1
GATE ECE 2022
MCQ (Single Correct Answer)
+2
-0.67

The function f(x) = 8loge x $$-$$ x2 + 3 attains its minimum over the interval [1, e] at x = __________.

(Here loge x is the natural logarithm of x.)

A
2
B
1
C
e
D
$${{1 + e} \over 2}$$
2
GATE ECE 2022
MCQ (Single Correct Answer)
+2
-0.67

Let $$\alpha$$, $$\beta$$ two non-zero real numbers and v1, v2 be two non-zero real vectors of size 3 $$\times$$ 1. Suppose that v1 and v2 satisfy $$v_1^T{v_2} = 0$$, $$v_1^T{v_1} = 1$$ and $$v_2^T{v_2} = 1$$. Let A be the 3 $$\times$$ 3 matrix given by :

A = $$\alpha$$v1$$v_1^T$$ + $$\beta$$v2$$v_2^T$$

The eigen values of A are __________.

A
0, $$\alpha$$, $$\beta$$
B
0, $$\alpha$$ + $$\beta$$, $$\alpha$$ $$-$$ $$\beta$$
C
0, $${{\alpha + \beta } \over 2},\sqrt {\alpha \beta } $$
D
0, 0, $$\sqrt {{\alpha ^2} + {\beta ^2}} $$
3
GATE ECE 2022
MCQ (More than One Correct Answer)
+2
-0.67

Consider the following series :

$$\sum\limits_{n = 1}^\infty {{{{n^d}} \over {{c^n}}}} $$

For which of the following combinations of c, d values does this series converge?

A
c = 1, d = $$-$$1
B
c = 2, d = 1
C
c = 0.5, d = $$-$$10
D
c = 1, d = $$-$$2
4
GATE ECE 2022
Numerical
+2
-0.67

The value of the integral

$$\int\!\!\!\int\limits_D {3({x^2} + {y^2})dx\,dy} $$,

where D is the shaded triangular region shown in the diagram, is ___________ (rounded off to the nearest integer).

GATE ECE 2022 Engineering Mathematics - Calculus Question 6 English

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