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The value of ${\sin }^2{5}^{\circ }+{\sin }^2{10}^{\circ }+{\sin }^2{15}^{\circ }+$..............

$+{\sin }^2{85}^{\circ }+{\sin }^2{90}^{\circ }$ is equal to

The shortest distance of curve $2x^2+5xy+2y^2=1$ from origin is

In how many ways 2 directors and 3 executives can be arranged for a meeting? If there are 6 chairs available two on one side and remaining four on the other side of the table and the two directors has to be together on one side and the executives on the other side.

If the vertices of a triangle are $(1,2),(4,-6)$ and $(3,5)$, then the area (in sq. units) of the triangle is

Find the unit vector in the direction of the resultant of vectors $\hat{i}+2\hat{j}+3\hat{k},-\hat{i}+2\hat{j}+\hat{k}$ and $3\hat{i}+\hat{j}$.

Find the equations of the tangent and
normal to the parabola y² = 4ax at the point
(at², 2at).

If the number of five digit numbers with distinct digits and $2$ at the $10\text{th}$ place is $336k$, then $k$ is equal to :

Find the equations of the circles which pass through the origin and cut off equal chords of $\sqrt{2}$ units from the lines y = x and y = -x.

A whistle emitting a sound of frequency $440\text{Hz}$ is tied to a string of $1.5\text{m}$ length and rotated with an angular velocity of $20\text{rad/s}$ in the horizontal plane. Then the range of frequencies heard by an observer at a large distance from the whistle will be (Speed of sound in air $v_s=330\text{m/s}$)

The shaded region in the given figure is

Let A ={a,b,c} and the relation R be defined on A as follows.
R ={(a,a),(b,c),(a,b)}

Then, write the ordered pairs to be added in R to make R reflexive and transitive.

For a question like A U B intersection C, if no brackets are specified, how do you approach. Is there a rule such as BODMAS for Operations on Sets also?

If $(3+x^{2008}+x^{2009}{)}^{2010}=a_0+a_{1}x+a_2{x}^2+\dots +a_n{x}^n$,

then the value of

$a_0-\frac{1}{2}{a}_1-\frac{1}{2}{a}_2+a_3-\frac{1}{2}{a}_4-\frac{1}{2}{a}_5+a_6-\cdots$upto $n$ terms is

Area bounded by curve y = k sin x between x = $\pi$ and x = $x=2\pi$ is

If the sum of all possible values of $x$ in $[0,\pi ]$ satisfying $2\sin \left({\cot }^{-1}\right(\cos x\left)\right)-\sqrt{3}=0$ is $\frac{k\pi }{2},$ then $k$ lies in the interval

Find the equation of the circle concentric with the circle
$x^2+y^2-6x+12y+15=0$
and double of its area.

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

$\left[\begin{array}{cc}2& -6\\ 1& -2\end{array}\right]$

The ratio of MPC and MPS is 4:1. The consumption at zero level of income is Rs. 40 crores.

(a) Frame a consumption equation

(b) Value of multiplier

(c) Break-even level of income

OR

The consumption equation is C = 125 + 0.5Y.
(a) Calculate break-even level of income.
(b) Calculate investment if the equilibrium level of income is Rs. 400 crores.

If the coefficient of $5^{\text{th}}$ term is numerically the greatest coefficient in the expansion of $(1-x{)}^n,$ then the positive integral value of $n$ is

Choose the function rule that matches the given system input with its output.

f:N-->N f(×)=x-(-1)^x then f is

One one

Onto

Many one

Into

The number of polynomials P(x) satisfying the equation P(x²) + 2x² + 10x = 2x. P(x + 1) + 3 is

If $f\left(x\right)=2\sqrt{x-1}+5\sqrt{1-x}+(x^2+x+1{)}^{3/2}$ exists, then domain of $f\left(x\right)$ is

The number of solutions of ${\sin }^{7}x+{\cos }^{7}x=1,x\in [0,4\pi ]$ is equal to:

The value of integral $\underset{0}{\overset{1}{\int }}[3x^2+1]dx$ is

(where $[.]$ denotes the greatest integer function)

Let $a_1,a_2,\cdots ,a_n$ be fixed real numbers and define a function $f\left(x\right)=(x-a_1)(x-a_2)\dots (x-a_n)$

What is $\underset{x\to a_1}{lim}f\left(x\right)?$ For some $a\ne a_1,a_2\dots .a_n$ compute $\underset{x\to a}{lim}f\left(x\right)$.

The locus of a point which divides the line segment joining the point $(0,-1)$ and a point on the parabola, $x^2=4y$, internally in the ratio $1:2$, is:

If $A=\underset{1}{\overset{4}{\int }}\{x-0.4\}dx$, where $\left\{x\right\}$ is a fractional part function, then the value of $A$ is