2025 MATHS SET1

If A = [1 0 0; 0 1 0; 0 0 1], then A-1 is:

If vector a = 3i + 2j - k and vector b = i - j + k, then which of the following is correct?

Integral from -1 to 1 of |x|/x dx, x != 0 is equal to

Which of the following is not a homogenous function of x and y?

If f(x) = |x| + |x - 1|, then which of the following is correct?

If A is a square matrix of order 2 such that det(A) = 4, then det(4 adj A) is equal to:

If E and F are two independent events such that P(E) = 2/3, P(F) = 2/7, then P(E/F) is equal to:

The absolute maximum value of function f(x) = x^3 - 3x + 2 in [0, 2] is:

Let A = [1 2 1; 0 4 1; 3 2 1], B = [2; 5; 7], C = [9 8 7], which of the following is defined?

If integral 1/x * 2^(x^2) dx = k * 2^(x^2) + C, then k is equal to

If a + b + c = 0, |a| = 3, |b| = 7 and |c| = 4, then angle between b and c is

The integrating factor of differential equation dy/dx + (x + 2y) = y is

If A = [7 0 x; 0 7 0; 0 0 y] is a scalar matrix, then yx is equal to

The corner points of the feasible region in graphical representation of a L.P.P. are (2, 72), (15, 20) and (40, 15). If Z = 18x + 9y be the objective function, then

If A and B are invertible matrices, then which of the following is not correct?

If the feasible region of a linear programming problem with objective function Z = ax + by is bounded, then which of the following is correct?

The area of the shaded region bounded by the curves y^2 = x, x = 4 and the x-axis is given by

Assertion: Let Z be the set of integers. A function f: Z -> Z defined as f(x) = 3x - 5, for all x in Z is bijective. Reason: A function is bijective if it is both surjective and injective.

Assertion: f(x) = {3x-8, x <= 5; 2k, x > 5} is continuous at x=5 for k=5/2. Reason: For a function f to be continuous at x=a, lim f(x) as x->a- = lim f(x) as x->a+ = f(a).

(a) Differentiate cos^2(x^2) w.r.t cos^2(x). OR (b) If tan^-1(x^2 + y^2) = a^2, then find dy/dx.

Evaluate: tan(1/2 * sin^-1(2x/(1+x^2)) + cos^-1((1-y^2)/(1+y^2)))

Evaluate: tan^-1(2sin(2cos^-1(1/2)))

The diagonals of a parallelogram are given by a = 2i - j + k and b = i + 3j - k. Find the area of the parallelogram.

Find the intervals in which function f(x) = 5x^3 - 3x^5 is (i) increasing (ii) decreasing.

(a) Two friends while flying kites from different locations, find the strings of their kites crossing each other. The strings can be represented by vectors a = 3i + j + 2k and b = 2i + 2j + 4k. Determine the angle formed between the kite strings. Assume there is no slack in the strings. OR (b) Find a vector of magnitude 21 units in the direction opposite to that of AB where A and B are the points A(2, 1, 3) and B(5, -1, 0) respectively.

The side of an equilateral triangle is increasing at the rate of 3 cm/s. At what rate its area increasing when the side of the triangle is 15 cm?

Solve the following linear programming problem graphically: Maximise Z = x + 2y Subject to the constraints: x - y >= 0, x - 2y >= -2, x >= 0, y >= 0

(a) Find integral of (x sin x) / (1 + cos x) dx. OR (b) Evaluate: integral from 0 to pi/4 of dx / (cos^2(x) + 2sin^2(x))

(a) Verify the lines given by r = (1-l)i + (l-2)j + (3-2l)k and r = (m+1)i + (2m-1)j + (2m+1)k are skew lines. Hence, find shortest distance between the lines. OR (b) During a cricket match, the position of the bowler, the wicket keeper and the leg slip fielder are in a line given by B = 2i + 8j, W = 6i + 12j and F = 12i + 18j respectively. Calculate the ratio in which the wicketkeeper divides the line segment joining the bowler and the leg slip fielder.

(a) Probability distribution: X=0,2,4,5; P(X)=p, 2p, 3p, p. (i) Calculate p. (ii) Calculate the mean. OR (b) 3000 candidates, 2/3 female, 1/3 male. Prob(male dist) = 0.4, Prob(female dist) = 0.35. Find prob a chosen candidate has distinction.

Sketch the graph of y = |x + 3| and find the area of the region enclosed by the curve, x-axis, between x = -6 and x = 6, using integration.

(a) If sqrt(1-x^2) + sqrt(1-y^2) = a(x-y), then prove that dy/dx = sqrt(1-y^2) / sqrt(1-x^2). OR (b) If x = a cos(theta + log(tan(theta/2))) and y = a sin(theta), find d^2y/dx^2 at theta = pi/4.

Find the absolute maximum and absolute minimum of function f(x) = 2x^3 - 15x^2 + 36x + 1 on [1, 5].

(a) Find the image A' of the point A(1, 6, 3) in the line (x-1)/1 = (y-2)/2 = (z-3)/3. Also, find the equation of the line joining A and A'. OR (b) Find a point P on the line (x+5)/1 = (y+3)/4 = (z-6)/-9 such that its distance from point Q(2, 4, -1) is 7 units. Also, find the equation of line joining P and Q.

A school wants to allocate students into three clubs: Sports, Music and Drama. Constraints: Sports = Music + Drama. Music = 0.5 * Sports + 20. Total = 180. Find the number of students allocated to different clubs using matrix method.

A technical company is designing a rectangular solar panel installation on a roof using 300 metres of boundary material. The design includes a partition running parallel to one of the sides dividing the area (roof) into two sections. Let the length of the side perpendicular to the partition be x metres and with parallel to the partition be y metres. (i) Write the equation for the total boundary material used in the boundary and parallel to the partition in terms of x and y. (ii) Write the area of the solar panel as a function of x. (iii) (a) Find the critical points of the area function. Use second derivative test to determine critical points at the maximum area. Also, find the maximum area. OR (iii) (b) Using first derivative test, calculate the maximum area the company can enclose with the 300 metres of boundary material, considering the parallel partition.

A class-room teacher is keen to assess the learning of her students the concept of 'relations' taught to them. She writes the following five relations each defined on the set A = {1, 2, 3} : R1 = {(2, 3), (3, 2)}; R2 = {(1, 2), (1, 3), (3, 2)}; R3 = {(1, 2), (2, 1), (1, 1)}; R4 = {(1, 1), (1, 2), (3, 3), (2, 2)}; R5 = {(1, 1), (1, 2), (3, 3), (2, 2), (2, 1), (2, 3), (3, 2)}. The students are asked to answer the following questions about the above relations: (i) Identify the relation which is reflexive, transitive but not symmetric. (ii) Identify the relation which is reflexive and symmetric but not transitive. (iii) (a) Identify the relations which are symmetric but neither reflexive nor transitive. OR (iii) (b) What pairs should be added to the relation R2 to make it an equivalence relation?

A bank offers loan to its customers on different types of interest namely, fixed rate, floating rate and variable rate. From the past data with the bank, it is known that a customer avails loan on fixed rate, floating rate or variable rate with probabilities 10%, 20% and 70% respectively. A customer after availing loan can pay the loan or default on loan repayment. The bank data suggests that the probability that a person defaults on loan after availing it at fixed rate, floating rate and variable rate is 5%, 3% and 1% respectively. Based on the above information, answer the following: (i) What is the probability that a customer after availing the loan will default on the loan repayment? (ii) A customer after availing the loan, defaults on loan repayment. What is the probability that he availed the loan at a variable rate of interest?