Verify whether the following statements are true or false. (i) The cu…

Question

Verify whether the following statements are true or false.

(i) The curve $y^2 = x^3$ has a node at the origin.

(ii) The curve $r = a \cos 2\theta$ has two leaves.

(iii) $\sin z = \sin x \cdot \cosh y + i \cos x \cdot \sinh y$.

(iv) $\sin x$ and $\sin 2x$ are independent solutions of $(D^4 + 1)y = 0$

(v) $(x^2 + y^2 + 2x) dx + 2y dy = 0$ is an exact differential equation.

(vi) The function $z = x \cdot \sin^{-1}\left(\frac{y}{x}\right)$ is homogeneous of degree 2.

(vii) The volume of a sphere of radius 'a' is $\frac{4}{3}\pi a^3$.

(viii) The value of $\int_{0}^{\pi/2} \sin^7 x dx$ is $\frac{16}{35}$.

(ix) $\cosh(x + y) = \cosh x \cdot \cosh y - \sinh x \cdot \sinh y$.

(x) The angle between the vectors $4\hat{i} - 2\hat{j} + \hat{k}$ and $\hat{i} + \hat{j} - 2\hat{k}$ is $\frac{\pi}{2}$.

(xi) $\text{div } \vec{r} = 3$, where $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$.

Related pages

• Maxima and Minima of Functions of Two Variables — topic
• Engineering Mathematics-I syllabus

More PYQs on Maxima and Minima of Functions of Two Variables

• (a) If $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$, then show that : $\nabla^2(r^m) = m(m - 1)r^{m-2}$. (b) Find the divergence and curl of the vector function…
• (a) Verify Rolle's theorem for $f(x) = e^x(\sin x - \cos x)$ in $\left[\frac{\pi}{4}, \frac{5\pi}{4}\right]$. (b) Examine the function $f = xy(a - x - y)$ for…
• (a) Find the angle between the surfaces $x^2 + y^2 + z^2 = 9$ and $x^2 + y^2 - z = 3$ at the point $(2, -1, 2)$. (b) Show that $\text{div}(\text{curl } \vec{F}…
• (a) Express $\tan^{-1} Z$ into real and imaginary parts. (b) If $e^{i\theta} = \sinh(x + i y)$, show that $\sinh^4 x = \cos^3 \theta = \cos^4 y$.