II PUC Mathematics Past Paper 2024-25 PDF

```markdown # Adhyayana Knowledge. Wisdom. Success. #999, 1st Main, 4th Cross, Hosahalli, Near Vasavi college, Vijayanagara, Bangalore ~ 560040. ## II PUC - Mathematics ## Part – E ### (MOST EXPECTED QUESTIONS) ### (Only for the Academic year 2024 – 25) ### SIX MARK QUESTIONS: **INTEGRALS:** 1. Pr...

```markdown # Adhyayana Knowledge. Wisdom. Success. #999, 1st Main, 4th Cross, Hosahalli, Near Vasavi college, Vijayanagara, Bangalore ~ 560040. ## II PUC - Mathematics ## Part – E ### (MOST EXPECTED QUESTIONS) ### (Only for the Academic year 2024 – 25) ### SIX MARK QUESTIONS: **INTEGRALS:** 1. Prove that $\int_{a}^{b} f(x)dx = \int_{a}^{c} f(x)dx + \int_{c}^{b} f(x)dx$ and hence evaluate (i) $\int_{1}^{3} |x-5|dx$, (ii) $\int_{2}^{3} |x^2-1|dx$. 2. Prove that $\int_{a}^{b} f(x)dx = \int_{a}^{b} f(a+b-x)dx$ and hence evaluate $\int_{0}^{\frac{\pi}{2}}\frac{1}{1+\sqrt{\tan x}}dx$. (A 21)(J 19)(J 15)(J 14) 3. Prove that $\int_{0}^{a} f(x)dx = \int_{0}^{a} f(a-x)dx$ and hence evaluate (i) $\int_{0}^{\frac{\pi}{2}}\frac{\cos^5 x}{\cos^5 x + \sin^5 x}dx$, (ii) $\int_{0}^{\frac{\pi}{2}}\frac{\sqrt{x}}{\sqrt{a-x}+\sqrt{x}}dx$, (iii) $\int_{0}^{\frac{\pi}{2}}(2\log\sin x - \log \sin 2x)dx$, (iv) $\int_{0}^{\frac{\pi}{4}}\frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}}dx$, (v) $\int_{0}^{\frac{\pi}{4}}\log(1+\tan x)dx$, (vi) $\int_{0}^{\frac{\pi}{2}}\frac{\sin x - \cos x}{1+\sin x\cos x}dx$, (vii) $\int_{0}^{\frac{\pi}{2}}\frac{\sin^2 x}{\sin^2 x + \cos^2 x}dx$. (J 24)(A 24)(M 24)(J 23)(M 23)(J 22)(M 22)(S 20)(M 19)(M 18)(M 17)(J 16)(M 14) 4. Prove that $\int_{0}^{2a} f(x)dx = 2\int_{0}^{a} f(x)dx$ if $f(2a-x) = f(x)$ and hence evaluate $\int_{0}^{2\pi}cos^5 x.dx$, if $f(2a-x) = -f(x)$ (M 16) 5. Prove that $\int_{-a}^{a} f(x)dx = 2\int_{0}^{a} f(x)dx$ if $f(x)$ is even and evaluate. $\int_{-1}^{0} f(x)dx= 0$ if $f(x)$ is odd (i) $\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} sin^2x.dx$. (ii) $\int_{-1}^{1} (x^2+x\cos x)dx$. (iii) $\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} sin^5x.\cos^2 x.dx$. (A 23)(M 20)(J 18)(J 17)(M 15) 6. **LINEAR PROGRAMMING PROBLEM:** 1. Maximize $Z = 4x + y$ (M 20)(M 22)(M 24) Subjected to constraints: $x + y \le 50, 3x + y \le 90, x \ge 0, y \ge 0$. 2. Maximize $Z = 3x + 2y$ (S 20)(J 23)(A 23) Subjected to constraints: $x + 2y \le 10, 3x + y \le 15, x, y\ge 0$. 3. Minimize $Z = 200x + 500y$ Subjected to constraints: $x + 2y \ge 10, 3x + 4y \le 24, x \ge 0, y \ge 0$. 4. Minimize $Z = -3x + 4y$ (J 17)(A 21)(J 22)(M 23) Subjected to constraints: $x + 2y \le 8, 3x + 2y \le 12, x \ge 0, y \ge 0$. 5. Minimize $Z = -3x + 4y$ (A 24) Subjected to constraints: $x + 2y \le 8, 3x + 2y \le 12, x \ge 0, y \ge 0$. 6. Minimize and Maximize $Z = 3x + 9y$ (M 18)(J 16)(J 24) Subjected to constraints: $x + 3y \le 60, x + y \ge 10, x \le y, x \ge 0, y \ge 0$. 7. Minimize and Maximize $Z = 600x + 400y$ (M 17) Subjected to constraints: $x + 2y \le 12, 2x + y \le 12, 4x + 5y \ge 20, x \ge 0, y \ge 0$. 8. Minimize and Maximize $Z = 5x + 10y$ (M 19)(M 16) Subjected to constraints: $x + 2y \le 120, x + y \ge 60, x - 2y \ge 0, x, y \ge 0$. 9. Minimize and Maximize $Z = x + 2y$ (J 14)(M 14) Subjected to constraints: $x + 2y \ge 100, 2x - y \le 0, 2x + y \le 200, x, y \ge 0$. 10. Minimize $Z = 3x + 5y$ Subjected to constraints: $x + 3y \ge 3, x + y \ge 2, x \ge 0, y \ge 0$. 11. Minimize $Z = 3x + 2y Prepared by: Linge Gowda A P(99160 23722) Subjected to constraints: $x + y \ge 8, 3x + 5y \le 15, x \ge 0, y \ge 0$. ## FOUR MARK QUESTIONS: **DETERMINANTS:** 1. Show that the matrix $A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}$ satisfies the equation $A^2 - 4A + I = 0$, where $I$ is 2x2 identity matrix and $O$ is 2x2 zero matrix. Using this equation, find $A^{-1}$. (A 21) 2. Show that the matrix $A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}$ satisfies the equation $A^2 - 5A + 7I = 0$, where $I$ is 2x2 identity matrix and $O$ is 2x2 zero matrix. Using this equation, find $A^{-1}$. (M 24)(A 24)(J 24) 3. Show that the matrix $A = \begin{bmatrix} 5 & 6 \\ 4 & 3 \end{bmatrix}$ satisfies the equation $A^2 - 8A - 9I = 0$, where $I$ is 2x2 identity matrix and $O$ is 2x2 zero matrix. Using this equation, find $A^{-1}$. (MQP) 4. For the matrix $A = \begin{bmatrix} 3 & 2 \\ 1 & 1 \end{bmatrix}$, find the numbers $a$ and $b$ such that $A^2 + aA + bI = 0$. 5. If $A = \begin{bmatrix} 2 & 3 \\ 1 & -4 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & -2 \\ -1 & 3 \end{bmatrix}$, then verify that $(AB)^{-1} = B^{-1}A^{-1}$. (MQP) 6. Let $A = \begin{bmatrix} 3 & 7 \\ 2 & 5 \end{bmatrix}$ and $B = \begin{bmatrix} 6 & 8 \\ 7 & 9 \end{bmatrix}$. Verify that $(AB)^{-1} = B^{-1}A^{-1}$. 7. Verify $A(adj A) = (adj A)A = |A|I$ for the matrix $A = \begin{bmatrix} 2 & 3 \\ -4 & -6 \end{bmatrix}$. 8. Find the inverse of the matrix $\begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\alpha & \sin\alpha \\ 0 & \sin\alpha & -\cos\alpha \end{bmatrix}$. 9. If $A = \begin{bmatrix} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{bmatrix}$, then verify that $A \ adj A = |A|I$. Also find $A^{-1}$. 10. If the matrix $A = \begin{bmatrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{bmatrix}$ satisfying $A^3 - 6A^2 + 9A - 4I = 0$, then evaluate $A^{-1}$. (MQP) **CONTINUITY AND DIFFERENTIABILITY:** 1. Find the value of $k$ so that the function $f(x) = \begin{cases} kx+1 & \text{if } x \le 5 \\ 3x-5 & \text{if } x > 5 \end{cases}$ at $x=5$ is a continuous function. (J 22)(M 19)(M 15)(MQP 2) 2. Find the value of $k$ so that the function $f(x) = \begin{cases} kx^2 & \text{if } x \le 2 \\ 3 & \text{if } x > 2 \end{cases}$ is continuous at $x=2$. (A 23)(J 23)(M 23)(J 18)(J 16) 3. Find the value of $k$, if $f(x) = \begin{cases} kx+1 & \text{if } x \le \pi \\ \cos x & \text{if } x > \pi \end{cases}$ is continuous at $x = \pi$. (M 20)(A 21)(A 22)(M 24)(A 24)(J 24) 4. Find the relationship between $a$ and $b$ so that the function $f$ defined by $f(x) = \begin{cases} ax+1 & \text{if } x \le 3 \\ bx+3 & \text{if } x > 3 \end{cases}$ is continuous at $x = 3$. (M 18)(MQP 1) 5. For what value of $\lambda$ is the function defined by $f(x) = \begin{cases} [\lambda^2x - 2x] & \text{if } x \le 0 \\ 4x+1 & \text{if } x > 0 \end{cases}$ continuous at $x=0$? (J 17) 6. If $f(x) = \begin{cases} \frac{k \cos x}{\pi - 2x} & \text{if } x < \frac{\pi}{2} \\ 3 & \text{if } x = \frac{\pi}{2} \end{cases}$ is continuous at $x = \frac{\pi}{2}$, find the value of $k$. (S 20)(J 19)(M 17)(J 14)(M 14) 7. Find the values of $a$ and $b$ such that $f(x) = \begin{cases} 5 & \text{if } x \le 2 \\ ax+b & \text{if } 2 < x < 10 \\ 21 & \text{if } x \ge 10 \end{cases}$ is a continuous function. (J 15) 8. Find all points of discontinuity of $f(x)$, where $f$ is defined by $f(x) = \begin{cases} x^3-3 & \text{if } x \le 2 \\ x^2+1 & \text{if } x > 2 \end{cases}$. (MQP 4) 9. Find the points of discontinuity of the function $f(x) = x - [x]$, where $[x]$ indicates the greatest integer not greater than $x$. Also write the set of values of $x$, where the function is continuous. (MQP 3) 10. Show that the function defined by $f(x) = \sin x^2$ is a continuous function. 11. Prove that the function $f$ given by $f(x) = |x-1|, x \in R$ is not differentiable at $x=1$. Prepared by: Linge Gowda A P(99160 23722) 12. Prove that the greatest integer function defined by $f(x) = [x], 0 < x < 3$ is not differentiable at $x=1$ and $x=2$. 13. Differentiate $(\sin x)^x + \sin^{-1}x$ with respect to $x$. 14. Find the derivative of the function given by $f(x)=(1+x)(1+x^2)(1+x^4)(1+x^8)$ and hence find $f'(1)$. 15. If $(\cos x)^y = (\cos y)^x$, then find $\frac{dy}{dx}$. 16. Differentiate $x^x - 2^{\sin x}$ with respect to $x$. 17. Differentiate $(\log x)^x + x^{\log x}$ with respect to $x$. ### Conduct coaching classes for ### I and II PUC (Science) ### (Boards + KCET/JEE/NEET) ### For more information Contact: ### +91 99160 23722 /+91 709 0000 283 ```