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Profit earned from Product X is shown below. Another Product Y also has the same profit but it started selling 1.5 years after start of sale of Product X. What will be the new set of input values for the Product Y?

If $\alpha \overrightarrow{a}+\beta \overrightarrow{b}+\gamma \overrightarrow{c}=\overrightarrow{0},$ then the resultant of $(\overrightarrow{a}\times \overrightarrow{b})\times \left\{\right(\overrightarrow{b}\times \overrightarrow{c})\times (\overrightarrow{c}\times \overrightarrow{a}\left)\right\}$ is

(where $\alpha ,\beta$ and $\gamma$ are not simultaneously zero.)

If $I=\int \frac{\sin x+{\sin }^{3}x}{\cos 2x}dx=P\cos x+Q\log \left|\frac{\sqrt{2}\cos x-1}{\sqrt{2}\cos x+1}\right|+R,$ then

A unit vector along normal to the curve $y=x^2{e}^x$ at origin, is

A pack of cards has one card missing. Two cards are drawn randomly and are found to be spades. The probability that the missing card is not a spade, is:

Total number of ways in which 256 identical balls can be placed in 16 numbered boxes (1,2,3,.,16) such that $r^{th}$ box contains at least r balls is $(1\le r\le 16)$

Let $f:R\to R$ be a function which satisfies $f(x+y)=f\left(x\right)+f\left(y\right)\forall x,y\in R$. If $f\left(1\right)=2$ and $g\left(n\right)=\sum _{k=1}^{(n-1)}f\left(k\right),n\in N$ then the value of $n$, for which $g\left(n\right)=20$, is:

1. 96

$f\left(x\right)=-3x^2+2x+5$ is concave at

Question 2

The number of diagonals in a septagon is

a) 21

b) 42

c) 7

d) 14

The mean and variance of $8$ observations are $10$ and $13.5$ respectively. If $6$ of these observations are $5,7,10,12,14,15,$ then the absolute diffrence of the remaining two observations is

$\int si{n}^{3}x.co{s}^{4}xdx$

If $u+iv=(x+iy{)}^3$ then $\frac{u}{x}+\frac{v}{y}=$

Let $n\ge 2$ be a natural number and $0<\theta <\frac{\pi }{2}$.
Then $\int \frac{{({\sin }^n\theta -\sin \theta )}^{\frac{1}{n}}\cos \theta }{{\sin }^{n+1}\theta }d\theta$ is equal to :
(where $C$ is constant of integration)

Which among the following function is not differentiable at $x=0$

Let $\alpha ,\beta$ be the roots of quadratic equation $a{x}^2+bx+c=0.$ If $1,\alpha +\beta ,\alpha \beta$ are in arithmetic progression and $\frac{1}{\alpha },\frac{1}{2},\frac{1}{\beta }$ are also in arithmetic progression, then the value of $\frac{{\alpha }^2+{\beta }^2-2{\alpha }^2{\beta }^2}{{\alpha }^2+{\beta }^2}$ is

If $\alpha ,\beta$ are the roots of the equation $\tan x+\sec x=2\cos x$ where $x\in [0,2\pi )$, then the value of $|\alpha -\beta |$ is

If $6^{83}+8^{83}$ is divided by $49,$ then the remainder is

Seven villages A, B, C, D E, F and G are situated as follows:

E is 2 km to the west of B

F is 2 km to the north of A

C is 1 km to the west of A

D is 2 km to the south of G

G is 2 km to the east of C

D is exactly in the middle of B and E.

Q68. Which two villages are the farthest from one another?

The equation of a circle whose radius is $8$ units and which touches both $x-$axis and $y-$axis is

Find the coordinates of focus, equation of directrix and length of the latus rectum for $y^2=-8x$.

If a,b,c are all different and $\left|\begin{array}{ccc}a& a^3& a^4-1\\ b& b^3& b^4-1\\ c& c^3& c^4-1\end{array}\right|=0,$ then the value of $abc(ab+bc+ca)$ is equal to:

The values of ‘a’ for which $(a^2-1)x^2+2(a-1)x+2$ is positive for any x are

Let $O$ be the origin, and $\overrightarrow{OX},\overrightarrow{OY},\overrightarrow{OZ}$ be three unit vectors in the directions of the sides $\overrightarrow{QR},\overrightarrow{RP},\overrightarrow{PQ},$ respectively, of a triangle $PQR.$



If the triangle $PQR$ varies, then the minimum value of

$\cos (P+Q)+\cos (Q+R)+\cos (R+P)$

Let $\alpha =\cos \frac{2\pi }{7}+\cos \frac{4\pi }{7}+\cos \frac{6\pi }{7}$ and $\beta ={\sin }^2\frac{\pi }{8}+{\sin }^2\frac{3\pi }{8}+{\sin }^2\frac{5\pi }{8}+{\sin }^2\frac{7\pi }{8}$. The angle between the two lines whose slopes are $\alpha ,\beta$ is

If ABC is an acute-angled triangle with circumcenter O and Orthocenter H. If AO = AH, then angle A = .

${lim}_{x\to \frac{\pi }{4}}\frac{\sqrt{2}cosx-1}{cotx-1}$ is equal to

1. 1