Learn more about the **Mathematics-III** solved **question paper** for the **BCA**. Improve your mathematics skills with advanced calculus and numerical techniques while gaining insightful knowledge that will help you succeed in school.

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## Section A: **Mathematics-III** **Very Short ****Question** Solutions

**Question**Solutions

**Q1. Show that **

**|z**_{1}** + z**_{2}**|**^{2}** + |z**_{1}** – z**_{2}**|**^{2}** = 2|z**_{1}**|**^{2}** + 2|z**_{2}**|**^{2}

**Ans. **We have,

**Q2. Define sequence with example. **

**Ans. **Let S be any non-empty set. A function whose domain is the set of N natural numbers and whose range is a subset of S, is called a sequence in the set S.

**Q3. If f(x, y, z) 3x**^{2}**y – y**^{3}**z**^{2}** then find grad fat point (1, -2, -1). **

**Sol. **The gradient is a vector:

**Q4. Solve**

**Sol. **Let x + y = z

**Q5. Solve**

**(D**^{3}** + 3D**^{2}** + 3D + 1)y = e**^{x}** + e**^{-x}** **

**Sol. **Given equation is

(D^{3} + 3D^{2} + 3D + 1)y = e^{x} + e^{-x}

∴ Auxillary equation is

D^{3} + 3D^{2} + 3D + 1 = 0

∴ (D+ 1)³ =0

∴ D = -1, -1, -1 are the roots.

∴ C.F. = (c_{1}x^{2} + c_{2}x + c_{3}) (e^{x} + e^{-x})

## Section B: **Mathematics-III** **Short ****Question** Solutions

**Question**

**Q6. Determine the regions defined by **

**|z-1| + |z + 1| ≤ 4 **

**Sol. **Here, |z_{1} -z_{2}| = |1 – (-1)| = 2

and a = 4

∴ |z_{1} – z_{2}| < a

Since |z – z_{0}| + |z – z_{1}| = c (constant) is the equation of ellipse with z_{0} and z_{1} as focii, so |z – 1| + |z + 1| ≤ 4 is the equation of all points lying inside and on an ellipse with focii (1, 0) and (-1,0).

Hence, region represented by |z – 1| + |z + 1| ≤ 4 is an interior and boundary of an ellipse.

**Q7. Solve**

**Ans. **The given equation can be written as

**Q8. Show **

**Sol. **Given,

## Section C: **Mathematics-III** **Detailed ****Question** Solutions

**Question**

**Q9. Show that**

**Sol.** Let

Similarly

Comparing eqs. (1) and (2), we get

S_{n+1} ≥ S_{n}, ∀n

The sequence 〈S_{n}〉 is monotonically increasing.

Now, from eq. (1), we have

⇒ The sequence 〈S_{n}〉 is bounded.

Thus, the sequence〈S_{n}〉, being a monotonically increasing sequence bounded above by 3 is convergent and equal to e where 2 < e < 3.

⇒ Limit of the sequence 〈S_{n}〉 lies between 2 and 3, i.e. = e.

**Q10. Test the convergence of following series. **

**Sol.** Here, we have

If x< 1,the series is convergent

If x > 1, the series is divergent

If x = 1,

Therefore, we have to use another test.

Hence, given series is divergent.

**Q11. Obtain the Fourier series of **

**in the interval (0, 2𝝅) and hence deduce **

**Sol. **Fourier series off(x) having period 2𝝅, 0

The formula of Fourier series is

a_{0} = 0

From Bernoulli rule

Apply Bernoulli rule

Substitute a_{0}, a_{n}, b_{n} in Fourier series, we get

**12. Solve**

**(1 + y**^{2}**)dx – (tan**^{-1}**y – x)dy = 0 **

**Sol. **The given differential equation can be written as

Thus, the solution of eq. (1) is

Which is the required solution.

**Q13. Solve**

**(D**^{2}** + 1)y = sin x sin 2x**

**Sol.** Here, the given equation is

(D^{2} + 1)y = sin x. sin2x

To find the C.F. of eq. (1), the auxiliary equation is

(m^{2} + 1) = 0 ⇒ m ± i

Hence, the complete solution of eq. (1) is given by

y = C.F.+ P.I.