PSEB CLASS 12 MATHEMATICS SAMPLE PAPER

 PSEB CLASS 12 MATHEMATICS SAMPLE PAPER  SET 1 

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Mathematics Paper
M.M. – 80
Time Allowed – 3 hrs

Section A (Each Question Carries one Mark)

Q1. MCQ (Choose the Correct Option)

(i) Let $f: \mathbb{N} \to \mathbb{N}$ be defined as $f(x) = x^2$, then
(a) $f$ is one-one only
(b) $f$ is onto only
(c) $f$ is both one and onto
(d) $f$ is neither one-one nor onto

(ii) If $L$ is a set of all straight lines in a plane and $R$ is a relation on $L$ defined as
$R = \{ (L_1, L_2): L_1 \parallel L_2 \}$, then relation $R$ is
(a) Reflexive only
(b) Symmetric only
(c) Transitive only
(d) Equivalence relation

(iii) Principal value of $\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)$ is
(a) $\frac{5\pi}{6}$
(b) $\frac{\pi}{6}$
(c) $-\frac{\pi}{6}$
(d) $-\frac{5\pi}{6}$

(iv) If $A$ is a matrix of order $2 \times 3$ and $B$ is a matrix of order $4 \times 2$, then order of $A^T B$ is
(a) $3 \times 4$
(b) $4 \times 3$
(c) $2 \times 2$
(d) $2 \times 4$

(v) Let $A$ be a square matrix of order $3 \times 3$ and $|2A| = 264$, then $|A|$ equals to
(a) 12
(b) 8
(c) 3
(d) 4

(vi) If $\begin{bmatrix} k & 4 \\ -4 & -k \end{bmatrix}$ is a singular matrix, then value of $k$ is
(a) 0
(b) $\pm 4$
(c) 4
(d) -4

(vii) If the function defined by $f(x) = \begin{cases} \dfrac{\sin 5x}{x}, & x \neq 0 \\ k+2, & x = 0 \end{cases}$ is continuous at $x=0$, then value of $k$ is
(a) 0
(b) 2
(c) -2
(d) -1

(viii) If $x = 2at, y = at^2$, then $\dfrac{dy}{dx}$ is equal to
(a) $t$
(b) $2at$
(c) $\dfrac{1}{t}$
(d) $2a$

(ix) If $f(x) = x^2 + 3x + 4$, then $f'(0)$ is
(a) 6
(b) 3
(c) 4
(d) 9

(x) The interval for which the function $f(x) = x^2 - 6x + 3$ is strictly increasing is
(a) (-3, 3)
(b) (3, ∞)
(c) (-3, ∞)
(d) (-∞, 3)

(xi) $\int_{-3}^3 (x^5 + \sin x) dx$ equals to
(a) $5^5 + \sin 3$
(b) 3
(c) 0
(d) 6

(xii) $\int \dfrac{(\log x)^2}{x} dx$ equals
(a) $\dfrac{1}{x} + C$
(b) $\log x + C$
(c) $\dfrac{(\log x)^2}{2} + C$
(d) $\dfrac{(\log x)^3}{3} + C$

(xiii) The number of arbitrary constants in the particular solution of a differential equation of fourth order is
(a) 0
(b) 4
(c) 3
(d) 2

(xiv) The inequality $|a+b| \leq |a| + |b|$ is called
(a) Cauchy–Schwartz Inequality
(b) Rolle’s Thm
(c) Triangular Inequality
(d) L.M.V. Thm

(xv) If $\vec{a}$ is any vector, then $\vec{a} \times \vec{a}$ is
(a) $\vec{0}$
(b) $a^2$
(c) $\vec{a}$
(d) $\hat{i}$

(xvi) If lines $\dfrac{x-1}{2} = \dfrac{y-3}{2} = \dfrac{z+6}{a}$ and $\dfrac{x-1}{1} = \dfrac{y-3}{2} = \dfrac{z+6}{2}$ are perpendicular, then value of $k$ is 
(a) 0
(b) 2
(c) 1
(d) -2
(e) 3

(XVII) The vector form of line 

$\dfrac{x-5}{3} = \dfrac{y+4}{-1} = \dfrac{z+8}{6}$ is:
(a) $\vec{r} = (3\hat{i} + 7\hat{j} + 6\hat{k}) + \lambda (5\hat{i} + 4\hat{j} + 8\hat{k})$
(b) $\vec{r} = (5\hat{i} - 4\hat{j} - 8\hat{k}) + \lambda (3\hat{i} + 7\hat{j} + 6\hat{k})$
(c) $\vec{r} = (5\hat{i} + 4\hat{j} + 8\hat{k}) + \lambda (3\hat{i} + 7\hat{j} + 6\hat{k})$
(d) $\vec{r} = (3\hat{i} + 7\hat{j} + 6\hat{k}) + \lambda (5\hat{i} - 4\hat{j} - 8\hat{k})$

(XVIII) The maximum value of $Z = 3x + 5y$ subject to constraints $x + y \leq 4, x, y \geq 0$ is
(a) 0
(b) 12
(c) 20
(d) 28

(XIX) The probability of obtaining an even prime number on each die, when a pair of dice is rolled is
(a) 0
(b) $\dfrac{1}{12}$
(c) $\dfrac{1}{6}$
(d) $\dfrac{1}{36}$

(XX) If $P(A) = \dfrac{7}{13}, P(B) = \dfrac{9}{13}, P(A \cap B) = \dfrac{4}{13}$, then $P(A \cup B)$ is
(a) $\dfrac{7}{13}$
(b) $\dfrac{7}{9}$
(c) $\dfrac{4}{9}$
(d) $\dfrac{4}{7}$

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Section – B (Each question carries two marks)

Q2. If $A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}$, find $f(A)$ where $f(x) = x^2 - 5x + 7$.

Q3. Differentiate $\cos^{-1}\left(\dfrac{1 - x^2}{1 + x^2}\right)$ w.r.t. $x$.

Q4. Evaluate: $\int e^x \left[\cot x + \log(\sin x)\right] dx$.

Q5. Find the area of the region bounded by the ellipse $\dfrac{x^2}{4} + \dfrac{y^2}{9} = 1$.

Q6. A man 2 m high walks at a uniform speed of 5 m/s away from a lamp post 6 m high. Find the rate at which the length of shadow increases.

Q7. A particle moves along the curve $6y = x^3 + 2$. Find the points on the curve at which the $y$-coordinate is changing 8 times as fast as the $x$-coordinate.

Q8. Find the projection of the vector $(\hat{i} + 3\hat{j} + \hat{k})$ on the vector $(7\hat{i} - \hat{j} + 8\hat{k})$.

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Section – C (Each question carries four marks)

Q9. Show that the function $f: \mathbb{R} \to \mathbb{R}, f(x) = \dfrac{3x - 6}{7}$ is one-one and onto.

Q10 (a). If $A = \begin{bmatrix} 4 \\ 3 \end{bmatrix}, B = \begin{bmatrix} -1 & 2 \end{bmatrix}$, verify that $(AB)^T = B^T A^T$.
(b) Using determinants, find the equation of line joining (1,2) and (3,6).

Q11. Find $\dfrac{dy}{dx}$ when $x^y + y^x = 2$.

Q12. If $x = \tan \left(\dfrac{y}{a} \log y\right)$, show that $(1 + x^2)\dfrac{dy}{dx} + (2x - a)\dfrac{dy}{dx} = 0$.

Q13. Evaluate: $\int \dfrac{1}{x^2 + 1} dx$.

Q13. (a) Evaluate: 

$\int_{2}^{5} |x-3| \, dx$

(b) Evaluate: $\int_{0}^{\frac{\pi}{2}} 2 \log \cos x - \log \sin 2x \, dx$

(c) Evaluate: $\int (\tan x + \sqrt{\cot x}) \, dx$

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Q14. Solve the LPP graphically:
Max and Min $Z = 8x + 5y - 2$
subject to constraints:

$$x+y \leq 10, \; x+2y \geq 6, \; 3x - y \leq 9, \; y \leq 9, \; x, y \geq 0$$

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Q15. A problem is given to three students whose chances of solving it are $\tfrac{1}{3}, \tfrac{1}{5}, \tfrac{1}{7}$ respectively. Find the probability that:
(i) the problem is solved
(ii) exactly one of them solves the problem.

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Q15 (alt). A factory has two machines A and B. Machine A produces 60% of the items of the output and machine B produces 40% of the items. Further, 2% of the items produced by machine A and 1% produced by machine B are defective. All of the items are put into one stockpile and then one item is chosen at random and is found to be defective. Find the probability that it was produced by machine B.

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Section – D (Each question carries 6 marks)

Q16. Solve the system of linear equations by Matrix Method:

$$5x + 2y + z = 3, \quad 2x + 3z = 1, \quad 3x - 2y + 4z = -1$$

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Q16 (alt).
(a) Express the matrix $\begin{bmatrix} 3 & 5 \\ 4 & 2 \end{bmatrix}$ as a sum of symmetric and skew-symmetric matrices.

(b) If $A = \begin{bmatrix} 3 & -1 \\ -1 & 2 \end{bmatrix}$, show that $A^2 - 5A + 7I = 0$. Hence find $A^{-1}$.

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Q17. (a) Find the general solution of differential equation:

$$\frac{dy}{dx} = (4 + y^2)(1 + 3x^2)$$

(b) Solve the differential equation:

$$\frac{dy}{dx} + 3y = \cos 2x$$

(c) Show that the height of a cylinder of maximum volume that can be inscribed in a sphere of radius $R$ is $\frac{2R}{\sqrt{3}}$. Also find the maximum volume.

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Q18. (a) Express $\vec{a} = 5\hat{i} - 2\hat{j} + 5\hat{k}$ as the sum of two vectors such that one is parallel to $\vec{b} = 3\hat{i} + \hat{k}$ and other is perpendicular to $\vec{b}$.

(b) Find the area of parallelogram whose diagonals are
$\vec{d}_1 = \hat{i} - 2\hat{j} + 3\hat{k}, \quad \vec{d}_2 = 3\hat{i} + 2\hat{j} + \hat{k}$.

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Q19. Find the equation of a line which passes through point $(1, 2, 3)$ and is parallel to the line $\dfrac{x-2}{1} = \dfrac{y+3}{7} = \dfrac{z-6}{3}$.

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(b) Find the shortest distance between lines:

$$\vec{r}_1 = ( \hat{i} + 2\hat{j} + 3\hat{k} ) + \lambda ( \hat{i} - 3\hat{j} + 2\hat{k} )$$

and

$$\vec{r}_2 = ( 4\hat{i} + 5\hat{j} + 6\hat{k} ) + \mu ( 2\hat{i} + 3\hat{j} + \hat{k} )$$

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