Class/Course  Engineering Entrance
Subject  Mathematics
Total Number of Question/s  3005
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1. Sets, Relations and Functions  Quiz
1. For real x, let f(x) = x^{3} + 5x + 1, then
a) f is oneone but not onto R
b) f is onto R but not oneone
c) f is oneone and onto R
d) f is neither oneone not onto R
2. If f : R → R satisfies f(x + y) = f(x) + f(y) , for all x,y ∈ R and f(1) = 7, then $\sum_{r = 1}^{n}f\left (r \right )$ is
a) $\frac{7n}{2}$
b) $\frac{7\left (n+1 \right )}{2}$
c) 7n(n+1)
d) $\frac{7n\left (n+1 \right )}{2}$

2. Complex Numbers and Quadratic Equations  Quiz
1. If $\left (\frac{1 + i}{1  i} \right )^{x}$ = 1 , then
a) x = 4n, where n is any positive integer
b) x = 2n, where n in any positive integer
c) x = 4n + 1, where n is any positive integer
d) x = 2n + 1, where n is any positive integer
2. Let α, β be real and z be a complex number. If z^{2} + 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)

3. Matrices and Determinants  Quiz
1. Let A = $\begin{bmatrix} 1 &2 \\ 3 &4 \end{bmatrix}$ and B = $\begin{bmatrix} a &0 \\ 0 &b \end{bmatrix}$, a, b ∈ N. Then
a) there exist more than one but finite number of B's such that AB = BA
b) there exist exactly one B such that AB = BA
c) there exists infinitely many B's such that AB = BA
d) there cannot exist any B such that AB = BA
2. If A and B are square matrices of size n x n such that A^{2}  B^{2} = (AB)(A+B), then which of the following will be always true?
a) AB = BA
b) either of A and B is a zero matrix
c) either of A or B is an identity matrix
d) A = B

4. Permutations and Combinations  Quiz
1. The set S = {1,2,3, .....,12} is to be partitioned into three sets A,B,C of equal size.
Thus, A ∪ B ∪ C = S,
A ∩ B = B ∩ C = A ∩ C = φ
The number of ways to partition S is
a) 12!/3!(4!)^{3}
b) 12!/3!(3!)^{4}
c) 12!/(4!)^{3}
d) 12!/(3!)^{4}
2. A student is to ansewr 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions. The number of choices available to him is
a) 140
b) 196
c) 280
d) 346

5. Mathematical Induction  Quiz
1. Statement I For every natural number n ≥ 2
$\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + .......... + \frac{1}{\sqrt{n}}$ > $\sqrt{n}$
statement II For every natural number $n \ge 2. \sqrt{n\left (n + 1 \right )} < + 1$
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) $A^{n}$ = $2^{n1}A + \left (n1 \right )I$
b) $A^{n}$ = $nA + \left (n1 \right )I$
c) A^{n} = $2^{n1}A\left (n1 \right )I$
d) A^{n} = nA  (n1)I

6. Binomial Theorem  Quiz
1. The coefficients of x^{n} in expansion of (1 + x)(1x)^{n} is
a) (n1)
b) (1)^{n}(1n)
c) (1)^{n1}(n1)^{2}
d) (1)^{n1}n
2. If the expansion in powers of x of the function $\frac{1}{\left (1ax \right )\left (1bx \right )}$ is $a_{0} + a_{1}x + a_{2}x^{2} + a_{3}x^{3} + ..... $ then a_{n} is
a) $\frac{a^{n}  b^{n}}{ba}$
b) $\frac{a^{n+1}  b^{n+1}}{ba}$
c) $\frac{b^{n+1}  a^{n+1}}{ba}$
d) $\frac{b^{n}  a^{n}}{ba}$

7. Sequences and Series  Quiz
1. The value of 2^{1/4}. 4^{1/8}, 8^{1/16} .... is
a) 1
b) 2
c) $\frac{3}{2}$
d) 4
2. Such term of a GP is 2, then the product of its 9 terms is
a) 256
b) 512
c) 1024
d) None of these

8. Limits, Continuity and Differentiabilty  Quiz
1. Let f : R → R be a function defined by f(x) = min {x + 1, x + 1}. Then which of the following is true?
a) f(x) ≥ 1 for all x ∈ R
b) f(x) is not differentiable at x = 1
c) f(x) is differentiable everywhere
d) f(x) is not differentiable at x = 0
2. If sin y = x sin (a + y), then $\frac{dy}{dx}$ is
a) $\frac{sin a}{sin^{2}\left (a + y \right )}$
b) $\frac{sin^{2}\left (a + y \right )}{sin a}$
c) sin a sin^{2} (a + y)
d) $\frac{sin^{2}\left (a  y \right )}{sin a}$

9. Integral Calculas  Quiz
1. $\int \frac{dx}{cosx  sinx}$ is equal to
a) $\frac{1}{\sqrt{2}}log \left tan \left (\frac{x}{2}  \frac{\pi}{8} \right ) \right + c$
b) $\frac{1}{\sqrt{2}}log\left  cot\left (\frac{x}{2} \right ) \right  + c$
c) $\frac{1}{\sqrt{2}}log \left tan \left (\frac{x}{2}  \frac{3\pi}{8} \right ) \right + c$
d) $\frac{1}{\sqrt{2}}log \left tan \left (\frac{x}{2} + \frac{3\pi}{8} \right ) \right + c$
2. Evaluate $\int_{0}^{\pi/2} \frac{\sqrt{sin x}}{\sqrt{sin x} + \sqrt{cos x}}dx$
a) $\frac{\pi}{4}$
b) $\frac{\pi}{2}$
c) 0
d) 1

10. Differential Equations  Quiz
1. Let I be the purchase value of an equipment and v(t) be the value after it has been used for t years . The value V(t) depreciates at a rate given by differential equation $\frac{dV\left (t \right )}{dt}$ = k(T t), where k > 0 is a constant and T is the total life in years of the equipment . Then , the scrap value V(T) of the equipment is
a) $I  \frac{kT^{2}}{2}$
b) $I  \frac{k\left (T  t \right )^{2}}{2}$
c) e^{kT}
d) $T^{2}  \frac{1}{k}$
2. The differential equation of all nonvertical lines in a plane is
a) $\frac{d^{2}y}{dx^{2}}$ = 0
b) $\frac{d^{2}x}{dy^{2}}$ = 0
c) $\frac{dy}{dx}$ = 0
d) $\frac{dx}{dy}$ = 0

11. Coordinate Geometry  Quiz
1. The line parallel to the xaxis ans passing through the intersectionn of the lines ax + 2 by + 3b = 0 and bx  2ay  3a = 0,
where (a,b) ≠ (0,0) is
a) abobe the xaxis at a distance of (2/3) from it
b) above the xaxis at a distance of (3/2) from it
c) below the xaxis at a distance of (2/3) from it
d) below the xaxis at a distance of (3/2) from it
2. The point diametricall opposite to the point P(1,0) on the circle x^{2} + y^{2} + 2x + 4y  3 = 0 is
a) (3,4)
b) (3,4)
c) (3,4)
d) (3,4)

12. Three Dimensional Geometry  Quiz
1. Two systems of rectangular axes have the same origin. If a plane cuts them at distance a,b,c and a',b', c' from the origin, then
a) $\frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}} + \frac{1}{a'^{2}} + \frac{1}{b'^{2}} + \frac{1}{c'^{2}}$ = 0
b) $\frac{1}{a^{2}} + \frac{1}{b^{2}}  \frac{1}{c^{2}} + \frac{1}{a'^{2}} + \frac{1}{b'^{2}}  \frac{1}{c'^{2}}$ = 0
c) $\frac{1}{a^{2}}  \frac{1}{b^{2}}  \frac{1}{c^{2}} + \frac{1}{a'^{2}}  \frac{1}{b'^{2}}  \frac{1}{c'^{2}}$ = 0
d) $\frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}}  \frac{1}{a'^{2}}  \frac{1}{b'^{2}}  \frac{1}{c'^{2}}$ = 0
2. The projections of a vector on the three coordinates axes are 6,3,2 respectively. The direction cosines of the vector are
a) 6,3,2
b) $\frac{6}{5}, \frac{3}{5}, \frac{2}{5}$
c) $\frac{6}{7}, \frac{3}{7}, \frac{2}{7}$
d) $\frac{6}{7}, \frac{3}{7}, \frac{2}{7}$

13. Vector Algebra  Quiz
1. If $\vec{a}, \vec{b}, \vec{c}$ are coplanar vectors and λ is a real number, then $\left [\lambda\left (\vec{a} + \vec{b} \right ) \lambda^{2}\vec{b} \lambda \vec{c} \right ]$ = $\left [\vec{a} \vec{b} + \vec{c} \vec{b} \right ]$
a) exactly two values of λ
b) exactly three values of λ
c) no values of λ
d) exactly one value of λ
2. Any three vectors such that $\vec{a}.\vec{b} \ne 0, \vec{b}. \vec{c} \ne 0$ then $\vec{a} and \vec{b}$ are
a) inclined at an angle of $\frac{\pi}{6}$ between them
b) perpendicular
c) parallel
d) inclined at an angle of $\frac{\pi}{3}$ between them

14. Statistics and Probabilty  Quiz
1. At a telephone enquiry sustem the number of phone calls , regarding relevent enquiry number of phone calls during 10 min time intervals. The probability that there is at the most one phone call during a 10 min time period is
a) $\frac{5}{6}$
b) $\frac{6}{55}$
c) $\frac{6}{e^{5}}$
d) $\frac{6}{5^{e}}$
2. Let A, B, C be pairwise independent events with p(C) > 0 and P(A ∩ B ∩ C) = 0. Then , P(A^{C} ∩ B^{C}C) equal to
a) P(A^{C})  P(B)
b) P(A)  P(B^{C})
c) P(A^{2}) + P(B^{C})
d) P(A^{C})  P(B^{C})

15. Trigonometry  Quiz
1. If in a δ ABC, the altitudes from the vertices A,B,C on opposite sides are in HP, then sin A, sib B, sin C are in
a) HP
b) ArithmeticoGeometric Progression
c) AP
d) GP
2. Let cos(α + β) = $\frac{4}{5}$ and let sin(aα  β) = $\frac{5}{13}$ where 0 ≤ α , β ≤ $\frac{\pi}{4}$ . Then tan 2α is equal to
a) $\frac{25}{16}$
b) $\frac{56}{33}$
c) $\frac{19}{12}$
d) $\frac{20}{7}$

16. Mathematical Reasoning  Quiz
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. Let S be a nonempty subset of R. Consider the following statement.
P : There is a rational number x ∈ S such that x > 0.
Which of the following statements is the negation of the statement P ?
a) There is a rational number x ∈ S such that x ≤ 0
b) There is no rational number x ∈ S such that x ≤ 0
c) Every rational number x ≤ S satisfies x ≤ 0
d) x ∈ S and x ≤ 0 ⇒ x is not rational