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Maths-

General

Easy

Question

The curve $y equals a x to the power of 3 end exponent plus b x to the power of 2 end exponent plus c x plus 5$ touches the $x$ ‐axis at P(-2,0) and cuts the y ‐axis at a point $Q left parenthesis O comma 5 right parenthesis$ where its gradient is 3 Then a+2b+c=-

$- 1 / 2$
$- 3 / 4$
$- 3$
1

The correct answer is: 1

Since the curve $y equals a x to the power of 3 end exponent plus b x to the power of 2 end exponent plus c x plus 5$ touches x‐ axis at $P left parenthesis negative 20 right parenthesis$ then x‐ axis is the tangent at $left parenthesis negative 20 right parenthesis$ The curve meets $y$ ‐axis in $left parenthesis 05 right parenthesis$ We have
$fraction numerator d y over denominator d x end fraction equals 3 a x to the power of 2 end exponent plus 2 b x plus c$

and $left parenthesis negative 20 right parenthesis$ lies on the curve then
$0 equals negative 8 a plus 4 b minus 2 c plus 5 blank equals 0 equals negative 8 a plus 4 b minus 1 blank c equals 3 right parenthesis$
$equals 8 a minus 4 b plus 1 equals 0$
$midline horizontal ellipsis midline horizontal ellipsis$ (3)
From (2) and (3) we get $a equals negative fraction numerator 1 over denominator 2 end fraction comma b equals negative fraction numerator 3 over denominator 4 end fraction$
Hence $a equals negative fraction numerator 1 over denominator 2 end fraction comma b equals negative fraction numerator 3 over denominator 4 end fraction$ $a n d c equals 3.$

Related Questions to study

If $open vertical bar table row alpha beta row gamma cell negative alpha end cell end table close vertical bar$ is to be square root of a determinant of the two rowed unit matrix, then $alpha comma beta comma$ $gamma$ should satisfy the relation

If $open vertical bar table row alpha beta row gamma cell negative alpha end cell end table close vertical bar$ is to be square root of a determinant of the two rowed unit matrix, then $alpha comma beta comma$ $gamma$ should satisfy the relation

Matrix Ais given by $A equals open curly brackets table attributes columnalign left end attributes row 3 11 row 2 8 end table close curly brackets$ then the determinant of $A to the power of 2011 end exponent minus 5 A to the power of 2010 end exponent$ is

Matrix Ais given by $A equals open curly brackets table attributes columnalign left end attributes row 3 11 row 2 8 end table close curly brackets$ then the determinant of $A to the power of 2011 end exponent minus 5 A to the power of 2010 end exponent$ is

If $A open parentheses table row 1 3 4 row 3 cell negative 1 end cell 5 row cell negative 2 end cell 4 cell negative 3 end cell end table close parentheses equals open parentheses table row 3 cell negative 1 end cell 5 row 1 3 4 row cell plus 4 end cell cell negative 8 end cell 6 end table close parentheses$ then $A equals negative times$

If $A open parentheses table row 1 3 4 row 3 cell negative 1 end cell 5 row cell negative 2 end cell 4 cell negative 3 end cell end table close parentheses equals open parentheses table row 3 cell negative 1 end cell 5 row 1 3 4 row cell plus 4 end cell cell negative 8 end cell 6 end table close parentheses$ then $A equals negative times$

parallel

If $open vertical bar table row cell a blank b blank 1 end cell row cell b blank c blank 1 end cell row cell c blank a blank 1 end cell end table close vertical bar equals 2010 text andif end text open vertical bar table row cell c minus a end cell cell c minus b end cell cell a b end cell row cell a minus b end cell cell a minus c end cell cell b c end cell row cell b minus c end cell cell b minus a end cell cell c a end cell end table close vertical bar minus open vertical bar table row cell c minus a end cell cell c minus b end cell cell c to the power of 2 end exponent end cell row cell a minus b end cell cell a minus c end cell cell a to the power of 2 end exponent end cell row cell b minus c end cell cell b minus a end cell cell b to the power of 2 end exponent end cell end table close vertical bar equals p$ then the number of positive divisors of the number p is

If $open vertical bar table row cell a blank b blank 1 end cell row cell b blank c blank 1 end cell row cell c blank a blank 1 end cell end table close vertical bar equals 2010 text andif end text open vertical bar table row cell c minus a end cell cell c minus b end cell cell a b end cell row cell a minus b end cell cell a minus c end cell cell b c end cell row cell b minus c end cell cell b minus a end cell cell c a end cell end table close vertical bar minus open vertical bar table row cell c minus a end cell cell c minus b end cell cell c to the power of 2 end exponent end cell row cell a minus b end cell cell a minus c end cell cell a to the power of 2 end exponent end cell row cell b minus c end cell cell b minus a end cell cell b to the power of 2 end exponent end cell end table close vertical bar equals p$ then the number of positive divisors of the number p is

The number of positive integral solutions of the equation $open vertical bar table row cell y to the power of 3 end exponent plus 1 end cell cell y to the power of 2 end exponent z end cell cell y to the power of 2 end exponent x end cell row cell y z to the power of 2 end exponent end cell cell z to the power of 3 end exponent plus 1 end cell cell z to the power of 2 end exponent x end cell row cell y x to the power of 2 end exponent end cell cell X to the power of 2 end exponent Z end cell cell x to the power of 3 end exponent plus 1 end cell end table close vertical bar equals 11$ is

The number of positive integral solutions of the equation $open vertical bar table row cell y to the power of 3 end exponent plus 1 end cell cell y to the power of 2 end exponent z end cell cell y to the power of 2 end exponent x end cell row cell y z to the power of 2 end exponent end cell cell z to the power of 3 end exponent plus 1 end cell cell z to the power of 2 end exponent x end cell row cell y x to the power of 2 end exponent end cell cell X to the power of 2 end exponent Z end cell cell x to the power of 3 end exponent plus 1 end cell end table close vertical bar equals 11$ is

If $A comma$ $B comma$ $C comma$ $D$ be real matrices (not necessasrily square) such that $A to the power of T end exponent equals B C D comma$ $B to the power of T end exponent equals C D A comma$ $C to the power of T end exponent equals D A B comma$ $D to the power of T end exponent equals A B C$ forthematrix $S equals A B C D$ consider the statements $I colon S to the power of 3 end exponent equals S$ $capital pi colon S to the power of 2 end exponent equals S to the power of 4 end exponent$

If $A comma$ $B comma$ $C comma$ $D$ be real matrices (not necessasrily square) such that $A to the power of T end exponent equals B C D comma$ $B to the power of T end exponent equals C D A comma$ $C to the power of T end exponent equals D A B comma$ $D to the power of T end exponent equals A B C$ forthematrix $S equals A B C D$ consider the statements $I colon S to the power of 3 end exponent equals S$ $capital pi colon S to the power of 2 end exponent equals S to the power of 4 end exponent$

parallel

If $l o g subscript 4 end subscript left parenthesis x plus 2 y right parenthesis plus l o g subscript 4 end subscript left parenthesis x minus 2 y right parenthesis equals 1$ , then
Statement $negative 1 colon left parenthesis vertical line x vertical line minus vertical line y vertical line right parenthesis subscript blank m i n blank end subscript equals square root of 3$ because
Statement $negative 2$ :A.M $greater or equal than G. M$ for $R to the power of plus end exponent.$

If $l o g subscript 4 end subscript left parenthesis x plus 2 y right parenthesis plus l o g subscript 4 end subscript left parenthesis x minus 2 y right parenthesis equals 1$ , then
Statement $negative 1 colon left parenthesis vertical line x vertical line minus vertical line y vertical line right parenthesis subscript blank m i n blank end subscript equals square root of 3$ because
Statement $negative 2$ :A.M $greater or equal than G. M$ for $R to the power of plus end exponent.$

If $alpha element of left parenthesis negative fraction numerator pi over denominator 2 end fraction comma 0 right parenthesis$ , then the value of $t a n to the power of negative 1 end exponent left parenthesis blank c o t blank alpha right parenthesis minus c o t to the power of negative 1 end exponent left parenthesis blank t a n blank alpha right parenthesis$ is equal to

If $alpha element of left parenthesis negative fraction numerator pi over denominator 2 end fraction comma 0 right parenthesis$ , then the value of $t a n to the power of negative 1 end exponent left parenthesis blank c o t blank alpha right parenthesis minus c o t to the power of negative 1 end exponent left parenthesis blank t a n blank alpha right parenthesis$ is equal to

The minimum value of $left parenthesis s e c to the power of negative 1 end exponent x right parenthesis to the power of 2 end exponent plus left parenthesis blank c o s blank e c to the power of negative 1 end exponent x right parenthesis to the power of 2 end exponent$ is

The minimum value of $left parenthesis s e c to the power of negative 1 end exponent x right parenthesis to the power of 2 end exponent plus left parenthesis blank c o s blank e c to the power of negative 1 end exponent x right parenthesis to the power of 2 end exponent$ is

parallel

If $5 f left parenthesis x right parenthesis plus 3 f left parenthesis fraction numerator 1 over denominator x end fraction right parenthesis equals x plus 2$ andy $equals x f$ (x)then $fraction numerator d y over denominator d x end fraction$ at x=1 is

If $5 f left parenthesis x right parenthesis plus 3 f left parenthesis fraction numerator 1 over denominator x end fraction right parenthesis equals x plus 2$ andy $equals x f$ (x)then $fraction numerator d y over denominator d x end fraction$ at x=1 is

Statement 1:Degree of differential equation $2 x minus 3 y plus 2 equals blank l o g blank left parenthesis fraction numerator d y over denominator d x end fraction 1$ is not defined
Statement 2: In the given D.E, the power of highest order derivative when expressed as a polynomials of derivatives is called degree

Statement 1:Degree of differential equation $2 x minus 3 y plus 2 equals blank l o g blank left parenthesis fraction numerator d y over denominator d x end fraction 1$ is not defined
Statement 2: In the given D.E, the power of highest order derivative when expressed as a polynomials of derivatives is called degree

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

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

parallel

Solution of the equation $fraction numerator d y over denominator d x end fraction equals fraction numerator y to the power of 2 end exponent minus y minus 2 over denominator x to the power of 2 end exponent plus 2 x minus 3 end fraction$ is $minus$ ( where c is arbitrary constant)

Solution of the equation $fraction numerator d y over denominator d x end fraction equals fraction numerator y to the power of 2 end exponent minus y minus 2 over denominator x to the power of 2 end exponent plus 2 x minus 3 end fraction$ is $minus$ ( where c is arbitrary constant)

Assertion (A) : If (2, 3) and (3,2) are respectively the orthocenters of $capital delta A B C$ and $capital delta D E F$ where D, E, F are mid points of sides of $capital delta A B C$ respectively then the centroid of either triangle is $Error converting from MathML to accessible text.$
Reason (R) :Circumcenters of triangles formed by mid‐points of sides of $capital delta A B C$ and pedal triangle of $capital delta A B C$ are same

Assertion (A) : If (2, 3) and (3,2) are respectively the orthocenters of $capital delta A B C$ and $capital delta D E F$ where D, E, F are mid points of sides of $capital delta A B C$ respectively then the centroid of either triangle is $Error converting from MathML to accessible text.$
Reason (R) :Circumcenters of triangles formed by mid‐points of sides of $capital delta A B C$ and pedal triangle of $capital delta A B C$ are same

Two metallic strings A and B of different materials are connected in series forming a joint. The strings have similar cross-sectional area. The length of A is $l subscript A end subscript equals 0.3 m$ and that of B is $l subscript B end subscript equals 0.75 m$ . One end of the combined string is tied with a support rigidly and the other end is loaded with a block of mass m passing over a frictionless pulley. Transverse waves are setup in the combined string using an external source of variable frequency. The total number of antinodes at this frequency with joint as node is (the densities of A and B are $6.3 cross times 10 to the power of 3 end exponent k g m to the power of negative 3 end exponent text and end text 2.8 cross times 10 to the power of 3 end exponent k g m to the power of negative 3 end exponent$ respectively)

Two metallic strings A and B of different materials are connected in series forming a joint. The strings have similar cross-sectional area. The length of A is $l subscript A end subscript equals 0.3 m$ and that of B is $l subscript B end subscript equals 0.75 m$ . One end of the combined string is tied with a support rigidly and the other end is loaded with a block of mass m passing over a frictionless pulley. Transverse waves are setup in the combined string using an external source of variable frequency. The total number of antinodes at this frequency with joint as node is (the densities of A and B are $6.3 cross times 10 to the power of 3 end exponent k g m to the power of negative 3 end exponent text and end text 2.8 cross times 10 to the power of 3 end exponent k g m to the power of negative 3 end exponent$ respectively)

physics-General

parallel

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