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If $z_1,z_2\in C,z_1^2+z_2^2\in R,z_1(z_1^2-3z_2^2)=2$ and $z_2(3z_1^2-z_2^2)=11$, then the value of $z_1^2+z_2^2$ is

Find the derivative of $y=sin(x^2-4)$

The CORRECT formula for the sentence, "not all rainy days are cold" is

If a hyperbola has length of its conjugate axis equal to $5$ and the distance between its foci is $13$, then the eccentricity of the hyperbola is :

Match the following with the parametric equation of the parabola:

$If|x|<1then\frac{d}{dx}(1-2x+3x^2-........)=$

The value of ${(}^{21}{C}_1{-}^{10}{C}_1)+{(}^{21}{C}_2{-}^{10}{C}_2)+{(}^{21}{C}_3{-}^{10}{C}_3)+{(}^{21}{C}_4{-}^{10}{C}_4)+...+{(}^{21}{C}_{10}{-}^{10}{C}_{10})$ is

In the quadratic equation p(x)=0 with real coefficients has purely imaginary roots. Then, the equation p[p(x)]=0 has

The number of ways in which 5 identical balls can be kept in 10 identical boxes, if not more than one can go into a box, is

In $△$ ABC, a sin (B - C) + b sin (C - A) + c sin (A-B) =

[ISM Dhanbad 1973]

If the semi-major axis of an ellipse is 3 and the latus rectum is 16/9, then the standard equation of the ellipse is

Determine the number of 5-card combinations out of a deck of 52 cards if each selection of 5 cards has exactly one king.

The general solution of $sinx=sin\alpha ,\alpha ϵ[-\frac{\pi }{2},\frac{\pi }{2}]$ is

In how many ways, can the letters of the word "HONESTY" be arranged ?

Also, in how many ways, can the letters be arranged when it starts with vowel and end with vowel.

Do you like jumbled letters of word honesty ? Why honesty is acquired in your life.

Suppose $A$ is any $3\times 3$ non-singular matrix and $(A-3I)(A-5I)=O$, where $I=I_3$ and $O=O_3$. If $\alpha A+\beta A^{-1}=4I$, then $\alpha +\beta$ is equal to :

Find the value of ${}^n{C}_0.(n+1)+n.{}^n{C}_1+(n-1){}^n{C}_2....1.{}^n{C}_n$

In a triangle ABC, Let , $\angle C=\frac{\pi }{2}$, if r is the inradius and R is the circumradius of the triangle ABC, then 2(r+R) equals

If $\tan x=n\tan y,n\in R^{+}$, then maximum value of ${\sec }^2(x-y)$ is

If the sum of the coefficients in the expansion of $(a+b{)}^n$ is 4096, then the greatest coefficient in the expansion is

Let P and Q be two finite sets such that n(P – Q) = 20, n(P U Q) =95, n(P ∩ Q) =35, then n(Q – P) is

If $6^{th}$ term in the expansion of $(\frac{3}{2}+\frac{x}{3}{)}^n$ is the numerically greatest term when x=3, then find the sum of all possible values of n

If the first term of a G.P. $a_1$, $a_2$, $a_3$,.........is unity such that

4$a_2$ + 5$a_3$ is least, then the common ratio of G.P. is

In
the First Past the Post system, that candidate is declared winner who

a.
Secures the largest number of postal ballots

b.
Belongs to the party that has highest number of votes in the country

c.
Has more votes than any other candidate in the constituency

d.
Attains first position by securing more than 50% votes

The acute angle between the curves $y^2=x\mathrm{and}{x}^2=y$ at (1, 1) is .

Let $△PQR$ be a right triangle, right angled at $R.$ If $\tan \left(\frac{P}{2}\right)$ and $\tan \left(\frac{Q}{2}\right)$ are the roots of the quadratic equation $a{x}^2+bx+c=0$, then which of the following is correct ?

The ratio of the speed of sound in nitrogen gas to that in helium gas, at 300 K is

[IIT 1999]

Find the value of other five trigonometric functions if tan x = $\frac{-5}{12}$, and x lies in second quadrant.

If $X=\{1,2,3,4,5\}$ and $Y=\{1,3,5,7,9\}$, then which of the following relation(s) is/are not a function from $X\to Y$?

If $\frac{4}{x}\ge 7$, then the range of $x$ is

Match the following (|x| < 1)

(1) ${(1+x)}^{-1}$ (P) 1 + 2x + 3 $x^2$ + 4 $x^3$.........

(2) ${(1-x)}^{-1}$ (Q) 1 - x + $x^2$ - $x^3$.........

(3) ${(1+x)}^{-2}$ (R) 1 + x + $x^2$ + $x^3$.........

(4) ${(1-x)}^{-2}$ (S)1 - 2x + 3 $x^2$ - 4 $x^3$.........

Express the following in standard form : (2 - 3i)²

Which one of the following predicate formulae is NOT logically valid?

Note that W is a predicate formula without any free occurrence of x.

If the lines $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ and $\frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}$ intersect, then the value of k is -

The sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one. Then the sides of the triangle are

If $f$ is a function such that $f\left(0\right)=2,$ $f\left(1\right)=3$ and $f(x+2)=2f\left(x\right)-f(x+1)$ for every real $x,$ then the value of $f\left(5\right)$ is

$If{x}^2+2ax+a<0forallxϵ[1,2]then:$

The coordinates of the point(s), which trisect(s) the line segment joining the points $(1,-2)$ and $(-3,4),$ are

Let $f\left(x\right)=x^2$ and $g\left(x\right)=3x+4$. Find the value of $\frac{(f+g)\left(x\right)}{(f-g)\left(x\right)}$ when $x=1$

Two fair dice are rolled simultaneously. It is found that one of the dice show odd prime number. The probability that the remaining dice also show an odd prime number, is equal to

Ten persons, amongst whom are A, B and C to speak at a function. The number of ways in which it can be done if A wants to speak before B and B wants to speak before C is

A line passes through (2,2) and is perpendicular in the line 3x+y = 3 its y-intercepts is ------------

Let $f\left(x\right)=\{\begin{array}{cc}{x}^2,& x\ge 0\\ ax,& x<0\end{array}$

The set of real values of $a$ such that $f\left(x\right)$ will have local minima at $x=0,$ is:

Suppose $x$ and $y$ are natural numbers, then the number of ordered pairs $(x,y)$ which satisfy $x+y\le 5$ is

Let $p,p_1$ be A.M. and G.M. between $a$ and $b$ respectively and $q,q_1$ be the A.M. and G.M. between $b$ and $c$ respectively where $a,b,c>0$. If $a,b,c$ are in A.P., then which of the following is CORRECT?