## 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) = x3 + 5x + 1, then
a) f is one-one but not onto R
b) f is onto R but not one-one
c) f is one-one and onto R
d) f is neither one-one 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 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)

• 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 A2 - B2 = (A-B)(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) $A^{n}$ = $2^{n-1}A + \left (n-1 \right )I$
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

1. The coefficients of xn in expansion of (1 + x)(1-x)n is
a) (n-1)
b) (-1)n(1-n)
c) (-1)n-1(n-1)2
d) (-1)n-1n

2. If the expansion in powers of x of the function $\frac{1}{\left (1-ax \right )\left (1-bx \right )}$ is $a_{0} + a_{1}x + a_{2}x^{2} + a_{3}x^{3} + .....$ then an is
a) $\frac{a^{n} - b^{n}}{b-a}$
b) $\frac{a^{n+1} - b^{n+1}}{b-a}$
c) $\frac{b^{n+1} - a^{n+1}}{b-a}$
d) $\frac{b^{n} - a^{n}}{b-a}$

• 7. Sequences and Series - Quiz

1. The value of 21/4. 41/8, 81/16 .... 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 sin2 (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 non-vertical 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 x-axis 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 x-axis at a distance of (2/3) from it
b) above the x-axis at a distance of (3/2) from it
c) below the x-axis at a distance of (2/3) from it
d) below the x-axis at a distance of (3/2) from it

2. The point diametricall opposite to the point P(1,0) on the circle x2 + y2 + 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(AC ∩ BC|C) equal to
a) P(AC) - P(B)
b) P(A) - P(BC)
c) P(A2) + P(BC)
d) P(AC) - P(BC)

• 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) Arithmetico-Geometric 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 non-empty 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