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If $2^x+2^y=2^{x+y}$, then $\frac{dy}{dx}$ is equal to

Find out the wrong number in the series given below :

$6,13,18,25,30,37,40$

If the $p^{th},q^{th}and{r}^{th}$ terms of a G.P. are a, b and c respectively, Prove that $a^{q-1}{b}^{r-p}{c}^{p-q}=1$

The period of y =sin 5x is

When $\frac{z+i}{z+2}$ is purely imaginary then z lies on

The terms of a G.P. are positive. If each term is equal to the sum of two terms that follow it, then the common ratio is ___.

Prove that sin${}^2\frac{\pi }{6}$ + cos ${}^2\frac{\pi }{3}$ - tan ${}^2\frac{\pi }{4}$ = -$\frac{1}{2}$.

If $\sin \theta =\frac{24}{25}$ and $\theta$ lies in the second quadrant, then $\sec \theta +\tan \theta =$

If the tangent at point P on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ meets the major axis at T, C is the centre and PN is the perpendicular on major axis, then the value of CN.CT is

The value of $cos\left(\frac{1}{2}co{s}^{-1}\left[cos\right(-\frac{14\pi }{5}\left)\right]\right)$

The minimum value of $3x^2+3x+9$ will be

(where $x\in R$)

Consider the parabola $(x-1{)}^2+(y-2{)}^2=\frac{(12x-5y+3{)}^2}{169}$

$\begin{array}{cc}ColumnI& ColumnII\\ \text{a. Locus of point of intersection of perpendicular tangent}& \text{p.}(12x-5y-2=0)\\ \text{b. Locus of foot of perpendicular from focus upon any tangent}& \text{q.}(5x+12y-29=0)\\ \text{c. Line along which minimum length of focal chord occurs}& \text{r.}(12x-5y+3=0)\\ \text{d. Line about which parabola is symmetrical}& \text{s.}(24x-10y+1=0)\end{array}$

If $lo{g}_{cosx}sinx+lo{g}_{sinx}cosx=2thenx=$

The value of $i^{1+3+5+......+(2n+1)}$ is

If $x^2+4+3\sin (ax+b)-2x=0$ has atleast one real solution, where $a,b\in [0,2\pi ]$, then the value of $a+b$ can be

Let $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ be three unit vectors such that $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0}$. If $\lambda =\overrightarrow{a}\cdot \overrightarrow{b}+\overrightarrow{b}\cdot \overrightarrow{c}+\overrightarrow{c}\cdot \overrightarrow{a}$ and $\overrightarrow{d}=\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}$, then the ordered pair $(\lambda ,\overrightarrow{d})$ is equal to:

Find the value of

$ta{n}^2\left(se{c}^{-1}\left(3\right)\right)$+$co{t}^2\left(cose{c}^{-1}\left(4\right)\right)$

The solution set for $2{\cos }^2\theta +\sin \theta \le 2,$where $\frac{\pi }{2}\le \theta \le \frac{3\pi }{2}$, is

$\frac{sin2A}{1+cos2A}$ . $\frac{cosA}{1+cosA}$ =

The interval(s) which satisfies the solution set of the inequality $4x-2>x-\frac{3-x}{4}>3$ is/are

If S be the sum, P be the product and R be the sum of the reciprocals of n terms in a GP, prove that $P^2={\left(\frac{S}{R}\right)}^n$

Sum of n terms of series 12 + 16 + 24 + 40 +......... will be

Find the sum of the products of the corresponding terms of finite geometrical progressions

2, 4, 8, 16, 32 and 128, 32, 8, 2, $\frac{1}{2}$

In how many ways the sum of upper faces of four distinct dies can be six.

Consider the first $10$ positive integers, if we multiply each number by $-1$ and then add $1$ to each number, then standard deviation of the numbers, so obtained is approximately equal to

Find the domain of the function f(x) = x |x|

The elements of power set of Set $A=\{-1,0,1\}$ are

Match the following (Figures are rough sketches)

(i)sinx

(ii) cosx

(iii) tanx

The characteristic roots of the two rowed orthogonal matrix $\left[\begin{array}{cc}\cos \theta & \sin \theta \\ \sin \theta & \cos \theta \end{array}\right]$ are $\lambda$ and $\overline{\lambda }$. ($\lambda$ lies in $I$ quadrant for $\theta \in (0,{90}^o))$, then ${\lambda }^5$ is equal to

The set of all $x$ in $(-\pi ,\pi )$ satisying $|4\cos x-1|<\sqrt{5}$ is

The value of the integral, $\underset{1}{\overset{3}{\int }}[x^2-2x-2]dx,$ where $\left[x\right]$ denotes the greatest integer less than or equal to $x,$ is

If $a$ and $b$ are the non-zero distinct roots of $x^2+ax+b=0$, then the least value of quadratic polynomial $x^2+ax+b$ is

The locus of the centres of the circles which cut the circles $x^2+y^2+4x-6y+9=0and{x}^2+y^2-5x+4y-2=0$ orthogonally is

The minimum value of $cos\theta +sin\theta$ is

[MNR 1976; Pb. CET 1996]

Number of positive integral solutions of the equation $xyz=90$ is

The sum of $10$ terms of the series $\mathrm{1.3.5}+\mathrm{3.5.7}+\mathrm{5.7.9}+\dots \dots$ is