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The value of $\stackrel{\pi /2}{\underset{-\pi /2}{\int }}\frac{x^2}{1+\tan x+\sqrt{1+{\tan }^{2}x}}dx$ is

The number of integers which do not satisfy the given equation $|x-1|+|x-2|+|x-3|\ge 6$ is

(a) Complete the table and by inspection of the table, find the
solution to the equation m + 10 = 16.

m	1	2	3	4	5	6	7	8	9	10	…
m + 10	−	−	−	−	−	−	−	−	−	−	−

(b) Complete the table and by inspection of the table, find the
solution to the equation 5t = 35.

t	3	4	5	6	7	8	9	10	11	…
5t	−	−	−	−	−	−	−	−	−	−

(c) Complete the table and find the solution of the equation z/3 = 4
using the table.

z	8	9	10	11	12	13	14	15	16	...
		3		−	−	−	−	−	−	−

(d) Complete
the table and find the solution to the equation m − 7 =
3

m	5	6	7	8	9	10	11	12	13	…
m − 7	−	−	−	−	−	−	−	−	−	−

If the sum of $n$ terms of an A.P. is given by $S_n=3n^2-4n,$ then its ${50}^{th}$ term is

In a factory 70% of the workers like oranges and 64% likes apples. If x% like both oranges and apples, then what are the possible values of x?

The parabola $x^2=py$ passes through (12, 16). Then the focal distance of the point is

The value of $\sin \frac{17\pi }{36}\cos \frac{11\pi }{36}\cos \frac{13\pi }{36}\sin \frac{11\pi }{36}\sin \frac{13\pi }{36}\cos \frac{17\pi }{36}$ is

For natural number n, $2^n$(n-1) ! < $n^n$, if

If a and b
are the roots of
are
roots of
,
where a, b, c, d, form a G.P. Prove that
(q + p): (q – p) = 17:15.

The locus of the points of trisection of the double ordinates of a parabola is a

Column Matching:



$$\begin{array}{cc}\text{Column (I)}& \text{Column (II)}\\ \begin{array}{cc}\text{(A)}& \text{In}{R}^2,\text{if the magnitude of the projection}\\ & \text{vector of the vector}\alpha \hat{i}+\beta \hat{j}\text{on}\sqrt{3}\hat{i}+\hat{j}\\ & \sqrt{3}\text{and if}\alpha =2+\sqrt{3}\beta ,\text{then possible}\\ & \text{value(s) of}|\alpha |\text{is (are)}\end{array}& \text{(P) 1}\\ & \\ \begin{array}{cc}\text{(B)}& \text{Let}a\text{and}b\text{be real numbers such that}\\ & \text{the function}\\ & f\left(x\right)=\{\begin{array}{c}-3a{x}^2-2,x<1\\ bx+a^2,x\ge 1\end{array}\\ & \text{is differentiable for all}x\in R.\text{Then}\\ & \text{possible value(s) of}a\text{is (are)}\end{array}& \text{(Q) 2}\\ & \\ \begin{array}{cc}\text{(C)}& \text{Let}\omega \ne 1\text{be a complex cube root of}\\ & \text{unity. If}(3-3\omega +2{\omega }^2{)}^{4n+3}+\\ & (2+3\omega -3{\omega }^2{)}^{4n+3}+(-3+2\omega +3{\omega }^2{)}^{4n+3}=0,\\ & \text{then possible value(s) of}n\text{is (are)}\end{array}& \text{(R) 3}\\ & \\ \begin{array}{cc}\text{(D)}& \text{Let the harmonic mean of two positive real}\\ & \text{numbers}a\text{and}b\text{be}4.\text{If}q\text{is a positive real}\\ & \text{number such that}a,5,q,b\text{is an arithmetic}\\ & \text{progression, then the value(s) of}|q-a|\text{is (are)}\end{array}& \text{(S) 4}\\ & \\ & \text{(T) 5}\\ & \end{array}$$

Option $\text{(D)}$ matches with which of the elements of right hand column?

$3^{2n+2}-8n-9$ is divisible by 8 for all n $ϵ$ N.

The value of a cos $\theta$ + b sin $\theta$ lies between

Evaluate :cos (tan^–1 x)

Examine if Rolle's theorem is applicable to any of the following functions. Can you say something about the converse of Rolle's theorem from these example ?

(i) $f\left(x\right)=\left[x\right]forxϵ[5,9]$

(i) $f\left(x\right)=\left[x\right]forxϵ[-2,2]$

(iii) $f\left(x\right)=x^2-1forxϵ[5,9]$.

The value of $\underset{0}{\overset{\pi /2}{\int }}\sin xdx+\underset{0}{\overset{1}{\int }}{\sin }^{-1}xdx$ is

The length of the perpendicular drawn from the point P(a, b, c) from z-axis is

The random variable X has a probability distribution P(X) of the following form, where k is some number
$\{\begin{array}{c}K,ifX=0\\ 2k,ifX=1\\ 3k,ifX=2\\ 0,ifotherwise\end{array}$

Find $P(X<2),P(X\le 2),P(X\ge 2)$

A function f(x) will have a local maximum at x = c if

[ h is positive and tends to zero]

An ellipse has eccentricity $\frac{1}{2}$ and one of the foci at the point $S(\frac{1}{2},1).$ One of the directrices of the ellipse is the common tangent to the circle $x^2+y^2=1$ and the hyperbola $x^2-y^2=1,$ corresponding to the focus $S.$ The equation of the ellipse in standard form is

A line makes the same angle $\theta$ with each of the $x$ and $z-$axes. If the angle $\beta ,$ which it makes with the $y-$axis, is such that ${\sin }^2\beta =3{\sin }^2\theta ,$ then ${\cos }^2\theta =$

Find the local maxima and local minima, if any of the following function. Also, find the local maximum and the local minimum values, as the case may be as follows.

$g\left(x\right)=\frac{x}{2}+\frac{2}{x},x>0$

Find the number of integral values of a for which the quadratic expression (x - a) (x - 10) + 1Can be factored as a product (x+α)(x+ β )of two factors and α, βϵI

Unit vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are coplanar. A unit vector $\overrightarrow{d}$ is perpendicular to them. If $(\overrightarrow{a}\times \overrightarrow{b})\times (\overrightarrow{c}\times \overrightarrow{d})=\frac{1}{6}\hat{i}-\frac{1}{3}\hat{j}+\frac{1}{3}\hat{k}$ and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is ${30}^{\circ }$ then $\overrightarrow{c}$ is equal to

Find the equation of the straight line passing through the point of intersection of the lines $5x-6y-1=0$ and $3x+2y+5=0$ and perpendicular to the line $3x-5y+1=0.$

Find the slope of the tangent to the
curve,
x ≠ 2 at x =
10.

Let $n_1$ and $n_2$ represent respectively the number of possible ordered and unordered triplets $(a,b,c)$ such that $abc=144,(a,b,c\in N)$, then

A point $P(x_1,y_1)$ lies on the same plane as that of the circle $x^2+y^2+2gx+2fy+c=0$. Also $T_1\Rightarrow x{x}_1+y{y}_1+g(x+x_1)+f(y+y_1)+c=0$

Statement (1) : $T_1$ represents a tangent if P is on the circle

Statement (2) : $T_1$ represents a chord of contact if p is outside the circle

The coefficient of $x^2$ in the expansion of ${1+x+x^2}^{10}$is