## Class/Course - Engineering Entrance

### Subject - Mathematics

#### Total Number of Question/s - 3005

Just Exam provide question bank for Engineering Entrance standard. Currently number of question's are 3005. We provide this data in all format (word, excel, pdf, sql, latex form with images) to institutes for conducting online test/ examinations. Here we are providing some demo contents. Interested person may contact us at info@justexam.in

• 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) = ax2 = ax2 + bx + c,
g(x) = a1x2 + b1 x + c1 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 |z-1| = |z+1| = |z-i| equals
a) 0
b) 1
c) 2
d) ∞

• 3. Matrices and Determinants - Quiz

1. If a1, a2 , ......., an, ....... , 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 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

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 - 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.

• 6. Binomial Theorem - Quiz

1. Statement I $\sum_{r = 0}^{n}\left (r + 1 \right ),^{n}C_{r}$ = $\left (n+2 \right )2^{n-1}$
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 x7 in the expansion of (1 - x - x2 + x3)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 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. 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) = $\left\{\begin{matrix} \left (x - 1 \right )sin\frac{1}{x-1}, & if \ x \ne 1 \\ 0, & if \ x = 1 \end{matrix}\right.$
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 = x3 + 6x2 + 6
b) Interval = $\left (-infty, \frac{1}{3} \right ]$, Function = 3x2 - 2x + 1
c) Interval = [2,∞), Function = 2x3 - 3x2 - 12x + 6
d) Interval = (-∞, ∞), Function = x3 - 3x2 + 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 y2dx + $\left (x - \frac{1}{y} \right )dy$ = 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 x-axis 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 x2 + y2 = p2 orthogonally, then the equation of the locus of its centre is
a) 2ax + 2by - (a2 - b2 + p2) = 0
b) x2 + y2 - 2ax - 3by + (a2 - b2 - p2) = 0
c) 2ax + 2by - (a2 - b2 + p2) = 0
d) x2 + y2 - 3ax - 4by + (a2 + b2 - p2) = 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 600 at the foot of the tower and the angles of elevation of the top of the tower A or B is 300. 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)