function application t plus 4 c o s invisible function application t right parenthesis d t comma ϕ left parenthesis x right parenthesis text end text$ increasing function in $open square brackets fraction numerator pi over denominator 6 end fraction comma fraction numerator pi over denominator 3 end fraction close square brackets text . end text$

parallel

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