(a) By Vector method prove that : $\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta$ **OR** (b) $(x^2+y^2)dy=xy\,dx$. It is given that $y(1)=1$ and $y(x_0)=e$. Find the value of $x_0$.

(a) By Vector method prove that : cos(α+β)=cosαcosβ-sinαsinβ **OR** (b) (x²+y²)dy=xydx. It is given that y(1)=

Question 162 of 162

\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta

(b)

(x^2+y^2)dy=xy\,dx

. It is given that

y(1)=1

and

y(x_0)=e

. Find the value of

x_0

Tamil Nadu DgeTamil Nadu HSC (DGE) Board 2026Subjective· 5mImportance★★★★★

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Concept understanding — Applications of Dot and Cross Product

The dot product $\vec a\cdot\vec b=|\vec a||\vec b|\cos\theta$ and cross product $\vec a\times\vec b$ (magnitude $|\vec a||\vec b|\sin\theta$ , direction perpendicular to both) are not just formulas — placed carefully, they reprove classical geometry theorems and compute two real mechanical quantities.

Vector proofs in geometry and trigonometry. The standard technique: put a convenient point (a triangle's vertex, a circle's centre) at the origin, write every other point as a position vector, and express every needed segment as a difference of position vectors. Then:

Perpendicularity is proved by showing a dot product is $0$ . E.g. if $O$ is a circle's centre and $M$ is the midpoint of chord $AB$ (position vectors $\vec a,\vec b$ with $|\vec a|=|\vec b|=r$ ), then $\vec{OM}=\tfrac{\vec a+\vec b}2$ and $\vec{AB}=\vec b-\vec a$ , and $\vec{OM}\cdot\vec{AB}=\tfrac12(|\vec b|^2-|\vec a|^2)=0$ .
Equal lengths / rectangles come from expanding $|\vec p\pm\vec q|^2=|\vec p|^2\pm2\vec p\cdot\vec q+|\vec q|^2$ and comparing.
Areas come from $\text{area of a parallelogram}=|\vec p\times\vec q|$ for adjacent sides $\vec p,\vec q$ ; a triangle is half that, and a general quadrilateral with diagonals $\vec d_1,\vec d_2$ has area $\tfrac12|\vec d_1\times\vec d_2|$ (proved by splitting along one diagonal, since the two triangles on either side add — same-sense cross products — instead of subtracting).
Compound-angle identities ( $\cos(\alpha\mp\beta)$ , $\sin(\alpha\pm\beta)$ ) drop out of dotting or crossing two unit vectors $\hat i\cos\alpha+\hat j\sin\alpha$ and $\hat i\cos\beta\pm\hat j\sin\beta$ placed at angles $\alpha,\pm\beta$ to the $x$ -axis. …

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(a) Represents $\hat a,\hat b$ as unit vectors at angles $\alpha,-\beta$ from the $x$ -axis and equates the two expressions for their dot product; (b) solves the homogeneous first-order ODE by the substitution $y=vx$ , fixes the constant from $y(1)=1$ , then solves for $x_0$ from $y(x_0)=e$ . Both alternatives answered below.

(a) Vector proof of $\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta$

1. Unit vectors. Let $\hat a=\cos\alpha\,\hat i+\sin\alpha\,\hat j$ (unit vector at angle $\alpha$ above the $x$ -axis) and $\hat b=\cos\beta\,\hat i-\sin\beta\,\hat j$ (unit vector at angle $\beta$ below the $x$ -axis, i.e. at angle $-\beta$ ).

2. Angle between $\hat a$ and $\hat b$ . Going from $\hat b$ (at angle $-\beta$ ) to $\hat a$ (at angle $\alpha$ ) sweeps an angle $\alpha-(-\beta)=\alpha+\beta$ . Since both are unit vectors:

$\hat a\cdot\hat b=|\hat a||\hat b|\cos(\alpha+\beta)=\cos(\alpha+\beta)$

3. Dot product by components.

$\hat a\cdot\hat b=(\cos\alpha)(\cos\beta)+(\sin\alpha)(-\sin\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta$

4. Equate the two expressions (steps 2 and 3), both being $\hat a\cdot\hat b$ :

$\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta$

as required — proved by the vector (dot-product) method.

(b) Solve $(x^2+y^2)dy=xy\,dx$ , with $y(1)=1$ , find $x_0$ where $y(x_0)=e$

1. Rewrite and identify as homogeneous. $\dfrac{dy}{dx}=\dfrac{xy}{x^2+y^2}$ — the right side is a function of $y/x$ alone.

2. Substitute $y=vx$ , so $\dfrac{dy}{dx}=v+x\dfrac{dv}{dx}$ :

$v+x\dfrac{dv}{dx}=\dfrac{x(vx)}{x^2+v^2x^2}=\dfrac{v}{1+v^2}$

3. Isolate $x\,dv/dx$ .

$x\dfrac{dv}{dx}=\dfrac v{1+v^2}-v=\dfrac{v-v(1+v^2)}{1+v^2}=\dfrac{-v^3}{1+v^2}$

4. Separate variables.

$\dfrac{1+v^2}{v^3}\,dv=-\dfrac{dx}x\ \Longrightarrow\ \left(\dfrac1{v^3}+\dfrac1v\right)dv=-\dfrac{dx}x$

…

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

CBSE 2019Set ANNUAL1 markMCQ
$\vec{a} + \vec{b} + \vec{c} = \vec{0}$
(a) $\dfrac{5\pi}{3}$
(b) $\dfrac{\pi}{2}$
(c) $\dfrac{\pi}{6}$
(d) $\dfrac{2\pi}{3}$
›Reveal solutionSolution
From $\vec a+\vec b+\vec c=\vec 0$ with $|\vec a|=3,|\vec b|=4,|\vec c|=5$ , the angle between $\vec a$ and $\vec b$ works out to $\dfrac{\pi}{2}$ .

From $\vec a+\vec b+\vec c=\vec0$ , we get $\vec c=-(\vec a+\vec b)$ .

Take magnitudes squared: $|\vec c|^2=|\vec a+\vec b|^2=|\vec a|^2+2\vec a\cdot\vec b+|\vec b|^2$ .

Substitute the given magnitudes: $5^2=3^2+2\vec a\cdot\vec b+4^2$ , i.e. $25=9+16+2\vec a\cdot\vec b=25+2\vec a\cdot\vec b$ .

So $2\vec a\cdot\vec b=0\Rightarrow \vec a\cdot\vec b=0$ . …
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CBSE 2017Set ANNUAL1 markMCQ
$\vec{PR} = 2\vec i + \vec j + \vec k$
(a) $5\sqrt3$
(b) $10\sqrt3$
(c) $\dfrac{5\sqrt3}{2}$
(d) $\dfrac{3}{2}$
›Reveal solutionSolution
The area of a quadrilateral in terms of its diagonal vectors $\vec d_1,\vec d_2$ is $\frac12|\vec d_1\times\vec d_2|$ ; here the diagonals are $\vec{PR}$ and $\vec{QS}$ , so compute their cross product and halve its magnitude.

For quadrilateral PQRS, the diagonals are $PR$ and $QS$ , and the standard vector-algebra result is $\text{Area}(PQRS) = \frac12\,|\vec{PR}\times\vec{QS}|.$

Given $\vec{PR}=2\vec i+\vec j+\vec k$ and $\vec{QS}=-\vec i+3\vec j+2\vec k$ .

Compute the cross product: $\vec{PR}\times\vec{QS}=\begin{vmatrix}\vec i & \vec j & \vec k\\ 2 & 1 & 1\\ -1 & 3 & 2\end{vmatrix}$

$\vec i$ -component: $(1)(2)-(1)(3) = 2-3=-1$ .

$\vec j$ -component: $-[(2)(2)-(1)(-1)] = -(4+1)=-5$ . …
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CBSE 2016Set ANNUAL1 markMCQ
$\vec{i} - \vec{j}$
(a) $0$
(b) $1$
(c) $-1$
(d) $2$
›Reveal solutionSolution
The vector $\vec i-\vec j$ has zero component along $\vec k$ , so its projection on the $z$ -axis is $0$ .

The projection of a vector $\vec r=x\vec i+y\vec j+z\vec k$ on the $z$ -axis equals $\vec r\cdot\vec k=z$ .

Here $\vec r=\vec i-\vec j=1\vec i-1\vec j+0\vec k$ .

So $\vec r\cdot\vec k=0$ . …
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CBSE 2016Set ANNUAL1 markMCQ
$3\vec i + \vec j - \vec k$
(a) $10\sqrt3$
(b) $6\sqrt{30}$
(c) $\dfrac{3}{2}\sqrt{30}$
(d) $3\sqrt{30}$
›Reveal solutionSolution
Recovering the second side from the diagonal and given side, then taking the cross product, gives an area of $3\sqrt{30}$ .

In a parallelogram with adjacent sides $\vec a$ and $\vec b$ from the same vertex, the diagonal from that vertex is $\vec d=\vec a+\vec b$ .

Given side $\vec a=\vec i-3\vec j+4\vec k$ and diagonal $\vec d=3\vec i+\vec j-\vec k$ , the other side is $\vec b=\vec d-\vec a=(3-1)\vec i+(1-(-3))\vec j+(-1-4)\vec k=2\vec i+4\vec j-5\vec k$ .

Area of the parallelogram $=|\vec a\times\vec b|$ .

Compute the cross product: $\vec a\times\vec b=\begin{vmatrix}\vec i&\vec j&\vec k\\1&-3&4\\2&4&-5\end{vmatrix}$ $i$ -component: $(-3)(-5)-(4)(4)=15-16=-1$ . $j$ -component: $-[(1)(-5)-(4)(2)]=-[-5-8]=13$ . …
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←Previous questionQ63 — Find the points where the straight line passes through (6, 7, 4) and (…→Next chapterCh 7Applications of Differential Calculus

Q.(a) By Vector method prove that : cos⁡(α+β)=cos⁡αcos⁡β−sin⁡αsin⁡β\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\betacos(α+β)=cosαcosβ−sinαsinβ OR

(a) Vector proof of cos⁡(α+β)=cos⁡αcos⁡β−sin⁡αsin⁡β\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\betacos(α+β)=cosαcosβ−sinαsinβ

(b) Solve (x2+y2)dy=xy dx(x^2+y^2)dy=xy\,dx(x2+y2)dy=xydx, with y(1)=1y(1)=1y(1)=1, find x0x_0x0​ where y(x0)=ey(x_0)=ey(x0​)=e

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Q.(a) By Vector method prove that : $\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta$ OR

(a) Vector proof of $\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta$

(b) Solve $(x^2+y^2)dy=xy\,dx$ , with $y(1)=1$ , find $x_0$ where $y(x_0)=e$

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