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Let the complex numbers $z_1,z_2$ and $z_3$ be the vertices of an equilateral triangle . Let $z_0$ be the circumcentre of the triangle , then $z_1^2+z_2^2+z_3^2=$

Solve the following systems of linear inequations graphically :

$\left(i\right)2x+3y\le 6,3x+2y\le 6,x\ge 0,y\ge 0\left(ii\right)2x+3y\le 6,x+4y\le 4,x\ge 0,y\ge 0\left(iii\right)x+y\ge 1,x+2y\le 8,2x+y\ge 2,x\ge 0,y\ge 0\left(iv\right)x+y\ge 1,7x+9y\le 63,y\le 5,x\ge 0,y\ge 0\left(v\right)2x+3y\le 35,y\ge 3,x\ge 2,x\ge 0,y\ge 0$

If $-3\le |x|<7,$ then $x$ can be represented on the number line by

There are 4 torn books in a lot consisting of 10 books. If 5 books are seleceted at random what is the probability of presence of 2 torn books among the selected?

A quadratic equation whose difference of roots is $3$ and the sum of the squares of the roots is $29,$ is given by

Two manufacturers Ram and Shyam manufacture only three wooden items; bed, table and sofa. The sales (in rupees) of these three items by both of manufacturers in months of June and July are given by the following matrices $A$ and $B$ respectively.



Sale in June

$\begin{array}{ccccccc}& & & \text{Bed}& \text{Table}& \text{Sofa}& \end{array}A=\left[\begin{array}{ccc}5000& 10000& 15000\\ 25000& 15000& 5000\end{array}\right]\begin{array}{c}\text{Ram}\\ \text{Shyam}\end{array}$



Sale in July

$\begin{array}{ccccccc}& & & \text{Bed}& \text{Table}& \text{Sofa}& \end{array}B=\left[\begin{array}{ccc}2500& 5000& 3000\\ 10000& 5000& 5000\end{array}\right]\begin{array}{c}\text{Ram}\\ \text{Shyam}\end{array}$



The combined sales in June and July for each manufacturer in each item is given by



[2 marks]

With initial condition x(1) = 0.5, the solution of the differential equation, t$\frac{dx}{dt}+x=t$ is

Let $P\left(k\right)=(1+cos\frac{\pi }{4k})(1+cos\frac{(2k+1)\pi }{4k})(1+cos\frac{(2k-1)\pi }{4k})(1+cos\frac{(4k-1)\pi }{4k})$ then which of the following is/are correct?

Solution of $\int \frac{x^4+3}{\sqrt{x^8+x^6-3x^4}}dx,x>0$ for arbitrary constant of integration $K$ is

The number of value(s) of $\theta \in [0,2\pi ]$ satisfying the equation $\left({\log }_{\sqrt{5}}\tan \theta \right)\sqrt{{\log }_{\tan \theta }5\sqrt{5}+{\log }_{\sqrt{5}}5\sqrt{5}}=-\sqrt{6}$ is

Identify ordinary differential equations

Show that the vectors $2\hat{i}-\hat{j}+\hat{k}$, $\hat{i}-3\hat{j}-5\hat{k}$ and $3\hat{i}-4\hat{j}-4\hat{k}$ form the vertices of a right angled triangle.

2 boys and 2 girls are in room P and 1 boy 3 girls are in room Q. Write the same space for the experiment in Which a room is selected and then a person

If $I_1=\underset{0}{\overset{\pi /2}{\int }}\cos \left(\sin x\right)dx,I_2=\underset{0}{\overset{\pi /2}{\int }}\sin \left(\cos x\right)dx$ and $I_3=\underset{0}{\overset{\pi /2}{\int }}\cos xdx$, then

If $\cot \frac{2x}{3}+\tan \frac{x}{3}=\text{cosec}\frac{kx}{3}$, then the value of $k$ is

Let $F_1(x_1,0)$ and $F_2(x_2,0),$ for $x_1<0$ and $x_2>0,$ be the foci of the ellipse $\frac{x^2}{9}+\frac{y^2}{8}=1.$ Suppose a parabola having vertex at the origin and focus at $F_2$ intersects the ellipse at point $M$ in the first quadrant and at point $N$ in the fourth quadrant.

If the tangents to the ellipse at $M$ and $N$ meet at $R$ and the normal to tha parabola at $M$ meets the $x-$axis at $Q$, then the ratio of area of the triangle $MQR$ to area of the quadrilateral $M{F}_{1}N{F}_2$ is

The sum of squares of the intercepts on the coordinate axes made by the tangent to $x^{1/3}+y^{1/3}=a^{1/3}at(\frac{a}{8},\frac{a}{8})$ is 2, and a > 0 then a = ____

Show that
the vectorsform
the vertices of a right angled triangle.

Which term of the progression 0.004, 0.02, 0.1, .... is 12.5 ?

Let $\frac{dy}{dx}+y=f\left(x\right)$ where $y$ is a continuous function of $x$ with $y\left(0\right)=1$ and $f\left(x\right)=\{\begin{array}{cc}{e}^{-x},& 0\le x\le 2\\ e^{-2},& x>2\end{array}$.
Which of the following hold(s) good?

If the sum ${\sum }_{k-1}^{\infty }\frac{1}{(k+2)\sqrt{k}+k\sqrt{k+2}}=\frac{\sqrt{a}+\sqrt{b}}{\sqrt{c}}$ where $a,b,cϵN$ and lie in [1, 15], then a+b+c equals to

If $\tan \theta +\sin \theta =m$ and $\tan \theta -\sin \theta =n,$ then which of the following relation between $m,n$ is correct?

An urn contains 7 red and 4 blue balls. Two balls are drawn at random with replacement. If the probability of getting 2 red balls is P(A), find $121\times P\left(A\right)$.

On a particular day, seven persons pick six different books, one each, from different counters at a public library. At the closing time, they arbitrarily put their books at vacant counters. The probability that at least four books are at their previous places is