SPSC Mathematics 25 years Past Papers

Q21: The value of lim x → 0 x Sin(15/a) is?

Zero

Infinity

15

None of these

Explanation

Since $lim_{x to 0} x sin(15/a)$ is of the form $x cdot text{constant}$ , and $x to 0$ , the whole expression tends to 0.

The sine function remains bounded, so multiplying by $x$ (which approaches zero) results in zero.

Q22: A group generated by one of its elements, is called?

A cyclic group

An abelian group

Both A and B

None of these

Q23: If 1 - I, is a root of the equation x² + ax + b = 0, where a, b, ∈ R, then the value of a - b is?

1

2

-4

None of these

Q24: What is the value of the given limit, lim 2/x x→∞?

0

2

1/2

None of these

Q25: Rolle's Theorem is special case of ______?

Taylor Series of a function

Leibniz theorem

Mean value theorem

None of these

Explanation

Rolle's Theorem is a special case of the Mean Value Theorem (MVT) where the function has equal values at two endpoints.

It ensuring the derivative is zero at some point in the interval.

It is derived from MVT by adding the condition $f(a) = f(b)$ .

Q26: Which of the following expressions represents the distributive law of Boolean algebra?

A(B + C) = AB + AC

A + (BC) = (A + B)(A + C)

AB + BC = A(B + C)

None of these

Q27: The probability distribution whose mean and standard deviation are equal is _____?

Log-normal distribution

Exponential distribution

Normal

None of these

Q28: As u = (2, -2, 3) and v = (1, -3, 4), then ||-2u + 2v|| is ______?

2√3

√2

√3

None of these

Calculation: First, compute -2u + 2v:−2u=(−4,4,−6)−2u=(−4,4,−6)2v=(2,−6,8)2v=(2,−6,8)−2u+2v=(−4+2,4−6,−6+8)=(−2,−2,2)−2u+2v=(−4+2,4−6,−6+8)=(−2,−2,2)

Explanation

Calculation: First, compute -2u + 2v:
- $- 2 u = (- 4, 4, - 6)$
- $2 v = (2, - 6, 8)$
- $- 2 u + 2 v = (- 4 + 2, 4 - 6, - 6 + 8) = (- 2, - 2, 2)$
Norm Calculation: The norm (magnitude) of $- 2 u + 2 v$ is:
$∣ ∣ - 2 u + 2 v ∣ ∣ = \sqrt{(- 2)^{2} + (- 2)^{2} + 2^{2}} = \sqrt{4 + 4 + 4} = \sqrt{12} = 2 \sqrt{3}$

Answer:

Q29: Solution of the separable differential equation (1+ x)dy - ydx = 0 is _____?

y = c(1 - x)

y = c^2 + (1 + x)

y = c(1 + x)

None of these

Q30: If Hessian Matric H is zero at any point, then the point is called?

Parabolic

Elliptic

Hyperbolic

None of these