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For all $nϵN,3\times 5^{2n+1}+2^{3n+1}$ is divisible by

If one root of the equation $a{x}^2+bx+c=0$ the square of the other, then a $(c-b{)}^3=cX,$ where X is

Two vertices of a triangle are $(-2,-1)$ and $(3,2)$ and third vertex lies on the line $x+y=5.$ If the area of the triangle is 4 square units, then the third vertex is

What are the points on X - axis whose perpendicular distance from the straight line $\frac{x}{a}+\frac{y}{b}=1$ is a ?

Is $g=\left\{\right(1,1),(2,3),(3,5),(4,7\left)\right\}$ a function? If g is described by $g\left(x\right)=\alpha x+\beta$, then what value should be assigned to $\alpha$ and $\beta$?

Find the general solutions of the following equations :
$\left(i\right)sin\theta =\frac{1}{2}\left(ii\right)cos\theta =\frac{-\sqrt{3}}{1}\left(iii\right)cosec\theta =-\sqrt{2}\left(iv\right)sec\theta =\sqrt{2}\left(v\right)tan\theta =-\frac{1}{\sqrt{3}}\left(vi\right)\sqrt{3}sec\theta =2$

Arithmetic mean of three numbers which are in G.P. is $\frac{14}{3}.$ By adding 1 to the first and second number, and subtracting 1 from the third number, resulting numbers are in A.P. Then sum of squares of original three numbers is

The focus of the parabola y=$2x^2+x$ is

The equation of the line which is at a distance of $3$ units from the origin and the perpendicular from the origin makes an angle of ${30}^{\circ }$ with positive $x-$axis is

In an examination, a student has to answer 4 questions out of 5 questions ; questions 1 and 2 are however compulsory, Determine the number of ways in which the student can make the choice.

If the sum of reciprocals of intercepts made by a straight line on the co-ordinate axes is a constant $K$, then the line will always pass through the point

If $\int f\left(x\right)\sin x\cos xdx=\frac{1}{2(b^2-a^2)}\log f\left(x\right)+c,$ where $c$ is the constant of integration, then $f\left(x\right)=$

Simplify : ( 3p+ pq) - ( p- q)

Evaluate $P(A\cup B),if2P\left(A\right)=P\left(B\right)=\frac{5}{13}andP\left(\frac{A}{B}\right)=\frac{2}{5}.$

By using properties of definite integrals, evaluate the integrals
${\int }_0^{1}x(1-x{)}^{n}dx.$

If there are two values of a which makes determinant, $\Delta =\left|\begin{array}{ccc}1& -2& 5\\ 2& a& -1\\ 0& 4& 2a\end{array}\right|=86$, then the sum of these numbers is

(a) 4
(b) 5
(c) - 4
(d) 9

Determine whether or not each of the definition of $\ast$ given below gives a binary operation. In the event that $\ast$ is not a binary operation, give justification for this.
(iv) On $Z^{+}$, defined $a\ast b=|a-b|$

Show that the vector $\hat{i}+\hat{j}+\hat{k}$ is equally inclined to the axes OX, OY and OZ.

Prove that $y=\frac{4sin\theta }{(2+cos\theta )}-\theta$ is an increasing on $\theta$ in $[0,\frac{\pi }{2}]$

The value of $\left|\begin{array}{ccc}x& x+y& x+2y\\ x+2y& x& x+y\\ x+y& x+2y& x\end{array}\right|$ is

(a) $9x^2(x+y)$
(b) $9y^2(x+y)$
(c) $3y^2(x+y)$
(d) $7x^2(x+y)$

The value of $\underset{x\to 0}{lim}[min(y^2+6y+17\left)\frac{\sin x}{x}\right]$ is
$\left(\right[.]$ denotes the greatest integer function$)$

DEFNITION - A set which is empty or consists of a different number of element is called a finite set? THIS DEFNITION IS WRITTEN IN NCERT BOOK. WHY WE USE THE WORD EMPTY THEIR?

If pair of straight lines $x^2-y^2+6x+4y+5=0$ are transverse and conjugate axes of hyperbola and perpendicular distance form origin to these lines represent the length of transverse and conjugate axis, then the eccentricity of hyperbola is:

In a triangle ABC , cos 3A+ cos 3B+ cos 3C = 1 ,then find any one angle.

For all real values of $a_0,a_1,a_2,a_3$ satisfying $a_0+\frac{a_1}{2}+\frac{a_2}{3}+\frac{a_3}{4}=0$, the equation $a_0+a_{1}x+a_2{x}^2+a_3{x}^3=0$ has a real root in the interval

If a-2b = 15 and find a²+ 4b² .

If x is real, then the minimum value of $x^2-8x+17$ is……

The incentre of the triangle whose vertices are $(1,\sqrt{3}),(0,0)$ and $(2,0)$ is

What is the value of tan 25^o?

If f:N$\to$N be a function defined by f(x) = $x^2$ + x, then find f(-2).f(2) + f(3).

If we cross AaBBCcDdeeFf ×AABBccDdEeFf . Then what will be the chance of genotype AaBBccddEeFf ?

If A(1, 2, 3), B(-1, -1, -1) be the points, then the distance AB is
[MP PET 2001; Pb. CET 2001]

How is N related to K?

I. N's brother is married to D.

II. D's sister is married to K.

$\text{The value of}(1+i)(1+i^2)(1+i^3)(1+i^4)\text{is}$

In a certain test $a_i$ students gave wrong answers to at least i questions where i = 1, 2, 3, ....k. No student gave more than k wrong answers. The total number of wrong answers given is

If x = $sec\theta -cos\theta$,$y$= $se{c}^{10}\theta -co{s}^{10}\theta$, and

$(x^2+4){\left(\frac{dy}{dx}\right)}^2$ = $k(y^2+4)$, then k is equal to

The differential equation of the family of curves $y=e^x(A\cos x+B\sin x)$ where $A,B$ are arbitary constants is