Range of Quadratic Expression

Q.

Let $\mathrm{I}$ be an identity matrix of order $2\times 2$ and $\mathrm{P}=\left[\begin{array}{cc}2& -1\\ 5& -3\end{array}\right]$. Then the value of $\mathrm{n}\in \mathrm{N}$ for which${\mathrm{P}}^{\mathrm{n}}=5\mathrm{I}-8\mathrm{P}$ is equal to

Q.

Find the value of the polynomial $5x-4x^2+3$ at $x=0$

Q. The value of $\lambda$ such that sum of the squares of the roots of the quadratic equation, $x^2+(3-\lambda )x+2=\lambda$ has the least value is:

1. $2$
2. $1$
3. $\frac{15}{8}$
4. $\frac{4}{9}$

Q. $f:R\to R,$ $f\left(x\right)=\frac{3x^2+mx+n}{x^2+1}.$ If the range of $f\left(x\right)$ is $[-4,3],$ then

1. $m=0$
2. $n=4$
3. $m=2$
4. $n=-4$

Q. Let $f\left(x\right)$ be a quadratic polynomial such that $f(-2)+f\left(3\right)=0.$ If one of the roots of $f\left(x\right)=0$ is $-1,$ then the sum of the roots of $f\left(x\right)=0$ is equal to:

Q.

The number of real solutions of the equation ${27}^{\frac{1}{x}}+{12}^{\frac{1}{x}}=2\left(8^{\frac{1}{x}}\right)$ is

1. One

2. Two

3. Infinite

4. Zero

Q. If a and b are the non-zero distinct roots of $x^2+ax+b=0$, then the least value of $x^2+ax+b$ is

1. 
2. 
3. 1
4. -

Q.

Let $a,b,c,d$ and $p$ be any non zero distinct real numbers such that$\left(a^2+b^2+c^2\right)p^2-2\left(ab+bc+ca\right)p+\left(b^2+c^2+d^2\right)=0$. then

1. $a,c,p$ are in G.P

2. $a,b,c,d$ are in G.P

3. $a,b,c,d$ are in A.P

4. $a,c,p$ are in A.P

Q. If the product of the roots of the equation $2x^2+6x+{\alpha }^2+1=0$ is –α, then the value of α is

यदि समीकरण $2x^2+6x+{\alpha }^2+1=0$ के मूलों का गुणनफल –α है, तब α का मान है

1. –1
2. 1
3. 2
4. –2

Q. The graph of $f\left(x\right)=a{x}^2+bx+c$ is shown below, such that $b^2-4ac=-4.$ If the length of segment $AB$ and $AC$ are $1$ and $4$ respectively, then the value of $(a+b+c)$ is equal to

Q.

The range of f(x) = $\frac{x-\left[x\right]}{1+x-\left[x\right]}$ where [] represents greatest integer function.

1. [0, 1]

2. [0, ½]

3. [0, ½)

4. None of these.

Q.

Find the middle term in the expansion of ${\left(\left(\frac{x}{3}\right)+9y\right)}^{10}$.

Q. The range of $f\left(x\right)=2x^2-3x+2$ if $x\in [0,2]$ is

1. $[\frac{7}{8},4]$
2. $[2,4]$
3. $[\frac{7}{8},\infty )$
4. $[2,\infty )$

Q. Let $P\left(x\right)=a{x}^2+bx+8$ is a quadratic polynomial. If the minimum value of $P\left(x\right)$ is $6$ at $x=2,$ then

1. $a=\frac{1}{2}$
2. $b=-\frac{1}{2}$
3. $a=2$
4. $b=-2$

Q.

Let $(x,y,z)$ be points with integer coordinates satisfying the system of homogeneous equations:
$3x-y-z=0-3x+z=0-3x+2y+z=0$
Then the number of such points for which
$x^2+y^2+z^2\le 100$ is ___

Q.

$1+i^2+i^4+i^6+..........+i^{2n}$ is

1. Positive

2. Negative

3. Zero

4. Cannot be determined

Q. If the roots of a quadratic equation $a{x}^2+bx+c=0$ (a > 0 and b, c are real numbers) are numerically equal and opposite in sign and p, q are roots of $e{x}^2-dx-e=0$(e ≠ 0 and d, e are real numbers), then the incorrect option is

यदि द्विघात समीकरण $a{x}^2+bx+c=0$ (a > 0 तथा b, c वास्तविक संख्याएं हैं) के मूल संख्यात्मक रूप से समान है तथा विपरीत चिन्ह वाले हैं और p, q समीकरण $e{x}^2-dx-e=0$(e ≠ 0 तथा d, e वास्तविक संख्याएं हैं) के मूल हैं, तब गलत विकल्प है

1. p and q are real and distinct
   
   p व q वास्तविक एवं भिन्न है
   
2. b = 0
   
3. c > 0
   
4. c < 0
   

Q. Suppose that $f\left(x\right)$ is a quadratic expression, positive for all real $x$. If $g\left(x\right)=f\left(x\right)+f^{\prime }\left(x\right)+f^{\prime \prime }\left(x\right)$, then
(where $f^{\prime }\left(x\right)$ and $f^{\prime \prime }\left(x\right)$ represent first and second derivative respectively)

1. $g\left(x\right)<0$ for all real values of $x$
2. $g\left(x\right)>0$ for all real values of $x$
3. $g\left(x\right)=0$ for some real values of $x$
4. $g\left(x\right)\le 0$ for some real values of $x$

Q.

The number of solutions of the equation $x^3$ + $x^2$ + 4x + 2sinx = 0 in 0 ≤ x ≤ 2$\pi$

1. 2

2. 4

3. 0

4. 1

Q.

Let $a,b,c,d$ and $p$be any non zero distinct real numbers such that $(a^2+b^2+c^2)p^2-2(ab+bc+cd)p+(b^2+c^2+d^2)=0$. Then

1. $a,c,p$ are in G.P.

2. $a,b,c,d$ are in G.P.

3. $a,b,c,d$ are in A.P.

4. $a,c,p$ are in A.P.

Q.

Which of the following sets are convex ?

1. {(x, y) : 3x² + 4y² ≥ 5}

2. {(x, y) : y ≤ 2. y ≤ 4}

3. {(x, y) : x² + y² ≥ 1}

4. {(x, y) : y² ≥ x}

Q. Range of the expression $f\left(x\right)=\frac{x^2-1}{x^2+1},x\in R$ is

1. $[-1,1]$
2. $[-1,1)$
3. $(-1,1)$
4. $(-1,1]$

Q.

The range of the function $f\left(x\right)=x^2+\frac{1}{[x^2+1]}$ is

1. $[1,\infty )$

2. $[2,\infty )$

3. $\left[\frac{3}{2},\infty \right]$

4. None of these

Q. If $\frac{z-1}{z+1}$ is a purely imaginary number $(z\ne -1)$, then find the value of $|z|$.

Q. (1) For any two complex numbers $z_1$, $z_2$ and any real numbers $a$, $b$, ${|a{z}_1-b{z}_2|}^2+{|b{z}_1+a{z}_2|}^2=$

(2) The value of $\sqrt{-25}\times \sqrt{-9}$ is

(3) The number $\frac{{(1-i)}^3}{1-i^3}$ is
equal to

(4) The sum of the series $i+i^2+i^3+......$ upto $1000$ terms is .

(5) Multiplicative inverse of $(1+i)$ is

(6) If $z_1$ and $z_2$ are complex numbers such that $z_1+z_2$ is real number then .

(7) $\text{arg}\left(z\right)+\text{arg}\left(\overline{z}\right)(\overline{z}\ne 0)$ is

(8) If $|z+4|\le 3$, then the greatest and least value of $|z+1|$ are and .

(9) If $\left|\frac{z-2}{z+2}\right|=\frac{\pi }{6}$, then the locus of $z$ is .

(10) If $\left|z\right|=4$ and $\text{arg}\left(z\right)=\frac{5\pi }{6}$, then $z=$

Q. If Rolle's theorem holds for the function $f\left(x\right)=2x^3+b{x}^2+cx,x\in [–1,1],$at the point $x=\frac{1}{2}$, then $2b+c$ equals :

1. $1$
2. $2$
3. $-1$
   
4. $-3$

Q. Consider the function $f\left(x\right)=x^2-4x+17.$ If $M$ and $m$ are the maximum and minimum values of $f$ in $[0,3]$ respectively, then the value of $2M-m$ is

Q. The range of $f\left(x\right)=\frac{1}{2x^2-6x+7}$ is

1. $(-\infty ,0)\cup [\frac{2}{5},\infty )$
2. $(-\infty ,0]\cup [\frac{2}{5},\infty )$
3. $(0,\frac{2}{5}]$
4. $[0,\frac{2}{5}]$

Q. The maximum value of $f\left(x\right)=\frac{8}{9x^2-6x+5}$ in its domain is

Q.

In a square matrix ‘A’ of order 3, ‘$a_{ii}$’s are the sum of the roots and ‘$a_{i,i+1}$’s are the product of the roots of the equation $x^2$ – 5x + 6 = 0; $a_{i,i-1}$’s are all unity and the rest of the elements are all zeros. The value of the determinant (A) is equal to ‘k’. The least prime number that divides it is___