If the first term of an AP is 2 and the sum of the first five terms is equal to one-fourth of the sum of the next five terms, then what is the sum of the first ten terms?
To get the value of any function at any point with respect to some other function, we need to manipulate the given function/ point in such a way that we get the function that is needed.
With the numbers 2, 4, 6, 8, all the possible determinants with these four different elements are constructed. What is the sum of the values of all such determinants?
The square of the length of any side of a given triangle is equal to the sum of the squares of the length of the other sides minus twice the product of the other two sides multiplied by the cosine of angle included between them.
Consider, a, b, and c are lengths of the side of a triangle ABC as shown, then;
cos(x)=2bcb2+c2−a2
cos(y)=2aca2+c2−b2
cos(z)=2aba2+b2−c2
Calculation:
Let m = n = 1 unit
Then,
m2+n2+mnRightarrow12+12+1=3
Using cosine rule;
cosθ=2×1×112+12−32
⇒ cos θ = -1/2
∴ θ = 120°
Now, the sum of the acute angles of the triangle = 180° - 120° = 60°
Question 19
What is the area of the triangle ABC with sides a = 10cm and c = 4cm angle B = 30°?
Equivalent set: Two sets A and B are said to be equivalent if they have the same number of elements (cardinality), regardless of what the elements are. , i.e n(A) = n(B).
Equal set: Two sets A and B are said to be equal if they have exactly the same elements. The repetition of elements does not matter in sets because a set is defined as a collection of distinct objects, so repeated elements are generally ignored.
If A = B, then n(A) = n(B) and for any x ∈ A, x ∈ B too.
Calculation:
A = {1, 3, 5} and B = {2, 4, 7} are equivalent sets. → True
Set A has three elements: 1, 3, and 5. Set B also has three elements: 2, 4, and 7. Since both sets have three elements, they are indeed equivalent sets.
A = {1, 5, 9} and B = {1, 5, 5, 9, 9} are equal sets. → True
Set A contains the elements 1, 5, and 9. Set B also contains the elements 1, 5, and 9 (despite some elements being repeated, duplicates are not considered in set theory). Since both sets have the same elements, they are equal.
Therefore, both statements are correct:
Statement 1 is correct: A and B are equivalent sets.
A function f from X to Y is onto (or surjective), if and only if for every element y ∈ Y there is an element x ∈ X with f(x) = y.
In words: “Each element in the co-domain of f has a pre-image”
Mathematical Description: f : X →Y is onto ⇔ ∀ y ∃x, f(x) = y
One-to-one Correspondence
A function f is a one-to-one correspondence (or bijection), if and only if it is both one-to-one and onto
In words: “No element in the co-domain of f has two (or more) pre images” (one-to-one) and “Each element in the co-domain of f has a pre-image” (onto).
Calculation:
1. A function f : Z → Z, defined by f(x) = x + 1, is one-one as well as onto.
f(x) = x + 1,
calculate f(x1) :
f(x1) = x1 + 1
calculate f(x2) :
f(x2) = x2 + 1
Now, f(x1) = f(x2)
⇒ x1 + 1 = x2 + 1
⇒ x1 = x2
So, f is one-one function.
Consider f(x) = y
y = x + 1
x = y - 1
f(y-1) = y - 1 + 1 = y
f is onto.
2. A function f : N → N, defined by f(x) = x + 1, is one-one but not onto.
f(x) = x + 1,
calculate f(x1) :
f(x1) = x1 + 1
calculate f(x2) :
f(x2) = x2 + 1
Now, f(x1) = f(x2)
⇒ x1 + 1 = x2 + 1
⇒ x1 = x2
So, f is one-one function.
Clearly, f(x) = x + 1 ≥ 2 for all x ∈ N
So, f(x) does not assume values 1.
f is not an onto function.
So, Both 1 and 2 are correct.
Question 24
Consider the following in respect of a complex number z:
General term: General term in the expansion of (x + y)n is given by
T(r;+;1)=;;nCr×xn−r×yr
Middle terms: The middle terms is the expansion of (x + y) n depends upon the value of n.
If n is even, then the total number of terms in the expansion of (x + y) n is n +1. So there is only one middle term i.e. (2n+1);th term is the middle term.
T(2n;+;1)=;;nC2n×x2n×y2n
If n is odd, then the total number of terms in the expansion of (x + y) n is n + 1. So there are two middle terms i.e. (2n;+;1)th;and (2n;+;3)th are two middle terms.
A ladder 6 m long reaches a point 6 m below the top of a vertical flagstaff. From the foot of the ladder, the elevation of the top of the flagstaff is 75°. What is the height of the Flagstaff?
Here, P = perpendicular, B = base & H = hypotenuses
And, H2 = B2+ P2
tan(A + B) = (tanA + tanB)/(1 - tanA.tanB)
Calculation:
Length of the ladder CD = 5 mFrom given data, the triangle diagram can be drawn as
DB = Height of flagstaff
AC = Ladder = 6 m = DA = 6 m.
∠DCB = 75°
∠ABC = 90°
Now, ∠BDC = 180° - (90° + 75°) = 15° .
Then, ∠ADC = ∠ACD { Since, ΔADC is isosceles triangle}
⇒ ∠ACD = 15°
Hence,
⇒ ∠ACB = ∠DCB - ∠ACD = 75° - 15° = 60°
Now Δ ABC can be separately drawn as,
Since, ∆ABC is right angle tringle
⇒ sin 60° = AB/AC
⇒ (√3/2) = AB/6
⇒ AB = 3√3 m .
Hence, Height of flagstaff = DB = BA + AD = 3√3 + 6 m
Question 44
The shadow of a tower is found to be x meter longer, when the angle of elevation of the sun changes from 60° to 45°. If the height of the tower is 5(3 + √3) m, then what is x equal to?
Let AB be the tower and let AC and AD be its shadows when the angles of elevation of the sum are 60o and 45o, respectively. Then, ∠ACB = 60o, ∠ADB = 45o, CD = x m
Let AB = 5(3 + √3) m and AC = y meter
In Δ CAB, We have,
tan60∘=ACAB
√3 = AC5(3+√3)
AC = √35(3+√3)
On rationalizing, we get
AC = √35(3+√3)(\times\)(\rm \frac{√3 }{√3 }\)
= 35(3√3+3)
In Δ DAB, We have,
tan45∘=ADAB
1 = ADAB
AB = AD
AD = 5(3 + √3)
x + y = 5(3 + √3)
x + 35(3√3+3) = 5(3 + √3)
x = 5(3 + √3) - 35(3√3+3)
x = 10 m
Question 45
If 3cosθ = 4sinθ, then what is the value of tan (45° + θ)?
If all the elements of a row (or column) of a determinant are multiplied by a non-zero constant, then the determinant gets multiplied by the same constant.
= \(\left| {\begin{array}{*{20}{c}} {{\ln:a_1}}&{{\ln r}}&{{\ln r }}\ {{\ln:a_4}}&{{\ln r }}&{{\ln r}}\ {{\ln:a_7}}&{{\ln r }}&{{\ln r}} \end{array}} \right|\) [Since apap+1=r and log ap + 1 - log ap = logapap+1=logr]
= 0 [Since C3 = C2]
Question 53
If the roots of the quadratic equation x2 + 2x + k = 0 are real, then
One property that is unique to matrices is the dimension property.
This property has two parts:
Part 1: The product of two matrices will be defined if the number of columns in the first matrix is equal to the number of rows in the second matrix.
Part 2: If the product is defined, the resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix.
Explanation:
Let A be a matrix of order [a × b] ----(i)
And B be a matrix of order [b × d] ----(ii)
Since we have to find the multiplication of A and B
So we let the column of A = Row of B [which is b]
From the properties of multiplication
The order of AB will be [a × d]
But, According to the question, the order of AB is [n × n]
i.e The value of a and d should be n
i.e a = d = n ----(iii)
Now, From (i), (ii), and (iii), we get
The order of A = [n × b]
The order of B = [b × n]
Here the order of A and B are not the same
But it can be the same when b = n
Here, We can say that to satisfy the above question the order of two matrices need not be the same.
Question 57
How many matrices of different orders are possible with elements comprising all prime numbers less than 30?
If A and B are square matrices of order 2 such that det(AB) = det(BA)
This property is true regardless of whether A and B are unit matrices or not. Therefore, the condition det(AB) = det(BA) does not imply that either A or B must be a unit matrix.
∴ Option 4 is correct
Additional Information Properties of determinants:
The determinant of a diagonal matrix is the product of the diagonal entries.
If A and B are both n×n matrices, then det(AB) = det(A) det(B).
For a n×n matrix A, det(kA) = kn det(A).
The determinant of a square matrix is the same as the determinant of its transpose.
Question 60
What is cot 2x cot 4x - cot 4x cot 6x - cot 6x cot 2x equal to
Median = 18 {As it is the middle value of the arrangement}
Mode = 18 {As it appears the most}
∴ Mean = Median = Mode
Option 2 is correct
Question 67
The mean of 12 observations is 75. If two observation are discarded, then the mean of the remaining observations is 65. What is the mean of the discarded observations?
∴ A and B are independent events and are not mutually exclusive events.
Only statement 1 is correct.
Question 71
The average of a set of 15 observations is recorded, but later it is found that for one observation, the digit in the tens place was wrongly recorded as 8 instead of 3. After correcting the observation, the average is
∴ The numerator must also be zero to have a certain value of the given limit, i.e.,
x→−1lim[f(x)+1]=0
x→−1limf(x) + 1 = 0
x→−1limf(x) = -1
Question 78
If the function \(\rm f\left( x \right) = \left{ {\begin{array}{*{20}{c}} {a + bx,;;}&{x < 1}\ {5,}&{x = 1}\ {b - ax,}&{x > 1} \end{array}} \right.\) is continuous, then what is the value of (a + b)?
The domain is the set of all possible value of x which have a finite value of f(x).
Calculation:
Given function f(x) = 3x
The function will have a finite value for all x ∈ (-∞, ∞)
Mistake PointsThe range of the given function will be from (0,∞). 0, when x = -∞, and ∞ when x = ∞.
Question 81
If the general solution of a differential equation is y2 + 2cy - cx + c2 = 0, where c is an arbitrary constant, then what is the order of the differential equation?
Order of a differential equation is the highest order of derivative that occurs in the differential equation.
Degree of a differential equation is the highest power of the highest order derivative that occurs in the equation, after all the derivatives are converted into rational and radical free form.
Calculation:
y2 + 2cy - cx + c2 = 0
⇒ (y + c)2 = cx
⇒ 2(y + c)y' = c
⇒ c = 1−2y′2yy′
Substituting this in the given equation, we get:
(y + 1−2y′2yy′)2 = 1−2y′2yy′x
⇒ y2 = 2xyy'(1 - 2y')
⇒ y2 - 2xyy' + 4xy(y')2 = 0
The highest derivative in it is y', therefore its order is 1.
And, the highest power of this derivative in the equation is 2, therefore its degree is 2.
Question 82
What is the degree of the following differential equation?
Order of a differential equation is the highest order of derivative that occurs in the differential equation.
Degree of a differential equation is the highest power of the highest order derivative that occurs in the equation, after all the derivatives are converted into rational and radical free form.
Calculation:
Getting rid of the radicals by raising both the sides to power 2 will give us:
x2=1+(dx2d2y)1
The highest derivative in it is dx2d2y, therefore its order is 2.
∴ The highest power of this derivative in the equation is 1, therefore its degree is 1.
Question 83
Which one of the following differential equations has the general solution y = aex + be-x?
Arithmetic Mean (A. M.): The Arithmetic Mean of a set of n numbers is given by:
A. M. = n∑ai.
Geometric Mean (G. M.): The Geometric Mean of a set of n numbers is given by:
G. M. = n∏ai.
Calculation:
Both the Arithmetic and Geometric Means use all the data for their calculations.
Question 87
The numbers of Science, Arts and Commerce graduates working in a company are 30, 70 and 50 respectively. If these figures are represented by a pie chart, then what is the angle corresponding to Science graduates?
When constructing a histogram with non-uniform (unequal) class widths, we must ensure that the areas of the rectangles are proportional to the class frequencies.
Remember that the histogram differs from a bar chart in that it is the area of the bar that denotes the value, not the height. This means that we would need to consider the widths in order to determine the height of each rectangle.
Calculation:
From the definition of histogram, it is clear that the frequency of a class is proportional to the area of the rectangle, and NOT to its height.
Question 89
The co-efficient of correlation is independent of:
In simple linear regression analysis, the co-efficient of correlation is a statistic which indicates an association between the independent variable and the dependent variable. The co-efficient of correlation is represented by "r" and its value lies between -1.00 and +1.00.
When the co-efficient of correlation is positive, such as +0.80, it means the dependent variable is increasing/decreasing when the independent variable is increasing/decreasing. A negative value indicates an inverse association; the dependent variable is increasing/decreasing when the independent variable is decreasing/increasing.
A co-efficient of correlation of +0.8 or -0.8 indicates a strong correlation between the independent variable and the dependent variable. An r of +0.20 or -0.20 indicates a weak correlation between the variables. When the co-efficient of correlation is 0.00, there is no correlation.
r = ∑(xi−xˉ)2∑(yi−yˉ)2∑(xi−xˉ)(yi−yˉ).
Calculation:
From the properties/nature of the co-efficient of correlation, we know that the correlation coefficient is independent of the choice of origin and scale.
Question 90
The following tables gives the frequency distribution of number of peas per pea pod of 198 pods:
Indeterminate Forms: Any expression whose value cannot be defined, like 00, ±∞∞, 00, ∞0 etc.
L'Hospital's Rule: For the differentiable functions f(x) and g(x), the x→climg(x)f(x), if f(x) and g(x) are both 0 or ±∞ (i.e. an Indeterminate Form) is equal to the x→climg′(x)f′(x) if it exists.
For the indeterminate form ∞ - ∞, first rationalize by multiplying with the conjugate and then divide the terms by the highest power of the variable to get terms so that x1 → 0 as x → ∞.
For the indeterminate form 00, first try to rationalize by multiplying with the conjugate, or simplify by cancelling some terms in the numerator and denominator. Else, use the L'Hospital's rule.
Calculation:
x→alimxx−aaax−xa
= aa−aaaa−aa
= 00, an indeterminate form.
Applying L'Hospital's rule, we get:
x→alimxx−aaax−xa
= x→alimxx(logx+1)axloga−axa−1
= aa(loga+1)aaloga−a⋅aa−1
= loga+1loga−1
According to the question, loga+1loga−1 = -1.
⇒ log a - 1 = - log a - 1
⇒ 2 log a = 0
⇒ a = 1.
Question 92
A particle starts from origin with a velocity (in m/s) given by the equation dtdx=x+1. The time (in seconds) taken by the particle to traverse a distance of 24 m is:
A function f(x) is said to be continuous at a point x = a in its domain, if x→alimf(x) exists or if its graph is a single unbroken curve at that point.
f(x) is continuous at x = a ⇔ x→a+limf(x)=x→a−limf(x)=x→alimf(x)=f(a).
Question 95
If ∫0a[f(x)+f(−x)]dx=∫−aag(x)dx, then what is g(x) equal to?
The area under the function y = f(x) from x = a to x = b and the x-axis is given by the definite integral ∫abf(x)dx, for curves which are entirely on the same side of the x-axis in the given range.
If the curves are on both the sides of the x-axis, then we calculate the areas of both the sides separately and add them.
Definite integral: If ∫ f(x) dx = g(x) + C, then ∫abf(x)dx=[g(x)]ab = g(b) - g(a).
∫a2−x2dx=2xa2−x2+2a2sin−1ax+C.
Calculation:
Let's first find the points where the curve meets the x-axis (y = 0).
⇒ y=16−x2=0
⇒ x = ±4
Now, since the curve y=16−x2 is entirely on one side of the x-axis in the given range x = -4 to x = 4, we have:
The three points A (x1, y1), B (x2, y2) and C (x3, y3) are said to be collinear if the area of Δ ABC is zero i.e \(\left| {\begin{array}{*{20}{c}} {{x_1}}&{{y_1}}&1\ {{x_2}}&{{y_2}}&1\ {{x_3}}&{{y_3}}&1 \end{array}} \right| = 0\)
CALCULATION:
Given: (p, p - 3), (q + 3, q) and (6, 3) are three points in a 2D plane.
Statement 1: The points lie on a straight line.
As we know that, if three points A (x1, y1), B (x2, y2) and C (x3, y3) are said to be collinear if the area of Δ ABC is zero i.e \(\left| {\begin{array}{*{20}{c}} {{x_1}}&{{y_1}}&1\ {{x_2}}&{{y_2}}&1\ {{x_3}}&{{y_3}}&1 \end{array}} \right| = 0\)
Let θ be the angle between two lines of slope m1 and m2, then the acute angle between the lines is given by:
tanθ=1+m1⋅m2m2−m1
Calculations:
Consider, the slope of the line (x - 2 = 0) is m1
So, m1 = ∞ .
And, the slope of the line (√3x - y - 2 = 0) is m2
So, m2 = √3.
Now, the angle between the given lines is θ.
⇒ tanθ=1+m1⋅m2m2−m1
⇒ tanθ=m11+m2m1m2−1
⇒ tanθ=31
⇒ θ = 30°
Hence, the correct option is 2.
Alternate Method
For line 1:x – 2 = 0Hence, x = 2
Since it is a vertical line, the slope is undefined.
Hence, θ1 = 90°
For line 2:
√3x - y - 2 = 0
Compare this equation with, y = mx + c
It becomes, y = √3x + 2
Hence, m = √3 = tan θ2
Hence, θ2 = tan−13 = 60°
Now, the graph of intersection between two lines can be drawn as
Hence, acute angle between the lines = θ1 – θ2 = 90° – 60° = 30°
Question 108
The point of intersection of diagonals of a square ABCD is at the origin and one of its vertices is at A(4, 2). What is the equation of the diagonal BD?
The constant ratio of distance of point lying on conic from the focus to its perpendicular distance from directrix is called the eccentricity of a conic section and is denoted by e.
For an ellipse: e < 1
For a parabola: e = 1
For a hyperbola: e > 1
For a circle: e = 0
For a pair of straight lines: e = ∞
CALCULATION:
As we know that, eccentricity of a circle is 0 i.e e = 0 for a circle.
So, statement 1 is correct.
We also know that, eccentricity of a parabola is equal to 1 i.e e = 1 for a parabola.
So, statement 2 is also correct.
We also know that, eccentricity of an ellipse is always less than 1 i.e e < 1 for an ellipse.
So, statement 3 is also correct.
Hence, correct option is 4.
Question 111
What is the angle between the two lines having direction ratios (6, 3, 6) and (3, 3, 0)?
The angle θ between two lines whose direction ratios are proportional to a1, b1, c1 and a2, b2, c2 respectively is given by: cosθ=a12+b12+c12;a22+b22+c22a1a2+b1b2+c1c2
CALCULATION:
Here, we have to find the angle between the two lines having direction ratios (6, 3, 6) and (3, 3, 0).
As we know that, if θ is the angle between two lines having direction ratios proportional to a1, b1, c1 and a2, b2, c2 is given by: cosθ=a12+b12+c12;a22+b22+c22a1a2+b1b2+c1c2
⇒ cosθ=62+32+62;32+32+026⋅3+3⋅3+6⋅0
⇒ cosθ=21
⇒ θ = π/4
Hence, correct option is 2.
Question 112
If l, m, n are the direction cosines of the line x - 1 = 2(y + 3) = 1 - z, then what is l4 + m4 + n4 equal to?
If a, b, c are the direction ratios of a line passing through the point (x1, y1, z1), then the equation of the line is given by: ax−x1=by−y1=cz−z1
If a, b, c are the direction ratios of a line then the direction cosine of the line is given by: l=a2+b2+c2am=a2+b2+c2b,n=a2+b2+c2c
Calculation
Given: l, m, n are the direction cosines of the line x - 1 = 2(y + 3) = 1 - z
The given equation of lines can be re-written as 1x−1=21y+3=−1z−1
So, by comparing the equation 1x−1=21y+3=−1z−1 with ax−x1=by−y1=cz−z1 we get
⇒ a = 1, b = 1/2 and c = - 1
As we know, if a, b, c are the direction ratios of a line, then the direction cosine of the line is given by: l=a2+b2+c2am=a2+b2+c2b,n=a2+b2+c2c
The direction cosine of the given line is:
⇒l=32,m=31,n=3−2
⇒ l4 + m4 + n4 = 8116+811+8116=8133=2711
Hence, the correct option is 2.
Question 113
What is the projection of the line segment joining A(1, 7, -5) and B(-3, 4, -2) on y-axis?
The projection of any point on (a, b, c) on x -axis is (a, 0, 0), y-axis is (0, b, 0) and z-axis is (0, 0, c).
Calculation:
The projection of A(1, 7, -5) on y-axis is (0, 7, 0) and projection of B(-3, 4, -2) on y-axis is (0, 4, 0).
Let C = (0 ,7, 0), D = (0, 4, 0) and O = (0, 0, 0) is the origin
⇒ OC = 7 and OD = 4
Let the projection of line segment AB is m = OD - OC.
⇒ m = OD - OC = 7 - 4 = 3
Hence, correct option is 3.
Question 114
What is the number of possible values of k for which the line joining the points (k, 1, 3) and (1, -2, k + 1) also passes through the point (15, 2, -4)?
The direction ratios of the line joining the points (x1, y1, z1) and (x2, y2, z2) is given by: a = x2 - x1, b = y2 - y1 and c = z2 - z1
If a, b, c are the direction ration ratios of a line passing through the point (x1, y1, z1), then the equation of line is given by: ax−x1=by−y1=cz−z1
CALCULATION:
Given: The line joining the points (k, 1, 3) and (1, -2, k + 1) also passes through the point (15, 2, -4)
As we know that, the direction ratios of the line joining the points (x1, y1, z1) and (x2, y2, z2) is given by: a = x2 - x1, b = y2 - y1 and c = z2 - z1
So, the direction ratios of the line joining the points (k, 1, 3) and (1, -2, k + 1) is: a = 1 - k, b = - 3 and c = k - 2
As we know, if a, b, and c are the direction ratio ratios of a line passing through the point (x1, y1, z1), then the equation of a line is given by: ax−x1=by−y1=cz−z1
So, the equation of the line with direction ratios a, b, c and passing through the point (k, 1, 3) is given by: 1−kx−k=−3y−1=k−2z−3
∵ It is given that, the line represented by 1−kx−k=−3y−1=k−2z−3 also passes through the point (15, 2, -4)
So, substitute x = 15, y = 2 and z = - 4 in the equation 1−kx−k=−3y−1=k−2z−3
⇒ 1−k15−k=−32−1=k−2−4−3
⇒ 1−k15−k=−31
⇒ - 45 + 3k = 1 - k
⇒ k = 23/2
⇒ −31=k−2−7
⇒ k - 2 = 21
⇒ k = 23
So, there are two possible values of k as shown above.
Hence, correct option is 3
Question 115
The foot of the perpendicular drawn from the origin to the plane x + y + z = 3 is
Perpendicular distance of a plane ax + by + cz + d = 0 from a point P (x1, y1, z1) is given by: d=a2+b2+c2ax1+by1+cz1+d
CALCULATION:
Let A(x, y, z) be the foot of the perpendicular drawn from the origin to the plane x + y + z = 3.
As we know that, the perpendicular distance of a plane ax + by + cz + d = 0 from a point P (x1, y1, z1) is given by: d=a2+b2+c2ax1+by1+cz1+d
So, the distance between the origin and the plane x + y + z - 3 = 0 is given by: d=12+12+120+0+0−3=33=3
So, this means that the length of the line joining the points origin and A is √3
As we can see that from the given options, if A = (1, 1, 1) then the distance between the points origin and A is √3
Hence, correct option is 3.
Question 116
A vector r=ai^+bj^ is equally inclined to both x and y axes. If the magnitude of the vector is 2 units, then what are the values of a and b respectively?
