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If $\left|\begin{array}{c}2a{x}_1{y}_1\\ 2b{x}_2{y}_2\\ 2c{x}_3{y}_3\end{array}\right|=\frac{abc}{2}\ne 0$, then the area of the triangle whose vertices are$(\frac{x_1}{a},\frac{y_1}{a}),(\frac{x_2}{b},\frac{y_2}{b}),(\frac{x_3}{c},\frac{y_3}{c})$ is

Write the number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines.

The value of $\underset{-\pi /2}{\overset{\pi /2}{\int }}\frac{{\cos }^{2}x}{1+3^x}dx$ is :

Which of the following is/are a function?

The angle between a pair of tangents drawn from a point $P$ to the circle $x^2+y^2+4x-6y+9{\sin }^2\alpha +13{\cos }^2\alpha =0$ is $2\alpha .$ The equation of the locus of the point $P$ is

Prove $x=\frac{n\pi }{2}$ or $x=(\frac{m\pi }{2}+\frac{3\pi }{8})$, where m, $n\in I$

The set of values of $x$ for which the function $f\left(x\right)={\log }_{[x+\frac{1}{2}]}|x^2-5x+6|$ is defined is
(where [.] denote the greatest integer function)

$\int \frac{dx}{x\sqrt{x^4-1}}=$

A hyperbola has its centre at the origin, passes through the point $(4,2)$ and has transverse axis of length $4$ along the $x$-axis. Then the eccentricity of the hyperbola is :

The length of transverse common tangent to two circles is $5\text{units}$ and a direct common tangent is $15\text{units}$, then the product of the radii of two circles is

$\int \frac{cosx-cos2x}{1-cosx}dx=$

${\int }_0^2(x^2+3)dx$

If the $A.M.$ between $p^{th}$ and $q^{th}$ terms of an $A.P.$ is equal to the $A.M.$ between $r^{th}$ and $s^{th}$ terms of the $A.P.$, then which of the following is true

If $f\left(1\right)=1,f^{\prime }\left(1\right)=2$, then write the value of $li{m}_{x\to 1}\frac{\sqrt{f\left(x\right)}-1}{\sqrt{x}-1}$

A point $P$ moves inside a square of area $4$ sq.units such that it is nearer to point of intersection of its diagonal than any vertex. Area of region traced by $P$ is-

Find the intervals in which the function given by $f\left(x\right)=4x^3-6x^2-72x+30$ is

i) Strictly increasing ii) Strictly decreasing

Among the following, the one which is not a differential equation is .

Any complex number in the polar form can be expressed in Euler's form as $\cos \theta +i\sin \theta =e^{i\theta }$. This form of the complex number is useful in finding the sum of series $\sum _{r=0}^n{}^n{C}_r(\cos \theta +i\sin \theta {)}^r$.
$\sum _{r=0}^n{}^n{C}_r(\cos r\theta +i\sin r\theta )=\sum _{r=0}^n{}^n{C}_r{e}^{ir\theta }=\sum _{r=0}^n{}^n{C}_r(e^{i\theta }{)}^r=(1+e^{i\theta }{)}^n$
Also, we know that the sum of binomial series does not change if $r$ is replaced by $n-r$. Using these facts, answer the following questions.

The value of $\sum _{r=0}^{100}{}^{100}{C}_r\left(\sin rx\right)$ is equal to

Write the remainder obtained when 1! + 2! + 3!+....+200! is divided by 14.

$\frac{c}{a-b}=\frac{tan\left(\frac{A}{2}\right)+tan\left(\frac{B}{2}\right)}{tan\left(\frac{A}{2}\right)-tan\left(\frac{B}{2}\right)}$

write the axis of symmetry of the parabola $y^2=x.$

Two vectors $\overline{a}\text{and}\overline{b}$ are at an angle of 60⁰ with each other. Their resultant makes an angle of 45⁰ with $\overline{a}$ . If $|\overline{b}|$ =2 units, then $|\overline{a}|$ is

The value of $\tan \left(\frac{\pi }{20}\right)\tan \left(\frac{3\pi }{20}\right)\tan \left(\frac{5\pi }{20}\right)\tan \left(\frac{7\pi }{20}\right)\tan \left(\frac{9\pi }{20}\right)$ is

If $\alpha ,\beta ,\gamma$ are the roots of $x^3+3x^2+4x+5=0$ , then which of the following is/ are true.

Let $A=\left[\begin{array}{cc}2& 4\\ 3& 2\end{array}\right],C=\left[\begin{array}{cc}-2& 5\\ 3& 4\end{array}\right]$.

Find the value of following:
3A-C

Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, -1) and (4, 3, -1)

A manufacturer of electronic circuits has a stock of 200 resistors, 120 transistors and 150 capacitors and is required to produce two types of circuits A and B. Type A requires 20 resitors, 10 transistors and 10 capacitors. Type B requires 10 resistors, 20 transistors and 30 capacitors. If the profit on type A circuit is Rs 50 and that on type B circuit is Rs 60, formulate this problem as a LPP, so that the manufacturer can maximise his profit.

The roots of the equation $t^3+3a{t}^2+3bt+c=0$ are $z_1,z_2,z_3$ which represent the vertices of an equilateral triangle, then

A pair of fair dice is tossed repeatedly until a sum of four or an odd sum appears. Then the probability that a sum of four appear first is

Given ellipse $x^2+4y^2=16$ and parabola $y^2-4x-4=0.$
The quadratic equation whose roots are the slopes of the common tangents to the parabola and the ellipse, is

If $\frac{1}{\sqrt{b}+\sqrt{c}},\frac{1}{\sqrt{c}+\sqrt{a}},\frac{1}{\sqrt{a}+\sqrt{b}}$ are in A.P., then the family of lines $ax+by+c=0$ will always passes through the fixed point:

If the locus of the circumcentre of a variable triangle having sides $y-$axis, $y=2$ and $lx+my=1$, where $(l,m)$ lies on the parabola $y^2=4ax$ is a curve $C$, then the length of smallest focal chord of this curve $C$ (in units) is

If $f:[-2,2]\to R$ is defined by $f\left(x\right)=\left\{\begin{array}{c}-1,for-2\le x\le 0then\\ x-1,for0\le x\le 2\end{array}\right\}$

The image of the point $(-8,12)$ with respect to the line mirror $4x+7y+13=0$ is

If $p=\frac{1}{{\log }_3\pi }+\frac{1}{{\log }_4\pi }+1$ then

The number of subsets of the set $A=\{1,2,3,\dots ,9\}$ containing at least one odd number is

The range of $f\left(x\right)=\frac{x^2-81}{x-9}$ is

Solve for x : $\frac{3}{x}$ < 5

$Ifsinx+si{n}^{2}x=1,thenthevalueofco{s}^{12}x+3co{s}^{10}x+3co{s}^{8}x+co{s}^{6}x-1isequalto$