(English Medium) ICSE Class 10 - CISCE Question Bank Solutions for Mathematics

If the polynomials ax³ + 4x² + 3x - 4 and x³ - 4x + a leave the same remainder when divided by (x - 3), find the value of a.

In the following two polynomials. Find the value of ‘a’ if x + a is a factor of each of the two:

x³ + ax² − 2x + a + 4

In the following two polynomials. Find the value of ‘a’ if x + a is a factor of each of the two:
x⁴ - a²x² + 3x - a.

In the following two polynomials, find the value of ‘a’ if x – a is a factor of each of the two:
x⁶ - ax⁵ + x⁴ - ax³ + 3a + 2

In the following two polynomials, find the value of ‘a’ if x – a is a factor of each of the two:
x⁵ - a²x³ + 2x + a + 1.

If x – 2 is a factor of each of the following three polynomials. Find the value of ‘a’ in each case:
x² - 3x + 5a

If x – 2 is a factor of each of the following three polynomials. Find the value of ‘a’ in each case:
x³ + 2ax² + ax - 1

If x – 2 is a factor of each of the following three polynomials. Find the value of ‘a’ in each case:
x⁵ - 3x⁴ - ax³ + 3ax² + 2ax + 4.

In the given figure, O is the centre of the circle and ∠PBA = 45°. Calculate the value of ∠PQB.

Show that x² - 9 is factor of x³ + 5x² - 9x - 45.

In the given figure, BAD = 65°, ABD = 70°, BDC = 45°.
(i) Prove that AC is a diameter of the circle.
(ii) Find ACB.

In Fig, Chord ED is parallel to the diameter AC of the circle. Given ∠CBE = 65°, Calculate ∠ DEC.

In the figure, ∠DBC = 58°. BD is diameter of the circle.

Calculate:

1. ∠BDC
2. ∠BEC
3. ∠BAC

In the figure given alongside, AD is the diameter of the circle. If ∠ BCD = 130°, Calculate: (i) ∠ DAB (ii) ∠ ADB.

Construct the rhombus ABCD whose diagonals AC and BD are of lengths 8 cm and 6 cm respectively. Construct the inscribed circle of the rhombus. Measure its radius.

Using ruler and compass construct a cyclic quadrilateral ABCD in which AC = 4 cm, ∠ ABC = 60°, AB 1.5 cm and AD = 2 cm. Also, write the steps of construction.

(a) Only the ruler and compass may be used in this question. All contraction lines and arcs must be clearly shown and be of sufficient length and clarity to permit assessment.
(i) Construct a ABC, such that AB = AC = 7 cm and BC = 5 cm.
(ii) Construct AD, the perpendicular bisector of BC.
(iii) Draw a circle with center A and radius 3 cm. Let this circle cut AD at P.
(iv) Construct another circle, to touch the circle with center A, externally at P, and pass through B and C.

Solve: 2cos²θ + sin θ - 2 = 0.

In each of the following, determine whether the given numbers are roots of the given equations or not; x² – x + 1 = 0; 1, – 1

In each of the following, determine whether the given numbers are roots of the given equations or not; x² – 5x + 6 = 0; 2, – 3