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Determine order and degree(if defined)
of differential equation

Let points $A(x,3,z),B(1,-1,1)$ and $C(\frac{5}{3},\frac{5}{3},-\frac{7}{3})$ are such that $\overrightarrow{OC}$ divides $\overrightarrow{AB}$ internally in $1:2,$ then which of the following is correct ?

(where $O$ is origin)

The value(s) of $x$ satisfying the equation $||x-3|-4|=3$, is/are

The range of the function ${\sin }^{2n}x+{\cos }^{2n}x;x\in R,n\in N$ is

Three numbers are choosen at random without replacement from {1, 2, 3, ....8}. The probability that their minimum is 3, given that their maximum is 6, is

Prove that
sin (n + 1)x sin (n + 2)x + cos (n
+ 1)x cos (n + 2)x = cos x

The number of positive integral values of $k$ for which the equation $k=|x+|2x-1||-|x-|2x-1||$ has exactly three real solution is

An analytic function of a complex variabel $z=x+iy$ is expressed as $f\left(x\right)=u(x,y)+iv(x,y)$ where $i=\sqrt{-1}$. If $u=xy,$ the expression for $v$ should be

In a $△ABC$, if $\cos A\cos B\cos C=\frac{\sqrt{3}-1}{8}$ and $\sin A\sin B\sin C=\frac{3+\sqrt{3}}{8}$, then

The value of $\tan A\tan B+\tan B\tan C+\tan C\tan A$ is

The mean
of the numbers obtained on throwing a die having written 1 on three
faces, 2 on two faces and 5 on one face is

(A) 1 (B) 2 (C) 5 (D)

If the tangent at the point P on the circle $x^2$ + $y^2$ + 6x + 6y = 2 meets the line 5x - 2y + 6 = 0 at a point Q on the y-axis, then length of PQ = _______

The number of $6$ digit numbers that can be formed using the digits $0,1,2,5,7$ and $9$ which are divisible by $11$ and no digit is repeated, is :

The number of numbers between $3000$ and $4000$ which are divisible by $5$, without repetition using digits $3,4,5,6,7,8$ is

If ${1690}^{2608}+{2608}^{1690}$ is divided by $7$, then the remainder is

The sum of rational terms in the expansion of ${(\sqrt{2}+3^{1/5})}^{10}$ is

If the points (-1, 3, 2), (-4, 2, -2) and $(5,5,\lambda )$ are collinear, then $\lambda =$

If $sin6\theta =32co{s}^5\theta sin\theta -32co{s}^3\theta sin\theta +3x,$ then x=

If $a\cos (x+y)=\sin (y-x)$ for $a\in R-\{\pm 1\},$ then $\frac{1}{1+a\sin 2x}+\frac{1}{1-a\sin 2y}=\frac{b}{c-a^2}.$ The value of $b+c$ is

There are $8$ events that can be scheduled in a week. Then total number of ways that these $8$ events are scheduled on exactly $6$ days of a week is given by $266\times k!,$ where $k\in N.$ The value of $k$ is

Find the slope of the tangent to the
curve y = x³ − 3x + 2 at the
point whose x-coordinate is 3.

If $10$ different balls has to placed in $4$ distinct boxes at random, then the probability that two of these boxes contain exactly $2$ and $3$ balls is :

Let $AD$ be a median of the $△ABC$. If $AE$ and $AF$ are medians of the triangle $ABD$ and $ADC$, respectively, and $AD=m_1,AE=m_2,AF=m_3$, then $\frac{a^2}{8}$ is equal to

A set $X$ representing all the multiples of $3$ greater than $3$ can be written in the roster form as .