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If the quadratic equations $a{x}^2+2cx+b=0$ and $a{x}^2+2bx+c=0,(b\ne c)$ have a common root, then the value of $a+4b+4c$ is

In $\Delta ABC$, if $\angle C=3\angle A,BC=27$ and $AB=48$. Then the value of $AC$ is

The value of $sin\left[2ta{n}^{-1}\left(\frac{1}{3}\right)\right]+cos\left[ta{n}^{-1}\right(2\sqrt{2}\left)\right]=$

Mjolis is a card game of Sweden.

Name a few indoor games played in your region. ‘Chopar’ could be an example.

How many of the following statements are correct?
1. If a point lies on the y-axis its x & z-coordinates are zero.
2. If a point lies on the x-z plane, its y-coordinate is zero.

3. The x-axis and y-axis taken together to determine a plane known as x-y plane.

Consdier $p\left(s\right)=s^3=a_2^2+a_{1}s+a_0$ with all realcoefficients. It is known that its derivative $p^{\prime }\left(s\right)$ has no real roots. The number of real rotos of $p\left(s\right)$ is

Find the sum of the series

${\sum }_n^{r=0}(-1{)}^r{}^n{C}_r[\frac{1}{2^r}+\frac{3^r}{2^{2r}}+\frac{7^r}{2^{3r}}+\frac{{15}^r}{2^{4r}}\cdots uptomterms]$

aRB if 2a+3b=30 check whether if it is reflexive, transitive and symmetric . Sir i need with steps plese

If $x,2y,3z$ are in A.P. where the distinct numbers $x,y,z$ are in G.P, then the common ratio of G.P. is

Differentiate the following functions with respect to x :

$\frac{secx-1}{secx+1}$

cos
4x
= 1 – 8sin²
x
cos²
x

The value of $\left(\right({\log }_{2}9{)}^2{)}^{\frac{1}{{\log }_2\left({\log }_{2}9\right)}}\times (\sqrt{7}{)}^{\frac{1}{{\log }_{4}7}}$ is

If $y=\sum _{r=1}^x{\tan }^{-1}\left(\frac{1}{1+r+r^2}\right),$ then $\frac{dy}{dx}$ at $x=2$ is equal to

${\sum }_{r=1}^n\left({\sum }_{k=0}^{r-1}{}^n{C}_r{}^r{C}_k{2}^k\right)$ is equal to

Let $a=3^{1/203}+1$ and $f\left(n\right)={}^n{C}_0{a}^{n-1}-{}^n{C}_1{a}^{n-2}+{}^n{C}_2{a}^{n-3}-\cdots +(-1{)}^{n-1}{}^n{C}_{n-1}{a}^0$ where $n\ge 3$. If $f\left(2030\right)+f\left(2031\right)=3^x(\frac{y}{a}-1)$, then the value of $\frac{x}{y}$ is

If the area of the bounded region

$R=\left\{\right(x,y):max\{0,{\log }_{e}x\}\le y\le 2^x,\frac{1}{2}\le x\le 2\}$ is, $\alpha ({\log }_{e}2{)}^{-1}+\beta ({\log }_{e}2)+\gamma ,$ then the value of $(\alpha +\beta -2\gamma {)}^2$ is equal to

Which of the following is the sketch of the graph $y=sinx\times cosecx$.

The range of values of $x$ for which the inequality $\frac{3x-2}{5x-3}\ge 4$ is satisfied is given by

The value of the integral $\underset{-1}{\overset{1}{\int }}{\log }_e(\sqrt{1-x}+\sqrt{1+x})dx$ is equal to

Let $S_k=\frac{1+2+3+....+k}{k}.$ If

$S_1^2+S_2^2+...+S_{10}^2=\frac{5}{12}A$, then $A$ is equal to:

The slope of aline is double of the slope of another line. If tangent of the angle between hem is $\frac{1}{3}$. Find the slopes of the lines.

The maximum value of $f\left(x\right)=(x+3)(4-x)+3$ is

The equation of the parabola whose axis is parallel to y – axis and passing through (4, 5), (–2, 11), (–4, 21) is

Let $P(8,4)$ be a point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. If the normal at point $P$ intersects the $x-$axis at $(12,0)$, then the value of eccentricity is

If the quadratic equation $a{x}^2+bx+c=0$ has two non-zero roots $\alpha \&\beta ,$, then find the valaue of $\frac{1}{\alpha }+\frac{1}{\beta }.$

The equation of the tangent to the parabola $y^2=8x$ inclined at ${30}^{\circ }$ to the x axis is

If P(x) be a polynomial of degree 4, with P(2)=-1, P'(2)=0, P”(2)=2, P”'(2)=-12 and P^ir(2) =24, then P”(1) is equal to

The abscissa of a point on the ellipse $\frac{x^2}{4}+\frac{y^2}{3}=1$ at a distance of $\frac{5}{4}$ unit from focus is

If ${1690}^{2608}+{2608}^{1690}$ is divided by $7$, then the remainder is