List of top Mathematics Questions

Let a unit vector which makes an angle of \(60^\circ\) with \( 2\hat{i} + 2\hat{j} - \hat{k} \) and an angle of \(45^\circ\) with \( \hat{i} - \hat{k} \) be \( \vec{C} \). Then \( \vec{C} + \left( -\frac{1}{2} \hat{i} + \frac{1}{3\sqrt{2}} \hat{j} - \frac{\sqrt{2}}{3} \hat{k} \right) \) is:

The range of the real valued function \( f(x) =\) \(\sin^{-1} \left( \frac{1 + x^2}{2x} \right)\) \(+ \cos^{-1} \left( \frac{2x}{1 + x^2} \right)\) is:

Given: Chords \( AB \) and \( CD \) of a circle with center \( P \) intersect at point \( E \).
To Prove: \( AE \times EB = CE \times ED \). 

In \( \triangle ABC \), ray \( BD \) bisects \( \angle ABC \), \( A - D - C \), and \( \text{seg } DE \parallel \text{side } BC \). If \( A - E - B \), then for showing \( \frac{AB}{BC} = \frac{AE}{EB} \), complete the following activity: 

In the figure below, \( m(\text{arc NS}) = 125^\circ \), \( m(\text{arc EF}) = 37^\circ \). Find the measure of \( \angle NMS \). 

In the figure given above, \( ABCD \) is a square, and a circle is inscribed in it. All sides of the square touch the circle. If \( AB = 14 \, \text{cm} \), find the area of the shaded region. 

Find the value of \( \sin^2 \theta + \cos^2 \theta \): 

In the above figure, \( \triangle ABC \) is inscribed in arc \( ABC \). If \( \angle ABC = 60^\circ \), find \( m\angle AOC \): 

If \( \triangle ABC \sim \triangle PQR \) and \( AB : PQ = 2 : 3 \), then find the value of \( \frac{A(\triangle ABC)}{A(\triangle PQR)} \):
\( AB \) is a chord of a circle with center \( O \), \( AOC \) is the diameter of the circle, and \( AT \) is a tangent at \( A \). Write the answers to the following questions:
[(a)] Draw the figure using the given information.
[(b)] Find the measures of \( \angle CAT \) and \( \angle ABC \) with reasons.
[(c)] Determine whether \( \angle CAT \) and \( \angle ABC \) are congruent. Justify your answer.
\( ABCD \) is a trapezium, \( AB \parallel CD \), and the diagonals of the trapezium intersect at point \( P \). Write the answers to the following questions:
[(a)] Draw the figure using the given information.
[(b)] Write any one pair of alternate angles and opposite angles.
[(c)] Write the names of similar triangles with a test of similarity.
Segment \( PM \) is a median of \( \triangle PQR \), \( PM = 9 \), and \( PQ^2 + PR^2 = 290 \). Find \( QR \).
\( \triangle ABC \sim \triangle LMN \). In \( \triangle ABC \), \( AB = 5.5 \, \text{cm} \), \( BC = 6 \, \text{cm} \), \( CA = 4.5 \, \text{cm} \). Construct \( \triangle ABC \) and \( \triangle LMN \) such that \( \frac{BC}{MN} = \frac{5}{4} \).
Determine whether the points \( A(1, -3), B(2, -5), C(-4, 7) \) are collinear.
Draw a circle with centre \( O \) having radius \( 3 \, \text{cm} \). Draw tangent segments \( PA \) and \( PB \) through the point \( P \) outside the circle such that \( \angle APB = 70^\circ \).
Prove that, "If a line parallel to a side of a triangle intersects the remaining sides in two distinct points, then the line divides the sides in the same proportion."
A cylinder of radius \( 12 \, \text{cm} \) contains water up to the height \( 20 \, \text{cm} \). A spherical iron ball is dropped into the cylinder, and thus the water level is raised by \( 6.75 \, \text{cm} \). What is the radius of the iron ball?
\( \frac{1}{\sin^2 \theta} - \frac{1}{\cos^2 \theta} - \frac{1}{\tan^2 \theta} - \frac{1}{\cot^2 \theta} - \frac{1}{\sec^2 \theta} - \frac{1}{\csc^2 \theta} = -3 \), then find the value of \( \theta \).
Radius of a sector of a circle is \( 3.5 \, \text{cm} \) and length of its arc is \( 2.2 \, \text{cm} \). Find the area of the sector.
Find the length of the hypotenuse of a right-angled triangle if the remaining sides are \( 9 \, \text{cm} \) and \( 12 \, \text{cm} \).
Find the surface area of a sphere of radius \( 7 \, \text{cm} \).
Find the slope of the line passing through the points \( A(2, 3) \) and \( B(4, 7) \).
A circle having radius \( 3 \, \text{cm} \), then the length of its largest chord is:
Two circles of radii \( 5 \, \text{cm} \) and \( 3 \, \text{cm} \) touch each other externally. Find the distance between their centres.
A line makes an angle of \( 45^\circ \) with the positive direction of the X-axis. Find the slope of the line.