List of practice Questions

If the roots of the equation \( 16x^3 - 44x^2 + 36x - 9 = 0 \) are in harmonic progression, then its greatest root is
The least value of \( n \) such that \( {}^{n-1}C_6 + {}^{n-1}C_7<{}^nC_8 \) is
Let \( A(-1, 1, 2), B(1, 2, 3), C(1, 1, 6) \) be three points. If \( l_1, m_1, n_1 \) are the direction cosines of \( AB \) and \( l_2, m_2, n_2 \) are the direction cosines of \( AC \), then \( |l_1 l_2 + m_1 m_2 + n_1 n_2| = \)
If \( Q \) is the inverse point of the point \( P(2, 3) \) with respect to the circle \( x^2 + y^2 - 2x - 2y + 1 = 0 \), then the circle with \( PQ \) as diameter is
If \[ S = 2x^2 + 2y^2 - 8x + 8y - 7 = 0 \] is the circle passing through the points of intersection of the circles \[ x^2 + y^2 - kx - ky + 1 = 0 \quad \text{and} \quad x^2 + y^2 - kx + ky - 2 = 0, \] then the length of the tangent drawn from the point \( (k, k) \) to the circle \( S \) is:
If \( \csc\theta \) and \( \cot\theta \) are the roots of \( cx^2 + bx + a = 0 \) where \( bc \neq 0 \), then evaluate \( b^2(b^2 - 4ac) \):
If \( \cos\theta \), \( \sin\theta \), and \( \cot\theta \) are in geometric progression, then: \[ \sin^9\theta + \sin^6\theta + 3\sin^5\theta + \sin^3\theta + \sin^2\theta = \]
Let \( a, b, c, d \) be non-zero numbers. If the point of intersection of the lines \( 4ax + 2ay + c = 0 \) and \( 5bx + 2by + d = 0 \) lies in the fourth quadrant and is equidistant from the two coordinate axes, then the relationship between \( a, b, c, d \) is:
The general solution of \( \frac{dy}{dx} = \cos^2(x - y - 1) \) is given by \( x = \):
If \( C \) is a point on the straight line joining the points \( A = -2+i \) and \( B = 3 - 4i \) in the Argand plane and } \[ \frac{AC}{CB} = \frac{1}{2}, \] then the argument of \( C \) is
The number of words that can be formed using the letters of the word VOWEL such that vowels always remain together is:
\( \cot 18^\circ \cdot \cot 36^\circ + 1 = \)
\(f(x)\) is a continuous function on \(\mathbb{R}\) and \(y = f(x)\) is a curve. If \((\alpha, \beta)\) is a point such that \(\beta = f(\alpha)\) and \(p\alpha + m\beta + n = 0\ (p \ne 0, m \ne 0)\), then which one of the following is True?
A team of 5 students is to be selected from 12 students. If two particular students are to be included in that team, then the number of ways that such team can be selected is:
If a real valued function \( f \) is defined by \( f(x) = \frac{ax + \sqrt{a^2 - x^2}}{bx} \), then \( f \) is:
In \( \triangle ABC \), if \( \sin B = \sin C \) and \( 3 \cos B = 2 \cos C \), then \( \triangle ABC \) is:
If a card is drawn at random from a well shuffled pack of playing cards, then the probability that it is either an Ace or a Spade card is
The modulus of the conjugate of \( z = \frac{-2 + i}{(1 - 2i)^2} \) is:
The coefficient of \( x^2 \) in the expansion of \( (1 - 3x)^{\frac{1}{3}} (1 + 2x)^{\frac{1}{2}} \) is:
If a line \( L \) makes angles \( \frac{\pi}{3} \) and \( \frac{\pi}{4} \) with the positive X-axis and positive Y-axis respectively, then the angle made by \( L \) with the positive direction of the Z-axis is:
Let \( C \) be the centre and \( A \) be one end of a diameter of the circle \( x^2 + y^2 - 2x - 4y - 20 = 0 \). If \( P \) is a point on \( AC \) such that \( CP : PA = 2 : 3 \), then the locus of \( P \) is:
If the mid-point of the chord intercepted by the circle \(x^2 + y^2 - 8x + 10y + 5 = 0\) on the line \(2x + y + 2 = 0\) is \((h, k)\), then \(k + 4h =\) ?
If the number of diagonals of a regular polygon of \( n \) sides is 104, then \( n = \)
If \( \overline{\mathbf{a}} = 2\overline{\mathbf{i}} + 3\overline{\mathbf{j}} + 4\overline{\mathbf{k}} \), \( \overline{\mathbf{b}} = 3\overline{\mathbf{i}} + 4\overline{\mathbf{k}} \), and \( \overline{\mathbf{c}} = 5\overline{\mathbf{i}} + 4\overline{\mathbf{k}} \) are three vectors, then a vector which is perpendicular to \( \overline{\mathbf{a}} \) and \( \overline{\mathbf{b}} \times \overline{\mathbf{c}} \) is
If eight coins are tossed simultaneously, then the probability of getting at least six heads is: