Mathematics Exam Paper 2025 - PDF

Here is the converted text from the images provided: ## SECOND PUC PREPARATORY EXAMINATION JANUARY - 2025 **Sub: MATHEMATICS (35)** **Time:** 3 Hrs **Total no. of questions:** 47 **Total Marks:** 80 **Instructions:** (i) The question paper has five parts namely A, B, C, D and E. Answer all the...

Here is the converted text from the images provided: ## SECOND PUC PREPARATORY EXAMINATION JANUARY - 2025 **Sub: MATHEMATICS (35)** **Time:** 3 Hrs **Total no. of questions:** 47 **Total Marks:** 80 **Instructions:** (i) The question paper has five parts namely A, B, C, D and E. Answer all the parts. (ii) Part- A has 15 multiple choice questions. 5 fill in the blanks of 1 mark each. (iii) Write the question numbers properly and use the graph sheet for the question on Linear Programming in Part - E. ### PART - A **I. Answer all the multiple choice questions: 15 x 1 = 15** 1. If a relation R on the set {1, 2, 3} be defined by R = {(1, 1)} then R is (a) Reflexive and symmetric (b) Reflexive and transitive (c) Symmetric and transitive (d) only symmetric 2. The range of $\sec^{-1} x$ is (a) $[-\frac{\pi}{2}, \frac{\pi}{2}]$ (b) $[-\frac{\pi}{2}, \frac{\pi}{2})$ (c) $[0,π]$ (d) $[0,π] - \{\frac{\pi}{2}\}$ 3. $\sin^{-1}(\sin[-600^\circ]) =$ (a) $\frac{\pi}{3}$ (b) $\frac{\pi}{3}$ (c) $\frac{2\pi}{3}$ (d) $-\frac{2\pi}{3}$ 4. If $A = \begin{bmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{bmatrix}$ then $A+A'= I$ if the value of $\alpha$ is (a) $\frac{\pi}{6}$ (b) $\frac{\pi}{3}$ (c) $\pi$ (d) $\frac{\pi}{2}$ 5. If A is a square matrix of order 3 and $|A| = 3$, then $|adj A|$ is (a) 3 (b) 9 (c) $\frac{1}{3}$ (d) 0 6. The function $f(x) = [x]$, where $[x]$ the greatest integer function is continuous at (a) 1.5 (b) 4 (c) 1 (d) -2 7. $\int e^x \{\frac{1}{\sqrt{x^2 + 9}} + \log (x + \sqrt{x^2 + 9}) \} dx =$ (a) $e^x \log(x + \sqrt{x^2 + 9}) + c$ (b) $\frac{e^x}{\sqrt{x^2 + 9}} + c$ (c) $e^x \cos \frac{x}{3} + c$ (d) $\frac{1}{\sqrt{x^2 + 9}}$ 8. The total revenue in rupees received from the sale of x units of a product is given by, $R(x) = 3x^2 + 36x + 5$. The marginal revenue, when $x = 15$ is (a) 116 (b) 96 (c) 90 (d) 126 9. $\int \frac{1}{\sqrt{x}} \cos \sqrt{x} dx$ (a) $2 \cos \sqrt{x} + c$ (b) $-2 \cos \sqrt{x} + c$ (c) $2 \sin \sqrt{x} + c$ (d) $\sin \sqrt{x} + c$ 10. The solution of $\frac{dy}{dx} + \sqrt{\frac{1-y^2}{1-x^2}} = 0$ is (a) $\cos^{-1}x - \cos^{-1}y = c$ (b) $\sin^{-1}x - \sin^{-1}y = c$ (c) $\sin^{-1}x + \sin^{-1}y = c$ (d) $\cos x + \cot y = c$ 11. If $\vec{a}, \vec{b}, \vec{c}$ are three vectors such that $\vec{a} + \vec{b} + \vec{c} = 0$ and $|\vec{a}| = 2, |\vec{b}| = 3, |\vec{c}| = 5$, then value of $\vec{a}.\vec{b} + \vec{b}.\vec{c} + \vec{c}.\vec{a}$ is (a) 0 (b) 1 (c) -19 (d) 38 12. The value of $\lambda$ for which the vectors $3\hat{i} - 6\hat{j} + \hat{k}$ and $2\hat{i} - 4\hat{j} + \lambda \hat{k}$ are parallel is (a) $\frac{2}{3}$ (b) $\frac{3}{2}$ (c) $\frac{5}{2}$ (d) $\frac{2}{5}$ 13. The angle between the straight lines $\frac{x+1}{2} = \frac{y-2}{5} = \frac{z+3}{4}$ and $\frac{x-1}{1} = \frac{y+2}{2} = \frac{z-3}{-3}$ is (a) $45^\circ$ (b) $30^\circ$ (c) $60^\circ$ (d) $90^\circ$ 14. If A and B are two events of a sample space S such that P(A) = 2.0, P(B) = 0.6 and P(A|B) = 0.5 then P(A'|B) = (a) $\frac{1}{2}$ (b) $\frac{3}{10}$ (c) $\frac{1}{3}$ (d) $\frac{2}{3}$ 15. If A and B are two independent events with P(A)=0.3 and P(A ∪ B) = 0.8 then P(B) is (a) $\frac{6}{7}$ (b) $\frac{5}{7}$ (c) $\frac{3}{7}$ (d) $\frac{4}{7}$ **II. Fill in the blanks by choosing appropriate answer from those given in the bracket.** 5 × 1 = 5 $[2, 4, 0, \frac{2}{3}, \frac{3}{2}]$ 16. The principal value of $\cos^{-1} (\cos(\frac{\pi}{k}))$, then the value of k = \_\_\_\_ 17. If $f(x) = \begin{cases} \frac{\sin(\frac{1}{x})}{k}, & x \neq 0 \\ k & x = 0 \end{cases}$ is continuous at $x = 0$, then K = \_\_\_\_ 18. If $\int_{0}^{\frac{\pi}{2}} \cos^2 x dx = \frac{\pi}{k}$, then the value of k = \_\_\_\_. 19. The projection on the y-axis of the vector $5\hat{i} + 2\hat{j} - 4\hat{k}$ is \_\_\_\_. 20. If P(A) = 0.3, P(B) = 0.5 and P(A/B) = 0.4 then P(B/A) is \_\_\_\_. ### PART - B **III. Answer any six questions: 6 x 2 = 12** 21. Find the area of the triangle with vertices (1, 0), (6, 0), (4, 3) using determinants. 22. Find $\frac{dy}{dx}$ if $y = \sin^{-1}(\frac{2x}{1+x^2})$. 23. An edge of a variable cube is increasing at the rate of 3 cm/sec. How fast is the volume of the cube increasing when the edge is 10cm, long? 24. Find the intervals in which the function f given by $f(x) = 2x^2 - 3x$ is strictly increasing. 25. Evaluate $\int_{1}^{e^{\pi/2}} \frac{e^{\tan^{-1}x}}{1+x^2} dx$. 26. In a bank, principal increases continuously at the rate of r% per year. Find the value of r, if Rs. 100 double itself in 10 years (log 2 = 0.6931). 27. Find a vector in the direction of vector $5\hat{i} - \hat{j} + 2\hat{k}$ which has magnitude 8 units. 28. Find the values of P so that the lines $\frac{1-x}{3P} = \frac{7y-14}{2} = \frac{z-3}{2}$ and $\frac{7-7x}{3P} = \frac{y-5}{1} = \frac{6-z}{5}$ are at right angles. 29. If A and B are two independent events, then prove that P(A U B) = 1-P(A').P(B'). ### PART - C **Answer any six questions: 6 x 3 = 18** 30. Determine whether the relation R in the set $A = \{1, 2, 3, 4, 5, 6\}$ as $R = \{(x, y) : y \text{ is divisible by } x\}$ is reflexive, symmetric and transitive. 31. Write the simplest form of $\tan^{-1} (\frac{\cos x}{1 - \sin x})$, $-\frac{3\pi}{2} < x < \frac{\pi}{2}$. 32. Prove that inverse of a square matrix, if it exists, is unique. 33. If $x = a (\theta - \sin \theta)$ and $y = a (1 + \cos \theta)$ then prove that $\frac{dy}{dx} = -\cot (\frac{\theta}{2})$. 34. An open topped box is to be constructed by removing equal squares from each corner of a 3 metre by 8 metre rectangular sheet of aluminium and folding up the sides. Find the volume of the larger such box. 35. Evaluate $\int \frac{dx}{x(x^n+1)}$. 36. If the vertices A, B and C of a triangle are (1, 2, 3), (-1, 0, 0) and (0, 1, 2) respectively then find [ABC]. 37. Find the shortest distance between the lines $\vec{r} = \hat{i} + \hat{j} + \lambda (2\hat{i} - \hat{j} + \hat{k})$ and $\vec{r} = 2\hat{i} + \hat{j} - \hat{k} + \mu (3\hat{i} - 5\hat{j} + 2\hat{k})$. 38. A man is known to speak truth 3 out of 4 times. He throws a die and reports that it is a six. Find the probability that it is actually a six. ### PART-D **Answer any Four questions 4 × 5 = 20** 39. Show that the function $f: N \rightarrow Y$ defined by $f(x) = 4x + 3$, where $Y = \{y = 4x + 3, y \in N, \text{for some } x \in N\}$ is invertible. Hence write the inverse of f. 40. If $A = \begin{bmatrix} 1 & 1 & -1 \\ 2 & 0 & 3 \\ 3 & -1 & 2 \end{bmatrix}$, $B = \begin{bmatrix} 1 & 3 \\ 0 & 2 \\ -1 & 4 \end{bmatrix}$ and $C = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 2 & 0 & -2 & 1 \end{bmatrix}$, find A (BC), (AB) C and that (AB)C = A (BC). 41. Solve the system of equations by matrix method $x - y + z = 4, 2x + y - 3z = 0, x + y + z = 2$. 42. If $y = (\tan^{-1} x)^2$ then show $(x^2+1)^2 \frac{d^2y}{dx^2} + 2x(x^2+1) \frac{dy}{dx} = 2$. 43. Find the integral of $\frac{1}{a^2 - x^2}$ with respect to x and hence evaluate $\int{\frac{1}{1 - x^2}} dx$. 44. Find the area of the region bounded by the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is by using integration. 45. Solve the differential equation $y.dx - (x + 2y^2).dy = 0$. ### PART-E **Answer the following questions** 46. Prove that $\int_{a}^{b} f(x) dx = \int_{a}^{c} f(x) dx + \int_{c}^{b} f(x)dx$ and hence evaluate $\int_{-1}^{2} |x + 2| dx$. **OR** Minimize and maximize $z = 3x + 9y$ subject to the constraints: $x + 3y \leq 60, x + y \geq 10, x \leq y, x \geq 0, y \geq 0$ by graphical method. 47. If $A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}$ satisfy the equation $A^2 - 5A + 7I = 0$ where $I$ is $2 \times 2$ identify matrix and 0 is $2 \times 2$ zero-matrix, using the equation find $A^{-1}$. **OR** Find the value of k, if $f(x) = \begin{cases} kx + 1 & \text{if } x \leq \pi \\ \cos x & \text{if } x > \pi \end{cases}$ is continuous at $x = \pi$.