List of practice Questions

The set of real values of \( \lambda \) for which the vectors \( \lambda\hat{i} - 3\hat{j} + 5\hat{k} \) and \( 2\lambda\hat{i} - \lambda\hat{j} + \hat{k} \) are perpendicular to each other is

The standard deviation of first 10 multiples of 4 is

The vector \( \vec{x} \) is perpendicular to the vectors \( \vec{a} = 3\hat{i} + 2\hat{j} + 2\hat{k} \), \( \vec{b} = 18\hat{i} - 22\hat{j} - 5\hat{k} \) and makes an obtuse angle with \( \hat{j} \). If \( |\vec{x}|=14 \), then \( \vec{x} = \)

\( \vec{a}, \vec{b} \) are two vectors such that \( |\vec{a}|=\sqrt{3}, |\vec{b}|=\sqrt{2} \). If \( \vec{x} \) is a unit vector satisfying \( \vec{x} \times \vec{a} = \vec{b} \) then \( \vec{x} = \)

\( \cos\theta (\csc\theta - \sec\theta) - \cot\theta = \)

In a triangle ABC, if a = 4, b = 5 and c = 7, then \( \sin(A/2) = \)

In a triangle ABC, let \(\angle C = \pi/2\). If r and R are respectively inradius and circumradius of ABC, then R + r =

In any triangle ABC, \( \frac{\cos 2A}{a^2} - \frac{\cos 2B}{b^2} = \)

Let ABC be an acute-angled triangle with area R. Then \( \sqrt{a^2b^2-4R^2} + \sqrt{b^2c^2-4R^2} + \sqrt{c^2a^2-4R^2} = \)

\(\tanh(\log x) = \) (Assuming \(\log x\) is natural logarithm \(\ln x\))

\( 2\cot^2\theta - \cot\theta - 3 = \) (The question implies factorization of the quadratic in \(\cot\theta\))

\( \frac{\sin x}{1+\cos x} + \frac{1+\cos x}{\sin x} = \)

\( \frac{x^3}{(x^2+1)(x^2+3)} = \)

If \( {}^n P_r = {}^{n-1}P_r + x \cdot {}^{n-1}P_{r-1} \quad \forall n,r \in N \text{ and } r \le n \), then x = .

The sum of all possible numbers that can be formed by using the digits 2,3,5,7 without repetition of digits is

The value of \( \sin(\frac{5\pi}{24}) \cdot \cos(\frac{\pi}{24}) \) 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 \( a^2+b^2+c^2=1, a,b,c \in R \), then the set of extreme values of \(ab+bc+ca\) is

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

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 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?

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

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

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