If A = {2, 3, 4, 5, 6}, then which of the following is not true?

(A) ∃ x ∈ A such that x + 3 = 8

(B) ∃ x ∈ A such that x + 2 < 5

(C) ∃ x ∈ A such that x + 2 < 9

(D) ∀ x ∈ A such that x + 6 ≥ 9

Concept: Mathematical Logic - Algebra of Statements

If 2x + y = 0 is one of the lines represented by 3x^{2} + kxy + 2y^{2} = 0, then the value of k is

A) `1/2`

B) `11/2`

C) `5/2`

D) `(-11)/2`

Concept: Pair of Straight Lines - Pair of Lines Passing Through Origin - Combined Equation

If a line is inclined at 60° and 30° with the X and Y-axes respectively, then the angle which it makes with Z-axis is

(A) 0

(B) `pi/4`

(C) `pi/2`

(D) `pi/6`

Concept: Acute Angle Between the Lines

If A = `[(1,2),(3,4)]` and AX = I then find X by using elementary transformations

Concept: Matrices - Elementary Transformation of a Matrix Revision of Cofactor and Minor

With usual notations, in ΔABC, prove that a(b cos C − c cos B) = b^{2} − c^{2}

Concept: Trigonometric Functions - Solution of a Triangle

Show that the equation of a tangent to the circle x^{2} + y^{2} = a^{2} at the point P(x_{1},y_{1}) on it is xx_{1} + yy_{1} = a^{2}

Concept: Circle - Tangents to a Circle from a Point Outside the Circle

Find k, if the line 2x − 3y + k = 0 touches the ellipse 5x^{2} + 9y^{2} = 45.

Concept: Circle - Tangent of a Circle - Equation of a Tangent at a Point to General Circle

Find the co-ordinates of the point, which divides the line segment joining the points A(2, − 6, 8) and B(− 1, 3, − 4) externally in the ratio 1 : 3.

Concept: Line - Distance of a Point from a Line

Using truth table, prove that ~p ∧ q ≡ (p ∨ q) ∧ ~ p

Concept: Mathematical Logic - Truth Tables of Compound Statements

Find the values of p and q, if the following equation represents a pair of perpendicular lines:

px^{2} − 8xy + 3y^{2} + 14x + 2y + q = 0.

Concept: Pair of Straight Lines - Condition for Perpendicular Lines

Find the equations of tangents to the parabola y^{2} = 12x from the point (2, 5).

Concept: Circle - Condition of tangency

The cost of 2 books, 6 notebooks and 3 pens is Rs 40. The cost of 3 books, 4 notebooks and 2 pens is Rs 35, while the cost of 5 books, 7 notebooks and 4 pens is Rs 61. Using this information and matrix method, find the cost of 1 book, 1 notebook and 1 pen separately.

Concept: Elementary Operation (Transformation) of a Matrix

Prove that `sin^(−1) (-1/2) + cos^(-1) (-sqrt3/2) = cos^(-1) (-1/2)`

Concept: Trigonometric Functions - Trigonometric equations

Show that the product of lengths of perpendicular segments drawn from the foci to any tangent to the hyperbola `x^2/25 + y^2/16 = 1` is equal to 16.

Concept: Conics - Tangents and normals - equations of tangent and normal at a point

Construct the new switching circuit for the following circuit with only one switch by simplifying the given circuit:

Concept: Mathematical Logic - Application - Introduction to Switching Circuits

Find the locus of a point, the tangents from which to the circle x^{2} + y^{2} = a^{2} are mutually perpendicular

Concept: Conics - Locus of Points from Which Two Tangents Are Mutually Perpendicular

Find the shortest distance between the lines

`(x+1)/7 = (y + 1)/(-6) = (z + 1)/1 and (x - 3)/1 = (y - 5)/(-2) = (z - 7)/1`

Concept: Pair of Straight Lines - Pair of Lines Passing Through Origin - Combined Equation

Find the angle between the line `(x - 1)/3 = (y + 1)/2 = (z + 2)/4` and the plane 2x + y − 3z + 4 = 0.

Concept: Angle Between Line and a Plane

Solve the following L. P. P. graphically:Linear Programming

Minimize Z = 6x + 2y

Subject to

5x + 9y ≤ 90

x + y ≥ 4

y ≤ 8

x ≥ 0, y ≥ 0

Concept: Graphical Method of Solving Linear Programming Problems

Find the volume of a tetrahedron whose vertices are A(−1, 2, 3), B(3, −2, 1), C(2, 1, 3) and D(−1, −2, 4).

Concept: Vectors - Collinearity and Coplanarity of Vectors

If x^{y} = e^{x−y} , then `dy/dx` = ______

A) `(1+x)/(1 + log x)`

B) `log x/(1 + log x)^2`

C) `(1 - log x)/(1 + log x)`

D) `(1-x)/(1 + log x)`

Concept: Continuity of Some Standard Functions - Trigonometric Function

`int 1/(1 + cos x)` dx = _____

A) `tan(x/2) + c`

B) `2 tan (x/2) + c`

C) -`cot (x/2) + c`

D) -2 `cot (x/2)` + c

Concept: Evaluation of Definite Integrals by Substitution

If X ~ B (n, p) and E(X) = 12, Var(X) = 4, then the value of n is _______

(A) 3

(B) 48

(C) 18

(D) 36

Concept: Bernoulli Trials and Binomial Distribution - Calculation of Probabilities

Find the equation of tangent to the curve y = 3x^{2} − x + 1 at P(1, 3).

Concept: Conics - Tangents and normals - equations of tangent and normal at a point

Evaluate: `int 1/(x(x-1)) dx`

Concept: Methods of Integration - Integration by Substitution

Solve the differential equation y − x = `dy/dx = 0`

Concept: Differential Equations - Applications of Differential Equation

In a bivariate data, n = 10, `bar x` = 25, `bary` = 30 and `sum xy` = 7900. Find cov(X,Y)

Concept: Statistics - Bivariate Frequency Distribution

A random variable X ~ N (0, 1). Find P(X > 0) and P(X < 0).

Concept: Random Variables and Its Probability Distributions

Examine the function for maximum and minimum f(x) = x^{3} − 9x^{2} + 24x.

Concept: Maxima and Minima - Introduction of Extrema and Extreme Values

If y = f(x) is a differentiable function of x such that inverse function x = f^{–1} (y) exists, then prove that x is a differentiable function of y and `dx/dy=1/((dy/dx)) " where " dy/dx≠0`

Concept: Derivative - Derivative of Inverse Function

The probability distribution of X, the number of defects per 10 metres of a fabric is given by

x | 0 | 1 | 2 | 3 | 4 |

P(X = x) | 0.45 | 0.35 | 0.15 | 0.03 | 0.02 |

Find the variance of X

Concept: Probability Distribution - Expected Value, Variance and Standard Deviation of a Discrete Random Variable

If `sqrt(1-x^2) + sqrt(1- y^2)` = a(x − y), show that dy/dx = `sqrt((1-y^2)/(1-x^2))`

Concept: Derivatives of Inverse Trigonometric Functions

Solve the differential equation `cos^2 x dy/dx` + y = tan x

Concept: General and Particular Solutions of a Differential Equation

Find the area of the region bounded by the curves y^{2} = 4x and 4x^{2} + 4y^{2} = 9 with x >= 0.

Concept: Pair of Straight Lines - Point of Intersection of Two Lines

Find the approximate value of tan^{−1} (1.001).

Concept: Trigonometric Functions - Trigonometric equations

Examine continuity of the function f(x) at x = 0, where

`f(x) = (10^x + 7^x - 14^x - 5^x)/(1-cos 4x) , " for " x != 0`

`= 10/7 , " for" x = 0`

Concept: Continuity - Continuity of a Function at a Point

The probability that a person who undergoes kidney operation will recover is 0.5. Find the probability that of the six patients who undergo similar operations,

(a) None will recover

(b) Half of them will recover.

Concept: Probability Distribution of a Discrete Random Variable

Prove that:

`int sqrt(a^2 +x^2)dx = x/2 sqrt(a^2 + x^2) + a^2/2 log |x + sqrt(x^2 + a^2)| + c`

Concept: Methods of Integration - Integration Using Partial Fractions

Find the volume of the solid generated, when the area between ellipse 4x^{2} + 9y^{2} = 36 and the chord AB, with A (3, 0), B (0, 2), is revolved about X-axis.

Concept: Circle - Tangent of a Circle - Equation of a Tangent at a Point to Standard Circle

Find Karl Pearson’s coefficient of correlation between the variables X and Y for the following data

X | 11 | 7 | 9 | 5 | 8 | 6 | 10 |

Y | 10 | 8 | 6 | 5 | 9 | 7 | 11 |

Concept: Statistics - Karl Pearsonâ€™s Coefficient of Correlation