List of practice Questions

If $\alpha,\beta$ are the roots of the equation \[ x^2-p(x+1)-c=0, \] then \[ \frac{\alpha^2+2\alpha+1}{\alpha^2+2\alpha+c} + \frac{\beta^2+2\beta+1}{\beta^2+2\beta+c} = \]

If $\alpha \neq \beta$ and $\alpha^2=5\alpha-3,\ \beta^2=5\beta-3$, then the equation whose roots are $\dfrac{\alpha}{\beta}$ and $\dfrac{\beta}{\alpha}$ is

If the roots of the equation \[ x^3-px^2+qx-s=0 \] are in geometric progression, then

If $\alpha,\beta,\gamma$ are the roots of \[ x^3-10x^2+7x+8=0, \] match List-I with List-II and choose the correct option.

Consider the quadratic expression \( f(x) = x^2 + (10-a)x - 10a \). The sum of all values of 'a', such that the roots of \( f(x)=3 \) are integers is:

The quadratic expression having zero's equal to the non-integral roots of \( ||3x-4|-6|=5 \) is:

If \(\alpha,\beta,\gamma\) are the roots of the equation \[ 24x^3-10x^2-3x+1=0, \] then \[ \frac{1}{\alpha^2}+\frac{1}{\beta^2}+\frac{1}{\gamma^2} = \]

A positive root of the equation \[ 18x^3-9x^2-5x+2=0 \] is twice another root of it. If among its three roots, the least value is \(a\) and the greatest value is \(b\), then \(2a+3b=\)

The expression \( \frac{n(n + 1)^2(n+2)}{12} \) for all \( n \in \mathbb{N} \) is always:

If the equation \[ x^4-10x^3+37x^2-60x+36=0 \] has two distinct real roots, where each one of them is a repeated root, then the sum of squares of all the roots of the given equation is

If the sum of two roots of the cubic equation $x^3 - 5x^2 - 2x + 24 = 0$ is $2$, then the roots of the equation are:

If $\alpha, \beta$ are the roots of the quadratic equation $x^2 - 2x + 4 = 0$, then the value of $\alpha^n + \beta^n$ is:

For a quadratic expression \( ax^{2}+bx+c \), if the minimum value \( \frac{49}{12} \) exists at \( x=\frac{-5}{6} \), then \( 12c-5b = \)

If a positive real \(x\) satisfies the following equation
\[ \log_2 x + \log_{\sqrt{2}} x = 48, \]
then the value of \(x\) is

Consider two distinct positive real numbers \(m, n\), with \(m > n\).
Let \(x = n^{\log_{10}(m)}\) and \(y = m^{\log_{10}(n)}\). The relation between \(x\) and \(y\) is ______.

Which of the following statements are correct:
A. Subgroup $H$ of a group $G$ is normal in $G$ if $xHx^{-1} \subseteq H$ $\forall x \text{ in } G$ B. A subgroup of an abelian group is not normal.
C. Centre $Z(G)$ of a group $G$ is normal.
D. Every subgroup of an abelian group is normal.
E. Centre $Z(G)$ of a group $G$ is not normal.
Choose the correct answer from the options given below:

Which of the following statements are correct:
A. Permutation group $S_n$ has order $n!$, $n > 1$.
B. Alternating group $A_n$ of degree $n$ has order $\frac{n!}{2}$, $n > 1$. C. Set of all even permutations forms a subgroup of $S_n$.
D. Every permutation in $S_n$, $n > 1$ is a product of 2-cycles.
E. $A_4$ has a subgroup of order 6.
Choose the correct answer from the options given below:

The generators of the set of integers $\mathbb{Z}$ under addition is:

Which of the following set forms a group under multiplication:

If a positive real \(x\) satisfies the following equation
\[ \log_2 x + \log_{\sqrt{2}} x = 48, \]
then the value of \(x\) is

At how many points will the curves \(y = x^2\) and \(y = -x^2 - 2x - 1\) intersect in the real \((x, y)\) plane?

In a group of students, 10 students like Mathematics, 12 students like English, 4 students like both Mathematics and English, and 6 students like neither Mathematics nor English. The number of students in the group is

Which of the following is an equation ?

If \(3x - 2y = 5\) and \(x + y = 5\), then the value of \(x\) is:

The sum of two numbers is 15 and the sum of their squares is 113. The product of the two numbers is: