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The range in meters of a projectile launched over a flat ground from the origin with positive velocity $V$ in $m/s$ at an angle $\theta$ given in radian is given by $R=\frac{V^2\sin \left(2\theta \right)}{g}$ where $g$ is a positive constant, assume $V=2m/s,g=10m/s^2$ and $\theta$ was measured to be $\frac{\pi }{12}$ radians. If there was a possible error in the measurement of $\theta$ of $\frac{1}{10\sqrt{3}}$ radians, estimate the corrosponding error in the computation of the range.

Find the 4th term from the end of the G.P. $\frac{1}{2},\frac{1}{6},\frac{1}{18},\frac{1}{54},.......,\frac{1}{4374}$.

If $S_1$ and $S_2$ are respectively the sets of local minimum and local maximum points of the function, $f\left(x\right)=9x^4+12x^3-36x^2+25,x\in R,$ then:

If $\alpha$ and $\beta$ are the roots of the quadratic equation $a{x}^2+bx+c=0,$ then $\underset{x\to \alpha }{lim}\frac{1-\cos (a{x}^2+bx+c)}{(x-\alpha {)}^2}$ is :

Equation of the straight line passing through point of intersection of the line $\frac{x}{a}+\frac{y}{b}=1$ and $\frac{x}{b}+\frac{y}{a}=1$ and is making an angle $\frac{\pi }{4}$ with the line $2x-2y+3=0$ is

${lim}_{x\to 1}\frac{\sqrt{x+8}}{\sqrt{x}}$

$\frac{6x-5}{4x+1}<0$

Solve $12+1\frac{5}{6}x\le 5+3x$ when

$\left(i\right)xϵN$

$\left(ii\right)xϵR$

Draw the graph of the solution set in each case.

sinx/cos3x+sin3x/cos9x+sin9x/cos27x=1/2(tan27x-tanx)

If $\underset{x\to a}{lim}\left[f\right(x)+g(x\left)\right]=10$ and $\underset{x\to a}{lim}f\left(x\right)=2$, then find the value of $\underset{x\to a}{lim}g\left(x\right)$, provided that $\underset{x\to a}{lim}f\left(x\right)$ and $\underset{x\to a}{lim}g\left(x\right)$ exists ___

The number of $7$-digit numbers formed by the digits $1,$ $2$ and $3$ only whose sum of the digits equals $10,$ is

The solution of the inequality ${\log }_{\frac{25-x^2}{16}}\left(\frac{24-2x-x^2}{14}\right)>1$ is

Let $f$ be a real valued function satisfying $f(x+y)=f\left(x\right)+f\left(y\right)$ for all $x,y.$ If $f\left(1\right)=\frac{1}{2}$, then the value of $f\left(16\right)$ is

Prove that $ta{n}^{-1}\frac{1}{4}+ta{n}^{-1}\frac{2}{9}=si{n}^{-1}\frac{1}{\sqrt{5}}.$

The number of all possible matrices of order $3\times 3$ with each entry 0 to 1 is
(a)27
(b)18
(c)81
(d)512

Find the equation of the parabola whose:

(i) focus is (3,0) and the directrix is 3x+4y=1

(ii) focus is (1,1) and the directrix is x+y+1=0

(iii) focus is (0,0) and the directrix is 2x-y-1=0

(iv) focus is (2,3) and the directrix is x-4y+3=0

The equation of the conjugate hyperbola of the hyperbola $x^2-2y^2-2\sqrt{5}x-4\sqrt{2}y-3=0$ is

Find the length of the chord of contact of the tangents drawn from the point (3, 2) to the hyperbola $x^2-9y^2=9$.

Find the value of sgn(-1) + sgn(0) + sgn(1), where sgn$\left(x\right)$ is the signum function

Let $f\left(x\right)=\frac{1}{3}x\sin x-(1-\cos x)$. The smallest positive integer $k$ such that $\underset{x\to 0}{lim}\frac{f\left(x\right)}{x^k}\ne 0$ is

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

The sides $AB,BC$ and $CA$ of $△ABC$ are marked with $3,4$ and $5$ interior points respectively. Number of triangles that can be constructed using these interior points as vertices is

Given the matrices $A$ and $B$ as $A=\left[\begin{array}{cc}1& -1\\ 4& -1\end{array}\right]$ and $B=\left[\begin{array}{cc}1& -1\\ 2& -2\end{array}\right]$. The two matrices $X$ and $Y$ are such that $XA=B$ and $AY=B$. Then $3(X+Y)$ is equal to

If a line makes angles 45^o, 150^o, 135^o, with x, y and z-axes respectively, find its direction cosines.

Let $\alpha ,\beta ,\gamma$ be the roots of the equation $x^3-12x^2+44x-48=0.$ If the coordinates of the vertices of a triangle are $A(\alpha ,\frac{1}{\alpha }),$ $B(\beta ,\frac{1}{\beta })$ and $C(\gamma ,\frac{1}{\gamma }),$ then the centroid of the $△ABC$ is

How many numbers are there between 100 and 1000 which exactly have one of their digits as 7?

Please answer efficiently.

Range of the expression $f\left(x\right)=\frac{x^2-1}{x^2+1},x\in R$ is

The minimum value of $\frac{{\sec }^4{\theta }_1}{{\tan }^2{\theta }_2}+\frac{{\sec }^4{\theta }_2}{{\tan }^2{\theta }_1}$, wherever defined, is
​​​​​​​(correct answer + 1, wrong answer - 0.25)

If $2sec2\alpha =tan\beta +cot\beta$, then one positive value of $\alpha +\beta$ is

Assuming the truth of P(k) and proving P(k + 1) to be true, for some integer k is known as the _______ .

Find the equations of a line which cuts the x-axis at a distance of 3 units to the left of the origin and has a slope equal to -2

If $2x_{1}x-3y_{1}y=2x_1^2-3y_1^2\&4x-3y=5$ are equation of same line, find the value of $x_1+y_1$.

If the curve $xy=R^2-16$ represents a rectangular hyperbola whose branches lie only in the quadrant in which abscissa and ordinate are opposite in sign, then

Let A and B be square matrices of order 3x3. Is (AB)²=A²B²? Give reasons.

If the value of determinant $\left|\begin{array}{ccc}x+1& \alpha & \beta \\ \alpha & x+\beta & 1\\ \beta & 1& x+\alpha \end{array}\right|$is equal to -8, then the value of x, is (where $\alpha ,\beta$ are non real cube roots of unity)

In a $△ABC,\angle A=\frac{\pi }{2}$, then $co{s}^{2}B+co{s}^{2}C$ equals

The number of possible tangent(s) drawn to the hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$, which is/are perpendicular to $5x+2y=10$, is