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The general solution of the differential equation $\frac{dy}{dx}=\frac{3x-4y+1}{4x+2y+3}$ is

(where $c$ is constant of integration)

The figure shows a portion of the graph $y=2x–4x^3.$ The line $y=c$ is such that the areas of the regions marked
I and II are equal. If $a,b$ are the $x-$coordinates of $A,B$ respectively, then $a+b$ equals

Range of the function $f\left(x\right)=2x^2+12x+16$ is

Prove that the greatest
integer function defined byis
not

differentiable at x
= 1 and x = 2.

Let f:
N → N be defined by

State
whether the function f is bijective. Justify your answer.

Prove that

Tangents are drawn to the hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1,$ parallel to the straight line $2x–y=1.$ The point of contact of the tangents on the hyperbola are

The sum of $10$ terms of the series $\mathrm{1.3.5}+\mathrm{3.5.7}+\mathrm{5.7.9}+\dots \dots$ is

If x, y, z are real and distinct, then $x^2+4y^2+9z^2-6yz-3zx-2xy$ =

Let the foot of perpendicular from a point $P(1,2,–1)$ to the straight line $L:\frac{x}{1}=\frac{y}{0}=\frac{z}{-1}$ be $N$. Let a line be drawn from $P$ parallel to the plane $x+y+2z=0$ which meets $L$ at point $Q$. If $\alpha$ is the acute angle between the lines $PN$ and $PQ,$ then $\cos \alpha$ is equal to

The area (in sq. units) of an equilateral triangle inscribed in the parabola $y^2=8x$, with one of its vertices on the vertex of this parabola, is:

Let $f:(0,\infty )\to (0,\infty )$ be a differentiable function such that $f\left(1\right)=e$ and $\underset{t\to x}{lim}\frac{t^2{f}^2\left(x\right)-x^2{f}^2\left(t\right)}{t-x}=0$. If $f\left(x\right)=1$, then $x$ is equal to

1. 3

If $x^2$ + 5 = 2x - 4 cos (a + bx) where a, b $ϵ$ (0, 5) is satisfied for alteast one real x, then the minimum value of a + bx is

The angle between the lines xy = 0 is

If $(x+iy{)}^5=p+iq$, then the value of $\frac{2(y+ix{)}^5}{q+ip}$ is

Let $\left[x\right]$ be the greatest integer less than or equal to $x$. Then, at which of the following point(s) the function $f\left(x\right)=x\cos \left(\pi \right(x+\left[x\right]\left)\right)$ is discontinuous?

Given the family of lines, a (3x + 4y + 6) + b(x + y + 2) = 0. The line of the family situated at the greatest distance from the point P(2, 3) has equation:

Let $\overrightarrow{A}$ and $\overrightarrow{B}$ be the two vectors of magnitude 10 unit each. If they are inclined to the X-axis at angles ${30}^0$ and ${60}^0$ respectively, find the resultant.

The equation of the ellipse, whose focus is the point $(-1,1)$, whose directrix is the straight line $x-y+3=0$ and whose eccentricity is $\frac{1}{2}$ is :

Solve the following inequalities and represent the solution
graphically on number line:

3x – 7 > 2(x – 6), 6 – x
> 11 – 2x

The sum of all natural numbers which are divisible by 3 and less than 100 is

The integration factor of differential equation $(1+x^2)\frac{dy}{dx}+2xy=4x^2$ is

Tangents are drawn from any point on the line $x+4a=0$ to the parabola $y^2=4ax$. Then the angle subtended by the chord of contact at the vertex will be .

The two roots of an equation $x^3-9x^2+14x+24=0$ are in the ration 3 : 2. The roots will be

If $g\left\{f\right(x\left)\right\}=|sinx|$ and $f\left\{g\right(x\left)\right\}=(sin\sqrt{x}{)}^2$, then

Two sides of a triangle are 2 and $\sqrt{3}$ and the included angle is ${30}^{\prime }$ then the in-radius r of the triangle is equal

Explain the conversion of angle in degrees, minutes and seconds into radian

Three ladies have brought one child each for admission to a school. The principal wants to interview the six persons one by one subject to the condition that no mother is interviewed before her child. The number of ways in which interviews can be arranged, is

If $\int \frac{{\sec }^2\theta \sin 2\theta d\theta }{\cos 3\theta }=f\left(\theta \right)+C$ for some arbitrary constant $C,$ then $f\left(\theta \right)$ is equal to