Multiple Choice Questions
Statement −1 is false, Statement −2 is true
Statement −1 is true, Statement −2 is true, Statement −2 is a correct explanation for Statement −1
Statement −1 is true, Statement −2 is true; Statement −2 is not a correct explanation for Statement −1.
Statement −1 is true, Statement −2 is true; Statement −2 is not a correct explanation for Statement −1.
Let p be the statement “x is an irrational number”, q be the statement “y is a transcendental number”, and r be the statement “x is a rational number iff y is a transcendental number”.
Statement –1: r is equivalent to either q or p
Statement –2: r is equivalent to ∼ (p ↔ ∼ q).
Statement −1 is false, Statement −2 is true
Statement −1 is true, Statement −2 is true, Statement −2 is a correct explanation for Statement −1
Statement −1 is true, Statement −2 is true; Statement −2 is not a correct explanation for Statement −1.
Statement −1 is true, Statement −2 is true; Statement −2 is not a correct explanation for Statement −1.
The statement p → (q → p) is equivalent to
p → (p → q)
p → (p ∨ q)
p → (p ∧ q)
p → (p ∧ q)
The quadratic equations x2 – 6x + a = 0 and x2 – cx + 6 = 0 have one root in common. The other roots of the first and second equations are integers in the ratio 4 : 3. Then the common root is
1
4
3
3
D.
3
Let α and 4β be roots of x2– 6x + a = 0 and
α, 3β be the roots of x2– cx + 6 = 0, then
α + 4β = 6 and 4αβ = a
α + 3β = c and 3αβ = 6.
We get αβ = 2 ⇒ a = 8
So the first equation is x2 – 6x + 8 = 0 ⇒ x = 2, 4
If α = 2 and 4β = 4 then 3β = 3
If α = 4 and 4β = 2, then 3β = 3/2 (non-integer)
∴ common root is x = 2.
How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two S are adjacent?
8 . 6C4 . 7C4
6 . 7 . 8C4
6 . 8 . 7C4
6 . 8 . 7C4
Let f : N → Y be a function defined as f (x) = 4x + 3, where Y = {y ∈ N : y = 4x + 3 for some x ∈ N}.Show that f is invertible and its inverse is
Let R be the real line. Consider the following subsets of the plane R × R.
S = {(x, y) : y = x + 1 and 0 < x < 2}, T = {(x, y) : x − y is an integer}. Which one of the following is true?
neither S nor T is an equivalence relation on R
both S and T are equivalence relations on R
S is an equivalence relation on R but T is not
S is an equivalence relation on R but T is not
Let A be a 2 × 2 matrix with real entries. Let I be the 2 × 2 identity matrix. Denote by tr (A), the sum of diagonal entries of A. Assume that A2= I.
Statement −1: If A ≠ I and A ≠ − I, then det A = − 1.
Statement −2: If A ≠ I and A ≠ − I, then tr (A) ≠ 0.
Statement −1 is false, Statement −2 is true
Statement −1 is true, Statement −2 is true, Statement −2 is a correct explanation for Statement −1
Statement −1 is true, Statement −2 is true; Statement −2 is not a correct explanation for Statement −1.
Statement −1 is true, Statement −2 is true; Statement −2 is not a correct explanation for Statement −1.
Let f(x) = 
Then which one of the following is true?
f is neither differentiable at x = 0 nor at x = 1
f is differentiable at x = 0 and at x = 1
f is differentiable at x = 0 but not at x = 1
f is differentiable at x = 0 but not at x = 1
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