Grade XII Applied Mathematics Worksheet

1. The determinant of a 3×3 matrix (A) is 5. What is the determinant of (3A)?

1. 15
2. 45
3. 135
4. None of the above

2. Find the inverse of the matrix ( begin{bmatrix} 2 & 3 \ 1 & 4 end{bmatrix} ).

1. ( begin{bmatrix} 4 & -3 \ -1 & 2 end{bmatrix} )
2. ( begin{bmatrix} 4 & -1 \ -3 & 2 end{bmatrix} )
3. ( begin{bmatrix} 2 & -1 \ 1 & 4 end{bmatrix} )
4. Non-invertible

3. What is the principal value of ( arcsin(-1) )?

1. (-pi)
2. (-frac{pi}{2})
3. 0
4. (frac{pi}{2})

4. Solve the differential equation ( frac{dy}{dx} + 2y = e^{-x} ).

1. (y = Ce^{-2x} + frac{1}{2}e^{-x})
2. (y = Ce^{-2x} – e^{-x})
3. (y = Ce^{2x} + e^{-x})
4. (y = Ce^{2x} – frac{1}{2}e^{-x})

5. Evaluate the integral ( int frac{dx}{x^2 + 1} ).

1. (arctan(x) + C)
2. (arcsin(x) + C)
3. (ln|x| + C)
4. (frac{1}{x} + C)

6. Which is the equation of a parabola with focus at (2,3) and directrix y=1?

1. ((x-2)^2 = 4(y-2))
2. ((y-3)^2 = 8(x-2))
3. ((y-2)^2 = 4(x-2))
4. ((x-2)^2 = 8(y-3))

7. Calculate the area enclosed by the curve (y=x^3) and the x-axis from x=0 to x=2.

1. 4 square units
2. 6 square units
3. 8 square units
4. None of the above

8. What is the volume of the solid obtained by rotating the area between (y=x^2) and (y=x) around the x-axis from x=0 to x=1?

1. (frac{pi}{10}) cubic units
2. (frac{pi}{30}) cubic units
3. (pi) cubic units
4. (frac{pi}{15}) cubic units

9. A linear programming problem has an objective function (F = 2x + 3y) to maximize, with constraints (x geq 0), (y geq 0), (x + y leq 10), and (2x + 3y leq 18). The maximum value of (F) is:

1. 36
2. 30
3. 48
4. 54

10. The solution set for the inequality (3x – 7 leq 2x + 8) is:

1. (x leq 15)
2. (x geq 15)
3. (x leq -15)
4. (x geq -15)