Class 9 CBSE – Paper 1
October 23, 2025 (Thursday)
Section A – 1 mark each
1. Write the decimal expansion of 1/7.
Divide 1 by 7: \( \frac{1}{7} = 0.\overline{142857} \)
The digits repeat infinitely.
Conclusion: Non-terminating recurring decimal ⇒ Rational number.
2. Find the degree of the polynomial \( 3x^4 - 5x^2 + 7 \).
The highest power of \( x \) is 4.
Answer: Degree = 4
3. What is the x-coordinate where the graph intersects the x-axis?
On the x-axis, \( y = 0 \).
Solve the equation for \( x \) when \( y = 0 \).
4. State Euclid’s first axiom.
“Things which are equal to the same thing are equal to one another.”
5. If two lines intersect, how many common points do they have?
Exactly one point.
Section B – 2 marks each
6. Rationalize \( \frac{1}{\sqrt{3} + \sqrt{2}} \).
Multiply numerator and denominator by the conjugate:
\[
\frac{1}{\sqrt{3} + \sqrt{2}} \cdot \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} - \sqrt{2}} = \frac{\sqrt{3} - \sqrt{2}}{1}
\]
7. Factorize \( x^2 - 5x + 6 \).
Find two numbers whose product is 6 and sum is –5:
\[
x^2 - 5x + 6 = (x - 2)(x - 3)
\]
8. Plot the point (3, –2) and mention the quadrant.
x > 0, y < 0 ⇒ Quadrant IV
9. Write the equation: “Sum of a number and 5 is 9.”
Let number be \( x \):
\[
x + 5 = 9
\]
10. If \( \angle A + \angle B = 180^\circ \), what is the relation between AB and CD?
Co-interior angles ⇒ AB ∥ CD
Section C – 3 marks each
11. Find \( a \) if \( x^2 + ax + 6 \) has (–2) as a zero.
Substitute \( x = -2 \):
\[
(-2)^2 + a(-2) + 6 = 0 \Rightarrow 4 - 2a + 6 = 0 \Rightarrow a = 5
\]
12. Show that \( \sqrt{5} \) is irrational.
Assume \( \sqrt{5} = \frac{p}{q} \Rightarrow p^2 = 5q^2 \).
This implies p divisible by 5 ⇒ contradiction.
Conclusion: Irrational.
13. Draw the graph of \( x + y = 4 \).
Choose values:
- x = 0 ⇒ y = 4
- x = 2 ⇒ y = 2
- x = 4 ⇒ y = 0
Plot: (0,4), (2,2), (4,0) and draw a line.
14. Prove vertically opposite angles are equal.
When two lines intersect, opposite angles share vertex and are formed by same lines.
Use angle properties:
\[
\angle A = \angle D, \quad \angle B = \angle C
\]
15. In triangle ABC, AB = AC and \( \angle B = 50^\circ \). Find \( \angle C \) and \( \angle A \).
Isosceles triangle ⇒ \( \angle C = 50^\circ \)
\[
\angle A = 180^\circ - (50^\circ + 50^\circ) = 80^\circ
\]
Section D – 4 marks each
16. Divide \( 2x^3 + 3x^2 - 5x + 6 \) by \( x - 2 \).
Use long division:
Quotient = \( 2x^2 + 7x + 9 \), Remainder = 24
17. Prove sum of angles in triangle is \( 180^\circ \).
Use angle sum property:
\[
\angle A + \angle B + \angle C = 180^\circ
\]
18. In triangle XYZ, XY = XZ and \( \angle Y = 40^\circ \). Find all angles.
Triangle XYZ is isosceles with XY = XZ, so angles opposite these sides are equal.
\[
\angle Y = \angle Z = 40^\circ
\]
Use angle sum property:
\[
\angle X = 180^\circ - (40^\circ + 40^\circ) = 100^\circ
\]
Final Answer: \( \angle X = 100^\circ, \angle Y = \angle Z = 40^\circ \)
19. Using Euclid’s axioms, prove: equals added to equals are equal.
Let \( a = b \) and \( c = d \).
According to Euclid’s axiom:
“If equals are added to equals, the wholes are equal.”
So:
\[
a + c = b + d
\]
Conclusion: The sum of equal quantities is equal.
20. Prove: In triangle, sides opposite equal angles are equal.
Let triangle ABC have \( \angle B = \angle C \).
By the converse of the Isosceles Triangle Theorem:
Sides opposite equal angles are equal.
So:
\[
AB = AC
\]
Conclusion: In any triangle, if two angles are equal, their opposite sides are equal.