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The parameter on which the value of the determinant

$\left|\begin{array}{ccc}1& a& a^2\\ \cos (p-d)x& \cos px& \cos (p+d)x\\ \sin (p-d)x& \sin px& \sin (p+d)x\end{array}\right|$

does not depend on

An urn contains 2 white and 2 blacks balls. A ball is drawn at random. If it is white it is not replaced into the urn. Otherwise it is replaced along with another ball of the same colour. The process is repeated. Find the probability that the third ball drawn is black.

The locus of the orthocentre of the triangle formed by the lines
$(1+p)x-py+p(1+p)=0,$ $(1+q)x-qy+q(1+q)=0$ and $4y=0,$ where $p\ne q$ is

Find the equation of the set of points which are equidistant from the points (1, 2 , 3) and (3, 2, -1)

If $\overrightarrow{PQ}=2\hat{i}+\hat{j}-\hat{k}$ and $\overrightarrow{P}=\hat{i}-\hat{j}+\hat{k}$ then $\overrightarrow{Q}$ is

The domain of $f\left(x\right)=\sqrt{x-4-2\sqrt{x-5}}-\sqrt{x-4+2\sqrt{x-5}}$ is

How is sinA+sinB+sinC=4 cosA/2 cosB/2 cosC/2

The shortest distance between line $y-x=1$ and the parabola $y^2=x$ is

If $y=\frac{(x-a)(x-c)}{x-b}$ assumes all real values for $x\in R-\left\{b\right\},$ then

Differentiate the following functions with respect to x

$cos\left(\sqrt{x}\right)$

The value of ${\cot }^{-1}\left(1\right)+{\cos }^{-1}\left(\frac{1}{\sqrt{2}}\right)$ is



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If $\alpha ,\beta$ are the roots of $x^2+7x+5=0$, then the equation whose roots are $\alpha -1,\beta -1$ is

Let $A(a,0),B(b,2b+1)$ and $C(0,b),b\ne 0,|b|\ne 1,$ be points such that the area of triangle $ABC$ is $1$ sq. unit, then the sum of all possible values of $a$ is

The inradius of pedal triangle of $\Delta ABC$ is:

Solution of the differentiable equation $\left(\frac{x+y-1}{x+y-2}\right)\frac{dy}{dx}=\left(\frac{x+y+1}{x+y+2}\right)$ such that, $x=1$ when $y=1,$ is

(where $\log$ is given with base ${}^{\prime }{e}^{\prime }$)

A man accepts a position with an initial salary of Rs. 5200 per month. It is understood that he will receive an automatic increase of Rs. 320 in the very next month and each month thereafter.

(i) Find his salary for the tenth month.

(ii) What is his total earnings during the first year?

If $n\in Z,$ then $(-\sqrt{-1}{)}^{4n+3}$ equals

Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^2\sin \theta -x(\sin \theta \cos \theta +1)+\cos \theta =0(0<\theta <{45}^{\circ })$, and $\alpha <\beta$. Then
$\sum _{\infty }^{n=0}({\alpha }^n+\frac{(-1{)}^n}{{\beta }^n})$ is equal to :

For the curve y = 4x³
− 2x⁵, find all the points at which the
tangents passes through the origin.

The pressure $p$ and the volume $v$ of a gas are connected by the relation $p{v}^{1.4}=$constant. The percentage error in $p$ corresponding to a decrease of $\frac{1}{2}\%$ in $v$.

If $\overrightarrow{a}=2\hat{i}+\hat{j}+3\hat{k},\overrightarrow{b}=3\hat{i}+2\hat{j}+\hat{k},\overrightarrow{c}=\hat{i}-\hat{j}-4\hat{k}$ and $\overrightarrow{d}=\hat{i}+2\hat{j}-\hat{k},$ then $(\overrightarrow{a}\times \overrightarrow{b})\times (\overrightarrow{c}\times \overrightarrow{d})=$

Simplified form of $\frac{(\sqrt{3}-i{)}^6}{(1+i{)}^8}$ is

The number of solutions of the equation $\cos 3x+\cos 2x=\sin \frac{3x}{2}+\sin \frac{x}{2}$ lying in the interval $[0,2\pi ]$ is

Number of solutions for the system of linear equations, will be,

x + y + z =6

x + 2y + 3z = 14

x + 4y + 2z =30