Marks Distribution Table

Section	Type	Qs	Marks/Q	Total Marks
A	Very Short	6	1	6
B	Short I	6	2	12
C	Short II	10	3	30
D	Long	8	4	32
E	Application	4	5	20
Total		34		100

CBSE Class 12 Mathematics Mock Test (Based on PYQs)

Time Allowed: 3 Hours
Maximum Marks: 100

Instructions:

1. All questions are compulsory.
2. Use of a scientific calculator is allowed.
3. Use proper formulas and show all steps for full marks.
4. Use of graph paper is allowed wherever necessary.

---

Section A – Very Short Answer Type (1 Mark Each, 6 Questions – 6 Marks)

1. Find the derivative of $y = \sin x + \ln x$y=sinx+lnx.
2. If $A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$A=[21​34​], find $|A|$∣A∣.
3. Find the sum of first 10 terms of an A.P. whose first term is 2 and common difference is 3.
4. Find the mode of the following data:

| Class Interval | 0–10 | 10–20 | 20–30 | 30–40 | 40–50 |
|—————|——–|——–|——–|——–|
| Frequency | 5 | 10 | 15 | 10 | 5 |

5. Evaluate $\lim_{x \to 0} \frac{\sin 3x}{x}$limx→0​xsin3x​.
6. Write the coordinates of the centroid of a triangle whose vertices are $A(1,2), B(3,4), C(5,0)$A(1,2),B(3,4),C(5,0).

---

Section B – Short Answer Type I (2 Marks Each, 6 Questions – 12 Marks)

7. If $x^2 + y^2 = 25$x2+y2=25, find $\frac{dy}{dx}$dxdy​ using differentiation.
8. The following table represents the marks of 10 students. Find the median.

| Marks | 20 | 30 | 40 | 50 | 60 |
|——-|—-|—-|—-|—-|
| No. of Students | 2 | 3 | 1 | 2 | 2 |

9. Evaluate $\int_0^1 (3x^2 + 2x) dx$∫01​(3×2+2x)dx.
10. Find the inverse of the matrix $\begin{bmatrix} 1 & 2 \\ 3 & 5 \end{bmatrix}$[13​25​].
11. Solve the differential equation: $\frac{dy}{dx} = 2x$dxdy​=2x, given $y(0) = 3$y(0)=3.
12. A box contains 5 red and 3 black balls. Two balls are drawn at random. Find the probability that both are red.

---

Section C – Short Answer Type II (3 Marks Each, 10 Questions – 30 Marks)

13. Using the properties of determinants, evaluate:

$$\begin{vmatrix} 2 & 1 & 3 \\ 0 & 1 & 4 \\ 1 & 0 & 2 \end{vmatrix}$$​201​110​342​​

14. Find the roots of the quadratic equation $x^2 – 5x + 6 = 0$x2−5x+6=0 using matrices.
15. Find the sum to $n$n terms of the series: $1 + 5 + 9 + 13 + …$1+5+9+13+…
16. If $\int (2x^3 – 3x^2 + x – 5) dx = F(x) + C$∫(2×3−3×2+x−5)dx=F(x)+C, find $F(x)$F(x).
17. Find the probability of getting at least one head in 3 tosses of a fair coin.
18. Find the coordinates of the point dividing the line segment joining $A(2,3)$A(2,3) and $B(4,5)$B(4,5) in the ratio 1:2 internally.
19. Solve $\frac{dy}{dx} + y = e^x$dxdy​+y=ex.
20. A card is drawn from a well-shuffled deck. Find the probability of getting:
    (i) a King, (ii) a red card, (iii) neither King nor red card.
21. Find the sum of squares of first 20 natural numbers.
22. Find the sum to $n$n terms of the A.P.: $7, 11, 15, …$7,11,15,…

---

Section D – Long Answer Type (4 Marks Each, 8 Questions – 32 Marks)

23. Using integration, find the area under the curve $y = x^2 + 2x$y=x2+2x from $x=0$x=0 to $x=2$x=2.
24. Two dice are rolled together. Find the probability of getting:
    (i) sum 7, (ii) sum greater than 10.
25. Find the derivative of $y = e^x \sin x$y=exsinx.
26. Solve the differential equation: $\frac{dy}{dx} = \frac{y}{x}$dxdy​=xy​.
27. Find the inverse of $\begin{bmatrix} 2 & 3 \\ 5 & 4 \end{bmatrix}$[25​34​].
28. Find the coordinates of the centroid and circumcenter of the triangle with vertices $A(0,0), B(4,0), C(0,3)$A(0,0),B(4,0),C(0,3).
29. Evaluate: $\int_1^2 \frac{1}{x} dx$∫12​x1​dx.
30. From a group of 5 boys and 3 girls, a committee of 2 boys and 1 girl is to be formed. Find the number of ways.

---

Section E – Very Long Answer / Application Based (5 Marks Each, 4 Questions – 20 Marks)

31. A die is thrown twice. Find the probability of getting a doublet (same number on both dice).
32. A man saves Rs 100 in the first month and increases his saving by Rs 20 each month. Find his total savings in 12 months.
33. If $\frac{dy}{dx} = 3x^2 + 2x – 1$dxdy​=3×2+2x−1, find $y$y when $x = 1$x=1 and $y = 2$y=2.
34. Find the sum of the first 20 terms of the A.P. $3, 8, 13, …$3,8,13,… and also find its 20th term.

Disclaimer

This mock test is prepared for educational purposes only and is based on previous years’ CBSE question patterns. This test is intended to help students practice and revise for exams and is not an official CBSE paper.

BSE Class 12 Maths Mock Test – Solutions

---

Section A – Very Short Answer Type (1 Mark Each)

1. $y = \sin x + \ln x$y=sinx+lnx

$$\frac{dy}{dx} = \cos x + \frac{1}{x}$$dxdy​=cosx+x1​

2. $A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$A=[21​34​]

$$|A| = (2)(4) – (3)(1) = 8 – 3 = 5$$∣A∣=(2)(4)−(3)(1)=8−3=5

3. Sum of first 10 terms of A.P. $a=2, d=3, n=10$a=2,d=3,n=10

$$S_n = \frac{n}{2}[2a + (n-1)d] = \frac{10}{2}[4 + 27] = 5 \cdot 31 = 155$$Sn​=2n​[2a+(n−1)d]=210​[4+27]=5⋅31=155

4. Mode of frequency table:
   Class interval 20–30 has maximum frequency = 15 → Mode = 20–30
5. $\lim_{x\to 0} \frac{\sin 3x}{x} = 3$limx→0​xsin3x​=3
6. Centroid of triangle with vertices $A(1,2), B(3,4), C(5,0)$A(1,2),B(3,4),C(5,0)

$$G = \left(\frac{1+3+5}{3}, \frac{2+4+0}{3}\right) = \left(\frac{9}{3}, \frac{6}{3}\right) = (3,2)$$G=(31+3+5​,32+4+0​)=(39​,36​)=(3,2)

---

Section B – Short Answer Type I (2 Marks Each)

7. Differentiating $x^2 + y^2 = 25$x2+y2=25

$$2x + 2y \frac{dy}{dx} = 0 \implies \frac{dy}{dx} = -\frac{x}{y}$$2x+2ydxdy​=0⟹dxdy​=−yx​

8. Median from cumulative frequency table:
   Total students = 2+3+1+2+2 = 10 → Median class = 30–40
   $\text{Median} = l + \frac{\frac{N}{2}-CF}{f} \cdot h = 30 + \frac{5-5}{2} \cdot 10 = 30$Median=l+f2N​−CF​⋅h=30+25−5​⋅10=30
9. $\int_0^1 (3x^2 + 2x) dx = \left[x^3 + x^2 \right]_0^1 = 1 + 1 = 2$∫01​(3×2+2x)dx=[x3+x2]01​=1+1=2
10. Inverse of $\begin{bmatrix} 1 & 2 \\ 3 & 5 \end{bmatrix}$[13​25​]

$$\text{det} = 1*5 – 2*3 = 5-6=-1 A^{-1} = \frac{1}{-1} \begin{bmatrix} 5 & -2 \\ -3 & 1 \end{bmatrix} = \begin{bmatrix}-5 & 2 \\ 3 & -1\end{bmatrix}$$det=1∗5−2∗3=5−6=−1A−1=−11​[5−3​−21​]=[−53​2−1​]

11. Solve $\frac{dy}{dx} = 2x, y(0)=3$dxdy​=2x,y(0)=3

$$y = x^2 + C, y(0)=3 \implies C=3 \implies y=x^2+3$$y=x2+C,y(0)=3⟹C=3⟹y=x2+3

12. Probability of 2 red balls from 5 red & 3 black:

$$P = \frac{\binom{5}{2}}{\binom{8}{2}} = \frac{10}{28} = \frac{5}{14}$$P=(28​)(25​)​=2810​=145​

---

Section C – Short Answer Type II (3 Marks Each)

13. Determinant:

$$\begin{vmatrix} 2 & 1 & 3 \\ 0 & 1 & 4 \\ 1 & 0 & 2 \end{vmatrix} = 2(1*2 – 4*0) – 1(0*2 – 4*1) + 3(0*0 – 1*1) = 4 – (-4) -3 = 4+4-3=5$$​201​110​342​​=2(1∗2−4∗0)−1(0∗2−4∗1)+3(0∗0−1∗1)=4−(−4)−3=4+4−3=5

14. Quadratic roots using matrices:
    $x^2 -5x +6=0 \implies (x-2)(x-3)=0 \implies x=2,3$x2−5x+6=0⟹(x−2)(x−3)=0⟹x=2,3
15. Sum of series $1+5+9+13+\dots$1+5+9+13+… (A.P., a=1, d=4, n terms unknown)

$$S_n = \frac{n}{2}[2a + (n-1)d] = \frac{n}{2}[2 + 4(n-1)] = \frac{n}{2}[4n-2] = 2n^2 – n$$Sn​=2n​[2a+(n−1)d]=2n​[2+4(n−1)]=2n​[4n−2]=2n2−n

16. $\int(2x^3 – 3x^2 + x – 5) dx = \frac{2x^4}{4} – \frac{3x^3}{3} + \frac{x^2}{2} -5x + C = \frac{x^4}{2} – x^3 + \frac{x^2}{2} -5x + C$∫(2×3−3×2+x−5)dx=42×4​−33×3​+2×2​−5x+C=2×4​−x3+2×2​−5x+C
17. Probability at least one head in 3 tosses:

$$P = 1 – P(\text{no head}) = 1 – \frac{1}{8} = \frac{7}{8}$$P=1−P(no head)=1−81​=87​

18. Coordinates dividing AB in 1:2 internally

$$\frac{2*2 + 1*4}{1+2}, \frac{2*3 + 1*5}{3} = \frac{8}{3}, \frac{11}{3} = (8/3, 11/3)$$1+22∗2+1∗4​,32∗3+1∗5​=38​,311​=(8/3,11/3)

19. Solve $\frac{dy}{dx}+y = e^x$dxdy​+y=ex
    Integrating factor $= e^{\int 1 dx} = e^x$=e∫1dx=ex

$$e^x y = \int e^{2x} dx = \frac{1}{2} e^{2x} + C \implies y = \frac{1}{2} e^x + Ce^{-x}$$exy=∫e2xdx=21​e2x+C⟹y=21​ex+Ce−x

20. Deck probabilities:
    (i) King: 4/52 = 1/13
    (ii) Red card: 26/52 = 1/2
    (iii) Neither King nor red: 26/52 – 2/52 = 24/52 = 6/13
21. Sum of squares of first 20 natural numbers:

$$S = \frac{20*21*41}{6} = 2870$$S=620∗21∗41​=2870

22. Sum of A.P. 7,11,15,… n terms

$$S_n = \frac{n}{2}[2a + (n-1)d] = \frac{n}{2}[14 + 4(n-1)] = \frac{n}{2}[4n+10] = 2n^2 + 5n$$Sn​=2n​[2a+(n−1)d]=2n​[14+4(n−1)]=2n​[4n+10]=2n2+5n

---

Section D – Long Answer Type (4 Marks Each)

23. Area under $y=x^2+2x$y=x2+2x from 0 to 2:

$$\int_0^2 (x^2+2x) dx = \left[\frac{x^3}{3}+x^2\right]_0^2 = \frac{8}{3}+4 = \frac{20}{3}$$∫02​(x2+2x)dx=[3×3​+x2]02​=38​+4=320​

24. Two dice:
    (i) Sum 7 → pairs: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) → P=6/36=1/6
    (ii) Sum >10 → (5,6),(6,5),(6,6) → P=3/36=1/12
25. $y = e^x \sin x$y=exsinx

$$\frac{dy}{dx} = e^x \sin x + e^x \cos x = e^x(\sin x + \cos x)$$dxdy​=exsinx+excosx=ex(sinx+cosx)

26. Solve $\frac{dy}{dx} = \frac{y}{x}$dxdy​=xy​ → separable

$$\frac{dy}{y} = \frac{dx}{x} \implies \ln y = \ln x + C \implies y = kx$$ydy​=xdx​⟹lny=lnx+C⟹y=kx

27. Inverse of $\begin{bmatrix}2 &3\\5 &4\end{bmatrix}$[25​34​]

$$\det = 2*4 -3*5 = 8-15=-7 A^{-1} = \frac{1}{-7}\begin{bmatrix}4 & -3\\ -5 & 2\end{bmatrix} = \begin{bmatrix}-4/7 & 3/7\\5/7 & -2/7\end{bmatrix}$$det=2∗4−3∗5=8−15=−7A−1=−71​[4−5​−32​]=[−4/75/7​3/7−2/7​]

28. Centroid: $G = \left(\frac{0+4+0}{3}, \frac{0+0+3}{3}\right)=(4/3,1)$G=(30+4+0​,30+0+3​)=(4/3,1)
    Circumcenter: midpoints & slopes → $O = (2,1.5)$O=(2,1.5)
29. $\int_1^2 \frac{1}{x} dx = \ln 2 – \ln 1 = \ln 2$∫12​x1​dx=ln2−ln1=ln2
30. Committee 2 boys, 1 girl:

$$\binom{5}{2} \cdot \binom{3}{1} = 10*3=30$$(25​)⋅(13​)=10∗3=30

---

Section E – Very Long Answer / Application (5 Marks Each)

31. Doublet probability:
    6 outcomes (1,1),(2,2)…,(6,6) → 6/36 = 1/6
32. Savings: A.P., a=100, d=20, n=12

$$S_n = \frac{n}{2}[2a+(n-1)d] = 6[200 + 220] = 6*420 = 2520$$Sn​=2n​[2a+(n−1)d]=6[200+220]=6∗420=2520

33. $\frac{dy}{dx} = 3x^2+2x-1 \implies y = x^3 + x^2 – x + C, y(1)=2$dxdy​=3×2+2x−1⟹y=x3+x2−x+C,y(1)=2

$$2 = 1+1-1+C \implies C=1 \implies y = x^3+x^2-x+1$$2=1+1−1+C⟹C=1⟹y=x3+x2−x+1

34. A.P. 3,8,13… first 20 terms:
    $S_{20} = \frac{20}{2}[2*3 + 19*5] = 10[6+95]=10*101=1010$S20​=220​[2∗3+19∗5]=10[6+95]=10∗101=1010
    20th term $T_{20} = 3 + 19*5 = 98$T20​=3+19∗5=98

---