RBSE Class 12 E-Mathematics Self Evaluation Test Papers in English

Students must start practicing the questions from RBSE 12th Maths Model Papers E-Mathematics Self Evaluation Test Papers in English Medium provided here.

RBSE Class 12 E-Mathematics Self Evaluation Test Paper 1 in English

Section – A

Question 1.
Multiple Choice
(i) If R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1,1), (4, 4), (I, 3), (3, 3), (3, 2)}; then: (1)
(a) R is reflexive and symmetric but not transitive,
(b) R is reflexive and transitive but not symmetric,
(c) R is symmetric and transitive but not reflexive,
(d) R is an equivalence relation.

(ii) The value of tan^-1[2sin(2cos^-1\(\frac{\sqrt{3}}{2}\))] (1)
(a) \(\frac{\pi}{2}\)
(b) \(\frac{\pi}{4}\)
(c) \(\frac{\pi}{3}\)
(d) \(\frac{\pi}{6}\)

(iii) If A = \(\left[\begin{array}{lll}
0 & 1 & 2 \\
1 & 2 & 3 \\
3 & a & 1
\end{array}\right]\), and A^-1 = \(\left[\begin{array}{rrr}
\frac{1}{2} & -\frac{1}{2} & \frac{1}{2} \\
-4 & 3 & c \\
\frac{5}{2} & -\frac{3}{2} & \frac{1}{2}
\end{array}\right]\), then: (1)
(a) a = 2, c = –\(\frac{1}{2}\)
(b) a = 1, c = -1
(c) a = -1, c = 1
(d) a = \(\frac{1}{2}\), c = \(\frac{1}{2}\)

(iv) If \(\left|\begin{array}{ccc}
3 x-8 & 3 & 3 \\
3 & 3 x-8 & 3 \\
3 & 3 & 3 x-8
\end{array}\right|\) = 0, then x = ________ (1)
(a) \(\frac{3}{2}, \frac{3}{11}\)
(b) \(\frac{3}{2}, \frac{11}{3}\)
(c) \(\frac{2}{3}, \frac{11}{3}\)
(d) \(\frac{2}{3}, \frac{3}{11}\)

(v) If xy + y² = tan x + y, then \(\frac{dy}{dx}\) is: (1)
(a) \(\frac{\sec ^{2} y-1}{x+y+2}\)
(b) \(\frac{\sec ^{2} x-y}{x+2 y-1}\)
(c) \(\frac{\sec ^{2} x-2}{x+2 y+3}\)
(d) \(\frac{\sec ^{2} x}{x+3 y-1}\)

(vi) If f(a + b + c) = f(x), then ∫^b _a x f(x) dx is equal to: (1)

(vii) The sum of order and the degree of the differential equation x²\(\frac{d^{2} y}{d x^{2}}\) = {1 + \(\left(\frac{d y}{d x}\right)^{2}\)}⁴ (1)
(a) 2
(b) 3
(c) 4
(d) 8

(viii) If \(\vec{a}\) and \(\vec{b}\) are two collinear vectors, then which of the following are incorrect: (1)
(a) \(\vec{b}\) = λ\(\vec{a}\), for some scalar λ
(b) \(\vec{a}=\pm \vec{b}\)
(c) the respective components of \(\vec{a}\) and \(\vec{b}\) are not proportional.
(d) both the vectors \(\vec{a}\) and \(\vec{b}\) have same direction but different magnitudes.

(ix) If P (A / B) > P (A), then which of the following is correct: (1)
(a) P (B/A) < P (B)
(b) P (A ∩ B) < P (A). P (B) (c) P (B/A) > P (B)
(d) P (B/A) P (B)

(x) If \(\) then the value of x is: (1)
(a) 7
(b) \(\frac{-2}{9}\)
(c) \(\frac{-3}{8}\)
(d) none of these

(xi) The value of constant k so that the given function is continuous at the indicated point, is: (1)

(a) 1
(b) 2
(c) 3
(d) 4

(xii) A unit vector in the direction of vector \(\overrightarrow{P O}\) where P and Q are the points (1, 3, 0) and (1)
(a) \(\frac{1}{2}\)î – \(\frac{1}{7}\)ĵ + \(\frac{5}{7}\)k̂
(b) \(\frac{3}{7}\)î + \(\frac{2}{7}\)ĵ + \(\frac{6}{7}\)k̂
(c) \(\frac{5}{7}\)î + \(\frac{6}{7}\)ĵ + \(\frac{1}{7}\)k̂
(d) \(\frac{3}{7}\)î – \(\frac{6}{7}\)ĵ + \(\frac{2}{7}\)k̂

Question 2.
Fill in the blanks :

(i) If A = [1, 2, 3], then the number of equivalence relation containing (1, 2) is ________ (1)

(ii) The value of cos^-1(cos\(\frac{13 \pi}{6}\)) is ________ (1)

(iii) If matrix A = \(\left[\begin{array}{cc}
1 & -1 \\
-1 & 1
\end{array}\right]\) and A² = kA, then the value of k is ________ (1)

(iv) If y = sin^-1(6x\(\frac{d y}{d x}\)), – \(\frac{1}{3 \sqrt{2}}\) < x < \(\frac{1}{3 \sqrt{2}}\), then \(\frac{d y}{d x}\) is ________ (1)

(v) The value of: (1)

is ________

(vi) The projection of the vector î – ĵ on the vector î – ĵ is ________ (1)

Question 3.
Very Short Answer Type Questions
(i) Prove that the function, f: N → N is defined by f(x) = x² + x + 1 is one-one. (1)

(ii) Show that: tan^-1 \(\frac{1}{2}\) + tan^-1 \(\frac{2}{11}\) = tan^-1\(\frac{3}{4}\) (1)

(iii) Construct a 2 × 2 matrix A = [a_ij] whose elements are given by a_ij = |(i)² – j|. (1)

(iv) For what value of the matrix \(\left[\begin{array}{cc}
6-x & 4 \\
3-x & 1
\end{array}\right]\) is a singular matrix. (1)

(v) Differentiate \(\sqrt{e^{\sqrt{x}}}\), x > 0 with respect to x. (1)

(vi) Find : ∫\(\frac{\sec ^{2} x}{\sqrt{\tan ^{2} x+4}}\)dx (1)

(vii) Find tht general solution of thc di(feentia1 equation \(\frac{d y}{d x}=\frac{1-\cos x}{1+\cos x}\). (1)

(viii) Find the unit vector in the direction of the sum of the vectors, \(\vec{a}\) = 2î + 2ĵ – 5k̂ and \(\vec{b}\) =2î + ĵ + 3k̂. (1)

(ix) If P(A) = \(\frac{7}{13}\), P(B) = \(\frac{9}{13}\) and P(A n B) = \(\frac{4}{13}\), evaluate P(A/B). (1)

(x) Using determinants find the equation of line joining the points (1,2) and (3, 6). (1)

(xi) Find the differential equation representing the family of curves y = ae^bx+3, where a and b are arbitrary constants. (1)

Section – B
Short Answer-Type Questions

Question 4.
Check whether the relation R in the set; N of natural numbers given by R = {(a, b): a is divisor of b} is reflexive, symmetric or transitive. (2)

Question 5.
If A = \(\left[\begin{array}{rrr}
2 & 0 & 1 \\
2 & 1 & 3 \\
1 & -1 & 0
\end{array}\right]\), then find the value of A² – 5A + 6I. (2)

Question 6.
Solve the following system of equations by matrix method : 4x – 3y = 3, 3x – 5y = 7. (2)

Question 7.
Show that f(x) = |x – 3| is continuous a x = 3. (2)

Question 8.
Find ∫\(\frac{(x-3) e^{x}}{(x-1)^{3}}\)dx. (2)

Question 9.
In a hostel, 60% of the students read Hindi newspaper, 40% read English newspaper and 20% read both Hindi and English newspapers. A student is selected at random.
(a) Find the probability that she reads neither Hindi nor English newspapers.
(b) If she reads Hindi newspaper, find the probability that she reads English newspaper. (2)

Question 10.
If \(\left[\begin{array}{cc}
2 x+1 & 5 x \\
0 & y^{2}+1
\end{array}\right]=\left[\begin{array}{cc}
x+3 & 10 \\
0 & 26
\end{array}\right]\) find the value of (x + y). (2)

Question 11.
If x = a(θ – sin θ) and y = a(1 + cosθ), then find \(\frac{dy}{dx}\) (2)

Question 12.
Using the property of determinants and without expanding, prove that: (2)
\(\left|\begin{array}{lll}
b+c & q+r & y+z \\
c+a & r+p & z+x \\
a+b & p+q & x+y
\end{array}\right|=2\left|\begin{array}{lll}
a & p & x \\
b & q & y \\
c & r & z
\end{array}\right|\)

Question 13.
Evaluate : (2)

Question 14.
Form a differential equation representing y = e^2x(a+bx) by eliminating arbitrary constants a and b. (2)

Question 15.
Let \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) be three vectors such that |\(\vec{a}\)| = 3, |\(\vec{b}\)| = 4. |\(\vec{c}\)| = 5 and each one of them being perpendicular to the sum of the other two, find \(|\vec{a}+\vec{b}+\vec{c}|\). (2)

Question 16.
A pair of dice is thrown 4 times. What will bethepobabi1ity lo gei maximum t.o 7? (2)

Section – C
Long Answer Type Questions

Question 17.
Find : tan[\(\frac{1}{2}\){sin^-1\(\left(\frac{2 x}{1+x^{2}}\right)\) + cos^-1\(\left(\frac{1-y^{2}}{1+y^{2}}\right)\)}], |x| < 1, y > 0 and xy > 1. (3)
Or
Prove that cot^-1\(\left[\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right]=\frac{x}{2}\), x ∈(0, \(\frac{π}{4}\)) (3)

Question 18.
If e^sinx + (tan x)^x, then find (3)
Or
Find th value of c in Rolle’s thcorem for the function f(x) = x³ – 3x in [-√3, 0]. (3)

Question 19.
Find : ∫e^x\(\left(\frac{1+\sin x}{1+\cos x}\right)\) dx. (3)
Or
Find : ∫[log (log x) + \(\frac{1}{(\log x)^{2}}\)] dx. (3)

Question 20.
Find the position vector oi a point R which divides the line joining two points P and Q; whose position vectors are (î + 2ĵ – k̂) and (- î + ĵ + k̂) respectively, in the ratio 2: 1 (i) internally (ii) externally. (3)
Or
If a unit vector \(\vec{a}\) makes angle \(\frac{\pi}{4}\) with with î\(\frac{\pi}{4}\) with ĵ and an açute angle θ with k̂, then find θ and hence the components of \(\vec{a}\). (3)

Section – D
Essay Type Questions

Question 21.
Find : (4)

Or
Find : (4)

Question 22.
Show that the foN i’ig’diffcreniiai equation is homogeneous and solve it y dx + x log\(\frac{y}{x}\)dy – 2x dy = 0. (4)
Or
Find the general solution of the thHoving differential:quation: cos²x +y tan x (o ≤ x ≤ \(\frac{\pi}{2}\)) (4)

Question 23.
Consider the experiment of tossing a coin. 1f the Coin shows head, toss it again hut if it shows tail, then throw a die. Find the conditional probability of the event that ‘the die shows a númber greater thrn 4’ given that ‘there is at letst one tail. (4)
Or
A black and a red dice are rolled.
(i) Find the conditional probability of obtaining a sum greater than 9, given that the black die resulted in a 5
(ii) Find the conditional probability of obtaining the sum 8, given that the red ðie resulted in a number less than 4. (4)

RBSE Class 12 E-Mathematics Self Evaluation Test Paper 2 in English

Section – A

Question 1.
Multiple Choice Questions
(i) If f(x) = log_e\(\left(\frac{1-x}{1+x}\right)\), |x| < 1,then f\(\left(\frac{2 x}{1+x^{2}}\right)\) is: (1) equal to:
(a) 2f(x)
(b) 2f(x²)
(c) (f(x))²
(d) -2 f(x)

(ii) The principal value of sec^-1(- 2) is: (1)
(a)
(b)
(c)
(d)

(iii) If 3A – B = \(\left[\begin{array}{ll}
5 & 0 \\
1 & 1
\end{array}\right]\) and B = \(\left[\begin{array}{ll}
4 & 3 \\
2 & 5
\end{array}\right]\), then the value of matrix A is: (1)
(a) \(\left[\begin{array}{ll}
3 & 1 \\
1 & 2
\end{array}\right]\)
(b) \(\left[\begin{array}{ll}
3 & 1 \\
1 & 2
\end{array}\right]\)
(c) \(\left[\begin{array}{ll}
1 & 2 \\
3 & 1
\end{array}\right]\)
(d) \(\left[\begin{array}{ll}
1 & 1 \\
2 & 3
\end{array}\right]\)

(iv) If A is a square matrix of order 3, such that A(adj A) 101, then I adj A I is equal to: (1)
(a) 1
(b) 10
(c) 100
(d) 101

(v) If y = A sin x + B cos x, then \(\frac{d^{2} y}{d x^{2}}\) + y is: (1)
(a) cos x
(b) sin x
(c) 1
(d) 0

(vi) ∫\(\frac{e^{2 x}-1}{e^{2 x}+1}\) dx (1)
(a) log(1 – e^-x) + C
(b) log(1 + e^-x) + C
(c) log|e^x + e^-x| + C
(d) log|e^x – e^-x| + C

(vii) Which of the following differential equatíms has y x as one of its particular solution? (1)
(a) \(\frac{d^{2} y}{d x^{2}}-x^{2} \frac{d y}{d x}\) + xy = x
(b) \(\frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}\) + xy = x
(c) \(\frac{d^{2} y}{d x^{2}}-x^{2} \frac{d y}{d x}\) + xy = 0
(d) \(\frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}\) + xy = 0

(viii) The angle between two vectors « and b with magnitudes √3 and 2, respectively having \(\vec{a} \cdot \vec{b}\) = √6. (1)
(a) \(\frac{\pi}{3}\)
(b) \(\frac{\pi}{4}\)
(c) \(\frac{\pi}{2}\)
(d) \(\frac{3 \pi}{4}\)

(ix) If A apd B are any two events such that P (A) + P (B) – P (A and B) – P (A), then (1)
(a) P (B/A) – 1
(b) P(A/B) = 1
(c) P(B/A) = 0
(d) P (A/B) = 0

(x) The number of matrices of order 3 × 3 matrices whose each entry is 0 or 1 will: (1)
(a) 27
(b) 18
(c) 81
(d) 512

(xi) If y = log (log x), then \(\frac{d y}{d x}\) is : (1)
(a) log x
(b) (log x)
(c) (x log x)^-1
(d) x^-1

(xii) If \(\vec{a} \cdot \vec{b}\) = 0 and \(\vec{a}+\vec{b}\) makes an angle of 60″ with \(\vec{a}\) then : (1)
(a) |\(\vec{a}\)| = √3|\(\vec{b}\)|
(b) \(\vec{a}\) = 2|\(\vec{b}\)|
(c) 2\(|\vec{a}|=|\vec{b}|\)
(d) √3\(|\vec{a}|=|\vec{b}|\)

Question 2.
Fill in the blanks
(i) If f: R → Randg :R → R are given f(x) = |x| and g(x) = |5x – 2|, then fog is _______________ (1)

(ii) The value of tan^-1cos^-1\(\left(\frac{1}{2}\right)\) – 2sin\(\left(\frac{-1}{2}\right)\) is _______________ (1)

(iii) If \(\left[\begin{array}{cc}
x+y & y \\
9 & x-y
\end{array}\right]=\left[\begin{array}{ll}
2 & 7 \\
9 & 4
\end{array}\right]\), then x, y _______________ (1)

(iv) The value of \(\frac{d}{dx}\)sin(cosx²) is _______________ (1)

(v) The value of ∫\(\frac{\sec ^{2} x}{{cosec}^{2} x}\)dx is _______________ (1)

(vi) The magnitude of \(\frac{1}{\sqrt{3}}\)î + \(\frac{1}{\sqrt{3}}\)ĵ – \(\frac{1}{\sqrt{3}}\)k̂ is _______________ (1)

Question 3.
Very Short Answer Type Questions
(i) Show that the function f: N → N, given by f(1) = f(2) = 1 and f(x) = x – 1, for every x > 2, is onto but not one-one. (1)

(ii) Solve the equation for x: 2 tan^-1(sin x) = tan^-1(2 sec x), x ≠ \(\frac{\pi}{2}\). (1)

(iii) Consuct a 2 × 2 matrix A = [a_ij] whose elerneiits e given by: a_ij = \(\frac{(2 i+j)^{2}}{2}\) (1)

(iv) If \(\left|\begin{array}{cc}
2 x & 5 \\
8 & x
\end{array}\right|=\left|\begin{array}{cc}
6 & -2 \\
7 & 3
\end{array}\right|\), then write the value of x. (1)

(v) Differentiate x^{sin x}, x > 0 w.r.t. x. (1)

(vi) Find: ∫cos 6x\(\sqrt{1+\sin 6 x}\) dx (1)

(vii) Show that the function y = A cos – B sin x, is a solution of the differential equation \(\frac{d^{2} y}{d x^{2}}\) + y = 0 (1)

(viii) Find the vector joining the points P(2, 3, 0) and Q (- 1, – 2, – 4) directed from P to Q. (1)

(ix) If (\(\frac{B}{A}\)) = 0.2 and P(A) = 0.8, then find P(A ∩ B). (1)

(x) Using determinants find the equation of line joining the points (3, 1) and (9, 3). (1)

(xi) Form the differential equation representing the family of ellipses having foci on x-axis and centre at the origin. (1)

(xii) If \(\vec{a}\) = 5î – ĵ – 3k̂ and \(\vec{b}\) = î + 3ĵ – 5k̂,thenshowthittIwectors \(\vec{a}+\vec{b}\) and \(\vec{a}-\vec{b}\) are perpendicular. (1)

Section – B
Short Answer Type Questions

Question 4.
Show that the relation R in the set A = {1, 2, 3, 4, 5, 6} given by R = -{(a, b) : |a – b| is even) is an equivalence relation. (2)

Question 5.
Find the matrix A such that A\(\left[\begin{array}{lll}
1 & 2 & 3 \\
4 & 5 & 6
\end{array}\right]=\left[\begin{array}{rrr}
-7 & -8 & -9 \\
2 & 4 & 6
\end{array}\right]\) (2)

Question 6.
For what value of k the system of linear equations
x + y + z = 2; 2x + y – z = 3 and 3x + 2y + kz = 4 has a unique solution? (2)

Question 7.

Determine the value of a so that f(x) is continuous at x = 0. (2)

Question 8.
Find : ∫\(\frac{x^{3}}{\left(x^{2}+1\right)\left(x^{2}+4\right)}\)dx (2)

Question 9.
Two candidates A and B had applied for a vacancy. Probability, that A is selected, is \(\frac{1}{3}\) and that B is selected, is \(\frac{1}{5}\). Find the probability that at least one is selected. (2)

Question 10.
Obtain the inverse of the following matrix using elementary operations: (2)
A = \(\left[\begin{array}{ccc}
2 & 1 & -3 \\
-1 & -1 & 4 \\
3 & 0 & 2
\end{array}\right]\)

Question 11.
If x = a(cos t + t sin t) and y = a(sint – t cos t) then find \(\frac{d^{2} y}{d x^{2}}\). (2)

Question 12.
Using properties of determinants, prove that dx
\(\left|\begin{array}{ccc}
1 & 1 & 1+3 x \\
1+3 y & 1 & 1 \\
1 & 1+3 z & 1
\end{array}\right|\) = 9(3xyz +xy + yz + zx). (2)

Question 13.
Evaluate:

Question 14.
Find the general solution of the differential equation \(\frac{d y}{d x}+\sqrt{\frac{1-y^{2}}{1-x^{2}}}\) = 0. (2)

Question 15.
Find \(|\vec{x}|\), if for a unit vector \(\vec{a},(\vec{x}-\vec{a}) \cdot(\vec{x}+\vec{a})\) = 12. (2)

Question 16.
A die is thrown twice and the sum of the numbers appealing is observed to be 6. What is the conditional probability that the number 4 has appeared at least once? (2)

Section – C
Long Answer Type Questions

Question 17.
Find the value of tan [sin^-1(\(\frac{3}{5}\)) + cot^-1(\(\frac{3}{2}\)) (3)
Or
Show that : sin^-1(\(\frac{3}{5}\)) – sin^-1(\(\frac{8}{17}\)) = cos^-1(\(\frac{84}{85}\)). (3)

Question 18.
If (tan^-1x)^y + y^{cot x} = 1, then find \(\frac{dy}{dx}\). (3)
Or
Verify Mean Value Theorem if f(x) = x³ – 5x² – 3x in the interval [a, b] where a = 1 and b = 3. Find all c ∈ (1, 3) for
f'(c) = 0. (3)

Question 19.
Find : ∫\(\frac{4 x+1}{\sqrt{2 x^{2}+x-3}}\) dx. (3)
Or
Find : ∫\(\frac{6 x+7}{\sqrt{(x-5)(x-4)}}\) dx. (3)

Question 20.
If \(\vec{a}\) ¡ + j ÷ k and \(\vec{b}\) = j k then find a vector \(\vec{c}\), such that \(\vec{a} \times \vec{c}=\vec{b}\) and \(\vec{a} \cdot \vec{c}\) = 3. (3)
Or
Find the unit vector in the direction of the vector PQ, where P and Q are the points (1, 2, 3) and (4, 5, 6) respectively. (3)

Section – D
Essay Type Questions

Question 21.
Evaluate : (4)

Or
Find : (4)

Question 22.
Show that the given differentiál equation is homogçneous and solve it: (x² + xy) dy = (x² + y²) dx. (4)
Or
Find the general solution of the differential equation: \(\frac{dy}{dx}\) = cos x. (4)

Question 23.
Given three identical boxes I, II and Ill, each coñtiung two coins. In box I, both coins are gold coins, in box II, both are silver coins and in the box ill, there is one gold and one silver coin. A person chooses a box at random aid takes out a coin. If the çoin is of gold, what is the probability that the other coin in the box is also of gold? (4)
Or
A random variable X has the following probability distribution: (4)

Determine
(i) k
(ii) P(X < 3) (iii) P(X > 6)
(iv) P(0 < X < 3)

RBSE Class 12 E-Mathematics Self Evaluation Test Paper 3 in English

Section – A

Question 1.
Multiple Choice Questions
(i) A relation R is defined in the set of integers I as follows (x, y) ∈ R if x² + y² = 9 which of the following is false? (1)
(a) R = {(0,3), (0, – 3), (3, 0), (-3,0)}
(b) Domain of R = {- 3. 0, 3}
(c) Range of R = {- 3, 0, 3}
(d) None of the above

(ii) The value of 2 cot^-1(\(\frac{1}{2}\)) – cot^-1(\(\frac{4}{3}\)) is : (1)
(a) \(-\frac{\pi}{8}\)
(b) \(\frac{3 \pi}{2}\)
(c) \(\frac{\pi}{4}\)
(d) \(\frac{\pi}{2}\)

(iii) If 2A + B + X = O, where A = \(\left[\begin{array}{rr}
-1 & 2 \\
3 & 4
\end{array}\right]\) and B = \(\left[\begin{array}{rr}
3 & -2 \\
1 & 5
\end{array}\right]\), then matrix X is:
(a) \(\left[\begin{array}{rr}
-1 & -2 \\
-7 & -13
\end{array}\right]\)
(b) \(\left[\begin{array}{cc}
-1 & 2 \\
-7 & 13
\end{array}\right]\)
(c) \(\left[\begin{array}{rr}
-1 & -2 \\
7 & 13
\end{array}\right]\)
(d) \(\left[\begin{array}{rr}
1 & 2 \\
7 & 13
\end{array}\right]\) (1)

(iv) If A = \(\left[\begin{array}{ccc}
1 & \sin \theta & 1 \\
-\sin \theta & 1 & \sin \theta \\
-1 & -\sin \theta & 1
\end{array}\right]\), where 0 ≤ θ ≤ 2π, then (1)
(a) Det (A) = 0
(b) Det (A) ∈ (2, ∞)
(c) Det (A) ∈ (2, 4)
(d) Det (A) ∈ [2, 4]

(v)

is continuous at x = 3, then value of n is: (1)
(a) 2.25
(b) 1.25
(c) -2.25
(d) -1.25

(vi) ∫\(\frac{\sqrt{x}}{\sqrt{a^{3}-x^{3}}}\) dx is equal to : (1)
(a) sin^-1\(\left(\frac{x^{3}}{a^{3}}\right)\) + C
(b) sin^-1\(\left(\sqrt{\frac{x^{3}}{a^{3}}}\right)\) + C
(c) \(\frac{2}{3}\)sin^-1\(\left(\sqrt{\frac{a^{3}}{x^{3}}}\right)\) + C
(d) \(\frac{2}{3}\)sin^-1\(\left(\sqrt{\frac{x^{3}}{a^{3}}}\right)\) + C

(vii) The degree of the differential equation x\(\left(\frac{d^{2} y}{d x^{2}}\right)^{3}\) + y\(\left(\frac{d y}{d x}\right)^{4}\) + x³ = 0, is: (1)
(a) 1
(b) 2
(c) 3
(d) 4

(viii) If |\(\vec{a}\)| =1, |\(\vec{b}\)| = 4. \(\vec{a} \cdot \vec{b}\) =2 and \(\vec{c}\) = 2\(\vec{a} \times \vec{b}-3 \vec{b}\), then the angle between \(\vec{b}\) and \(\vec{c}\) is: (1)
(a) \(\frac{\pi}{3}\)
(b) \(\frac{3 \pi}{6}\)
(c) \(\frac{5 \pi}{6}\)
(d) \(\frac{\pi}{6}\)

(ix) A die is thrown once. Let A be the event that the number obtained is greater than 3. Let B be the event that the number obtained is less than 5. Then P(A ∪ B) is: (1)
(a) \(\frac{2}{5}\)
(b) \(\frac{3}{5}\)
(c) 0
(d) 1

(x) Find the value of x, for the following: [1 × 1]\(\left[\begin{array}{ccc}
1 & 3 & 2 \\
2 & 5 & 1 \\
15 & 3 & 2
\end{array}\right]\left[\begin{array}{l}
1 \\
2 \\
x
\end{array}\right]\) = 0 is: (1)
(a) – 2
(b) 3
(c) – 4
(d) – 5

(xi) If x = a sec θ and y = b tanθ ,then \(\frac{dy}{dx}\) is: (1)
(a) cosec θ
(b) sin θ
(c) \(\frac{a}{b}\) sin θ
(d) \(\frac{b}{a}\) cosec θ

(xii) If \(\vec{a}\) = î – 2ĵ + 3k̂ and \(\vec{b}\) is a vector such that \(\vec{a} \cdot \vec{b}=|\vec{b}|^{2}\) and \(|\vec{a}-\vec{b}|=\sqrt{7}\),then \(|\vec{b}|\) is equal to: (1)
(a) 7
(b) 3
(c) √7
(d) √3

Question 2.
Fill in the blanks:
(i) If f(x) = \(\frac{(4 x+3)}{(6 x-4)}\), x ≠ \(\frac{2}{3}\) then,f0f(x) = x, ________________ (1)

(ii) The value of tan(2 tan^-1\(\frac{1}{5}\)) is ________________ (1)

(iii) If \(\left[\begin{array}{ll}
1 & 3 \\
4 & 5
\end{array}\right]\left[\begin{array}{l}
x \\
2
\end{array}\right]=\left[\begin{array}{l}
5 \\
6
\end{array}\right]\), then x = ________________ (1)

(iv) If y = x cos x, then \(\frac{dy}{dx}\) = ________________ (1)

(v) The value of ∫¹_-1 dx is ________________ (1)

(vi) The direction cosines of vector equally inclined to the axes OX, OY and OZ are ________________ (1)

Question 3.
Very Short Answer Type Questions one-one function.
(i) Let f: N → N be a function defined asf(x) = 9x² + 6x – 5. Show that f: N → S, where S s the range of f, is one-one function. (1)

(ii) If tan^-1\(\left(\frac{x-1}{x-2}\right)\) + tan^-1\(\left(\frac{x+1}{x+2}\right)\) = \(\frac{\pi}{4}\), then find the value of x. (1)

(iii) Using elementary row transformations, find inverse of matrix A = \(\left[\begin{array}{ll}
6 & 5 \\
5 & 4
\end{array}\right]\) (1)

(iv) Find the equation of the line joining A (1, 3) and B (0, 0) using determinants and find k if D (k, 0) is a point such that area of ∆ABD is 3 sq. units. (1)

(v) If y = cos^-1\(\left(\frac{2 x}{1+x^{2}}\right)\), – 1 < x < 1, then find \(\frac{dy}{dx}\). (1)

(vi) Evaluate : ∫\(\frac{\sqrt{16+(\log x)^{2}}}{x}\)dx. (1)

(vii) Form the differential equation of the family of circles touching they-axis at origin, (1)

(viii) Find the values of x and y so that the vectors 2î + 3ĵ and xî + yĵ are equal. (1)

(ix) From die set {1, 2, 3, 4, 5} two numbers a and b (a ≠ b) are chosen at random. Find the probability that \(\frac{a}{b}\) is an integer. (1)

(x) If points (2, λ), (3, 2λ) and (7, 3λ) are collinear, then find A. (1)

(xi) Find the general solution of the differential equation y dx – (x + 2y²)dy = 0. (1)

(xii) Write the direction ratio’s of the vector \(\vec{a}\) = î + ĵ – 2k̂ and hence calculate its direction cosines. (1)

Section – B
Short Answer Type Questions

Question 4.
Let A = R – {2} and B = R – {1}. If f: A → B is a function defined by f(x) = \(\frac{x-1}{x-2}\), show that f is one-one and onto Find f^-1. (2)

Question 5.
Show that th element on the main diagonal of skew symmetric matrix all zero. (2)

Question 6.
If A = \(\left[\begin{array}{rrr}
2 & -3 & 5 \\
3 & 2 & -4 \\
1 & 1 & -2
\end{array}\right]\), find A^-I. Using A^-1 solve the following system of equations:
2x – 3y + 5z = 11; 3x + 2y -4: = -5 and x + y – 2z = – 3 . (2)

Question 7.
Verify Rolle’s theorem for the function f(x) = x² – 4x + 3 on the interval [1, 3]. (2)

Question 8.
Evaluate:∫\(\frac{3 x+5}{x^{2}+3 x-18}\)dx (2)

Question 9.
A card from a pack of 52 playing cards is lost. From the remaining cards of the pack three cards are drawn at random (without replacement) and are found to be in 1 spades.’ Find the probability of the lost card being a spade. (2)

Question 10.
Find the values of x and y from the following equation: \(\left[\begin{array}{cc}
x & 5 \\
7 & y-3
\end{array}\right]+\left[\begin{array}{cc}
3 & -4 \\
1 & 2
\end{array}\right]=\left[\begin{array}{cc}
7 & 6 \\
15 & 14
\end{array}\right]\) (2)

Question 11.
If x^y = e^x-y then prove that \(\frac{d y}{d x}=\frac{\log x}{(1+\log x)}\) (2)

Question 12.
Prove that : \(\left|\begin{array}{lll}
a^{3} & 2 & a \\
b^{3} & 2 & b \\
c^{3} & 2 & c
\end{array}\right|\) = 2(a – b)(b – c)(c – a)(a + b + c) (2)

Question 13.
Find : ∫²₀ x\(\sqrt{x+2}\) dx (2)

Question 14.
Find the differential equation representing the family of curves V = \(\frac{A}{r}\) + B, where A and B are arbitrary constants. (2)

Question 15.
Find a vector in the direction of vector 5î – ĵ + 2k̂ which has magnitude 8 units. (2)

Section – C
Long Answer Type Questions

Question 17.
Prove that: tan^-1\(\left[\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}\right]=\frac{\pi}{4}-\frac{1}{2}\)cos^-1x, \(\) ≤ x ≤ 1. (3)
Or
Prove that cot^-1\(\left[\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right]=\frac{x}{2}\), x ∈ (0, \(\frac{π}{4}\)) (3)

Question 18.
If y = x^x, then prove that \(\left[\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right]=\frac{x}{2}\) = 0 (3)
Or
Examine if Rolle’s theorem is applicable to any of the following functions. Can you say something about the converse of Rolle’s theorem from these examples? (3)
(i) f(x)= [x] for x ∈ [5, 9] (ii) f(x) = [x] for x ∈ [- 2, 2] (iii) f(x) = x² – 1 for x ∈ [1, 2],

Question 19.
Evaluate : ∫\(\frac{\sqrt{x^{2}+1}\left[\log \left|x^{2}+1\right|-2 \log |x|\right]}{x^{4}}\) dx (3)
Or
Evaluate : ∫(x + 3)\(\sqrt{3-4 x-x^{2}}\)dx (3)

Question 20.
Find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5). (3)
Or
Find the position Vector of a point R which dicides the lines joining two points P and Q whose position vectors are P(2 \(\vec{a}+\vec{b}\)) and Q( \(\vec{a}-3 \vec{b}\)) externally in tlw ratio 1: 2. Also; show that P is the mid joint of the line segment RQ. (3)

Section – D
Essay Type Questions

Question 21.
Find : (4)

Or
Find: (4)

Question 22.
Show that the given differential equation is he nogeneous and solve it: (x² – y²) dx + 2xy dy = 0 (4)
Or
For th following diFferential equation, luid th general solution: \(\frac{dy}{dx}\) + (sec x)y = tan x (o ≤ x < \(\frac{π}{2}\)) (4)

Question 23.
Assume that each born child is equally li1’ y to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls given that (j) the youngest is a girl, (ii) at least one is a girl? (4)
Or
Bag I contains 3 red and 4 black balls while another Bag II contains 5 red and 6 black balls. One ball is drawn at random from one of the bags and it is found to be red. Find the probabilty that it was drawn from Bag II. (4)

RBSE Class 12 E-Mathematics Self Evaluation Test Paper 4 in English

Section – A

Question 1.
Multiple Choice Questions
(i) Let A = {1, 2, 3}. Then number of relations containing (1,2) and (1, 3) which are reflexive and symmetric but not transitive is : (1)
(a) 1
(b) 2
(c) 3
(d) 4

(ii) sin{cot^-1(tan cos^-1 x)} (1)
(a) x
(b) \(\sqrt{1-x^{2}}\)
(c) 1/x
(d) – x

(iii) If A = \(\left[\begin{array}{ll}
1 & -1 \\
2 & -1
\end{array}\right]\) B = \(\left[\begin{array}{rr}
a & 1 \\
b & -1
\end{array}\right]\) and (A + B)² = A² + B², the values of a and b are: (1)
(a) a = 1,b = 4.
(b) a = 2, b = 3
(c) a = 3, b = 4
(d) a = 1, b = 2

(iv) Let A = \(\left[\begin{array}{cc}
200 & 50 \\
10 & 2
\end{array}\right]\) and B = \(\left[\begin{array}{cc}
50 & 40 \\
2 & 3
\end{array}\right]\), then |AB| is equal to: (1)
(a) 460
(b) 2000
(c) 3000
(d) -7000

(v) If f(x) = \(\frac{2-(256+5 x)^{\frac{1}{8}}}{(5 x+32)^{\frac{1}{5}}-2}\) (x ≠ 0), then forf to be continuous every where f(0) is equal to: (1)
(a) \(\frac{2}{7}\)
(b) \(\frac{-7}{32}\)
(c) \(\frac{7}{64}\)
(d) \(\frac{-7}{64}\)

(vi) If f(x) = ∫^x ₀ t sin t dt, then f'(x) is: (1)
(a) cot x + x sin x
(b) x sin x
(c) x cos x
(d) sin x + x cos x

(vii) The degree of the differential equation \(\left(\frac{d^{2} y}{d x^{2}}\right)^{3}+\left(\frac{d y}{d x}\right)^{2}\) + sin\(\left(\frac{d y}{d x}\right)\) + 1 = 0. (1)
(a) 3
(b) 2
(c) 1
(d) not defined

(viii) Area of a rectangle having vertices A, 8, Cand D with position vectors.
-î + \(\frac{1}{2}\)ĵ + 4k̂, î + \(\frac{1}{2}\)ĵ + 4k̂, î – \(\frac{1}{2}\)ĵ + 4k̂ and -î – \(\frac{1}{2}\)ĵ + 4k̂ respectively is: (1)
(a) \(\frac{1}{2}\)
(b) 1
(c) 2
(d) 4

(ix) A bag contains 3 white, 4 black and 2 red balls. 112 balls are drawn at random (without replacement), then tite probability that both the balls are white, is: (1)
(a) \(\frac{1}{18}\)
(b) \(\frac{1}{36}\)
(c) \(\frac{1}{12}\)
(d) \(\frac{1}{24}\)

(x) If the matrix A = \(\left[\begin{array}{rrr}
0 & a & -3 \\
2 & 0 & -1 \\
b & 1 & 0
\end{array}\right]\) a skew-synimetri matrix then the values of ‘a’ and ‘b’ are: (1)
(a) a = 3, b = 1
(b) a = 3, b = -2
(c) a = 2, b = 3
(d) a = 2, b = 3

(xi) If x = a (2θ – sin 2θ) and y = a(1 – cos 2θ), then \(\frac{d y^{\prime}}{d x}\),when θ = \(\frac{\pi}{3}\).
(a) \(\frac{1}{\sqrt{3}}\)
(b) \(\frac{1}{\sqrt{2}}\)
(c) \(\frac{1}{\sqrt{7}}\)
(d) \(\frac{1}{\sqrt{5}}\)

(xii) Suppose \(\vec{a}\) = λî – 7ĵ + 3k̂, \(\vec{b}\) = λî + ĵ + 2λk̂. If the angle between \(\vec{a}\) and \(\vec{b}\) is greater than 90°, then λ satisfies the inequality:
(a) λ > 1
(b) – 7 < λ < 1
(c) – 5 < λ < 1
(d) 1 < λ < 7

Question 2.
Fill in the blanks:
(i) Let R be a relation in N defined by R = {(1 + x, 1 + x²): x ≤ 5, x ∈ N}, then R = ____________ . (1)

(ii) The value of tan^-1 \(\left(\tan \frac{3 \pi}{4}\right)\) is ____________ . (1)

(iii) If A = \(\left[\begin{array}{rr}
1 & 0 \\
-1 & 7
\end{array}\right]\) and A² = 8A + kI², then k = ____________ . (1)

(iv) x³ + x²y + xy² + y³ = 81, then \(\frac{d y}{d x}\) is ____________ . (1)

(v)

is equal to A ____________ . (1) .

(vi) The direction cosines of the vector î + 2ĵ + 3k̂ is ____________ . (1)

Question 3.
Very Short Answer Type Questions
(i) Consider the binary operation * on the set {1, 2, 3, 4, 5} defined by a * b = min {a, b}. Write operation table of operation *. (2)

(ii) Find the value of: cos^-1\(\left(\frac{1}{2}\right)\) + 2 sin^-1\(\left(\frac{1}{2}\right)\). (2)

(iii) For the matrix A = \(\left[\begin{array}{ll}
2 & 3 \\
5 & 7
\end{array}\right]\), find A + A^T and verify that it is a symmetric matrix.

(iv) Given, A = \(\left[\begin{array}{cc}
2 & -3 \\
-4 & 7
\end{array}\right]\), compute A^-1 and show that 2A^-1 = 9I – A.

(v) If sin²y + cos xy = k, then find \(\frac{d y}{d x}\)

(vi) Find : ∫x sec²x dx

(vii) Find the general solution of the differential equation \(\frac{d y}{d x}+\frac{1}{x}=\frac{e^{y}}{x}\)

(viii) Find \(|\vec{a}-\vec{b}|\), if two vectors \(\vec{a}\) and \(\vec{b}\) are such,that |\(\vec{a}\)| = 2, |\(\vec{b}\)| = 3 and \(\vec{a} \cdot \vec{b}\) = 4.

(ix) If a leap ycar.is selected at random, what is the chance that it will contain 53 Tuesdays?

(x) Find values of k if area of triangle is 4 sq. units, and vertices are (- 2, 0), (0, 4) and (0, k).

(xi) Find the differential equation representing the family of curves y = ae^x + be^-x + x², where a and b are arbitrary constants.

(xii) If \(\vec{a}\) is a unit vector \((\vec{x}-\vec{a}) \cdot(\vec{x}+\vec{a})\) = 8

Section – B
Short Answer Type Questions

Question 4.
Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} a R = {(a, b): b = a + 1} is reflexive, symmetric or transitive. (2)

Question 5.
Let A = \(\left[\begin{array}{rr}
2 & -1 \\
3 & 4
\end{array}\right]\), B = \(\left[\begin{array}{ll}
5 & 2 \\
7 & 4
\end{array}\right]\), C = \(\left[\begin{array}{ll}
2 & 5 \\
3 & 8
\end{array}\right]\), find a matrix D such that CD – AB = O.

Question 6.
Solve the following equations: (2)
\(\frac{2}{x}+\frac{3}{y}+\frac{10}{z}\) = 4, \(\frac{4}{x}-\frac{6}{y}+\frac{5}{z}\) = 1 and \(\frac{6}{x}+\frac{9}{y}-\frac{20}{z}\) = 2

Question 7.
Find all the points_ot discontinuity off del irwd byf(x) = |x| – |x + 1|. (2)

Question 8.
Evaluate: ∫\(\sqrt{x^{2}-x+1}\) dx .

Question 9.
Ten cards numbered 1 to 10 are placed in a box, mixed up thoroughly and then one card is drawn randomly.
If it is known that the number on the drawn card is more than 3, what is the probability that it is an even number? (2)

Question 10.
If A = \(\left[\begin{array}{ccc}
1 & 3 & 2 \\
2 & 0 & -1 \\
1 & 2 & 3
\end{array}\right]\), then show that A³ – 4A² – 3A + 111 = 0. Hence, find A^-1. (2)

Question 11.
If y = sin (sin x), then prove that: \(\frac{d^{2} y}{d x^{2}}\) + tan x \(\frac{d y}{d x}\)+ y cos² x = 0 (2)

Question 12.
If x, y, z are different and \(\left|\begin{array}{ccc}
x & x^{2} & 1+x^{3} \\
y & y^{2} & 1+y^{3} \\
z & z^{2} & 1+z^{3}
\end{array}\right|\) = 0, then using properties of determinants show that 1 + xyz = 0. (2)

Question 13.
Find (2)

Question 14.
Find the particular solution of differential equation \(\frac{d y}{d x}\) = – \(\frac{x+y \cos x}{1+\sin x}\) given that q = 1, when x = 0. (2)

Question 15.
For given vectors \(\vec{a}\) = 2î – ĵ + 2k̂ and \(\vec{b}\) = – î + ĵ – k̂, find the unit vector in the direction of the vector \(\vec{a}+\vec{b}\). (2)

Question 16.
In a shop X, 30 tins of ghee of type A and 40 tins of ghee of type B which look alike, are kept for sale. While in shop y, similar 50 tins of ghee of type A and 60 tins of ghee of type B are there. One tin of ghee is purchased from one of the randomly selected shop and is found to be of type B. Find the probability that it is purchased from shot y. (2)

Section – C
Long Answer Type Questions

Question 17.
prove that: tan^-1\(\left(\frac{63}{16}\right)\) = sin^-1\(\left(\frac{5}{13}\right)\) + cos^-1\(\left(\frac{3}{5}\right)\)
Or
Prove that: sin^-1\(\left(\frac{8}{17}\right)\) + sin^-1\(\left(\frac{3}{5}\right)\) = tan^-1\(\left(\frac{77}{36}\right)\)

Question 18.
If y = tan^-1\(\left(\frac{a}{x}\right)\) + log\(\sqrt{\frac{x-a}{x+a}}\), prove that \(\frac{d y}{d x}=\frac{2 a^{3}}{x^{4}-a^{4}}\)
Or
Show that the function f defined by f(x) = |1 – x + |x||, where x is any real number, is a continuous function. (3)

Question 19.
Find: ∫\(\frac{1}{\sqrt{(x-1)(x-2)}}\)dx
Or
Find: ∫\(\frac{(x+3)}{x^{2}-2 x-5}\)dx

Question 20.
The scalar product of the vector î + ĵ + k ,with a unit vector.along the sum of vectors 2î + 4ĵ – 5k̂ and λî + 2ĵ + 3k̂ is equal to one. Find the value of λ. (3)
Or
Find a vector of magnitude 5 units, and parallel to the resultant of the vectors \(\vec{a}\) = 2î + 3ĵ – k̂ and \(\vec{b}\) = î – 2ĵ + k̂

Section – D
Essay Type Questions 

Question 21.
Find: (4)

Or
Find: (4)

Question 22.
For the following differential equation, find the particular solution satisfying the given condition: (4)
(x + y) dy + (x – y) dx = 0; y = 1 when x = 1.
Or
Show that the given differential equation is homogeneous and solve it : y’ = \(\frac{x+y}{x}\) (4)

Question 23.
A man is known to speak truth 3 out of 4 times. He throws a die and reports that it iš a six. Find the probability that it is actually a six. (4)
Or
An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of an accident are 0.01, 0.03 and 0.15 respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver? (4)