Total Number of Question/s - 3005

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  • 1. Sets, Relations and Functions - Quiz


    SAMPLE QUESTIONS


    1. 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
    a) g(y) = $\frac{y-3}{4}$
    b) g(y) = $\frac{3y+4}{3}$
    c) g(y) = $4 + \frac{y+3}{4}$
    d) g(y) = $\frac{y+3}{4}$

    2. Let f be a function defined by f(x) = (x-1)2 + 1, (x ≥ 1)
    Statement I
    The set { x : f(s) = f-1(x)} = {1,2}
    Statement II
    f = f is bijection and f-1(s) = 1 + $\sqrt{x - 1}$, x ≥ 1
    a) Statement I is false, Statement II is true.
    b) Statement I is true , Statement II is true; Statement II is a correct explanation for Statement I.
    c) Statement I is true, Statement II is true, Statement II is not a correct explanation for explanation I.
    d) Statement I is true, Statement I is true, Statement II is false.

  • 2. Complex Numbers and Quadratic Equations - Quiz


    SAMPLE QUESTIONS


    1. Let α, β be real and z be a complex number. If z2 + az + β = 0 has two distinct roots on the line Re z = 1, then it is necessary that
    a) β ∈ (-1,0)
    b) |β| = 1
    c) β ∈ (1,∞)
    d) β ∈ (0,1)

    2. If ω is an imaginary cube root of unity, then (1 + ω - ω2)7 equals
    a) 128ω
    b) -128ω
    c) 128ω2
    d) -128ω2

  • 3. Matrices and Determinants - Quiz


    SAMPLE QUESTIONS


    1. If ω ≠ 1 is the complex cube root of unity and matrix H - $\begin{bmatrix} \omega & 0\\ 0& \omega \end{bmatrix}$ , then H70 is equal to
    a) H
    b) 0
    c) -H
    d) H2

    2. Let A be a 2 x 2 matrix with real entries. Let I be the 2 x identity matrix. Denote by tr(A), the sum of diagonal entries of A. Assume that A2 = I.
    Statement I
    If A ≠ I and A ≠ , then det(A) = -1.
    Statement II
    If A ≠ I and A ≠ -I, then tr(A) ≠ 0.
    a) Statement I is false, Statement II is true
    b) Statement I is true, Statement II is false;Statement II is a correct explanation for Statement I
    c) Statement I is true, Statement II is true; Statement II is not a correct explanation for Statement I
    d) Statement I is true, Statement II is false

  • 4. Permutations and Combinations - Quiz


    SAMPLE QUESTIONS


    1. If the letters of the word SACHIN are arranged in all possible ways and these words are written out as in dictionary , then the word SACHIN appears at serial number
    a) 602
    b) 603
    c) 600
    d) 601

    2. In a shop there are five types of ice-creams available. A child buys six ice-cream available. A child buys six ice-creams
    Statement I The number of different ways the child can buy the six ice-creams is 10C5.
    Statement II The number of different ways the child can buy the six ice-creams is equal to the number of different ways of arranging 6A's and 4B's in a row.
    a) Statement I is false, Statement II is true
    b) Statement I is true, Statement II is true; Statement II is a correct explanation for Statement I
    c) Statement I is true, Statement II is true; Statement II is not a correct explanation for Statement I
    d) Statement I is true, Statement II is false

  • 5. Mathematical Induction - Quiz


    SAMPLE QUESTIONS


    1. Statement I For each natural number n,(n+1)7 - n7 - 1 is divisible by 7.
    Statement II For each natural number n, n7 - n is divisible by 7.
    a) Statement I is false , Statement II is true.
    b) Statement I is true, Statement II is true; Statement II is correct explanation for Statement I.
    c) Statement I is true, Statement II is true; Statement II is not a correct explanation for Statement I.
    d) Statement I is true, Statement II is false.

    2. If A = $\begin{bmatrix} 1 &0 \\ 1& 1 \end{bmatrix}$ and I = $\begin{bmatrix} 1 &0 \\ 0& 1 \end{bmatrix}$, then which one of the following holds for all n ≥ 1, by the principle of mathematical induction?
    a) $A^{n}$ = $2^{n-1}A + \left (n-1 \right )I$
    b) $A^{n}$ = $nA + \left (n-1 \right )I$
    c) An = $2^{n-1}A-\left (n-1 \right )I$
    d) An = nA - (n-1)I

  • 6. Binomial Theorem - Quiz


    SAMPLE QUESTIONS


    1. If the coefficient of x7 in $\left [ax^{2} + \frac{1}{bx} \right ]^{11}$ equals the coefficient of x-7 in $\left [ax^{2} - \frac{1}{bx} \right ]^{11}$, then a and b satisfy the relation
    a) ab = 1
    b) $\frac{a}{b}$ = 1
    c) a + b = 1
    d) a - b = 1

    2. Let S1 = $\sum_{j=1}^{10}j\left (j-1 \right )^{10}C_{j}, S_{2}$ = $\sum_{j=1}^{10} j^{10}C_{j} \ and \ S_{3}$ = $\sum_{j=1}^{10}$ j2 10Cj
    Statement I S3 = 55 × 29.
    Statement II S1 = 90 x 28 and S2 = 10 x 28.
    a) Statement I is false, Statement II is true.
    b) Statement I is true, Statement II is true; Statement II is a correct explanations of Statement I.
    c) Statement I is true, Statement II is true, Statement II is not a correct explanation for Statement I.
    d) Statement I is true, Statement II is false.

  • 7. Sequences and Series - Quiz


    SAMPLE QUESTIONS


    1. A person is to count 4500 currently notes. Let a11, denotes the number of notes he counts in the nth minute . If a1 = a2 = .... = a10 = 150 and a10, a11,.... are in AP with common difference -2, then the common difference -2, then the time taken by him to count to count all notes, is
    a) 24 min
    b) 34 min
    c) 125 min
    d) 135 min

    2. In p and q are positive real numbers such that p2 + q2 =1, then the maximum value of (p + q) is
    a) 2
    b) $\frac{1}{2}$
    c) $\frac{1}{\sqrt{2}}$
    d) $\sqrt{2}$

  • 8. Limits, Continuity and Differentiabilty - Quiz


    SAMPLE QUESTIONS


    1. Let f : R → [0,∞) be such that $lim_{x \rightarrow 5}f\left (x \right )$ exists and $lim_{x \rightarrow 5 }\frac{\left [f\left (x \right ) \right ]^{2} - 9}{\sqrt{\left | x - 5 \right |}}$ = 0. Then, $lim_{x \rightarrow 5}f\left (x \right )$ equals to
    a) 3
    b) 0
    c) 1
    d) 2

    2. If xmyn = (x + y)m+n, then $\frac{dy}{dx}$ is
    a) $\frac{x + y}{xy}$
    b) xy
    c) $\frac{x}{y}$
    d) $\frac{y}{x}$

  • 9. Integral Calculas - Quiz


    SAMPLE QUESTIONS


    1. The value of the integral I = $\int_{0}^{1}x\left (1 - x \right )^{n}dx$ is
    a) $\frac{1}{n + 1}$
    b) $\frac{1}{n + 2}$
    c) $\frac{1}{n + 1}$ - $\frac{1}{n + 21}$
    d) $\frac{1}{n + 1}$ + $\frac{1}{n + 2}$

    2. The value of $\int_{1}^{a}$[x]f'(x) dx , a > 1 where [x] denotes the greatest integer not exceeding x, is
    a) [a]f(a) - {f(1) + f(2) + ....... + f([a])}
    b) [a]f([a]) - {f(1) + f(2) + ...... + f(a)}
    c) af([a]) - {f(1) + f(2) + ..... + f(a)}
    d) af(a) - {f(1) + f(2) + ...... + f([a])}

  • 10. Differential Equations - Quiz


    SAMPLE QUESTIONS


    1. The differential equation for the family of curves x2 + y2 - 2ay = 0, where a is an arbitrary constant, is
    a) 2(x2 - y2)y' = xy
    b) 2(x2 + y2)y' = xy
    c) (x2 - y2)y' = 2xy
    d) (x2 + y2)y' = 2xy

    2. The differential equaltion which represents the family of curve y = c1ec2x, where c1 and c2 are arbitrary constants is
    a) y' = y2
    b) y" = y'y
    c) yy" = y'
    d) yy" = (y')2

  • 11. Coordinate Geometry - Quiz


    SAMPLE QUESTIONS


    1. If the circles x2 + y2 + 2ax + cy + a = 0 and x2 + y2 - 3ax + dy - 1 = 0 intersect in two distinct points P and Q, then the line 5x + by - a = 0 passes through P and Q for
    a) exactly two values of a
    b) infinitely many value of a
    c) no value of a
    d) exactly one value of a

    2. A parabola has the origin as its focus and the line x = 2 as the directrix. Then , the vertex of the parabola is at
    a) (2,0)
    b) (0,2)
    c) (1,0)
    d) (0,1)

  • 12. Three Dimensional Geometry - Quiz


    SAMPLE QUESTIONS


    1. Statement I The point A(3,1,6) is the mirror image of the point B(1,3,4) in the plane x - y + z = 5
    Statement II The plane x - y + z = 5 bisects the line segments joining A(3,1,6) and B(1,3,4)
    a) Statement I is true, Statement II is a correct explanation for Statement I.
    b) Statement I is true, Statement II is true, Statement II is not a correct explanation for Statement I.
    c) Statement I is true, Statement II is false.
    d) Statement I is false, Statement II is true.

    2. The two lines x = ay + b, z = cy + d and x = a'y + b', z = c'y + d' are perpendicular to each other, if
    a) aa' + cc' = 1
    b) $\frac{a}{a'} + \frac{c}{c'}$ = -1
    c) $\frac{a}{a'} + \frac{c}{c'}$ = 1
    d) aa' + cc' = -1

  • 13. Vector Algebra - Quiz


    SAMPLE QUESTIONS


    1. If the vectors $\vec{a}, \vec{b}$, and $\vec{c}$ from the sides BC, CA and AB respectively of a triangle ABC, then
    a) $\vec{a}.\vec{b}$ = $\vec{b}.\vec{c}$ = $\vec{c}, \vec{b}$ = 0
    b) $\vec{a}\times \vec{b}$ = $\vec{b}\times \vec{c}$ = $\vec{c} \times \vec{a}$ = 0
    c) $\vec{a}.\vec{b}$ = $\vec{b}.\vec{c}$ = $\vec{c}, \vec{a}$ = 0
    d) $\vec{a}\times \vec{a}$ = $\vec{a}\times \vec{c}$ = $\vec{c} \times \vec{a}$ = 0

    2. The distance between the line $\vec{r}$ = $2\vec{i} + 2\vec{j} + 3\vec{k} + \lambda \left ( \vec{i} - \vec{j} + 4 \vec{k} \right )$ and the plane $\vec{r}. \left (\hat{i} + 5\hat{j} + \hat{k} \right )$= 5 is
    a) $\frac{10}{3}$
    b) $\frac{3}{10}$
    c) $\frac{10}{3\sqrt{3}}$
    d) $\frac{10}{9}$

  • 14. Statistics and Probabilty - Quiz


    SAMPLE QUESTIONS


    1. If C and D are two events such that the C ⊂ D and P(D) ≠ 0, then the correct statement among the following is
    a) P(C|D) ≥ P(C)
    b) P(C|D) < P(C)
    c) P(C|D) = $\frac{P\left (D \right )}{P\left (C \right )}$
    d) P(C|D) = P(C)

    2. Statement I The varience of first n even natural numbers is $\frac{n^{2} - 1}{4}$.
    Statement II The sum first n natural numbers is $\frac{n\left (n + 1 \right )}{2}$ and the sum of squares of first n natiural numbers is $\frac{n\left (n + 1 \right )\left (2n + 1 \right )}{6}$
    a) Statement I is true, Statement II is true; Statement II is a correct explanation for Statement I.
    b) Statement I is true, Statement II is true; Statement II is not a correct explanation for Statement I.
    c) Statement I is true, Statement II is false.
    d) Statement I is false, Statement II is false.

  • 15. Trigonometry - Quiz


    SAMPLE QUESTIONS


    1. The upper $\left (\frac{3}{4} \right )th$ portion of a vertical pole substends an angle $tan^{-1}\left (\frac{3}{5} \right )$ at a point in the horizontal plane through its foot and at a distance 40m from the foot. A possible height of the vertical pole is
    a) 20 m
    b) 40m
    c) 60 m
    d) 80 m

    2. The value of $\frac{1 - tan^{2}15^{0}}{1 + tan^{2}15^{0}}$ is
    a) 1
    b) $\sqrt{3}$
    c) $\frac{\sqrt{3}}{2}$
    d) 2

  • 16. Mathematical Reasoning - Quiz


    SAMPLE QUESTIONS


    1. Consider the following statements
    P : Suman is brilliant.
    Q : Suman is rich.
    R : Suman is honest.
    The negative of the statement. Suman is brilliant and dishonest if and only if Suman is rich can be expressed as
    a) ∼ (Q ↔ (O P ∼ R)
    b) ∼ Q ↔ P ^ R
    c) ∼ (P ^ ∼ R) ↔ Q
    d) ∼ P ^ (Q ↔ ∼ R)

    2. The only statement among the following that is a tautology is
    a) B → ∧ ( A → B)]
    b) A ∧ (A ∨ B)
    c) A ∨ (A ∧ B)
    d) [A ∧ (A → B)] → B