Higher Order Equations

Q. The sum of the roots of the equation $x+1-2{\log }_2(2^x+3)+2{\log }_4(10-2^{-x})=0$ is

1. ${\text{log}}_{2}11$
2. ${\text{log}}_{2}12$
3. ${\text{log}}_{2}13$
4. ${\text{log}}_{2}14$

Q.

The set of all real values of $\lambda$ for which the quadratic equations, $({\lambda }^2+1)x^2-4\lambda x+2=0$ always have exactly one root in the interval $(0,1)$ is

1. $(-3,-1)$

2. $(2,4)$

3. $(1,3)$

4. $(0,2)$

Q. If $x=2+2^{2/3}+2^{1/3}$, then the value of $x^3-6x^2+6x$ is

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

Q. The equation $x^{\frac{3}{4}(lo{g}_{2}x{)}^2+lo{g}_{2}x-\frac{5}{4}}=\sqrt{2}$ has

1. atleast one real solution
2. exactly three real solutions
3. exactly one irrational solution
4. complex roots

Q. If $\alpha$ and $\beta$ are the roots of the equation $x^2+3x+1=0$, then the value of ${\left(\frac{\alpha }{1+\beta }\right)}^2+{\left(\frac{\beta }{\alpha +1}\right)}^2$ is equal to

1. $18$
2. $19$
3. $20$
4. $21$

Q. If $x=2+2^{2/3}+2^{1/3}$, then the value of $x^3-6x^2+6x$ is

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

Q.

If $l,m,n$ are real, $l\ne m$, then the roots by the equation: $(l-m)x^2-5(l+m)x-2(l-m)=0$ are:

1. Complex

2. Real and Equal

3. Real and unequal.

4. None of these

Q.

The product of all real roots of the equation $x^2-|x|-6=0$ is

1. -9

2. 6

3. 9

4. 36

Q. The number of real solutions of $(x-1)(x+1)(2x+1)(2x-3)=15$ is

1. $0$
2. $2$
3. $3$
4. $4$

Q. If $x^2-3x+2$ is a factor of $x^4-a{x}^2+b$ then the equation whose roots are $a,b$ is

1. $x^2-9x-20=0$
2. $x^2-9x+20=0$
3. $x^2+9x-20=0$
4. $x^2+9x+20=0$

Q. The product of the real roots of the equation $(x-1{)}^4+(x-5{)}^4=82$ is

1. $-25$
2. $8$
3. $-8$
4. $25$

Q. If $x=\sqrt{2}+\sqrt{3}+\sqrt{6}$ is a root of $x^4+a{x}^3+b{x}^2+cx+d=0$ where $a,b,c,d$ are integers, what is the value of $|a+b+c+d|$ ?

Q. The value of $x$ satisfying the equation $6\sqrt{\frac{x}{x+4}}-2\sqrt{\frac{x+4}{x}}=11$ is

1. $-\frac{4}{35}$
2. $\frac{16}{3}$
3. None of these.
4. $\frac{4}{35}$

Q.

If both the roots of $(2a-4)9^x-(2a-3)3^x+1=0$ are non-negative, then

1. 2 < a <

2. a <

3. a > 3

4. 0 < a < 2

Q. Polynomial $P\left(x\right)$ contains only terms of odd degree. When $P\left(x\right)$ is divided by $(x-3)$, the remainder is $6$. If $P\left(x\right)$ is divided by $(x^2-9)$, then the remainder is $g\left(x\right)$. Then the value of $g\left(2\right)$ is

Q. If the sum of two roots of $x^4-2x^3+4x^2+6x-21=0$ is zero, then which of the following is/are true?

1. one of the roots of the equation is $1+i\sqrt{6}$
2. all roots of the equation are real
3. the equation has only two real roots
4. sum of all the real roots of the equation is $0$

Q. Let $f:R\to R$ be defined by $f\left(x\right)=\frac{(e^x-e^{-x})}{2}$.
The inverse of the given function is:

1. $f^{-1}\left(x\right)={\log }_e(x+\sqrt{x^2+1})$
2. $f^{-1}\left(x\right)={\log }_e(x-\sqrt{x^2+1})$
3. $f^{-1}\left(x\right)={\log }_e(x+\sqrt{x^2-1})$
4. $f^{-1}\left(x\right)={\log }_e(x-\sqrt{x^2-1})$

Q. Suppose the quadratic polynomial $P\left(x\right)=a{x}^2+bx+c$ has positive coefficients $a,b,c$ in arithmetic progression in that order. If $P\left(x\right)=0$ has integer roots $\alpha$ and $\beta$, then $\alpha +\beta +\alpha \beta$ equals

1. $5$
2. $3$
3. $7$
4. $14$

Q. The product of two of the four roots of the quadratic equation $x^4-{18}^3+k{x}^2+200x-1984=0$ is $-32.$ Determine the value of $k.$
(correct answer + 3, wrong answer 0)

Q. If $\sqrt{2}$ and $3i$ are two roots of a biquadratic equation with rational coefficients, then its equation is, (where $i^2=-1$)

1. $x^4+7x^2-18=0$
2. $x^4-7x^2-18=0$
3. $x^4+7x^2+18=0$
4. $x^4-7x^2+18=0$

Q. Let $f\left(x\right)=x^4+a{x}^3+b{x}^2+ax+1$ be a polynomial, where $a,b\in R$. If $b=-1$, then the range of $a$ for which $f\left(x\right)=0$ does not have real roots is

1. $(-\frac{1}{2},\frac{1}{2})$
2. $(-\frac{5}{2},\frac{5}{2})$
3. $(-1,1)$
4. $(-3,3)$

Q. The number of solutions of $\sqrt{4-x}+\sqrt{x+9}=5$ is

1. $0$
2. $1$
3. $2$
4. $3$

Q. If $x$ is a whole number, than $x^2(x^2-1)$ is always divisible by

1. $24$
2. $12-x$
3. $12$
4. $12(x-1)$

Q. The number of distinct positive real roots of the equation $(x^2+6{)}^2-35x^2=2x(x^2+6)$ is

1. $4$
2. $3$
3. $2$
4. $0$

Q. If the product of two roots of the equation $x^4-5x^3+5x^2+5x-6=0$ is $3$, then which of the following is/are correct?

1. The equation has only one negative root.
2. The equation has three negative roots.
3. The product of all positive roots will be $6.$
4. The product of all negative roots will be $6.$

Q. If the equation $(a-2)(x-[x]{)}^2+2(x-\left[x\right])+a^2=0,a\in R$ has no integral solution and has exactly one solution in $[2,3)$, then $a$ lies in the interval
(where $\left[x\right]$ denotes the greatest integer function)

1. $(0,1)$
2. $(-1,0)$
3. $(2,3)$
4. $(-1,2)$

Q. For the equation $|x-4{|}^{\left(\frac{x^2-10x+24}{x-3}\right)}=1$, which among the following statement(s) is/are true?

1. Sum of the real roots is $18.$
2. Sum of the real roots is $11.$
3. Product of the real roots is $90.$
4. Product of the real roots is $30.$

Q. If $f\left(x\right)=\frac{t+3x-x^2}{x-4},$ where $t$ is a parameter and $f\left(x\right)$ has exactly one minimum and one maximum, then the range of values of $t$ is

1. $(0,4)$
2. $(0,\infty )$
3. $(-\infty ,4)$
4. $(4,\infty )$

Q.

Let , $f\left(x\right)=x^2+5x+6$, then the number of real roots of $\left(f\right(x){)}^2+5f(x)+6-x=0$ is

1. 0

2. 1

3. 2

4. 3

Q. Total number of values of $x$ satisfying $(\sqrt{3}+1{)}^{2x}+(\sqrt{3}-1{)}^{2x}=2^{3x}$ is