NDA Mathematics 14th September 2025
DEFENCE 2025 Previous Year
3 hDuration
300Total Marks
120Questions
1Sections
Instructions
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Paper Structure
Mathematics
Mathematics
Q1.
mcq single
+2.5 / 0.83
The frequency distribution of the marks obtained by students in a Science examination is given below:
Marks
5-15
15-25
25-35
35-45
Number of
students
20
30
30
20
What is the arithmetic mean?
Q2.
mcq single
+2.5 / 0.83
The standard deviation of 100 observations is 10. If 20 is added to each observation, then what will be the new standard deviation?
Q3.
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Consider the following statements:
I. Mean and variance have the same unit of measurement.
II. Mean deviation and standard deviation have the same unit of measurement.
Which of the statements given above is/are correct?
Q4.
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In a class of 45 students, 34 like to play cricket and 26 like to play football. Further, each student likes to play at least one of the two games. How many students like to play exactly one game?
Q5.
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If A and B are non-empty subsets of a set, and $A ^ c$ and $ B ^ c$ represent their complements, then which of the following is/are correct?
I. A - B = $B ^ c - A ^ c$
II. $A - B ^ c = A ^ c - B$
Select the answer using the code given below.
Q6.
mcq single
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Let A = {- 3, - 2, - 1, 0, 1, 2, 3} and B = {0, 1, 4, 9} How many elements does the subset of AB corresponding to the relation R = {(x, y) : |x| < y} have, where x $\in$ A and y $\in$ B?
Q7.
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Set X contains 3n elements and set Y contains 2n elements, and they have n elements in common. How many elements does (X - Y) × (Y-X) have?
Q8.
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**Passage:**
For the following three (03) items:
Consider the function f(x) = x|x|
---
What is $\lim _{x\rightarrow -1}f(x)$ equal to?
Q9.
mcq single
+2.5 / 0.83
**Passage:**
For the following two (02) items:
Let
$f(x)=\begin{cases}\frac{1-\cos 2x}{x^{2}}&,x<0\\ 9&,x=0\\ \frac{\sqrt{x}}{\sqrt{(16+\sqrt{x})}-4}&,x>0\end{cases}$
---
What is $\lim _{x\rightarrow 0^{-}}f(x)$ equal to?
Q10.
mcq single
+2.5 / 0.83
**Passage:**
For the following three (03) items:
Consider the function f(x) = x|x|
---
Consider the following statements:
I. The function is increasing in the interval (-∞,∞),
II. The function is differentiable at x = 0
Which of the statements given above is/are correct?
Q11.
mcq single
+2.5 / 0.83
**Passage:**
For the following two (02) items:
Let
$f(x)=\begin{cases}\frac{1-\cos 2x}{x^{2}}&,x<0\\ 9&,x=0\\ \frac{\sqrt{x}}{\sqrt{(16+\sqrt{x})}-4}&,x>0\end{cases}$
---
What is $\lim _{x\rightarrow 0^{-}}f(x)$ equal to?
Q12.
mcq single
+2.5 / 0.83
**Passage:**
For the following two (02) items:
Consider the function
$f(x)=\begin{cases}4(5^{x}),&x<0\\ 8k+x,&x\ge 0\end{cases}$
---
If the function is continuous, then what is the value of k?
Q13.
mcq single
+2.5 / 0.83
**Passage:**
For the following two (02) items:
Let the function f(x) = |x - 3| + |x - 4| be defined on the interval [0, 5].
---
Consider the following statements:
I. The function is differentiable at x = 3
II. The function is differentiable at x = 4
Which of the statements given above is/are correct?
Q14.
mcq single
+2.5 / 0.83
**Passage:**
For the following two (02) items:
Let ( 6 + 10 + 14 +... up to m terms) =( 1 + 3 + 5 + 7 +... up to n terms)
where m < 25 and n < 25
---
What is the relation between m and n?
Q15.
mcq single
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If and $ (10+\log _{10}x), (10+\log _{10}y)$ are $(10 + log_{10}z)$ in AP, then consider the following statements:
I. The GM of x and z is $y ^ 2$
II. The AM of $ \log _{10}x$ and $ \log _{10}z$ is $log_{10} y $
Which of the statements given above is/are correct?
Q16.
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How many terms of the series 1 + 3 + 5 + 7 +... amount to a sum equal to 12345678987654321?
Q17.
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**Passage:**
For the following two (02) items:
Let ( 6 + 10 + 14 +... up to m terms) =( 1 + 3 + 5 + 7 +... up to n terms)
where m < 25 and n < 25
---
How many values of m are possible?
Q18.
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What is the sum of the binary numbers $(10110110) _2$ and $( 10001 1) _2$
Q19.
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How many terms are identical in the two APs 19, 21, 23,... up to 110 terms and 19, 22, 25, 28,... up to 75 terms?
Q20.
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What is the harmonic mean of the numbers C(10,3) C(10,4) C(10,5) C(10,6) and C(10,7) ?
Q21.
mcq single
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If $p ^ x = q ^ y = r ^ z$ where x, y and z are in GP, then consider the following statements:
I. p, q and rare in AP.
II. In p, In q and Inrare in GP.
Which of the statements given above is/are correct?
Q22.
mcq single
+2.5 / 0.83
What is the area of the parallelogram whose sides are represented by the vectors $\hat{i}+2\hat{j}+3\hat{k}$ and $2\hat{i}+\hat{j}+2\hat{k}$ ?
Q23.
mcq single
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Let $\vec{p}=\vec{a}-\vec{b},\vec{q}=\vec{a}+\vec{b}$. If $|\vec{a}|=|\vec{b}|=2$ and $\vec{a}\cdot \vec{b}=2$, then what is the value of $|\vec{p}\times \vec{q}|$?
Q24.
mcq single
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The position vectors of the vertices A, B, Cand D of a quadrilateral ABCD are given by $3\hat{i}+4\hat{j}-2\hat{k}$ , $4\hat{i}-4\hat{j}-3\hat{k}$, $2\hat{i}-3\hat{j}+2\hat{k}$ and $6\hat{i}-2\hat{j}+\hat{k}$ respectively.
What is the angle between the diagonals AC and BD of the quadrilateral?
Q25.
mcq single
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A force$\vec{F} = 2\hat{i} - \lambda\hat{j} + 5\hat{k}$ is applied at the point A(1, 2, 5) If its moment about the point B(- 1, - 2, 3) is $16\hat{i} - 6\hat{j} + 2\lambda\hat{k}$, then what is the value of $\lambda$ ?
Q26.
mcq single
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How many of the following can be a vector perpendicular to both the vectors $2\hat{i} - \hat{j} + \hat{k}$ and $\hat{i} + \hat{j} + 3\hat{k}$ ?
I. $4\hat{i}+5\hat{j}-3\hat{k}$
II. $-8\hat{i}-10\hat{j}+6\hat{k}$
III. $\frac{1}{50}(-4\hat{i}-5\hat{j}+3\hat{k})$
Select the correct answer.
Q27.
mcq single
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What is$\cot ^{-1}9+\>\mathrm{cosec}\>{}^{-1}\left(\frac{\sqrt{41}}{4}\right)$
equal to?
Q28.
mcq single
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What is the value of $\tan \left[\frac{1}{2}\sec ^{-1}\left(\frac{2}{\sqrt{3}}\right)\right]?$
Q29.
mcq single
+2.5 / 0.83
Under which of the following conditions may binomial distribution be used?
I. The number of trials is infinite and not fixed.
II. The trials are independent.
III. Each trial has two possible outcomes.
Select the correct answer using the code given below.
Q30.
mcq single
+2.5 / 0.83
Three faces of a die are black, two faces are white and one face is red. The die is tossed three times. What is the probability that the colours black, white and red appear in the first, second and third tosses respectively?
Q31.
mcq single
+2.5 / 0.83
The probability that a family owns a laptop is 0.68; that it also owns a desktop is 0.56. If the probability that it owns both is 0.48, then what is the probability that a randomly selected family owns a laptop or a desktop?
Q32.
mcq single
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An urn contains 10 white and 5 red balls. If two balls are drawn at random, then what is the probability that both the balls are red?
Q33.
mcq single
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Let X be a random variable following binomial distribution with parameters n = 5 and p = k Further, P(X = 1) = 0.4096 and P(X = 2) = 0.2048 . What is the value of k?
Q34.
mcq single
+2.5 / 0.83
If P(A) = 1/3 , P(B) = 1/2 and $ P(A\cap B)=1/4$ , then what is the value of $P(\overline{A}\cup B)$ ?
Q35.
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In a throw of three dice, what is the probability of getting one prime number, one composite number and one number which is neither prime nor composite?
Q36.
mcq single
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An integer is chosen at random from the first 50 integers. What is the probability that the integer is neither divisible by 5 nor 9?
Q37.
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The probability that a student passes Physics test is 2/3 and the probability that he passes both Physics test and English test is 11/15. The probability that he passes at least one test is 4/5 What is the probability that he passes English test?
Q38.
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A person X speaks the truth 4 out of 5 times and person Y speaks the truth 5 out of 6 times. What is the probability that they will contradict each other in stating the fact?
Q39.
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An event X can happen with probability pand event Y can happen with probability q. Further, X and Y are independent events. Which of the following statements is/are correct?
1. The probability that exactly one of the events happens is p + q - pq
II. The probability that at least one of the events happens is p + q - 2pq
Select the answer using the code given below.
Q40.
mcq single
+2.5 / 0.83
An urn contains 5 white, 6 red and 4 blue balls. Three balls are drawn at random. What is the probability that a white ball, a red ball and a blue ball are drawn?
Q41.
mcq single
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A fair coin is tossed 4 times. What is the probability that two heads do not occur consecutively?
Q42.
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Out of 50 consecutive natural numbers, two integers are chosen at random. What is the probability that their sum is odd?
Q43.
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If P(A) = 0.3 , P(B) = 0.4 and P( A |B)=0.5, then what is the value P( B |A) ?
Q44.
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In a sample survey of a village, the probability that a farmer is in debt is 0.60. What is the probability that three randomly selected farmers are all in debt (assume independence of events)?
Q45.
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Which one of the following is the perpendicular form of the straight line $\sqrt{3}x+2y=7$ ?
Q46.
mcq single
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For different values of m, the equation 4y=mx-m+ 2 represents
Q47.
mcq single
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What is the value of sin $\theta $ if $\theta $ is the acute angle between the lines whose equations are px + qy = p + q and p(x - y) + q(x + y) = 2q
Q48.
mcq single
+2.5 / 0.83
What is the sum of the intercepts of the line
$\frac{x}{a^{2}}+\frac{y}{b^{2}}=\frac{2}{a^{2}+b^{2}} $ on the coordinate axes?
Q49.
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What is the area of the region bounded by |x| <= 2k and |y| <= k where k is a positive real number?
Q50.
mcq single
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If the vertices Band D of a square ABCD are (2, 3) and (4, 1) respectively, then what is the area of the square?
Q51.
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The equation of the locus of a point equidistant from the points (a, b) and (c, d) is (a - c) x + (b - d) y + k = 0 What is the value of k?
Q52.
mcq single
+2.5 / 0.83
**Passage:**
For the following two (02) items:
Let sin A + sin B = p and cos A + cos B = q
---
What is $\frac{p^{2}-q^{2}}{p^{2}+q^{2}}$ equal to?
Q53.
mcq single
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If $0\le x\le \frac{\pi }{2}$ then what is the number of values of x satisfying the equation $\tan x+\sec x=2\cos x$
Q54.
mcq single
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**Passage:**
For the following two (02) items:
Let p = cosec 20° and q = cosec 70°.
---
What is $\frac{p^{2}+q^{2}}{p^{2}q^{2}}$ equal to?
Q55.
mcq single
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How many values of $\theta $, where$- \pi < \theta < \pi$ satisfy both the equations $\cot \theta =-\sqrt{3}$ and $\csc \theta =-2$ simultaneously?
Q56.
mcq single
+2.5 / 0.83
**Passage:**
For the following two (02) items:
Let $\cos (2x+3y)=\frac{1}{2}$ and $\cos (3x+2y)=\frac{\sqrt{3}}{2}$ where $-\pi <(2x+3y)<\pi $ and $-\pi <(3x+2y)<\pi .$
---
How many values does (x + y) have?
Q57.
mcq single
+2.5 / 0.83
**Passage:**
For the following two (02) items:
Let p = cosec 20° and q = cosec 70°.
---
What is $(\frac{\sqrt{3}p}{4}-\frac{q}{4})$ equal to?
Q58.
mcq single
+2.5 / 0.83
**Passage:**
For the following two (02) items:
Let sin A + sin B = p and cos A + cos B = q
---
What is p/q equal to?
Q59.
mcq single
+2.5 / 0.83
If $x+\frac{1}{x}=2\cos \theta$ then what is $x^{3}+\frac{1}{x^{3}}$ equal to?
Q60.
mcq single
+2.5 / 0.83
**Passage:**
For the following two (02) items:
Let $\cos (2x+3y)=\frac{1}{2}$ and $\cos (3x+2y)=\frac{\sqrt{3}}{2}$ where $-\pi <(2x+3y)<\pi $ and $-\pi <(3x+2y)<\pi .$
---
How many values does (y-x) have?
Q61.
mcq single
+2.5 / 0.83
**Passage:**
For the following two (02) items:
Consider the function $f(x)=1-\sqrt[3]{(x-1)^{2}}$
---
The function has
Q62.
mcq single
+2.5 / 0.83
Consider the following statements:
Statement-I: The function $f(x)=\frac{x^{3}+128}{x}$ has a minimum value 48 at x = 4
Statement-II: As x increases through 4, f'(x) changes sign from positive to negative.
Which one of the following is correct in respect of the above statements?
Q63.
mcq single
+2.5 / 0.83
A wire of length 20 cm is to be bent into a rectangle. Which of the following statements is/are correct?
I. The rectangle of the largest area is the square.
II. It is possible to form a rectangle of an area of$ 27c m ^ 2$
Select the answer using the code given below.
Q64.
mcq single
+2.5 / 0.83
**Passage:**
For the following three (03) items:
Let $y=f(x)=\frac{x\sin ^{-1}x}{\sqrt{1-x^{2}}}+\ln \sqrt{1-x^{2}}.$
---
What is the slope of the tangent to the curve y = f(x) at x = 0.5
Q65.
mcq single
+2.5 / 0.83
Consider the following statements regarding the function f(x) = 1/(x - 5)
Statement-I: f(x) is decreasing on the intervals x < 5 and x > 5
Statement-II: f''(x) > 0 for all x ≠ 5.
Which one of the following is correct in respect of the above statements?
Q66.
mcq single
+2.5 / 0.83
**Passage:**
For the following three (03) items:
Consider the function f(x) = x|x|
---
What is the area bounded by the curve f(x) the x-axis and the lines x = - 2 and x = 1
Q67.
mcq single
+2.5 / 0.83
**Passage:**
For the following two (02) items:
A plane P is parallel to the line having direction ratios $\langle1, 3, 2\rangle $ and contains the line of intersection of the planes 6x + 4y - 5z = 2 and x - 2y + 3z = 0
---
What is the equation of the plane P?
Q68.
mcq single
+2.5 / 0.83
**Passage:**
For the following two (02) items:
A plane P is parallel to the line having direction ratios $\langle1, 3, 2\rangle $ and contains the line of intersection of the planes 6x + 4y - 5z = 2 and x - 2y + 3z = 0
---
Which of the following are the direction ratios of the line of intersection of the given planes?
Q69.
mcq single
+2.5 / 0.83
**Passage:**
For the following two (02) items:
Let A(1, - 1, 0) B(- 2, 1, 8) and C(- 1, 2, 7) are three consecutive vertices of a parallelogram ABCD.
---
If angle BCD is $\theta $ then what is $ cos^2 \theta$ equal to?
Q70.
mcq single
+2.5 / 0.83
**Passage:**
For the following two (02) items:
Suppose S is the sphere with the smallest radius that passes through the points A(1, 0, 0) B(0, 1, 0) and C(0, 0, 1)
---
On which one of the following planes does the centre of S lie?
Q71.
mcq single
+2.5 / 0.83
**Passage:**
For the following two (02) items:
Suppose S is the sphere with the smallest radius that passes through the points A(1, 0, 0) B(0, 1, 0) and C(0, 0, 1)
---
What is the radius of S?
Q72.
mcq single
+2.5 / 0.83
**Passage:**
For the following two (02) items:
Let A(1, - 1, 0) B(- 2, 1, 8) and C(- 1, 2, 7) are three consecutive vertices of a parallelogram ABCD.
---
What is the fourth vertex D?
Q73.
mcq single
+2.5 / 0.83
Consider the following statements in respect of the equation $x ^ 2 + 3y = 0$
I. The equation represents the equation to parabola that opens upwards.
II. The axis of the parabola is x = 0
III. The equation of the latus rectum is 4y - 3 = 0
How many of the statements given above are correct?
Q74.
mcq single
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Which one of the following is the greatest coefficient in the expansion of $(1 + x) ^ {100}$ ?
Q75.
mcq single
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What is the remainder when $5 ^ {99}$ is divided by 13?
Q76.
mcq single
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**Passage:**
For the following two (02) items:
Let $\alpha$ and $\beta $ be the roots of the quadratic equation
$(x^{2}+(\log _{0.5}(\alpha ^{2}))x+(\log _{0.5}(\alpha ^{2}))^{4}=0$
where $a ^ 2 \ne1$ and $\log _{0.5}(\alpha ^{2})>0$ Further, $\beta ^{2}=\alpha (\log _{\alpha ^{2}}(0.5))$
---
What is ẞ equal to?
Q77.
mcq single
+2.5 / 0.83
**Passage:**
For the following two (02) items:
Let $p=\sum_{j=1}^{n}\log_{10}2^{j} and q=\sum_{j=1}^{n}\log_{10}5^{j}$
---
If p + q = 66 then which one of the following is correct?
Q78.
mcq single
+2.5 / 0.83
**Passage:**
For the following two (02) items:
Let $p=\sum_{j=1}^{n}\log_{10}2^{j} and q=\sum_{j=1}^{n}\log_{10}5^{j}$
---
If p + q = 15 then what is equal to? q - p
Q79.
mcq single
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Let 1, ω, $\omega ^ 2$ be three cube roots of unity. If x = a + b $y=a\omega +b\omega ^{2}, z=a\omega ^{2}+b\omega$ then what is $ x ^ 2 + y ^ 2 + z ^ 2$ equal to?
Q80.
mcq single
+2.5 / 0.83
If $\alpha =\frac{-1+\sqrt{-3}}{2}$ then what is the value of $(1+\alpha ^{19}-\alpha ^{35})^{100}-(1-3\alpha ^{25}+\alpha ^{38})^{50}? $
Q81.
mcq single
+2.5 / 0.83
If $(\frac{1-i}{1+i})^{2m}(\frac{1+i}{1-i})^{2n}=1$where $i=\sqrt{-1}$ then what is the smallest positive value of (m - n)?
Q82.
mcq single
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What is the distance between the foci of the hyperbola $x ^ 2 - 4y ^ 2 = 1$
Q83.
mcq single
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**Passage:**
For the following two (02) items:
Consider the function
$f(x)=\frac{x}{1-x}(x>0,x\ne 1)$
---
What is $\frac{f(x)}{f(x+1)}$ equal to?
Q84.
mcq single
+2.5 / 0.83
**Passage:**
For the following two (02) items:
Consider the function $f(x)=\frac{10^{x}-10^{-x}}{10^{x}+10^{-x}}$
---
What is the inverse of the function?
Q85.
mcq single
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**Passage:**
For the following two (02) items:
Consider the function $f(x)=\frac{10^{x}-10^{-x}}{10^{x}+10^{-x}}$
---
What is$f\circ f\circ f\circ f\circ f(0)$ equal to?
Q86.
mcq single
+2.5 / 0.83
**Passage:**
For the following two (02) items:
Consider the function $f(x)=1-\sqrt[3]{(x-1)^{2}}$
---
What is the domain of the function?
Q87.
mcq single
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**Passage:**
For the following two (02) items:
Consider the function
$f(x)=\frac{x}{1-x}(x>0,x\ne 1)$
---
What is $ (1-x)f(\sqrt{x})+xf(\sqrt{x}+1)$ equal to?
Q88.
mcq single
+2.5 / 0.83
Consider the following statements:
I. $y=xe^{2x}$ is the solution of $\frac{dy}{dx}=y(2+\frac{1}{x})$
II. $y=x\ln |x|+cx$ is the solution of $\frac{dy}{dx}=\frac{x+y}{x}$
Which of the statements given above is/are correct?
Q89.
mcq single
+2.5 / 0.83
What is the degree of the differential equation
$(\frac{d^{2}y}{dx^{2}})^{\frac{3}{2}}=(\frac{dy}{dx})^{\frac{5}{2}}$
Q90.
mcq single
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If k is an arbitrary constant, then what is the general solution of the equation $(x+y)^{2}\frac{dy}{dx}=k^{2}$
Q91.
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The circle $ x ^ 2 + y ^ 2 - 2kx - 2ky + k ^ 2 = 0$ touches the x-axis at P and y-axis at Q. What is PQ equal to?
Q92.
mcq single
+2.5 / 0.83
**Passage:**
For the following two (02) items:
Consider the equation abx² + bcx + ca = cax² + abx + bc
---
If the roots of the equation are equal, then which one of the following is correct?
Q93.
mcq single
+2.5 / 0.83
**Passage:**
For the following two (02) items:
Let $\alpha$ and $\beta $ be the roots of the quadratic equation
$(x^{2}+(\log _{0.5}(\alpha ^{2}))x+(\log _{0.5}(\alpha ^{2}))^{4}=0$
where $a ^ 2 \ne1$ and $\log _{0.5}(\alpha ^{2})>0$ Further, $\beta ^{2}=\alpha (\log _{\alpha ^{2}}(0.5))$
---
What is the relation between α and β
Q94.
mcq single
+2.5 / 0.83
**Passage:**
For the following two (02) items:
Let $f(x) = a x ^ 2 + bx + c$ be a quadratic polynomial such that f(1) = f(4) = 2 Further, 2 is a root of f(x) = 0
---
What is (a + b + c) equal to?
Q95.
mcq single
+2.5 / 0.83
**Passage:**
For the following two (02) items:
Consider the equation abx² + bcx + ca = cax² + abx + bc
---
If the roots of the equation are equal, then a, b, c are in
Q96.
mcq single
+2.5 / 0.83
**Passage:**
For the following two (02) items:
Let $f(x) = a x ^ 2 + bx + c$ be a quadratic polynomial such that f(1) = f(4) = 2 Further, 2 is a root of f(x) = 0
---
What is the other root of f(x) = 0
Q97.
mcq single
+2.5 / 0.83
**Passage:**
For the following two (02) items:
Consider the function
$f(x)=\begin{cases}4(5^{x}),&x<0\\ 8k+x,&x\ge 0\end{cases}$
---
What is f'(-1) equal to?
Q98.
mcq single
+2.5 / 0.83
**Passage:**
For the following two (02) items:
Let the function f(x) = |x - 3| + |x - 4| be defined on the interval [0, 5].
---
What is $\frac{dy}{dx}$ at x = 3.5 equal to?
Q99.
mcq single
+2.5 / 0.83
**Passage:**
For the following three (03) items:
Let $y=f(x)=\frac{x\sin ^{-1}x}{\sqrt{1-x^{2}}}+\ln \sqrt{1-x^{2}}.$
---
What is $\frac{d^{2}y}{dx^{2}}$ at x = 0 equal to?
Q100.
mcq single
+2.5 / 0.83
**Passage:**
For the following three (03) items:
Let $y=f(x)=\frac{x\sin ^{-1}x}{\sqrt{1-x^{2}}}+\ln \sqrt{1-x^{2}}.$
---
If x = sin $\theta$ then what is $\frac{dy}{dx}$ equal to?
Q101.
mcq single
+2.5 / 0.83
Let y=x! and z = (2x)! If (z / y) = 120 then what is the value of (3x)!?
Q102.
mcq single
+2.5 / 0.83
**Passage:**
For the following two (02) items:
There are 8 points on a plane out of which 4 points are collinear.
---
How many quadrilaterals can be formed by joining these points?
Q103.
mcq single
+2.5 / 0.83
How many 4-digit numbers that are divisible by 4 can be formed using the digits 1, 2, 3 and 4 (repetition of digits is not allowed)?
Q104.
mcq single
+2.5 / 0.83
**Passage:**
For the following two (02) items:
There are 8 points on a plane out of which 4 points are collinear.
---
How many triangles can be formed by joining these points?
Q105.
mcq single
+2.5 / 0.83
Let n be a natural number. The number of consecutive zeros at the end of the expansion of n! is exactly 2. How many values of n are possible?
Q106.
mcq single
+2.5 / 0.83
What is $\int _{n}^{n+1}(x-[x])dx$, where [.] is the greatest integer function and n is natural number?
Q107.
mcq single
+2.5 / 0.83
If $I_{1}=\int _{e}^{e^{2}}\frac{dx}{ln~x}$ and $I_{2}=\int _{1}^{2}\frac{e^{x}}{x}dx$ then which one of the following is correct?
Q108.
mcq single
+2.5 / 0.83
Consider the following in respect of non-singular matrices A and B:
I. $(AB) ^ {- 1} = A ^ {- 1} B ^ {- 1}$
II. $ (BA)(AB)^{-1}=I$ = I where I is the identity matrix
III. $ (AB)^{T}=A^{T}B^{T} $
How many of the above are correct?
Q109.
mcq single
+2.5 / 0.83
Consider the following statements:
Statement-l: If X is an nn matrix, then det(mX) = $m ^ n$ det(X), where m is a scalar.
Statement-II: If Y is a matrix obtained from X by multiplying any row or column by a scalar m, then det (Y) = m det (X).
Which one of the following is correct in respect of the above statements?
Q110.
mcq single
+2.5 / 0.83
The value of the determinant $\left|\begin{matrix}a&b&c\\ l&m&n\\ p&q&r\end{matrix}\right|$ is equal to
Q111.
mcq single
+2.5 / 0.83
The system of equations 15y - 10x + 50 = 0, 2x - 3y - 5 = 0
Q112.
mcq single
+2.5 / 0.83
**Passage:**
For the following two (02) items:
Let
$A=\left[\begin{matrix}\cos \theta &\sin \theta \\ -\sin \theta &\cos \theta \end{matrix}\right] $
---
What is the value of the determinant of the matrix $A ^ 4$
Q113.
mcq single
+2.5 / 0.83
In obtaining the solution of the system of equations x + y + z = 7 x + 2y + 3z = 16 and x + 3y + 4z = 22 by Cramer's rule, the value of y is obtained by dividing D by $D_{2}$ where
$D=\left|\begin{matrix}1&1&1\\ 1&2&3\\ 1&3&4\end{matrix}\right|$ What is the value of the determinant $D_{2}$ ?
Q114.
mcq single
+2.5 / 0.83
If a, b, c are the sides of a triangle ABC and p is the perimeter of the triangle, then what is equal to? det $\left|\begin{matrix}p+c&a&b\\ c&p+a&b\\ c&a&p+b\end{matrix}\right|$
Q115.
mcq single
+2.5 / 0.83
What is the value of the determinant of the inverse of the matrix $\begin{bmatrix} -4 & -5 \\ 2 & 2 \end{bmatrix} ?$
Q116.
mcq single
+2.5 / 0.83
**Passage:**
For the following two (02) items:
Let
$A=\left[\begin{matrix}\cos \theta &\sin \theta \\ -\sin \theta &\cos \theta \end{matrix}\right] $
---
What is $[adj A ]^ {-1}$ equal to?
Q117.
mcq single
+2.5 / 0.83
Consider the following statements about
the matrix $M=\left[\begin{matrix}71&23&48\\ 57&28&29\\ 65&17&48\end{matrix}\right]$
Statement-I: The inverse of M does not exist.
Statement-II: M is non-singular.
Which one of the following is correct in respect of the above statements?
Q118.
mcq single
+2.5 / 0.83
**Passage:**
For the following two (02) items:
Let $u=\int e^{x}cos~x~dx $ and $v=\int e^{x}sin~x~dx.$
---
Consider the following:
I. $\frac{du}{dx}=-v $
II. $\frac{dv}{dx}=-u$
Which of the above is/are correct?
Q119.
mcq single
+2.5 / 0.83
**Passage:**
For the following two (02) items:
Let $u=\int e^{x}cos~x~dx $ and $v=\int e^{x}sin~x~dx.$
---
What is u + v equal to?
Q120.
mcq single
+2.5 / 0.83
What is $\int \frac{dx}{10^{x}+10^{-x}}$ equal to?