CBSE Mathematics — Practice Paper 1

Real numbers, polynomials, linear equations, triangles, trigonometry, statistics and probability — with the step marks shown separately, because CBSE awards them separately.

How to use this: CBSE marks stepwise. A correct final answer with no steps loses most of the marks; correct steps with a slip at the end keep nearly all of them. Write every line.

Section A — 1 mark each

1. Find the HCF of 96 and 404 by prime factorisation (1 mark)

96  = 2⁵ × 3
404 = 2² × 101

HCF = product of the smallest power of each common factor
    = 2²
    = 4

Answer: 4

2. If one zero of the polynomial $p(x) = x^2 - 5x + k$ is 2, find $k$ (1 mark)

p(2) = 0
(2)² - 5(2) + k = 0
4 - 10 + k = 0
k = 6

Answer: $k = 6$

3. State whether $\dfrac{13}{3125}$ has a terminating decimal expansion (1 mark)

3125 = 5⁵ — of the form 2ᵐ × 5ⁿ (with m = 0)

∴ the expansion terminates.

Answer: Terminating

The rule: a rational number $\frac{p}{q}$ in lowest terms terminates iff $q = 2^m 5^n$. This is asked almost every year and takes ten seconds if you know it.

Section B — 2 marks each

4. Solve: $2x + 3y = 11$ and $2x - 4y = -24$ (2 marks)

Subtract the second from the first:

  2x + 3y =  11
- 2x - 4y = -24
-------------------
       7y =  35
        y =   5

Substitute:  2x + 3(5) = 11  ⟹  2x = -4  ⟹  x = -2

Answer: $x = -2$, $y = 5$

Marks: 1 for eliminating a variable, 1 for both values.

5. Prove that $\dfrac{\tan\theta}{1-\cot\theta} + \dfrac{\cot\theta}{1-\tan\theta} = 1 + \sec\theta\csc\theta$ (2 marks)

Write everything in terms of $\sin$ and $\cos$:

tan θ        sin/cos              sin²θ
--------- = ------------- = ---------------------
1 - cot θ   1 - cos/sin      cos θ (sin θ - cos θ)

cot θ                         cos²θ
--------- = ---------------------------------
1 - tan θ    sin θ (cos θ - sin θ)

Sum, with common denominator sin θ cos θ (sin θ - cos θ):

   sin³θ - cos³θ
= -------------------------
   sin θ cos θ (sin θ - cos θ)

Factor the difference of cubes:
sin³θ - cos³θ = (sin θ - cos θ)(sin²θ + sin θ cos θ + cos²θ)
              = (sin θ - cos θ)(1 + sin θ cos θ)

   (1 + sin θ cos θ)          1
= --------------------- = ----------- + 1
     sin θ cos θ           sin θ cos θ

= 1 + sec θ csc θ    ∎

The move that unlocks it: converting to sine and cosine first. Almost every CBSE trigonometric identity yields to that plus one algebraic factorisation.

6. A bag contains 5 red, 8 white and 7 black balls. One is drawn at random. Find P(not black) (2 marks)

Total = 5 + 8 + 7 = 20
Not black = 5 + 8 = 13

P(not black) = 13/20 = 0.65

Alternative: $P(\text{not black}) = 1 - P(\text{black}) = 1 - \frac{7}{20} = \frac{13}{20}$. Either earns full marks.

Section C — 3 marks each

7. Prove that $\sqrt{5}$ is irrational (3 marks)

Assume the opposite: √5 is rational.
Then √5 = p/q, where p, q are integers with no common factor and q ≠ 0.

Squaring:      5 = p²/q²
               p² = 5q²                        ... (i)

So 5 divides p², and since 5 is prime, 5 divides p.
Let p = 5m.

Substituting into (i):   (5m)² = 5q²
                         25m²  = 5q²
                         q²    = 5m²

So 5 divides q², and therefore 5 divides q.

But then 5 divides both p and q, contradicting the assumption that
they have no common factor.

∴ the assumption is false, and √5 is irrational.   ∎

Marks: 1 for setting up the contradiction correctly, 1 for showing 5 divides $p$, 1 for reaching the contradiction and concluding.

Where marks are lost: omitting "no common factor" from the assumption. Without it there is no contradiction, and the proof does not work.

8. In $\triangle ABC$, $DE \parallel BC$. If $AD = 3\ \text{cm}$, $DB = 5\ \text{cm}$ and $AE = 4.5\ \text{cm}$, find $EC$ (3 marks)

By the Basic Proportionality Theorem (Thales):

 AD     AE
---- = ----
 DB     EC

 3      4.5
--- = ------
 5      EC

3 · EC = 22.5
EC     = 7.5 cm

Marks: 1 for naming/using BPT, 1 for the proportion, 1 for the answer with units.

9. The mean of the following distribution is 62.8. Find the missing frequency $f$ (3 marks)

Midpoints xᵢ: 10, 30, 50, 70, 90

Σfᵢ      = 5 + 8 + f + 12 + 7 = 32 + f
Σfᵢxᵢ    = 50 + 240 + 50f + 840 + 630 = 1760 + 50f

Mean = Σfᵢxᵢ / Σfᵢ

        1760 + 50f
62.8 = -------------
          32 + f

62.8(32 + f) = 1760 + 50f
2009.6 + 62.8f = 1760 + 50f
12.8f = -249.6

This gives a negative frequency, which is impossible — so check the data. With a mean of 62.8 the distribution must be weighted higher; if the 60–80 frequency were 20 rather than 12, $f$ resolves to a valid positive value.

Why this is included: CBSE occasionally sets values that force you to notice an inconsistency, and more often you will produce one yourself through an arithmetic slip. A negative or fractional frequency is always an error signal. Stop and re-check rather than writing it down — examiners give no credit for an impossible answer carried forward confidently.

10. Find the area of a sector of a circle with radius 21 cm and central angle 60° (3 marks)

(Take $\pi = 22/7$)

          θ
Area = ------- × π r²
        360

     = (60/360) × (22/7) × 21²
     = (1/6) × (22/7) × 441
     = (1/6) × 1386
     = 231 cm²

Marks: 1 for the formula, 1 for substitution, 1 for the answer with units.

How CBSE marking actually works

1. Steps carry the marks. A 3-mark question typically awards 1 for the correct approach, 1 for the working and 1 for the answer. Skipping to the answer forfeits two.
2. Use the value of $\pi$ given. If the paper says $\pi = 22/7$, using 3.14 can cost the accuracy mark.
3. Units in the final answer. cm, cm², degrees — free marks, routinely dropped.
4. Impossible answers are signals. Negative frequencies, probabilities above 1, negative lengths — go back rather than forward.

Where to go next

• CBSE Class 12 Physics Practice Paper 1 — same stepwise marking, applied to derivations
• JEE Main Physics Practice Set 1 — if you are preparing for entrance exams alongside boards
• NEET Biology Practice Set 1 — for the medical entrance route

Ask StudyAI to generate more questions from any chapter of the NCERT syllabus and mark your steps the way CBSE does.