Chemistry-
General
Easy
Question
Graph between log
and log P is a straight line at angle 0 45 with intercept OA as shown Hence ,
at a pressure of 2 atm is
![](data:image/png;base64,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)
- 2
- 4
- 8
- 1
The correct answer is: 4
Related Questions to study
Maths-
If
and
then
If
and
then
Maths-General
Maths-
Assertion : For
the volume of the parallel piped formed by vectors
and
is maximum (The vectors form a right-handed system)
Reason: The volume of the parallel piped having three coterminous edges
and ![stack c with ‾ on top](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAAZCAYAAADjRwSLAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAY00gKyAAAAGxJREFUeNpjYICA/zjwKBgFOAAPEHcB8VMg/gXE64BYEF3BOSBuBWI+IGYD4hIg9kFW1AHEk/BZwwTEr9GNRgemQHyYkIO9oI7EC4yB+AYx3r8A9RkL1HcFQGyNrkgWiA8C8R8gvgTEySSFMAC4kxWWWazxUwAAAHd0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW92ZXIgYWNjZW50PSJ0cnVlIj48bWk+YzwvbWk+PG1pPiYjeDIwM0U7PC9taT48L21vdmVyPjwvbWF0aD6siPvzAAAAAElFTkSuQmCC)
For such questions, we should know the formula of a parallelepiped. We should also know how to take a scalar product.
Assertion : For
the volume of the parallel piped formed by vectors
and
is maximum (The vectors form a right-handed system)
Reason: The volume of the parallel piped having three coterminous edges
and ![stack c with ‾ on top](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAAZCAYAAADjRwSLAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAY00gKyAAAAGxJREFUeNpjYICA/zjwKBgFOAAPEHcB8VMg/gXE64BYEF3BOSBuBWI+IGYD4hIg9kFW1AHEk/BZwwTEr9GNRgemQHyYkIO9oI7EC4yB+AYx3r8A9RkL1HcFQGyNrkgWiA8C8R8gvgTEySSFMAC4kxWWWazxUwAAAHd0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW92ZXIgYWNjZW50PSJ0cnVlIj48bWk+YzwvbWk+PG1pPiYjeDIwM0U7PC9taT48L21vdmVyPjwvbWF0aD6siPvzAAAAAElFTkSuQmCC)
Maths-General
For such questions, we should know the formula of a parallelepiped. We should also know how to take a scalar product.
Maths-
Assertion (A): Let
. If
such that
is collinear with
and
is perpendicular to
is possible, then
.
Reason (R): If
and
are non-zero, non-collinear vectors, then
can be expressed as
where
is collinear with
and
is perpendicular to ![stack a with rightwards arrow on top](data:image/png;base64,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)
Assertion (A): Let
. If
such that
is collinear with
and
is perpendicular to
is possible, then
.
Reason (R): If
and
are non-zero, non-collinear vectors, then
can be expressed as
where
is collinear with
and
is perpendicular to ![stack a with rightwards arrow on top](data:image/png;base64,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)
Maths-General
Maths-
Statement
If
are distinct non-negative numbers and the vectors â
and
are coplanar then
is arithmetic mean of
and
.
Statement -2: Parallel vectors have proportional direction ratios.
Statement
If
are distinct non-negative numbers and the vectors â
and
are coplanar then
is arithmetic mean of
and
.
Statement -2: Parallel vectors have proportional direction ratios.
Maths-General
Maths-
If
and
is a vector satisfying
.
Assertion
can be expressed in terms of
and
.
Reason ![left parenthesis R right parenthesis colon r with ‾ on top equals 1 over 7 left parenthesis 7 i plus 5 j minus 9 k plus a with ‾ on top cross times b with ‾ on top right parenthesis](data:image/png;base64,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)
If
and
is a vector satisfying
.
Assertion
can be expressed in terms of
and
.
Reason ![left parenthesis R right parenthesis colon r with ‾ on top equals 1 over 7 left parenthesis 7 i plus 5 j minus 9 k plus a with ‾ on top cross times b with ‾ on top right parenthesis](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAN0AAAAoCAYAAAB6m/+wAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAc1CXO0QAABOxJREFUeNrtm1Fk1VEcx/+umT1k9DAzPVySJJORJEliZpKZuNJDD4lJJpnoYWaSMTM99JZJkhkzkySRmSSJZKaHRHpKJnpIcl2X9Tv6Xf079/zO//zvPed//7t9P/zY/d//Ped/fud8z++c3/8sigAAjbItGAAAAAAAAFgDAwDagv1kU2QbcAUmdZAND8nG0BEAtGZGBABAdABAdAAAiA4AiA4AANEBANEBIHKFbDblb2b5d+0sukb8AkCiPg6RvW6w4Ff8+3YUXTN+AcCqD/XFgPCjMg/UL2RvyJ6S9cS+P0z2coeILe0RId0v2wlm4hdZIXA7ktrj+xkUJ7nfy1z+CtneDOvPAy7tMupjgAeXiX1R/VnFGbI57dpLLrydsPlFp0S2IPhvs8XL4xDPMET2nuwID7oOsktkH8h6M6g/D6RpV50+lIDGhZtHyRa1a+fI7mvXrrbhvsfmlzi97NSujMSUh0TQWyGqlQwTshpDSxlOMlmRpl11+nhGdkK4eZrsunbtJtk17doxsjVHZ6kyb5Bd5s4r8+e8YfNLnMeWpfl0iraFEt10AP9Whesq6r2z1K8i4gOy04FEV+DxuUVW4VVa0XL/JFva72rtmuT6vvI4XifrM9xbp48fZJ1CwctkI7HP3cISopPLcXHWMu8P5y17AJf9U+gZ0OaXGmrimLJ8v8wzYitFl+YZXPkkbCeKvM8x1d/Ff58KGOmekN0h280CdGm7SVxJgqu16xNPKLX6bgvRr04fVUvBH1m5aoYa5lmsJNxbcXSMKvPsDlheVhO+L3JiqSOhrXsCiK7CnaiSWhNkuzw9gytqDHzmZEqB7QyPj4qh/n5eERwNuLw8z0KI4xpV4yJzEVytXQe1a92G4GPUh22pUNUii2255SK6gmEmDMF2g5ZGdC8SZu2ktjYbyTs42kzx6uOAcE+oZzjJy6ky2zInF34ZfPCaheer70ys8zJO76OiY71KaM8dBSf1bZer6KRl1FFuSHxdOicU2GmpzFZmFHBQhlxeljjK+GprswmDIR5gpr3EepQdXRz9dR+oKHQvcCJFT9+rv3+mqCeN6KS+NeU2jPpQFR0XwrWepXwuzBxSZS5l5hXJLwWOLAOe29rsJFLOgb9V5L8l1D/Pe+BQovumfT5uSOr4Wl5Kfp0Q9vh1+pBS42pDOmb48QPDveOWKKiXeWmHiE7yy6gQVZptazOi6+dNvUsfhmRG2z/q9UsTmQ+/fIn+fW2jBv9KSsG5JlJUuy4aotlmZM5e1ulDegm8KmxC1wwbSNPL8e0UZeYRyS9LBodHHtrqOrhWefKrJTCGWXCm5NSjQP5e4/pqy+89PMhHEnywm5effQH8oiLsXfaJWo0tOmwBGn1lsMqJlMFY+1ct48J4eERtdPXzYd8j8wvfY1pjpGNgJmdJZeYVk1+2ePAkEaqtJe7wKtehEhhHMn6GQd7TVHmvshSZzxea6u9n0YZ4rkn2xxSXvxX5f11SGwMjsX7YsNQjHpNs5wPPzSRjdvKB514eFP8zPTl4Bqs+1CY37VGuuYz3DL4jxoLDfY34pdUUOHExEYFW5wXG4Ia/UcDHecm8cSH6k8X8GeF/AEHOsJ2XBAB4Jum8JADAIy7nJQEAHkk6LwkA8IjLeUkAgCdcz0sCADzhel4SAOAJ1/OSAABPuJ6XBCAovwHm96FWkK2boQAAAbl0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+KDwvbW8+PG1pPlI8L21pPjxtbz4pPC9tbz48bW8+OjwvbW8+PG1vdmVyIGFjY2VudD0idHJ1ZSI+PG1pPnI8L21pPjxtaT4mI3gyMDNFOzwvbWk+PC9tb3Zlcj48bW8+PTwvbW8+PG1mcmFjPjxtbj4xPC9tbj48bW4+NzwvbW4+PC9tZnJhYz48bW8+KDwvbW8+PG1uPjc8L21uPjxtaT5pPC9taT48bW8+KzwvbW8+PG1uPjU8L21uPjxtaT5qPC9taT48bW8+LTwvbW8+PG1uPjk8L21uPjxtaT5rPC9taT48bW8+KzwvbW8+PG1vdmVyIGFjY2VudD0idHJ1ZSI+PG1pPmE8L21pPjxtaT4mI3gyMDNFOzwvbWk+PC9tb3Zlcj48bW8+JiN4RDc7PC9tbz48bW92ZXIgYWNjZW50PSJ0cnVlIj48bWk+YjwvbWk+PG1pPiYjeDIwM0U7PC9taT48L21vdmVyPjxtbz4pPC9tbz48L21hdGg+0mMygAAAAABJRU5ErkJggg==)
Maths-General
Maths-
If
are non-zero vectors such that
then
Assertion
: Least value of
is ![2 square root of 2 minus 1](data:image/png;base64,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)
Reason (R): The expression
is least when magnitude of
is ![square root of 2 t a n invisible function application open parentheses fraction numerator pi over denominator 8 end fraction close parentheses end root](data:image/png;base64,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)
If
are non-zero vectors such that
then
Assertion
: Least value of
is ![2 square root of 2 minus 1](data:image/png;base64,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)
Reason (R): The expression
is least when magnitude of
is ![square root of 2 t a n invisible function application open parentheses fraction numerator pi over denominator 8 end fraction close parentheses end root](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFcAAAArCAYAAAANKBTWAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAaPUZr5AAABIpJREFUeNrtW1FIFUEUHeQh8XgfhYiKiRAiISJChUmJChIifogYZiEUhURU9CFkiJhI0EdQmBgRJiIimWhQRBAhEiH9WAmVFNFHRMgDjRIVC+xePA+2YfftzOiqA+/Agfd2d2b2HWfvnHt3FEIdEeJqglpURj7xq0ggEFQTxy2753PE7oD67kL/G4IW4k2LhC0hviWGAuqf+50mlm5EZ3eIZy0RNowQVhjwOLFQGVlvRy+JZZaI20ns2aSxujDeujBPTLdA2FTiT2LmJo2XQfyFcY2wm/jDklnbShzQXPQWfSzVZZ8+eoltpjd8mDhpibgfsJipop+YRRwh1jiOj8AhqeAQccb0hs8T+ywQNpcYNWz7BSLHwAtVmkb7WeJek4HZgjVbIO4p4qChrVqUvi9o9jGI8bUxRqyzQNx+Q7tYLCVI/P2FZh+nMb42PptO+U3GE2KVQbsGKezVGwhVSXxqctN/iDssEDdqaIl6pEe60yBtjpjE+3zMXK9VcgQ+bwXp5oktFJfjZpLhYubM5q4RnxNzkO2pIEmK20pge/LY49wEHqlY+pdHfIVjMnJwLkisGrTh3zclHTuACTOg2deK7uDsEm5rXJ+NgoaMWsOVPGhxt3R8DvRnNNssS9/b42Q8/Dh1wCfGQku2S/tm+M4+PH5zxIu2i/uGuF/T1rg9/sNSBuRc4bn4sQtCu103DCFZ2FJcl4k/YthmcX8Tdypey47iNRY6GTMuM7IBwsleVbZTn4hNLn3y7E2xVVyugs0rXssz7xHxiMZKOo6ZLi+S2VLbBYWsSuXHBf1eTEvcMrFWx/XDHgib43G+CKL5WSc3IYuE++ulYpfjVs3ckwoFG87c7vn4wQaPjCfq4punfLKneMe3rbj5HgWbS3E6S0PM9HtPdUusVdZkfJcyvzYkJU50IW8XCse3pbgVOCEvOJw8VPvk8io1B04le12Oc5p5F+EgGz5Yzs9HPeoFbse3nbgVWLRWXWbpRx/xVAN/HmYpH0+WzrVi9rdhFs9KVmzOo67hdnzbiduC2Tkjpbn8qC+J4F5NeyF1HW3/GrYLI8wswDs/E2bv4DzT3/vi/x01B5FA2ATTwk0vijUhPFnX4dd1EIpXuDkuKX+M+NAycb/Db+ti2cWX6xZhUkSckmMIMSOWjV0FbQIvsJUG7TgcRCRxv2n2USV8iuWc6l7A5yHMXpvAvtfkPVaXVATiBOWGZh9NfjnBe0euP4m4axMahX4NllGAMLCCTJL3aJRo9jHk94d1LmpLwo5XO07kwsrpIOavMxw2cZ9YeztRoNFP1M/zN0LUdGHPDhsZ0y7FoHgY9RCxXKhvmS0XCptCYrvH64U9O2xkNHtkg6pOQfWcExyKrqh6xQeIITaCLZHORjze/uRWzctTfHqz4DZSVAZ7h9nbLOxFu1B/NV4LgcthwZLwmWOuygYTrgh2qN7YMMStsVjcsOaCdFSslTpXEAomFH9/odDc/FwHcQuF3eDacJDb9pOxeGpZtjSIGxH2g419UDvMu4XhvzGMiQQCQ11CguAQSkiwfvwDLz5wlOKho/8AAADSdEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1zcXJ0Pjxtbj4yPC9tbj48bWk+dDwvbWk+PG1pPmE8L21pPjxtaT5uPC9taT48bW8+JiN4MjA2MTs8L21vPjxtZmVuY2VkIHNlcGFyYXRvcnM9InwiPjxtZnJhYz48bWk+JiN4M0MwOzwvbWk+PG1uPjg8L21uPjwvbWZyYWM+PC9tZmVuY2VkPjwvbXNxcnQ+PC9tYXRoPkTivuYAAAAASUVORK5CYII=)
Maths-General
Maths-
Statement- 1: If
and
and
, then there exist real numbers
,
such that
![alpha stack b with rightwards arrow on top plus beta stack c with rightwards arrow on top plus gamma d](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGUAAAAWCAYAAADZylKgAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAATRJrTQAAAAtpJREFUeNrtWE9kHFEcHisiqkLEqorIpSpyKhUR1UOoqlW1Qg49RESo6KmirKqqqFJVVatCRNWqKlGjh4peImJVbhGrqkL1UBFVonqoqmX6Pb6p1583k9nNvNlt+j4+OzszO+/37/2+36znpYun4FHPoa2wBG6DIy4U2eMqGMSwDp50YWqfnfIRHHahaB+0u6Y4zXOa5+A0j/gB5lqw7iz4DvwJVsH+Q6J5B47nCbDWAsPvgI+oDcqBB+DzQ6ApqcSzCL5IwZigQcM3xbmJBpJi2z4r8ewFK2wLFVFVFQYgxG3wJjgP7vI3a+Bxi07fIHVU9xFu5cN9cAf8BfpgjwX7lEZtRVzLcxcMaTaVwW+M20vwMViSP+wE34IFHq/xRoUBQ4Uusz+X6KRqJQ+b2D2NJGWZFaUwCK5QX+ISouy+C3bTr+vgRUv2LYKT4lwHC2eM33OM7TSPu1nggebbH9wSVXiKQffYHs6K+7e1zIdQC3y3mBS1Zh8/AwZ8KOb+e6zIrNrXoEEXFsA5EedSjG9/4bNBBNfB83ywjhwnBYmuBEkJEtAEuWaeVb/HYJju/9pgqzqIfSFUtznD43HwVYI4G+N5mltMYgbcMDxkhFtQYhRctVSJUWvOR+yG4QifbAv9LKfDPDWmV3Sf9aS+TURoQYWZl7jM8VJijtvThtNRa6oe/sRwvkBRzzopx9j2VewuiGtq5zxL6tu4wYElilGVo6iOMq/JQaFmcfoqR4j6Klusafd/aNFIXIsolCKHFQmVqCueQaA/cazrYdYWeW3FUP0+hekcv/fx3LRFp306O6q9s/gRuyfEFievDvp4Tev3NpOywPUkjoDv2cbCwnnNcb1getAA+9qmeB+ZMkwUX8BLTEydzhctv1ztcY0dzvYbtC0O/ezhdfowk9GL4Jt92puv+TBG37r+tf/YOpsYtVtp6673HyAt0Xa2poipFF4Cs8Kk9k9IU/gNCI7PnKz60+0AAAD9dEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1pPiYjeDNCMTs8L21pPjxtb3ZlciBhY2NlbnQ9InRydWUiPjxtaT5iPC9taT48bW8+JiN4MjE5Mjs8L21vPjwvbW92ZXI+PG1vPis8L21vPjxtaT4mI3gzQjI7PC9taT48bW92ZXIgYWNjZW50PSJ0cnVlIj48bWk+YzwvbWk+PG1vPiYjeDIxOTI7PC9tbz48L21vdmVyPjxtbz4rPC9tbz48bWk+JiN4M0IzOzwvbWk+PG1pPmQ8L21pPjwvbWF0aD7aEMwVAAAAAElFTkSuQmCC)
Statement- 2:
are four vectors in a 3 - dimensional space. If
are non-coplanar, then there exist real numbers
such that ![stack a with rightwards arrow on top equals alpha stack b with rightwards arrow on top plus beta stack c with rightwards arrow on top plus gamma stack d with rightwards arrow on top](data:image/png;base64,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)
Statement- 1: If
and
and
, then there exist real numbers
,
such that
![alpha stack b with rightwards arrow on top plus beta stack c with rightwards arrow on top plus gamma d](data:image/png;base64,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)
Statement- 2:
are four vectors in a 3 - dimensional space. If
are non-coplanar, then there exist real numbers
such that ![stack a with rightwards arrow on top equals alpha stack b with rightwards arrow on top plus beta stack c with rightwards arrow on top plus gamma stack d with rightwards arrow on top](data:image/png;base64,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)
Maths-General
Maths-
Statement-
:If
are non-collinear points. Then every point
in the plane of
, can be expressed in the form ![open parentheses fraction numerator k x subscript 1 end subscript plus l x subscript 2 end subscript plus m x subscript 3 end subscript over denominator k plus l plus m end fraction comma fraction numerator k y subscript 1 end subscript plus l y subscript 2 end subscript plus m y subscript 3 end subscript over denominator k plus l plus m end fraction close parentheses](data:image/png;base64,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)
Statement-
:The condition for coplanarity of four points
is that there exists scalars
not all zeros such that
where
.
Statement-
:If
are non-collinear points. Then every point
in the plane of
, can be expressed in the form ![open parentheses fraction numerator k x subscript 1 end subscript plus l x subscript 2 end subscript plus m x subscript 3 end subscript over denominator k plus l plus m end fraction comma fraction numerator k y subscript 1 end subscript plus l y subscript 2 end subscript plus m y subscript 3 end subscript over denominator k plus l plus m end fraction close parentheses](data:image/png;base64,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)
Statement-
:The condition for coplanarity of four points
is that there exists scalars
not all zeros such that
where
.
Maths-General
Maths-
Assertion (A): The number of vectors of unit length and perpendicular to both the vectors.
and
is zero Reason
(R):
and
are two non-zero and non-parallel vectors it is true that
is perpendicular to the plane containing
and ![stack b with ‾ on top](data:image/png;base64,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)
Assertion (A): The number of vectors of unit length and perpendicular to both the vectors.
and
is zero Reason
(R):
and
are two non-zero and non-parallel vectors it is true that
is perpendicular to the plane containing
and ![stack b with ‾ on top](data:image/png;base64,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)
Maths-General
Maths-
The value of
for which the straight lines
and
are coplanar is
For such questions, we should know the condition for two lines to be coplanar. We should also know about the scalar triple product.
The value of
for which the straight lines
and
are coplanar is
Maths-General
For such questions, we should know the condition for two lines to be coplanar. We should also know about the scalar triple product.
Maths-
The position vector of the centre of the circle ![vertical line r with rightwards arrow on top vertical line equals 5 comma r with rightwards arrow on top times left parenthesis i with ˆ on top plus j with ˆ on top plus k with ˆ on top right parenthesis equals 3 square root of 3](data:image/png;base64,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)
The position vector of the centre of the circle ![vertical line r with rightwards arrow on top vertical line equals 5 comma r with rightwards arrow on top times left parenthesis i with ˆ on top plus j with ˆ on top plus k with ˆ on top right parenthesis equals 3 square root of 3](data:image/png;base64,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)
Maths-General
Maths-
Let
where
are three noncoplanar vectors. If
is perpendicular to
, then minimum value of ![x squared plus y squared](data:image/png;base64,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)
Let
where
are three noncoplanar vectors. If
is perpendicular to
, then minimum value of ![x squared plus y squared](data:image/png;base64,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)
Maths-General
Chemistry-
Which of the following represents Westrosol?
Which of the following represents Westrosol?
Chemistry-General
Chemistry-
Which of the following sequences would yield -nitro chlorobenzene (Z) from benzene?
Which of the following sequences would yield -nitro chlorobenzene (Z) from benzene?
Chemistry-General
Chemistry-
For the preparation of chloroethane
For the preparation of chloroethane
Chemistry-General