List of top Mathematics Questions

Evaluate the integral \( \displaystyle \int \frac{x^2}{(x^2 + 1)(x^2 + 4)} \, dx \) using partial fractions.

Find the value of \( x \) if \( \begin{bmatrix} -5 & 6 \\ 2 & 3 \end{bmatrix}^{T} = \begin{bmatrix} 9y & 6z \\ 2x & 3 \end{bmatrix}. \)

If \( |\vec{a}| = \sqrt{26},\ |\vec{b}| = 7,\ \text{and}\ |\vec{a} \times \vec{b}| = 35 \), find the angle between \( \vec{a} \) and \( \vec{b} \).

Find the principal value of \( \sin^{-1}\!\left(-\frac{\sqrt{3}}{2}\right) + \cosec^{-1}\left(-\frac{2}{\sqrt{3}}\right). \)\!

Show that the relation \( R \) in the set of natural numbers \( N \times N \) defined by \( (a, b)\, R\, (c, d) \) if \( a + d = b + c \) is an equivalence relation.

Find the shortest distance between the skew lines whose vector equations are given.

Using determinants, find the value of \( \lambda \) if the points \( (1, -5) \), \( (-4, 5) \), and \( (\lambda, 7) \) are collinear.

Prove that the parallelogram circumscribing a circle is a rhombus.

A drone is flying at a height of 100 m above the ground. It observes on its right two stationary cars on a highway at angles of depression 45° and 30°. On the basis of above information, answer the following questions: If the drone rises to 150 m, find the tangent of the angle of depression of each car at the new height.

If \( A \) is a skew-symmetric matrix of odd order, write the value of \( |A| \).

Two years ago, father was thrice as old as his daughter, and 6 years later he will be 4 years older than twice her age. How old are they now?

Prove the identity \( \sec^2 \theta = 1 + \tan^2 \theta \) for any right-angled triangle and use it to show that: \[ \frac{\sin \theta - \cos \theta + 1}{\sin \theta + \cos \theta - 1} = \frac{1}{\sec \theta - \tan \theta}. \]

A drone is flying at a height of 100 m above the ground. It observes on its right two stationary cars on a highway at angles of depression 45° and 30°. On the basis of above information, answer the following questions: Find the distance between the two cars.

Find a point on the Y-axis which is equidistant from the points \( A(6, 5) \) and \( B(-4, 3) \).

Find the coordinates of the points that trisect the line segment AB joining the points \( A(-1, 2) \) and \( B(2, 8) \).

A drone is flying at a height of 100 m above the ground. It observes on its right two stationary cars on a highway at angles of depression 45° and 30°. On the basis of above information, answer the following questions: Find the distance of each car from the point on the highway just below the drone.

A pole has to be erected at a point on the boundary of a circular park of diameter 13 metres in such a way that the differences of its distances from two diametrically opposite fixed gates A and B on the boundary is 7 metres. Is it possible to do so? If yes, at what distances from the two gates should the pole be erected?

Show graphically that the linear equations \( x - y = 8 \), \( 3x - 3y = 16 \) are inconsistent, i.e. it has no solution.

If AP and DQ are medians of triangles ABC and DEF respectively, where \( \triangle ABC \sim \triangle DEF \), prove that: \[ \frac{AB}{DE} = \frac{AP}{DQ} \]

Find the area of a circle with maximum area that can be inscribed in a square of side 7 cm.

Two coins are tossed simultaneously, what is the probability of getting at least one head?

Find the value of K for which the given system of equations has infinitely many solutions: \[ Kx + 3y = K-3 \quad \text{(1)} \] \[ 12x + Ky = K \quad \text{(2)} \]

Form a quadratic equation, one of whose zero is \( 2 + \sqrt{5} \) and the sum of zeros is 4.

What do you understand by irrational number? Prove that \( 5 - 3\sqrt{2} \) is an irrational number.

The A.P. \( 8, 10, 12, \ldots \) has 60 terms. Find the sum of the last 20 terms.