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Cos15^o – sin15^o =?

Find the mean deviation about the mean for the data.

The value of $\left|\begin{array}{ccc}a+x& y& z\\ x& a+y& z\\ x& y& a+z\end{array}\right|$ is equal to

The parametric equations of a parabola are $x=t^2+1,y=2t+1$.The cartesian equation of its directrix is

For $x\in (-\pi ,\pi ),tanx>0$ for

Let (x, y) be a pair of real number satisfying
$56x+33y=\frac{-y}{x^2+y^2}$ and
$33x–56y=\frac{x}{x^2+y^2}$. If |x| + |y| = $\frac{p}{q}$ (where p and q are relatively prime), then (6p – q)is ___

Describe the following sets in Roster form :

(i) {x : x is a letter before e in the English alphabet}.

(ii) {$xϵN:x^2<25$}

(iii) {$xϵN$: x is a prime number, 10 < x < 20}.

(iv) $xϵN:x=2n,nϵN$.

(v) $xϵR:x>x$.

(vi) {x : x is a prime number which is a divisor of 60}

(vii) {x : x ix a two digit number such that the sum of its digits is 8}.

(viii) The set of all letters in the word ' Trigonometry'.

(ix) The set of all letters in the word 'Better'.

For any real numbers $\alpha$ and $\beta ,$ Let $y_{\alpha ,\beta }\left(x\right),x\in R,$ be the solution of the differential equation $\frac{dy}{dx}+\alpha y=x{e}^{\beta x},y\left(1\right)=1$. Let $S=\left\{y_{\alpha ,\beta }\right(x):\alpha ,\beta \in R\}.$ Then which of the following functions belong(s) to the set $S$ ?

In a simultaneous throw of a pair of dice, find the probability &getting :

(i) 8 as the sum

(ii) a doublet

(iii) a doublet of prime numbers

(iv) a doublet of odd numbers

(v) a sum greater than 9

(vi) an even number on first

(vii) an even number on one and a multiple of 3 on the other

(vii) neither 9 nor 11 as the sum of thenuntbers on the faces

(ix) a sum less than 6

(x) a sum less than 7

(xi) a sum more than 7

(xii) neither a doublet nor a total of 10

(xiii) odd number on the first and 6 on the second

(xiv) a number greater than 4 on each dice

(xv) a total of 9 or 11

(xvi) a total greater than 8.

What will be the next number in the following sequence?



5, 7, 11, 13, 17, 19, __

If θ1, θ2, θ3........,θn are in A.P., whose common difference is d, then,

sec θ1 sec θ2+sec θ2 sec θ3+..........+sec θn−1 sec θn=

Tangents are drawn to the hyperbola $4x^2-y^2=36$ at the points $P$ and $Q$. If these tangents intersect at the point $T(0,3)$, then the area (in sq.units) of $\Delta PTQ$ is

The fundamental period of the function $f\left(x\right)={\sin }^{3}x+{\cos }^{2}x$ is

A trust fund has Rs 30000 that must be invested in two different types of bonds. The first bond pays 5% interest per year and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide R 30000 amoung the two types of bonds, if the trust fund must obtain an annual total interest of
(a) Rs 2000

A and B are finite sets such that n(A)=17 and n(B)=29. If A is a subset of B, then, $n\left(\right(A\cup B)-(A\cap B\left)\right)$= ___ .

The domain of $f\left(x\right)=\sqrt{(x^2-3x-10)\left[\ln \right(x-3){]}^2}$ is

Let $\overrightarrow{p},\overrightarrow{q},\overrightarrow{r}$ be three mutually perpendicular vectors of the same magnitude. If a vector $\overrightarrow{x}$ satisfies the equation

$\overrightarrow{p}\left[\right(\overrightarrow{x}-\overrightarrow{q})\times \overrightarrow{p}]+\overrightarrow{q}\times \left[\right(\overrightarrow{x}-\overrightarrow{r})\times \overrightarrow{q}]+\overrightarrow{r}\times \left[\right(\overrightarrow{x}-\overrightarrow{p})\times \overrightarrow{r}]=\overrightarrow{0}$, then $\overrightarrow{x}$ is given by

Consider f:
R₊
→ [−5, ∞)
given by f(x)
= 9x²
+ 6x −
5. Show that f
is invertible with.

Find the 5th term from the end in the expansion of $(3x-\frac{1}{x^2})$

If $\frac{si{n}^{4}A}{a}+\frac{co{s}^{4}A}{b}=\frac{1}{a+b},$ then the value of $\frac{si{n}^{8}A}{a^3}+\frac{co{s}^{8}A}{b^3}$ is equal to

Are the following pairs of statements are negation of each other.

(i) The number x is not a rational number.

The number x is not an irrational number.

(ii) The number x is not a rational number.

The number x is an irrational number.

A and B are two events such that P(A) = 0.25 and P(B) = 0.50. The pobability of both happening together is 0.14. The probability of both A and B not happening is

Find sets A, B and C such that $A\cap B$, $A\cap C$ and $B\cap C$ are non-empty sets and $A\cap B\cap C=\varphi$

Let $f\left(x\right)$ be a non-negative continuous function such that the area bounded by the curve $y=f\left(x\right),x-$axis and the ordinates $x=\frac{\pi }{4}$ and $x=\beta >\frac{\pi }{4}$ is $(\beta \sin \beta +\frac{\pi }{4}\cos \beta +\sqrt{2}\beta ).$ Then $f\left(\frac{\pi }{2}\right)$ is:

Consider the equation $(m^2+1)x^2-3x+(m^2+1{)}^2=0.$ Let $p$ be the least value of product of roots and $q$ be the greatest value of sum of roots of the equation. Then the sum of an infinitely decreasing G.P. whose first term is equal to $p+2$ and the common ratio is $\frac{2}{q},$ is

The value of $\underset{0}{\overset{\pi /2}{\int }}\ln \left(\frac{4+3\sin x}{4+3\cos x}\right)dx$ is

Find the equation of the straight line perpendicular to $2x-3y=5$ and cutting off an intercept 1 on the positive direction of the x-axis.

Using elementary transformations, find the inverse of the followng matrix.

$\left[\begin{array}{ccc}1& 3& -2\\ -3& 0& -5\\ 2& 5& 0\end{array}\right]$