Mathematics Question Sample Paper – Class 12

 

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Mathematics Question Paper – Class 12

Maximum Marks: 80
Time Allowed: 3 Hours

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Section-A (Each question carries one mark)

Q1. MCQ (Choose the correct option):

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

(ii) Let $L =$ set of all straight lines in plane and R is a relation on L defined as
$R = \{ (l_1, l_2) \mid l_1 \parallel l_2 \}$. Then relation R is
(a) Reflexive only
(b) Symmetric only
(c) Transitive only
(d) Equivalence relation

(iii) Principal value of $\sin^{-1}(-1/2)$ is
(a) 5π/6
(b) π/6
(c) -π/6
(d) -5π/6

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

(v) Let A be a square matrix of order 3×3 and $|2A| = 24$. 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) ±4
(c) 4
(d) -4

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

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

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

(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) $\int_{0}^{3} (x^2 + \sin x) dx$ equals to
(a) 3³ + sin 3
(b) 3
(c) 0
(d) 6

(xii) $\int \dfrac{(\log x)^2}{x} dx$ equals
(a) 1/x + C
(b) log x + C
(c) (log x)²/2 + C
(d) (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 Th.
(c) Triangular Inequality
(d) L.M.V Th.

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

(xvi) If lines $\dfrac{x-1}{2} = \dfrac{y-3}{3} = \dfrac{z+6}{7}$ and $\dfrac{x-1}{3} = \dfrac{y-3}{2} = \dfrac{z+6}{2}$ are perpendicular, then value of k is
(a) 2
(b) 1
(c) -2
(d) 3

(xvii) The vector form of a line $\dfrac{x-5}{3} = \dfrac{y+4}{7} = \dfrac{z+8}{6}$ 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) 1/36
(c) 1/12
(d) 1/6

(xx) If $P(A) = 7/13, P(B) = 9/13, P(A \cap B) = 4/13$, then $P(A \cup B)$ is
(a) 13/13
(b) 12/13
(c) 4/9
(d) 4/4

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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: $\int e^x (\cot x + \log (\sin x)) dx$.

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 $f: R \to R, f(x) = \dfrac{3x-6}{7}$ is one-one and onto.

Q10.
(a) If $A = \begin{bmatrix} 1 & -1 \\ -4 & 3 \end{bmatrix}, B = \begin{bmatrix} -1 & 2 \\ 0 & 1 \end{bmatrix}$, 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 $x^y + y^x = a$.

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

Q13.
(a) Evaluate $\int_{2}^{5} |x-3| dx$.
(b) Evaluate $\int_{0}^{\pi/2} \log \cos x - \log \sin 2x \, dx$.
(c) Evaluate $\int (\tan x + \cot x) dx$.

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Section-D (Each question carries six marks)

Q14. Solve the LPP graphically:
Maximize or Minimize $Z = 8x + 5y - 2$ subject to constraints:
$x+y \leq 10, \, x+y \geq 6, \, 3x-y \leq 9, \, y \leq 9, \, x,y \geq 0$.

Q15. A problem is given to three students whose chances of solving it are 1/2, 1/3, and 1/5 respectively. Find the probability that:
(i) The problem is solved.
(ii) Exactly one of them solves the problem.

(Alternative Q15) A factory has two machines A and B. Machine A produces 60% of the items and machine B produces 40%. Further, 2% of A’s items and 1% of B’s items are defective. One item is chosen at random and is found defective. Find the probability it was produced by B.

Q16. Solve the system of equations by Matrix Method:
$5x+2y+z=3, \, 2y-y+3z=1, \, 3x-2y+4z=-1$.

Q17.
(a) Express $\begin{bmatrix} 3 & 1 \\ 4 & 2 \end{bmatrix}$ as sum of symmetric and skew-symmetric matrices.
(b) If $A = \begin{bmatrix} 3 & 1 \\ -3 & 2 \end{bmatrix}$, show that $A^2 - 5A + 7I = 0$. Hence find $A^{-1}$.
(c) Solve the differential equation $\dfrac{dy}{dx} + 3y = \cos 2x$.
(d) Show that height of a cylinder of maximum volume inscribed in a sphere of radius R is $\dfrac{2R}{\sqrt{3}}$. Also find maximum volume.

Q18.
(a) Express $\vec{a} = 5\hat{i} - 2\hat{j} + 5\hat{k}$ as sum of two vectors, one parallel to $\vec{b} = 3\hat{i} + \hat{k}$ and other perpendicular to $\vec{b}$.
(b) Find area of parallelogram whose diagonals are $\vec{a} = \hat{i} - 2\hat{j} + 3\hat{k}$ and $\vec{b} = 3\hat{i} + 2\hat{j} + \hat{k}$.

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

Q20. 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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