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If $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$, then show that $\frac{dy}{dx}.\frac{dx}{dy}=1$

The integrating factor of the differential equation $(x+3y^2)\frac{dy}{dx}=y$ is

Let $A$ be a fixed point $(0,6)$ and $B$ be a moving point $(2t,0)$. Let $M$ be the mid-point of $AB$ and the perpendicular bisector of $AB$ meets the $y$-axis at $C$. The locus of the mid-point $P$ of $MC$ is

Let $\alpha$ and $\beta$ be the roots of $x^2-6x-2=0$ with $\alpha >\beta$. If $a_n={\alpha }_n-{\beta }_n$ for $n\ge 1$, then the value of $\frac{a_{10}-2a_8}{2a_9}$ is

The value of $\underset{0}{\overset{\alpha }{\int }}\sqrt{\frac{1+\cos 2x}{2}}dx,$ where $\alpha \in (\frac{\pi }{2},\pi )$ is

Determine order and degree(if defined)
of differential equation

If $y=(1+x^2){\tan }^{-1}x-x,$ then $\frac{dy}{dx}$ is equal to

The value of $\frac{1}{\cos {290}^{\circ }}+\frac{1}{\sqrt{3}\sin {250}^{\circ }}$ is

If 12 identical balls are to be placed in 3 different boxes, then the probability that a particular box contains exactly 3 balls, is?

If the difference between mean and mode is $63$, then the difference between mean and median is

If integrating factor of $x(1-x^2)dy+(2x^{2}y-y-a{x}^3)dx=0$ is $e^{\underset{}{\overset{}{\int }}Pdx},$ then $P$ is equal to:

Let $f_1:R\to R,$ $f_2:[0,\infty )\to R,$ $f_3:R\to R$ and $f_4:R\to [0,\infty )$ be defined by

$f_1\left(x\right)=\{\begin{array}{c}|x|\text{if}x<0,\\ e^x\text{if}x\ge 0;\end{array}$



$f_2\left(x\right)=x^2;$



$f_3\left(x\right)=\{\begin{array}{c}\sin x\text{if}x<0,\\ x\text{if}x\ge 0;\end{array}$



and



$f_4\left(x\right)=\{\begin{array}{c}{f}_2\left(f_1\right(x\left)\right)\text{if}x<0,\\ f_2\left(f_1\right(x\left)\right)-1\text{if}x\ge 0.\end{array}$



$\begin{array}{cc}ListI& ListII\\ \text{P.}{f}_4\text{is}& \text{1. onto but not one-one}\\ \text{Q.}{f}_3\text{is}& \text{2. neither continuous nor one-one}\\ \text{R.}{f}_2\circ f_1\text{is}& \text{3. differentiable but not one-one}\\ \text{S.}{f}_2\text{is}& \text{4. continuous and not one-one}\end{array}$



which of the following option is correct ?

The distances of the roots of the equation $|\sin {\theta }_1|z^3+|\sin {\theta }_2|z^2+|\sin {\theta }_3|z+|\sin {\theta }_4|=3$, from $z=0$, are

For non zero, a, b, c if $\Delta =\left|\begin{array}{ccc}1+a& 1& 1\\ 1& 1+b& 1\\ 1& 1& 1+c\end{array}\right|=0$, then the value of $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=$

Find the sum of the n terms of the series $1+3+7+15+31+\cdots n$ terms.

Solve the following quadratic equations :

(i)$x^2-(3\sqrt{2}+2i)x+6\sqrt{2}i=0$

(ii)$x^2-(5-i)x+(18+i)=0$

(iii)$(2+i)x^2-(5-i)x+2(1-i)=0$

(iv)$x^2-(2+i)x-(1-7i)=0$

(v)$i{x}^2-4x-4i=0$

(vi)$x^2+4ix-4=0$

(vii)$2x^2+\sqrt{15}ix-i=0$

(viii)$x^2-x+(1+i)=0$

(ix)$i{x}^2-x+12i=0$

(x)$x^2-(3\sqrt{2}-2i)x-\sqrt{2}i=0$

(xi)$x^2-(\sqrt{2}+i)x+\sqrt{2}i=0$

(xii)$2x^2-(3+7i)x+(9i-3)=0$

Let $S_1$ and $S_2$ be circles of radii 1 and r (r > 1) respectively touching the coordinate axes.
Column-1: Conditions between circles $S_1$ and $S_2$
Column-2: Values of r for conditions in Column-1.
Column-3: Number of common tangents between S1 and S2 for conditions in column-1.
$\begin{array}{cccccc}& \text{Column 1}& & \text{Column 2}& & \text{Column 3}\\ \text{(I)}& S_2\text{passes through the centre}& \text{(i)}& 3& \text{(P)}& 1\\ & \text{of}{S}_1.& & & & \\ \text{(II)}& S_1\text{and}{S}_2\text{touch each other}& \text{(ii)}& 2+\sqrt{2}& \text{(Q)}& 2\\ \text{(III)}& S_1\text{and}{S}_2\text{are orthogonal}& \text{(iii)}& 2+\sqrt{3}& \text{(R)}& 3\\ \text{(IV)}& S_1\text{and}{S}_2\text{have longest}& \text{(iv)}& 3+2\sqrt{2}& \text{(S)}& 4\\ & \text{common chord}& & & & \end{array}$

Which of the following options is the only CORRECT combination?

$3x–5y=4$

$9x=2y+7$

Solve the above equations by elimination method.

Which of the following is TRUE for positive real values of $x$

verify mean value theorem for function 2sinx+sin2x on [0,pi]

Shape of the feasible region formed by the following constraints is

2x + 3y ≥ 6, 2x + 3y ≤ 12, x ≥ 0, y ≥ 0

The value of $\int |e^x+1-\sin x|dx$ is

(where $C$ is constant of integration)

Given that, for all real 'x', the expression $\frac{x^2-2x+4}{x^2+2x+4}$ lies between $\frac{1}{3}$ and 3. The values between which the expression $\frac{{9.3}^{2x}+{6.3}^x+4}{{9.3}^{2x}-{6.3}^x+4}$ lies are

Prove the following by using the principle of mathematical induction for all n ∈ N:

The number of noncongruent integer-sided triangles whose sides belong to the set $\{10,11,12,\dots .,22\}$ is

If $(2x-1{)}^{20}-(ax+b{)}^{20}={(x^2+px+q)}^{10}$ holds true for all $x\in R$ where $a,b,p$ and $q$ are real numbers, then the value of $|b|$ is

The number of pairs (x,y) where both x and y are real satisfying x² + y² + 2 = ( 1+x )(1 + y) is

The equation of the plane containing the line 2x - 5y + z =3, x + y + 4z = 5 and parallel to the plane x + 3y + 6z =1 is