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Let $P$ and $Q$ be points on the circle $x=a\cos \theta ,y=a\sin \theta$, such that the value of their parameters $\left(\theta \right)$ differs by $\frac{\pi }{6}$. The locus of the point of intersection of tangents to the circle at $P$ and $Q$ is

The angle between the asymptotes of the hyperbola $x^2–3y^2=3$ is

If tangents are drawn to the parabola $(x-3{)}^2+(y+4{)}^2=\frac{(3x-4y-6{)}^2}{25}$ at the extremities of the chord $2x-3y-18=0$, then angle between tangents is

Solution of the equation tan ^-1 (x - 1) + tan ^-1 x + tan ^-1 (x+1)=tan ^-1 3x is

${lim}_{x\to 0}\frac{xcosx+2sinx}{x^2+tanx}$

If $si{n}^x\theta +co{s}^x\theta \ge 1$, $0<\theta <\frac{\pi }{2}$, then

The coefficient of the term independent of x in ${[\sqrt{\frac{x}{3}}+\sqrt{\frac{3}{2x^2}}]}^{10}$ is

If ${\int }_{ln2}^x\frac{du}{(e^u-1{)}^{\frac{1}{2}}}=\frac{\pi }{6}$, then $e^x$ equals

A laboratory blood test is 99% effective in detecting a certain diesease when it is fact present. However, the test also yields a false positive result for 0.5% of the healthy person tested (i.e., if a healthy person is tested, then with probability 0.005, the test will imply he has the disease). If 0.1% of the population actually has the disease, what is the probability that a person has diease given that his test result is positive?

If $\alpha$ and $\beta$ are the roots of the equation $a{x}^2+bx+c=0;a,b,c\in I^{+}$, such that $\Delta =\left|\begin{array}{ccc}3& 1+\alpha +\beta & 1+{\alpha }^2+{\beta }^2\\ 1+\alpha +\beta & 1+{\alpha }^2+{\beta }^2& 1+{\alpha }^3+{\beta }^3\\ 1+{\alpha }^2+{\beta }^2& 1+{\alpha }^3+{\beta }^3& 1+{\alpha }^4+{\beta }^4\end{array}\right|$, then $\Delta$ is always divisible by

The number of real solutions of $(x-1)(x+1)(2x+1)(2x-3)=15$ is

Find the equation of the straight line which passes through the point (-3, 8) and cuts off positive intercepts on the coordinate axes whose sum is 7.

Write the domain and range of function f(x) given by $f\left(x\right)=\frac{1}{\sqrt{x-\left[x\right]}}$.

Let $A$ and $B$ be any two points on the lines represented by $4x^2-9y^2=0.$ If the area of triangle $OAB$ is $5$ $\left(O\text{is origin}\right)$ then which of the following is the possible equation of the locus of midpoint of $AB?$

Prove that following identities:

$tanA+tan({60}^{\circ }+A)-tan({60}^{\circ }-A)=3tan3A$

The mean and standard deviation of a group of 100 observations were found to be 20 and 3 respectively. Later on it was found that three observations were incorrect, which were recorded as 21, 22 and 18. Find the mean and standard deviation if the incoorect observations were omitted.

Find the conntuiuty of following function.

$f\left(x\right)=\{\begin{array}{c}|x|+3,ifx\le -3\\ -2x,if-3<x<3\\ 6x+2ifx\ge 3\end{array}$

Solve $-4\le \frac{3x-2}{5}\le 2$

2x-5\5x+2=3\22

Acute angle between the line $\frac{x-5}{2}=\frac{y+1}{-1}=\frac{z+4}{1}$ and the plane $3x–4y–z+5=0$ is

$\frac{x-1}{3}+4<\frac{4-5}{5}-2$

Prove that the greatest integer function $f:R\to R$, given by f(x) =[x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.

Prove that the determinant $\left|\begin{array}{ccc}x& sin\theta & cos\theta \\ -sin\theta & -x& 1\\ cos\theta & 1& x\end{array}\right|$ is independent of $\theta$.

The value of ${\int }_{-1}^{2}f\left(x\right)\mathrm{dx},\mathrm{where}f\left(x\right)=\left(x+1\right|+\left(x\right|+\left(x-1\right|$ is

A man rides his motorcyle at the speed of 50 km/h. He has to spend Rs 2 per km on petrol. If he rides it at a faster speed of 80 km/h, the petorl cost increases to Rs 3 per km. He has atmost Rs 120 to spend on petrol and one hour's time. He wishes to find the maximum distance that he can travel. Express this problem as a linear programming problem.

A coin is tossed. If it shows a tail, we draw a ball from a box which contains 2 red and 3 black balls; if it shows a head, we throw a die. Find the sample space for this experiment.

The set of solutions for $x+\frac{1}{x}\ge 2$ is

How many 7-digit numbers can be formed using the digits1,2,0,2,4,2,4?

show that the relation in the set A={x€Z:0<x<12} given by R={(a,b):|a-b| is a multiple of 4} is an equivalence relation. Find the set of all elements related to 1.

Please explain the answer in detail

The value of $(1-i{)}^4$ is

Let A, B are two non-singular matrices of order 3 with real entries such that $\text{adj}\left(A\right)=3B$ and $\text{adj}\left(B\right)=2A$ then -

find domain and range for y=√(9x-x^2 )

Number of $4$ letter words starting with vowel that can be formed using letters of the word ${}^{\prime }CHEMISTR{Y}^{\prime }$ are

If n is an odd natural number , then $\sum _{r=0}^n\frac{(-1{)}^r}{{}^r{C}_r}$ equals :

In $8$ people, there are more girls than boys. How many girl could be there?

The Boolean expression $(p\wedge \sim q)\vee q\vee (\sim p\wedge q)$ is equivalent to

Let f(x) = $x^2$ -bx+c,b is odd positive integer, f(x) = 0 have two prime

numbers as roots and b+c=35.Then the global minimum value of f(x) is

In a GP, if the (m + n)thterm is P and the (m – n)th term is q, then its mth term is

The opposite of the word ‘dependent’ will take the prefix

If $\alpha ,\beta$ are the roots of the equation $8x^2-3x+27=0$ then the value of ${\left(\frac{{\alpha }^2}{\beta }\right)}^{\frac{1}{3}}+{\left(\frac{{\beta }^2}{\alpha }\right)}^{\frac{1}{3}}$ is.

Assume that each childborn is equally likely to be a boy or a girl .if a family has three children, then conditional probability that two children are girls given that family has at least one girl child is

The volume of the tetrahedron with vertices at (1,2,3), (4,3,2), (5,2,7), (6,4,8) is

Let $n\left(U\right)=700$, $n\left(A\right)=200$, $n\left(B\right)=300$ and $n(A\cap B)=100$, Then $n(A^C\cap B^C)=$