Maths-

General

Easy

Question

The polar equation of $x to the power of 3 end exponent plus y to the power of 3 end exponent equals 3$ axy is

$r ({C o s}^{3} θ + {S i n}^{3} θ) = 3 a C o s θ S i n θ$
$r ({C o s}^{3} θ + {S i n}^{3} θ) = a C o s 2 θ$
$r ({C o s}^{3} θ - {S i n}^{3} θ) = 3 a C o s θ S i n θ$
$r ({C o s}^{3} θ - {S i n}^{3} θ) = a C o s 2 θ$

The correct answer is: $r open parentheses C o s cubed space theta plus S i n cubed space theta close parentheses equals 3 a C o s space theta S i n space theta$

If x, y, z are integers and x $greater or equal than$ 0, y $greater or equal than$ 1, z $greater or equal than$ 2, x + y + z = 15, then the number of values of the ordered triplet (x, y, z) is -

greater or equal than

0, p

greater or equal than

0, q

greater or equal than

0 and x + y + z = 15

rightwards double arrow

x + p + q = 12

therefore

The reqd. number of values of (x, y, z) and hence of (x, p, q)
= No. of non-negative integral solutions of x + p + q= 12
= Coeff. of x¹² in (x⁰ + x¹ + x² + ……)³
= Coeff. of x¹² in (1 – x)^–3
= Coeff. of x¹² in [²C₀ + ³C₁ x + ⁴C₂ x² + ….]
= ¹⁴C₁₂ =

fraction numerator 14 factorial over denominator 2 factorial 12 factorial end fraction

fraction numerator 14 cross times 13 over denominator 2 end fraction

= 91.

If x, y, z are integers and x $greater or equal than$ 0, y $greater or equal than$ 1, z $greater or equal than$ 2, x + y + z = 15, then the number of values of the ordered triplet (x, y, z) is -

maths-General

Let y = p + 1 and z = q + 2.
Then x

greater or equal than

0, p

greater or equal than

0, q

greater or equal than

0 and x + y + z = 15

rightwards double arrow

x + p + q = 12

therefore

fraction numerator 14 factorial over denominator 2 factorial 12 factorial end fraction

fraction numerator 14 cross times 13 over denominator 2 end fraction

= 91.

General

Maths-

If $alpha not equal to beta comma alpha 2 equals 5 alpha minus 3 comma beta 2 times equals 5 beta minus 3 comma$ then the equation whose roots are $alpha divided by beta straight & beta divided by alpha$

Maths-General

General

Maths-

Let p, q $element of$ {1, 2, 3, 4}. Then number of equation of the form px² + qx + 1 = 0, having real roots, is

element of

{1, 2, 3, 4}, the equation given is : px² + qx + 1 = 0

Now we know that for real roots, the discriminant is always greater than or equal to

0, so we have:

D=b²-4ac, applying this, we get:

q^{2} - 4 p \geq 0 \Rightarrow q^{2} \geq 4 p

Now the set includes 4 terms, putting each, we get:

For

p = 1, q^{2} \geq 4

q = 2, 3, 4

For

p = 2, q^{2} \geq 8

q = 3, 4

For

p = 3, q^{2} \geq 12

q = 4

For

p = 4, q^{2} \geq 16

q = 4

So here we can see that total 7 seven solutions are possible so

7

equations can be formed.

Let p, q $element of$ {1, 2, 3, 4}. Then number of equation of the form px² + qx + 1 = 0, having real roots, is

Maths-General

element of

{1, 2, 3, 4}, the equation given is : px² + qx + 1 = 0

Now we know that for real roots, the discriminant is always greater than or equal to

0, so we have:

D=b²-4ac, applying this, we get:

q^{2} - 4 p \geq 0 \Rightarrow q^{2} \geq 4 p

Now the set includes 4 terms, putting each, we get:

For

p = 1, q^{2} \geq 4

q = 2, 3, 4

For

p = 2, q^{2} \geq 8

q = 3, 4

For

p = 3, q^{2} \geq 12

q = 4

For

p = 4, q^{2} \geq 16

q = 4

So here we can see that total 7 seven solutions are possible so

7

equations can be formed.

General

Maths-

ax² + bx + c = 0 has real and distinct roots null. Further a > 0, b < 0 and c < 0, then –

A quadratic equation, or sometimes just quadratics, is a polynomial equation with a maximum degree of two. It takes the following form:
ax² + bx + c = 0
where a, b, and c are constant terms and x is the unknown variable.
Now we have given a > 0, b < 0 and c < 0, the equation is ax² + bx + c = 0.
Let the roots be α and β, where β>α, then:
α + β = -b/a

> 0

a > 0, b < 0

.
αβ = c/a as

a > 0, c < 0

.
Now that the roots are of opposite signs, so β > 0 and α < 0.
So:

∣ α ∣ < β

as α

+ β > 0

.
So therefore: 0 < ∣α∣ < β

ax² + bx + c = 0 has real and distinct roots null. Further a > 0, b < 0 and c < 0, then –

Maths-General

> 0

a > 0, b < 0

.
αβ = c/a as

a > 0, c < 0

.
Now that the roots are of opposite signs, so β > 0 and α < 0.
So:

∣ α ∣ < β

as α

+ β > 0

.
So therefore: 0 < ∣α∣ < β

General

Maths-

The cartesian equation of $r squared c o s space 2 theta equals a squared$ is

r squared c o s space 2 theta equals a squared

.
Now we know that:

cos space 2 theta equals cos squared theta minus sin squared theta

.
So applying this, we get:

r squared left parenthesis cos squared theta minus sin squared theta right parenthesis equals a squared

Now lets substitute x=

r cos θ and y = r sin θ, we get:

r squared cos squared theta minus r squared sin squared theta equals a squared x squared minus y squared equals a squared

The cartesian equation of $r squared c o s space 2 theta equals a squared$ is

Maths-General

r squared c o s space 2 theta equals a squared

.
Now we know that:

cos space 2 theta equals cos squared theta minus sin squared theta

.
So applying this, we get:

r squared left parenthesis cos squared theta minus sin squared theta right parenthesis equals a squared

Now lets substitute x=

r cos θ and y = r sin θ, we get:

r squared cos squared theta minus r squared sin squared theta equals a squared x squared minus y squared equals a squared

General

Maths-

The castesian equation of $r c o s invisible function application left parenthesis theta minus alpha right parenthesis equals p$ is

r c o s invisible function application left parenthesis theta minus alpha right parenthesis equals p

.
Now we know that:

cos space left parenthesis space theta space minus space alpha space right parenthesis equals cos space theta space cos space alpha space plus space sin space theta space sin space alpha

.
So applying this, we get:

r left parenthesis cos space theta space cos space alpha space plus space sin space theta space sin space alpha right parenthesis equals p

Now lets substitute x=

r cos θ and y = r sin θ, we get:

r cos space theta space cos space alpha space plus space r sin space theta space sin space alpha right parenthesis equals p x space cos space alpha space plus space y space sin space alpha space equals space p

The castesian equation of $r c o s invisible function application left parenthesis theta minus alpha right parenthesis equals p$ is

Maths-General

r c o s invisible function application left parenthesis theta minus alpha right parenthesis equals p

.
Now we know that:

cos space left parenthesis space theta space minus space alpha space right parenthesis equals cos space theta space cos space alpha space plus space sin space theta space sin space alpha

.
So applying this, we get:

r left parenthesis cos space theta space cos space alpha space plus space sin space theta space sin space alpha right parenthesis equals p

Now lets substitute x=

r cos θ and y = r sin θ, we get:

r cos space theta space cos space alpha space plus space r sin space theta space sin space alpha right parenthesis equals p x space cos space alpha space plus space y space sin space alpha space equals space p

General

maths-

The equation of the directrix of the conic whose length of the latusrectum is 5 and eccenticity is 1/2 is

maths-General

General

maths-

The equation of the directrix of the conic $r C o s to the power of 2 end exponent invisible function application open parentheses fraction numerator theta over denominator 2 end fraction close parentheses equals 5$ is

maths-General

General

maths-

A small source of sound moves on a circle as shown in the figure and an observer is standing on $O.$ Let $n subscript 1 end subscript comma n subscript 2 end subscript$ and $n subscript 3 end subscript$ be the frequencies heard when the source is at $A comma blank B blank$ and $C$ respectively. Then

At point

A comma

source is moving away from observer so apparent frequency

n subscript 1 end subscript less than n

(actual frequency) At point

B

source is coming towards observer so apparent frequency

n subscript 2 end subscript greater than n

and point

C

source is moving perpendicular to observer so

n subscript 3 end subscript equals n

Hence

n subscript 2 end subscript greater than n subscript 3 end subscript greater than n subscript 1 end subscript

A small source of sound moves on a circle as shown in the figure and an observer is standing on $O.$ Let $n subscript 1 end subscript comma n subscript 2 end subscript$ and $n subscript 3 end subscript$ be the frequencies heard when the source is at $A comma blank B blank$ and $C$ respectively. Then

physics-General

At point

A comma

source is moving away from observer so apparent frequency

n subscript 1 end subscript less than n

(actual frequency) At point

B

source is coming towards observer so apparent frequency

n subscript 2 end subscript greater than n

and point

C

source is moving perpendicular to observer so

n subscript 3 end subscript equals n

Hence

n subscript 2 end subscript greater than n subscript 3 end subscript greater than n subscript 1 end subscript

General

Maths-

In a triangle ABC, if a : b : c = 7 : 8 : 9, then cos A : cos B equals to

Maths-General

General

physics-

Which of the following curves represents correctly the oscillation given by $y equals y subscript 0 end subscript sin invisible function application open parentheses omega t minus ϕ close parentheses comma blank w h e r e blank 0 less than ϕ less than 90$

Given equation

y equals y subscript 0 end subscript sin invisible function application left parenthesis omega t minus ϕ right parenthesis

t equals 0 comma blank y equals negative y subscript 0 end subscript sin invisible function application ϕ

This is case with curve marked

D

Which of the following curves represents correctly the oscillation given by $y equals y subscript 0 end subscript sin invisible function application open parentheses omega t minus ϕ close parentheses comma blank w h e r e blank 0 less than ϕ less than 90$

physics-General

Given equation

y equals y subscript 0 end subscript sin invisible function application left parenthesis omega t minus ϕ right parenthesis

t equals 0 comma blank y equals negative y subscript 0 end subscript sin invisible function application ϕ

This is case with curve marked

D

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The polar equation of axy is

The correct answer is:

Related Questions to study

If x, y, z are integers and x 0, y 1, z 2, x + y + z = 15, then the number of values of the ordered triplet (x, y, z) is -

If x, y, z are integers and x 0, y 1, z 2, x + y + z = 15, then the number of values of the ordered triplet (x, y, z) is -

If then the equation whose roots are

If then the equation whose roots are

Let p, q {1, 2, 3, 4}. Then number of equation of the form px2 + qx + 1 = 0, having real roots, is

Let p, q {1, 2, 3, 4}. Then number of equation of the form px2 + qx + 1 = 0, having real roots, is

ax2 + bx + c = 0 has real and distinct roots null. Further a > 0, b < 0 and c < 0, then –

ax2 + bx + c = 0 has real and distinct roots null. Further a > 0, b < 0 and c < 0, then –

The cartesian equation of is

The cartesian equation of is

The castesian equation of is

The castesian equation of is

The equation of the directrix of the conic whose length of the latusrectum is 5 and eccenticity is 1/2 is

The equation of the directrix of the conic whose length of the latusrectum is 5 and eccenticity is 1/2 is

The equation of the directrix of the conic is

The equation of the directrix of the conic is

The equation of the circle touching the initial line at pole and radius 2 is

The equation of the circle touching the initial line at pole and radius 2 is

The equation of the circle passing through pole and centre at (4,0) is

The equation of the circle passing through pole and centre at (4,0) is

The polar equation of the circle with pole as centre and radius 3 is

The polar equation of the circle with pole as centre and radius 3 is

(Area of GPL) to (Area of ALD) is equal to

(Area of GPL) to (Area of ALD) is equal to

A small source of sound moves on a circle as shown in the figure and an observer is standing on Let and be the frequencies heard when the source is at and respectively. Then

A small source of sound moves on a circle as shown in the figure and an observer is standing on Let and be the frequencies heard when the source is at and respectively. Then

In a triangle ABC, if a : b : c = 7 : 8 : 9, then cos A : cos B equals to

In a triangle ABC, if a : b : c = 7 : 8 : 9, then cos A : cos B equals to

Which of the following curves represents correctly the oscillation given by

Which of the following curves represents correctly the oscillation given by

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The polar equation of $x to the power of 3 end exponent plus y to the power of 3 end exponent equals 3$ axy is

The correct answer is: $r open parentheses C o s cubed space theta plus S i n cubed space theta close parentheses equals 3 a C o s space theta S i n space theta$

If x, y, z are integers and x $greater or equal than$ 0, y $greater or equal than$ 1, z $greater or equal than$ 2, x + y + z = 15, then the number of values of the ordered triplet (x, y, z) is -

If x, y, z are integers and x $greater or equal than$ 0, y $greater or equal than$ 1, z $greater or equal than$ 2, x + y + z = 15, then the number of values of the ordered triplet (x, y, z) is -

If $alpha not equal to beta comma alpha 2 equals 5 alpha minus 3 comma beta 2 times equals 5 beta minus 3 comma$ then the equation whose roots are $alpha divided by beta straight & beta divided by alpha$

If $alpha not equal to beta comma alpha 2 equals 5 alpha minus 3 comma beta 2 times equals 5 beta minus 3 comma$ then the equation whose roots are $alpha divided by beta straight & beta divided by alpha$

Let p, q $element of$ {1, 2, 3, 4}. Then number of equation of the form px² + qx + 1 = 0, having real roots, is

Let p, q $element of$ {1, 2, 3, 4}. Then number of equation of the form px² + qx + 1 = 0, having real roots, is

ax² + bx + c = 0 has real and distinct roots null. Further a > 0, b < 0 and c < 0, then –

ax² + bx + c = 0 has real and distinct roots null. Further a > 0, b < 0 and c < 0, then –

The cartesian equation of $r squared c o s space 2 theta equals a squared$ is

The cartesian equation of $r squared c o s space 2 theta equals a squared$ is

The castesian equation of $r c o s invisible function application left parenthesis theta minus alpha right parenthesis equals p$ is

The castesian equation of $r c o s invisible function application left parenthesis theta minus alpha right parenthesis equals p$ is

The equation of the directrix of the conic $r C o s to the power of 2 end exponent invisible function application open parentheses fraction numerator theta over denominator 2 end fraction close parentheses equals 5$ is

The equation of the directrix of the conic $r C o s to the power of 2 end exponent invisible function application open parentheses fraction numerator theta over denominator 2 end fraction close parentheses equals 5$ is

(Area of $increment$ GPL) to (Area of $increment$ ALD) is equal to

(Area of $increment$ GPL) to (Area of $increment$ ALD) is equal to

Which of the following curves represents correctly the oscillation given by $y equals y subscript 0 end subscript sin invisible function application open parentheses omega t minus ϕ close parentheses comma blank w h e r e blank 0 less than ϕ less than 90$

Which of the following curves represents correctly the oscillation given by $y equals y subscript 0 end subscript sin invisible function application open parentheses omega t minus ϕ close parentheses comma blank w h e r e blank 0 less than ϕ less than 90$