List of top Mathematics Questions

If one of the zeroes of the quadratic polynomial \((\alpha - 1)x^2 + \alpha x + 1\) is \(-3\), then the value of \(\alpha\) is:
The diameter of a circle is of length 6 cm. If one end of the diameter is \((-4, 0)\), the other end on \(x\)-axis is at:
The value of \(k\) for which the pair of linear equations \(5x + 2y - 7 = 0\) and \(2x + ky + 1 = 0\) don't have a solution, is:
Two dice are rolled together. The probability of getting a doublet is:
Problem Figure
Assertion (A): If the PA and PB are tangents drawn to a circle with center O from an external point P, then the quadrilateral OAPB is a cyclic quadrilateral.
Reason (R): In a cyclic quadrilateral, opposite angles are equal.
Assertion (A): Zeroes of a polynomial \(p(x) = x^2 − 2x − 3\) are -1 and 3.
Reason (R): The graph of polynomial \(p(x) = x^2 − 2x − 3\) intersects the x-axis at (-1, 0) and (3, 0).
\(D\) is a point on the side \(BC\) of \(\triangle ABC\) such that \(\angle ADC = \angle BAC\). Show that:
\(AC^2 = BC \times DC\)
Problem Figure
Solve the following pair of linear equations for \(x\) and \(y\) algebraically:
\(x + 2y = 9 \quad \text{and} \quad y - 2x = 2\)
Check whether the point \((-4, 3)\) lies on both the lines represented by the linear equations:
\(x + y + 1 = 0 \quad \text{and} \quad x - y = 1\)
Prove that \(6 - 4\sqrt{5}\) is an irrational number, given that \(\sqrt{5}\) is an irrational number.
Show that \(11 \times 19 \times 23 + 3 \times 11\) is not a prime number.
Evaluate: \(\sin A \cos B + \cos A \sin B\); if \(A = 30^\circ\) and \(B = 45^\circ\).
A bag contains 4 red, 5 white, and some yellow balls. If the probability of drawing a red ball at random is \(\frac{1}{5}\), then find the probability of drawing a yellow ball at random.
Two alarm clocks ring their alarms at regular intervals of 20 minutes and 25 minutes respectively. If they first beep together at 12 noon, at what time will they beep again together next time?
The greater of two supplementary angles exceeds the smaller by \(18^\circ\). Find the measures of these two angles.
Find the co-ordinates of the points of trisection of the line segment joining the points \((-2, 2)\) and \((7, -4)\).
In two concentric circles, the radii \(OA = r\) cm and \(OQ = 6\) cm, as shown in the figure. Chord $CD$ of the larger circle is a tangent to the smaller circle at $Q$. $PA$ is tangent to the larger circle. If $PA$ = $16$ cm and $OP = 20$ cm, find the length of $CD$.
Problem Figure
In the given figure, two tangents PT and QT are drawn to a circle with centre O from an external point T. Prove that \(\angle\) PQT = \(2\angle\) OPQ.
Problem Figure
A solid is in the form of a cylinder with hemispherical ends of the same radii. The total height of the solid is 20 cm and the diameter of the cylinder is 14 cm. Find the surface area of the solid.
A juice glass is cylindrical in shape with a hemispherical raised-up portion at the bottom. The inner diameter of the glass is 10 cm and its height is 14 cm. Find the capacity of the glass. (use π = 3.14)
Prove that: \((\cot \theta - \csc \theta)^2 = \frac{1 - \cos \theta}{1 + \cos \theta}.\)
If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then prove that the other two sides are divided in the same ratio.
Sides $AB$ and $BC$ and median $AD$ of a $△ABC$ are respectively proportional to sides $PQ$ and $PR$ and median $PM$ of $△PQR$. Show that $△ABC ∼ △PQR$.
How many terms of the $A.P.$ $27, 24, 21, . . .$ must be taken so that their sum is 105? Which term of the A.P. is zero?
The shadow of a tower standing on a level ground is found to be 40 m longer when the Sun’s altitude is 30° than when it was 60°. Find the height of the tower and the length of the original shadow. (use \(\sqrt{ 3}\) = 1.73)