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In $\Delta ABC,ccos(A-\alpha )+\alpha cos(C+\alpha )=$

Define histogram and construct a histogram from given data :

$\begin{array}{ccccccc}\text{Age in Month}& 40-60& 60-80& 80-100& 100-120& 120-140& 140\text{and more}\\ \text{No. of subject about mortality}& 11& 15& 13& 7& 7& 2\end{array}$

Find the locus of the point which is at a distance of 3 units from the origin.

Energy due to the position of a particle is given by, $U=\frac{\alpha \sqrt{y}}{y+\beta }$, where $\alpha$ and $\beta$ are constants, $y$ is distance. The dimensions of $(\alpha \times \beta )$ is:

$\alpha ,\beta ,\gamma$ are real numbers satisfying $\alpha +\beta +\gamma =\pi$.

The value of the given expression

$sin\alpha +sin\beta +sin\gamma$ is

Given that $g\left(x\right)=\left[f\right(x)-1{]}^2$. Find the domain of f(x) = 1 - 2x, given that $0\le g\left(x\right)<4$.

If ${\cos }^6\alpha +{\sin }^6\alpha +k{\sin }^{2}2\alpha =1\forall \alpha \in \left(0,\pi /2\right)$, then k is

Let $y$ be an implicit function of $x$ defined by $x^{2x}-2x^{x}coty-1=0.$ Then $y^{\prime }\left(1\right)$ equals

If p, q, n are three positive real numbers and p > q then which of the following is correct.

The value of $x$, if ${\log }_4(3x^2+11x)=1$

The coefficient of middle term in the expansion of $(1+x{)}^{10}$ is:

In the given figure of cuboid if the coordinates of point E is (3, 2 ,1) and one of the corners as the origin.

How many of the following coordinates are correct?

A(0, 1, 0) B(3, 0, 1) C(3, 0, 0) D(2, 3, 0)

E(3, 2, 1) F(0, 2, 0) G(0, 2, 1) H(0, 0, 0)

There are $5$ multiple choice questions (only one correct option) in a test. If the first three questions have $4$ choices each and the next two have $5$ choices each, then number of possible ways in which a student can answers all the question is

Find the value of ${}^n{C}_1+2^n{C}_2+3^n{C}_3.........n^n{C}_n$

Find the sum of $1.n^2+2(n-1{)}^2+3(n-2{)}^2+......n{.1}^2$

If $\pi <2\theta <\frac{3\pi }{2}$,then

1. $\sqrt{co{s}^2\theta }=cos\theta$

2. $\sqrt{si{n}^2\theta }=sin\theta$

Which of the above statement is/are correct?

Let $f$ and $g$ be differentiable funcitons on $R$, such that $fog$ is the identity funciton. If for some $a,b\in R,g^{\prime }\left(a\right)=5$ and $g\left(a\right)=b$, then $f^{\prime }\left(b\right)$ is equal to :

Mean of numbers $\frac{{}^{50}{C}_0}{1},\frac{{}^{50}{C}_2}{3},\frac{{}^{50}{C}_4}{5},\cdots ,\frac{{}^{50}{C}_{50}}{51}$ is

If A is a square matrix of order n and A=kB, where k is a scalar, then |A|=

$\text{The value of}\underset{x\to \infty }{lim}[\sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x}]\text{is}.$

Two systems of rectangular axes have the same origin. If a plane cuts the two sets of axes at distances a, b, c and a', b', c' from the origin, then:

If the system of linear equations

$x+y+z=5x+2y+2z=6x+3y+\lambda z=\mu ,$

$(\lambda ,\mu \in R),$ has infinitey many solutions, then the value of $\lambda +\mu$ is :

With usual notations, is a $△ABC\frac{b^2-c^2}{asecc}+\frac{c^2-a^2}{bsecc}+\frac{a^2-b^2}{csecc}$ is equal to

Find the numerically greatest term in the expansion of $(2+3x{)}^9$, when x=$\frac{3}{2}$

If $\alpha$ and $\beta$ are the roots of the quadratic equation $(x-2)(x-3)+(x-3)(x+1)+(x+1)(x-2)=0$, then the value of $\frac{1}{(\alpha +1)(\beta +1)}+\frac{1}{(\alpha -2)(\beta -2)}+\frac{1}{(\alpha -3)(\beta -3)}$ is

If, $I=\int \frac{dx}{sin(x-\frac{\pi }{3})cosx}$ then I equals

If m = $\frac{sinA}{sinB}$ find $\frac{m+1}{m-1}$

If $f\left(x\right)+2f(1-x)=x^2+1\forall xϵR$ then f(x) is

The value of $\int \frac{e^x+9\cos x-2\sin x+7}{e^x+7\sin x+11\cos x+14}dx$ is

(where $c$ is the constant of integration)

A box $B_1$ contains $1$ white ball, $3$ red balls, and $2$ black balls. Another box $B_2$ contains $2$ white balls, $3$ red balls, and $4$ black balls. A third box $B_3$ contains $3$ white balls, $4$ red balls, and $5$ black balls.
If $2$ balls are drawn (without replacement) from a randomly selected box and one of the balls is white and the other ball is red, the probability that these $2$ balls are drawn from box $B_2$ is

Box I contain three cards bearing numbers 1,2,3; box II contains five cards bearing numbers 1,2,3,4,5; and box III contains seven cards bearing numbers 1,2,3,4,5,6,7. A card is drawn from each of the boxes. Let $x_i$ be the number on the card drawn from the $i^{th}$ box $i=1,2,3.$

The probability that $x_1+x_2+x_3$ is odd, is ?

If $n^{th}$ of a sequence is given by $T_n=2n+1$, then the sum of $4$ terms is

Find the principal and general solutions of the following equation.
sec x=2

The range of the function $f\left(x\right)={\log }_2(3-2x-x^2)$ is

The obtuse angle between the lines $x-\sqrt{3}y=5$ and $\sqrt{3}x-y=7$ is

Find the value of ${\sec }^{2}x$ - ${\text{cosec}}^{2}x$.

$\mathrm{If}f\left(x\right)=\left(\begin{array}{ccc}x& \sin x& \cos x\\ x^2& \tan x& -x^3\\ 2x& \sin 2x& 5x\end{array}\right|,\mathrm{then}\underset{x\to 0}{\lim }\frac{f\text{'}\left(x\right)}{x}\mathrm{equals}$

If sinA+sinB=C,cosA+cosB=D, then the value of sin(A+B)=

[MP PET 1986]

If α, β, γ are the roots of the equation $x^3+4x+1=0$,then $(\alpha +\beta {)}^{-1}+(\beta +\gamma {)}^{-1}+(\gamma +\alpha {)}^{-1}=$

If two distinct chords drawn from the point (p, q) on the circle $x^2+y^2-px-qy=0(wherepq\ne 0)$ are bisected by the x-axis, then

Find the modulus and argument of the complex number $\frac{1+2i}{1-3i}$

The digits of a three-digit positive integer are in A.P. and their sum is $15$. The number obtained by reversing the digits is $594$ less than the original number. Then the unit place of the number is