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Consider a determinant $\Delta =\left|\begin{array}{ccc}ax-by-cz& ay+bx& cx+az\\ ay+bx& by-cz-ax& bz+cy\\ cx+az& bz+cy& cz-ax-by\end{array}\right|,\text{then}$

The differential equation representing the family of ellipses having foci either on the x-axis or on the y-axis, centre at the origin and passing through the point (0, 3) is :

The equation sin^–1x = 3sin^–1α has a solution, then the set of exhaustive values of ‘α’ is



समीकरण sin^–1x = 3sin^–1α का एक हल है, तब ‘α’ के सम्पूर्ण मानों का समुच्चय है

$\frac{a^2-c^2}{b^2}=\frac{sin(A-C)}{sin(A+C)}$

A vector $\overrightarrow{a}$ has components 2p and 1 with respect to a rectangular Cartesian system. This system is rotated through a certain angle about the origin in the counter-clockwise sense. If with respect to the new system, $\overrightarrow{a}$ has components (p+1) and 1, then

The values of $k$ for which the point of minimum of the function $f\left(x\right)=4+k^{2}x-2x^3$ satisfies the inequality $\frac{x^2+2x+4}{x^2+6x+8}<0$ is

Let $f\left(x\right)=x+\sin x$ and $x\in [0,\frac{\pi }{2}]$, then which of the following(s) equation(s) have atleast one real solution

If ABCD is a square having coordinates A (4, 5), B (4, 1) and C (8, 1), the coordinates of D are

If $\int \frac{1+e^{7\ln x}-e^{3\ln x}}{e^{2\ln x}-1}dx=a\ln \left|\frac{x+b}{x-b}\right|+g\left(x\right)+C,g\left(0\right)=0.$ Then which of the following option(s) is/are correct

(where $a,b$ are fixed constants and $C$ is constant of integration)

Find the real value of a for which $3i^3-2a{i}^2+(1-a)i+5$ is real.

The product of sum expression of a Boolean function $F(A,B,C)$ of three variables is given by $F(A+B+C)=(A+B+\overline{C}).(A+\overline{B}+\overline{C}).(\overline{A}+B+C).(\overline{A}+\overline{B}+\overline{C}).$ The canonical sum of product expression of $F(A,B,C)$ is given by

If $\underset{x\to 0}{lim}\frac{\sin 3x+a\sin 2x}{x^3}=\lambda$ and $\lambda$ is finite non zero real number, then $\lambda =$

If A = {1, 2}, form the set $A\times A\times A$

If $S$ is the set of all real $x$ such that $\frac{2}{x^2-x+1}-\frac{1}{x+1}-\frac{2x-1}{x^3+1}\ge 0$, then $S$ contains

If $A=\left[\begin{array}{ccc}\frac{2}{3}& 1& \frac{5}{3}\\ \frac{1}{3}& \frac{2}{3}& \frac{4}{3}\\ \frac{7}{3}& 2& \frac{2}{3}\end{array}\right]andB=\left[\begin{array}{ccc}\frac{2}{5}& \frac{3}{5}& 1\\ \frac{1}{5}& \frac{2}{5}& \frac{4}{5}\\ \frac{7}{5}& \frac{6}{5}& \frac{2}{5}\end{array}\right]$, then compute 3A -5B.

Solution of differential equation $x^2\frac{dy}{dx}+y^{2}e\frac{x(y-x)}{y}=2y(x-y)$ be given by

If $P=\left[\begin{array}{cc}5& -3\\ 111& 336\end{array}\right]$ and $det(-3P^{2013}+P^{2014})={\alpha }^{\alpha }{\beta }^2(1+\gamma +{\gamma }^2)$ where $\alpha ,\beta ,\gamma$ are natural numbers, then

Cos 4x cos 8x - cos 5x cos 9x = 0 if

If a convex polygon has $35$ diagonals, then the number of triangles in which exactly one side is common with that of polygon is

If a tangent to the ellipse $x^2+4y^2=4$ meets the tangents at the extremities of its major axis at $B$ and $C$, then the circle with $BC$ as diameter passes through the point

If the ellipse $\frac{x^2}{4}+y^2=1$ meets the ellipse $x^2+\frac{y^2}{a^2}=1$ in four distinct points and $a=b^2-5b+7,$ then $b$ does not lie in

By using
properties of determinants, show that:

The length of a rectangle is decreasing at the rate of $3\text{cm}/\text{min}$ and its width is increasing at the rate of $2\text{cm}/\text{min}.$ If length is $10\text{cm},$ width is $6\text{cm}$ and $P,A$ represent the perimeter and area of the rectangle respectively, then which of the following is/are true

If $P$ is a point $(x,y)$ on the line $y=-3x$ such that $P$ and the point $(3,4)$ are on the opposite sides of the line $3x-4y=8$, then

Let a vertical tower $AB$ have its end $A$ on the level ground. Let $C$ be the mid-point of $AB$ and $P$ be a point on the ground such that $AP=2AB$. If $\angle BPC=\beta$ , then $\tan \beta$ is equal to:

If $\int \frac{(3\sin \varphi -2)\cos \varphi }{5-{\cos }^2\varphi -4\sin \varphi }d\varphi =3\ln (2-\sin \varphi )+\frac{k}{2+m\sin \varphi }+C$, then which of the following is/are true:

(where $C$ is integration constant)

Let $f\left(x\right)=\{\begin{array}{c}x+1,\text{if}x>0\\ x-1,\text{if}x<0\end{array}$ Prove that $li{m}_{x\to 0}$ f(x)does not exist.

The locus of a point which moves such that the sum of squares of its distance from the point $A(1,2,3),B(2,-3,5)$ and $C(0,7,4)$ is equal to $120,$ is

Consider $f\left(x\right)=\frac{(x+a{)}^2}{(a-b)(a-c)}+\frac{(x+b{)}^2}{(b-a)(b-c)}+\frac{(x+c{)}^2}{(c-a)(a-b)}$ (where a, b, c are distinct real number). If ‘p’ denotes the number of natural number in the range of f(x), then unit digit of $(p+8{)}^{2015}$ is ___

The value of ${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}{2}^{sinx}dx+{\int }_{\frac{5}{2}}^{4}si{n}^{-1}lo{g}_2(x-2)dx$ is equal to