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RS is the chord of contact of the point P on a circle center at O. if $m_1=slopeof\overline{RS}and{m}_2=slopeof\overline{OP}$

Draw the graph of sgn [x], sgn is the signum function

The vertices of a triangle are (2,0) (0,2) (4,6) then the equation of the median through the vertex (2,0) is

For f function y = f(x) if we have f^'(c) = 0 and f^" > 0 then x = c is a point of

If m^2 + m'^ 2 + 2mm' cos theta = 1 , n^2+ n' ^2 + 2nn' cos theta = 1 and mn + m'n' + (mn' + m'n) cos theta = 0, prove that m^2 + n^2 = cosec^2 theta .

Consider a family of circles which are passing through the point (-1, 1) and are tangent to the x-axis. If (h, k) are the coordinates of the centre of the circles, then the set of values of k is given by the interval

The number of real values of m for which the equation $z^3+(3+i)z^2-3z-(m+i)=0$ has at least one real root is

A(1, 0) and B(0, 1) are two fixed points on the circle $x^2+y^2=1$. C is a variable point of this circle. As C moves, the locus of orthocenter of the triangle ABC is

$\overrightarrow{a}$ and $\overrightarrow{c}$ are unit vectors and $|\overrightarrow{b}|=4.$ The angle between $\overrightarrow{a}and\overrightarrow{c}isco{s}^{-1}\left(\frac{1}{4}\right).$
If $\overrightarrow{b}-2\overrightarrow{c}=\lambda \overrightarrow{a}$, then $\lambda$ is

The value of $\underset{x\to 0}{lim}\frac{1n(1+\{x\left\}\right)}{\left\{x\right\}}$ is (where {x} denotes the fractional part of x)

If $tan{\theta }_1,tan{\theta }_2,tan{\theta }_3$ and $tan{\theta }_4$ are the roots of the equation $x^4-x^{3}sin2\beta +x^{2}cos2\beta -xcos\beta -sin\beta =0$ then $tan({\theta }_1+{\theta }_2+{\theta }_3+{\theta }_4)=$

If $y={\tan }^{-1}\left(\frac{\log \frac{e}{x^3}}{\log \left(e{x}^3\right)}\right)+{\tan }^{-1}\left(\frac{3+3\log x}{1-9\log \left(x\right)}\right)$,then $\frac{d^{2}y}{d{x}^2}=$

Let $a_1,a_2,a_3,\dots ,a_n$ be the integers which are not included in the range of $f\left(x\right)=\frac{40}{(x-5)(x+1)},$ where $a_i>a_{i+1}$ for each $i=1,2,3,\dots ,n$. If $\sum _{i=1}^n$ ${}^{5-a_i}{C}_{i-1}={}^x{C}_y$, then $x+y$ can be

Which of the following is/are correct regarding their fundamental period?

Let $\frac{\sin x}{a}=\frac{\cos x}{b}=\frac{\tan x}{c}=2,$ where $0<x<\frac{\pi }{2}$ and $R=bc+\frac{1}{2c}+\frac{2a}{1+2b}.$ Then the minimum value of $R$ is

A sector $OABO$ of central angle $\theta$ is constructed in a circle with centre $O$ and radius $6.$ The radius of the circle that is circumscribed about the triangle $OAB,$ is

If P = $\left(\begin{array}{ccc}1& \propto & 3\\ 1& 3& 3\\ 2& 4& 4\end{array}\right)$ is the adjoint of a 3x3 matrix A and det(A)=4,then α is equal to

The tangents drawn from the origin to the circle $x^2+y^2-2rx-2hy+h^2=0$ are perpendicular then sum of all possible values of $\frac{h}{r}$ is ___

If the function $g\left(t\right)={\int }_{2t}^{t^2}co{t}^{-1}\left|\frac{1+x}{(1+t{)}^2-x}\right|dx,then\frac{g\left(5\right)}{g\left(3\right)}$ is equal to ___.

1. 5

   
   

Minimum distance between the curve $y^2=4x$ and $x^2+y^2-12x+31=0$ is

${lim}_{x\to a}\frac{sin\sqrt{x}-sin\sqrt{a}}{x-a}$

The vertices of a triangle are A(10,4), B(-4,9), and (-2,-1). Find the equation of its altitude which passes through (10,4)

If two lines having slopes $m_1$ and $m_2$ are parallel to each other, then which of the following is correct?

The value of integral ${\int }_0^{\log 5}\frac{e^x\sqrt{e^x-1}}{e^x+3}dx=$

If the area (in sq. units) bounded by the parabola $y^2=4\lambda x$ and the line $y=\lambda x,\lambda >0$, is $\frac{1}{9}$ , then $\lambda$ is equal to :

Match the following by appropriately matching the lists based on the information given in Column I and Column II​​​​​​
$\begin{array}{cccc}& ColumnI& & ColumnII\\ a.& \text{If}a,b,c\text{are in}G.P.,\text{then}& \text{p.}& A.P.\\ & {\log }_{a}10,{\log }_{b}10,{\log }_{c}10\text{are in}& & \\ b.& \text{If}\frac{a+b{e}^x}{a-b{e}^x}=\frac{b+c{e}^x}{b-c{e}^x}=\frac{c+d{e}^x}{c-d{e}^x},& \text{q.}& H.P.\\ & \text{then}a,b,c,d& & \\ c.& \text{If}a,b,c\text{are in}A.P.;& \text{r.}& G.P.\\ & a,x,b\text{are in}G.P.\text{and}& & \\ & b,y,c\text{are in}G.P.,& & \\ & \text{then}{x}^2,b^2,y^2\text{are in}& & \end{array}$

If the coefficients of three consecutive terms in the expansion of $(1+x{)}^n$ be 76, 95 and 76, find n.

If $y=y\left(x\right)$ is the solution of the differential equation $\frac{dy}{dx}=(\tan x-y){\sec }^{2}x,$ $x\in (-\frac{\pi }{2},\frac{\pi }{2}),$ such that $y\left(0\right)=0,$ then $y(-\frac{\pi }{4})$ is equal to :

If the coefficients of $2^{nd}$, $3^{rd}$ and $4^{th}$ terms in the expansion of $(1+x{)}^n,n\in N$ are in A.P., then n =

Let $A$ be a non-empty set such that $A\times A$ has $9$ elements among which $(-2,0),(0,2)$ are found. Then $A$ is equal to

Prove that:

$\sqrt{2+\sqrt{2+2cos4\theta }}=2cos\theta$

prove that (1+cotA+tanA)(sinA-cosA)=(secA/cosec^2A)-(cosecA/sec^2A)

Prove that tan A + tan÷ tan a - tan b= sin a + b ÷ Sin A - b

The smallest positive root of the equation $\sqrt{sin(1-x)}=\sqrt{cosx}$ is

For the line segment joining the points A(x1, y₁, z₁) and B(x₂, y₂, z₂)

are

One card is drawn at random from a pack of 52 cards. What's the probability that it is a king or queen ?

The value of ${\sin }^2\frac{\pi }{16}+{\sin }^2\frac{3\pi }{16}+{\sin }^2\frac{5\pi }{16}+{\sin }^2\frac{7\pi }{16}$
is

The value of k for which the points (2,-3), (k,-1) and (0,4) are collinear, is .

If $|z-3i|\le 5$ then the minimum value of $|z+2|$ will be