KEAM 2026 Engineering Question Paper for April 19 is available for download here. CEE Kerala conducted KEAM 2026 Engineering exam on April 19 in session 2 from 2 PM to 5 PM. KEAM 2026 Engineering exam is an online CBT with a total of 150 questions carrying a maximum of 600 marks.

  • The KEAM Engineering exam is divided into 3 subjects- Physics (45 questions), Chemistry (30 questions) and Mathematics (75 questions).
  • 4 marks are given for every correct answer and 1 mark is deducted for every incorrect answer

Candidates can download KEAM 2026 April 19 Engineering Question Paper with Solution PDF from the links provided below.

KEAM 2026 Engineering April 19 Question Paper with Solution PDF

KEAM 2026 Engineering Question Paper April 19 Download PDF Check Solution
KEAM 2026 Engineering April 19 Question Paper with Solutions

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Question 1:

Let \(X = \{a_1, a_2, a_3, \ldots, a_n\}\) be a set consisting of \(n\) elements. The relation \(R = \{(a_1,a_1),(a_2,a_2),(a_3,a_3),\ldots,(a_n,a_n)\}\) on the set \(X\) is:

  • (A) reflexive, symmetric but not transitive
  • (B) reflexive, transitive but not symmetric
  • (C) transitive, symmetric but not reflexive
  • (D) reflexive, symmetric and transitive
  • (E) reflexive, not symmetric and not transitive

Question 2:

Let \(X = \{a,b,c,d,e,f\}\) and \(Y = \{7,8,9,10,11\}\) be two sets. Which one of the following is true?

  • (A) \(\{(a,8),(b,7),(c,9),(d,10),(e,11)\}\) is one-to-one function from \(X\) to \(Y\)
  • (B) \(\{(a,7),(b,11),(c,8),(d,10),(e,9),(f,11)\}\) is one-to-one function from \(X\) to \(Y\)
  • (C) \(\{(a,7),(b,8),(c,9),(d,10),(e,11)\}\) is one-to-one function from \(X\) to \(Y\)
  • (D) \(\{(a,11),(b,10),(c,9),(d,8),(e,7),(f,9)\}\) is one-to-one function from \(X\) to \(Y\)
  • (E) one-to-one function cannot be defined from \(X\) to \(Y\)

Question 3:

Let \(f(x) = \dfrac{2x+3}{x-2}, \, x \in \mathbb{R}, \, x \neq 2\) and \(h(x) = f(f(x))\). Then \(h(h(10))\) is equal to:

  • (A) \(100\)
  • (B) \(20\)
  • (C) \(10\)
  • (D) \(1000\)
  • (E) \(1\)

Question 4:

The inverse of the function \(f(x) = x^2 + 4x + 4, \, x \leq -2\) is \(f^{-1}(x) =\)

  • (A) \(-2 - \sqrt{x}, \, x \geq 0\)
  • (B) \(-2 - \sqrt{x-1}, \, x \geq 1\)
  • (C) \(-2 - \sqrt{x}, \, x \geq 2\)
  • (D) \(-2 - \sqrt{x}, \, x \geq 4\)
  • (E) \(-2 - \sqrt{x}, \, x \geq 5\)

Question 5:

Given that \(i^2 = -1\). Then \(i^{13} + i^{14} + i^{15} + \ldots + i^{2026}\) is equal to

  • (A) \(i - 2\)
  • (B) \(i + 2\)
  • (C) \(2i + 1\)
  • (D) \(i - 1\)
  • (E) \(-i + 1\)

Question 6:

Let \(x\) and \(y\) be real numbers. If \((3+i)x + y + (1-i)y + 3i - 4 = (2x+1)i + (x-y+2)i\), where \(i=\sqrt{-1}\), then the pair \((x,y)\) is equal to

  • (A) \((1,2)\)
  • (B) \((0,2)\)
  • (C) \((0,-2)\)
  • (D) \((3,2)\)
  • (E) \((-1,-2)\)

Question 7:

Let \(z_1 = \dfrac{5+7i}{7-5i}, \, z_2 = \dfrac{3+2i}{3-2i}\) and \(z_3 = \dfrac{1+11i}{11-i}\). Then \(z_1\overline{z_1} + z_2\overline{z_2} + z_3\overline{z_3}\) is equal to

  • (A) \(2\)
  • (B) \(1+2i\)
  • (C) \(1\)
  • (D) \(3\)
  • (E) \(1-2i\)

Question 8:

The value of \(\dfrac{(1+i)^n}{(1-i)^{n-4}}\), where \(i=\sqrt{-1}\) and \(n\) is an integer, is

  • (A) \(\dfrac{i^n}{4}\)
  • (B) \(4i^n\)
  • (C) \(-4i^n\)
  • (D) \(-1\)
  • (E) \(1\)

Question 9:

The number of terms in the sequence \(2,6,18,\ldots,1458\) is

  • (A) \(14\)
  • (B) \(12\)
  • (C) \(10\)
  • (D) \(8\)
  • (E) \(7\)

Question 10:

Let \(t_1, t_2, t_3, \ldots, t_{2n}\) be in G.P. with common ratio \(r\). Then

  • (A) \(t_1,t_3,t_5,\ldots,t_{2n-1}\) are in G.P. with common ratio \(r\)
  • (B) \(t_1,t_4,t_7,\ldots,t_{2n-1}\) are in G.P. with common ratio \(r^2\)
  • (C) \(t_1,t_3,t_5,\ldots,t_{2n-1}\) are in G.P. with common ratio \(r^2\)
  • (D) \(t_2,t_4,t_6,\ldots,t_{2n}\) are in G.P. with common ratio \(r^3\)
  • (E) \(t_2,t_4,t_6,\ldots,t_{2n}\) are in G.P. with common ratio \(r^5\)

Question 11:

If \(\dfrac{4^{n+1} + 16^{n+1}}{4^n + 16^n}\) is the Geometric Mean between \(4\) and \(16\), then the value of \(n\) is

  • (A) \(\frac{1}{2}\)
  • (B) \(\frac{3}{2}\)
  • (C) \(10\)
  • (D) \(-\frac{1}{2}\)
  • (E) \(8\)

Question 12:

The first and last term of a G.P. are 7 and 448 respectively. If the sum is 889, then the common ratio is

  • (A) \(4\)
  • (B) \(2\)
  • (C) \(\frac{1}{2}\)
  • (D) \(\frac{1}{4}\)
  • (E) \(3\)

Question 13:

There are two main entrances to a building with five floors. Each entrance leads to three lifts and each lift can stop at all the five floors. A person enters the building and reaches a floor. The number of possible ways that the person can reach the floor, is

  • (A) \(15\)
  • (B) \(25\)
  • (C) \(10\)
  • (D) \(30\)
  • (E) \(50\)

Question 14:

If \({}^9P_5 = (504)({}^6P_r)\), then the value of \(r\) is equal to

  • (A) \(3\)
  • (B) \(2\)
  • (C) \(1\)
  • (D) \(4\)
  • (E) \(5\)

Question 15:

The sum of all 3-digit numbers that can be formed using \(1,2,3,4\) without repetitions is

  • (A) \(6668\)
  • (B) \(8886\)
  • (C) \(12486\)
  • (D) \(9876\)
  • (E) \(6660\)

Question 16:

A box contains 24 identical balls of which one ball is black and the remaining balls are green. Three balls are taken simultaneously and randomly. The number of ways of getting only green balls, is

  • (A) \(1765\)
  • (B) \(1764\)
  • (C) \(1763\)
  • (D) \(1771\)
  • (E) \(1864\)

Question 17:

The coefficient of \(\frac{1}{x^2}\) in the binomial expansion of \(\left(3x - \frac{1}{3x}\right)^4\) is

  • (A) \(\frac{4}{7}\)
  • (B) \(\frac{3}{8}\)
  • (C) \(\frac{2}{9}\)
  • (D) \(\frac{4}{9}\)
  • (E) \(-\frac{4}{9}\)

Question 18:

If \((x \;\; 3 \;\; -1) \begin{pmatrix} 1 & 1 & 1
-1 & 0 & 1
1 & 0 & -1 \end{pmatrix} \begin{pmatrix} 2
3
1 \end{pmatrix} = 0\), then the values of \(x\) are

  • (A) \(-2\)
  • (B) \(-\frac{1}{3}\)
  • (C) \(-3\)
  • (D) \(\frac{2}{3}\)
  • (E) \(-\frac{2}{3}\)

Question 19:

Let \(P = \begin{pmatrix} 1 & 0 & 0
0 & 1 & 0
10 & 100 & -1 \end{pmatrix}\). Then \(P^{4052}\) is equal to

  • (A) \(P\)
  • (B) \(P^T\)
  • (C) \(I\), the unit matrix of order 3
  • (D) \(-P^T\)
  • (E) \(2P^T\)

Question 20:

Evaluate the determinant \(\begin{vmatrix} 11 & 1 & 1
1 & 21 & 1
1 & 1 & 31 \end{vmatrix}\)

  • (A) \(7100\)
  • (B) \(6800\)
  • (C) \(7300\)
  • (D) \(6900\)
  • (E) \(6700\)

Question 21:

If \(A = \begin{pmatrix} 0 & 1
-1 & 0 \end{pmatrix}\) and \((\alpha I + \beta A)^2 = A\), where \(I\) is \(2 \times 2\) unit matrix, then \(\alpha^2 - \beta^2 =\)

  • (A) \(2\)
  • (B) \(-2\)
  • (C) \(-1\)
  • (D) \(1\)
  • (E) \(0\)

Question 22:

Let \(x\) be a real number such that \(5 < |x - 1| < 15\). Then

  • (A) \(-18 < x < -3\) or \(3 < x < 19\)
  • (B) \(-14 < x < -3\) or \(6 < x < 17\)
  • (C) \(-16 < x < -2\) or \(6 < x < 20\)
  • (D) \(-14 < x < -4\) or \(6 < x < 16\)
  • (E) \(-10 < x < -1\) or \(3 < x < 18\)

Question 23:

Let \(x\) be a real number such that \(\frac{x-3}{x-2} \geq 1\). Then the solution set of the inequality is

  • (A) \((-\infty,3)\)
  • (B) \((-\infty,2)\)
  • (C) \([0,\infty)\)
  • (D) \((-9,\infty)\)
  • (E) \((0,8)\)

Question 24:

If \(\sin \theta \cos \theta > 0\), then \(\theta\) lies

  • (A) only in the first quadrant
  • (B) only in the second quadrant
  • (C) in the first quadrant or in the fourth quadrant
  • (D) in the second quadrant or in the fourth quadrant
  • (E) in the first quadrant or in the third quadrant

Question 25:

If \(4\sin^2 x - 2(1+\sqrt{3})\sin x + \sqrt{3} = 0\) and \(15^\circ < x < 150^\circ\), then the values of \(x\) are

  • (A) \(30^\circ,45^\circ,90^\circ\)
  • (B) \(45^\circ,100^\circ,120^\circ\)
  • (C) \(30^\circ,60^\circ,120^\circ\)
  • (D) \(35^\circ,45^\circ,90^\circ\)
  • (E) \(30^\circ,45^\circ,130^\circ\)

Question 26:

If \(\tan \alpha = \frac{5}{6}\) and \(\tan \beta = \frac{1}{11}\), where \(0 < \alpha,\beta < \frac{\pi}{2}\) then \(\alpha + \beta =\)

  • (A) \(\frac{\pi}{6}\)
  • (B) \(\frac{\pi}{2}\)
  • (C) \(\frac{\pi}{3}\)
  • (D) \(\frac{\pi}{4}\)
  • (E) \(\frac{2\pi}{3}\)

Question 27:

The value of \(\sin6^\circ \cos36^\circ \sin66^\circ + \cos12^\circ \sin42^\circ \sin18^\circ\) is equal to

  • (A) \(\frac{1}{12}(\sin18^\circ + \cos36^\circ)\)
  • (B) \(\frac{1}{3}(\sin18^\circ + \cos36^\circ)\)
  • (C) \(\frac{1}{16}(\sin18^\circ + \cos36^\circ)\)
  • (D) \(\frac{1}{4}(\sin18^\circ + \cos36^\circ)\)
  • (E) \(\frac{1}{2}(\sin18^\circ + \cos36^\circ)\)

Question 28:

The domain of the function \(f(x) = 2\sin^{-1}(2x-1) - \frac{\pi}{4}\) is

  • (A) \([-1,1]\)
  • (B) \([0,1]\)
  • (C) \([0,2]\)
  • (D) \([2,5]\)
  • (E) \([-2,2]\)

Question 29:

The value of \(\sin^{-1}\left(\sin \frac{5\pi}{9} \cos \frac{\pi}{9} + \sin \frac{\pi}{9} \cos \frac{5\pi}{9}\right)\) is equal to

  • (A) \(\frac{2\pi}{3}\)
  • (B) \(\frac{\pi}{2}\)
  • (C) \(\frac{\pi}{6}\)
  • (D) \(\frac{\pi}{3}\)
  • (E) \(\frac{\pi}{9}\)

Question 30:

The value of \(\sin\left(2\sin^{-1}\frac{3}{5}\right)\) is equal to

  • (A) \(\frac{23}{25}\)
  • (B) \(\frac{21}{25}\)
  • (C) \(\frac{22}{25}\)
  • (D) \(\frac{24}{25}\)
  • (E) \(\frac{18}{25}\)

Question 31:

Let \(P = \left(\frac{15}{2}(\csc \theta + \sin \theta), \; 8(\csc \theta - \sin \theta)\right)\), where \(\theta\) is a variable parameter. Then the locus of \(P\) is

  • (A) \(\frac{x^2}{15} - \frac{y^2}{16} = 1\)
  • (B) \(\frac{x^2}{256} - \frac{y^2}{225} = 1\)
  • (C) \(\frac{x^2}{225} + \frac{y^2}{256} = 1\)
  • (D) \(\frac{x^2}{225} - \frac{y^2}{256} = 1\)
  • (E) \(\frac{x^2}{16} + \frac{y^2}{30} = 1\)

Question 32:

A straight line makes \(y\)-intercept of 5. If the angle made by the line with \(y\)-axis is \(60^\circ\) and the line intersects \(x\)-axis in the negative direction, then the equation of the line is

  • (A) \(x + \sqrt{3}y + 5\sqrt{3} = 0\)
  • (B) \(x - \sqrt{3}y + 5\sqrt{3} = 0\)
  • (C) \(\sqrt{3}x - y + 5 = 0\)
  • (D) \(\sqrt{3}x + y + 5 = 0\)
  • (E) \(\sqrt{3}x - y + 5\sqrt{3} = 0\)

Question 33:

The perpendicular drawn from the origin to the straight line \(\sqrt{3}x + y - 24 = 0\) makes an angle \(\alpha\) with the positive direction of x-axis. Then \(\alpha\) is equal to

  • (A) \(120^\circ\)
  • (B) \(45^\circ\)
  • (C) \(135^\circ\)
  • (D) \(60^\circ\)
  • (E) \(30^\circ\)

Question 34:

If the one end of a diameter of the circle \(x^2 + y^2 + 3x + y - 6 = 0\) is at \((-4,-2)\), then the other end of the diameter is at

  • (A) \((4,-2)\)
  • (B) \((1,-1)\)
  • (C) \((1,1)\)
  • (D) \((-1,-1)\)
  • (E) \((1,-2)\)

Question 35:

The vertex of a parabola is at \((2,-5)\) and the focus is at \((5,-5)\). The equation of the parabola is

  • (A) \(y^2 + 10y - 10x + 49 = 0\)
  • (B) \(y^2 + 10y - 12x + 49 = 0\)
  • (C) \(y^2 + 10y - 12x + 46 = 0\)
  • (D) \(y^2 + 8y - 12x + 49 = 0\)
  • (E) \(y^2 + 10y - 18x + 48 = 0\)

Question 36:

Let \(R(-2,-2)\) be a point and let \(\dfrac{(x-3)^2}{25} + \dfrac{(y+2)^2}{16} = 1\) be an ellipse. If \(S\) and \(T\) are the foci of the ellipse, then \(RS + RT\) is equal to

  • (A) \(128\)
  • (B) \(61\)
  • (C) \(12\)
  • (D) \(10\)
  • (E) \(124\)

Question 37:

The equation of the latus rectum of the parabola \(y^2 + 8x + 4y + 12 = 0\) is

  • (A) \(x + 3 = 0\)
  • (B) \(y + 3 = 0\)
  • (C) \(x + 1 = 0\)
  • (D) \(y + 2 = 0\)
  • (E) \(x + 2 = 0\)

Question 38:

Let \(O\) be the origin. Let \(\overrightarrow{OA} = \vec{a}\) and \(\overrightarrow{OB} = \vec{b}\) be the position vectors of the points \(A\) and \(B\) respectively. A point \(P\) divides the line segment \(AB\) internally in the ratio \(m:n\). Then \(\overrightarrow{AP}\) is equal to

  • (A) \(\frac{2n(\vec{b}-\vec{a})}{m+n}\)
  • (B) \(\frac{n(\vec{b}+\vec{a})}{m+n}\)
  • (C) \(\frac{n(\vec{b}-\vec{a})}{m-n}\)
  • (D) \(\frac{m(\vec{b}-\vec{a})}{m+n}\)
  • (E) \(\frac{n(\vec{b}-\vec{a})}{m+n}\)

Question 39:

If \(2\hat{i} - \hat{j} + \hat{k} = s(3\hat{i} - 4\hat{j} - 4\hat{k}) + t(\hat{i} - 3\hat{j} - 5\hat{k})\), where \(s\) and \(t\) are scalars, then \(3s + 5t\) is equal to

  • (A) \(2\)
  • (B) \(-4\)
  • (C) \(-2\)
  • (D) \(6\)
  • (E) \(14\)

Question 40:

Let \(\vec{a} = 2\hat{i} - 2\hat{j} + 4\hat{k}\), \(\vec{b} = -5\hat{i} - \hat{j} + 8\hat{k}\) and \(\vec{c} = 3\hat{i} + \hat{j} - \lambda \hat{k}\). If \(\vec{a} + \vec{b} + \vec{c}\) and \(\vec{a} - \vec{b} + \vec{c}\) are perpendicular, then the values of \(\lambda\) are

  • (A) \(4\) and \(-12\)
  • (B) \(-2\) and \(12\)
  • (C) \(-6\) and \(14\)
  • (D) \(-3\) and \(12\)
  • (E) \(-4\) and \(12\)

Question 41:

If \(|\vec{a}| = \sqrt{26}, \; |\vec{b}| = \sqrt{3}\) and \(\vec{a} \times \vec{b} = 5\hat{i} + \hat{j} - 4\hat{k}\), then \(\vec{a} \cdot \vec{b} =\)

  • (A) \(\pm 12\)
  • (B) \(\pm 4\)
  • (C) \(\pm 10\)
  • (D) \(\pm 8\)
  • (E) \(\pm 6\)

Question 42:

Consider the straight line \(\vec{r} = (5\hat{i} + 2\hat{j} - 3\hat{k}) + t(4\hat{i} + 6\hat{j} - 7\hat{k}), \; t \in \mathbb{R}\). Which one of the following points is a point on the straight line?

  • (A) \((21,24,-31)\)
  • (B) \((17,20,-22)\)
  • (C) \((1,-4,5)\)
  • (D) \((25,32,-38)\)
  • (E) \((45,66,-36)\)

Question 43:

The equation of a line passing through \((-1,2,-4)\) and parallel to the straight line \(\frac{-x-1}{4} = \frac{2y+1}{-1} = \frac{-z+4}{3}\), is

  • (A) \(\vec{r} = (-\hat{i} + 2\hat{j} - 4\hat{k}) + t(4\hat{i} + 6\hat{j} - 7\hat{k}), \; t \in \mathbb{R}\)
  • (B) \(\vec{r} = (-\hat{i} + 2\hat{j} - 4\hat{k}) + t(3\hat{i} + 5\hat{j} - 2\hat{k}), \; t \in \mathbb{R}\)
  • (C) \(\vec{r} = (-\hat{i} + 2\hat{j} - 4\hat{k}) + t(8\hat{i} + \hat{j} + 6\hat{k}), \; t \in \mathbb{R}\)
  • (D) \(\vec{r} = (-\hat{i} + 2\hat{j} - 4\hat{k}) + t(7\hat{i} + 6\hat{j} + 6\hat{k}), \; t \in \mathbb{R}\)
  • (E) \(\vec{r} = (-\hat{i} + 2\hat{j} - 6\hat{k}) + t(8\hat{i} + \hat{j} + 6\hat{k}), \; t \in \mathbb{R}\)

Question 44:

A straight line passes through the point whose position vector is \(\hat{k}\). The straight line also passes through the point of intersection of the lines \(\vec{r} = \hat{j} + \lambda \hat{i}, \lambda \in \mathbb{R}\) and \(\vec{r} = \hat{i} + s\hat{j}, s \in \mathbb{R}\). Then the equation of the straight line is

  • (A) \(\vec{r} = \hat{k} + t(\hat{i} + \hat{j} - \hat{k}), \; t \in \mathbb{R}\)
  • (B) \(\vec{r} = \hat{k} + t(\hat{i} - \hat{j} - \hat{k}), \; t \in \mathbb{R}\)
  • (C) \(\vec{r} = \hat{k} + t(\hat{i} - \hat{j} + \hat{k}), \; t \in \mathbb{R}\)
  • (D) \(\vec{r} = \hat{k} + t(-\hat{i} + \hat{j} + 2\hat{k}), \; t \in \mathbb{R}\)
  • (E) \(\vec{r} = \hat{k} + t(-\hat{i} + 2\hat{j} - \hat{k}), \; t \in \mathbb{R}\)

Question 45:

The shortest distance between the lines \(\vec{r} = -\hat{i} + t\hat{k}, \; t \in \mathbb{R}\) and \(\vec{r} = -\hat{j} + s\hat{i}, \; s \in \mathbb{R}\) is

  • (A) \(8\)
  • (B) \(5\)
  • (C) \(3\)
  • (D) \(4\)
  • (E) \(1\)

Question 46:

The mean deviation about the mean for the data: \(5, 6, 14, 15\) is

  • (A) \(3.5\)
  • (B) \(4.5\)
  • (C) \(4.2\)
  • (D) \(3.8\)
  • (E) \(4.0\)

Question 47:

The variance for the data: \(65, 70, 75\) is

  • (A) \(\frac{50}{3}\)
  • (B) \(\frac{55}{3}\)
  • (C) \(\frac{50}{6}\)
  • (D) \(\frac{50}{2}\)
  • (E) \(70\)

Question 48:

A fair die is rolled once. Which one of the following is not true?

  • (A) \(\{1,3\}\) and \(\{2,4,6\}\) are mutually exclusive events
  • (B) \(\{1,5\},\{2,4\}\) are \(\{3,6\}\) mutually exclusive and exhaustive events
  • (C) \(\{1,2,4,3,6,5\}\) is sure event
  • (D) \(\{1\},\{2\}\) and \(\{6\}\) are elementary events
  • (E) \(\{1,3,2\}\) and \(\{2,4,6\}\) are mutually exclusive events

Question 49:

Let \(A, B, C\) be all the three possible mutually exclusive events of a random experiment. Which one of the following is not permissible in terms of their probabilities?

  • (A) \(P(A)=\frac{7}{19}, \; P(B)=\frac{4}{19}, \; P(C)=\frac{8}{19}\)
  • (B) \(P(A)=\frac{18}{95}, \; P(B)=\frac{29}{95}, \; P(C)=\frac{48}{95}\)
  • (C) \(P(A)=\frac{81}{190}, \; P(B)=\frac{41}{190}, \; P(C)=\frac{68}{190}\)
  • (D) \(P(A)=\frac{21}{95}, \; P(B)=\frac{42}{95}, \; P(C)=\frac{32}{95}\)
  • (E) \(P(A)=\frac{77}{190}, \; P(B)=\frac{47}{190}, \; P(C)=\frac{67}{190}\)

Question 50:

The value of \(\lim_{x \to 0} \dfrac{\sin^2 x}{1 - \cos x}\) is equal to

  • (A) \(4\)
  • (B) \(2\)
  • (C) \(\frac{1}{2}\)
  • (D) \(\frac{1}{4}\)
  • (E) \(0\)

Question 51:

The value of \(\lim_{x \to 1} \dfrac{x - 1}{3\sqrt{x} - 1}\) is equal to

  • (A) \(3\)
  • (B) \(\frac{1}{3}\)
  • (C) \(2\)
  • (D) \(\frac{1}{2}\)
  • (E) \(0\)

Question 52:

If the function \(f(x)= \begin{cases} \dfrac{2x^2+3x-5}{x-1}, & x \ne 1
k, & x=1 \end{cases} \) is continuous at \(x=1\), then the value of \(k\) is

  • (A) \(6\)
  • (B) \(8\)
  • (C) \(-6\)
  • (D) \(7\)
  • (E) \(-7\)

Question 53:

The value of \(\lim_{x \to 0} \dfrac{\sqrt{1 - \cos(x^2)}}{1 - \cos x}\) is equal to

  • (A) \(\frac{1}{\sqrt{2}}\)
  • (B) \(\sqrt{2}\)
  • (C) \(\frac{1}{2\sqrt{2}}\)
  • (D) \(2\sqrt{2}\)
  • (E) \(0\)

Question 54:

The domain of the function \(f(x) = \dfrac{\log_2 (x - 5)}{x^2 + 3x - 4}\) is

  • (A) \((1,\infty)\)
  • (B) \((10,\infty)\)
  • (C) \((5,\infty)\)
  • (D) \(\mathbb{R} \setminus \{-4\}\)
  • (E) \(\mathbb{R} \setminus \{-4,1\}\)

Question 55:

Which one of the following is not true?

  • (A) \(f(x)=x|x|\) is differentiable in \((-1,1)\)
  • (B) \(g(x)=\sqrt{|x|}\) is differentiable in \((4,5)\)
  • (C) \(h(x)=|x-2|+|x+3|\) is differentiable in \((3,2)\)
  • (D) \(k(x)=|x+1|+|x-6|\) is differentiable in \((-1,6)\)
  • (E) \(t(x)=x+[x]\), where \([x]\) is greatest integer function, is differentiable at \(x=0\)

Question 56:

Let \(y = \dfrac{3x^3 - 2x^2 + x}{|x|}, \; x \ne 0\). Then \(\frac{dy}{dx}\) at \(x=-2\) is equal to

  • (A) \(14\)
  • (B) \(-12\)
  • (C) \(-14\)
  • (D) \(12\)
  • (E) \(10\)

Question 57:

If \((3 + 5x)e^{\frac{y}{x}} = x\), then \(\frac{dy}{dx}\) is equal to

  • (A) \(\log\left|\frac{x}{3+5x}\right| + \frac{3}{3+5x}\)
  • (B) \(\log\left|\frac{x}{3+5x}\right| + \frac{5x-3}{3+5x}\)
  • (C) \(\log\left|\frac{x}{3+5x}\right| + \frac{5x-2}{3+5x}\)
  • (D) \(\log\left|\frac{3x}{3+5x}\right| + \frac{10x+3}{3+5x}\)
  • (E) \(\log\left|\frac{x}{3+5x}\right| + \frac{3-10x}{3+5x}\)

Question 58:

If \(y = e^{-x^2}\), then at \(\frac{d^2y}{dx^2} + 2x\frac{dy}{dx} =\)

  • (A) \(2y\)
  • (B) \(-2y\)
  • (C) \(-\frac{y}{2}\)
  • (D) \(-y\)
  • (E) \(y\)

Question 59:

Let \(f(x)\) and \(g(x)\) be two differentiable functions such that \(f'(x)=g(x)\) and \(g'(x)=-f(x)\). Let \(h(x)=(f(x))^2+(g(x))^2\) and \(h(3)=100\). Then \(h(100)\) is equal to

  • (A) \(100\)
  • (B) \(10\)
  • (C) \(50\)
  • (D) \(200\)
  • (E) \(300\)

Question 60:

Let \(f\) and \(g\) be differentiable real valued functions on \([0,\infty)\). If \(f\) is increasing, \(g\) is decreasing and \(h(x)=f(g(x))\), then \(h(2026)-h(2025)\) is

  • (A) greater than 1000 but less than 2000
  • (B) greater than or equal to 0
  • (C) less than or equal to 0
  • (D) greater than 2025
  • (E) greater than 2026

Question 61:

Let \(f(x)=10x^2+ax,\; x\in \mathbb{R}\) be such that \(a^2-400<0\). Let \(g(x)=f(x)+f'(x)+f''(x)\). Then \(g(x)\) is

  • (A) greater than 100 but less than 200
  • (B) greater than 10 but less than 100
  • (C) less than 10
  • (D) greater than 0
  • (E) less than 1

Question 62:

The minimum of \(f(x) = \dfrac{x^{100} - 1}{x^{100} + 1}, \; x \in \mathbb{R}\) is

  • (A) \(-5\)
  • (B) \(-1.5\)
  • (C) \(-1\)
  • (D) \(-2\)
  • (E) \(-3\)

Question 63:

Let \(f(x) = 1 + x\log\left(x + \sqrt{x^2+1}\right) - \sqrt{x^2+1}, \; x \geq 0\). Then

  • (A) \(f(x)\) is increasing on \((0,\infty)\)
  • (B) \(f(x)\) is increasing only on \((10,\infty)\)
  • (C) \(f(x)\) is increasing only on \((0,e)\)
  • (D) \(f(x)\) is decreasing on \((0,\infty)\)
  • (E) \(f(x)\) is decreasing only on \((100,\infty)\)

Question 64:

\(\displaystyle \int \frac{\sin(\cot^{-1}x)}{1+x^2} \, dx\) is equal to

  • (A) \(-\cos(\cot^{-1}x) + C\)
  • (B) \(\cos(\cot^{-1}x) + C\)
  • (C) \(\frac{\cos(\cot^{-1}x)}{1+x^2} + C\)
  • (D) \(\frac{\cos(\cot^{-1}x)}{2} + C\)
  • (E) \(-\frac{\cos(\cot^{-1}x)}{1+x^2} + C\)

Question 65:

\(\displaystyle \int \sqrt{1 + \sin\left(\frac{x}{8}\right)} \, dx =\)

  • (A) \(16\sin\left(\frac{x}{32}\right) - 16\cos\left(\frac{x}{32}\right) + C\)
  • (B) \(16\sin\left(\frac{x}{16}\right) - 16\cos\left(\frac{x}{16}\right) + C\)
  • (C) \(16\sin\left(\frac{x}{32}\right) + 16\cos\left(\frac{x}{32}\right) + C\)
  • (D) \(16\sin\left(\frac{x}{16}\right) + 16\cos\left(\frac{x}{16}\right) + C\)
  • (E) \(8\sin\left(\frac{x}{16}\right) - 8\cos\left(\frac{x}{16}\right) + C\)

Question 66:

\(\displaystyle \int \left(\frac{1}{(1+x)^2} - \frac{2}{(1+x)^3}\right)e^x \, dx\) is equal to

  • (A) \(-\frac{2}{(1+x)^2} + C\)
  • (B) \(\frac{1}{(1+x)^2} + C\)
  • (C) \(\frac{2}{(1+x)^2} + C\)
  • (D) \(-\frac{2e^x}{(1+x)^2} + C\)
  • (E) \(\frac{e^x}{(1+x)^2} + C\)

Question 67:

\(\displaystyle \int \frac{x^4 - 1}{x + 1} \, dx\) is equal to

  • (A) \(\frac{x^4}{4} + \frac{x^3}{3} + \frac{x^2}{2} + x + C\)
  • (B) \(\frac{x^4}{4} + \frac{x^3}{3} + \frac{x^2}{2} + 2x + C\)
  • (C) \(\frac{x^4}{4} - \frac{x^3}{3} + \frac{x^2}{2} - x + C\)
  • (D) \(\frac{x^4}{4} - \frac{x^3}{3} + \frac{x^2}{2} - 2x + C\)
  • (E) \(-\frac{x^4}{4} - \frac{x^3}{3} - \frac{x^2}{2} - x + C\)

Question 68:

\(\displaystyle \int \frac{\sin t + \cos t}{13 + 36\sin^2 t} \, dt\) is equal to

  • (A) \(\frac{1}{84}\log\left|\frac{7 + 6(\sin t - \cos t)}{7 - 6(\sin t - \cos t)}\right| + C\)
  • (B) \(\frac{1}{81}\log\left|\frac{7 + 6(\sin t - \cos t)}{7 - 6(\sin t - \cos t)}\right| + C\)
  • (C) \(\frac{1}{84}\log\left|\frac{7 - 6(\sin t - \cos t)}{7 + 6(\sin t - \cos t)}\right| + C\)
  • (D) \(\frac{1}{48}\log\left|\frac{7 + 6(\sin t - \cos t)}{7 - 6(\sin t - \cos t)}\right| + C\)
  • (E) \(\frac{1}{64}\log\left|\frac{7 + 6(\sin t - \cos t)}{7 - 6(\sin t - \cos t)}\right| + C\)

Question 69:

\(\displaystyle \int_{-6}^{0} \left[t^3 + 9t^2 + 27t + 29 + (t+3)\cos(t+3)\right] dt\) is equal to

  • (A) \(6\)
  • (B) \(12\)
  • (C) \(18\)
  • (D) \(4\)
  • (E) \(24\)

Question 70:

If \(I = \displaystyle \int_{-1}^{1} \frac{x^4}{1 - x^4} \cos^{-1}\left(\frac{2x}{1+x^2}\right) dx\), then \(2I\) is equal to

  • (A) \(\pi \int_{-1}^{1} \frac{x^4}{1 - x^4} dx\)
  • (B) \(2\pi \int_{-1}^{1} \frac{x^4}{1 - x^4} dx\)
  • (C) \(\int_{-1}^{1} \frac{x^4}{1 - x^4} dx\)
  • (D) \(\pi \int_{-1}^{1} \frac{x^4}{1 + x^4} dx\)
  • (E) \(-\pi \int_{-1}^{1} \frac{x^4}{1 - x^4} dx\)

Question 71:

\(\displaystyle \int_{0}^{1} \left[\tan^{-1}\left(\frac{1}{1+x+x^2+x^3}\right) + \tan^{-1}(1+x+x^2+x^3)\right] dx\) is equal to

  • (A) \(\frac{\pi}{4}\)
  • (B) \(2\pi\)
  • (C) \(\frac{\pi}{2}\)
  • (D) \(1\)
  • (E) \(4\pi\)

Question 72:

The value of \(\displaystyle \int_{0}^{1} x(1-x)^4 \, dx\) is equal to

  • (A) \(\frac{1}{60}\)
  • (B) \(\frac{1}{15}\)
  • (C) \(\frac{1}{30}\)
  • (D) \(\frac{1}{45}\)
  • (E) \(\frac{1}{20}\)

Question 73:

Elimination of arbitrary constants \(A\) and \(B\) from \(y = Ae^x + Be^{-2x}\) gives the differential equation

  • (A) \(\frac{d^2y}{dx^2} + \frac{dy}{dx} - 2y = 0\)
  • (B) \(\frac{d^2y}{dx^2} + \frac{dy}{dx} + 2y = 0\)
  • (C) \(\frac{d^2y}{dx^2} - \frac{dy}{dx} - 2y = 0\)
  • (D) \(\frac{d^2y}{dx^2} - 2\frac{dy}{dx} + y = 0\)
  • (E) \(\frac{d^2y}{dx^2} + 2\frac{dy}{dx} + y = 0\)

Question 74:

The solution of the differential equation \((x + 2y)dx + (2x - y)dy = 0\) is

  • (A) \(x^2 - y^2 + 6xy = C\)
  • (B) \(x^2 - y^2 - 4xy = C\)
  • (C) \(x^2 - y^2 + 4xy = C\)
  • (D) \(x^2 - y^2 + 3xy = C\)
  • (E) \(2x^2 - y^2 + 4xy = C\)

Question 75:

Consider the Linear Programming Problem (LPP): Maximize \(z = 30x + 60y\) subject to constraints \(x + 2y \leq 12\), \(2x + y \leq 12\), \(4x + 5y \geq 20\), \(x \geq 0\), \(y \geq 0\). Then the number of corner points of the feasible region is

  • (A) \(8\)
  • (B) \(6\)
  • (C) \(3\)
  • (D) \(4\)
  • (E) \(5\)

Question 76:

The physical quantity that doesn’t have appropriate unit is

  • (A) Compressibility -- \(N^{-1}m^2\)
  • (B) Latent heat -- \(J\,kg^{-1}\)
  • (C) Intensity -- \(Wm^{-2}\)
  • (D) Energy density -- \(Jm^{-2}\)
  • (E) Impulse -- \(Ns\)

Question 77:

In the equation \(A = \dfrac{B}{CD^2}\), if \(B, C\) and \(D\) have the dimensions of inductive reactance, capacitive reactance and angular frequency respectively, then the dimensions of \(A\) are

  • (A) \(M^0L T^{-2}\)
  • (B) \(ML^0T^{-2}\)
  • (C) \(M^0L^0T^{2}\)
  • (D) \(M^{-1}L^0T^{-2}\)
  • (E) \(M^0L^0T^{-2}\)

Question 78:

The position of a particle moving along \(y\)-axis is given as \(y = t^2 + 2t + 3\). The average acceleration of the particle between \(t=3s\) and \(t=6s\) (in \(ms^{-2}\)) is

  • (A) \(2\)
  • (B) \(5\)
  • (C) \(4\)
  • (D) \(3\)
  • (E) \(6\)

Question 79:

The ratio of distances traversed by a freely falling body in successive intervals of time is

  • (A) \(3:4:6:9\)
  • (B) \(1:3:6:9\)
  • (C) \(1:2:4:6\)
  • (D) \(2:5:7:9\)
  • (E) \(1:3:5:7\)

Question 80:

If the scalar product of two vectors \(x\hat{i} + 3\hat{j} + 2\hat{k}\) and \(2\hat{i} - 3\hat{j} + 4\hat{k}\) is 9, then the value of \(x\) is

  • (A) \(9\)
  • (B) \(5\)
  • (C) \(6\)
  • (D) \(1\)
  • (E) \(2\)

Question 81:

Among the following the INCORRECT statement is

  • (A) Newton’s second law relates the net external force to its acceleration.
  • (B) Impulse is equal to the change in momentum.
  • (C) Newton’s second law is inconsistent with Newton’s first law.
  • (D) Same force acting on different bodies for the same time brings the same change in momentum.
  • (E) Action and reaction act on different bodies.

Question 82:

A ball of \(200g\) mass moving with a speed of \(5\,ms^{-1}\) collides with a wall and bounces back with the same speed. If the force exerted on the wall is \(1N\), then the ball is in contact with the wall for

  • (A) \(2s\)
  • (B) \(1s\)
  • (C) \(0.5s\)
  • (D) \(1.5s\)
  • (E) \(0.75s\)

Question 83:

A block of \(10\,kg\) mass moving on a frictionless surface with speed \(5\,ms^{-1}\) compresses a spring by \(5cm\) and comes to rest. The force constant of the spring (in \(Nm^{-1}\)) is

  • (A) \(2 \times 10^5\)
  • (B) \(2.5 \times 10^5\)
  • (C) \(3 \times 10^5\)
  • (D) \(1.5 \times 10^5\)
  • (E) \(1 \times 10^5\)

Question 84:

If a gun fires 25 bullets in one second, each of \(10g\) mass with a velocity of \(20\,ms^{-1}\), then the recoil force on the gun in \(N\) is

  • (A) \(50\)
  • (B) \(5\)
  • (C) \(15\)
  • (D) \(10\)
  • (E) \(20\)

Question 85:

The angular momentum of a uniform rod of mass \(m\) and length \(l\), rotating in a horizontal circle about one of its ends with an angular velocity \(\omega\) is

  • (A) \(ml^2\omega\)
  • (B) \(\frac{1}{4}ml^2\omega\)
  • (C) \(2ml^2\omega\)
  • (D) \(\frac{1}{3}ml^2\omega\)
  • (E) \(\frac{1}{2}ml^2\omega\)

Question 86:

The moment of inertia of a solid sphere of radius \(20cm\) about its diameter is same as that of a solid cylinder of same mass about its axis, then the radius of the cylinder in \(cm\) is

  • (A) \(3\sqrt{5}\)
  • (B) \(5\sqrt{5}\)
  • (C) \(2\sqrt{5}\)
  • (D) \(8\sqrt{5}\)
  • (E) \(7\sqrt{5}\)

Question 87:

The maximum and minimum distances of a satellite revolving in an elliptical orbit are in the ratio \(3:1\). If the speed of the satellite at the nearest distance is \(v\), then the speed at the farthest distance is

  • (A) \(\frac{v}{6}\)
  • (B) \(3v\)
  • (C) \(\frac{v}{3}\)
  • (D) \(6v\)
  • (E) \(9v\)

Question 88:

The ratio of the magnitudes of gravitational potential energy to that of kinetic energy of an earth satellite of mass \(m\) revolving in any orbit is

  • (A) \(1:2\)
  • (B) \(2:1\)
  • (C) \(2:3\)
  • (D) \(1:3\)
  • (E) \(3:1\)

Question 89:

Which one of the following materials has the highest modulus of elasticity?

  • (A) steel
  • (B) aluminium
  • (C) copper
  • (D) glass
  • (E) brass

Question 90:

Two capillary tubes of radii in the ratio \(1:2\) are dipped in the same liquid. The ratio of heights through which the liquid will rise in the tubes is

  • (A) \(1:1\)
  • (B) \(1:2\)
  • (C) \(1:4\)
  • (D) \(4:1\)
  • (E) \(2:1\)

Question 91:

The relative viscosity of blood \(\left(\frac{\eta}{\eta_{water}}\right)\) is constant between

  • (A) \(10^\circ C\) and \(47^\circ C\)
  • (B) \(0^\circ C\) and \(47^\circ C\)
  • (C) \(0^\circ C\) and \(37^\circ C\)
  • (D) \(30^\circ C\) and \(40^\circ C\)
  • (E) \(20^\circ C\) and \(47^\circ C\)

Question 92:

There is no change in internal energy of an ideal gas in an

  • (A) isothermal process
  • (B) adiabatic process
  • (C) isobaric process
  • (D) isochoric process
  • (E) both in adiabatic process and isobaric process

Question 93:

Which one is not an extensive variable?

  • (A) total mass
  • (B) internal energy
  • (C) volume
  • (D) density
  • (E) workdone \((PdV)\)

Question 94:

Experimental P-V curves and theoretically predicted P-V curves are in good agreement at

  • (A) high temperature and high pressure
  • (B) high temperature and low pressure
  • (C) low temperature and high pressure
  • (D) low temperature and atmospheric pressure
  • (E) low temperature and low pressure

Question 95:

The mass of one molecule of water is approximately

  • (A) \(8 \times 10^{-26}\) kg
  • (B) \(6.5 \times 10^{-26}\) kg
  • (C) \(3.5 \times 10^{-28}\) kg
  • (D) \(2.5 \times 10^{-28}\) kg
  • (E) \(3 \times 10^{-26}\) kg

Question 96:

If the instantaneous displacement of a wave is \(y = 2(\sin 2\pi t + \sqrt{3}\cos 2\pi t)\,cm\), then the amplitude of the wave in \(cm\) is

  • (A) \(4\)
  • (B) \(3\)
  • (C) \(5\)
  • (D) \(2\)
  • (E) \(6\)

Question 97:

The equation of a transverse wave in a string, \(y = 3\sin 2\pi (25t + 0.4x)\) m. The wavelength of the wave is

  • (A) \(4.5\,m\)
  • (B) \(3\,m\)
  • (C) \(2.5\,m\)
  • (D) \(3.5\,m\)
  • (E) \(6.5\,m\)

Question 98:

If a spherical conductor of \(10\,cm\) radius contains \(5 \times 10^6\) electrons, then the electric field on its surface (in \(NC^{-1}\)) is

  • (A) \(0.86\)
  • (B) \(0.36\)
  • (C) \(0.45\)
  • (D) \(1.44\)
  • (E) \(0.72\)

Question 99:

When a capacitor of \(9\,pF\) is connected to a battery, the electrostatic energy stored in the capacitor is \(18 \times 10^{-8}J\). The quantity of charge stored in the capacitor is

  • (A) \(1.2\,nC\)
  • (B) \(1.8\,nC\)
  • (C) \(2.7\,nC\)
  • (D) \(3.6\,nC\)
  • (E) \(2.4\,nC\)

Question 100:

If the electric potential is given by \(V = 3x^2 + 4x\) volt, then the magnitude of the electric field at the point \(x = 1m\) is

  • (A) \(6Vm^{-1}\)
  • (B) \(4Vm^{-1}\)
  • (C) \(8Vm^{-1}\)
  • (D) \(10Vm^{-1}\)
  • (E) \(12Vm^{-1}\)

Question 101:

Which one of the following statements is CORRECT?

  • (A) The molecules of air are non-polar.
  • (B) In a polar molecule, centres of positive and negative charges coincide.
  • (C) A molecule of water is an example of a non-polar molecule.
  • (D) Dielectrics are conducting substances.
  • (E) In a non-polar molecule, centres of positive and negative charges coincide.

Question 102:

If the drift velocity of electrons in a copper wire of cross-sectional area \(2\,mm^2\) carrying current \(I\) is \(v_1\) and that in another copper wire of cross-sectional area \(1.5\,mm^2\) carrying current \(2I\) is \(v_2\), then the ratio \(v_1 : v_2\) is

  • (A) \(3:8\)
  • (B) \(2:4\)
  • (C) \(8:3\)
  • (D) \(4:2\)
  • (E) \(1:3\)

Question 103:

A uniform metallic wire of radius \(r\) and length \(l\) is heated by passing a current through it. The heat produced can be made 8 times if

  • (A) \(l\) is doubled
  • (B) both \(l\) and \(r\) are halved
  • (C) \(l\) is doubled and \(r\) is halved
  • (D) \(r\) is doubled
  • (E) both \(l\) and \(r\) are doubled

Question 104:

Two cells each of \(2V\) and internal resistance \(0.1\,\Omega\) are connected in parallel combination. This combination is equivalent to a single cell with emf and internal resistance of

  • (A) \(1V\) and \(0.05\,\Omega\)
  • (B) \(2V\) and \(0.05\,\Omega\)
  • (C) \(2V\) and \(0.1\,\Omega\)
  • (D) \(4V\) and \(0.05\,\Omega\)
  • (E) \(4V\) and \(0.1\,\Omega\)

Question 105:

When a bar magnet placed parallel to the magnetic field is rotated by \(45^\circ\), the amount of work done is \(2.07J\). The amount of work to be done to rotate the magnet further by \(45^\circ\) is

  • (A) \(2.07J\)
  • (B) \(3J\)
  • (C) \(4.41J\)
  • (D) \(5J\)
  • (E) \(6.21J\)

Question 106:

Two charged particles \(2q\) and \(q\) having equal momentum enter a uniform magnetic field in a direction perpendicular to the magnetic field. Then their respective radii of circular paths \(r_1\) and \(r_2\) are in the ratio

  • (A) \(2:1\)
  • (B) \(1:4\)
  • (C) \(1:2\)
  • (D) \(1:3\)
  • (E) \(4:1\)

Question 107:

A wire of certain length carrying current \(I\), when bent into a circular coil of single turn produces a magnetic field \(B\) at its centre. If the same wire is bent into a circular coil of 3 turns and it carries the same current, then the magnetic field at the centre of the coil is

  • (A) \(12B\)
  • (B) \(3B\)
  • (C) \(6B\)
  • (D) \(9B\)
  • (E) \(15B\)

Question 108:

For an electron of mass \(m_e\) and charge \(e\) revolving around the nucleus of an atom, the ratio of its angular momentum to magnetic moment is

  • (A) \(\frac{e}{m_e}\)
  • (B) \(\frac{e}{2m_e}\)
  • (C) \(\frac{2e}{m_e}\)
  • (D) \(\frac{e}{4m_e}\)
  • (E) \(em_e\)

Question 109:

If an air core solenoid with self-inductance of \(0.5\,mH\) is filled with soft iron of relative permeability of \(1500\), its self-inductance becomes

  • (A) \(0.5\,H\)
  • (B) \(1.5\,H\)
  • (C) \(0.25\,H\)
  • (D) \(1.25\,H\)
  • (E) \(0.75\,H\)

Question 110:

The r.m.s current of an alternating current given by, \(i = 4\sqrt{2}\sin \omega t + 3\sqrt{2}\cos \omega t\) is

  • (A) \(5A\)
  • (B) \(3A\)
  • (C) \(5\sqrt{2}A\)
  • (D) \(2.5A\)
  • (E) \(7\sqrt{2}A\)

Question 111:

The law that is a symmetrical counterpart of Faraday’s law of electromagnetic induction is

  • (A) Ampere-Maxwell law
  • (B) Gauss law
  • (C) Lenz’s law
  • (D) Ampere’s circuital law
  • (E) Coulomb’s law

Question 112:

A real object is placed at distance \(f\) in front of a convex mirror of focal length \(f\). The image will be formed at a distance

  • (A) \(2f\)
  • (B) \(\frac{f}{8}\)
  • (C) \(f\)
  • (D) \(\frac{f}{4}\)
  • (E) \(\frac{f}{2}\)

Question 113:

The magnifying power of a simple microscope can be increased by using

  • (A) diverging lens of large radius of curvature
  • (B) converging lens of small focal length
  • (C) diverging lens of small focal length
  • (D) converging lens of large focal length
  • (E) diverging lens of large focal length

Question 114:

The intensity of the transmitted light after passing through a first polaroid \(P_1\) is \(I_0\). If the second polaroid \(P_2\) is rotated through an angle of \(45^\circ\) with respect to \(P_1\), then the change in the intensity of the transmitted light after passing through the second polaroid is

  • (A) \(\frac{I_0}{2}\)
  • (B) \(\frac{I_0}{4}\)
  • (C) \(\frac{3I_0}{4}\)
  • (D) \(\frac{I_0}{3}\)
  • (E) \(\frac{3I_0}{2}\)

Question 115:

The order of the electric field required to pull out electrons from a metal by field emission (in \(Vm^{-1}\)) is

  • (A) \(10^2\)
  • (B) \(10^8\)
  • (C) \(10^5\)
  • (D) \(10^6\)
  • (E) \(10^4\)

Question 116:

If the threshold wavelengths of two metals are in the ratio \(1:3\), then the work functions of these metals are in the ratio

  • (A) \(1:3\)
  • (B) \(2:1\)
  • (C) \(3:1\)
  • (D) \(1:2\)
  • (E) \(3:2\)

Question 117:

The radius of innermost orbit of an electron in the hydrogen atom is \(0.53\,\AA\). Then, the radius of the 3rd electron orbit is

  • (A) \(1.59\,\AA\)
  • (B) \(2.38\,\AA\)
  • (C) \(0.53\,\AA\)
  • (D) \(4.77\,\AA\)
  • (E) \(9.54\,\AA\)

Question 118:

The energy released by \(2.35\,g\) of \(^{235}U\) by fission in a nuclear reactor (in \(MeV\)) is (Average energy released per fission is \(200\,MeV\))

  • (A) \(1.2 \times 10^{24}\)
  • (B) \(0.4 \times 10^{24}\)
  • (C) \(0.6 \times 10^{24}\)
  • (D) \(0.8 \times 10^{24}\)
  • (E) \(2.4 \times 10^{24}\)

Question 119:

In a silicon crystal containing \(N\) atoms, at absolute zero the \(4N\) energy states of

  • (A) both the valence band and conduction band are completely occupied
  • (B) the valence band is completely occupied and the conduction band is completely empty
  • (C) both the valence band and conduction band are completely empty
  • (D) the valence band is completely occupied and the conduction band is partially occupied
  • (E) the valence band is completely empty and the conduction band is completely occupied

Question 120:

The rate of fall of the voltage across the capacitor in a filter used in a diode rectifier depends

  • (A) upon the difference of the capacitive reactance and load resistance
  • (B) inversely only on the capacitance
  • (C) directly on the product of the capacitance and load resistance
  • (D) upon the sum of the capacitive reactance and load resistance
  • (E) on the inverse of the product of the capacitance and effective load resistance

Question 121:

What is the volume of methanol needed for making \(2\,L\) of \(0.4\,M\) solution? (Density of methanol \(= 0.64\,kg\,L^{-1}\) and molar mass \(= 32\,g\,mol^{-1}\))

  • (A) \(20\,mL\)
  • (B) \(4\,mL\)
  • (C) \(40\,mL\)
  • (D) \(10\,mL\)
  • (E) \(80\,mL\)

Question 122:

The minimum energy required to remove an electron from sodium atom is \(3.313 \times 10^{-19}J\). What is the maximum wavelength of radiation that will eject photoelectron from sodium metal? (\(h = 6.626 \times 10^{-34}Js\))

  • (A) \(400\,nm\)
  • (B) \(500\,nm\)
  • (C) \(700\,nm\)
  • (D) \(200\,nm\)
  • (E) \(600\,nm\)

Question 123:

Which of the following statements are correct about the postulates of quantum mechanical model of an atom?

(i) The energy of electron in atom is quantized.

(ii) The existence of quantized electronic energy level is a result of the particle property of electrons.

(iii) The path of the electron can be determined accurately.

(iv) In a multi electron atom, the electrons are filled in various orbitals in the order of increasing energy.

(v) The probability of finding an electron at a point within an atom is proportional to the square of the orbital wave function.

  • (A) (i), (iv), (v)
  • (B) (i), (ii), (iv)
  • (C) (i), (iii), (v)
  • (D) (ii), (iii), (v)
  • (E) (i), (ii), (v)

Question 124:

Which of the following pair of elements have greater ability to form \(p\pi - p\pi\) multiple bonds?

  • (A) Carbon and Oxygen
  • (B) Boron and Aluminium
  • (C) Nitrogen and Phosphorus
  • (D) Fluorine and Chlorine
  • (E) Carbon and Silicon

Question 125:

Which of the following statements are true about electronegativity?

(i) Electronegativity generally increases across a period and decreases down a group.

(ii) The electronegativity of a given element is constant.

(iii) The electronegativity values decrease with the increase in atomic radii.

(iv) Electronegativity is directly related to the metallic property of the elements.

(v) Electronegativity is inversely related to the non-metallic property of the elements.

  • (A) (i) and (iv)
  • (B) (ii) and (iii)
  • (C) (i) and (iii)
  • (D) (iv) and (v)
  • (E) (iii) and (v)

Question 126:

Which of the following statements are correct about the \(PCl_5\) molecule?

(i) It has trigonal bipyramidal geometry.

(ii) It has three equatorial and two axial bonds.

(iii) The equatorial bond pairs suffer more repulsive interaction from the axial bond pair.

(iv) The equatorial bonds are slightly weaker than axial bonds.

(v) The hybridization involved in the molecule is \(sp^3d\).

  • (A) (i), (ii), (v)
  • (B) (ii), (iii), (iv)
  • (C) (i), (ii), (iii)
  • (D) (ii), (iv), (v)
  • (E) (i), (ii), (v)

Question 127:

The geometry of a molecule of type \(AB_3E_2\) with 3 bonding pairs and 2 lone pairs is

  • (A) T-shape
  • (B) trigonal pyramidal
  • (C) trigonal bi-pyramidal
  • (D) square pyramidal
  • (E) see-saw

Question 128:

The enthalpy of combustion of benzene, graphite and dihydrogen at \(298\,K\) are \(-3260\), \(-390\) and \(-290\,kJ\,mol^{-1}\) respectively. Enthalpy of formation of benzene is

  • (A) \(-50\,kJ\,mol^{-1}\)
  • (B) \(+50\,kJ\,mol^{-1}\)
  • (C) \(+60\,kJ\,mol^{-1}\)
  • (D) \(-60\,kJ\,mol^{-1}\)
  • (E) \(+80\,kJ\,mol^{-1}\)

Question 129:

Choose the correct pair

Bond \hspace{2cm} Mean single bond enthalpy (kJ mol\(^{-1}\))

(a) C-H \hspace{2.5cm (i) 464

(b) O-H \hspace{2.5cm (ii) 569

(c) F-H \hspace{2.5cm (iii) 293

(d) Si-H \hspace{2.2cm (iv) 414

  • (A) (a)-(iv), (b)-(i), (c)-(ii), (d)-(iii)
  • (B) (a)-(ii), (b)-(iii), (c)-(i), (d)-(iv)
  • (C) (a)-(ii), (b)-(iv), (c)-(iii), (d)-(i)
  • (D) (a)-(iv), (b)-(i), (c)-(ii), (d)-(iii)
  • (E) (a)-(i), (b)-(iii), (c)-(iv), (d)-(ii)

Question 130:

For the equilibrium, \(X_2(g) + O_2(g) \rightleftharpoons 2XO(g)\), the equilibrium concentrations of \(X_2(g)\) and \(O_2(g)\) are \(4 \times 10^{-3}M\) and \(8 \times 10^{-3}M\) respectively. What is the equilibrium concentration of \(XO(g)\)? (\(K_c = 0.5\))

  • (A) \(4 \times 10^{-3}M\)
  • (B) \(6 \times 10^{-3}M\)
  • (C) \(5 \times 10^{-3}M\)
  • (D) \(2 \times 10^{-3}M\)
  • (E) \(8 \times 10^{-3}M\)

Question 131:

The concentration of hydrogen ions in a hydrochloric acid solution is \(3 \times 10^{-3}\) M. Its pH value is about (\(\log 3 = 0.4771\))

  • (A) 2.32
  • (B) 2.52
  • (C) 2.47
  • (D) 3.47
  • (E) 5.52

Question 132:

What is the emf of the cell at \(298\,K\) in which the following reaction takes place? \[ Ni(s) + 2Ag^+(0.002M) \rightarrow Ni^{2+}(0.04M) + 2Ag(s) \]
(\(E^\circ_{cell} = 1.05V\))

  • (A) 1.16 V
  • (B) 0.93 V
  • (C) 0.73 V
  • (D) 0.83 V
  • (E) 1.32 V

Question 133:

Which of the following metals are normally used in the preparation of dihydrogen in the laboratory?

  • (A) Na and Ca
  • (B) Ca and Ba
  • (C) Fe and Na
  • (D) Zn and Mg
  • (E) Na and Ba

Question 134:

What is the mass of ethanoic acid required to prepare \(0.5\,m\) solution containing \(100\,g\) of water? (Molar mass of ethanoic acid = \(60\,g\,mol^{-1}\))

  • (A) 3 g
  • (B) 6 g
  • (C) 0.3 g
  • (D) 7.5 g
  • (E) 2 g

Question 135:

In a first order reaction, \(N_2O_5(g) \rightarrow 2NO_2(g) + \frac{1}{2}O_2(g)\), the initial concentration of \(N_2O_5\) was \(1.6 \times 10^{-3} mol\,lit^{-1}\) at \(300\,K\). The concentration of \(N_2O_5\) after 23 minutes was \(0.8 \times 10^{-3} mol\,lit^{-1}\). (\(\log 2 = 0.3010\)). Find the rate constant.

  • (A) \(0.04\,min^{-1}\)
  • (B) \(0.06\,min^{-1}\)
  • (C) \(0.3\,min^{-1}\)
  • (D) \(0.6\,min^{-1}\)
  • (E) \(0.03\,min^{-1}\)

Question 136:

Which of the following reactions are complex reactions?

(i) Oxidation of ethane

(ii) Thermal decomposition of HI on gold surface

(iii) Saponification of methyl acetate

(iv) Nitration of phenol

(v) Decomposition of \(NH_3\) on hot Pt surface

  • (A) (i) and (iii)
  • (B) (ii) and (iv)
  • (C) (i) and (iv)
  • (D) (ii) and (v)
  • (E) (i) and (v)

Question 137:

Which of the following transition metal has more than one metallic structure at normal temperature?

  • (A) Chromium
  • (B) Nickel
  • (C) Manganese
  • (D) Vanadium
  • (E) Copper

Question 138:

In chemotherapy, the ligand used to remove the excess of copper is

  • (A) ethylenediamine
  • (B) D-penicillamine
  • (C) cupron
  • (D) ethylenediamine
  • (E) \(\alpha\)-nitroso-\(\beta\)-naphthol

Question 139:

Some transition metal ions given below contain spin only magnetic moment (BM). Which of the following is not correctly matched?

  • (A) \(Ni^{2+}\) (Z=28) \hspace{0.5cm} 4.73
  • (B) \(Ti^{2+}\) (Z=22) \hspace{0.5cm} 2.84
  • (C) \(Mn^{2+}\) (Z=25) \hspace{0.5cm} 5.92
  • (D) \(Fe^{2+}\) (Z=26) \hspace{0.5cm} 4.90
  • (E) \(Co^{2+}\) (Z=27) \hspace{0.5cm} 3.87

Question 140:

In Carius method, \(0.40\,g\) of an organic compound gave \(0.188\,g\) of \(AgBr\). The percentage of bromine in the compound is (Atomic mass of Ag = 108 g mol\(^{-1}\) and Br = 80 g mol\(^{-1}\))

  • (A) 30%
  • (B) 25%
  • (C) 35%
  • (D) 24%
  • (E) 20%

Question 141:

Which of the following are carcinogenic hydrocarbons?

(i) 1,2-Benzanthracene

(ii) Pent-1-yne

(iii) 1,2-Benzpyrene

(iv) Cyclohexane

(v) 3-Methylcholanthrene

  • (A) (i), (ii), (v)
  • (B) (ii), (iii), (iv)
  • (C) (i), (ii), (v)
  • (D) (i), (iii), (v)
  • (E) (ii), (iii), (iv)

Question 142:

IUPAC name of \((CH_3)_3C-CH_2Br\) is

  • (A) 1-Bromotrimethylpropane
  • (B) neo-pentyl bromide
  • (C) 1-Bromo-2,2-dimethylpropane
  • (D) 2,2-dimethylethylenediamine
  • (E) 3-bromo-2,2-dimethylpropane

Question 143:

In Swarts reaction, Freon-12 is manufactured from

  • (A) dichloromethane
  • (B) chloromethane
  • (C) trichloromethane
  • (D) methane
  • (E) tetrachloromethane

Question 144:

IUPAC name of \(CH_3-CH(OH)-CH_2-CH(OH)-CH(C_2H_5)-CH_2CH_3\) is

  • (A) 3-Ethylheptane-4,6-diol
  • (B) 3-Ethylheptan-4,6-diol
  • (C) 5,5-Diethylpentane-2,4-diol
  • (D) 5-Ethylpentane-2,4-diol
  • (E) 2-Ethylheptane-4,6-diol

Question 145:

Pyridinium chlorochromate is a complex of

  • (A) chromic acid with pyridine and \(Cl_2\)
  • (B) potassium chromate with pyridine and \(KCl\)
  • (C) chromium trioxide with pyridine and HCl
  • (D) potassium dichromate with pyridine and HCl
  • (E) chromic trioxide with pyrrolidine and HCl

Question 146:

The reagent used for the conversion of decanol into decanoic acid is

  • (A) Tollens’s reagent
  • (B) Jones reagent
  • (C) Grignard reagent
  • (D) Fehling’s reagent
  • (E) DIBAL-H

Question 147:

An organic compound with molecular formula \(C_5H_{10}O\) does not reduce Tollens’ reagent but forms an addition compound with sodium hydrogen sulphite and gives positive iodoform test. On vigorous oxidation, it gives ethanoic and propanoic acids. The compound is

  • (A) pentan-3-one
  • (B) pentanal
  • (C) pentan-2-one
  • (D) ethoxy ethane
  • (E) pentanol

Question 148:

The increasing order of boiling point of the following amines is

  • (A) \(C_2H_5N(CH_3)_2 < (C_2H_5)_2NH < C_4H_9NH_2\)
  • (B) \(C_4H_9NH_2 < C_2H_5N(CH_3)_2 < (C_2H_5)_2NH\)
  • (C) \(C_2H_5NH(CH_3) < C_4H_9NH_2 < (C_2H_5)_2NH\)
  • (D) \((C_2H_5)_2NH < C_4H_9NH_2 < C_2H_5N(CH_3)_2\)
  • (E) \((C_2H_5)_2NH < C_2H_5N(CH_3)_2 < C_4H_9NH_2\)

Question 149:

Which of the following amines does not form carbylamine?

  • (A) Ethanamine
  • (B) Benzenamine
  • (C) Propan-2-amine
  • (D) Propan-1-amine
  • (E) N-Methylethanamine

Question 150:

The vitamin present in vegetable oils and its deficiency causes muscular weakness is

  • (A) vitamin-A
  • (B) vitamin-E
  • (C) vitamin-B6
  • (D) vitamin-B12
  • (E) vitamin-D

KEAM 2026 Exam Pattern

Particulars Details
Paper Engineering
Mode of Exam Online CBT
Subjects Physics- 45 questions
Chemistry- 30 questions
Mathematics- 75 questions
Type of Question Objective Type
Total Number of questions 150
Marks are awarded for each correct answer 4 marks
Marks are awarded for each incorrect answer 1 marks
KEAM total marks for Engineering 600 marks
Duration of KEAM Engineering exam 3 hours

KEAM 2026 Final Revision