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Which of the following statements is a tautology?

Which of the following statements is a tautology? a. ~( ∧ ~ ) → ( ∨ ) b. (~ ∨ ~ ) → ( ∧ ) c. ∨ (~ ) → ( ∧ ) d. ~( ∨ ~ ) → ( ∨ )... View Article

The length of the perpendicular from the origin, on the normal to the curve, ² + 2 − 3 ² = 0 at the point (2, 2) is:

a. 2 b. 2√2 c. 4√2 d. √2 Solution: Given curve: x 2 + 2xy − 3y 2 = 0 ⇒ x 2 + 3xy − xy − 3y 2 = 0 ⇒ (x + 3y)(x − y) = 0 Equating we get, x... View Article

The mirror image of the point (1, 2, 3) in a plane is (− 7/3, − 4/3 , − 1/ 3 ). Which of the following points lies on this plane

a. (1, −1, 1) b. (−1, −1, 1) c. (1, 1, 1) d. (−1, −1, −1) Solution: Image of point P(1, 2, 3) w.r.t. a plane x + y + + = 0 is (− 7/3... View Article

Find the roots of the quadratic equation

Let = (−1+ √3)/2 . If , then and are the roots of the quadratic equation: a. x 2 + 101x + 100 = 0 b. x 2 + 102x + 101 = 0 c. x 2 −... View Article

If a line, = + is a tangent to the circle

If a line, y = x + is a tangent to the circle, (x − 3) 2 + y 2 = 1 and it is perpendicular to a line 1, where 1 is the tangent to the... View Article

Given limit is equal to:

a. 0 b. 1/10 c. – 1/10 d. − 1/5 Solution: Applying L - Hospital's Rule: = Answer: (a)... View Article

If a hyperbola passes through the point (10, 16) and it has vertices at (±6, 0), then the equation of the normal at is:

a. 3x + 4y = 94 b. x + 2y = 42 c. 2x + 5y = 100 d. x + 3y = 58 Solution: Vertex of hyperbola is (± , 0) ≡ (±6, 0) ⇒ = 6 Equation of... View Article

If and be the coefficients of ⁴ and ²respectively in the expansion of [latex](x+sqrt{x^{2}-1})^{6}+(x-sqrt{x^{2}-1})^{6}[/latex] , then

a. α + β = −30 b. α − β = −132 c. α + β = 60 d. α − β = 60 Solution: = 2[6𝐶0x 6 + 6𝐶2x 4 (x2 − 1) + 6𝐶4x 2(x 2 − 1) 2 + 6𝐶6 (x2 − 1)3] =... View Article

Let : (1,3) → be a function defined by ( ) = [ ] /( ²+1) , where [ ] denotes the greatest integer ≤ . Then the range of is

a. ( 2/5 , 3/5 ] U ( 3/4 , 4/5 ) b. ( 2/5 , 4/5 ] c. ( 3/5 , 4/5 ) d. ( 2/5 , 1/2 ) U ( 3/5 , 4/5 ] Solution: f(x) = x[x]/(x2+1) Answer: (d)... View Article

Let a and b be two vectors. If c is a vector such that….

Let ⃗ = ̂− 2 ̂+ ̂, ⃗ = ̂− ̂+ ̂ be two vectors. If ⃗ is a vector such that ⃗× ⃗= ⃗× ⃗and ⃗. ⃗= 0 then ⃗. ⃗ is equal to: a. 1/2 b.... View Article

The mean and variance of 20 observations are found to be 10 and 4, respectively. On rechecking, it was found that an observation 9 was incorrect and the correct observation was 11. Then the correct variance is:

a. 4.01 b. 3.99 c. 3.98 d. 4.02 Solution: Mean = 10 ⇒ ∑ x /20 = 10 ⇒ ∑ x = 200 Variance = 4 ⇒ ∑ x 2/20 − 100 = 4 ⇒ ∑ x 2 = 2080 New... View Article

Let be the set of all real roots of the equation, 3^x (3^x − 1) + 2 = |3^x − 1| + |3^x − 2|. Then S:

a. is a singleton. b. is an empty set. c. contains at least four elements d. contains exactly two elements. Solution: 3x (3x − 1) + 2 = |3 x... View Article

Let and be two events such that the probability that exactly one of them occurs is 2/5 and the probability that or occurs is 1/2, then the probability of both of them occur together is

a. 0.10 b. 0.20 c. 0.01 d. 0.02 Solution: P(A or B) = P(A U B) = 1/2 P(exactly one of A or B)= 2/5 ⇒ P(A) − P(A ∩ B) + P(B) − P(A ∩ B) =... View Article

M	T	W	T	F	S	S
			1	2	3	4
5	6	7	8	9	10	11
12	13	14	15	16	17	18
19	20	21	22	23	24	25
26	27	28	29	30	31

Archives

Which of the following statements is a tautology?

The length of the perpendicular from the origin, on the normal to the curve, ² + 2 − 3 ² = 0 at the point (2, 2) is:

The mirror image of the point (1, 2, 3) in a plane is (− 7/3, − 4/3 , − 1/ 3 ). Which of the following points lies on this plane

Find the roots of the quadratic equation

If a line, = + is a tangent to the circle

Given limit is equal to:

If a hyperbola passes through the point (10, 16) and it has vertices at (±6, 0), then the equation of the normal at is:

If and be the coefficients of ⁴ and ²respectively in the expansion of [latex](x+sqrt{x^{2}-1})^{6}+(x-sqrt{x^{2}-1})^{6}[/latex] , then

Let : (1,3) → be a function defined by ( ) = [ ] /( ²+1) , where [ ] denotes the greatest integer ≤ . Then the range of is

Let a and b be two vectors. If c is a vector such that….

The mean and variance of 20 observations are found to be 10 and 4, respectively. On rechecking, it was found that an observation 9 was incorrect and the correct observation was 11. Then the correct variance is:

Let be the set of all real roots of the equation, 3^x (3^x − 1) + 2 = |3^x − 1| + |3^x − 2|. Then S:

Let and be two events such that the probability that exactly one of them occurs is 2/5 and the probability that or occurs is 1/2, then the probability of both of them occur together is

If the distance between the plane, 23x-10y-2z+48 = 0 and the plane containing the lines

The number of terms common to the A.P.’s 3,7,11,…407 and 2,9,16,…709 is

If the curves x² -6x +y² +8 = 0 and x²-8y+y²+16-k = 0, (k>0) touch each other at a point, then the largest value of k is

If C_r = ²⁵C_r and C₀+5⋅C₁+9⋅C₂ +⋯+101. C₂₅ =2²⁵ .k then k is equal to

If vector a is perpendicular to vector b times vector c, then vector a times vector b times vector c is equal to

What is the ordered pair (λ, f(θ))

A random variable X has the following probability distribution:

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