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Solve 5x - 3 < 7 when

(i) x is an integer

(ii) x is a real number

Find the second order derivative of the given functions.

$e^{x}sin5x$

For the following question verify that the given function (explicit or implicit) is a solution of the corresponding differential equation.

$y=\sqrt{a^2-x^2}andx+y\frac{dy}{dx}=0(y\ne 0).$

Let $f:R\to R$ be defined as f(x)=3x. Choose the correct answer.
(a)f is one-one onto
(b)f is many -one onto
(c) f in one-one but not onto
(d) f is neither one-one nor onto

Find the perpendicular distance of $(1,1)$ from $x-y+\sqrt{2}=0$.

If log 2 = 0.3010, then the value of log50 is

How many ways can Adarsh distribute 17 identical apples among 6 students if he can also keep some apples for himself and each student should get at least one apple and not more than 6?

The perimeter of a triangle is 20 and the points (-2, -3) and (-2, 3) are two of the vertices of it. Then the locus of third vertex is :

The length of the transverse axis of a hyperbola is 7 and it passes through the point (5, -2). The equation of the hyperbola is

Let $AB$ be a line segment with midpoint $C$, and $D$ be the midpoint of $AC$. Let $C_1$ be the circle with diameter $AB$, and $C_2$ be the circle with diameter $AC$. Let $E$ be a point on $C_1$ such that $EC$ is perpendicular to $AB$. Let $F$ be a point on $C_2$ such that $DF$ is perpendicular to $AB$, and $E$ and $F$ lie on opposite sides of $AB$. Then the value of $\sin \angle FEC$ is

Match the following:

$$\begin{array}{cc}\text{Column A}& \text{Column B}\\ \text{1. {x: x}\in \text{R, -5 < x}\le \text{15 }}& \text{A. 1}\\ 2.A=ThepowersetP\left(A\right)has\underset{–}{}elements& B.8\\ 3.UniversasetofRationalNumbers\&IrrationalNumbers& C.RealNumbers\\ 4.No.ofelementsinPowerSetofA=\{a,b,c\}& D.Integers\\ & E.(-5,15]\\ & F.[-5,15)\end{array}$$

Consider the letters of the word $MATHEMATICS$.
Possible number of words in which no two vowels are together is

If $lo{g}_{10}x=ythenlo{g}_{1000}{x}^2$ is equal to

If

${\Delta }_1$, $\left|\begin{array}{ccc}1& 1& 1\\ a& b& c\\ a^2& b^2& c^2\end{array}\right|$, ${\Delta }_2$ = $\left|\begin{array}{ccc}1& bc& a\\ 1& ca& b\\ 1& ab& c\end{array}\right|$ then

If $I_n=\int (lnx{)}^{n}dxthen{I}_n+n{I}_{n-1}$

${lim}_{0\to 0}\frac{sin4\theta }{tan3\theta }$

Let the functions $f\left(x\right)$ and $g\left(x\right)$ be inverse function of each other. If both th functions are differentiable at $x=c$, then $g^{\prime }\left(f\right(c\left)\right)$ =

The general solution of differential equation $\frac{dy}{dx}=logx$ is

Evaluate:
$\left|\begin{array}{ccc}a+x& y& z\\ x& a+y& z\\ x& y& a+z\end{array}\right|$

If $\tan A,\tan B,\tan C$ are the roots of the equation $x^3-3x^2-2x+1=0,$ then the value of ${\sin }^2(A+B+C)+\sin (A+B+C)\cos (A+B+C)$ is

If $f\left(x\right)$ is a quadratic polynomial such that graph of $y=f\left(x\right)$ touches at $(4,0)$ and intersects the positive $y-$axis at $4$, then which of the following is/are correct?

The equation x-$\frac{2}{x-1}$=1 -$\frac{2}{x-1}$ has

The relation on the set $A=\{x:1<|x|\le 4,x\in Z\}$ is defined by $R=\left\{\right(x,y):y=|x|\}.$ Then the number of elements in the power set of $R$ is

If the coefficients of $x^3$ and $x^4$ in the expansion of $(1+ax+b{x}^2$) $(1–2x{)}^{18}$ in powers of $x$ are both zero, then $(a,b)$ is equal to

TanX+Tan2X+TanX.Tan2X =1

The general solution of the differential equation $(1+e^{\frac{x}{y}})dx+(1-\frac{x}{y})e^{\frac{x}{y}}dy=0$ is ($c$ is an arbitary constant)

By what factor does the capacitance of a metal sphere increase if it's volume is trippled?

If $\frac{a+bx}{a-bx}$ = $\frac{b+cx}{b-cx}$ = $\frac{c+dx}{c-dx}$(x ≠ 0), then a, b, c, d are in:

Let $z=x+iy$ be a complex number, where $x$ and $y$ are real numbers. Let $A$ and $B$ be the sets defined by $A=z:|z|\le 4$ and $B=z:(z+\overline{z})-i(z-\overline{z})\ge 8$. The area of region $A\cap B$ is

Find the ratio in which the line segment joining the points (4, 8, 10) and

(6, 10, -8) is divided by the yz-plane.

Equation $6x^2-5xy+y^2=0$ represents____________________

The latus rectum of an ellipse is 10 and the length of the minor axis is equal to the distance between the foci. The equation of the ellipse is

If the function$f\left(x\right)=\{\begin{array}{c}\frac{kcosx}{\pi -2x},ifx\ne \frac{\pi }{2}\\ 3,ifx=\frac{\pi }{2}\end{array}$

is continues at $x=\frac{\pi }{2}$

$2cos\frac{\pi }{13}cos\frac{9\pi }{13}+cos\frac{3\pi }{13}+cos\frac{5\pi }{13}=$

If $y=mx+4$ is a tangent to both the parabolas, $y^2=4x$ and $x^2=2by,$ then $b$ is equal to

The general solution of the differential equation $x\frac{dy}{dx}+xy=e^{-x}$ is

The range of the function $f\left(x\right)=\frac{x^2-x}{x^2+2x}$ is

If $e_1$ and $e_2$ are the eccentricities of the ellipse , $\frac{x^2}{18}+\frac{y^2}{4}=1$ and the hyperbola, $\frac{x^2}{9}-\frac{y^2}{4}=1$ respectively and $(e_1,e_2)$ is a point on th ellipse , $15x^2+3y^2=k$. Then $k$ is equal to :