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State whether the following statement are true or false. Justify
(i) for an arbitrary binary operation $\ast$ on a set N, $a\ast a=a\forall a\in N$.

(ii) If $\ast$ is a commutative binary operation on N, then $a\ast (b\ast c)=(c\ast b)\ast a.$

Let $f$ is a continuous function for all $x\in R$ and $f\left(x\right)=f(x+T),T>0.$ If $I=\underset{0}{\overset{T}{\int }}f\left(x\right)dx,$ then $\underset{2}{\overset{2+10T}{\int }}f\left(\frac{x}{2}\right)dx$ is equal to

A variable straight line $AB$ divides the circumference of the circle $x^2+y^2=25$ in the ratio $1:2.$ If a tangent $CD$ is drawn to the smaller arc parallel to $AB,$ such that $ABCD$ is a rectangle $($as shown in the figure$),$ then locus of $C$ and $D$ is

Line passing through the points $2\overrightarrow{a}+3\overrightarrow{b}-\overrightarrow{c},3\overrightarrow{a}+4\overrightarrow{b}-2\overrightarrow{c}$ intersects the line through the points$\overrightarrow{a}-2\overrightarrow{b}+3\overrightarrow{c},\overrightarrow{a}-6\overrightarrow{b}+6\overrightarrow{c}$ at $P$. Then position vector of $P$ is

For the hyperbola $\frac{x^2}{co{s}^2\alpha }-\frac{y^2}{si{n}^2\alpha }=1$

Which one of the following remain constant with change of $\alpha$ ?

If $|Z_1+Z_2|$ = $\left|Z_1\right|+\left|Z_2\right|$, then find the value of arg $\left(\frac{Z_1}{Z_2}\right)$

Two ships A and B are sailing straight away from the fixed point O along the routes such that $\angle AOB={60}^{\circ }$ always.At a certain instance, OA = 4 km, OB = 3km and ship A is sailing at the rate of 20km/hr and ship B is sailing at the rate of 30km/hr. Then the distance between A and B is changing at the rate (in km/hr) :

Let $\alpha >0,\beta >0$ be such that ${\alpha }^3+{\beta }^2=4.$ If the maximum value of the term independent of $x$ in the binomial expansion of ${(a{x}^{\frac{1}{9}}+\beta x^{\frac{1}{6}})}^{10}$ is $10k$, then $k$ is equal to:

The modulus of the complex number $\left(\frac{3+4i}{1-2i}\right)$ is

The number of ways in which we can get a sum of $11$ by throwing three dice is :

Let $a_1,a_2,\dots ,a_{30}$ be in A.P., $S=\sum _{i=1}^{30}{a}_i$ and $T=\sum _{i=1}^{15}{a}_{2i-1}$. If $a_5=27$ and $S-2T=75$, then the value of $a_{10}$ is

If $si{n}^{-1}(a-\frac{a^2}{3}+\frac{a^3}{9}-......\infty )+co{s}^{-1}(1+b+b^2+....\infty )=\frac{\pi }{2}$ then

The corner points of the feasible region determined by the system of linear constraints are (0, 0), (0, 40), (20, 40), (60, 20), (60, 0). The objective function is Z = 4x + 3y. Compare the quantity in column A and column B.

$\begin{array}{cc}\text{Column A}& \text{Column B}\\ \text{Maximum of Z}& 325\end{array}$
(a) The qunatity in column A is greater
(b) The qunatity in column b is greater
(c) The two quantities are equal
(d) The relationship cannot be determined on the basis of the information supplied.

If roots of the equation $a{x}^3+b{x}^2+cx+d=0$ remain unchanged by increasing each coefficient by one unit, then

Let a tangent be drawn to the ellipse $\frac{x^2}{27}+y^2=1$ at $(3\sqrt{3}\cos \theta ,\sin \theta )$ where $\theta \in (0,\frac{\pi }{2}).$ Then the value of $\theta$ such that the sum of intercepts on axes made by tangent is minimum is equal to :

Let $f\left(x\right)=x^2,x\in R.$ For any $A\subseteq R$, define $g\left(A\right)=\{x\in R:f(x)\in A\}$. If $S=[0,4]$, then which one of the following statements is not true ?

Find the values of $\alpha$ so that the point $P({\alpha }^2,\alpha )$ lies inside or on the triangle formed by the lines $x-5y+6=0$, $x-3y+2=0$ and $x-2y-3=0$

Let $r$ and $R$ be the inradius and the circumradius of a $△ABC$. Let $\theta$ be the angle between the line joining the incentre and the circumcentre of the $△ABC$ and $BC$. Then $\theta$ is equal to

Show that

In a triangulara $ABC,\frac{b^2\sin 2C+c^2\sin 2B}{\Delta }$ is always equal to

If the point $P(x_1+t(x_2-x_1),y_1+t(y_2-y_1\left)\right)$ divides the line segment joining the points $A(x_1,y_1)$ and $B(x_2,y_2)$ internally, then the range of $t$ is

Find the set of real values of x for which ${\log }_{(x+3)}$ ($x^2$ - x) < 1________.

Let $A=\{1,2,3,4,...,40\}$ be the set of registration number of students of class $12$. Students having registration number as multiple of $4$ play cricket but do not play football, students having even registration number play either cricket or football or both and students having registration number as multiple of $10$ play both cricket and football. Find the number of students who play football.

If $f\left(x\right)=\{\begin{array}{cc}\frac{(1-\cos 2x)(3+\cos x)}{x\tan ax}& x<0;\\ \frac{x(e^x-1)}{1-\cos x}& x>0;\end{array}$ then find the value of $a$ such that $\underset{x\to 0}{lim}f\left(x\right)$ exists

If x₁, x₂, x₃ as well as y₁, y₂, y₃ are in G.P. with the same common ratio, then the points A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃)

Determine order and degree(if defined)
of differential equation

If B = {1, 3, 5, 7, 9}, the set-builder representation of B is