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in GP why three terms that we take is different as when there is product it is a/r,a,ar ² and when it is sum it is a, ar , ar ² . in ap it is same all the time whether it is product or sum. please clarify my doubt.

Let $f\left(x\right)=x-x^2$ and $g\left(x\right)=ax$. If the area bounded by $y=f\left(x\right)$ and $y=g\left(x\right)$ is equal to the area bounded by the curves $x=3y-y^2$ and $x+y=3$, then the number of possible value(s) of $a$ is

Let g(x) a polynomial of degree 3 passing through origin and have a local maximum at $x=-\frac{1}{2\sqrt{2}}$. Also g’(x) has a local minimum at x = 0 and g(1) = 5.

Let f(x) = sgn(x) (where sgn(x) denotes signum function of x), then which of the following statement is incorrect for f(g(x))?

The two circles $x^2+y^2-4y=0$ and $x^2+y^2-8y=0$

Prove the following by using the principle of mathematical induction for all n ∈ N: 10^{2n – 1} + 1 is divisible by 11.

Let
U be the set of all triangles in a plane. If A is the set of all
triangles with at least one angle different from 60°,
what is?

$Ina\Delta ABC,ifcosC=\frac{sinA}{2sinB},\text{prove that the triangle is isosceles.}$

Let $A$ and $B$ be two events with $P\left(A^c\right)=0.55,P\left(B\right)=0.36,P(A\cup B)=\mathrm{0.60.}$ Then $P\left(A|B^c\right)$ is

In a parallelogram OABC, vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are respectively the position vectors of vertices A, B, C with reference to O is origin. A point $\overrightarrow{E}$ is taken on the side BC which divides it in the ratio of 2 : 1. Also, the line segment AE intersects the line bisecting the angel O internally in point P. If CP, when extended meets AB in point F. Then

The position vector of point F is

If $f\left(x\right)=|x^2-4|,$ then the value of derivative of $f\left(x\right)$ at $x=-1$ is



[1 mark]

The number of $5$ letters palindrome words that can be formed only using vowels is

Binomial coefficient of $7^{th}$ term in the expansion of $(2x-3y{)}^{20}$ is/are

Which of the following is a superset of the set $X=\{k:k\text{is a factor of}6\}?$

If $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right],$ where $a,b,c,d$ are the roots of the equation $x^4-13x^3+p{x}^2+qx-64=0,$ $p,q\in R$ whose three roots are positive and one is negative, then the minimum positive value of $\text{det}\left(A\right)$ is

In an acute angle $△ABC,2\cos A=\frac{\sin B}{\sin C}$​ and $2^{{\tan }^{2}B}$ is a solution of the equation $x^2-9x+8=0,$ then the distance between incentre to the circumcentre of $△ABC$ is

Number of points of discontiuity of the function $f\left(x\right)=\frac{1}{2\sin x-1}$ in $[0,2021\pi ]$ is

The length of the latus rectum of the parabola $25\left[\right(x-2{)}^2+(y-4{)}^2]=(4x-3y+12{)}^2$ is

If $sin\theta$ and $cos\theta$ are the roots of $a{x}^2+bx+c=0,$ then $co{s}^{-1}(a^2-b^2+2ac)=$

Explain,
giving reasons, which of the following sets of quantum numbers are
not possible.

A capillary tube is immersed vertically in water and the height of the water column is x. When this arrangement is taken into a mine of depth d, the height of the water column is y. If R is the radius of the earth, the ratio $\frac{x}{y}$ is

The equation of tangent to the parabola $y^2=x$ at point $(4,2)$ is

The arithmetic mean of two numbers $a$ and $b$ $(a<b)$ is $6.$ If the geometric mean $G$ and harmonic mean $H$ of the two numbers satisfy the relation $G^2+3H=48,$ then the value of ${\log }_{b}a$ is

By using properties of definite integrals, evaluate the integrals
${\int }_0^{\pi }log(1+cosx)dx.$

If $f\left(x\right)=\frac{x^2-8}{x-3}$ is an increasing function, then

Number of ordered pairs $(x,y)$ which satisfies $\frac{x^4+1}{8x^2}={\sin }^{2}y{\cos }^{2}y;$ where $y\in [0,2\pi ]$ is

If $\sum _{r=1}^n{t}_r=\sum _{k=1}^n\sum _{j=1}^k\sum _{i=1}^j\left(2\right),$ then $\sum _{r=1}^n\frac{1}{t_r}$ is

The value of $\underset{x\to 0}{lim}\frac{1}{x^3}{\int }_0^x\frac{t\ln (1+t)}{t^4+4}dt$ is

Show that $\left|\frac{z-2}{z-3}\right|=2$ represents a circle. Also, find its centre and radius.

Or

If arg (z - 1) = arg (z + 3 i), find (x - 1) : y, where z = x + iy.

Let $A$ be a $3\times 3$ matrix with $\text{det}\left(A\right)=4.$ Let $R_i$ denote the $i^{\text{th}}$ row of $A.$ If a matrix $B$ is obtained by performing the operation $R_2\to 2R_2+5R_3$ on $2A,$ then $\text{det}\left(B\right)$ is equal to:

The equation of the circle in the first quadrant which touches each axis at a distance 5 from the origin is

What is the inverse of the matrix $A=\left[\begin{array}{cc}3& 1\\ 7& 9\end{array}\right]$

The sum of $1^3$ + $2^3$ + $3^3$ + $4^3$ + ........... + ${15}^3$ is