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Mathematics 2023 Paper I 50 marks Solve

Paper I — Q4

Find the rank of the matrix A = 1 & 2 & -1 & 0 -1 & 3 & 0 & -4 2 & 1 & 3 & -2 1 & 1 & 1 & -1 by reducing it to row-reduced…

Find the rank of the matrix

A = 1 & 2 & -1 & 0 -1 & 3 & 0 & -4 2 & 1 & 3 & -2 1 & 1 & 1 & -1

by reducing it to row-reduced echelon form. 15 marks

(b)

Trace the curve y²(x² - 1) = 2x - 1. 20 marks

(c)

Prove that the locus of a line which meets the lines y = mx, z = c; y = -mx, z = -c and the circle x² + y² = a², z = 0 is c²m²(cy - mzx)² + c²(yz - cmx)² = a²m²(z² - c²)². 15 marks

हिंदी में प्रश्न पढ़ें

आव्यूह A = 1 & 2 & -1 & 0 -1 & 3 & 0 & -4 2 & 1 & 3 & -2 1 & 1 & 1 & -1

का पंक्ति समानतित सोपानक रूप में समान्यन करके उसकी कोटि ज्ञात कीजिए। 15

(b)

वक्र y²(x² - 1) = 2x - 1 को अनुरेखित कीजिए। 20

(c)

सिद्ध कीजिए कि रेखाओं y = mx, z = c; y = -mx, z = -c और वृत्त x² + y² = a², z = 0 से मिलने वाली रेखा का विद्यु-पथ c²m²(cy - mzx)² + c²(yz - cmx)² = a²m²(z² - c²)² है। 15

Q4 of the 2023 UPSC Mains Mathematics Paper I, as printed
The question as printed in the 2023 Mathematics paper
Evaluate your answer to this question → See all 2023 Mathematics questions

Model answer

Written by UPSC Answer Check against this question's marking rubric, to the expected length. UPSC does not publish answers for Mains — this is one way to score well, not an official key.

(a) Use Gaussian elimination to reduce A to row-reduced echelon form.

A = [1 2 -1 0; -1 3 0 -4; 2 1 3 -2; 1 1 1 -1].

Apply:

  • R2 → R2 + R1 = (0,5,-1,-4)
  • R3 → R3 − 2R1 = (0,-3,5,-2)
  • R4 → R4 − R1 = (0,-1,2,-1)

Now swap R2 and R4: [1 2 -1 0; 0 -1 2 -1; 0 -3 5 -2; 0 5 -1 -4].

Then R2 → −R2: [1 2 -1 0; 0 1 -2 1; 0 -3 5 -2; 0 5 -1 -4].

Eliminate the second column:

  • R1 → R1 − 2R2 = (1,0,3,-2)
  • R3 → R3 + 3R2 = (0,0,-1,1)
  • R4 → R4 − 5R2 = (0,0,9,-9)

So: [1 0 3 -2; 0 1 -2 1; 0 0 -1 1; 0 0 9 -9].

Now R3 → −R3 = (0,0,1,-1). Eliminate the third column:

  • R1 → R1 − 3R3 = (1,0,0,1)
  • R2 → R2 + 2R3 = (0,1,0,-1)
  • R4 → R4 − 9R3 = (0,0,0,0)

Hence the row-reduced echelon form is [1 0 0 1; 0 1 0 -1; 0 0 1 -1; 0 0 0 0].

There are 3 non-zero rows. Therefore rank(A) = 3.

(b) The curve is y²(x² − 1) = 2x − 1.

Since only y² appears, the curve is symmetric about the x-axis.

Write y² = (2x − 1)/(x² − 1).

Intercepts:

  • On the x-axis, y = 0 gives 2x − 1 = 0, so x = 1/2. Point: (1/2,0).
  • On the y-axis, x = 0 gives −y² = −1, so y = ±1. Points: (0,±1).

Asymptotes:

  • x = 1 and x = −1 make the denominator zero while the numerator is non-zero, so these are vertical asymptotes.
  • As x → +∞, y² = (2x − 1)/(x² − 1) → 0, so y = 0 is a horizontal asymptote in the positive x-direction.

Region of existence: y² ≥ 0. The quotient is positive only for −1 < x ≤ 1/2 or x > 1. There is no real curve for x < −1 or 1/2 < x < 1.

Monotonicity: Let q(x) = (2x − 1)/(x² − 1). Then q′(x) = [2(x² − 1) − (2x − 1)2x]/(x² − 1)² = (−2x² + 2x − 2)/(x² − 1)² = −2(x² − x + 1)/(x² − 1)². But x² − x + 1 = (x − 1/2)² + 3/4 > 0, so q′(x) < 0 wherever defined. Thus |y| decreases on each interval.

Trace:

  • For −1 < x ≤ 1/2, the curve comes from x = −1 with y → ±∞, passes through (0,±1), and meets the x-axis at (1/2,0) with a vertical tangent.
  • For x > 1, the curve comes from x = 1 with y → ±∞ and tends to the x-axis y = 0 as x → +∞.
  • The upper and lower halves are mirror images about the x-axis.
  • The vertical lines x = −1 and x = 1 are asymptotes, and the x-axis is an asymptote for the right branch as x → +∞.

Thus the curve consists of two symmetric branches, one between x = −1 and x = 1/2, and the other for x > 1.

(c) Let the variable line meet L1: y = mx, z = c at P = (p, mp, c) and L2: y = −mx, z = −c at Q = (q, −mq, −c).

Let R = (x,y,z) be any point on the line PQ. Then R = P + t(Q − P).

Since z = c(1 − 2t), we get t = (c − z)/(2c), 1 − t = (c + z)/(2c).

Put S = p + q and D = p − q. Then 2cx = cS + zD, 2cy/m = zS + cD.

Solving these two equations: S = 2c(cmx − zy)/(m(c² − z²)), D = 2c(cy − mzx)/(m(c² − z²)).

The line meets the circle x² + y² = a², z = 0 at the point where t = 1/2. At z = 0, the coordinates of intersection are x₀ = S/2, y₀ = mD/2. Since this point lies on the circle, (S/2)² + (mD/2)² = a², so S² + m²D² = 4a².

Substitute the values of S and D: 4c²(cmx − zy)²/[m²(c² − z²)²] + 4c²(cy − mzx)²/(c² − z²)² = 4a².

Multiplying by m²(c² − z²)² gives c²(cmx − zy)² + c²m²(cy − mzx)² = a²m²(c² − z²)².

Since (cmx − zy)² = (yz − cmx)² and (c² − z²)² = (z² − c²)², this becomes c²m²(cy − mzx)² + c²(yz − cmx)² = a²m²(z² − c²)².

This is exactly the required locus equation.

What "Solve" is asking you to do

Choose the method, then carry it through to a final answer. Identifying what kind of problem this is and why that method applies is the first thing marked; a correct figure arrived at invisibly earns almost nothing.

Structure that answers it

Given data and what is required → method chosen, with the reason it applies → set-up (equation, circuit, free body, trial balance) → working, step by step → answer with units and any condition of validity

Where marks are lost

Doing the middle steps mentally and writing only the result. In mathematics papers, a further loss comes from giving a decimal where the exact value in surds or fractions was wanted, or from skipping the justification a part explicitly asks for.

All UPSC directive words, compared →

How this answer will be evaluated

Approach

(a) calculate: given > formula > substitution > result with units > interpretation | (b) trace: start point > the stages in sequence > end point > what changed | (c) justify: claim > 3-4 reasons > evidence > conclusion Full marks: Complete working, correct results, clear presentation, and verification steps.

Key points expected

  • Perform elementary row operations to reach RREF
  • Show intermediate matrices or steps clearly
  • Count non-zero rows to determine rank
  • State the final rank explicitly
  • Determine domain of x (where 2x-1 ≥ 0)
  • Find x and y intercepts
  • Analyze symmetry and asymptotes
  • Sketch the curve with key features labeled

Evaluation rubric

Each sub-part is marked on its own, against the marks and word limit printed on the paper.

  1. (a) Determine the rank of the given 4x4 matrix via row-reduced echelon form. 15 marks

    calculate— given → formula → substitution → result with units → interpretation

    Must cover

    • Perform elementary row operations to reach RREF
    • Show intermediate matrices or steps clearly
    • Count non-zero rows to determine rank
    • State the final rank explicitly

    Loses marks

    • Skipping intermediate row operation steps
    • Arithmetic errors in row reduction
    • Stating rank without showing RREF

    Earns more

    • Verify rank by checking minors of that order
    • Identify pivot columns correctly

    Extra mark

    • Mention determinant is zero for rank < 4
  2. (b) Analyze and sketch the curve defined by y²(x² - 1) = 2x - 1. 20 marks

    trace— start point → the stages in sequence → end point → what changed

    Must cover

    • Determine domain of x (where 2x-1 ≥ 0)
    • Find x and y intercepts
    • Analyze symmetry and asymptotes
    • Sketch the curve with key features labeled

    Loses marks

    • Missing domain analysis
    • Incorrect or missing sketch
    • Failing to identify asymptotes

    Earns more

    • Calculate turning points via differentiation
    • Analyze behavior at infinity

    Extra mark

    • Provide a neat, labeled diagram
  3. (c) Prove the equation of the locus of a line meeting two given lines and a circle. 15 marks

    justify— claim → 3-4 reasons → evidence → conclusion

    Must cover

    • Define the general equation of the line
    • Apply intersection conditions with the two given lines
    • Apply intersection condition with the circle
    • Derive the final locus equation algebraically

    Loses marks

    • Skipping algebraic derivation steps
    • Incorrect application of intersection conditions
    • Failing to reach the exact given equation

    Earns more

    • Use vector or parametric form for the line
    • Clearly state the geometric conditions used

    Extra mark

    • Geometric interpretation of the locus

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