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Prove that:
(i) $\frac{cos(A+B+C)+cos(-A+B+C)+cos(A-B+C)+cos(A+B-C)}{sincos(A+B+C)+sincos(-A+B+C)+sincos(A-B+C)-sin\left(cos\right(A+B-C\left)\right)}$
(ii) $sin(B-C)cos(A-D)+sin(C-A)cos(B-D)+sin(A-B)cos(C-D)=0$

Equation of the line perpendicular to the line $6x+2y+7=0$ and which dIvides the line segment joining the points $(5,-3)$ and $(0,2)$ internally in the ratio $7:4$ is

Prove that:

$co{t}^{2}A-ta{n}^{2}A=4cot2Acosec2A$

Let ABCD be tetrahedron with AB = 41, AC = 7, AD = 18, BC = 36, BD = 27 and CD = 13 as shown in figure. Let d be the distance between the midpoints of edges AB and CD, then $\left[\frac{d}{2}\right]$ is
(where [.] denotes greatest integer functions) ___

The function defined by f(x) = $max.\{x^2,(x–1{)}^2,2x(1–x),0\le x\le 1\}$

Let $S$ be the circle in the $xy$-plane defined by the equation $x^2+y^2=4$.



Let $P$ be a point on the circle $S$ with both coordinates being positive. Let the tangent to $S$ at $P$ intersect the coordinate axes at the points $M$ and $N$. Then, the mid-point of the line segment $MN$ must lie on the curve

The rifleman is firing at a distant target and has only $10\%$ chance of hitting it. The least number of rounds he must fire to have more than $50\%$ chance of hitting it at lease once is

If the two lines $(a+b)x-4y+5=0$ and $x+(a-b)y+10=0$ are perpendicular to each other then

The mean deviation from the median is

Examine the applicable of MVT for all three functions.

$f\left(x\right)=\left[x\right]forxϵ[5,9]$

$f\left(x\right)=\left[x\right]forxϵ[-2,2]$

$f\left(x\right)=1-x^{2}forxϵ[1,2]$

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

$y=\sqrt{1+x^2}and{y}^{\prime }=\frac{xy}{1+x^2}$

The solution of the differential equation $\frac{dy}{dx}+\frac{3x^2}{1+x^3}y=\frac{{\text{sin}}^{2}x}{1+x^3}$ is

Solve the following systems of inequalities graphically:

$3x+2y\le 12,x\ge 1,y\ge 2$

Find all pairs of consecutive even positive integers, both of which are larger than 5 such that their sum is less than 23.

Two lines $4x+2y=10$ and $2x-y=20$ are touching a circle whose radius is $\sqrt{5}$ units. Then the equation of the circle which is nearest to the $x$-axis, is

If $z=\sqrt{3}+i$ and $w=3i$
Then $\arg \left(zw\right)$ and $\arg \frac{w}{z}$ is

If g:R -->R be two functions defined as f(x)=|x|+x and g(x)=|x|-x for all x belongs to R. Then find fog and gof.

In a G.P., the first term and common ratio is $a$ and $r$ respectively. If $S_n$ denotes the sum of $n$ terms and $U_n=\sum _{n=1}^n{S}_n$, then $r{S}_n+(1-r)U_n$ is equal to

Let l(n) = 2cosnx, n belong to natural number then l(1) l(n+1) l(n)

The equation of the tangent to the parabola $y^2=4x$ inclined at an angle $\frac{\pi }{4}$ to the positive direction of x – axis is

$\alpha ,\beta$ are the roots of the equation $x^2$ + bx + c = 0 and $\alpha +2,\beta +2$ are the roots of the equation $x^2$ + px + q=0. If the minimum value of the expression $x^2$ + bx + c is +2, find the minimum value of $x^2$ + px + q

Let $f:R\to R$ be the Signum function defined as $f\left(x\right)=\{\begin{array}{cc}1,& x>0\\ 0,& x=0\\ -1,& x<0\end{array}$ and $g:R\to R$ be the

greatest integer function given by g(x) =[x] is greatest integer less than or equal to x. Then, fog and gof coincide in (0,1].

Find range of

f(x)=3cosx+4sinx+10

Find the equation of the circle with
Centre $(\frac{1}{2},\frac{1}{4})$ and radius $\frac{1}{12}$

A hyperbola has focus at origin, its eccentricity is $\sqrt{2}$ and corresponding directrix is $x+y+1=0$. The equation of its asymptotes is/are:

If the equation of the normal is y = mx + c to the parabola $y^2=4ax$, then find the value of 'c' in terms of a and m.

$C_1,C_2$ are two circles of radii a, b (a < b) touching both the coordinate axes and have their centres in the first quadrant. Then the true statements among the following are

$\underset{x\to \infty }{lim}\frac{(2x+1{)}^{40}(4x-1{)}^5}{(2x+3{)}^{45}}$

A coin is tossed five times and outcomes are recorded. How many possible outcomes are there?

How $12x$ square $+7xy-12y$ square equal to $(3x+4y)\times (4x-3)$?

Find the equation of a circle which passes through the origin and cuts off intercepts -2 and 3 from the x-axis and the y-axis respectively.

Find the value of $x^2+y^2$ for which the complex numbers $-3-i{x}^{2}y$ and $x^2+y+4i$ are equal , where x and y are real numbers .

The vertices of a triangle are (2, 7), (10, 8) and (1, 1), the area of the triangle is:

The sum of all the coefficient of those terms in the expansion of $(a+b+c+d{)}^8$ which contains b but not c

What is the value of sin 35⁰-sin55⁰?

The number of ways that a volley ball team of $6$ can be selected out of $10$ players so that $2$ particular players are always included, is

A can cultivate ${\frac{2}{5}}^{th}$ of a field in 6 days and B can cultivate ${\frac{1}{3}}^{rd}$ of the same field in 10 days. Working together A and B can cultivate ${\frac{4}{5}}^{th}$ of the field in -

The value of $\int \frac{(x^4-x{)}^{\frac{1}{4}}}{x^5}dx$ is

If $\frac{\sqrt{5+x}+\sqrt{5-x}}{\sqrt{5+x}-\sqrt{5-x}}=4$, then value of $x$ is