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Let $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$ and $B=A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ are two matrices such that AB = BA and $c\ne 0$, then value of $\frac{a-d}{3b-c}$ is :

If $\alpha \in [\frac{\pi }{2},\pi ],$ then the value of $\sqrt{1+\sin \alpha }-\sqrt{1-\sin \alpha }$ is

If ${\log }_2\left(\frac{x-1}{x-2}\right)>0$, then

If the sum and product of the first three terms in an $A.P.$ are $33$ and $1155$, respectively, then the value of its ${11}^{\text{th}}$ term is :

Solve

$|x-1|+|x-2|\ge 4,xϵR$

If two circles which pass through the points (0, a) and (0, -a) cut each other orthogonally and touch the straight line $y=mx+c$, then

The set of values of $b$ for which $f\left(x\right)=2x^4+b{x}^3+3x^2,$ has a point of inflection

Let f = {(1, 1), (2, 3), (0, -1), (-1, -3)} be a function from Z to Z defined by f(x) = ax + b for some integer a, b. Determine a, b.

The bulk modulus of a spherical object is $B$. If it is subjected to uniform pressure $P$, the fractional decrease in radius is

$z_1$ and $z_2$ are the roots of 3$z^2$ + 3z + b = 0. If O(0), A($z_1$)B($z_2$) is an equilateral triangle, then

If $A=\{1,2,3,4\}$ and $B=\{5,7,9\}$, then the number of onto function from $A$ to $B$ is

Find the general solution of the equation 4 sin x cos x + 2 sin x + 2 cos x +1 = 0.

If one of the foci of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)$ coincide with the focus of the parabola $y^2=8x$ and they intersect at a point where the ordinate is double the abscissa, then the value of $\left[b^2\right]$ is

(where [.] represents greatest integer function)

${}^n{C}_0\times {}^{n+1}{C}_n+{}^n{C}_1{}^n{C}_{n-1}+{}^n{C}_2\times {}^{n-1}{C}_{n-2}+.....{}^n{C}_n$ =

Positive numbers $x,y$ and $z$ satisfy

$xyz$ = ${10}^{81}$ and $\left({\log }_{10}x\right)\left({\log }_{10}yz\right)+\left({\log }_{10}y\right)\left({\log }_{10}z\right)=468$, then the value of $({\log }_{10}x{)}^2+({\log }_{10}y{)}^2+({\log }_{10}z{)}^2$ is -

If $k=sin\frac{\pi }{18}.sin\frac{5\pi }{18}.sin\frac{7\pi }{18}$ then the numerical value of k is

$If\overrightarrow{u}=\overrightarrow{a}-\overrightarrow{b}and\overrightarrow{v}=\overrightarrow{a}+\overrightarrow{b}and|\overrightarrow{a}|=|\overrightarrow{b}|=2,then|\overrightarrow{u}\times \overrightarrow{v}|=$

If $[x{]}^2-7[x]+10>0$, then $x$ lies in the interval

(Here, $[.]$ denotes the greatest integer function)

The sum of (n+1) terms of $\frac{1}{1}+\frac{1}{1+2}+\frac{1}{1+2+3}+..........$ is

Given that $f\left(x\right)=\frac{sinx}{x}$. Then f'(x)=.

The value of $\sum _{r=2}^{10}{}^r{C}_2\cdot {}^{10}{C}_r$ is not divisible by

If $2^{\frac{2\pi }{{\sin }^{-1}x}}-2(a+2)2^{\frac{\pi }{{\sin }^{-1}x}}+8a<0$ for atleast one real $x$, then

$\frac{1}{2-|x|}\ge 1,xϵR-\{-2,2\}$

If the coefficient of $x^7$ in $(a{x}^2+\frac{1}{bx}{)}^{11}$ is equal to the coefficient of $x^{-7}$ in $(ax-\frac{1}{b{x}^2}{)}^{11}$, then ab =

A scientist is weighing each of 30 fishes. Their mean weight worked out is 30 gm and a standard deviaiton of 2 gm. Later, it was found that the measuring scale was misaligned and always under-reported every fish weight by 2gm. The correct mean and standard deviation (in gm) of fishes are respectively:

The points (0, $\frac{8}{3}$), (1, 3) and (82, 30) are the vertices of

Total number of ordered pairs (x,y) satisfying $|x|+|y|=2,sin\left(\frac{\pi x^2}{3}\right)=1$ is

The $n^{th}$ term of a sequence of numbers is $a_n$ and given by the formula $a_n=a_{n-1}+2n$ for $n\ge 2$ and $a_1=1$.



Using the above information $a_n$ will be

The standard deviation of 17 numbers is zero. Then

A block is fastened at one end of a wire and is rotated in a vertical circle of radius $R$. Determine the ratio of change in length of the wire at the lowest point to that at the highest point of the circle. Assume that speed of the block at highest and lowest points is the same ($v$).

If $\theta$ lies in the second quadrant, then the value of $\sqrt{\frac{1-sin\theta }{1+sin\theta }}$ + $\sqrt{\frac{1+sin\theta }{1-sin\theta }}$.

If the function $f\left(x\right)=\begin{array}{cc}(1+|sinx|{)}^{\frac{a}{|sinx|}},& -\frac{\pi }{6}<x<0\\ b,& x=0\\ e^{\frac{tan2x}{tan3x}},& 0<x<\frac{\pi }{6}\end{array}$, is continuous at x = 0, then

If $2x-y+1=0$ is a tangent to the hyperbola$\frac{x^2}{a^2}-\frac{y^2}{16}=1$, then which of the following CANNOT be sides of a right triangle?

From the data given below, find the no of items (N) :

$\sum$xy=120, r = 0.5, standard deviation of Y = 8, $\sum x^2=90,$ where x and y are deviations from arithmetic mean.

The focus of the parabols $x^2=-16y$ is

Number of real solutions of the equation $|x-3{|}^{3x^2-10x+3}=1$ is

The following information are extract from the trial balance of M/s Nisha traders on 31 December, 2005.

$\begin{array}{cc}\text{Sundry Debtors}& 80,500\\ \text{Bad debts}& 1,000\\ \text{Provision for bad debts}& 5,000\\ \text{Additional Information}& \\ \text{Bad Debts}& \text{Rs. 500}\end{array}$

Provision is to be maintained at 2 % of Debtors.

Prepare bad debts account, Provision for bad debts account and Profit and Loss account.

If $co{s}^4$ x + a $co{s}^{2}x$ + 1 = 0 has atleast one solution then

If x = 2 + 5i and $\frac{2}{9!}$ + $\frac{2}{3!7!}$ = $\frac{2^a}{b!}$, then $x^3-5x^2$ + 33x - 19 =

Let $f\left(x\right)=\{\begin{array}{cc}{x}^2,& x\ge 0\\ ax,& x<0\end{array}$

The set of real values of $a$ such that $f\left(x\right)$ will have local minima at $x=0,$ is:

If two sets $A$ and $B$ are such that $(A-B)=A$, then $A\cap B=$

Prove that ${\text{sin}}^{2}6x-{\text{sin}}^{2}4x$ = $\text{sin}10x\text{sin}2x$