NDA Mathematics 16 April 2023

Q1. mcq single +2.5 / 0.83

Let P be the median, Q be the mean and R be the mode of observations x~1, ~x~2,~ x~3~, .....x~n~. Let $S=\sum\limits_{i=1}^n\left(2 x_i-a\right)^2$ S takes minimum value, when a is equal to

Q2. mcq single +2.5 / 0.83

Consider the following frequency distribution : x 1 2 3 5 f 4 6 9 7 What is the value of median of the distribution ?

Q3. mcq single +2.5 / 0.83

The observations 4, 1, 4, 3, 6, 2, 1, 3, 4, 5, 1, 6 are outputs of 12 dices thrown simultaneously. If m and M are means of lowest 8 observations and highest 4 observations respectively, then what is (2m + M) equal to ?

Q4. mcq single +2.5 / 0.83

The central angles p, q, r and s (in degrees) of four sectors in a Pie Chart satisfy the relation 9p = 3q = 2r = 6s. What is the value of 4p - q ?

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The mean and variance of five observations are 14 and 13.2 respectively. Three of the five observations are 11, 16 and 20. What are the other two observations ?

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Let P, Q, R represent mean, median and mode. If for some distribution $5 P=4 Q=\frac{R}{2}$ then what is $\frac{P+Q}{2 P+0.7 R}$ equal to ?

Q7. mcq single +2.5 / 0.83

A die is thrown 10 times and obtained the following outputs : 1, 2, 1, 1, 2, 1, 4, 6, 5, 4 What will be the mode of data so obtained ?

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For data -1, 1, 4, 3, 8, 12, 17, 19, 9, 11; if M is the median of first 5 observations and N is the median of last five observations, then what is the value of 4M - N ?

Q9. mcq single +2.5 / 0.83

A mapping f : A → B defined as $f(x)=\frac{2 x+3}{3 x+5}, x \in A$ If f is to be onto, then what are A and B equal to ?

Q10. mcq single +2.5 / 0.83

What is the binary number equivalent to decimal number 1011 ?

Q11. mcq single +2.5 / 0.83

What is $\lim\limits_{x \rightarrow 1} \frac{x^9-1}{x^3-1}$ equal to ?

Q12. mcq single +2.5 / 0.83

What is $\lim\limits_{x \rightarrow 5} \frac{5-x}{|x-5|}$ equal to ?

Q13. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next three (03) items that follow : Let f(x) = $\left|\begin{array}{ccc} \cos x & x & 1 \\ 2 \sin x & x^2 & 2 x \\ \tan x & x & 1 \end{array}\right|$ --- What is $\lim _{x \rightarrow 0} \frac{f(x)}{x}$ equal to ?

Q14. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow Let f(x) = $\left \{ \begin{matrix} x + 6, x \le 1 \\\ px + q, 1 < x < 2 \\\ 5x , x \ge 2 \end{matrix} \right.$ and f(x) is continuous --- What is the value of q ?

Q15. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next three (03) items that follow : Let f(x) = $\left|\begin{array}{ccc} \cos x & x & 1 \\ 2 \sin x & x^2 & 2 x \\ \tan x & x & 1 \end{array}\right|$ --- What is $\lim _{x \rightarrow 0} \frac{f(x)}{x^2}$ equal to ?

Q16. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow Let f(x) = $\left \{ \begin{matrix} x + 6, x \le 1 \\\ px + q, 1 < x < 2 \\\ 5x , x \ge 2 \end{matrix} \right.$ and f(x) is continuous --- What is the value of p ?

Q17. mcq single +2.5 / 0.83

In an AP, the first term is x and the sum of the first n terms is zero. What is the sum of next m terms ?

Q18. mcq single +2.5 / 0.83

p, q, r and s are in AP such that p + s = 8 and qr = 15. What is the difference between largest and smallest numbers ?

Q19. mcq single +2.5 / 0.83

If G is the geometric mean of numbers 1, 2, 2^(2), 2^(3),.....2^(n-1), then what is the value of 1 + 2log~2~G ?

Q20. mcq single +2.5 / 0.83

If $2^{\frac{1}{c}}, 2^{\frac{b}{a c}}, 2^{\frac{1}{a}}$ are in GP, then which one of the following is correct ?

Q21. mcq single +2.5 / 0.83

If log~x~a, a^(x) and log~b~x are in GP, then what is x equal to ?

Q22. mcq single +2.5 / 0.83

The first and the second terms of an AP are $\frac{5}{2}$ and $\frac{23}{12}$ respectively. If n^(th) term is the largest negative term, what is the value of n ?

Q23. mcq single +2.5 / 0.83

**Passage:** Consider the following for the item that follow : Let a vector $\vec{a}=4 \hat{i}-8 \hat{j}+\hat{k}$ make angles α, β, γ with the positive directions of x, y, z axes respectively. --- What is cosα equal to ?

Q24. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : Let $\vec{a}=3 \hat{i}+3 \hat{j}+3 \hat{k} $ and $ \vec{c}=\hat{j}-\hat{k} \text {. Let } \vec{b}$ be such that $\vec{a} \cdot \vec{b}=27 $ and $ \vec{a} \times \vec{b}=\overrightarrow{9 c}$ --- What is $\vec{b}$ equal to ?

Q25. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : Let a vector $\vec{a}=4 \hat{i}-8 \hat{j}+\hat{k}$ make angles α, β, γ with the positive directions of x, y, z axes respectively. --- What is cos2β + cos2γ equal to ?

Q26. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : The position vectors of two points A and B are î - ĵ and ĵ + k̂ respectively. --- Consider the following points : 1. (-1, -3, 1) 2. (-1, 3, 2) 3. (-2, 5, 3) Which of the above points lie on the line joining A and B ?

Q27. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : Let $\vec{a}=3 \hat{i}+3 \hat{j}+3 \hat{k} $ and $ \vec{c}=\hat{j}-\hat{k} \text {. Let } \vec{b}$ be such that $\vec{a} \cdot \vec{b}=27 $ and $ \vec{a} \times \vec{b}=\overrightarrow{9 c}$ --- What is the angle between $(\vec{a}+\vec{b})$ and $\vec{c}$ ?

Q28. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : The position vectors of two points A and B are î - ĵ and ĵ + k̂ respectively. --- What is the magnitude of $\overrightarrow{A B}$ ?

Q29. mcq single +2.5 / 0.83

One bag contains 3 white and 2 black balls, another bag contains 2 white and 3 black balls. Two balls are drawn from the first bag and put it into the second bag and then a ball is drawn from the second bag. What is the probability that it is white ?

Q30. mcq single +2.5 / 0.83

Let A and B be two independent events such that P(A̅) = 0.7, P(B̅) = k, P(A ∪ B) = 0.8, what is the value of k ?

Q31. mcq single +2.5 / 0.83

Three dice are thrown. What is the probability that each face shows only multiples of 3 ?

Q32. mcq single +2.5 / 0.83

Let X and Y be two random variables such that X + Y = 100. If X follows Binomial distribution with parameters n = 100 and p = $\frac{4}{5}$, what is the variance of Y?

Q33. mcq single +2.5 / 0.83

A biased coin with the probability of getting head equal to $\frac{1}{4}$ is tossed five times. What is the probability of getting tail in all the first four tosses followed by head ?

Q34. mcq single +2.5 / 0.83

A coin is biased so that heads comes up thrice as likely as tails. In four independent tosses of the coin, what is probability of getting exactly three heads ?

Q35. mcq single +2.5 / 0.83

A natural number n is chosen from the first 50 natural numbers. What is the probability that $n+\frac{50}{n}<50 $ ?

Q36. mcq single +2.5 / 0.83

What is the probability that the month of December has 5 Sundays ?

Q37. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : A flagstaff 20 m long standing on a pillar 10 m high subtends an angle tan^(-1)(0.5) at a point P on the ground. Let θ be the angle subtended by the pillar at this point P. --- What is a possible value of tan θ ?

Q38. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : A flagstaff 20 m long standing on a pillar 10 m high subtends an angle tan^(-1)(0.5) at a point P on the ground. Let θ be the angle subtended by the pillar at this point P. --- If x is the distance of P from the bottom of the pillar, then consider the following statements : 1. x can take two values which are in the ratio 1 : 3 2. x can be equal to the height of the flagstaff Which of the statements given above is/are correct?

Q39. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : A quadrilateral is formed by the lines x = 0, y = 0, x + y = 1 and 6x + y = 3. --- What is the equation of diagonal through origin ?

Q40. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : A quadrilateral is formed by the lines x = 0, y = 0, x + y = 1 and 6x + y = 3. --- What is the equation of other diagonal ?

Q41. mcq single +2.5 / 0.83

If two lines of regression are x + 4y + 1 = 0 and 4x + 9y + 7 = 0, then what is the value of x when y = -3 ?

Q42. mcq single +2.5 / 0.83

A bivariate data set contains only two points (-1, 1) and (3, 2). What will be the line of regression of y on x ?

Q43. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : Let $x=\frac{\sin ^2 A+\sin A+1}{\sin A}$ where 0 < A ≤ $\frac{\pi}{2}$ --- What is the minimum value of x ?

Q44. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : Let sin β be the GM of sin α and cos α; tan γ be the AM of sin α and cos α. --- What is cos 2β equal to ?

Q45. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : Given that sinx + cosx + tanx + cotx + secx + cosecx = 7 --- The given equation can be reduced to

Q46. mcq single +2.5 / 0.83

If f(x) = x(4x^(2) - 3), then what is f(sinθ) equal to ?

Q47. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : Given that m(θ) = cot^(2)θ + n^(2)tan^(2)θ + 2n, where n is a fixed positive real number. --- Under what condition does m attain the least value ?

Q48. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : Let sin β be the GM of sin α and cos α; tan γ be the AM of sin α and cos α. --- What is the value of sec^(2)γ?

Q49. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : Given that sinx + cosx + tanx + cotx + secx + cosecx = 7 --- If sin2x = a - b√c, where a and b are natural numbers and c is prime number, then what is the value of a - b + 2c ?

Q50. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : Let $x=\frac{\sin ^2 A+\sin A+1}{\sin A}$ where 0 < A ≤ $\frac{\pi}{2}$ --- At what value of A does x attain the minimum value ?

Q51. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : Given that m(θ) = cot^(2)θ + n^(2)tan^(2)θ + 2n, where n is a fixed positive real number. --- What is the least value of m(θ) ?

Q52. mcq single +2.5 / 0.83

Consider the following statements : 1. f(x) = In x is increasing in (0, ∞) 2. $g(x)=e^x+e^{\frac{1}{x}} $ is decreasing in (0, ∞) Which of the statements given above is/are correct ?

Q53. mcq single +2.5 / 0.83

What is the derivative of sin^(2) x with respect to cos^(2)x ?

Q54. mcq single +2.5 / 0.83

For what value of m with m < 0, is the area bounded by the lines y = x, y = mx and x = 2 equal to 3 ?

Q55. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : Let A(1, -1, 2) and B(2, 1, -1) be the end points of the diameter of the sphere x^(2) + y^(2) + z^(2) + 2ux + 2vy + 2wz - 1 = 0. --- What is u + v + w equal to ?

Q56. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : Consider two lines whose direction ratios are (2, -1, 2) and (k, 3, 5). They are inclined at an angle $\frac{\pi}{4}$ --- What is the value of k?

Q57. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : Let A(1, -1, 2) and B(2, 1, -1) be the end points of the diameter of the sphere x^(2) + y^(2) + z^(2) + 2ux + 2vy + 2wz - 1 = 0. --- P(x, y, z) is any point on the sphere, then what is PA^(2) + PB^(2) equal to ?

Q58. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : Consider two lines whose direction ratios are (2, -1, 2) and (k, 3, 5). They are inclined at an angle $\frac{\pi}{4}$ --- What are the direction ratios of a line which is perpendicular to both the lines ?

Q59. mcq single +2.5 / 0.83

If the sixth term in the binomial expansion of $\left(x^{-\frac{8}{3}}+x^2 \log _{10} x\right)^8$ is 5600, then what is the value of x ?

Q60. mcq single +2.5 / 0.83

Consider the following statements for a fixed natural number n: 1. C(n, r) is greatest if n = 2r 2. C(n, r) is greatest if n = 2r - 1 and n = 2r + 1 Which of the statements given above is/are correct ?

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How many terms are there in the expansion of (3x - y)^(4)(x + 3y)^(4) ?

Q62. mcq single +2.5 / 0.83

If z is a complex number such that $\frac{z-1}{z+1}$ is purely imaginary, then what is |z| equal to ?

Q63. mcq single +2.5 / 0.83

If 2 - i√3 where i = √-1 is a root of the equation x2 + ax + b = 0, then what is the value of (a + b)?

Q64. mcq single +2.5 / 0.83

If ω is a non-real cube root of 1, then what is the value of $\left|\frac{1-\omega}{\omega+\omega^2}\right| ?$

Q65. mcq single +2.5 / 0.83

If α and β are the distinct roots of equation x2 - x + 1 = 0, then what is the value of $\left|\frac{\alpha^{100}+\beta^{100}}{\alpha^{100}-\beta^{100}}\right|$ ?

Q66. mcq single +2.5 / 0.83

If $z=\frac{1+i √{3}}{1-i √{3}}$ where i = √-1 then what is the argument of z ?

Q67. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : Let f(x) = sin[π^(2)]x + cos[-π^(2)]x where [.] is a greatest ‘integer function --- What is $f\left(\frac{\pi}{4}\right)$ equal to ?

Q68. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : Consider the function f(x) = |x - 2| + |3 - x| + |4 - x|, where x ∈ R. --- What is the minimum value of the function ?

Q69. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next three (03) items that follow : Let f(x) = $\left|\begin{array}{ccc} \cos x & x & 1 \\ 2 \sin x & x^2 & 2 x \\ \tan x & x & 1 \end{array}\right|$ --- What is f(0) equal to ?

Q70. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : Let f(x) = sin[π^(2)]x + cos[-π^(2)]x where [.] is a greatest ‘integer function --- What is $f\left(\frac{\pi}{2}\right)$ equal to ?

Q71. mcq single +2.5 / 0.83

What is the range of the function f(x) = x + |x| if the domain is the set of real numbers ?

Q72. mcq single +2.5 / 0.83

If f(x) = x^(2) + 2 and g(x) = 2x - 3, then what is (fog)(1) equal to ?

Q73. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : Consider the function f(x) = |x - 2| + |3 - x| + |4 - x|, where x ∈ R. --- At what value of x does the function attain minimum value ?

Q74. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : Suppose E is the differential equation representing family of curves y^(2) = 2cx + 2c√c where c is a positive parameter. --- What is the degree of the differential equation ?

Q75. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : Suppose E is the differential equation representing family of curves y^(2) = 2cx + 2c√c where c is a positive parameter. --- What is the order of the differential equation ?

Q76. mcq single +2.5 / 0.83

A solution of the differential equation $\left(\frac{d y}{d x}\right)^2-x \frac{d y}{d x}=0 $ is

Q77. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : The line y = x partitions the circle (x - a)^(2) + y^(2) = a^(2) in two segments. --- What is the area of minor segment ?

Q78. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : The line y = x partitions the circle (x - a)^(2) + y^(2) = a^(2) in two segments. --- What is the area of major segment ?

Q79. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : A quadratic equation is given by (3 + 2√2)x^(2) - (4 + 2√3)x + (8 + 4√3) = 0 --- What is the HM of the roots of the equation ?

Q80. mcq single +2.5 / 0.83

How many real numbers satisfy the equation |x - 4| + |x - 7| = 15 ?

Q81. mcq single +2.5 / 0.83

For how many integral values of k, the equation x^(2) - 4x + k = 0, where k is an integer has real roots and both of them lie in the interval (0, 5) ?

Q82. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : A quadratic equation is given by (3 + 2√2)x^(2) - (4 + 2√3)x + (8 + 4√3) = 0 --- What is the GM of the roots of the equation ?

Q83. mcq single +2.5 / 0.83

α and β are distinct real roots of the quadratic equation x^(2) + ax + b = 0. Which of the following statements is/are sufficient to find α ? 1. α + β = 0, α^(2) + β^(2) = 2 2. αβ^(2) = -1, a = 0 Select the correct answer using the code given below :

Q84. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : P(x, y) is any point on the ellipse x^(2 )+ 4y^(2) = 1. Let E, F be the foci of the ellipse. --- What is PE + PF equal to ?

Q85. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : P(x, y) is any point on the ellipse x^(2 )+ 4y^(2) = 1. Let E, F be the foci of the ellipse. --- Consider the following points : 1. $\left(\frac{\sqrt{3}}{2}, 0\right)$ 2. $\left(\frac{\sqrt{3}}{2}, \frac{1}{4}\right)$ 3. $\left(\frac{\sqrt{3}}{2},-\frac{1}{4}\right)$ Which of the above points lie on latus rectum of ellipse ?

Q86. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next three (03) items that follow : Let f(x) = |ln x|, x ≠ 1 --- What is the derivative of fof(x), where 1 < x < 2 ?

Q87. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next three (03) items that follow : Let f(x) = |ln x|, x ≠ 1 --- What is the derivative of f(x) at x = 2 ?

Q88. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next three (03) items that follow : Let f(x) = |ln x|, x ≠ 1 --- What is the derivative of f(x) at x = 0.5 ?

Q89. mcq single +2.5 / 0.83

What is the derivative of cosec(x°) ?

Q90. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next three (03) items that follow : Let f(x) = Pe^(x) + Qe^(2x) + Re^(3x) where P, Q, R are real numbers. Further f(0) = 6, f'(ln 3) = 282 and $\int_0^{\ln 2} f(x) d x=11$ --- What is f'(0) equal to ?

Q91. mcq single +2.5 / 0.83

Consider the following statements : 1. (25)! + 1 is divisible by 26 2. (6)! + 1 is divisible by 7 Which of the above statements is/are correct ?

Q92. mcq single +2.5 / 0.83

m parallel lines cut n parallel lines giving rise to 60 parallelograms. What is the value of (m + n) ?

Q93. mcq single +2.5 / 0.83

What is the number of 6-digit numbers that can be formed only by using 0, 1, 2, 3, 4 and 5 (each once); and divisible by 6 ?

Q94. mcq single +2.5 / 0.83

5-digit numbers are formed using the digits 0, 1, 2, 4, 5 without repetition. What is the percentage of numbers which are greater than 50,000 ?

Q95. mcq single +2.5 / 0.83

Let x be the number of permutations of the word ‘PERMUTATIONS’ and y be the number of permutations of the word ‘COMBINATIONS’. Which one of the following is correct ?

Q96. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : Consider the sum S = 0! + 1! + 2! + 3! + 4! +... .+ 100! --- If the sum S is divided by 60, what is the remainder ?

Q97. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : Consider the sum S = 0! + 1! + 2! + 3! + 4! +... .+ 100! --- If the sum S is divided by 8, what is the remainder ?

Q98. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next three (03) items that follow : Let $I_1=\int_0^\pi \frac{x}{1+\cos ^2 x} d x$ and $I_2=\int_0^\pi \frac{1}{1+\sin ^2 x} d x $ --- What is the value of $8 I_1^2$

Q99. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next three (03) items that follow : Let $I_1=\int_0^\pi \frac{x}{1+\cos ^2 x} d x$ and $I_2=\int_0^\pi \frac{1}{1+\sin ^2 x} d x $ --- What is the value of $\frac{I_1+I_2}{I_1-I_2} $ ?

Q100. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next three (03) items that follow : Let $I_1=\int_0^\pi \frac{x}{1+\cos ^2 x} d x$ and $I_2=\int_0^\pi \frac{1}{1+\sin ^2 x} d x $ --- What is the value of I~2~ ?

Q101. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : Let $l=\int_a^b \frac{|x|}{x} d x, $ a < b --- What is l equal to when a < b < 0 ?

Q102. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next three (03) items that follow : Let f(x) = Pe^(x) + Qe^(2x) + Re^(3x) where P, Q, R are real numbers. Further f(0) = 6, f'(ln 3) = 282 and $\int_0^{\ln 2} f(x) d x=11$ --- ‘What is the value of Q ?

Q103. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next three (03) items that follow : Let f(x) = Pe^(x) + Qe^(2x) + Re^(3x) where P, Q, R are real numbers. Further f(0) = 6, f'(ln 3) = 282 and $\int_0^{\ln 2} f(x) d x=11$ --- What is the value of R ?

Q104. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : Let $l=\int_a^b \frac{|x|}{x} d x, $ a < b --- What is l equal to when a < 0 < b ?

Q105. mcq single +2.5 / 0.83

Let A and B be symmetric matrices of same order, then which one of the following is correct regarding (AB - BA) ? 1. Its diagonal entries are equal but nonzero 2. The sum of its non-diagonal entries is zero Select the correct answer using the code given below :

Q106. mcq single +2.5 / 0.83

The system of linear equations x + 2y + z = 4, 2x + 4y + 2z = 8 and 3x + 6y + 3z = 10 has

Q107. mcq single +2.5 / 0.83

Consider the following statements in respect of square matrices A, B, C each of same order n : 1. AB = AC ⇒ B = C if A is non-singular 2. If BX = CX for every column matrix X having n rows then B = C Which of the statements given above is/are correct ?

Q108. mcq single +2.5 / 0.83

Let AX = B be a system of 3 linear equations with 3-unknowns. Let X1 and X2 be its two distinct solutions. If the combination aX1 + bX2 is a solution of AX = B; where a, b are real numbers, then which one of the following is correct ?

Q109. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : In the triangle ABC, a^(2) + b^(2) + c^(2) = ac + √3 bc --- If c = 8, what is the area of the triangle ?

Q110. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : The perimeter of a triangle ABC is 6 times the AM of sine of angles of the triangle. Further BC = √3 and CA = 1 --- What is the perimeter of the triangle ?

Q111. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : The perimeter of a triangle ABC is 6 times the AM of sine of angles of the triangle. Further BC = √3 and CA = 1 --- Consider the following statements : 1. ABC is right angled triangle 2. The angles of the triangle are in AP Which of the statements given above is/are correct ?

Q112. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : In a triangle POR, P is the largest angle and cosP = $\frac{1}{3}$. Further the in-circle of the triangle touches the sides PQ, QR and RP at N, L and M respectively such that the lengths PN, QL and RM are n, n + 2, n + 4 respectively where n is an integer. --- What is the value of n ?

Q113. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : In the triangle ABC, a^(2) + b^(2) + c^(2) = ac + √3 bc --- What is the nature of the triangle ?

Q114. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : In a triangle POR, P is the largest angle and cosP = $\frac{1}{3}$. Further the in-circle of the triangle touches the sides PQ, QR and RP at N, L and M respectively such that the lengths PN, QL and RM are n, n + 2, n + 4 respectively where n is an integer. --- What is the length of the smallest side ?

Q115. mcq single +2.5 / 0.83

Let A be a matrix of order 3 × 3 and |A| = 4. If |2adj(3A)| = 2^(α )3^(β), then what is the value of (α + β)?

Q116. mcq single +2.5 / 0.83

What is the sum of the roots of the equation $\left|\begin{array}{ccc} 0 & x-a & x-b \\ 0 & 0 & x-c \\ x+b & x+c & 1 \end{array}\right|=0 $ ?

Q117. mcq single +2.5 / 0.83

If a, b, c are in AP, then what is $\left|\begin{array}{lll} x+1 & x+2 & x+3 \\ x+2 & x+3 & x+4 \\ x+a & x+b & x+c \end{array}\right|$ equal to ?

Q118. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : Let Δ(a, b, c, α) = $\left|\begin{array}{ccc} a & b & a \alpha+b \\ b & c & b \alpha+c \\ a \alpha+b & b \alpha+c & 0 \end{array}\right|$ --- If Δ(a, b, c, α) = 0 for every α > 0, then which one of the following is correct ?

Q119. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : Let Δ(a, b, c, α) = $\left|\begin{array}{ccc} a & b & a \alpha+b \\ b & c & b \alpha+c \\ a \alpha+b & b \alpha+c & 0 \end{array}\right|$ --- If Δ(7, 4, 2, α) = 0, then α is a root of which one of the following equations ?

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