Question:
If
\[
\Delta_r =
\begin{vmatrix}
1 & n & n \\
2r & n^2+n+1 & n^2+n \\
2r-1 & n^2 & n^2+n+1
\end{vmatrix}
\]
and \( \sum_{r=1}^{n} \Delta_r = 56 \), then \( n \) is
If
\[
\Delta_r =
\begin{vmatrix}
1 & n & n \\
2r & n^2+n+1 & n^2+n \\
2r-1 & n^2 & n^2+n+1
\end{vmatrix}
\]
and \( \sum_{r=1}^{n} \Delta_r = 56 \), then \( n \) is
Show Hint
Use linearity of determinant to simplify summation problems.
Updated On: Apr 23, 2026
- 4
- 6
- 7
- 8
Show Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
Concept:
Determinant linearity with respect to rows.
Step 1: Expand determinant.
$\Delta_r$ becomes linear in $r$.
Step 2: Use summation formula.
\[ \sum r = \frac{n(n+1)}{2} \]
Step 3: Simplify total sum.
Expression reduces to polynomial in $n$.
Step 4: Solve equation.
\[ \sum \Delta_r = 56 \Rightarrow n = 7 \] Conclusion:
$n = 7$
Step 1: Expand determinant.
$\Delta_r$ becomes linear in $r$.
Step 2: Use summation formula.
\[ \sum r = \frac{n(n+1)}{2} \]
Step 3: Simplify total sum.
Expression reduces to polynomial in $n$.
Step 4: Solve equation.
\[ \sum \Delta_r = 56 \Rightarrow n = 7 \] Conclusion:
$n = 7$
Was this answer helpful?
0
0
Top MET Mathematics Questions
- $\int\limits_{0}^{8}| x -5| d x$ is equal to
- The equation $3^{3x + 4} = 9^{2x - 2},\ x>0$ has the solution
- A parallelogram is constructed on the vectors $\mathbf{a} = 3\alpha - \beta$, $\mathbf{b} = \alpha + 3\beta$. If $|\alpha| = |\beta| = 2$ and the angle between $\alpha$ and $\beta$ is $\dfrac{\pi}{3}$, then the length of a diagonal of the parallelogram is
- Every group of order 7 is
- Let $U_{n} = 2 + 2^{3} + 2^{5} + \cdots + 2^{2n+1}$ and $V_{n} = 1 + 4 + 4^{2} + \cdots + 4^{n-1}$. Then $\displaystyle\lim_{n \to \infty} \dfrac{U_n}{V_n}$ is equal to
View More Questions
Top MET Properties of Determinants Questions
- If $A$ is a square matrix of order 3 and $|A| = 5$, then $|adj A|$ is:
- The value of \[ \begin{vmatrix} 1 & a & b + c \\ 1 & b & c + a \\ 1 & c & a + b \end{vmatrix} \] is:
- If \( x, y, z \) are all positive and are the \( p \)th, \( q \)th and \( r \)th terms of a geometric progression respectively, then the value of the determinant \( \begin{vmatrix} \log x & p & 1 \\ \log y & q & 1 \\ \log z & r & 1 \end{vmatrix} \) equals
- If \(A+B+C=\pi\), then \[ \begin{vmatrix} \sin(A+B+C) & \sin B & \cos C -\sin B & 0 & \tan A \cos(A+B) & -\tan A & 0 \end{vmatrix} \] is equal to
- The value of \[ \left| \begin{array}{ccc} (a+1)^2 & (b+1)^2 & (c+1)^2 (a-1)^2 & (b-1)^2 & (c-1)^2 \end{array} \right| \] is:
View More Questions
Top MET Questions
- A coil of 100 turns and area $2 \times 10^{-2}m^2 $ is pivoted about a vertical diameter in a uniform magnetic field and carries a current of 5A. When the coil is held with its plane in north-south direction, it experiences a couple of 0.33 Nm. When the plane is east-west, the corresponding couple is 0.4 Nm, the value of magnetic induction is [Neglect earth's magnetic field]
- $\int\limits_{0}^{8}| x -5| d x$ is equal to
- Radiation, with wavelength 6561 $?$ falls on a metal surface to produce photoelectrons. The electrons are made to enter a uniform magnetic field of $3 \times 10^{-4} T$. If the radius of the largest circular path followed by the electrons is 10 mm, the work function of the metal is close to :
- In intrinsic semiconductor at room temperature number of electrons and holes are
- Which of the following product is formed in the reaction $ CH_3MgBr {->[(i)CO_2][(ii)H_2O]} ? $
View More Questions