List of top Mathematics Questions

The probability that it will rain tomorrow in cities A, B and C is \(60\%\), \(70\%\) and \(80\%\) respectively. Assuming these events are independent, the probability that it will rain tomorrow in at least one of the cities is:

The region represented by the system of linear inequalities \( 3x + y \ge 3 \), \( 2x - y \ge -5 \), and \( x, y \ge 0 \) is:

The sum of the order and the degree of the differential equation \(\left[\frac{d^2y}{dx^2}\right]^2 + \left[\frac{dy}{dx}\right]^4 + 3\left(\frac{dy}{dx}\right) + 1 - y^2 = 0\) is:

In a linear programming problem, the feasible region representing the maximization of the objective function \( Z = px + qy \), where \( p, q > 0 \), is a shaded polygonal region. If all points on the line segment joining the corner points \( A(0, 3) \) and \( B(3, 2) \) give the maximum value of \( Z \), then which of the following is true?

Three points \( A(0, 1, 1) \), \( B(2, 0, -1) \) and \( C(1, 0, 3) \) form \(\Delta ABC\). The area of \(\Delta ABC\) is:

The area of the region bounded by the curve \(y = x\) and the x-axis, between the boundaries \(x = 0\) and \(x = 2\) is:

The greatest integer function, \(f(x) = [x]\), evaluated across the open domain boundary \(0 < x < 3\), is not differentiable at how many points?

If the distance travelled by a particle in \(t\) seconds is given by \(S = 72t + 3t^2 - t^3\), then the time taken by the particle to come to rest is:

The indefinite integral \(\int \frac{dx}{\sqrt{25 - 16x^2}}\) is equal to:

The differential of \(e^{e^x}\) with respect to \(x\) is:

If \(\int_{0}^{1} \frac{dx}{e^x + e^{-x}} = \tan^{-1}e + k\), then the constant value of \(k\) is:

The graph of an inverse trigonometric function passes through the point $(0, \frac{\pi}{2})$ with asymptotes at $y = 0$ and $y = \pi$. This graph represents the function:

If the points \((2, 3)\), \((0, 4)\) and \((p, 2)\) are collinear, then the value of \(p\) is:

Let the arithmetic mean of \(\frac{1}{a}\) and \(\frac{1}{b}\) be \(\frac{5}{16}\), where \(a>2\). If \(a,4,b\) are in A.P., then the equation \[ ax^2-ax+2(a-2b)=0 \] has:

Case Study - 1 : A model of Leafy Ball Fountain is made to be kept on the tabletop. Water gently cascades down the ball into a decorative cylindrical pool where it is recycled. The diameter of spherical ball is 21 cm. Cylindrical pool - Outer diameter is 50 cm and inner diameter is 40 cm. Height of solid base is 14 cm. Height of water filled is 7 cm. Observe the figure and answer the following questions : (i) Determine the total height of the fountain. (ii) Find the volume of the ball. (iii) (a) If one-third of the ball is submerged in the water, find the volume of the water filled in the pool. OR (iii) (b) Find the sum of the outer curved surface area of the cylindrical part and surface area of the ball.

The area of the rectangle formed by the tangents drawn at the ends of both major and minor axes of an ellipse is 24. If the eccentricity of the ellipse is \(\frac{1}{4}\), then the equation of the ellipse is:

Case Study - 2 : During a theatre drama, a backdrop of building arches was used. The shape of the curve shown below can be represented by the polynomial p(x) = -x\(^2\) + 2x + 8, where x is the length (in feet) on stage level. Based on the figure, answer the following questions : (i) Determine the height of the arch. (ii) (a) Find zeroes of the polynomial p(x). Which points on the graph represent the zeroes? OR (ii) (b) Find the span of the arch on the stage floor. (iii) Write the coordinates of the point of intersection of the above curve with the y-axis.

Case Study - 3 : A group of friends wanted to play cards with two identical packs together. While shuffling the cards, three cards are dropped. Rest of the cards are shuffled and one card is drawn at random. Assuming that the dropped cards were a queen of hearts, a ten of spades and an ace of clubs, answer the following questions : (i) Find the probability that the drawn card is a face card. (ii) Find the probability that the drawn card is either a king or a queen. (iii) (a) Do you think that the probability of getting a queen was higher if none of the cards were dropped? Justify your answer. OR (iii) (b) Find the probability that the drawn card is a jack. Compare it with the probability when none of the cards were dropped. In which case is the probability of getting a jack higher?

The angle of elevation of the top of a building from a point A, on the ground, is 30\(^{\circ}\). On moving a distance of 24 m towards its base to the point B, the angle of elevation changes to 60\(^{\circ}\). Find the height of the building and distance of point A from the base of the building. (Take \(\sqrt{3}\) = 1.73)

In the given figure, chord AB subtends an angle of 120\(^{\circ}\) at the centre of the circle with radius 7 cm. Find (i) perimeter of major sector OACB, and (ii) area of the shaded segment, if area of \(\Delta\) OAB = 21.2 cm\(^2\).

Find the mean and the mode of the following frequency distribution :

A tower stands vertically on the ground. A man standing at the top of the tower observes his friend at an angle of depression of 30\(^{\circ}\), who is approaching the foot of the tower with a uniform speed. 30 seconds later, the angle of depression changes to 60\(^{\circ}\). Find the time taken by his friend to reach the foot of the tower from this point.

In \(\Delta\) ABC, AD is a median. X is a point on AD such that AX : XD = 2 : 3. BX is extended so that it intersects AC at Y. Prove that BX = 4 XY.

Solve the following system of equations graphically: x - 2y = 3, 3x - 8y = 7.