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The value of $\left\{\frac{2^{4n}}{15}\right\},n\in N$ is
(where {.} represents fractional part function)

For $2\le r\le n$, $\left(\begin{array}{c}n\\ r\end{array}\right)+2\left(\begin{array}{c}n\\ r-1\end{array}\right)+\left(\begin{array}{c}n\\ r-2\end{array}\right)$ is equal to

Construct $a_{2\times 2}$ matrix, where

$a_{ij}=|-2i+3j|$

Find the slope of the normal to the
curve x = 1 − a sin θ, y = b
cos²θ at
.

Let $T_1,T_2,T_3,\dots$ be terms of an A.P. If $S_1=T_1+T_2+T_3+\cdots +T_n$ and $S_2=T_2+T_4+T_6+\cdots +T_{n-1}$, where $n$ is odd, then the value of $\frac{S_1}{S_2}$ is

For a frequency distribution consisting of 18 observations, the mean and the standard deviation were found to be 7 and 4 respectively. But on comparison with the original data, it was found that a figure 12 was miscopied as 21 in calculations. Find the correct mean and standard deviation.___

Let $z_1=10+6i$ and $z_2=4+6i$. If $z$ is any complex number such that the argument of $(z-z_1)/(z-z_2)$ is $\pi /4$, then $|z-7-9i|$ is

Sum of the series ${}^n{C}_1+2\cdot 5{}^n{C}_2+3\cdot 5^2{}^n{C}_3+\cdots$ upto $n$ terms is

In $\Delta ABC$, if the sides are $a=3,b=5$ and $c=4$, then

$\sin \frac{B}{2}+\cos \frac{B}{2}$ is equal to

Differentiable but not continous

For any integer $n,$ the argument of $z=\frac{(\sqrt{3}+i{)}^{4n+1}}{(1-i\sqrt{3}{)}^{4n}}$ is

If the equation $||x-1|+a|=4$ has a real soltuion then $a$ belongs to

A circle is inscribed in an equilateral triangle of side a, the area of any square inscribed in the circle is

The difference between the two acute angles of a right - angled triangle is $\frac{2\pi }{5}$ radians. Express the angles in degrees.

Which of the following is the least?

Solve 3x + 8 > 2 when

(i) x is integer

(it) x is a real number

Let $\alpha =3^{{\log }_{4}5}-5^{{\log }_{4}3}+2.$ If $p$ and $q$ are the roots of the equation ${\log }_{\alpha }x+{\log }_x\alpha =\frac{10}{3},$ where $p>q,$ then the value of $p+q^3$ is

Let $S_1=\{(i,j,k):i,j,k\in \{1,2,\dots ,10\}\},$



$S_2=\{(i,j):1\le i<j+2\le 10,i,j\in \{1,2,\dots ,10\}\},$



$S_3=\{(i,j,k,l):1\le i<j<k<l,i,j,k,l\in \{1,2,\dots ,10\}\},$



and $S_4=\{(i,j,k,l):i,j,k\text{and}l\text{distinct elements in}\{1,2,\dots ,10\}\}.$



If the total number of elements in the set $S_r\text{is}{n}_r,r=1,2,3,4$, then which of the following statements is(are) TRUE?

Prove that
the determinant
is
independent of θ.

The arcs of the same length in two circles subtend angles of ${28}^{\circ }$ and ${35}^{\circ }$ at their centres. Then the ratio of their respective radii is

If $2x^2+5x+2b=0$ and $2x^3+7x^2+5x+1=0$ have atleast one common root for three values of $b$, then the sum of all three values of $b$ is

Number of ways of selecting $6$ shoes, out of $6$ pair of shoes, having exactly two pairs is

In $\Delta ABC$, if $3\tan \frac{A}{2}\tan \frac{C}{2}=1$, then $a,b,c$ are in:

The following table shows distribution of workforce in India for the year 1972-73. Analyse it and give reasons for the nature of work force distribution.

$\begin{array}{cc}\text{Place of Residence}& \text{Workforce (in millions)}\\ & \text{Male Female Total}\\ \text{Rural}& 12569195\\ \text{Urban}& 32739\end{array}$

Find the sum total of $7+7+7$

The function $f\left(x\right)=ta{n}^{-1}(sinx+cosx),x>0$ is always an increasing function on the interval