(English Medium) ICSE Class 10 - CISCE Important Questions

Using the remainder theorem, find the remainders obtained when x³ + (kx + 8 )x + k is divided by x + 1 and x − 2. 

Hence, find k if the sum of the two remainders is 1.

If x – 2 is a factor of x³ – kx – 12, then the value of k is ______.

Find the value of 'a' if x – a is a factor of the polynomial 3x³ + x² – ax – 81.

Factorize completely using factor theorem:

2x³ – x² – 13x – 6

What must be subtracted from the polynomial x³ + x² – 2x + 1, so that the result is exactly divisible by (x – 3)?

The polynomial 3x³ + 8x² – 15x + k has (x – 1) as a factor. Find the value of k. Hence factorize the resulting polynomial completely.

A polynomial in ‘x’ is divided by (x – a) and for (x – a) to be a factor of this polynomial, the remainder should be ______.

Find the sum of first 51 terms of an AP whose second and third terms are 14 and 18 respectively.

For what value of n, the n^th terms of the arithmetic progressions 63, 65, 67, ... and 3, 10, 17, ... equal?

In a Arithmetic Progression (A.P.) the fourth and sixth terms are 8 and 14 respectively. Find that:
(i) first term
(ii) common difference
(iii) sum of the first 20 terms. 

The sum of the first three terms of an Arithmetic Progression (A.P.) is 42 and the product of the first and third term is 52. Find the first term and the common difference.

The sum of first 15 terms of an A.P. is 750 and its first term is 15. Find its 20^th term.

The 5^th term and the 9^th term of an Arithmetic Progression are 4 and – 12 respectively.

Find:

1. the first term
2. common difference
3. sum of 16 terms of the AP.

Which term of the Arithmetic Progression (A.P.) 15, 30, 45, 60...... is 300?

Hence find the sum of all the terms of the Arithmetic Progression (A.P.)

The n^th term of an Arithmetic Progression (A.P.) is given by the relation T_n = 6(7 – n)..

Find:

1. its first term and common difference
2. sum of its first 25 terms

The point (3, 0) is invariant under reflection in:

Use graph sheet to Solution this question. Take 2 cm = 1 unit alogn both the axes.

1. Plot A, B, C where A(0, 4), B(1, 1) and C(4, 0)
2. Reflect A and B on the x-axis and name them as E and D respectively.
3. Reflect B through the origion and name it F. Write down the coordinates of F.
4. Reflect B and C on the y-axis and name them as H and G respectively.
5. Join points A, B, C, D, E, F, G, H and A in order and name the closed figure formed.

Use graph sheet for this question. Take 2 cm = 1 unit along the axes.

1. Plot A(0, 3), B(2, 1) and C(4, –1).
2. Reflect point B and C in y-axis and name their images as B' and C' respectively. Plot and write coordinates of the points B' and C'.
3. Reflect point A in the line BB' and name its images as A'.
4. Plot and write coordinates of point A'.
5. Join the points ABA'B' and give the geometrical name of the closed figure so formed.

Study the graph and answer each of the following:

1. Write the coordinates of points A, B, C and D.
2. Given that, point C is the image of point A. Name and write the equation of the line of reflection.
3. Write the coordinates of the image of the point D under reflection in y-axis.
4. Whats the name given to a point whose image is the point itself?
5. On joining the points A, B, C, D and A in order, a figure is formed. Name the closed figure.

The coordinates of the vertices of ΔABC are respectively (–4, –2), (6, 2), and (4, 6). The centroid G of ΔABC is ______.