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# NDA II 2016 Mathematics

This test contains 120 question. Each question comprises four responses (answers). You need to select only ONE response for each question.

All questions carry equal marks.

Each question for which a wrong answer has been marked, one-third of the marks assigned to that question will be deducted as penalty.

If a candidate gives more than one answer, it will be treated as a wrong answer even if one of the given answers happens to be correct and there will be same penalty as above to
that question.

If a question is left blank, i.e., no answer is given by the candidate, there will be no penalty for that question.
• Question 1
Let S be the set of all distinct numbers of the form $\frac{\mathit{p}}{\mathit{q}}$ where p, qWhat is the cardinality of the set S?
Option A: 21
Option B: 23
Option C: 32
Option D: 36
VIEW SOLUTION
• Question 2
If c > 0 and 4a + c < 2b, then $a{x}^{2}-bx+c=0$ has a root in which one of the following intervals?
Option A: (0, 2)
Option B: (2, 3)
Option C: (3, 4)
Option D: (–2, 0)
VIEW SOLUTION
• Question 3
If A = $\left\{x\in \mathbit{R}:{x}^{2}+6x-7<0\right\}$ and B = $\left\{x\in \mathbit{R}:{x}^{2}+9x+14>0\right\}$, then which of the following is /are correct?
(I) $A\cap B=\left\{x\in \mathbit{R}:-2
(II) A\B $\left\{x\in \mathbit{R}:-7
Select the correct answer using the code given below :
Option A: 1 only
Option B: 2 only
Option C: Both 1 and 2
Option D: Neither 1 nor 2
VIEW SOLUTION
• Question 4
If A is a square matrix of order 3 and det A = 5, then what is det $\left[2{\left(\mathrm{A}\right)}^{-1}\right]$ equal to?
Option A: $\frac{1}{10}$
Option B: $\frac{2}{5}$
Option C: $\frac{8}{5}$
Option D: $\frac{1}{40}$
VIEW SOLUTION
• Question 5
What is ${\omega }^{\mathrm{100}}+{\omega }^{\mathrm{200}}+{\omega }^{\mathrm{300}}$ equal to, where ω is the cube root of unity?
Option A: 1
Option B:
Option C: 2
VIEW SOLUTION
• Question 6
If $\mathrm{Re}\left(\frac{z-1}{z+1}\right)=0$ , where z = x + iy is a complex number, then which one of the following is correct?
Option A: z = 1 + i
Option B: |z| = 2
Option C: z = 1 – i
Option D: |z| = 1
VIEW SOLUTION
• Question 7
What is [x y z$\left[\begin{array}{ccc}\mathit{a}& \mathit{h}& \mathit{g}\\ \mathit{h}& \mathit{b}& \mathit{f}\\ \mathit{g}& \mathit{f}& \mathit{c}\end{array}\right]$ equal to?
Option A: $\left[\begin{array}{ccc}ax+hy+gz& h+b+f& g+f+c\end{array}\right]$
Option B: $\left[\begin{array}{ccc}\mathit{a}& \mathit{h}& \mathit{g}\\ \mathit{h}\mathit{x}& \mathit{b}\mathit{y}& \mathit{f}\mathit{z}\\ \mathit{g}& \mathit{f}& \mathit{c}\end{array}\right]$
Option C: $\left[\begin{array}{l}ax+hy+gz\\ hx+by+fz\\ gx+fy+cz\end{array}\right]$
Option D: $\left[\begin{array}{ccc}ax+hy+gz& hx+by+fz& gx+fy+cz\end{array}\right]$
VIEW SOLUTION
• Question 8
Out of 15 points in a plane, n points are in the same straight line. 445 triangles can be formed by joining these points. What is the value of n?
Option A: 3
Option B: 4
Option C: 5
Option D: 6
VIEW SOLUTION
• Question 9
If ${z=\left(\frac{\sqrt{3}}{2}+\frac{i}{2}\right)}^{107}+{\left(\frac{\sqrt{3}}{2}-\frac{i}{2}\right)}^{107},$ then what is the imaginary part of z equal to?
Option B: $\frac{1}{2}$
Option C: $\frac{\sqrt{3}}{2}$
Option D: 1
VIEW SOLUTION
• Question 10
If both the roots of the equation ${x}^{2}-2kx+{k}^{2}-4=0$ lie between –3 and 5, then which one of the following is correct?
Option A: $-2
Option B: $-5
Option C: $-3
Option D: $-1
VIEW SOLUTION
• Question 11
What is the number of distinct solutions of the equation ${z}^{2}+|z|=0$ (where z is a complex number)?
Option A: One
Option B: Two
Option C: Three
Option D: Five
VIEW SOLUTION
• Question 12
How many geometric progressions is/are possible containing 27, 8 and 12 as three of its/their terms?
Option A: One
Option B: Two
Option C: Four
Option D: Infinitely many
VIEW SOLUTION
• Question 13
Let R be a relation from A = (1, 2, 3, 4) to B = (1, 3, 5) such that R = [(a, b) : a < b, where a ∈ A and b ∈ B].
What is RoR–1 equal to?
Option A:
Option B:
Option C:
Option D:
VIEW SOLUTION
• Question 14
A five-digit number divisible by 3 is to be formed using the digits 0, 1, 2, 3 and 4 without repetition of digits. What is the number of ways this can be done?
Option A: 96
Option B: 48
Option C: 32
Option D: No number can be formed
VIEW SOLUTION
• Question 15
What is ${}^{47}{C}_{4}{+}^{51}{C}_{3}+{\sum _{j=2}^{5}}^{52-j}{C}_{3}$ equal to?
Option A: ${}^{52}{C}_{4}$
Option B: ${}^{51}{C}_{5}$
Option C: ${}^{53}{C}_{4}$
Option D: ${}^{52}{C}_{5}$
VIEW SOLUTION
• Question 16
Let a, x, y, z, b be in AP, where x + y + z = 15. Let a, p, q, r, b be in HP, where ${p}^{-1}+{q}^{-1}+{r}^{-1}=5/3.$

What is the value of ab?
Option A: 10
Option B: 9
Option C: 8
Option D: 6
VIEW SOLUTION
• Question 17
Let a, x, y, z, b be in AP, where x + y + z = 15. Let a, p, q, r, b be in HP, where ${p}^{-1}+{q}^{-1}+{r}^{-1}=5/3.$

What is the value of xyz?
Option A: 120
Option B: 105
Option C: 90
Option D: Cannot be determined
VIEW SOLUTION
• Question 18
Let a, x, y, z, b be in AP, where x + y + z = 15. Let a, p, q, r, b be in HP, where ${p}^{-1}+{q}^{-1}+{r}^{-1}=5/3.$

What is the value of pqr?
Option A: $\frac{35}{243}$
Option B: $\frac{81}{35}$
Option C: $\frac{243}{35}$
Option D: Cannot be determined
VIEW SOLUTION
• Question 19
The sixth term of an AP is 2 and its common difference is greater than 1.

What is the common difference of the AP so that the product of the first, fourth and the fifth terms isthe greatest?
Option A: 8/5
Option B: 9/5
Option C: 2
Option D: 11/5
VIEW SOLUTION
• Question 20
The sixth term of an AP is 2 and its common difference is greater than 1.

What is the first term of the AP, so that the product of the first,fourth and the fifth terms is the greatest?
Option A:  –4
Option B: –6
Option C: –8
Option D: –10
VIEW SOLUTION
• Question 21
$\mathrm{Let}\mathrm{ }a{x}^{3}+b{x}^{2}+cx+d=\left|\begin{array}{l}x+1\\ 2x+3\\ 2-x\end{array}\begin{array}{cc}2x& 3x\\ x+1& x\\ 3x+4& 5x-1\end{array}\right|$, then

What is the value of c?
Option A: –1
Option B: 34
Option C: 35
Option D: 50
VIEW SOLUTION
• Question 22
$\mathrm{Let}\mathrm{ }a{x}^{3}+b{x}^{2}+cx+d=\left|\begin{array}{l}x+1\\ 2x+3\\ 2-x\end{array}\begin{array}{cc}2x& 3x\\ x+1& x\\ 3x+4& 5x-1\end{array}\right|$, then

What is the value of a + b + c + d?
Option A: 62
Option B: 63
Option C: 65
Option D: 68
VIEW SOLUTION
• Question 23
The interior angles of a polygon of n sides are in AP. The smallest angle is 120° and the common difference is 5°.

How many possible values can n have
Option A: One
Option B: Two
Option C: Three
Option D: Infinitely many
VIEW SOLUTION
• Question 24
The interior angles of a polygon of n sides are in AP. The smallest angle is 120° and the common difference is 5°.

What is the largest interior angle of the polygon?
Option A: 160° only
Option B: 195° only
Option C: Either 160° or 195°
Option D: Neither 160° nor 195°
VIEW SOLUTION
• Question 25
If $m=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$  and $n=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right],$ then what is the value of the determinant of m cosθ – n sinθ?
Option A: –1
Option C: 1
Option D: 2
VIEW SOLUTION
• Question 26
If , then which of the following are correct?

1. $f\left(\theta \right)×f\left(\varphi \right)=f\left(\theta +\varphi \right).$
2. The value of the determinant of the matrix $f\left(\theta \right)×f\left(\varphi \right)$ is 1.
3. The determinant of f(x) is an even function.
Select the correct answer using the code given below :
Option A: 1 and 2 only
Option B: 2 and 3 only
Option C: 1 and 3 only
Option D: 1, 2 and 3
VIEW SOLUTION
• Question 27
Which of the following are correct in respect of the system of equations x + y + z = 8, xy + 2z = 6 and 3xy + 5z = k?
1. They have no solution, if k = 15.
2. They have infinitely many solutions, if k = 20.
3. They have unique solution, if k = 25.
Select the correct answer using the code given below :
Option A: 1 and 2 only
Option B: 2 and 3 only
Option C: 1 and 3 only
Option D: 1, 2 and 3
VIEW SOLUTION
• Question 28
If  and  then which of the following is/are correct?

1. $AB\left({A}^{-1}{B}^{-1}\right)$ is a unit matrix.
2. ${\left(AB\right)}^{-1}={A}^{-1}{B}^{-1}.$
Select the correct answer using the code given below :
Option A: 1 only
Option B: 2 only
Option C: Both 1 and 2
Option D: Neither 1 nor 2
VIEW SOLUTION
• Question 29
If ${x}^{\mathrm{In} \left(\frac{y}{z}\right)}\cdot y{ }^{\mathrm{In}\mathrm{ }{\left(x z\right)}^{2}}\cdot {z}^{\mathrm{In} \left(\frac{x}{y}\right)}=y{ }^{4 \mathrm{In}\mathrm{ }y}$ for any x >1, y > 1 and z > 1, then which one of the following is correct?
Option A: ln y is the GM of ln x, ln x, ln x and ln z
Option B: ln y is the AM of ln x, ln x, ln x and ln z
Option C: ln y is the HM of ln x, ln x, ln x and ln z
Option D: ln y is the AM of ln x, ln x, ln z and ln z
VIEW SOLUTION
• Question 30
If the number 235 in the decimal system is converted into the binary system, then what is the resulting number?
Option A: (11110011)2
Option B: (11101011)2
Option C: (11110101)2
Option D: (11011011)2
VIEW SOLUTION
• Question 31
Let α and β be the roots of the equation

${x}^{2}-\left(1-2{a}^{2}\right)x+\left(1-2{a}^{2}\right)=0$.

Under what condition, does the above equation have real roots?
Option A: ${a}^{2}<\frac{1}{2}$
Option B: ${a}^{2}>\frac{1}{2}$
Option C: ${a}^{2}\le \frac{1}{2}$
Option D: ${a}^{2}\ge \frac{1}{2}$
VIEW SOLUTION
• Question 32
Let α and β be the roots of the equation

${x}^{2}-\left(1-2{a}^{2}\right)x+\left(1-2{a}^{2}\right)=0$.

Under what condition is $\frac{1}{{\alpha }^{2}}+\frac{1}{{\beta }^{2}}<1?$
Option A: ${a}^{2}<\frac{1}{2}$
Option B: ${a}^{2}>\frac{1}{2}$
Option C: ${a}^{2}>1$
Option D: ${a}^{2}\in \left(\frac{1}{3},\frac{1}{2}\right)$ only
VIEW SOLUTION
• Question 33
What is $\sqrt{\frac{1+{\omega }^{2}}{1+\omega }}$ equal to, where ω is the cube root of unity?
Option A: 1
Option B: ω
Option C: ω2
Option D:
VIEW SOLUTION
• Question 34
In an examination, 70% students passed in Physics, 80% students passed in Chemistry, 75% students passed in Mathematics and 85% students passed in Biology, and x% students failed in all the four subjects. What is the minimum value of x?
Option A: 10
Option B: 12
Option C: 15
Option D: None of the above
VIEW SOLUTION
• Question 35
For the system of linear equations, 2x + 3y + 5z = 9, 7x + 3y – 2z = 8 and 2x + 3y + λz = µ

Under what condition, does the above system of equations have infinitely many solutions?
Option A: λ = 5 and µ ≠ 9
Option B: λ = 5 and µ = 9
Option C: λ = 9 and µ = 5
Option D: λ = 9 and µ ≠ 5
VIEW SOLUTION
• Question 36
For the system of linear equations, 2x + 3y + 5z = 9, 7x + 3y – 2z = 8 and 2x + 3y + λz = µ

Under what condition, does the above system of equations have unique solutions?
Option A: λ = 5 and µ = 9
Option B: λ ≠ 5 and µ = 7 only
Option C: λ ≠ 5 and µ has any real value
Option D: λ has any real value and µ ≠ 9
VIEW SOLUTION
• Question 37
What is the number of odd integers between 1000 and 9999 with no digit repeated?
Option A: 2100
Option B: 2120
Option C: 2240
Option D: 3331
VIEW SOLUTION
• Question 38
What is the greatest value of the positive integer n satisfying the condition $1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...+\frac{1}{{2}^{n-1}}<2-\frac{1}{1000}$?
Option A: 8
Option B: 9
Option C: 10
Option D: 11
VIEW SOLUTION
• Question 39
$2{x}^{2}+3x-\alpha =0$ has roots –2 and β while the equation ${x}^{2}-3mx+2{m}^{2}=0$ has both the roots positive, where α > 0 and β > 0.

What is the value of α?
Option A: 1/2
Option B: 1
Option C: 2
Option D: 4
VIEW SOLUTION
• Question 40
$2{x}^{2}+3x-\alpha =0$ has roots –2 and β while the equation ${x}^{2}-3mx+2{m}^{2}=0$ has both the roots positive, where α > 0 and β > 0.

If β, 2, 2m is in GP, then what is the value of $\beta \sqrt{m}$?
Option A: 1
Option B: 2
Option C: 4
Option D: 6
VIEW SOLUTION
• Question 41
Sin A + 2 sin 2A + sin 3A is equal to which of the following?

1.

2.
3.
Select the correct answer using the code given below:
Option A: 1 and 2 only
Option B: 2 and 3 only
Option C: 1 and 3 only
Option D: 1, 2 and 3
VIEW SOLUTION
• Question 42
If x = sin 70° · sin 50° and y = cos 60° · cos 80°, then what is xy equal to?
Option A: 1/16
Option B: 1/8
Option C: 1/4
Option D: 1/2
VIEW SOLUTION
• Question 43
If $\mathrm{sin}{\theta }_{1}+\mathrm{sin}{\theta }_{2}+\mathrm{sin}{\theta }_{3}+\mathrm{sin}{\theta }_{4}=4,$ then what is the value of $\mathrm{cos}{\theta }_{1}+\mathrm{cos}{\theta }_{2}+\mathrm{cos}{\theta }_{3}+\mathrm{cos}{\theta }_{4}?$
Option B: 1
Option C: 2
Option D: 4
VIEW SOLUTION
• Question 44
What is the value of
Option A: $\frac{1}{2}$
Option B: $\frac{1}{2}+\frac{1}{2\sqrt{2}}$
Option C: $\frac{1}{2}-\frac{1}{2\sqrt{2}}$
Option D: $\frac{1}{8}$
VIEW SOLUTION
• Question 45
If , then what is the value of
Option A: ${x}^{2}+{y}^{2}-{z}^{2}$
Option B: ${x}^{2}-{y}^{2}-{z}^{2}$
Option C: ${x}^{2}-{y}^{2}+{z}^{2}$
Option D: ${x}^{2}+{y}^{2}+{z}^{2}$
VIEW SOLUTION
• Question 46
What is the value of cos
Option A: 0.81
Option B: 0.56
Option C: 0.48
Option D: 0.28
VIEW SOLUTION
• Question 47
The top of a hill when observed from the top and bottom of a building of height h is at angles of elevation p and q, respectively. What is the height of the hill?
Option A:
Option B:
Option C:
Option D:
VIEW SOLUTION
• Question 48
If  then what is the value of sin 81°?
Option A: $\frac{\sqrt{3+\sqrt{5}}+\sqrt{5-\sqrt{5}}}{4}$
Option B: $\frac{\sqrt{3+\sqrt{5}}+\sqrt{5+\sqrt{5}}}{4}$
Option C: $\frac{\sqrt{3-\sqrt{5}}+\sqrt{5-\sqrt{5}}}{4}$
Option D: $\frac{\sqrt{3+\sqrt{5}}-\sqrt{5-\sqrt{5}}}{4}$
VIEW SOLUTION
• Question 49
A moving boat is observed from the top of a cliff of 150 m height. The angle of depression of the boat changes from 60° to 45° in 2 minutes. What is the speed of the boat in meter per hour?
Option A: $\frac{4500}{\sqrt{3}}$
Option B: $\frac{4500\left(\sqrt{3}-1\right)}{\sqrt{3}}$
Option C: $4500\sqrt{3}$
Option D: $\frac{4500\left(\sqrt{3}+1\right)}{\sqrt{3}}$
VIEW SOLUTION
• Question 50
What is  equal to?
Option A: $\sqrt{3}$
Option B: $-\sqrt{3}$
Option C: $\sqrt{2}-1$
Option D: $1-\sqrt{2}$
VIEW SOLUTION
• Question 51
An equilateral triangle has one vertex at (0, 0) and another at $\left(3,\sqrt{3}\right).$ What are the coordinates of the third vertex?
Option A:  Only
Option B: $\left(3,-\sqrt{3}\right)$ Only
Option C:
Option D: Neither  nor $\left(3,-\sqrt{3}\right)$
VIEW SOLUTION
• Question 52
What is the equation of the right bisector of the line segment joining (1, 1) and (2, 3)?
Option A: $2x+4y-11=0$
Option B: $2x-4y-5=0$
Option C: $2x-4y-11=0$
Option D: $x-y+1=0$
VIEW SOLUTION
• Question 53
What is the radius of the circle passing through the point (2, 4) and having centre at the intersection of the lines $x-y=4$ and $2x+3y+7=0?$
Option A: 3 units
Option B: 5 units
Option C: $3\sqrt{3}$ units
Option D: $5\sqrt{2}$ units
VIEW SOLUTION
• Question 54
What is the equation of the hyperbola having latus rectum and eccentricity 8 and $\frac{3}{\sqrt{5}}$ respectively?
Option A: $\frac{{x}^{2}}{25}-\frac{{y}^{2}}{20}=1$
Option B: $\frac{{x}^{2}}{40}-\frac{{y}^{2}}{20}=1$
Option C: $\frac{{x}^{2}}{40}-\frac{{y}^{2}}{30}=1$
Option D: $\frac{{x}^{2}}{30}-\frac{{y}^{2}}{25}=1$
VIEW SOLUTION
• Question 55
If the point (a, a) lies between the lines $|x+y|=2$, then which one of the following is correct?
Option A: $|a|<2$
Option B: $|a|<\sqrt{2}$
Option C: $|a|<1$
Option D: $|a|<\frac{1}{\sqrt{2}}$
VIEW SOLUTION
• Question 56
What is the equation of the straight line which passes through the point of intersection of the straight lines x + 2y = 5 and 3x + 7y = 17 and is perpendicular to the straight line 3x + 4y = 10?
Option A: 4x + 3y + 2 = 0
Option B: 4xy + 2 = 0
Option C: 4x – 3y – 2 = 0
Option D: 4x – 3y + 2 = 0
VIEW SOLUTION
• Question 57
If (a, b) is at unit distance from the line 8x + 6y + 1 = 0, then which of the following conditions are correct?
1. 3a – 4b – 4 = 0
2. 8a + 6b + 11 = 0
3. 8a + 6b – 9 = 0
Select the correct answer using the code given below :
Option A: 1 and 2 only
Option B: 2 and 3 only
Option C: 1 and 3 only
Option D: 1, 2 and 3
VIEW SOLUTION
• Question 58
If the ellipse $9{x}^{2}+16{y}^{2}=144$ intercepts the line 3x + 4y = 12, then what is the length of the chord so formed?
Option A: 5 units
Option B: 6 units
Option C: 8 units
Option D: 10 units
VIEW SOLUTION
• Question 59
A straight line cuts off an intercept of 2 units on the positive direction of x-axis and passes through the point (–3, 5). What is the foot of the perpendicular drawn from the point (3, 3) on this line?
Option A: (1, 3)
Option B: (2, 0)
Option C: (0, 2)
Option D: (1, 1)
VIEW SOLUTION
• Question 60
What is the eccentricity of rectangular hyperbola?
Option A: $\sqrt{2}$
Option B: $\sqrt{3}$
Option C: $\sqrt{5}$
Option D: $\sqrt{6}$
VIEW SOLUTION
• Question 61
Let Q be the image of the point P (–2, 1, –5) in the plane 3x – 2y + 2z + 1 = 0.

Consider the following :
1. The coordinates of Q are (4, –3,–1).
2. PQ is of length more than 8 units.
3. The point (1, –1, –3) is the mid-point of the line segment PQ and lies on the given plane.
Which of the above statements are correct?
Option A: 1 and 2 only
Option B: 2 and 3 only
Option C: 1 and 3 only
Option D: 1, 2 and 3
VIEW SOLUTION
• Question 62
Let Q be the image of the point P (–2, 1, –5) in the plane 3x – 2y + 2z + 1 = 0.

Consider the following :
1. The direction ratios of the line segment PQ are < 3, –2, 2 >.
2. The sum of the squares of direction cosines of the line segment PQ is unity.
Which of the above statements is/are correct?
Option A: 1 only
Option B: 2 only
Option C: Both 1 and 2
Option D: Neither 1 nor 2
VIEW SOLUTION
• Question 63
A line L passes through the point P(5, –6, 7) and is parallel to the planes x + y + z = 1 and 2xy – 2z = 3.

What are the direction ratios of the line of intersection of the given planes?
Option A: < 1, 4, 3 >
Option B: < –1, –4, 3 >
Option C: < 1, –4, 3 >
Option D: < 1, –4, –3 >
VIEW SOLUTION
• Question 64
A line L passes through the point P(5, –6, 7) and is parallel to the planes x + y + z = 1 and 2xy – 2z = 3.

What is the equation of the line L?
Option A: $\frac{x-5}{-1}=\frac{y+6}{4}=\frac{z-7}{-3}$
Option B: $\frac{x+5}{-1}=\frac{y-6}{4}=\frac{z+7}{-3}$
Option C: $\frac{x-5}{-1}=\frac{y+6}{-4}=\frac{z-7}{3}$
Option D: $\frac{x-5}{-1}=\frac{y+6}{-4}=\frac{z-7}{-3}$
VIEW SOLUTION
• Question 65
Let  and $\stackrel{\to }{b}=\stackrel{\to }{c}+\stackrel{\to }{d}$, where $\stackrel{\to }{c}$ is parallel to $\stackrel{\to }{a}$ and $\stackrel{\to }{d}$ is perpendicular to $\stackrel{\to }{a}$.

What is $\stackrel{\to }{c}$ equal to?
Option A:
Option B: $\frac{2\left(\stackrel{^}{i}+\stackrel{^}{j}\right)}{3}$
Option C: $\frac{\left(\stackrel{^}{i}+\stackrel{^}{j}\right)}{2}$
Option D: $\frac{\left(\stackrel{^}{i}+\stackrel{^}{j}\right)}{3}$
VIEW SOLUTION
• Question 66
Let  and $\stackrel{\to }{b}=\stackrel{\to }{c}+\stackrel{\to }{d}$, where $\stackrel{\to }{c}$ is parallel to $\stackrel{\to }{a}$ and $\stackrel{\to }{d}$ is perpendicular to $\stackrel{\to }{a}$.

If $\stackrel{\to }{d}=x\stackrel{^}{i}+y\stackrel{^}{j}+z\stackrel{^}{k}$ then which of the following equations is/are correct?

1. yx = 4

2. 2z – 3 = 0

Select the correct answer using the code given below :
Option A: 1 only
Option B: 2 only
Option C: Both 1 and 2
Option D: Neither 1 nor 2
VIEW SOLUTION
• Question 67
Let and $\stackrel{\to }{c}$ be three vectors such that $\stackrel{\to }{a}+\stackrel{\to }{b}+\stackrel{\to }{c}=\stackrel{\to }{0}$, and  and $\left|\stackrel{\to }{c}\right|=14$.

What is $\stackrel{\to }{a}·\stackrel{\to }{b}+\stackrel{\to }{b}·\stackrel{\to }{c}=+\stackrel{\to }{c}·\stackrel{\to }{a}$ equal to?
Option A: –332
Option B: –166
Option D: 166
VIEW SOLUTION
• Question 68
Let and $\stackrel{\to }{c}$ be three vectors such that $\stackrel{\to }{a}+\stackrel{\to }{b}+\stackrel{\to }{c}=\stackrel{\to }{0}$, and  and $\left|\stackrel{\to }{c}\right|=14$.

What is the angle between $\stackrel{\to }{a}$ and $\stackrel{\to }{b}$ ?
Option A: 30°
Option B: 45°
Option C: 60°
Option D: 75°
VIEW SOLUTION
• Question 69
In a right-angled triangle ABC, if the hypotenuse AB = p, then what is  equal to?
Option A: p
Option B: p2
Option C: 2p2
Option D: $\frac{{p}^{2}}{2}$
VIEW SOLUTION
• Question 70
A force $\stackrel{\to }{F}=3\stackrel{^}{i}+2\stackrel{^}{j}-4\stackrel{^}{k}$ is applied at the point (1, –1, 2). What is the moment of the force about the point (2, –1, 3)?
Option A: $\stackrel{^}{i}+4\stackrel{^}{j}+4\stackrel{^}{k}$
Option B: $2\stackrel{^}{i}+\stackrel{^}{j}+2\stackrel{^}{k}$
Option C: $2\stackrel{^}{i}-7\stackrel{^}{j}-2\stackrel{^}{k}$
Option D: $2\stackrel{^}{i}+4\stackrel{^}{j}-\stackrel{^}{k}$
VIEW SOLUTION
• Question 71
What is the domain of the function $f\left(x\right)=\frac{1}{\sqrt{|x|-x}}?$
Option A: (–∞, 0)
Option B: (0, ∞)
Option C: 0 < x < 1
Option D: x > 1
VIEW SOLUTION
• Question 72
Consider the following in respect of the function $f\left(x\right)=\left\{\begin{array}{cl}2+x,& x\ge 0\\ 2-x,& x<0\end{array}\right\$
1.  $\underset{x\to 1}{\mathrm{lim}}f\left(x\right)$ does not exist.

2.  f(x) is differentiable at x = 0.

3. f(x) is continuous at x = 0.

Which of the above statement is/are correct?
Option A: 1 only
Option B: 3 only
Option C: 2 and 3 only
Option D: 1 and 3 only
VIEW SOLUTION
• Question 73
Let f : A → R where  A = R\[0] is such that $f\left(x\right)=\frac{x+|x|}{x}$. On which one of the following sets is f(x) continuous?
Option A: A
Option B: B = {xR : x ≥ 0}
Option C: C = {xR : x ≤ 0}
Option D: D = R
VIEW SOLUTION
• Question 74
Which one of the following statements is correct with respect to the function
Option A: It has local maximum at x = 0.
Option B: It has local minimum at x = 0.
Option C: It has neither maximum nor minimum at x = 0.
Option D: It has maximum value 1.
VIEW SOLUTION
• Question 75
What is the area bounded by the curves $|y|=1-{x}^{2}?$
Option A: 4/3 square units
Option B: 8/3 square units
Option C: 4 square units
Option D: 16/3 square units
VIEW SOLUTION
• Question 76
$f\left(x\right)=\left\{\begin{array}{lr}3{x}^{2}+12x-1,& -1\le x\le 2\\ 37-x,& 2

Which of the following statements is /are correct?
1. f(x) is increasing in the interval [–1, 2].
2. f(x) is decreasing in the interval (2, 3].
Select the correct answer using the code given below :
Option A: 1 only
Option B: 2 only
Option C: Both 1 and 2
Option D: Neither 1 nor 2
VIEW SOLUTION
• Question 77
$f\left(x\right)=\left\{\begin{array}{lr}3{x}^{2}+12x-1,& -1\le x\le 2\\ 37-x,& 2

Which of the following statements are correct?
1.  f(x) is continuous at, x = 2.
2.  f(x) attains greatest value at x = 2.
3.  f(x) is differentiable at x = 2.
Select the correct answer using the code given below :
Option A: 1 and 2 only
Option B: 2 and 3 only
Option C: 1 and 3 only
Option D: 1, 2 and 3
VIEW SOLUTION
• Question 78
Let $f\left(x\right)={\left[\left|x\right|-\left|x-1\right|\right]}^{2}$.

What is f’(x) equal to when x >1?
Option B: 2x – 1
Option C: 4x – 2
Option D: 8x – 4
VIEW SOLUTION
• Question 79
Let $f\left(x\right)={\left[\left|x\right|-\left|x-1\right|\right]}^{2}$.

What is f’(x) equal to when 0 < x < 1?
Option B: 2x – 1
Option C: 4x –2
Option D: 8x – 4
VIEW SOLUTION
• Question 80
Let $f\left(x\right)={\left[\left|x\right|-\left|x-1\right|\right]}^{2}$.

Which of the following equations is/are correct?
1. f(–2) = f(5)
2. f” (–2) + f” (0·5) + f” (3) = 4
Select the correct answer using the code given below :
Option A: 1 only
Option B: 2 only
Option C: Both 1 and 2
Option D: Neither 1 nor 2
VIEW SOLUTION
• Question 81
Let f(x) = [x], where [·] is the greatest integer function and g(x) = sin x be two real valued functions over .

Which of the following statements is correct?
Option A: Both f(x) and g(x) are continuous at x = 0.
Option B: f(x) is continuous at x = 0 but g(x) is not continuous at x = 0.
Option C: g(x) is continuous at x = 0 but f(x) is not continuous at x = 0.
Option D: Both f(x) and g(x) are discontinuous at x = 0.
VIEW SOLUTION
• Question 82
Let f(x) = [x], where [·] is the greatest integer function and g(x) = sin x be two real valued functions over .

Which one of the following statements is correct?
Option A: $\underset{x\to 0}{\mathrm{lim}}\left(fog\right)\left(x\right)\mathrm{exists}.$
Option B: $\underset{x\to 0}{\mathrm{lim}}\left(gof\right)\left(x\right)\mathrm{exists}.$
Option C: $\underset{x\to {0}^{-}}{\mathrm{lim}}\left(fog\right)\left(x\right)=\underset{x\to {0}^{-}}{\mathrm{lim}}\left(gof\right)\left(x\right)$
Option D: $\underset{x\to {0}^{-}}{\mathrm{lim}}\left(fog\right)\left(x\right)=\underset{x\to {0}^{+}}{\mathrm{lim}}\left(gof\right)\left(x\right)$
VIEW SOLUTION
• Question 83
Let f(x) = [x], where [·] is the greatest integer function and g(x) = sin x be two real valued functions over .

Which of the following statements are correct?
1. (fof) (x) = f(x).
2. (gog) (x) = g(x) only when x = 0.
3. (go(fog))(x) can take only three values.
Select the correct answer using the code given below.
Option A: 1 and 2 only
Option B: 2 and 3 only
Option C: 1 and 3 only
Option D: 1, 2 and 3
VIEW SOLUTION
• Question 84
Let $f\left(x\right)=\left\{\begin{array}{cc}\frac{{e}^{x}-1}{x},& x>0\\ 0,& x=0\end{array}\right\$be a real valued function.
Which one of the following statements is correct?
Option A: f(x) is a strictly decreasing function in (0, x).
Option B: f(x) is a strictly increasing function in (0, x).
Option C: f(x) is neither increasing nor decreasing in (0, x).
Option D: f(x) is not decreasing in (0, x).
VIEW SOLUTION
• Question 85
Let $f\left(x\right)=\left\{\begin{array}{cc}\frac{{e}^{x}-1}{x},& x>0\\ 0,& x=0\end{array}\right\$be a real valued function.

Which of the following statements is/are correct?
1.  f(x) is right continuous at x = 0.
2. f(x) is discontinuous at x = 1.
Select the correct answer using the code given below.
Option A: 1 only
Option B: 2 only
Option C: Both 1 and 2
Option D: Neither 1 nor 2
VIEW SOLUTION
• Question 86
Consider the parabola $y={x}^{2}+7x+2$ and the straight line y = 3x – 3.

What are the coordinates of the point on the parabola which is closest to the straight line?
Option A: (0, 2)
Option B: (–2, –8)
Option C: (–7, 2)
Option D: (1, 10)
VIEW SOLUTION
• Question 87
Consider the parabola $y={x}^{2}+7x+2$ and the straight line y = 3x – 3.

What is the shortest distance from the above point on the parabola to the line?
Option A: $\frac{\sqrt{10}}{2}$
Option B: $\frac{\sqrt{10}}{5}$
Option C: $\frac{1}{\sqrt{10}}$
Option D: $\frac{\sqrt{5}}{4}$
VIEW SOLUTION
• Question 88
Let $f\left(x\right)=\left\{\begin{array}{cc}-2,& -3\le x\le 0\\ x-2,& 0 and $g\left(x\right)=f\left(\left|x\right|\right)+\left|f\left(x\right)\right|$

Which of the following statements is/are correct?
1. g(x) is differentiable at x = 0.
2. g(x) is differentiable at x = 2.
Select the correct answer using the code given below :
Option A: 1 only
Option B: 2 only
Option C: Both 1 and 2
Option D: Neither 1 nor 2
VIEW SOLUTION
• Question 89
Let $f\left(x\right)=\left\{\begin{array}{cc}-2,& -3\le x\le 0\\ x-2,& 0 and $g\left(x\right)=f\left(\left|x\right|\right)+\left|f\left(x\right)\right|$

What is the value of the differential coefficient of g(x) at x = –2?
Option A: –1
Option C: 1
Option D: 2
VIEW SOLUTION
• Question 90
Let $f\left(x\right)=\left\{\begin{array}{cc}-2,& -3\le x\le 0\\ x-2,& 0 and $g\left(x\right)=f\left(\left|x\right|\right)+\left|f\left(x\right)\right|$

Which of the following statements are correct?
1. g(x) is continuous at x = 0.
2. g(x) is continuous at x = 2.
3. g(x) is continuous at x = –1.
Select the correct answer using the code given below :
Option A: 1 and 2 only
Option B: 2 and 3 only
Option C: 1 and 3 only
Option D: 1, 2 and 3
VIEW SOLUTION
• Question 91
Let f(x) be a function such that What is  equal to?
Option A: 2f(1)
Option C: 2f (–1)
Option D: 4 f(1)
VIEW SOLUTION
• Question 92
What is$\int \frac{{x}^{4}-1}{{x}^{2}\sqrt{{x}^{4}+{x}^{2}+1}}dx$ equal to?
Option A: $\sqrt{\frac{{x}^{4}+{x}^{2}+1}{x}}+C$
Option B: $\sqrt{{x}^{4}+2-\frac{1}{{x}^{2}}}+C$
Option C: $\sqrt{{x}^{2}+\frac{1}{{x}^{2}}+1}+C$
Option D: $\sqrt{\frac{{x}^{4}-{x}^{2}+1}{x}}+C$
VIEW SOLUTION
• Question 93
What is the degree and order, respectively, of the differential equation satisfying ${e}^{y\sqrt{1-{x}^{2}}+x\sqrt{1-{y}^{2}}}=c{e}^{x},$
Option A: 1, 1
Option B: 1, 2
Option C: 2, 1
Option D: 2, 2
VIEW SOLUTION
• Question 94
What is the curve that passes through the point (1, 1) and whose slope is $\frac{2y}{x}?$
Option A: Circle
Option B: Parabola
Option C: Ellipse
Option D: Hyperbola
VIEW SOLUTION
• Question 95
If xdy = ydx + y2dy, y > 0 and y (1) = 1, then what is y (–3) equal to?
Option A: 3 only
Option B: –1 only
Option C: Both –1 and 3
Option D: Neither –1 nor 3
VIEW SOLUTION
• Question 96
What is the order of the differential equation
Option A: 1
Option B: 2
Option C: 3
Option D: Cannot be determined
VIEW SOLUTION
• Question 97
Which one of the following differential equations represents the family of straight lines, which are at unit distance from the origin?
Option A: ${\left(y-x\frac{dy}{dx}\right)}^{2}=1-{\left(\frac{dy}{dx}\right)}^{2}$
Option B: ${\left(y+x\frac{dy}{dx}\right)}^{2}=1+{\left(\frac{dy}{dx}\right)}^{2}$
Option C: ${\left(y-x\frac{dy}{dx}\right)}^{2}=1+{\left(\frac{dy}{dx}\right)}^{2}$
Option D: ${\left(y+x\frac{dy}{dx}\right)}^{2}=1-{\left(\frac{dy}{dx}\right)}^{2}$
VIEW SOLUTION
• Question 98
What is $\int {e}^{\mathrm{sin}x}\frac{x{\mathrm{cos}}^{3}x-\mathrm{sin}x}{{\mathrm{cos}}^{2}x}dx$ equal to?
Option A:
Option B:
Option C:
Option D:
VIEW SOLUTION
• Question 99
If , then what is the value of k?
Option A: 1/4
Option B: 1/2
Option C: 1
Option D: 2
VIEW SOLUTION
• Question 100
What is ${\int }_{1}^{3}\left|1-{x}^{4}\right|dx$ equal to?
Option A: –232/5
Option B: –116/5
Option C: 116/5
Option D: 232/5
VIEW SOLUTION
• Question 101
A special dice with numbers 1, –1, 2, –2, 0 and 3 is thrown thrice. What is the probability that the sum of the numbers occurring on the upper face is zero?
Option A: 1/72
Option B: 1/8
Option C: 7/72
Option D: 25/216
VIEW SOLUTION
• Question 102
There is 25% chance that it rains on any particular day. What is the probability that there is at least one rainy day within a period of 7 days?
Option A: $1-{\left(\frac{1}{4}\right)}^{7}$
Option B: ${\left(\frac{1}{4}\right)}^{7}$
Option C: ${\left(\frac{3}{4}\right)}^{7}$
Option D: $1-{\left(\frac{3}{4}\right)}^{7}$
VIEW SOLUTION
• Question 103
A salesman has a 70% chance to sell a product to any customer. The behaviour of successive customers is independent. If two customers A and B enter, what is the probability that the salesman will sell the product to customer A or B?
Option A: 0.98
Option B: 0.91
Option C: 0.70
Option D: 0.49
VIEW SOLUTION
• Question 104
A student appears for tests I, II and III. The student is considered successful if he passes in tests I, II or I, III or all the three. The probabilities of the student passing in tests I, II and III are m, n and 1/2 respectively. If the probability of the student to be successful is 1/2, then which one of the following is correct?
Option A: m (1 + n) = 1
Option B: n (1 + m) = 1
Option C: m = 1
Option D: mn = 1
VIEW SOLUTION
• Question 105
Three candidates solve a question. Odds in favour of the correct answer are 5 : 2, 4 : 3 and 3 : 4 respectively for three candidates. What is the probability that at least two of them solve the question correctly?
Option A: 134/343
Option B: 149/343
Option C: 60/343
Option D: 209/343
VIEW SOLUTION
• Question 106
Consider the following statements :
1. The mean and median are equal in symmetric distribution.
2. The range is the difference between the maximum value and the minimum value in the data.
3. The sum of the areas of the rectangles in the histogram is equal to the total area bounded by the frequency polygon and the horizontal axis.
Which of the above statements are correct?
Option A: 1 and 2 only
Option B: 2 and 3 only
Option C: 1 and 3 only
Option D: 1, 2 and 3
VIEW SOLUTION
• Question 107
The scores of 15 students in an examination were recorded as 10, 5, 8, 16, 18, 20, 8, 10, 16, 20, 18, 11, 16, 14 and 12. After calculating the mean, median and mode, an error is found. One of the values is wrongly written as 16 instead of 18. Which of the following measures of central tendency will change?
Option A: Mean and median
Option B: Median and mode
Option C: Mode only
Option D: Mean and mode
VIEW SOLUTION
• Question 108
For 10 observations on price (x) and supply (y), the following data was obtained :

and $\sum xy=3467.$

What is the line of regression of y on x?
Option A: y = 0.91x + 8.74
Option B: y = 1.02x + 8.74
Option C: y = 1.02x – 7.02
Option D: y = 0.91x – 7.02
VIEW SOLUTION
• Question 109
In a study of two groups, the following results were obtained :
 Group A Group B Sample size 20 25 Sample mean 22 23 Sample standard deviation 10 12

Which of the following statements is correct?
Option A: Group A is less variable than Group B because Group A’s standard deviation is smaller.
Option B: Group A is less variable than Group B because Group A’s sample size is smaller.
Option C: Group A is less variable than Group B because Group A’s sample mean is smaller.
Option D: Group A is less variable than Group B because Group A’s coefficient of variation is smaller.
VIEW SOLUTION
• Question 110
Consider the following statements with respect to the class intervals of grouped frequency distribution.
1. Class-intervals need not be mutually exclusive
2. Class-intervals should be exhaustive
3. Class-intervals need not be of equal width
Which of the above statements are correct?
Option A: 1 and 2 only
Option B: 2 and 3 only
Option C: 1 and 3 only
Option D: 1, 2 and 3
VIEW SOLUTION
• Question 111
A medicine is known to be 75% effective to cure a patient. If the medicine is given to 5 patients, what is the probability that at least one patient is cured by this medicine?
Option A: $\frac{1}{1024}$
Option B: $\frac{243}{1024}$
Option C: $\frac{1023}{1024}$
Option D: $\frac{781}{1024}$
VIEW SOLUTION
• Question 112
For two events A and B, it is given that P(A) = $\frac{3}{5},$ P(B) = $\frac{3}{10}$ and $\mathrm{P}\left(\mathrm{A}\left|\mathrm{B}\right\right)=\frac{2}{3}$. If $\overline{)\mathrm{A}}$ and $\overline{)\mathrm{B}}$ are the complementary events of A and B , then what is $\mathrm{P}\left(\overline{)\mathrm{A}}\left|\overline{)\mathrm{B}}\right\right)$ equal to?
Option A: $\frac{3}{7}$
Option B: $\frac{3}{4}$
Option C: $\frac{1}{3}$
Option D: $\frac{4}{7}$
VIEW SOLUTION
• Question 113
A machine had three parts A, B and C, whose chances of being defective are 0.02, 0.10 and 0.05, respectively. The machine stops working if any one of the parts becomes defective. What is the probability that the machine will not stop working?
Option A: 0.06
Option B: 0.16
Option C: 0.84
Option D: 0.94
VIEW SOLUTION
• Question 114
Three independent events, A1, A2 and A3 occur with probabilities  What is the probability that at least one of the three events occurs?
Option A: $\frac{1}{4}$
Option B: $\frac{2}{3}$
Option C: $\frac{3}{4}$
Option D: $\frac{1}{24}$
VIEW SOLUTION
• Question 115
Two variance, x and y are uncorrelated and have standard deviations σx and σy respectively. What is the correlation coefficient between x + y and xy?
Option A: $\frac{{\sigma }_{x}\sigma y}{{\sigma }_{x}^{2}+{\sigma }_{y}^{2}}$
Option B: $\frac{{\sigma }_{x}+{\sigma }_{y}}{2{\sigma }_{x}+{\sigma }_{y}}$
Option C: $\frac{{\sigma }_{x}^{2}-{\sigma }_{y}^{2}}{{\sigma }_{x}^{2}+{\sigma }_{y}^{2}}$
Option D: $\frac{{\sigma }_{y}-{\sigma }_{x}}{{\sigma }_{x}{\sigma }_{y}}$
VIEW SOLUTION
• Question 116
A random sample of 20 people is classified in the following table according to their ages :
 Age Frequency 15 – 25 2 25 – 35 4 35 – 45 6 45 – 55 5 55 – 65 3

What is the mean age of this group of people?
Option A: 41.0
Option B: 41.5
Option C: 42.0
Option D: 42.5
VIEW SOLUTION
• Question 117
If the covariance between x and y is 30, variance of x is 25 and variance of y is 144, then what is the correlation coefficient?
Option A: 0·4
Option B: 0·5
Option C: 0·6
Option D: 0·7
VIEW SOLUTION
• Question 118
A coin is tossed three times. Consider the following events :
B : Exactly one head appears
C : At least two heads appear

Which one of the following is correct?
Option A: $\left(A\cup Β\right)\cap \left(A\cup C\right)=B\cup C$
Option B: $\left(A\cap Β\text{'}\right)\cup \left(A\cap C\text{'}\right)=B\text{'}\cup C\text{'}$
Option C: $Α\cap \left(B\text{'}\cup C\text{'}\right)=A\cup B\cup C$
Option D: $Α\cap \left(B\text{'}\cup C\text{'}\right)=B\text{'}\cap C\text{'}$
VIEW SOLUTION
• Question 119
In a series of 3 one-day cricket matches between terms A and B of a college, the probability of team A winning or drawing are 1/3 and 1/6, respectively. If a win, loss or draw gives 2, 0 and 1 point, respectively, then what is the probability that team A will score 5 points in the series?
Option A: $\frac{17}{18}$
Option B: $\frac{11}{12}$
Option C: $\frac{1}{12}$
Option D: $\frac{1}{18}$
VIEW SOLUTION
• Question 120
Let the random variable X follow B (6, p). If 16 P(X = 4) = P(X = 2), then what is the value of p?
Option A: $\frac{1}{3}$
Option B: $\frac{1}{4}$
Option C: $\frac{1}{5}$
Option D: $\frac{1}{6}$
VIEW SOLUTION
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