MCA Entrance Previous Year Question Bank

If $A = \begin{bmatrix} \cos\alpha & \sin\alpha \\ -\sin\alpha & \cos\alpha \end{bmatrix}$, then what is $A{A}^T$ equal to (where $A^T$ is the transpose of $A$)?

If $a\ne b\ne c$, then one value of $x$ which satisfies the equation $$</span><spanstyle="font-family{:}^{\prime }arialblac{k}^{\prime },sans-serif;font-size:14pt;">\left|\begin{array}{ccc}</span><spanstyle="font-family{:}^{\prime }arialblac{k}^{\prime },sans-serif;font-size:14pt;">0& x-a& x-b\\ </span><spanstyle="font-family{:}^{\prime }arialblac{k}^{\prime },sans-serif;font-size:14pt;">x+a& 0& x-c\\ </span><spanstyle="font-family{:}^{\prime }arialblac{k}^{\prime },sans-serif;font-size:14pt;">x+b& x+c& 0</span><spanstyle="font-family{:}^{\prime }arialblac{k}^{\prime },sans-serif;font-size:14pt;">\end{array}\right|</span><spanstyle="font-family{:}^{\prime }arialblac{k}^{\prime },sans-serif;font-size:14pt;">=0</span><spanstyle="font-family{:}^{\prime }arialblac{k}^{\prime },sans-serif;font-size:14pt;">$$ is given by

If B is a non-singular matrix and A is a square matrix, then the value of det (B^-1AB) is equal to

If $A=\left[\begin{array}{cc}\alpha & 2\\ 2& \alpha \end{array}\right]$ and $\det (A^3)=125$, then $\alpha$ is equal to

The expansion of (x - y)ⁿ , n ≥ 5 is done in the descending powers of x. If the sum of the fifth and sixth terms is zero, then x/y is equal to

If S = {x : x² + 1 = 0, x is real}, then S is

In the binary equation (1_p101)₂ + (10_q1)₂ = (100r00)₂ where p, q and r are binary digits, what are the possible values of p, q and r respectively?

Consider the following in respect of sets A and B: (1) (A - B) ∪ B = A (2) ( A - B) ∪ A = A (3) (A - B) ∩ B = φ (4) A ⊆ B ⇒ A ∪ B = B Which of the above are correct ?

The sum of the first $n$ terms of the series $\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+\cdots$ is equal to