Date: October 2012

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

Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic

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

`1/2`

`11/2`

`5/2`

`(-11)/2`

Chapter: [0.04] Pair of Straight Lines

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`

Chapter: [0.04] Pair of Straight Lines

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

Chapter: [0.02] Matrices

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

Chapter: [0.03] Trigonometric Functions

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}

Chapter: [0.05] Circle

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

Chapter: [0.05] 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.

Chapter: [0.016] Line and Plane [0.09] Line

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

Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic

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.

Chapter: [0.04] Pair of Straight Lines

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

Chapter: [0.05] Circle

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.

Chapter: [0.012] Matrics [0.02] Matrices

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

Chapter: [0.03] Trigonometric Functions

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.

Chapter: [0.06] Conics

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

Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic

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

Chapter: [0.06] Conics

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`

Chapter: [0.04] Pair of Straight Lines

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

Chapter: [0.1] 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

Chapter: [0.017] Linear Programming [0.11] 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).

Chapter: [0.07] 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)`

Chapter: [0.12] Continuity

`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

Chapter: [0.15] Integration

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

Chapter: [0.2] Bernoulli Trials and Binomial Distribution

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

Chapter: [0.06] Conics

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

Chapter: [0.023] Indefinite Integration [0.15] Integration

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

Chapter: [0.17] Differential Equation

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

Chapter: [0.18] Statistics

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

Chapter: [0.027000000000000003] Probability Distributions [0.19] Probability Distribution

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

Chapter: [0.14] Applications of Derivative

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`

Chapter: [0.13] Differentiation

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

Chapter: [0.19] Probability Distribution

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

Chapter: [0.13] Differentiation

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

Chapter: [0.17] Differential Equation

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

Chapter: [0.17] Differential Equation

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

Chapter: [0.13] Differentiation

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`

Chapter: [0.12] Continuity

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.

Chapter: [0.19] Probability Distribution

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`

Chapter: [0.023] Indefinite Integration [0.15] Integration

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.

Chapter: [0.05] 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 |

Chapter: [0.18] Statistics

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