Class/Course  Engineering Entrance
Subject  Mathematics
Total Number of Question/s  3005
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1. Sets, Relations and Functions  Quiz
1. Let f : (1,1) → B, be a function defined by f(x) = $tan^{1}\left (\frac{2x}{1  x^{2}} \right )$, the f is both one one and onto when B is in the interval
a) $\left (\frac{\pi}{2}, \frac{\pi}{2} \right )$
b) $[\frac{\pi}{2}, \frac{\pi}{2}]$
c) $[0,\frac{\pi}{2})$
d) $(0,\frac{\pi}{2})$
2. Let for a ≠ a1 ≠ 0, f(x) = ax^{2} = ax^{2} + bx + c,
g(x) = a_{1}x^{2} + b_{1} x + c_{1} and p(x) = f(x)  g(x) , If p(x) = 0 only for x = 1 and p(2) = 2, then the value of p(2) is
a) 18
b) 3
c) 9
d) 6

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. The number of complex numbers z such that z1 = z+1 = zi equals
a) 0
b) 1
c) 2
d) ∞

3. Matrices and Determinants  Quiz
1. If a_{1}, a_{2} , ......., a_{n}, ....... , are in GP, then the determinant
Δ = $\begin{vmatrix} log a_{n} & loga_{n+1} &log a_{n+2} \\ log a_{n} + 3& loga_{n + 4} & loga_{n + 5} \\ log a_{n+6} & loga_{n+7} & loga_{n + 8} \end{vmatrix}$
is equal to
a) 2
b) 4
c) 0
d) 1
2. If D = $\begin{vmatrix} 1 & 1 & 1\\ 1 & 1 + x & 1 \\ 1 & 1 & 1 + y \end{vmatrix}$ for x ≠ 0, y ≠ 0, then D is
a) divisible by neither x not y
b) divisible by both x and y
c) divisible by x but not y
d) divisible by y but not x

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. In a shop there are five types of icecreams available. A child buys six icecream available. A child buys six icecreams
Statement I The number of different ways the child can buy the six icecreams is ^{10}C_{5}.
Statement II The number of different ways the child can buy the six icecreams 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
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. Statement I For each natural number n,(n+1)^{7}  n^{7}  1 is divisible by 7.
Statement II For each natural number n, n^{7}  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.

6. Binomial Theorem  Quiz
1. Statement I $\sum_{r = 0}^{n}\left (r + 1 \right ),^{n}C_{r}$ = $\left (n+2 \right )2^{n1}$
Statement II $\sum_{r=0}^{n}\left (r+1 \right )^{n}C_{r}.x^{r}$ = $\left (1+x \right )^{n} + nx\left (1+x \right )^{n+1}$
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.
2. The coefficient of x^{7} in the expansion of (1  x  x^{2} + x^{3})^{6} is
a) 132
b) 144
c) 132
d) 144

7. Sequences and Series  Quiz
1. A person is to count 4500 currently notes. Let a_{11}, denotes the number of notes he counts in the nth minute . If a_{1} = a_{2} = .... = a_{10} = 150 and a_{10}, a_{11},.... 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. 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(x) = $\left\{\begin{matrix} \left (x  1 \right )sin\frac{1}{x1}, & if \ x \ne 1 \\ 0, & if \ x = 1 \end{matrix}\right.$
Then which one of the following is true?
a) f is differentiable at x = 1 but not at x = 0
b) f is neither differentiable at x = 0 nor at x = 1
c) f is differentiable at x = 0 and at x = 1
d) f is differentiable at x = 0 but not at x = 1
2. A function is matched below against an interval where it is supposed to be increasing . Which of the following pairs is incorrectly matched?
a) Interval  (∞, 4), Function = x^{3} + 6x^{2} + 6
b) Interval = $\left (infty, \frac{1}{3} \right ]$, Function = 3x^{2}  2x + 1
c) Interval = [2,∞), Function = 2x^{3}  3x^{2}  12x + 6
d) Interval = (∞, ∞), Function = x^{3}  3x^{2} + 3x + 3

9. Integral Calculas  Quiz
1. The value of $\int_{2}^{3}\left  1  x^{2} \right dx$ is
a) $\frac{28}{3}$
b) $\frac{14}{3}$
c) $\frac{7}{3}$
d) $\frac{1}{3}$
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
1. Consider the differential equation y^{2}dx + $\left (x  \frac{1}{y} \right )dy$ = 0. If y(1) = 1, then x is given by
a) $1  \frac{1}{y} + \frac{\frac{1}{e^{y}}}{e}$
b) $4  \frac{2}{y} + \frac{\frac{1}{e^{y}}}{e}$
c) $3  \frac{1}{y} + \frac{\frac{1}{e^{y}}}{e}$
d) $1 + \frac{1}{y}  \frac{\frac{1}{e^{y}}}{e}$
2. Solution of the differential equation cos x dy = y(sinx  y)dx, 0 < x < $\frac{\pi}{2}$ , is
a) sec x = (tan + c) y
b) y sec x = tan x + c
c) y tanx = secx + c
d) tan x = (sec x + c) y

11. Coordinate Geometry  Quiz
1. A circle touches the xaxis and also touches the circle with centre at (0,3) and radius 2. The locus of the centre of the circle is
a) a parabola
b) a hyperbola
c) a circle
d) an ellipse
2. If a circle passes through the point (a,b) and cuts the circle x^{2} + y^{2} = p^{2} orthogonally, then the equation of the locus of its centre is
a) 2ax + 2by  (a^{2}  b^{2} + p^{2}) = 0
b) x^{2} + y^{2}  2ax  3by + (a^{2}  b^{2}  p^{2}) = 0
c) 2ax + 2by  (a^{2}  b^{2} + p^{2}) = 0
d) x^{2} + y^{2}  3ax  4by + (a^{2} + b^{2}  p^{2}) = 0

12. Three Dimensional Geometry  Quiz
1. 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
2. Distance between two parallel planes 2x + y + 2z = 8 and 4x + 2y + 4z + 5 = 0 is
a) $\frac{3}{2}$
b) $\frac{5}{2}$
c) $\frac{7}{2}$
d) $\frac{9}{2}$

13. Vector Algebra  Quiz
1. If $\vec{a} $= $\frac{1}{^{10}}\left ( 3\hat{i} +\hat{k}\right )$ and $\vec{b} $= $\frac{1}{7}\left ( 2\hat{i} + 3\hat{k}  6\hat{k}\right )$then the value of $\left ( 2\vec{a}\vec{b} \right )\cdot \left [ \left ( \vec{a}\times \vec{b} \right ) \times \left ( \vec{a}+2\vec{b} \right ) \right ]$ is
a) 3
b) 5
c) 3
d) 5
2. If the vectors $\vec{\epsilon}$ = $\hat{i}  \hat{j} + 2\hat{k}, \vec{b}$ = $2\hat{i} + 4\hat{j} + \hat{k}$ and $\vec{c}$ = $\lambda \hat{i} + \hat{j} + \mu\hat{k}$ are mutually orthogonal, then (λ, μ) is equal to
a) (3,2)
b) (2,3)
c) (2,3)
d) (3,2)

14. Statistics and Probabilty  Quiz
1. For two two data sets, each of size, 5 the variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4 , respectively. The variance of the combined data set is
a) $\frac{5}{2}$
b) $\frac{11}{2}$
c) 6
d) $\frac{13}{2}$
2. If the mean deviations about the median of the number a, 2a, .......5a is 50, than a equals to
a) 3
b) 4
c) 5
d) 2

15. Trigonometry  Quiz
1. If in a triangle ABC
$a \ cos^{2}\left (\frac{C}{2} \right ) + c \ cos^{2}\left (\frac{A}{2} \right )$ = $\frac{3b}{2}$
then the sides,a , b and c
a) are in AP
b) are in GP
c) are in HP
d) satisfy a + b = c
2. A tower stands at the centre of a circular park. A and B are two points on the boundary of the park such that AB = ( = a) subtends an angle of 60^{0} at the foot of the tower and the angles of elevation of the top of the tower A or B is 30^{0}. The height of the tower is
a) $\frac{2a}{\sqrt{3}}$
b) $2a\sqrt{3}$
c) $\frac{a}{\sqrt{3}}$
d) $\sqrt{3}$

16. Mathematical Reasoning  Quiz
1. The statement p → (q → p) is equivalent to
a) p → (p ↔ q)
b) p → (p → q)
c) p → (p ∨ q)
d) p → (p ∧ q)
2. 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)