Elementor #1027

Solution

Let \(a\) be the first natural number and \(n\) be the number of consecutive natural numbers whose sum is \(200\).

Then,

\[ a+(a+1)+(a+2)+\cdots +(a+n-1)=200 \]

Using the sum formula for an arithmetic progression,

\[ \frac{n}{2}\big[a+(a+n-1)\big]=200 \]

\[ \frac{n}{2}(2a+n-1)=200 \]

\[ n(2a+n-1)=400 \]

Therefore, \(n\) must be a positive divisor of \(400\).

Now, solve for \(a\).

\[ 2a+n-1=\frac{400}{n} \]

\[ 2a=\frac{400}{n}-n+1 \]

\[ \boxed{a=\frac12\left(\frac{400}{n}-n+1\right)} \]

We need to find all positive integers \(n\) for which \(a\) is a natural number.

Checking possible values of \(n\)

1. Put \(n=1\):

\[ a=\frac12(400-1+1)=200 \] Hence, \(a=200\) is valid.

2. Put \(n=2\):

\[ a=\frac12(200-2+1)=\frac{199}{2} \] This is not a natural number.

3. Put \(n=4\):

\[ a=\frac12(100-4+1)=\frac{97}{2} \] This is not a natural number.

4. Put \(n=5\):

\[ a=\frac12(80-5+1)=38 \] Hence, \(a=38\) is valid.

5. Put \(n=8\):

\[ a=\frac12(50-8+1)=\frac{43}{2} \] This is not a natural number.

6. Put \(n=10\):

\[ a=\frac12(40-10+1)=\frac{31}{2} \] This is not a natural number.

7. Put \(n=20\):

\[ a=\frac12(20-20+1)=\frac12 \] This is not a natural number.

Valid examples:

\[ n=1,\ a=200 \] \[ n=5,\ a=38 \] For \(n=5\), the consecutive natural numbers are \(38,39,40,41,42\), and their sum is \(200\).

Note: The values in this worked solution have been corrected mathematically. The handwritten work uses \(200\) in an intermediate equation where the correct value should be \(400\), because the factor \(2\) must be accounted for.

Matrices and Determinants

Previous Year Questions — 2017

Q1. If \(A\) and \(B\) are two matrices such that \(AB=B\) and \(BA=A\), then \(A^2=\)

(a) \(2AB\)

(b) \(2BA\)

(d) \(B\)

Q2. The matrix below is:

\[ A=\begin{bmatrix} i & 1-2i\\ -1-2i & 0 \end{bmatrix} \]

(a) Symmetric

(b) Skew-symmetric

(d) Skew-Hermitian

Q3. If \(\frac13\) and \(-\frac12\) are eigenvalues of a non-singular matrix \(A\) and \(|A|=4\), then the eigenvalues of \(\operatorname{adj}(A)\) are:

(a) \( \frac43,-2 \)

(b) \( \frac43,-1 \)

(d) \(12,8\)

Q4. If \(A\) is a square matrix, then \(A-A^T\) is a:

(a) Unit matrix

(b) Zero matrix

(d) Skew-symmetric matrix

Here, \(A^T\) denotes the transpose of \(A\).

Q5. The value of the following determinant is:

\[ \begin{vmatrix} \frac1a & a^2 & bc\\ \frac1b & b^2 & ca\\ \frac1c & c^2 & ab \end{vmatrix} \]

(a) \(1\)

(b) \(0\)

(d) \(abc\)

Q6. The equations \(2x+y=4,\ 3x+2y=2,\ x+y=-2\) have:

(a) No solution

(b) One solution

(d) Infinitely many solutions

Q7. If \(A^T=-A\), where \(A\) is a \(3\times3\) matrix and \(A^T\) denotes its transpose, then \(|A|=\)

(a) \(1\)

(b) \(0\)

(d) \(2\)

Q8. If \(A\) is a singular matrix, then \(\operatorname{adj}(A)\) is necessarily:

(a) Singular

(b) Non-singular

(d) A null matrix