List of practice Questions

If \( C \) is the midpoint of line segment \( \overline{AB} \), and \( P \) is any point outside the line \( AB \), then:
Let \( \vec{a}, \vec{b} \) be non-collinear vectors, with \( |\vec{a}| = 2\sqrt{2} \), \( |\vec{b}| = 3 \), and the angle between them is \( 45^\circ \). Then the lengths of the diagonals of the parallelogram whose adjacent sides are \( \vec{5a} + \vec{2b} \) and \( \vec{a} - \vec{3b} \) are:
Two vertices of a triangle are at \( -\vec{i} + 3\vec{j} \) and \( 2\vec{i} + 5\vec{j} \). Its orthocenter is at \( \vec{i} + 2\vec{j} \). If the position vector of the third vertex is \( a\vec{i} + b\vec{j} \), then \( (a, b) = \):
If \( \vec{a}, \vec{b}, \vec{c} \) are vectors such that \( \vec{a} \ne 0 \), \( \vec{a} \times \vec{b} = 2(\vec{a} \times \vec{c}) \), \( |\vec{a}| = 1 \), \( |\vec{c}| = 1 \), \( |\vec{b}| = 4 \), and the angle between \( \vec{b} \) and \( \vec{c} \) is \( \cos^{-1} \left( \frac{1}{4} \right) \). If \( \vec{b} - 2\vec{c} = \lambda \vec{a} \), then \( \lambda = \):
The range of the data set: \( 35, 12, 21, 24, 15, 7, 16, 12, 30, 32, 13, 17 \) is:
If \( A \) and \( B \) are two independent events such that \( P(A) = 0.3 \), \( P(B) = x \), and \( P(A \cup B) = 0.44 \), then \( x = \):
If \( \sin\left(x + \frac{\pi}{3}\right) + \sin\left(x - \frac{\pi}{3}\right) = 1 \), then the value of \( x \) in the interval \( [0, \pi] \) is:
In \( \triangle ABC \), if \( 3\sin A + 4\cos B = 6 \) and \( 4\sin B + 3\cos A = 1 \), then the angle \( C \) is:
If \( \cosh(x - \log 3) = \sinh x \), then \( x = \):
The sides of a triangle \( ABC \) are given by \( a = 3 \), \( b = 5 \), and \( c = 3 \). Then \( \cos A = \)
In triangle \( ABC \), if \( a = 2 \), \( b = 3 \) and \( \sin A = \frac{2}{3} \), then \( \angle B = \):
In a triangle \( ABC \), if \( a : b : c = 4 : 5 : 6 \), the ratio of radius of the circumcircle to that of the incircle is:
PQRS is a quadrilateral and \( \vec{PQ} = \vec{a} \), \( \vec{QR} = \vec{b} \), \( \vec{SP} = \vec{a} - \vec{b} \).
\( M \) is the midpoint of \( QR \), and \( X \) lies on \( \vec{SM} \) such that \( \vec{SX} = \frac{4}{5} \vec{SM} \).
If \( \vec{SM} = m(4\vec{a} - \vec{b}) \), and \( \vec{SX} = n(4\vec{a} - \vec{b}) \), then \( m + n = \):
Let \( P_1, P_2, \ldots, P_{15} \) be 15 points on a circle. The number of distinct triangles formed by points \( P_i, P_j, P_k \) such that \( i + j + k \ne 15 \), is:
What is the sum of the first \( n \) terms of the series, whose \( k \)-term is \( k! \cdot k \)?
If \( \frac{4x^3 + 16x + 7}{(x^2 + 4)^2} = \frac{Ax + B}{x^2 + 4} + \frac{Cx + D}{(x^2 + 4)^2} \), then the number of non-zero values among \( A, B, C, D \) is:
Evaluate: \( 1 + \cot^2 30^\circ - \sec^2 45^\circ \)
The smallest integer \( n \) such that \[ \frac{1}{\sin 45^\circ \sin 46^\circ} + \frac{1}{\sin 47^\circ \sin 48^\circ} + \cdots + \frac{1}{\sin 133^\circ \sin 134^\circ} = \frac{1}{\sin(n^\circ)} \]
Simplify: \( \sin(x + y) \sec x \sec y = \)
If \( 1, \omega, \omega^2 \) are the cube roots of unity, then evaluate: \[ (2 - \omega)^2 (2 - \omega^2)^2 (2 - \omega^{10})^2 (2 - \omega^{11})^2 \]
Let \( f(x) = x^2 + \frac{1}{x^2} \) and \( g(x) = x - \frac{1}{x} \) for \( x \in \mathbb{R} \setminus \{-1, 0, 1\} \). Then, the local minimum of \( \frac{f(x)}{g(x)} \) is:
The range of the function \( f(x) = \frac{x^2 + x + 1}{x^2 - x + 1} \) is:
Which of the following quadratic equations whose real roots \( x_1, x_2 \) satisfy the conditions \( x_1^2 + x_2^2 = 5 \), \( 3(x_1^5 + x_2^5) = 11(x_1^3 + x_2^3) \)?
The remainder when the polynomial \( 2x^5 - 3x^4 + 5x^3 - 3x^2 + 7x - 9 \) is divided by \( x^2 - x - 3 \) is:
If \( ^nC_{r-1} = 36 \), \( ^nC_r = 84 \), and \( ^nC_{r+1} = 126 \), then the value of \( nr^2 \) is: