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Find the values of

If $\sin (x^2+x)=\cos (x-x^2),x\in [0,\frac{\pi }{2}],$ then the range of the function $f\left(x\right)=5{\tan }^{4}x-4{\tan }^{3}x+3{\tan }^{2}x-2\cot x+1$ is

What is the distance between point B and point C ?

Let $z_1$ and $z_2$ be two complex numbers such that $\left|\frac{z_1-2z_2}{2-z_1\overline{z_2}}\right|=1$ and $|z_2|\ne 1$. Then the value of $|z_1|$ is

Which of the following will be an imaginary number?

The eccentricity of the ellipse, whose end points of major axis and minor axis are $(\pm \sqrt{5},0)$ and $(0,\pm 1)$ respectively, is

$IfF\left(x\right)=\left[\begin{array}{ccc}cosx& -sinx& 0\\ sinx& cosx& 0\\ 0& 0& 1\end{array}\right],thenF\left(x\right).F\left(y\right)=$

The solution of $\frac{dy}{dx}+\frac{x}{y}=x^2$ is

A wheel makes 360 revolutions per minute. Through how many radians does it turn in 1 second?

Find the equations of the pair of lines through the origin which are perpendicular to the lines represented $6x^2-5xy+y^2=0$

A line (with constant term $k$) perpendicular to $4x+3y+2=0$, is tangent to circle with integral radius (radius as integer). If the ratio of minimum to maximum possible distance from origin to such tangents is $1:3$, then possible value of $k$ such that the radius of circle is minimum

Number of $4$ digit numbers using digits $0,1,2,3,4,5$ which are divisble by $8$, when each digit is used at most once is

Tangent to the ellipse $\frac{x^2}{4}+y^2=1$ at the point $P(\sqrt{2},\frac{1}{\sqrt{2}})$ touches the circle $x^2+y^2=r^2$ at the point $Q.$ Then the length of $PQ$ is

If $\left(\cot {\alpha }_1\right)\left(\cot {\alpha }_2\right)...\left(\cot {\alpha }_n\right)=1$ and $0<{\alpha }_1,{\alpha }_2,...,{\alpha }_n<\frac{\pi }{2}$, then the maximum value of $\left(\cos {\alpha }_1\right)\left(\cos {\alpha }_2\right)...\left(\cos {\alpha }_n\right),$ is

Given that for each $a\in (0,1),$

$\underset{h\to 0^{+}}{lim}\underset{h}{\overset{1-h}{\int }}{t}^{-a}(1-t{)}^{a-1}dt$

exists. Let this limit be $g\left(a\right)$. In addition, it is given that the function $g\left(a\right)$ is differentiable on $(0,1)$.



The value of $g^{\prime }\left(\frac{1}{2}\right)$ is

If $z=4-3i$ is rotated by ${180}^{\circ }$ in the clockwise direction about origin and stretched three times of its original length, then the new complex number is

Hour hand of a clock is near about 4 and minute hand is at 6. How many second divisions are there between the hands in clock wise direction from hour hand?

If the derivative of f(x) with respect to x is $\frac{\frac{1}{2}-si{n}^{2}x}{f\left(x\right)}$ then period of f(x) is

If the coordinates of the vertices of an equilateral triangle with sides of length 'a' are $(x_1,y_1),(x_2,y_2)$ and $(x_3,y_3)$, then
${\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|}^2=\frac{3a^4}{4}$

The equation of the curve passing through $(\frac{{\pi }^2}{4},1),$ which has a solution of the equation as $y^{2}cos\sqrt{x}dx-2\sqrt{x}{e}^{\frac{1}{y}}dy=0$

Consider the family of circles $x^2+y^2-2x-2\lambda y-8=0$ passing through two fixed points A and B. Then the distance between the points A and B, is $\underset{–}{}$

1. 1

In the given figure of a cube,

i) Which edge is the intersection of faces EFGH and BCGF?

ii) Which vertex of the cube do we get by the intersection of the edges EF, FG and FB?

Let $\overrightarrow{a}$ and $\overrightarrow{c}$ be unit vectors and $|\overrightarrow{b}|=4$ with $\overrightarrow{a}\times \overrightarrow{b}=2\overrightarrow{a}\times \overrightarrow{c}.$ The angle between $\overrightarrow{a}$ and $\overrightarrow{c}$ is ${\cos }^{-1}\left(\frac{1}{4}\right).$ If $\overrightarrow{b}-2\overrightarrow{c}=\lambda \overrightarrow{a},$ then $\lambda$ is equal to

If $x\in (-1,\infty )$, then the value of $x$ satisfying ${\tan }^{-1}\frac{1-x}{1+x}=\frac{1}{2}{\tan }^{-1}x$ is

Find the
principal and general solutions of the equation

The cartesian equation of the plane passing through the point $(3,-2,-1)$ and parallel to the vectors $\overrightarrow{b}=\hat{i}-2\hat{j}+4\hat{k}$ and $\overrightarrow{c}=3\hat{i}+2\hat{j}-5\hat{k},$ is

A pair of straight lines through $A(2,7)$ is drawn to intersect the line $x+y=5$ at $C$ and $D.$ If angle between the pair of straight lines is $\frac{\pi }{2},$ then the locus of incentre of $△ACD$ is

The common tangent to the circles $x^2+y^2=4$ and $x^2+y^2+6x+8y-24=0$ intersects the coordinate axes at $A$ and $B$ respectively. If $OA$ and $OB$ are equal to half of the length of the major and minor axes of an ellipse respectively, where $O$ is the origin, then the eccentricity of the ellipse is

Five cards are drawn successively with replacement from a well-shuffled deck of $52$ cards. Then the probability that all the five cards are spades is: