List of practice Questions

If \(\alpha, \beta\) are the roots of \(ax^2+bx+c=0\), then the quadratic equation whose roots are \( \sqrt{5}\alpha, \sqrt{5}\beta \) is

If \( a^2+b^2+c^2=1, a,b,c \in R \), then the set of extreme values of \(ab+bc+ca\) is

If 2, 3, 6 are the roots of the polynomial \(f(x) = x^3+ax^2+bx+c\), where \(a,b,c \in C\). Then the value of a--c is

If \( y^2+z^2=3yz, z^2+x^2=8zx, x^2+y^2=4xy \), then the value of \( \frac{x^2}{yz} + \frac{y^2}{zx} + \frac{z^2}{xy} \) is

The English alphabets have 5 vowels and 21 consonants. How many words with two different vowels and two different consonants can be formed from the alphabet?

In a plane there are 37 straight lines of which 13 pass through point A and 11 pass through point B. Moreover, no three lines (apart from the lines passing through A and B) pass through same point and no two lines are parallel. What is the number of points of intersection of the straight lines?

The rank of the matrix \( A = \begin{bmatrix} 2 & 1 & 2 \\ 1 & 0 & 1 \\ 4 & 1 & 4 \end{bmatrix} \) is

If \( A = \begin{bmatrix} 1 & -2 & 2 \\ -2 & -6 & 5 \\ 0 & 0 & 4 \end{bmatrix} \) then Adj A =

\( \frac{(1+i)x-2i}{3+i} + \frac{(2-3i)y+i}{3-i} = i \implies x+y = \)

Let z and w be two distinct non-zero complex numbers if \(|z|^2w - |w|^2z = z - w\), then

If 1, \(\omega, \omega^2\) denote the cube roots of unity, then, the value of \( (1-\omega+\omega^2)^5 + (1+\omega-\omega^2)^5 \) is

The values of \(\theta\), for which \( \frac{3+2i\sin\theta}{1-2i\sin\theta} \) is real are

If f is a relation from set of positive real numbers to the set of positive real numbers defined by \(f(x) = 3x^2-2\) then f is

Let \( A = \begin{bmatrix} b^2+c^2 & a^2 & a^2 \\ b^2 & c^2+a^2 & b^2 \\ c^2 & c^2 & a^2+b^2 \end{bmatrix} \). If \(a = \sin \pi/6, b = \cos \pi/4\) and \(c = \cot \pi/2\) then A is

If a matrix A satisfies the equation \( A^3 - 6A^2 + 11A - 6I = 0 \), then \( A^{-1} \) can be

Let \(f: R \rightarrow R\) be defined by \(f(x) = 2x+3\). If \(\alpha, \beta\) are the roots of the equation \(f(x^2) - 2f(\frac{x}{2}) - 1 = 0\) then \(\alpha^2 + \beta^2 = \)

Given below are half-cell reactions:
\(MnO_4^- + 8H^+ + 5e^- → Mn^{2+} + 4H_2O\),
\(E°_{(Mn^{+2}/MnO_4^-)} = –1.510 V\)
\((\frac{1}{2})O_2 + 2H^+ + 2e^- → H_2O,\)
\(E°(O_2/H_2O) = +1.223 V\)
Will the permanganate ion, \(MnO_4^-\) liberate \(O_2\) from water in the presence of an acid?

Young's double slit experiment is first performed in air and then in a medium other than air. It is found that the 9th bright fringe in the medium lies where the 5th dark fringe lies in the air, then the refractive index of the medium is:

What mass of 95% pure \(CaCO_3\) will be required to neutralise 50 mL of 0.5 M \(HCl\) solution according to the following reaction?
\(CaCO_3(s) + 2HCl(aq) → CaCl_2(aq) + CO_2(g) + 2H_2O(l)\)
[Calculate upto second place of decimal point]

If the charge on a capacitor is increased by 2 C, the energy stored in it increases by 44%. The original charge on the capacitor is (in C)

Let A and B be two non zero square matrices and AB and BA both are defined. It means

If $ AX = D $ represents the system of simultaneous linear equations: $$ \begin{aligned} x + y + z &= 6 \\ 5x - y + 2z &= 3 \\ 2x + y - z &= -5 \end{aligned} $$ Then compute: $ (\text{Adj } A) D $

If $$ A = \begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix} $$ then find: $$ \det(A^6 + B^6) $$

If $$ A = \begin{bmatrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{bmatrix} $$ then $ AA^T $ is: