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(a + b) + c = a + (b + c) describes ___ .

In design of a concrete mix by Fineness modulus method the value of F.M. of coarse aggregate is $7.3$, the value of fine aggregate is $2.8$, and value of F.M. of combined aggregates is $6.4$. The percentage of fine aggregate to combined aggregate is

If $A=\left[\begin{array}{cc}2& -3\\ -4& 1\end{array}\right],$ then adj$(3A^2+12A)$ is equal to:

The equation of the plane containing the two lines $\frac{x-1}{2}=\frac{y+1}{-1}=\frac{z}{3}$ and $\frac{x}{2}=\frac{y-2}{-1}=\frac{z+1}{3}$ is

If $a_1,a_2,a_3,\cdots ,a_n;(n\ge 2)$ are real and $(n-1)a_1^2-2n{a}_2<0,$ then atleast roots of the equation $x^n+a_1{x}^{n-1}+a_2{x}^{n-2}+\cdots +a_n=0$, are imaginary.

Evaluate $\int \frac{(1+cosx)}{x+sinx}dx$

The integrating factor of the differential equation $x\frac{dy}{dx}+y-x+xy\cot x=0,x\ne 0$ can be

If $f$ is an even function, then the value of $\underset{0}{\overset{\pi /2}{\int }}f\left(\cos 2x\right)\cos xdx$ can be equal to

Which of the two numbers is greater, $2^{300}$ or $3^{200}$?

If the value of $e^{-2qi\left({\tan }^{-1}p\right)}{\left(\frac{i-p}{i+p}\right)}^q=R$, where $q,p\in I\&i=\sqrt{-1}$. Then the value of $R^3$ is :

If A,B,C,D be the angles of a cyclic quadrilateral, taken in order, prove that: $cos({180}^{\circ }-A)+cos({180}^{\circ }+B)-sin({90}^{\circ }+D)=0$

For any two complex number $z_1$ and $z_2$ and any real numbers a and b;

$|(a{z}_1-b{z}_2){|}^2$ + $|(b{z}_1-a{z}_2){|}^2$ =

Find the maximum and minimum values, if any, of the following
functions given by

(i) f(x)
= |x + 2| − 1 (ii) g(x) = − |x
+ 1| + 3

(iii) h(x)
= sin(2x) + 5 (iv) f(x) = |sin 4x + 3|

(v) h(x)
= x + 4, x
(−1, 1)

Let $A(2\hat{i}+3\hat{j}+5\hat{k}),$ $B(-\hat{i}+3\hat{j}+2\hat{k})$ and $C(\lambda \hat{i}+5\hat{j}+\mu \hat{k})$ be vertices of a triangle and the median through $A$ is equally inclined to the positive direction of coordinate axes. Then the value of $\lambda +3\mu$ is

${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\sqrt{\frac{1}{2}(1-cos2x)}dx=$

If $f\left(x\right)=\{\begin{array}{cc}\frac{\sin 5x}{x^2+2x},& x\ne 0\\ k+\frac{1}{2},& x=0\end{array}$ is continuous at $x=0$, then the value of $k$ is

If $A,B$ and $C$ are three mutually exclusive and exhaustive events such that $P\left(A\right)=\frac{4}{5}P\left(C\right)$ and $P\left(B\right)=\frac{3}{5}P\left(C\right),$ then the value of $P\left(\overline{C}\right)$ is

A firm has to transport 1200 packages using large vans which can carry 200 packages each and small vans which can take 80 packages each. The cost for engaging each large van is Rs 400 and each small van is Rs 200. Not more than Rs 3000 is to be spent on the job and the number of large vans cannot exceed the number of small vans. Formulate this problem as a LPP given that the objective is to minimise cost.

Any chord passing through the focus (ae, 0) of the hyperbola $x^2-y^2=a^2$is conjugate to the line

In a triangle $ABC,$ if $|\overrightarrow{BC}|=8,|\overrightarrow{CA}|=7,|\overrightarrow{AB}|=10$, then the projection of the vector $\overrightarrow{AB}$ on $\overrightarrow{AC}$ is equal to:

Sum of the roots of $3^{sin2x+2co{s}^{2}x}+3^{1-sin2x+2si{n}^{2}x}=28$ in $[-2\pi ,2\pi ]$ is

A variable chord $PQ$ of the parabola $y^2=4ax$ is drawn parallel to line $y=x$. Then the locus of point of intersection of normals at $P$ and $Q$ is:

The relation R is defined on the set of natural numbers as {(a,b) : a = 2b}. Then $R^{-1}$ is given by

The function $f\left(x\right)=(3x-7)x^{2/3},x\in R$ is increasing for all $x$ lying in