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The value of $\theta$ for which the area of the triangle formed by the line $x\sin \theta +y\cos \theta =4$ and the coordinate axis is minimum is

Prove that:

$cos4\theta -cos4\alpha =8(cos\theta -cos\alpha )(cos\theta +cos\alpha )(cos\theta -sin\alpha )(cos\theta -sin\alpha )$

Determine
the volume contraction of a solid copper cube, 10 cm on an edge, when
subjected to a hydraulic pressure of 7.0 ×10⁶
Pa.

Let $f:R\to R$ be a positive increasing function with $\underset{x\to \infty }{lim}\frac{f\left(7x\right)}{f\left(x\right)}=1.$ Then which of the following(s) is(are) correct?

A hag contains 7 white, 5 black and 4 red balls. If two balls are drawn at random, find the probability that :

(i) both the balls are white

(ii)one ball is black and the other red

(iii) both the balls are of the same colour.

The angle between the plane $x-2y+3z=5$ and the line

$\frac{x-1}{1}=\frac{y-1}{-1}=\frac{z+1}{1}$

is ${\sin }^{-1}\left(\sqrt{\alpha }\right).$ Then 7$\alpha$ is equal to

Consider the Karnaugh map given below,



The function represented by this map can be simplified to the minimal form as

By using properties of deteminants.

$\left|\begin{array}{ccc}0& a& -b\\ -a& 0& -c\\ b& c& 0\end{array}\right|=0$

The solution of the differential equation $\frac{d^{2}y}{d{x}^2}+6\frac{dy}{dx}+9y=9x+6$ with $C_1$ and $C_2$ as constant is

The common region determined by all the constraints including non-negative constraints x, y ≥ 0 of a linear programming problem is called the ……...

A triangular open channel has a vertex angle of ${90}^o$ and carries flow at a critical depth of 0.30 m. The discharge in the channel is

If the tangent at $(1,7)$ to the curve $x^2=y-6$ touches the circle $x^2+y^2+16x+12y+c=0,$ then the value of $c$ is

The maximum value of $f\left(x\right)=x^{\frac{1}{x}}$ is

Let $y=y\left(x\right)$ be the solution of the differential equation $\frac{dy}{dx}=(y+1)((y+1)e^{x^2/2}-x),$ $0<x<2.1,$ with $y\left(2\right)=0.$ Then the value of $\frac{dy}{dx}$at $x=1$ is equal to:

Distance(in $\text{units}$) between the parallel planes $2x-3y+4z-1=0$ and $4x-6y+8z+7=0$ is

Find $A^{-1}$, if $A=\left|\begin{array}{ccc}0& 1& 1\\ 1& 0& 1\\ 1& 1& 0\end{array}\right|$ and show that $A^{-1}=\frac{A^2-3I}{2}$.

The 4th and 7th terms of a G.P. are $\frac{1}{27}$ and $\frac{1}{792}$ respectively. Find the sum of n terms of the G.P.

The approximate value of $(1.0002{)}^{3000}$ is

If the locus of the midpoint of contact of tangent drawn to the parabola $y^2=8x$ and foot of perpendicular drawn from its focus to the tangents is a conic then length of latus rectum of this conic is

Let $[.]$ denote the greatest integer function.

Consider $f\left(x\right)=\{\begin{array}{cc}2-\left|x\right|,& -1\le x\le 1\\ |x-2|-x,& 1<x\le 3\end{array}$ and $g\left(x\right)=\{\begin{array}{cc}\sin x-1,& 0\le x<\frac{\pi }{2}\\ \left[x\right]-\cos (x-2),& \frac{\pi }{2}\le x\le \pi \end{array}.$



Which of the following statements is/are CORRECT?

Consider a sequence $\left\{a_n\right\}$ with $a_1=2$ and $a_n=\frac{a_{n-1}^2}{a_{n-2}}$for all $n\ge 3$, terms of the sequence being distinct. Given that $a_2$ and $a_5$ are positive integers and $a_5\le 162$ then the possible value(s) of $a_5$ can be

The locus of the orthocenter of the triangle formed by the focal chord of the parabola $y^2=4ax$ and the normal drawn at its extermities is

1. 2

A vector $\overrightarrow{a}=\alpha \hat{i}+2\hat{j}+\beta \hat{k}(\alpha ,\beta \in R)$ lies in the plane of the vectors, $\overrightarrow{b}=\hat{i}+\hat{j}$ and $\overrightarrow{c}=\hat{i}-\hat{j}+4\hat{k}$. If $\overrightarrow{a}$ bisects the angle between $\overrightarrow{b}$ and $\overrightarrow{c},$ then

If $f\left(\frac{x-4}{x+2}\right)=2x+1,$ where $x\in R-\{1,-2\},$ then$\int f\left(x\right)dx$ is equal to

(where $C$ is constant of integration)

Six points in a plane be joining in all possible way by staright lines, and if no two of them be coincident or parallel, and no three pass through the same point (with the exception of the original $6$ points). The number of distinct points of intersection is equal to

${lim}_{x\to 0}\frac{ta{n}^{3}x-si{n}^{3}x}{x^5}=$