List of top Mathematics Questions asked in CBSE Class IX

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

(i) p(x) = x² + x + k

(ii) p(x) = 2x² + kx +√2

(iii) p(x) = kx² – \(\sqrt{2x}\) + 1

(iv) p(x) = kx² – 3x + k

Use the Factor Theorem to determine whether g(x) is a factor of p(x) in each of the following cases:
(i) p(x) = 2x³ + x² – 2x – 1, g(x) = x + 1
(ii) p(x) = x³ + 3x² + 3x + 1, g(x) = x + 2
(iii) p(x) = x³ – 4x² + x + 6, g(x) = x – 3

Determine which of the following polynomials has (x + 1) a factor :

(i) x³ + x² + x + 1

(ii) x⁴ + x³ + x²+ x + 1

(iii) x ⁴ + 3x³ + 3x² + x + 1

(iv) x³ – x² – ( 2 + √2 )x + √2

Verify whether the following are zeroes of the polynomial, indicated against them.

(i) p(x) = 3x + 1, x = - \(\frac{1}{3}\)

(ii) p(x) = 5x – π, x = \(\frac{4}{5}\)

(iii) p(x) = x² – 1, x = 1, –1

(iv) p(x) = (x + 1) (x – 2), x = – 1, 2

(v) p(x) = x² , x = 0

(vi) p(x) = lx + m, x = – \(\frac{m}{l}\)

(vii) p(x) = 3x² – 1, x = -\(\frac{1 }{ √3} ,\frac{2}{ √3} \)

(viii) p(x) = 2x + 1, x = \(\frac{1}{2}\)

Find p(0), p(1) and p(2) for each of the following polynomials:

(i) p(y) = y 2 – y + 1

(ii) p(t) = 2 + t + 2t 2 – t 3

(iii) p(x) = x 3

(iv) p(x) = (x – 1) (x + 1)

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

(i) 2 + x² + x

(ii) 2 – x² + x 3

(iii) \(\frac{π }{ 2}\) x² + x

(iv) √2 x -1

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

Number of letters	Number of surnames
1 − 4	6
4 − 6	30
6 − 8	44
8 − 12	16
12 − 20	4

(i) Draw a histogram to depict the given information.
(ii) Write the class interval in which the maximum number of surnames lie.

A random survey of the number of children of various age groups playing in a park was found as follows:

Age (in years)	Number of children
1 − 2	5
2 − 3	3
3 − 5	6
5 − 7	12
7 − 10	9
10 − 15	10
15 − 17	4

Draw a histogram to represent the data above.

The runs scored by the two teams A and B in the first 60 balls in a cricket match are given below:

Number of balls	Team A	Team B
1 − 6	2	5
7 − 12	1	6
13 − 18	8	2
19 − 24	9	10
25 − 30	4	5
31 − 36	5	6
37 − 42	6	3
43 − 48	10	4
49 − 54	6	8
55 − 60	2	10

Represent the data of both the teams on the same graph by frequency polygons.
[Hint: First make the class intervals continuous.]

Simplify : (i) 2 \(\frac{2}{3}\) . 2 \(\frac{1}{5}\) (ii) (\(\frac{1}{33}\))7 (iii) 11 \(\frac{1}{2}\) / 11\(\frac{1}{4}\) (iv) 7 \(\frac{1}{2}\) . 8 \(\frac{1}{2}\)

The length of 40 leaves of a plant is measured correct to one millimeter and the obtained data is represented in the following table:

Length (in mm)	Number of leaves
118 − 126	3
127 − 135	5
136 − 144	9
145 − 153	12
154 − 162	5
163 − 171	4
172 − 180	2

(i) Draw a histogram to represent the given data.
(ii) Is there any other suitable graphical representation for the same data?
(iii) Is it correct to conclude that the maximum number of leaves are 153 mm long? Why?

ABC is an isosceles triangle with AB=AC. Draw AP⊥BC to show that ∠B = ∠C.

BE and CF are two equal altitudes of a triangle ABC. Using RHS congruence rule, prove that the triangle ABC is isosceles.

Two sides AB and BC and median AM of one triangle ABC are respectively equal to sides PQ and QR and median PN of ∆ PQR (see Fig. 7.40). Show that:

(i) ∆ BM≅∆ PQN

(ii) ∆ ABC≅∆ PQR

Two sides AB and BC and median AM of one triangle ABC

AD is an altitude of an isosceles triangle ABC in which AB = AC. Show that

(i) AD bisects BC

(ii) AD bisects ∠A.

∆ ABC and ∆ DBC are two isosceles triangles on the same base BC and vertices A and D are on the same side of BC (see Fig.). If AD is extended to intersect BC at P, show that

(i) ∆ABD ≅ ∆ACD

(ii) ∆ABP≅ ∆ACP

(iii) AP bisects ∠A as well as ∠D.

(iv) AP is the perpendicular bisector of BC.

two isosceles triangles on the same base BC and vertices A and D

Show that the angles of an equilateral triangle are 60° each.

ABC is a right angled triangle in which ∠A = 90° and AB = AC. Find ∠B and ∠C.

ABC is an isosceles triangle in which altitudes BE and CF are drawn to equal sides AC and AB respectively (see Fig. 7.31). Show that these altitudes are equal.

Given below are the seats won by different political parties in the polling outcome of a state assembly elections:

Political Party	A	B	C	D	E	F
Seats Won	75	55	37	29	10	37

(i) Draw a bar graph to represent the polling results.
(ii) Which political party won the maximum number of seats?

What are the possible expressions for the dimensions of the cuboids whose volumes are given below?

(i) Volume : 3x2 – 12x

(ii) Volume : 12ky² + 8ky – 20k

Give possible expressions for the length and breadth of each of the following rectangles, in which their areas are given:

(i) Area : 25a² – 35a + 12

(ii) Area : 35y² + 13y –12

Without actually calculating the cubes, find the value of each of the following:

(i) (–12)³ + (7)³ + (5)³

(ii) (28)³ + (–15)³ + (–13)³

If x+y+z=0, show that x³+y³+z³= 3xyz.

Verify that x³ + y³ + z³ – 3xyz = \(\frac{1}{2}\) (x + y + z) [(x - y)² + (y - z)² + (z - x)²].