Grade XII Applied Mathematics Practice Questions

1. Which equation represents the eigenvalues of the matrix ( begin{bmatrix} 2 & 1 \ 1 & 2 end{bmatrix} )?

1. (lambda^2 – 4lambda + 3 = 0)
2. (lambda^2 – 4lambda + 4 = 0)
3. (lambda^2 – 3lambda + 2 = 0)
4. (lambda – 3 = 0)

2. The area of the region bounded by the curves ( y = x^2 ) and ( y = x ) is:

1. 0.1667 square units
2. 0.3333 square units
3. 0.5 square units
4. 1 square unit

3. Solve the linear system using matrix inversion method: ( x + 2y = 5 ) and ( 3x + 4y = 11 ).

1. (x = 1, y = 2)
2. (x = 2, y = 1.5)
3. (x = 3, y = 1)
4. (x = 7, y = -1)

4. What is the Fourier transform of ( f(t) = e^{-|t|} )?

1. (frac{2}{1 + omega^2})
2. (frac{1}{1 + omega^2})
3. (frac{2}{omega^2 – 1})
4. (frac{2omega}{1 + omega^2})

5. A function ( f(x) ) is defined as ( f(x) = left{
begin{array}{ll}
x^2 & quad x < 1 \ 2x + 1 & quad x geq 1 end{array} right. ). Find the limit ( lim_{x to 1} f(x) ).

1. 1
2. 2
3. 3
4. Does not exist

6. Calculate the derivative of the function ( g(x) = ln(x^2 + 1) ).

1. (frac{2x}{x^2 + 1})
2. (frac{2}{x^2 + 1})
3. (frac{x}{x^2 + 1})
4. (frac{1}{2x(x^2 + 1)})

7. The integral ( int frac{x^2}{x^2 + 1} dx ) can be solved by:

1. Partial fractions
2. Substitution
3. Integration by parts
4. Direct integration

8. A probability function ( P(x) ) is defined for ( x = 0, 1, 2, ldots ). If ( E(X) = 1 ) and ( operatorname{Var}(X) = 2 ), what is the distribution of ( X )?

1. Poisson distribution
2. Binomial distribution
3. Normal distribution
4. Exponential distribution

9. The solution to the differential equation ( frac{d^2y}{dx^2} – 4frac{dy}{dx} + 4y = 0 ) is:

1. (y = (C_1 + C_2x)e^{2x})
2. (y = C_1 e^{2x} + C_2 e^{-2x})
3. (y = C_1 e^{4x} + C_2 e^{-4x})
4. (y = C_1 cos(2x) + C_2 sin(2x))

10. The length of the curve ( y = ln(cos x) ) from ( x = 0 ) to ( x = frac{pi}{4} ) is:

1. (frac{pi}{4})
2. (frac{1}{4})
3. (ln(2))
4. Cannot be determined