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If $\frac{si{n}^3\theta -co{s}^3\theta }{sin\theta -cos\theta }-\frac{cos\theta }{\sqrt{1+co{t}^2\theta }}-2tan\theta cot\theta =-1,\theta \in [0,2\pi ],$ then

If $f:R\to R$ and $g:R\to R$ are given by f(x) = |x| and g(x) = [x], then $g\left(f\right(x\left)\right)\le f\left(g\right(x)$ is true for -

In a particular family, each boy has as many brothers as sisters but each girl has twice as many brothers as that of sisters. How many siblings are there in the family?

Find the area of the region enclosed by the parabola $x^2=y$ and the line $y=x+2$.

The value of $\underset{x\to 0}{lim}\frac{(1-e^x)\sin x}{x^2+x^3}=$

The function

$f\left(x\right)=max\{x^2,(1-x{)}^2,2x(1-x)\}$ where $0\le x\le 1$

then area of the region bounded by the curve $y=f\left(x\right),x-$axis and $x=0,x=1$ is equals

A biased ordinary die is loaded in such a way that probability of getting an even outcome is five times the probability of getting an odd outcome. This die is rolled two times. The probability that the sum of outcome will be a prime number is equal to:

Using binomial theorem, indicate which is larger $(1.1{)}^{10000}or1000$.

If the mean and variance of the following data :

$6,10,7,13,a,12,b,12$ are $9$ and $\frac{37}{4}$ respectively, then $(a-b{)}^2$ is equal to

If all the letters of the word $RANDOM$ are rearranged to form $6$ letter words and arranged in ascending order as in a dictionary, then the rank of the word $RANDOM$ is

If $a<b<c<d$ and $x\in R,$ then the least value of the function $f\left(x\right)=|x-a|+|x-b|+|x-c|+|x-d|$ is

A man alternately tosses a coin and throws a die beginning with coin. The probability that he gets a head in the coin before he gets 5 or 6 on the die is

Find the mean deviation from the median for the following data :

(i) $\begin{array}{cccccc}{x}_i& 15& 21& 27& 30& 35\\ f_i& 3& 5& 6& 7& 8\end{array}$

(ii) $\begin{array}{cccccccc}{x}_i& 74& 89& 42& 54& 91& 94& 35\\ f_i& 20& 12& 2& 4& 5& 3& 4\end{array}$

(iii) $\begin{array}{cccccc}\text{Marks obtained}& 10& 11& 12& 14& 15\\ \text{No. of students}& 2& 3& 8& 3& 4\end{array}$

An electric dipole is made up of two particles, one having charge $+1\mu \text{C}$ and mass $1\text{kg}$ and the other with charge $-1\mu \text{C}$ and mass $1\text{kg}$ separated by distance $1\text{m}$. It is in equilibrium in a uniform electric field of $20\times {10}^3\text{V}/\text{m}$. If the dipole is deflected through an angle $2^{\circ }$, then time taken by it to come again at equilibrium position is

What is the probability of guessing a true or false question correctly?Give the answer up to 1 decimal place.

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Find the following integrals.
$\int {(\sqrt{x}-\frac{1}{\sqrt{x}})}^{2}dx.$

If $\theta$ is an angle between the lines given by the equation $6x^2+5xy-4y^2+7x+13y-3=0$, then equation of the line passing through the point of intersection of these lines and making an angle $\theta$ with the positive x - axis is

Let $z_1=10+6i$ and $z_2=4+6i,$ where $i=\sqrt{-1}.$ If $z$ is any complex number such that $\arg \left(\frac{z-z_1}{z-z_2}\right)=\frac{\pi }{4},$ then the value of $|z-7-9i|$ is

Let R be a relation on $N\times N$ defined by $(a,b)R,(c,d)\iff a+d=b+c$ for all $(a,b),(c,d)ϵN\times N$

Show that :

(i) (a, b) R (a, b) for all $(a,b)ϵN\times N$

(ii) $(a,b)R(c,d)\Rightarrow (c,d)R(a,b)$ for all $(a,b),(c,d)ϵN\times N$

(iii) (a, b) R (c, d) and (c, d) R (e, f) $\Rightarrow$ (a, b) R (e, f) for all (a, b), (c, d), (e, f) $ϵN\times N.$

X power n - y power n is divisible by x-y where X and Y belongs to n and X is not equal to y

The auxiliary circle of family of ellipses, passes through origin and makes intercept of $8$ and $6$ units on the $x-$ axis and the $y-$ axis respectively. If eccentricity of all such family of ellipse is $\frac{1}{2}$, then the locus of focus of ellipse will be

A line passing through $P(2,-3)$ is making an angle ${135}^{\circ }$ in anticlockwise direction with $x-$ axis and is intersecting another line $x+2y-3=0$ at $Q$. Then the length of $PQ$ is

While measuring length of an object, it was observed that the zero of the Vernier lies between $1.4$ and $1.5$ of the main scale and the fifth Vernier division coincides with a main scale division. If the length of the object measured is $l$, then the value of $(l-1.4)$ in terms of the least count $C$ of the instrument is

Let $f$ be a twice differentiable function on $(-1,4).$ If $f\left(0\right)=2,f^{\prime }\left(0\right)=0,f^{\prime }\left(x\right)\ge 1$ and $f^{\prime \prime }\left(x\right)\ge 3\forall x\in (-1,4),$ then which of the following options is/are true

If x = 9 is the chord of contact of the hyperbola $x^2-y^2=9$, then the equation of the corresponding pair of tangents is

Simplified form of the algebraix expression $(0.5u^3-u^2)(0.5u^3-u^2)=$$u^6-u^5+$

Given 5 flags of different colours, how many different signals can be generated if each signal requires the use of 2 flags, one below the other?

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

(i) f(x) = (2x − 1)² +
3 (ii) f(x) = 9x² + 12x + 2

(iii) f(x) = −(x − 1)² +
10 (iv) g(x) = x³ + 1

In the given figure, what is the angle subtended by chord AB at the point C on the circle?