Which of the following statements is a tautology?
Which of the following statements is a tautology? a. ~( ∧ ~ ) → ( ∨ ) b. (~ ∨... View Article
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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... 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... View Article
Find the roots of the quadratic equation
Let = (−1+ √3)/2 . If \(\begin{array}{l}a= (1+\alpha )\sum_{k=0}^{100}a^{2k}\end{array} \) and \(\begin{array}{l}b= \sum_{k=0}^{100}a^{3k}\end{array} \) , then and are the roots... 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... View Article
Given limit is equal to:
a. 0 b. 1/10 c. – 1/10 d. − 1/5 Solution: \(\begin{array}{l}\lim_{x\rightarrow 0}\frac{\int_{0}^{x}tsin (10t)dt}{x}\end{array} \) Applying L – Hospital’s Rule:... 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... View Article
If and be the coefficients of 4 and 2 respectively 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... 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. (... 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... 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 ⇒ ∑... View Article
Let be the set of all real roots of the equation, 3x (3x − 1) + 2 = |3x − 1| + |3x − 2|. Then S:
a. is a singleton. b. is an empty set. c. contains at least four elements d. contains exactly two elements.... 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... View Article
If the distance between the plane, 23x-10y-2z+48 = 0 and the plane containing the lines
If the distance between the plane, 23x-10y-2z+48 = 0 and the plane containing the lines \(\begin{array}{l}\frac{x+1}{2}=\frac{y-3}{4}=\frac{z+1}{3}\end{array} \) and \(\begin{array}{l}\frac{x+3}{2}=\frac{y+2}{6}=\frac{z-1}{\lambda }\end{array}... View Article
The number of terms common to the A.P.’s 3,7,11,…407 and 2,9,16,…709 is
Solution: First common term is 23 Common difference = LCM(7,4) = 28 23+(n−1)28 ≤ 407 n−1 ≤ 13.71 n =... View Article
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
Solution: Two circles touch each other if C1C2 = ǀr1± r2ǀ √k +1 = 5 or ǀ√k-1ǀ = 5 k... View Article
If Cr = 25Cr and C0+5⋅C1+9⋅C2 +⋯+101. C25 =225 .k then k is equal to
Solution: S = 25C0 + 525C1 + 925C2 + ….+9725C24 +10125C25 = 225k ..(i) Reverse and apply property nCr =... View Article
If vector a is perpendicular to vector b times vector c, then vector a times vector b times vector c is equal to
Let \(\begin{array}{l}\vec{a}\end{array} \), \(\begin{array}{l}\vec{b}\end{array} \), and \(\begin{array}{l}\vec{c}\end{array} \) be three vectors such that \(\begin{array}{l}\vec{a}=\sqrt{3}\end{array} \) , \(\begin{array}{l}\vec{b}=5\end{array} \) , \(\begin{array}{l}\vec{b}.\vec{c}=10\end{array}... View Article
What is the ordered pair (λ, f(θ))
If \(\begin{array}{l}\int \frac{d\theta }{\cos ^{2}\theta (\tan 2\theta +\sec 2\theta )}= \lambda \tan \theta +2\: log_{e}\left | f(\theta ) \right |+C\end{array}... View Article
A random variable X has the following probability distribution:
A random variable X has the following probability distribution: X 1 2 3 4 5 P(X) K2 2K K 2K... View Article
