List of top Questions asked in BSE Manipur Class 10

Show that\( a^3 + b^3 + c^3 - 3abc = \dfrac{1}{2}(a+b+c)\left\{(a-b)^2 + (b-c)^2 + (c-a)^2\right\}\)

Prove that \(x^n - y^n\) is divisible by \(x+y\) only when \(n\) is even.
If three consecutive vertices of a parallelogram are \( A(1,-2) \), \( B(3,6) \) and \( C(5,10) \), find its fourth vertex.
If \( \cos A = \dfrac{3}{5} \), calculate \( \sin A \) and \( \tan A \).
If \(x, y, z\) are real numbers, \(x \neq 0\), and \(xy = xz\), prove that \(y = z\).
A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height \(h\). At a point on the plane, the angle of elevation of the bottom of the flagstaff is \(\alpha\) and that of the top of the flagstaff is \(\beta\). Prove that the height of the tower is \(\dfrac{h\tan\alpha}{\tan\beta-\tan\alpha}\).
Show that the square of an odd integer is of the form \( 8k + 1 \).
A metallic sphere of radius \(9\) cm is melted and recast to form a cylinder of radius \(3\) cm. Find the curved surface area of the cylinder.
Find the area of the quadrilateral whose vertices are \( (1,1) \), \( (3,4) \), \( (5,-2) \) and \( (4,-7) \) taken in order.
Write down the formula to find the area of a triangle whose vertices are \((x_1,y_1)\), \((x_2,y_2)\) and \((x_3,y_3)\).
The ratio of incomes of two persons is \(9:7\) and the ratio of their expenditures is \(4:3\). If each of them saves Rs. \(5000\) per month, find their monthly incomes.
Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.
State and prove Basic Proportionality Theorem.
Find the area of a circle whose radius is 7 cm.
Define mode of a frequency distribution.
Prove that the area of a sector of sectorial angle \( \theta \) and radius \( r \) is \( \dfrac{\pi r^2 \theta}{360} \).
Write the full form of SCORES.
Write down the quadratic equation whose roots are 2 and -3.
How many tangents can be drawn to a circle through a point lying outside the circle?
In a right triangle \( ABC \), right angled at \( B \), prove that \( \sin A = \cos (90^\circ - A) \) and \( \cos A = \sin (90^\circ - A) \).
Write the statement of Euclid's Division Lemma.
What is the first term of the quotient when \(2x^3 + x^2 - 3x + 5\) is divided by \(1 - 3x + x^2\)?
The remainder when \(4x^3 + 4x^2 + x - 4\) is divided by \(2x - 1\) is
Solve the quadratic equation \(ax^2 + bx + c = 0\), \((a \neq 0)\) by the method of completing square.
A point P is at a distance of 13 cm from the centre of a circle. If the radius of the circle is 5 cm, the length of the tangent from P to the circle is