Q1 50M Compulsory prove Linear algebra and calculus
(a) If $A=\begin{bmatrix} 1 & -1 & 1 \\ 2 & -1 & 0 \\ 1 & 0 & 0 \end{bmatrix}$, then show that $A^2 = A^{-1}$ (without finding $A^{-1}$). (10 marks)
(b) Find the matrix associated with the linear operator on $V_3(R)$ defined by $T(a, b, c) = (a+b, a-b, 2c)$ with respect to the ordered basis $B = \{(0, 1, 1), (1, 0, 1), (1, 1, 0)\}$. (10 marks)
(c) Given: $$\Delta(x)=\begin{vmatrix} f(x+\alpha) & f(x+2\alpha) & f(x+3\alpha) \\ f(\alpha) & f(2\alpha) & f(3\alpha) \\ f'(\alpha) & f'(2\alpha) & f'(3\alpha) \end{vmatrix}$$ where $f$ is a real valued differentiable function and $\alpha$ is a constant. Find $\displaystyle\lim_{x \to 0} \frac{\Delta(x)}{x}$. (10 marks)
(d) Show that between any two roots of $e^x \cos x = 1$, there exists at least one root of $e^x \sin x - 1 = 0$. (10 marks)
(e) Find the equation of the cylinder whose generators are parallel to the line $x = -\frac{y}{2} = \frac{z}{3}$ and whose guiding curve is $x^2 + 2y^2 = 1$, $z = 0$. (10 marks)
हिंदी में पढ़ें
(a) यदि $A=\begin{bmatrix} 1 & -1 & 1 \\ 2 & -1 & 0 \\ 1 & 0 & 0 \end{bmatrix}$ है, तो $A^{-1}$ को ज्ञात किए बिना दर्शाइए कि $A^2 = A^{-1}$। (10 अंक)
(b) क्रमित आधारक $B = \{(0, 1, 1), (1, 0, 1), (1, 1, 0)\}$ के सापेक्ष $V_3(R)$ पर परिभाषित रैखिक संकारक : $T(a, b, c) = (a+b, a-b, 2c)$ से संबंधित आव्यूह ज्ञात कीजिए। (10 अंक)
(c) दिया गया है : $$\Delta(x)=\begin{vmatrix} f(x+\alpha) & f(x+2\alpha) & f(x+3\alpha) \\ f(\alpha) & f(2\alpha) & f(3\alpha) \\ f'(\alpha) & f'(2\alpha) & f'(3\alpha) \end{vmatrix}$$ जहाँ $f$ एक वास्तविक-मान अवकलनीय फलन है तथा $\alpha$ एक अचर है। $\displaystyle\lim_{x \to 0} \frac{\Delta(x)}{x}$ को ज्ञात कीजिए। (10 अंक)
(d) दर्शाइए कि $e^x \cos x = 1$ के किन्हीं दो मूलों के बीच में $e^x \sin x - 1 = 0$ का कम से कम एक मूल विद्यमान है। (10 अंक)
(e) उस बेलन का समीकरण ज्ञात कीजिए जिसके जनक, रेखा : $x = -\frac{y}{2} = \frac{z}{3}$ के समानांतर हैं तथा जिसका निर्देशक-वक्र $x^2 + 2y^2 = 1$, $z = 0$ है। (10 अंक)
Answer approach & key points
Prove the required results across all five sub-parts with rigorous mathematical reasoning. For (a), compute A² and show A³=I implying A²=A⁻¹ without explicit inversion; for (b), construct the change-of-basis matrix and apply T to basis vectors; for (c), use Taylor expansion and determinant properties; for (d), apply Rolle's theorem to the appropriate auxiliary function; for (e), use the standard cylinder equation with given generator direction. Allocate approximately 15% time to (a), 20% to (b), 20% to (c), 25% to (d), and 20% to (e), reflecting the analytical depth required for the calculus and analysis components.
- For (a): Compute A² by matrix multiplication, then verify A³ = I₃ (identity), hence A² = A⁻¹ without computing inverse explicitly
- For (b): Apply T to each basis vector of B, express results as linear combinations of B, and assemble coefficients as columns of the matrix representation
- For (c): Expand f(x+kα) = f(kα) + xf'(kα) + O(x²), substitute into determinant, identify leading term as x times a 3×3 determinant involving f and f' values
- For (d): Define g(x) = e⁻ˣ - cos x, note roots of g correspond to roots of eˣcos x = 1, apply Rolle's theorem to g between consecutive roots
- For (e): Use direction ratios (1, -2, 3) from generator line, write cylinder as locus of points at fixed distance from axis with given guiding curve x² + 2y² = 1, z = 0
- For (c): The limit equals the determinant with rows [f'(α), f'(2α), f'(3α)], [f(α), f(2α), f(3α)], [f'(α), f'(2α), f'(3α)] which simplifies appropriately
- For (d): Show g'(x) = -e⁻ˣ + sin x = 0 implies eˣsin x = 1, completing the proof via intermediate value property
- For (e): Final equation eliminates parameter to give 13(x² + 2y²) - (x - 2y + 3z)² = 13 or equivalent standard form
Q2 50M prove Analytical geometry and multivariable calculus
(a) Show that the planes, which cut the cone $ax^2 + by^2 + cz^2 = 0$ in perpendicular generators, touch the cone $\frac{x^2}{b+c} + \frac{y^2}{c+a} + \frac{z^2}{a+b} = 0$. (20 marks)
(b) Given that $f(x,y) = |x^2 - y^2|$. Find $f_{xy}(0,0)$ and $f_{yx}(0,0)$. Hence show that $f_{xy}(0,0) = f_{yx}(0,0)$. (15 marks)
(c) Show that $S = \{(x, 2y, 3x) : x, y$ are real numbers$\}$ is a subspace of $R^3(R)$. Find two bases of $S$. Also find the dimension of $S$. (15 marks)
हिंदी में पढ़ें
(a) दर्शाइए कि वे समतल, जो कि शंकु $ax^2 + by^2 + cz^2 = 0$ को लंब जनकों में काटते हैं, शंकु $\frac{x^2}{b+c} + \frac{y^2}{c+a} + \frac{z^2}{a+b} = 0$ को स्पर्श करते हैं। (20 अंक)
(b) दिया गया है : $f(x,y) = |x^2 - y^2|$, तब $f_{xy}(0,0)$ तथा $f_{yx}(0,0)$ ज्ञात कीजिए। अतः दर्शाइए कि $f_{xy}(0,0) = f_{yx}(0,0)$। (15 अंक)
(c) दर्शाइए कि $S = \{(x, 2y, 3x) : x, y$ वास्तविक संख्याएँ हैं$\}$ $R^3(R)$ का एक उपसमष्टि है। $S$ के दो आधार ज्ञात कीजिए। $S$ की विमा भी ज्ञात कीजिए। (15 अंक)
Answer approach & key points
Prove the three mathematical statements systematically, allocating approximately 40% of effort to part (a) given its 20 marks, and 30% each to parts (b) and (c). Begin each part with clear statement of what is to be proved, develop the proof through logical steps with proper mathematical notation, and conclude with explicit verification of the required result. For (a), establish the condition for perpendicular generators first; for (b), carefully handle the absolute value through case analysis; for (c), verify all three subspace axioms before finding bases.
- Part (a): Derive condition for perpendicular generators of cone ax² + by² + cz² = 0 using direction cosines and orthogonality condition l₁l₂ + m₁m₂ + n₁n₂ = 0
- Part (a): Show that tangent plane condition leads to the reciprocal cone x²/(b+c) + y²/(c+a) + z²/(a+b) = 0 using the determinant condition for tangency
- Part (b): Analyze f(x,y) = |x² - y²| in four quadrants/regions to compute partial derivatives fx and fy near origin
- Part (b): Calculate mixed partial derivatives f_xy(0,0) and f_yx(0,0) using limit definition, showing both equal zero despite |x²-y²| not being C²
- Part (c): Verify S is subspace of R³ by checking: (i) non-empty/contains zero, (ii) closed under addition, (iii) closed under scalar multiplication
- Part (c): Express S as span{(1,0,3), (0,2,0)} = span{(1,0,3), (0,1,0)}, verify linear independence, conclude dim(S) = 2 with two distinct bases
Q3 50M solve Calculus, Linear Algebra and Analytical Geometry
(a)(i) If $u = x^2 + y^2$, $v = x^2 - y^2$, where $x = r\cos\theta$, $y = r\sin\theta$, then find $\frac{\partial(u,v)}{\partial(r,\theta)}$. (7 marks)
(a)(ii) If $\int\limits_{0}^{x} f(t)\,dt = x + \int\limits_{x}^{1} tf(t)\,dt$, then find the value of $f(1)$. (5 marks)
(a)(iii) Express $\int\limits_{a}^{b} (x-a)^m (b-x)^n\,dx$ in terms of Beta function. (8 marks)
(b) A sphere of constant radius $r$ passes through the origin $O$ and cuts the axes at the points $A, B$ and $C$. Find, the locus of the foot of the perpendicular drawn from $O$ to the plane $ABC$. (15 marks)
(c)(i) Prove that the eigen vectors, corresponding to two distinct eigen values of a real symmetric matrix, are orthogonal. (8 marks)
(c)(ii) For two square matrices A and B of order 2, show that trace (AB) = trace (BA). Hence show that AB - BA ≠ I₂, where I₂ is an identity matrix of order 2. (7 marks)
हिंदी में पढ़ें
(a)(i) यदि $u = x^2 + y^2$, $v = x^2 - y^2$, जहाँ पर $x = r\cos\theta$, $y = r\sin\theta$ है, तब $\frac{\partial(u,v)}{\partial(r,\theta)}$ ज्ञात कीजिए। (7 अंक)
(a)(ii) यदि $\int\limits_{0}^{x} f(t)\,dt = x + \int\limits_{x}^{1} tf(t)\,dt$ है, तो $f(1)$ का मान ज्ञात कीजिए। (5 अंक)
(a)(iii) $\int\limits_{a}^{b} (x-a)^m (b-x)^n\,dx$ को बीटा-फलन के रूप में व्यक्त कीजिए। (8 अंक)
(b) अचर त्रिज्या $r$ का एक गोला मूल-बिंदु $O$ से गुजरता है तथा अक्षों को $A, B, C$ बिंदुओं पर काटता है। $O$ से समतल $ABC$ पर खींचे गए लंब-पाद का बिंदुपथ ज्ञात कीजिए। (15 अंक)
(c)(i) सिद्ध कीजिए कि एक वास्तविक सममित आव्यूह के दो भिन्न अभिलक्षणिक मानों के संगत अभिलक्षणिक सदिश, लंबिक हैं। (8 अंक)
(c)(ii) दो वर्ग आव्यूह A तथा B जिनकी कोटि, 2 है के लिए दर्शाइए कि अनुरेख (AB) = अनुरेख (BA)। अतैव दर्शाइए कि AB - BA ≠ I₂ जहाँ I₂ एक 2-कोटि का तत्समक आव्यूह है। (7 अंक)
Answer approach & key points
Solve all six sub-parts systematically, allocating approximately 35% time to part (b) (15 marks) as the highest-weighted component, 30% to part (a) (20 marks across three items), and 35% to part (c) (15 marks across two items). Begin each sub-part with clear statement of given conditions, show complete working with proper mathematical notation, and conclude with boxed final answers. For (a)(iii) and (c)(i), explicitly state theorems being applied (Beta function definition, spectral theorem for symmetric matrices).
- For (a)(i): Correct application of chain rule for Jacobians, computing ∂(u,v)/∂(x,y) and ∂(x,y)/∂(r,θ) separately, then multiplying to get final result 8r³sinθcosθ or equivalent simplified form
- For (a)(ii): Differentiation under integral sign using Leibniz rule, establishing f(x) = 1 - xf(x), solving to get f(x) = 1/(1+x), hence f(1) = 1/2
- For (a)(iii): Substitution x = a + (b-a)t to transform limits to 0 and 1, identifying parameters p = m+1, q = n+1, final answer as (b-a)^(m+n+1) B(m+1, n+1)
- For (b): Setting up sphere equation x²+y²+z²-2ux-2vy-2wz=0 with center (u,v,w), using |OA|=|OB|=|OC|=2r condition, finding plane ABC as x/u + y/v + z/w = 2, deriving foot of perpendicular coordinates and eliminating parameters to get locus x⁻² + y⁻² + z⁻² = r⁻²
- For (c)(i): Using definition of eigenvectors AX=λX, AY=μY with λ≠μ, exploiting symmetry A=Aᵀ to show λXᵀY = μXᵀY, hence XᵀY=0 proving orthogonality
- For (c)(ii): Direct computation of trace(AB) and trace(BA) showing equality via ΣΣaᵢⱼbⱼᵢ, then using trace(AB-BA)=0 while trace(I₂)=2 to establish contradiction
Q4 50M solve Linear Algebra, Calculus and Three Dimensional Geometry
(a)(i) Reduce the following matrix to a row-reduced echelon form and hence also, find its rank:
A = [1 3 2 4 1
0 0 2 2 0
2 6 2 6 2
3 9 1 10 6] (10 marks)
(a)(ii) Find the eigen values and the corresponding eigen vectors of the matrix
A = (0 -i
i 0), over the complex-number field. (10 marks)
(b) Show that the entire area of the Astroid : x^(2/3) + y^(2/3) = a^(2/3) is (3/8)πa². (15 marks)
(c) Find equation of the plane containing the lines
$$\frac{x+1}{3} = \frac{y+3}{5} = \frac{z+5}{7},$$
$$\frac{x-2}{1} = \frac{y-4}{3} = \frac{z-6}{5}.$$
Also find the point of intersection of the given lines. (15 marks)
हिंदी में पढ़ें
(a)(i) निम्नलिखित आव्यूह का पंक्ति-समानीत सोपानक रूप में समान्यन कीजिए एवं अतैव इसकी कोटि भी ज्ञात कीजिए।
A = [1 3 2 4 1
0 0 2 2 0
2 6 2 6 2
3 9 1 10 6] (10 अंक)
(a)(ii) सम्मिश्र संख्या क्षेत्र पर आव्यूह A = (0 -i
i 0) के अभिलक्षणिक मान तथा संगत अभिलक्षणिक सदिशों को ज्ञात कीजिए। (10 अंक)
(b) दर्शाइए कि ऐस्ट्रॉइड : x^(2/3) + y^(2/3) = a^(2/3) का पूरा क्षेत्रफल (3/8)πa² है। (15 अंक)
(c) रेखाओं
(x+1)/3 = (y+3)/5 = (z+5)/7,
(x-2)/1 = (y-4)/3 = (z-6)/5
को अंतर्विष्ट करने वाले समतल का समीकरण ज्ञात कीजिए। दी गई रेखाओं के प्रतिच्छेद बिंदु को भी ज्ञात कीजिए। (15 अंक)
Answer approach & key points
Solve this multi-part numerical problem by allocating approximately 40% time to part (a) covering matrix reduction and eigenvalue computation (20 marks), 30% to part (b) deriving the astroid area using parametric integration (15 marks), and 30% to part (c) finding the plane equation and intersection point of skew lines (15 marks). Begin with clear statement of given data, proceed with systematic computational steps showing all row operations/integrations/vector calculations, and conclude with boxed final answers for each sub-part.
- For (a)(i): Correct application of elementary row operations to reduce 4×5 matrix to row-reduced echelon form; identification of pivot positions and correct rank determination (rank = 3)
- For (a)(ii): Setting up and solving characteristic equation det(A - λI) = 0; obtaining eigenvalues λ = ±1; finding normalized eigenvectors (1, i) and (1, -i) or equivalent scalar multiples
- For (b): Parametric representation x = a cos³θ, y = a sin³θ; correct area integral setup using symmetry and Jacobian; evaluation yielding (3/8)πa²
- For (c): Verification that lines are coplanar via condition [a₂-a₁, b₁, b₂] = 0; calculation of plane normal via cross product of direction vectors; point of intersection by solving parametric equations simultaneously
- Correct handling of complex numbers in (a)(ii) and proper geometric interpretation of astroid as hypocycloid with four cusps