Discriminant

Q. Let $f:R-\left\{\frac{\alpha }{6}\right\}\to R$ be defined by $f\left(x\right)=\frac{5x+3}{6x-\alpha }.$
Then the value of $\alpha$ for which $\left(fof\right)\left(x\right)=x,$ for all
$x\in R-\left\{\frac{\alpha }{6}\right\},$ is:

1. $5$
2. $8$
3. No such $\alpha$ exists
4. $6$

Q. Find the domain of the function $f\left(x\right)=\frac{x^2+2x+1}{x^2-8x+12}$.

Q. Roots of the quadratic equation $(x^2-4x+3)+\lambda (x^2-6x+8)=0,\lambda \in R$ will be

1. always real
2. always imaginary
3. real only when lis positive
4. real only when l is negative

Q.

If $x^2+3x+5=0anda{x}^2+bx+c=0$ have a common root and a, b, c $ϵ$ N then minimum value of a + b + c is equalto :

1. 3

2. 9

3. 6

4. 12

Q. The discriminant of the quadratic equation $2x^2-4x+3=0$ is

1. $2$
2. $-10$
3. $-8$
4. $0$

Q. A quadratic equation with rational coefficients if one of its roots is ${\cot }^2{18}^{\circ }$ is

1. $x^2+10x-5=0$
2. $x^2+10x+5=0$
3. $x^2-10x-5=0$
4. $x^2-10x+5=0$

Q.

The least value of k which makes the roots of the equation

$x^2+5x+k=0$imaginary is

1. 4

2. 5

3. 6

4. 7

Q. Let $f\left(x\right)=\sqrt{x}andg\left(x\right)=x$ be two functions defined in the domain $R^{+}\cup \left\{0\right\}$. Find

(i) $(f+g)\left(x\right)$

(ii) $(f-g)\left(x\right)$

(iii) $\left(fg\right)\left(x\right)$

(iv) $\left(\frac{f}{g}\right)\left(x\right)$

Q. If $S=\{z\in C:\frac{z-i}{z+2i}\in R\}$, then

1. $S$ contains exactly two elements
2. $S$ is a circle in the complex plane
3. $S$ is a straight line in the complex plane
4. $S$ contains only one element

Q. The number of real roots of $3^{2x^2-7x+7}=9$ is :

1. zero
2. 2
3. 1
4. 4

Q.

The equation $a_8{x}^8+a_7{x}^7+a_6{x}^6+...+a_0=0$ has all its roots positive and real $(where{a}_8=1,a_7=-4,a_0={\frac{1}{2}}^8)$, then

1. 

2. 

3. 

4. 

Q.

If a < b < c < d, then the roots of the equation (x-a)(x-c) + 2(x-b)(x-d) = 0 are

1. Real and equal

2. Imaginary

3. one real and one imaginary

4. Real and distinct

Q. The domain for which the functions defined by $f\left(x\right)=3x^2–1$ and $g\left(x\right)=3+x$ are equal is

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

Q.

If $a+b+c=0$, then the equation $3a{x}^2+2bx+c=0$ has, in the interval $\left(0,1\right)$

1. At least one root

2. At most one root

3. No root

4. Exactly one root exists

Q. For real x, the function $\frac{(x-a)(x-b)}{(x-c)}$ will assume all real values provided

1. a > b > c
2. $a\le c\le b$
3. a < b < c
4. a > b < c

Q.

The value of $p$, for which both the roots of the equation $4x^2-20px+25p^2+15p-66=0$ are less than $2$, is

1. $\left(\frac{4}{5},2\right)$

2. $\left(-1,\frac{-4}{5}\right)$

3. $\left(2,\infty \right)$

4. $\left(-\infty ,-1\right)$

Q. The domain of the function f given by $f\left(x\right)=\frac{x^2+2x+1}{x^2-x-6}$

1. $R–\{3,–2\}$
2. $R–\{3,–2\}$
3. $R–\{-3,–2\}$
4. $R–\{3,–2\}$

Q.

$\frac{ax+b}{cx+d}$
$\left(\text{Differentiate with respect to}x,\text{using first principle}\right)$

Q.

$1)$ Functions $f,g:R\to R$ are defined, respectively, by $f\left(x\right)=x^2+3x+1,g\left(x\right)=2x–3,$ find

$\left(i\right)$ $fog$

$2)$ Functions $f,g:R\to R$ are defined, respectively, by $f\left(x\right)=x^2+3x+1,g\left(x\right)=2x–3,$ find

$\left(ii\right)$ $gof$

$3)$ Functions $f,g:R\to R$ are defined, respectively, by $f\left(x\right)=x^2+3x+1,g\left(x\right)=2x–3,$ find

$\left(iii\right)$ $fof$

$4)$ Functions $f,g:R\to R$ are defined, respectively, by $f\left(x\right)=x^2+3x+1,g\left(x\right)=2x–3,$ find

$\left(iv\right)$ $gog$

Q. Let $f:R\to R,g:R\to R$ and $h:R\to R$ be differentiable functions such that $f\left(x\right)=x^3+3x+2,g\left(f\right(x\left)\right)=x$ and $h\left(g\right(g\left(x\right)\left)\right)=x$ for all $x\in R$. Then

1. $h^{\prime }\left(1\right)=666$
2. $h\left(0\right)=16$
3. $h\left(g\right(3\left)\right)=36$
4. $g^{\prime }\left(2\right)=\frac{1}{15}$

Q.

If $A=\left[35\right]$ , $B=\left[73\right]$ , then find a non-zero matrix C such that AC=BC

Q.

Solve the equation $\mathrm{kx}(\mathrm{x}-2)+6=0$ for the value of $\mathrm{x}$ if roots of this equation are real and equal.

Q.

If one root of the equation $x^2+ax+12=0$ is $4$ while the equation $x^2+ax+b=0$ has equal roots, then the value of $b$ is

1. $\frac{4}{49}$

2. $\frac{49}{4}$

3. $\frac{7}{4}$

4. $\frac{4}{7}$

Q.

The number of solutions of the equations of the equation $x^2+\left[x\right]-4x+3=0$ is Where [ ] denotes G.I.F.

1. 1

2. 2

3. 3

4. 0

Q.

If $x^2$+px + q = 0 is the quadratic equation whose roots are a – 2 and b – 2 where a and b are the roots of $x^2$ - 3x + 1 =0, then

1. p = 1 , q = 5

2. p = 1, q = - 5

3. p = -1, q = 1

4. None of these

Q.

The values of $k$, for which the equations $x^2-kx-21=0$ and $x^2-3kx+35=0$will have a common roots, is

1. $k=\pm 4$

2. $k=\pm 1$

3. $k=\pm 3$

4. $k=0$

Q.

If $x^2+2ax+10-3a>0$ for all x ∈ R, then

1. -5 < a < 2

2. a < -5

3. a > 5

4. 2 < a < 5

Q.

The common roots of the equations $z^3+2z^2+2z+1=0$ and $z^{1985}+z^{100}+1=0$ are

1. $\omega ,{\omega }^2$

2. $\omega ,{\omega }^3$

3. ${\omega }^2,{\omega }^3$

4. None of these

Q.

Solve the following quadratic equations :

(i)$x^2-(3\sqrt{2}+2i)x+6\sqrt{2}i=0$

(ii)$x^2-(5-i)x+(18+i)=0$

(iii)$(2+i)x^2-(5-i)x+2(1-i)=0$

(iv)$x^2-(2+i)x-(1-7i)=0$

(v)$i{x}^2-4x-4i=0$

(vi)$x^2+4ix-4=0$

(vii)$2x^2+\sqrt{15}ix-i=0$

(viii)$x^2-x+(1+i)=0$

(ix)$i{x}^2-x+12i=0$

(x)$x^2-(3\sqrt{2}-2i)x-\sqrt{2}i=0$

(xi)$x^2-(\sqrt{2}+i)x+\sqrt{2}i=0$

(xii)$2x^2-(3+7i)x+(9i-3)=0$

Q. The number of real roots of $3^{2x^2-7x+7}=9$ is :

1. 4
2. zero
3. 2
4. 1