List of top Mathematics Questions

Use the Factor Theorem to determine whether g(x) is a factor of p(x) in each of the following cases:
(i) p(x) = 2x ³ + x ² – 2x – 1, g(x) = x + 1
(ii) p(x) = x ³ + 3x ² + 3x + 1, g(x) = x + 2
(iii) p(x) = x ³ – 4x ² + x + 6, g(x) = x – 3

Find the value of k, if x – 1 is a factor of p(x) in each of the following cases: 

(i) p(x) = x ² + x + k 

(ii) p(x) = 2x ² + kx +√2 

(iii) p(x) = kx² – \(\sqrt{2x}\) + 1 

(iv) p(x) = kx² – 3x + k

Use suitable identities to find the following products: 

(i) (x + 4) (x + 10) 

(ii) (x + 8) (x – 10) 

(iii) (3x + 4) (3x – 5) 

(iv) \((y^ 2 + \frac{3 }{ 2}) (y^ 2 – \frac{3 }{ 2}) \)

(v) (3 – 2x) (3 + 2x)

Evaluate the following products without multiplying directly: 

(i) 103 × 107 (ii) 95 × 96 (iii) 104 × 96

Factorise the following using appropriate identities: 

(i) 9x ² + 6xy + y 2 

(ii) 4y ² – 4y + 1 

(iii) x ² – \(\frac{y^2 }{ 100}\)

Expand each of the following, using suitable identities: 

(i) (x + 2y + 4z) 2 (ii) (2x – y + z) ² (iii) (–2x + 3y + 2z) ² 

(iv) (3a – 7b – c) ² (v) (–2x + 5y – 3z) ² (vi) [ \(\frac{1 }{ 4}\) a - \(\frac{1 }{ 2}\) b + 1]²

Write the coefficients of x 2 in each of the following: 

(i) 2 + x ² + x 

(ii) 2 – x ² + x 3 

(iii) \(\frac{π }{ 2}\) x² + x 

(iv) √2 x -1

Find the value of the polynomial 5x – 4x ² + 3 at 

(i) x = 0 (ii) x = –1 (iii) x = 2

Factorise: 

(i) 4x ² + 9y ² + 16z ² + 12xy – 24yz – 16xz 

(ii) 2x ² + y ² + 8z ² – 2√2 xy + 4√2 yz – 8xz

Write the following cubes in expanded form: 

(i) (2x + 1)³ (ii) (2a – 3b) ³ (iii) [\(\frac{3}{2}\) x + 1]³ (iv) [x - \(\frac{2 }{ 3} \)y]³

Factorise each of the following: 

(i) 8a ³ + b ³ + 12a 2b + 6ab² 

(ii) 8a ³ – b ³ – 12a 2b + 6ab2 

(iii) 27 – 125a ³ – 135a + 225a ² 

(iv) 64a ³ – 27b ³ – 144a 2b + 108ab2 

(v) 27p ³ – \(\frac{1}{ 216}\) – \(\frac{9 }{ 2}\) p² + \(\frac{1 }{4}\) p

Factorise each of the following: 

(i) 27y ³ + 125z ³ 

(ii) 64m³ – 343n ³ 

[ Hint : See Question 9. ]

The following frequency distribution gives the monthly consumption of electricity of 68 consumers of a locality. Find the median, mean and mode of the data and compare them

A life insurance agent found the following data for distribution of ages of 100 policy holders. Calculate the median age, if policies are given only to persons having age 18 years onwards but less than 60 year.

The lengths of 40 leaves of a plant are measured correct to the nearest millimetre, and the data obtained is represented in the following table :

Find the median length of the leaves. 
(Hint : The data needs to be converted to continuous classes for finding the median, since the formula assumes continuous classes. The classes then change to 117.5 - 126.5, 126.5 - 135.5, . . ., 171.5 - 180.5.)

The following table gives the distribution of the life time of 400 neon lamps :

Find the median life time of a lamp.

100 surnames were randomly picked up from a local telephone directory and the frequency distribution of the number of letters in the English alphabets in the surnames was obtained as follows:

Determine the median number of letters in the surnames. Find the mean number of letters in the surnames? Also, find the modal size of the surnames.

The distribution below gives the weights of 30 students of a class. Find the median weight of the students.

A student noted the number of cars passing through a spot on a road for 100 periods each of 3 minutes and summarised it in the table given below. Find the mode of the data :

The given distribution shows the number of runs scored by some top batsmen of the world in one-day international cricket matches.

Find the mode of the data.