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Find $\frac{dy}{dx}$, if x and y are connected parametrically by the equations given in questions without eliminating the parameter.

$x=asec\theta ,y=btan\theta .$

If a hyperbola passes through the point P $(\sqrt{2},\sqrt{3})$ and has foci $(\pm 2,0),$ then the tangent to this hyperbola at P also passes through the point

The number of natural numbers less than ${10}^7,$ whose sum of digits is equal to $6,$ is

Given parabola $y^2=4ax$, find the equation of Normal which will interest the normal at (8, 8) and the parabola at the same point.

The normal a point $(b{t}_1^2,2b{t}_1)$ on a parabola meets the parabola again in the point $(b{t}_2^2,2b{t}_2)$ then

Let $f\left(n\right)=[\frac{1}{4}+\frac{n}{25}]n$, where $\left[x\right]$ denotes the greatest integer less than or equal to $x$, then $\sum _{n=1}^{50}f\left(n\right)$ is equal to

1)if y=e^3logx, then show that dy\dx=3x²

The length of the latus-rectum of the parabola $169\left\{\right(x-1{)}^2+(y-3{)}^2\}=(5x-12y+17{)}^2$ is

The equation of one of the curve passing through $(1,4)$ and satisfying the differential equation ${\left(\frac{dy}{dx}\right)}^2=\frac{-x-(x+y)\frac{dy}{dx}}{y}$ is

Let the angle between the two curves $y=2\sqrt{x}$ and $x=2\sqrt{y}$ is $\theta$. If $\theta \ne \frac{\pi }{2}$ and it is acute. then $\sqrt{sin\theta +sin3\theta +sin5\theta }$ =

$\begin{array}{cccc}& \text{Column I}& & \text{Column II}\\ \text{(a)}& \text{The minimum value of}{9}^3{27}^{\text{cos}2x}{81}^{\text{sin}2x}\text{is}& \text{(p)}& 1\\ \text{(b)}& \text{Number of solutions of the equation}co{s}^{7}x+si{n}^{4}x=1,xϵ[0,2\pi ]& \text{(q)}& 2\\ \text{(c)}& \text{Value of a for which the equation}{a}^2-2a+se{c}^2\pi (a+x)=0\text{has a solution}& \text{(r)}& 3\\ \text{(d)}& \text{If cos (Psin x) = sin (P cos x), then the minimum possible value of}\frac{4\sqrt{2}}{\pi }\text{P is}& \text{(s)}& 4\end{array}$

Which of the following is correct?

The values of x which satisfy the inequation $2^{\frac{1}{co{s}^{2}x}}\sqrt{y^2-y+\frac{1}{2}}\le 1$ is

If P is related to Q and S is related to T in a certain way, to which of the following would V be related to following the same pattern?

Which of the following is incorrect?

The set of positive real values of the parameter 'a' for which the equation |sin2x|-|x|-a=0 does not have any real solution is

${lim}_{x\to a}\frac{x^{\frac{5}{7}}-a^{\frac{5}{7}}}{x^{\frac{2}{7}}-a^{\frac{2}{7}}}$

If the system of linear equations
$x+y+z=5$
$x+2y+3z=9$
$x+3y+\alpha z=\beta$
has infinitely many solutions, then $\beta -\alpha$ equals:

$$\begin{array}{cc}coloumn1& coloumn2\\ a& p)x\\ b& q)x^3\\ c& r)x^5\end{array}$$

If X and y are two variates connected by the relation $Y=\frac{aX+b}{c}$ and Var(X) = \sigma ^2\), then wrote the expression for the standard deviation of Y.

A $\equiv$ (cos $\theta$, sin $\theta$), B $\equiv$ (sin $\theta$, – cos $\theta$) are two points. The locus of the centroid of $\Delta$OAB, where ‘O’ is the origin is

If the angles of a triangle are in A.P., then the measures of one of the angles in radians is

Using properties of determinants prove the following questions.

$\left|\begin{array}{ccc}sin\alpha & cos\alpha & cos(\alpha +\delta )\\ sin\beta & cos\beta & cos(\beta +\delta )\\ sin\gamma & cos\gamma & cos(\gamma +\delta )\end{array}\right|=0$

If $\frac{si{n}^{4}x}{2}+\frac{co{s}^{4}x}{3}=\frac{1}{5}$, then

$\text{Solve the following equations :}\left(i\right)sin\theta +cos\theta =\sqrt{2}\left(ii\right)\sqrt{3}cos\theta +sin\theta =1\left(iii\right)sin\theta +cos\theta =1\left(iv\right)cosec\theta =1+cos\theta \left(v\right)(\sqrt{3}-1)cos\theta +(\sqrt{3}+1)sin\theta =2$

If a set contains n elements, then write the number of elements in its power set.

If a and b denote the sum of the coefficients of $x^n$ in the expansions of $(1-3x+10x^2{)}^n$ and $(1+x^2{)}^n$ respectively, then write the relation between a and b.

The value of $\underset{n\to \infty }{lim}\frac{1}{n}\{{\sec }^2\frac{\pi }{4n}+{\sec }^2\frac{2\pi }{4n}+.......+{\sec }^2\frac{n\pi }{4n}\}$ is

The sum of roots of ${\sin }^{2}x-5\sin x\cos x+2=0$, where $x\in [0,2\pi ]$ is

The negation of the statement $\sim p\wedge (p\vee q)$ is

The solution set of $\frac{3}{x+1}-\frac{2}{x-1}<0$ is

The distance between the foci of the ellipse $3x^2+4y^2=48$ is

Let $z_1=10+6i$ and $z_2=4+6i$. If $z$ is any complex number such that the argument of $(z-z_1)/(z-z_2)$ is $\pi /4$, then $|z-7-9i|$ is

The sum of the series $1+3x+5x^2+7x^3+\dots$ upto $n$ terms is

The expression $\frac{x^n-y^n}{x^2-y^2}$ is equal to
(where $x^2\ne y^2$ and $n$ is even natural number greater than $6$)

Three numbers which are relatively prime to each other has a product of $900,$ then their sum is

If $\sum _{r=1}^n{t}_r=\sum _{k=1}^n\sum _{j=1}^k\sum _{i=1}^j\left(2\right)$, then $t_5$ is

Which of the following is "CORRECT" option?
$$\begin{array}{cc}\text{List I}& \text{List II}\\ \text{(a) The coordinates of a point on the line}& \\ x=4y+5,z=3y-6\text{at a distance}3\text{from}& \\ \text{the point}(5,3,-6)\text{is/are}& \text{(p)}(-1,-2,0)\\ \text{(b) The plane containing the lines}& \\ \frac{x-2}{3}=\frac{y+3}{5}=\frac{z+5}{7}& \\ \text{and parallel to}\hat{i}+4\hat{j}+7\hat{k}\text{has}& \\ & \text{(q)}(5,0,-6)\\ \text{(c) A line passes through two points}A(2,-3-,1)& \\ \text{and}B(8,-1,2)\text{. The coordinates of a point}& \\ \text{on this line farthest to the origin and at a}& \\ \text{distance of}14\text{units from}A\text{is}& \text{(r)}(2,5,7)\\ \text{(d) The coordinates of the foot of the perpendicular}& \\ \text{from the point}(3,-1,11)\text{on the line}& \\ \frac{x}{2}=\frac{y-2}{3}=\frac{z-3}{4}\text{is/are}& \text{(s)}(14,1,5)\end{array}$$

Find the number of (i) diagonals (ii) Triangles formed in a decagon.

If A(-1,1) and B(2,3) are two fixed points, find the locus of a point P so that the area of $\Delta PAB$ =8sq.units.